AN ABSTRACT OF THE THESIS OF
RONALD LYNN ADAMS for the degree of DOCTOR OF PHILOSOPHY in MECHANICAL ENGINEERING presented on 3 urr.a..
VIII
Title: AN ANALYTICAL MODEL OF HEAT TRANSFER TO A
HORIZONTAL CYLINDER IMMERSED IN A GAS FLUIDIZED
BED
Abstract approved:
James R. Welty/
A steady gas convection model of heat transfer to a horizontal cylinder immersed in a gas fluidized bed is presented.
Contributions attached bubbles as well as the interstitial voids are included.
The interstitial flow is approximated as flow within a series of double cusped channels and the resulting three-dimensional boundary layer flow is analyzed using a Stokes approximation for the corner flow which is simply matched to a two-dimensional integral analysis of the central region.
Effects of interstitial turbulence are included but gas property variations are neglected. Radiation heat transfer from the hot particle surfaces is included so that results at combustion temperatures can be obtained.
A computer program is developed and used to obtain results for the case of a horizontal cylinder immersed in a bubbling twodimensional atmospheric pressure bed. The interstitial gas flow for

this case is obtained using complex analysis to determine the pressure field near the cylinder with bubbles present.
Generally, the presence of a single bubble, having a diameter equal to the cylinder diameter, has a relatively small effect on the total heat transfer but significantly affects the local Nusselt number distribution.
The heat transfer coefficients calculated using the model were found to be within the range of experimental results obtained by others, but more detailed experimental work is required to completely validate the model.
The assumptions of the model are expected to be valid for mean particle diameters greater than 2-3 mm.

An Analytical Model of Heat Transfer to a Horizontal Cylinder
Immersed in a Gas Fluidized Bed by
Ronald Lynn Adams
A THESIS submitted to
Oregon State University in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
Completed June 1977
Commencement June 1978

ERRATA
Equation 3.3.2.6, page 64, should read
C
R
=
/Tw /
1
+ 110K/TB
\TB/
\Tw/TB + 110K/TB
This error is also present in the computer program listing on page 146.
Line
11 of the left-hand column of the listing should read
CR
= TWTINF**0.5*(1. + ATINF)/(TWTINF + ATINF)
This error does not affect the qualitative conclusions obtained from the computations presented.
However, quantitative errors are present in the hot bed results presented in Figures 4.3, 4.4, 4.10, 4.13, 4.16, 4.19, 4.20, and 4.22
as well as Tables 4.2 and 4.3.
The magnitude of these errors vary from a few percent differences in Nusselt number ratios shown in Table 4.3 and Figures
4.20 and 4.22 to as much as 50% in local values of Nusselt number distributions shown in Figures 4.4, 4.10, 4.13, 4.16, and 4.19.
Additionally, total Nusselt numbers presented in Table 4.2 for 3mm and 6mm hot parameters are in error by, at most, 16%.
Results obtained after correcting Eq. 3.3.2.6 are to be published in the AIChE Journal as part of the paper entitled, "A Gas Convection
Model of Heat Transfer in Large Particle Fluidized Beds," by Ronald L. Adams and James R. Welty.

APPROVED:
Profess of Mechanical Engineering in charge of major
Redacted for Privacy
Department of Mechanical Engineering
Redacted for Privacy
Dean of Graduate School
Date thesis is presented___
Typed by Clover Redfern for Ronald Lynn Adams

ACKNOWLEDGMENTS
It is with pleasure that I acknowledge the contributions of the following:
Dr. James R. Welty, my major professor, provided direction for my research and financial support during the course of my graduate studies at Oregon State University.
Dr. Thomas J. Fitzgerald and Dr. Robert E. Wilson provided helpful suggestions and physical insight which were invaluable inputs to the development of the model.
The Oregon State University Computer Center provided computer time through unsponsored research grants which allowed me to carry out the computations reported herein.
Mr. Nozar Jafarey developed the print plotting routine which I used to obtain convective Nusselt number plots and Mrs. Clover
Redfern typed the manuscript.
Judy, Wendi and Ronnie's unending patience and understanding made all of this possible.

TABLE OF CONTENTS
Chapter
I.
INTRODUCTION
II.
MODEL DESCRIPTION
III.
ANALYTICAL DEVELOPMENT
3.1 Average Interstitial Gas Flow
3.1.1 Average Interstitial Gas Flow in the Vicinity of Two-Dimensional Bubbles
3.1.2 Average Gas Flow Inside an Attached Two
Dimensional Bubble
3.2 Analysis of the Two-Dimensional Boundary Layers
3.3 Analysis of Interstitial Corner Regions
3.3.1 Validation of Stokes Flow Model
3.3.2 Application of Stokes Flow Model to Cusped
Corner s
3.3.3 Determination of the Mapping Function
3.4 Heat Transfer Due to Particle Radiation
3.5 Composite Model
IV. RESULTS
V. CONCLUSIONS AND RECOMMENDATIONS
BIBLIOGRAPHY
APPENDIX: Computer Codes
Page
1
8
62
79
97
100
104
33
38
55
57
132
135
141
17
18
23

LIST OF FIGURES
Figure
1. 1.
Fluidized bed.
1. 2.
1.3.
2. 1.
2. 2.
2. 3 .
Bubble formation.
Packing near horizontal cylinder.
Transient cooling of spherical limestone particles.
Interstitial channel.
Assumed variation of local interstitial velocity.
Coordinate systems.
3. 1.
3. 1.
1.
1.
3. 1.
1.
2.
Bubble parameters.
Pressure field, Case No. 27.
3. 1.
1.
3.
Pressure field, Case No. 49.
3. 1.
2.
1.
Average gas velocity inside bubble.
3. 2.
1.
3. 2.
2.
Shape factor at stagnation point.
Effect of free stream turbulence on Nu
D point for flow past cylinder.
at stagnation
3.3.
1.
Stokes flow regions.
3.3.
1. 1.
Flow along a right angle corner.
3. 3.
1.
2.
Skin friction coefficient for flow along a right angle corner.
3. 3.
2.
1.
Stokes region specification and boundary conditions.
3.3.
2. 2.
Stokes region total energy parameter.
1.
Mapping sequence.
2.
Channel geometry parameters.
Page
3
5
7
11
14
16
19
25
31
32
35
51
54
56
58
61
63
78
81
87

Figure
3.3.3.3.
ln(ri(4)) for mapping exterior of unit circle onto exterior of square.
3.3.3.4.
Integral equation solution for mapping function.
3.3.3.5.
Stokes region matching function.
3.3.3.6.
Average mapping function.
3.3.3.7.
Temperature profile on cusped wall.
3.4.1.
Enclosures for radiative exchange calculation.
4.1.
4.2.
Surface voidage distribution.
Average interstitial velocity at surface, no bubble.
4.3.
4.4.
Location of edge of Stokes region, 3 mm hot parameters, no bubble.
Convective Nusselt No. distribution 3 mm hot parameters, no bubble.
4.5.
4.6.
4.7.
4.8.
4. 9.
4.10.
4.11.
4.12.
put
89
93
94
95
96
98
106
107
108
109
Location of edge of Stokes region, 6 mm cold parameters, no bubble.
Convective Nusselt No. distribution, 6 mm cold parameters, no bubble.
110
111
Bubble configurations.
Pressure field, configuration No. 39.
113
114
Average velocity distribution bubble configuration 39.
115
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 39.
116
Pressure field, configuration No. 40.
117
Average velocity distribution, bubble configuration 40.
118

Figure
4.13.
4.14.
4.15.
4.16.
4.17.
4.18.
4.19.
4.20.
4.21.
4.22.
Page
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 40.
Pressure field, configuration No. 41.
119
120
Average velocity distribution, bubble configuration 41.
121
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 41.
Pressure field, configuration No. 42.
122
123
Average velocity distribution, bubble configuration 42.
124
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 42.
125
{(NuD)conv. /(NuD)conv.
bubble} for 3 mm hot no parameters/6 mm cold parameters.
128
Bubble trajectories.
Convective heat transfer vs. time for 3 mm hot parameters.
130
131

LIST OF TABLES
Table
Plotting symbols.
4.1.
Baseline parameters, D = 0.0508 m, p = 1 ATM.
4.2.
4.3.
Total heat transfer for 3 mm and 6 mm hot and cold parameters, no bubble.
Effect of single bubble on convective Nusselt number, RB = D/2.
page
34
105
126
127

NOMENCLATURE
Symbols Description a a
1
,a
2
,a
3
,a
4 a
H
A2, A3
Cylinder radius
Boundary layer velocity profile coefficients
Hypergeometric function parameter
Areas used in radiative exchange calculation b+
Width used in average bubble region velocity calculation b b
2' b
3' b
4
BiR
Boundary layer temperature profile coefficients
Radiation-convection parameter cf cf2D
Skin friction coefficient
Two-dimensional skin friction coefficient
CH c n
, c
0 c
Hypergeometric function parameter
Coefficients in Fourier series representation of Stokes region temperature solution
Specific heat of gas at constant pressure cR
D
DB d e eW eB
Chapman-Rubesin constant
Cylinder diameter
Bubble diameter
Particle diameter
Particle region emissivity
Wall emissivity
Bubble surface emissivity
Unit vectors in X and Y directions, respectively

h hmax h0
H12 i
H2V
, k
KB
K g
,Kg
KT
F1-2 g,t ge
G
GE
Symbols f fD fs
G
2H
GHV
GN, GN
Description
Drag force per particle
Mapping function
Dimensionless boundary layer velocity profile
Velocity profile shape factor function
Gravitational acceleration
Dimensionless boundary layer temperature profile
Complex pressure due to a single bubble
Euler' s constant
Enthalpy thickness/thermal boundary layer thickness
Boundary layer thickness ratio function
Complex pressure due to N bubbles and its complex conjugate, respectively
Heat transfer coefficient
Maximum bed-wall heat transfer coefficient
Magnification at cylinder wall
Displacement thickness/momentum thickness
Momentum thickness/velocity boundary layer thickness
Unit imaginary number, 4:4-
Summation index
Single bubble rise velocity parameter
Gas thermal conductivity
Turbulence parameter

Symbols
KR
L
Lc
Description
Drag coefficient
Characteristic length
Average particle spacing
Stokes region length along upper wall of cusped corner
L6
Summation index
Free stream turbulence length scale
Summation index m, n
M m+
N n
Boundary layer momentum parameter
Dimensionless mass flow rate
Number of terms in series and number of bubbles
Number density of particles
Nu
1,
Nu
D 2 rt
Effective radiative Nusselt numbers
Unit vector normal to bubble boundary
(Nu
)
D ave
NHV
Average Nusselt number for channel region
Boundary layer thickness ratio
(NuD)Stokes Nusselt number at edge of Stokes region at xs
Null Nusselt number based upon cylinder diameter
Nusselt number without turbulence
(Nu
D
)0
(Nu
D
)2D
Pr
P
Two-dimensional Nusselt number
Prandtl number

Symbols Description
Average interstitial gas velocity
(Qg)ave
Qg
Q ql
Gas velocity
Solid particle velocity
Net radiation leaving surface
1
Net radiation leaving surface 2
Wall heat transfer q2 qW
Qgx,QgY
(Q )00
X and Y components of gas velocity
Ambient interstitial gas velocity r r, R
Re
D
Radial coordinate
Reynolds number based on umf, D, and gas properties at bed temperature and pressure
Re z r
1
R c
Re
Reynolds number based on z
Channel boundary radial coordinate
Channel boundary parameter
Reynolds number based on particle diameter
RB
0
R1
Bubble radius
Radial coordinate of bubble center
Inner radius of Stokes region on circle plane
Rs
1,
R s 2
,Rs
3
Mapping function parameters s
Half distance between particle centers
S,
SO
Thermal conductivity-temperature parameters

U ao v tit tie uH uenet u
Orel uB
0
LIBR us us
1
TW u ug umf
Symbols t s, sO
TB
T
Description
Distance along bubble boundary
Time
Bed temperature
Temperature
Wall temperature
Velocity, velocity component
Gas velocity (superficial)
Minimum fluidizing superficial velocity
Superficial velocity at distributor
Single bubble rise velocity
Real part of complex mapping function
Real part of complex mapping function evaluated on unit circle boundary
Turbulence intensity
Tangential gas velocity at cylinder surface
Integration variable
Net tangential gas velocity
Relative tangential gas velocity
Bubble rise velocity
Velocity at edge of boundary layer
Complex variable
Velocity component

ZD zh zH zs zc z xH
X0, Y0 xl, x2 x26
Z
Symbols vs
W w x, y, z
X, Y x
Description
Imaginary part of complex mapping function
Boundary layer energy parameter
Complex variable
Rectangular coordinates
Stokes region edge location
Hypergeometric function parameter
Bubble position coordinates
Position coordinates on half plane
Stokes region edge location on half plane
Complex variable
Bubble center location
Complex variable on half plane
Fourier coefficient parameter
Complex variable on semicircle plane
Complex variable on circle plane
Complex variable on physical plane
Greek Symbols a
B
Polar angle relative to bubble centered coordinate system a
H a
Exponent
Plank mean absorption coefficient

E
Symbols Description
13142
RT
Drag parameters
Eddy diffusivity parameter
RTB
Eddy diffusivity parameter for bubble region
13ls ,132s, P3s,R4s,P5s
Stokes region temperature parameters
Polar angle on physical plane
VC r( )
Ak
5v
6H
51
62
AZ
62D
Arc center location
Gamma function kth angle increment
Transformed velocity boundary layer thickness
Transformed thermal boundary layer thickness
Displacement thickness
Momentum thickness
Enthalpy thickness
Two-dimensional boundary layer thickness
EH
Too
(ceH)
OB1, OB2
Voidage
Hyper geometric function parameter
Eddy diffusivity
Eddy diffusivity at edge of boundary layer
Rieman zeta function
Velocity profile similarity parameter
Temperature profile similarity parameter
Bubble boundary bounds

p g
Ps p o-
E e
(a
H) g
la n
T v v
B
Symbols
0
0
8
Os
T
0
Description
Bubble center location
Polar angle
Temperature difference, T-T
TB-TW
Eigenvalue
Dynamic viscosity
Power law parameter for temperature variation on circle plane
Kinematic viscosity at edge of boundary layer
Bed kinematic viscosity
Kinematic viscosity
Boundary layer transformed normal coordinate
Gas density
Radial coordinate on semicircle plane
Gas density
Stefan-Boltzmann constant

Symbols
4p s's 0 coc s's
1
4) 1 4) 2 I
11)
3
Is
B
41
H
Description
Angular coordinate on circle plane
Angular extent of Stokes region on circle plane
Cusped corner location on circle plane
Angular coordinate
Mapping parameters
Bubble velocity parameter
Stokes region matching parameter
Confluent hypergeometric function
Integration variable
Velocity profile shape factor
1-2V
Superscripts
)+
)*
)i
Dimensionless parameter
Parameter /cylinder radius
Value at ith iteration

AN ANALYTICAL MODEL OF HEAT TRANSFER TO A
HORIZONTAL CYLINDER IMMERSED IN A GAS
FLUIDIZED BED
I.
INTRODUCTION
Presently, major research and development efforts are underway in the United States with the objective of demonstrating fluidized bed combustion of high sulfur coal (47).
The major technical achieve ment of this combustion concept is the reduction of SO2 emissions through the addition of limestone or dolomite to the combustion chamber.
Also, combustion temperatures compatible with efficient removal of SO2 are lower than conventional boiler temperatures and this results in reduced emissions of nitrogen oxides (NO x).
Furthermore, a fluidized bed is a more efficient heat transfer medium than combustion gases in a conventional boiler, so reduction in heat exchanger size is possible (57). Accordingly, an adequate understanding of heat transfer to surfaces immersed in a fluidized bed boiler is of considerable importance in fluid bed boiler design.
Fluidization as an industrial process technique, allows a broad spectrum of operations involving two or three material phases.
Examples of the use of the process technique range from catalytic
CaCO
3
1 The absorption reaction for limestone is
1
+ SO2 + 2 02 = CaSO4 + CO2 .

2 cracking of petroleum to reduction of iron ore and the first industrial application was the Winkler coal gas generator invented in 1922 (41).
Because of the variety of uses of the fluidization process technique, a large number of books have been written on the subject, for example
A brief summary of the main features of a gas fluidized bed is presented below.
A gas fluidized bed typically consists of a slender tank partially filled with crushed solid material as shown in Fig. 1.1.
The solid particles in the bed are fluidized by introducing gas through a distributor plate at the bottom of the tank.
As the gas flow is increased from zero, the solid material experiences a transition from a packed, stationary condition ("packed bed", Fig.
1. 1A) to a loose, fluidized state (Fig.
1: 1B).
The bed is said to be fluidized when the aerodynamic drag and gravitational forces acting on the solid particles balance.
Also, the total pressure drop across the bed at minimum fluidizing conditions is equal to the weight of bed material per unit cross sectional area, the superficial velocity at the distributor is designated umf and the bed voidage is designated emf.
Further increases in gas velocity beyond umf result in the formation of large gas voids or "bubbles" (Fig.
1. 1C).
These bubbles generally originate at the distributor plate and their formation has been described by Zenz in Ref. 22 Briefly, bubbles are formed as a result of penetration and expansion of a high velocity jet at a

Gas
A. Packed bed
Gas
B. Fluidized bed
Figure 1. 1. Fluidized bed.
Gas
C. Bubbling bed

distributor opening as shown in Fig.
1. 2.
Since the superficial velocity at the distributor is larger than umf, the aerodynamic drag force will be larger than the gravitational force acting on the material adjacent to the opening and the material will be levitated. As the jet expands and displaces bed material, the jet velocity decreases so that the velocity of the gas at the boundary becomes umf.
Now,
4 the bed material flows inward and upward at the base and "pinches off" a roughly spherical bubble.
Once formed, the gas bubbles rise through the bed with nearly the same velocity as bubbles of equal size rising in a liquid (22).
In fact the vertical velocity of a single bubble rising in a fluidiz%d bed is u
BR cc vriDB
Also, the vertical velocity of a "swarm" of bubbles is u.B = (u0 -umf) + KB gDB (1. 2)
Thus, when superficial velocity u
0 is near umf, velocity is relatively independent of gas velocity.
the bubble rise
So, large particle systems with high minimum fluidizing velocities may contain relatively slow moving bubbles.
The motions of gas and solid material in a fluidized bed are altered by the presence of immersed surfaces. In the case of a horizontal cylinder, a relatively stationary stack of solid material has

u
0
> urrif
A B
Figure 1. Z.
Bubble formation.
C
U,

been observed to form on the leeward or top of the cylinder while a bubble is often attached to the windward or bottom side as shown in
Fig. 1.3 (18). The solid material is more closely packed and less dynamic in the lee stack than on the sides but is removed and replenished as bubbles pass the cylinder. Also, because of local accelerations of the gas flow due to the presence of the cylinder, bubble formation can occur on the sides of the cylinder and in fact has been observed experimentally by Glass and Harrison (29).
The heat transfer model described in this thesis has been developed on the basis of the expected operating parameters for an atmospheric pressure fluidized bed boiler (57).
At atmospheric pressure, the combustion temperature for efficient SO2 removal is about 1100 K and the volume flow of air required necessitates solid
of bed material is expected to be limestone or dolomite; approximately 10% or less will be coal depending upon bed depth.
Heat exchanger designs are expected to be based upon 0.0508 m (2 in.) diameter tubes and horizontal arrangements seemto be the most practical (57).
Accordingly, the base line geometry for this analysis will be a 0.0508 m diameter horizontal cylinder.
6

0?0
o '16
00 0b
08°0
0
00
0 o 0 o 00
C) 000 0 c002o
00 u 0
0o
CP8
0 0
.11,44
VI liai
III, it iVii
..). 4hip ghli .64-4!,...
0
....
0 ss_gs
PolPeAsb 0 4i4t 6 9
*4- 0 u''
-*, flo.:
-t.: ,
4,44. ,
,
1, r,
92)0(P%
.
oo oo° o
0
0
0
0 o
L)0 o°
OC
00
Co
0
0
0
000
00,0co)
604,, cooc,
Bubble
.
n
6
EC34334
OU
,,,,c, 0e e,a-g79
0 0 opoe os003808,00
,(00 (90 os
CO
00
Gas flow
Figure 1. 3. Packing near hor izontal cylinder.
7

8
II.
MODEL DESCRIPTION
In a high temperature gas fluidized bed, heat is transferred to an immersed surface via conduction and thermal radiation from the gas adjacent to the surface and thermal radiation from the adjacent solid particle surfaces. Since particle contact area at the surface is negligibly small (zero for perfectly spherical particles), negligible heat transfer occurs by conduction at particle contact points.
The heat conducted by the gas is dependent upon the gas temperature gradient at the surface and this gradient is affected by the presence and motion of the solid particles as well as the motion of the gas.
The solid particles contain most of the thermal mass in a gas fluidized bed and act as a continual source of energy which is conducted to the sur face through the gas. Consequently, the most fundamental analytical treatment of fluid bed heat transfer involves consideration of the unsteady flow of gas adjacent to the immersed surface and adjacent to solid particles in the vicinity of the surface as well as unsteady conduction within the solid particles themselves. An approach of this nature is both analytically unwieldy and computationally impractical, particularly when the stochastic nature of the fluidized bed is considered.
However certain limiting cases can be treated analytically using more practical approaches. These are the small particle limit in which unsteady effects are dominant and gas velocity has a

9 negligible effect, and the large particle limit in which unsteady effects are negligible but gas velocity has a dominant effect on the heat transfer.
As will be shown below, the latter limit is expected to be appropriate for coal combustion near minimum fluidizing conditions.
A number of analytical models have been developed to describe the small particle limit and are generally accepted as models of fluid bed heat transfer (9).
The fundamental assumption common to these models is that a packet of bed material is swept to the immersed surface and exchanges energy with the surface by unsteady conduction during its characteristic residence time.
Differences in the models are primarily differences in modeling the characteristics of the packet.
The packet was treated as a continuum by Mickley and
Fairbanks (46) and their model was refined and modified by a number of investigators, for example, Yoshida, Kunii, and Levenspiel (69),
Wasan and Ahluwalia (65), Chung, Fan, and Hwang (19), Broughton and
Kubie (15) and others (9).
A model based upon discrete particles dispersed throughout the gas was developed by Botterill and Williams
(14) as well as Ziegler, Koppel, and Brazelton (74) and Basu (6).
Gabor (26) developed a model based upon unsteady conduction in alternate slabs of solid and gas. All of these models require information regarding the residence time distribution of the packet and the continuum approaches require experimentally determined thermal properties of the emulsion.

10
Very little analytical work has been directed toward the large particle limit, though a few gas convection based models have been developed.
One of the first of these was based upon "scouring" of the gas boundary layer at particle contact points (a two-dimensional view of the interstitial boundary layer development) and was developed by Levenspiel and Walton (43). Also Baskakov, Berg, et al.
(4) have developed an empirical model for this regime based upon experimental data (5). Recently, Botterill and Denloye (12) extended a packed bed model, based upon one-dimensional flow and an effective conductivity, to estimate heat transfer to a vertical cylinder due to gas convection.
However, none of these models is considered adequate for heat transfer to an immersed horizontal cylinder because they do not include the effect of the packing shown in Fig.
1. 3.
Therefore, an analytical model has been developed for the gas convection dominant regime, with consideration of the expected inter action of an immersed horizontal cylinder with a bubbling fluid bed boiler. Typically, a bubble is expected to be in contact with the windward side of the cylinder as discussed in Chapter I, so that three characteristic regions are present as shown in Fig.
1. 3.
In the regions of particle contact with the cylinder (sides and lee stack), the solid particles are expected to be isothermal.
This isothermal behavior is due to the combined effects of large particle size and relatively short residence time at the surface.
Figure 2. 1 shows the

2
T
T
8
4
6
TB--T
TB-TW at surface
T
0. 2 h h max
Tr
0.1
0
0 2
Diameter, mm
4 6
Figure 2. 1.
Transient cooling of spherical limestone particles.
11

12 time required to convectively cool spherical limestone particles so that temperature difference (T -T) at the surface changes by 10% and 20%.
This estimate was obtained from transient conduction charts given in Ref. 51 and physical property data from Ref. 's 60 and 66.
The convective heat transfer coefficient was based upon maximum values given by Botter ill (9) for the particle sizes shown.
The elapsed time for 10% reduction in temperature 'difference is generally greater than anticipated residence times for particle diameters
Because of the isothermal behavior of the solid particles the mechanisms of heat transfer are convection due to flow of gas within bubbles contacting the cylinder and within the interstitial voids bounded by the cylinder wall as well as the isothermal particle surfaces and thermal radiation emitted by the hot particles.
The effect of combustion of coal particles adjacent to the heat transfer surface is not considered since the coal content is expected to be low.
This effect has been estimated as a function of coal content by Basu (6) and found to be small for coal content less than about 10%. The gas is expected to be optically thin and negligible heat transfer occurs by radiation from the gas.
In fact, the characteristic length for absorption in CO2 (52) is about three cylinder diameters and the radiative term in the gas energy equation is 0( 1
Re
D
) as shown in Sec.
3. Z.
Unsteady effects due to particle motion produced by passing bubbles are neglected since

13 bubble velocity is expected to be small relative to interstitial velocity.
However, the influence of interstitial turbulence on the heat transfer is considered. Thus, the gas convection dominant limit provides a compatible description of the heat transfer process.
The convective heat transfer due to flow in the interstitial regions of the sides and lee stack is modeled by considering the flow of gas inside the channel shown in Fig. 2.2.
This flow channel is bounded below by the cylinder wall and on the sides by surfaces approximately defining the circulating gas trapped between adjacent particles.
The geometry of the channel is further specified by requiring the width and length at the base to be equal to the average distance between particles as determined by the voidage at the cylinder wall.
The thickness of the gas boundary layer which forms along the lower" surface of the channel (cylinder wall) is expected to be much smaller than the height of the channel, so the gas flow in the central core of the channel is assumed to be inviscid.
Furthermore, the gas is assumed to be at bed temperature in the core.
The core velocity variation is estimated through consideration of the nature of the interstitial flow in general.
This flow has been described by Galloway and
Sage (28) as a series of jets, wakes, and stagnant regions with rapid changes in velocities near particle surfaces.
Detailed analytical determination of this flow is extremely complicated, so a simple

Figure 2.2.
Interstitial channel.
14

15 model is used in which the core velocity is assumed to vanish when a solid particle is encountered, then increase linearly over the length of the channel until another particle is encountered as shown in Fig.
2. 3.
The actual magnitude of the velocity at the end of the channel,
(u co
)max is established from analytical determination of the average interstitial velocity.
This velocity variation will result in a thinning of the boundary layer due to acceleration and is consistent with the expected physical behavior of the three-dimensional flow.
The gas flow in the boundary layer portion of the interstitial channel will be three-dimensional due to the cusped corner formed by the free surface and cylinder wall and this will produce a threedimensional temperature field as well. However, the flow in these corner regions is assumed to be Stokes like so that the convective terms in the momentum and energy equations are neglected. Also, the boundary layer in the central region of the channel away from the corners is assumed to be two-dimensional.
The merits of this approach are tested in Sec. 3. 3. 1 where it is applied to constant property flow along a right angle corner.

*I Lc 14
2(Qg )ave
,....
,...
)
....
_,.."
/
Streamwise position
Figure 2. 3. Assumed variation of local interstitial velocity.
16

17
III.
ANALYTICAL DEVELOPMENT
The qualitative model of heat transfer to a horizontal cylinder immersed in a gas fluidized bed which was described in Chapter II will now be reduced to a mathematical model from which a set of operational equations will be obtained. A necessary element in determination of the heat transfer is the local average gas velocity adjacent to the cylinder wall.
There are no experimental data for this parameter, so an approximate two-dimensional model which includes the presence of bubbles will be developed.
This model will be used to obtain the average interstitial gas velocity as well as the average gas velocity within an attached bubble.
Next, the equations governing the flow in the boundary layers adjacent to the cylinder wall will be reduced to a set of nonlinear ordinary differential equations using the von Karman-
Pohlhausen integral technique (see, e. g. , Schlichting (50)) modified to account for the presence of interstitial turbulence.
Solution of these equations will provide the local Nusselt number within the attached bubble as well as the two-dimensional portion of the interstitial channels. Then the corner region of the interstitial channels will be analyzed and a simple matching procedure developed which will allow specification of the extent of this region.
Finally an approximation for the heat transfer due to thermal radiation emitted by the iso thermal particles is included so that complete specification of the heat

18 transfer at combustion temperatures is achieved.
The coordinate systems used in the analysis are summarized in
Fig. 3. 1.
3. 1
Avera e Interstitial Gas Flow
The equations governing the average flow of gas and solid material in a fluidized bed have been developed by a number of investigators (22) by treating the motion of gas and solids as if they are interpenetrating continua.
In these developments, point fluidmechanical variables are replaced with averages over regions involving several particles. The set of equations developed by Anderson and Jackson (22) for this purpose are aE at
+ v (E Q ) 0
8(1-)+v[(1-)61= at
0
E p { g aQ at g + Q g
Voc--5- } = -E Vp + E div g
+ E p npfD
P (1-E
Ca' at
+ Q s vQ } =
( 1 div -7 s s
+ (1-E )pt- n f p p D
(3. 1.1)
(3. 1.2)
(3. 1.3)
(3. 1. 4) where fD is the gas/solid interaction force per unit volume.
The dependence of the interaction force, fD on local bed properties is assumed not to be affected by the presence of immersed

Stagnation point
Channel cross section
Figure 3.1.
Coordinate systems.
19

objects so that pressure drop correlations for a bed without internals can be used.
Under these conditions, the gas momentum equation reduces to
20
-p g
Vp -
E
(3. 1. 5)
The particular correlation used in this analysis is adopted from that for fixed beds given by Kunii and Levenspiel (41) and is npf
D
E
= (Q -Q ) 150( g s
1-E 2
(4)
P d
P
)
2
+ 1.75 p g E
T-L1 1 -01-
4)
P d
P g
si
(3. 1. 6)
This expression is linearized by replacing the nonlinear term with the ambient average interstitial velocity, form of the model is u mf
/e co' so that a linear n p 13,
E
(Q -Q ) 150[ (1-E ]2u, g S
(I) d
P P
1.75p
gumf (1-E)
E (1:1 d
P P
Now, the gas momentum equation becomes
(3.1.7) p g
{-
at g vc) g
_vp div rr-r g
(6 -Id g s p g
(1-E)
150[
E (I) d
P P
]
211+1.75p
E
(3. 1. 8)
E
(1-E)
C') d
P P

21
The equations governing the average motion of the gas are further reduced by introducing dimensionless variables and investigating the relative orders of magnitude of the various terms. For this purpose, the following dimensionless parameters are introduced.
Q+ _ g unif
(3. 1. 9)
(3. 1. 10)
P
P t
+ tu
= 7"
2
(3. 1. 11)
(3. 1. 12)
T
+ T
D2 mf
(3. 1. 13)
The resulting dimensionless equations governing the gas flow are where u +
B aE+ at
+ v. (E Q ) 0
LIB art" at+
+ Q v Q = -Vp _ g g
(Q -Q)
131 fs
Re
R
1' 2
'
+ u
2 mf
+
1
Re
D div T
(3. 1. 14)
(3. 1. 15)

13
=
150[(1
E (I) p
)]2
I. 75 1-E
00 p pa umf D
ReD
[1, p gumfdp
Re
22
(3.
1.
16)
(3.
1.
17)
(3.
1.
18)
(3.
1.
19)
Examination of the various terms appearing in Eq. 's 3. 1. 14 and
15 leads to the following observations: a) The magnitude of the unsteady terms depends upon the relative magnitude of the bubble velocity. In this analysis, superficial velocities near umf are assumed so this velocity is expected to be small relative to umf and certainly small relative to the interstitial velocity, Q .
Accordingly, the unsteady terms will be neglected.
b) The particle velocity, Qs, is expected to be the order of magnitude of the bubble velocity and, hence small relative to
Q
.
Therefore Qs will also be neglected.
c) Changes in voidage,
E will substantially complicate the analysis.
Hence
E will be assumed to be constant so that the drag coefficient will be constant.
Later, when the gas velocity is calculated from the pressure gradient, local

23 values of
E will be used in determining the drag coefficient
(a "zeroth order'? correction).
d) The momentum equation is dominated by the drag term to order Did
.
Therefore, the pressure gradient will be balanced by the drag force.
These simplifications lead to the following reduced forms of Eq. s
3. 1. 14 and 15:
-.+
V Q = 0 g
Vp
+
+ dp
13 1
Re
2
The divergence of Eq. 3. 1. 21 is
2 +
= 0 (3. 1.22)
Thus the average gas velocity can be obtained by first solving
Laplace's equation for the pressure field, then using Eq. 3. 1. 21 to determine the velocity.
3. 1. 1
Average Interstitial Gas Flow in the Vicinity of
Two-Dimensional Bubbles
There have been a number of analytical solutions of the equations of motion for the average gas and solids motion in the vicinity of both two and three dimensional single bubbles. Most of these are summarized by Jackson (22). The simplest solution was obtained by

24
Davidson (22) based upon Darcy's Law and has been found to agree well with experimental observation (37). Davidson's model is based upon solution of Eq. 3. 1.22 so that the assumptions discussed above are reasonable. However, there are no published solutions of the problem of a small number of bubbles in the vicinity of a cylinder, though some work has been done on bubble-bubble interactions, for example Clift and Grace (21).
A particular solution of Eq. 3.1.22 for the case of two -dimensional bubbles can be obtained by using complex analysis and the method of images. This solution will be used to obtain an approximation for the gas motion and calculated results will thus be limited to the two-dimensional case.
For the case of a single bubble in the vicinity of a cylinder the boundary conditions for Eq. 3. I. 22 are (see Fig. 3. 1. 1. 1) p+
= const on bubble boundary (3. 1. 1. 1) ap
8R+
= 0 at the cylinder surface (no flow through (3. 1. 1. 2) the cylinder) vp
= ( dp dY
+
)00 ambient gradient at R+ >> 1 (3. 1. 1. 3)
Taking p to be the real part of a complex pressure and considering an isolated bubble located at Z
0
, the complex pressure is

Figure 3. 1. 1. 1.
Bubble parameters.
25

G = dY+
) co
{Z+
+2
RB
Z
+ + o
+
On the bubble surface, Z Z
+
0 iaB
+
= RB e
, so p
+
G = -i( d
) dY+
B e iaB
+e-ia
B
Z
+
0
R+ dp +
B dY
2 cos a
ZO
B+ +
RB
}
26
(3. 1. 1.4)
(3. 1. 1. 5)
(3. 1. 1. 6)
Real part (G)
= p+
Y_
0+( !i
-P-T)00= const.
dY
(3. 1. 1. 7)
Thus the boundary conditions are met.
Furthermore, the potential for
N bubbles is (by superposition)
GN dY
+100
N +2
RB
J
+ +
.
Oi
(3. 1. 1. 8)
Note that the simple addition used here will result in distortions in the pressure fields in the vicinity of individual bubbles. A more rigorous approach would involve repeated application of the image method until bubble-bubble perturbations are small.

27
Now consider the horizontal cylinder in the presence of N bubbles.
The complex pressure, GN, does not satisfy the boundary condition at the cylinder surface, so GN must be modified.
The boundary condition at the cylinder surface is satisfied when the imaginary part of the complex pressure is constant.
This is achieved by placing an image of each bubble inside the cylinder and mathematically reduces to the following expression for the complex pressure for N bubbles in the vicinity of the cylinder (for a discussion of the image method or circle theorem, see Batchelor (7));
FN = GN + GN
(a
+2
+
)
(3. 1. 1. 9) dY
+/co +
Z+
N j=1
+2
B
1 1
Z+-Z+ a
Oj -.4 --2
+
- Zu
J
(3. 1. 1. 10)
The relative pressure in the bed is obtained by taking the real part of Eq. 3. 1. 1. 10 and is
= R ((I+
1
)sin 0+
R''2 Bj
2
(3. 1. 1. 11) x
( *
sin 0
Oj
R
+ (-sine +R R sin°
) /(1
Oj Oj
*2
+
R
2
2)/(R
R "R0. cos(0-0 .))
Oj
2
ROJ
.c o s(0-O ))
Oj

28
R
, R
*
=
R a
Oj a
Oj
*
Bj
R
_131 a
,
O
,
00 are polar coordinates of the jth bubble.
Also, the pressure gradient obtained by taking derivatives of Eq. 3. 1. 1. 11 is
Vp aP
ex + aP ay
(3..1..1. 12) with ap a x a
(
+
+ dp+ dY
)
*
2X Y
*
R*2
N
*2
RBi x
(
-2(Y
*
Oj
* * *
,
O j
*2 *2
03
.+R -2(X
*
Oj
* *
* 2
ZY
03
* *2
/(1+R
Oi
12.
*2
* * *
-2(X X .+Y Y
))
Oj Oj
*
- 2(YOj
2
R*-Y* )(X*
*2
ROj
Oj
)/(1+R
*2 *2
03.
* *
R -2(X X
Oj
* * 2
+Y Y .))
(3. 1. 1. 13) ap
+
+ a
( dp+ dY
)00
- 1+ 1
R
*2
*2
2Y ^2+
R*4 j=1
*2
Bj
/ (R.
*2 *2 *
+R ". 2(X X
0
*
Y6C
.+Y YO.))
+.
* 2 * *2
2(Y -Y /(R R -2(X
Oj Oi
2
-F
03
*
+y
*
Oj
*
))
2
+ (2Y
Oj
*
-1)/(1+R
*2 *2
Oj
-2(X
*
Oj
* * *
+
Oj

29
+2(Y -Y
03
2
)(R
*2
03
-Y )/(1+R
Oj 03
2 *2 * *
-2(X X .+Y Y .))
03 03
2
(3. 1. 1. 14)
The pressure gradient along the wall of the cylinder, required for determination of the local velocity there, is ap+ ae a
+
( dp dY
+)00
2 cos 0+
N
*2
Bj
-2 cos 0
*2 *
1+ROj -2ROj cos(0-0
)
Oj
4R0j sin(0-0
(1+R
*2
03
03
.)(R
*
-2R
03
03
.
sin 00-sin cos(0-0 .))
03
2
(3. 1. 1. 15)
Note, that from Eq. 3. 1.21, the ambient pressure gradient is
( dp+
+)°,0 dY p
Re
P 000
) p
+ 132(E00 (3.
1.
1.
16)
1 d
+E
P
(1(E00)
Re
+ (32(E00) (3.
1. 1.
17) and the local velocity is

30 d+
g r1
Re p
+132
Vp (3. 1. 1. 18)
Thus the average gas velocity is specified.
At the surface of the cylinder, the local values of voidage in the stack and side regions are unequal so that
131 and
P2 are not constant over the surface of the cylinder This variation has not been included in the analysis so far, but will be accounted for by using local values of voidage in the final determination of velocity at the cylinder surface using Eq. 3. 1.21.
The solution for the pressure field obtained using the image method as described above may result in distortions of the bubble geometry. Some distortion can be tolerated since bubbles in a fluidized bed are not exactly spherical in reality.
However, for some bubble positions close to the cylinder wall, a closed isobar defining the bubble surface was not obtained. An example of this result is shown in Fig. 3. 1. 1.2, a "print plots' of the relative pressure field.
Also, the lower stagnation point may not lie on the cylinder wall
(neither inside nor outside the bubble), but somewhere off the wall as shown in Fig. 3. 1. 1. 3.
This qualitative result,obtained from the print plot,was supported by the tangential pressure gradient variation along the cylinder wall as well as calculated average velocities inside

SCALE.
DELTA X= .1000E4.00
DELTA Y= .1000E1.00
CASE NO.
27
.-1-.9118-8 ------- C-..C-C-C- - -
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- - - - CCC 313-0- - - - - - - 0 -0 - 0 - 0 - 0 - 0 - - - - - - - - - C-
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.
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HHHHHHHHHHHHHHHHHHH
IIIIIIIIIIIIIIII
IIIIIIIIIIIIIIIIIIIIIIIIIII.
L L L L
LLL LL LLLLLLL
IC...,K
LL LLL LILLILLLL
LL ILL
K,KK
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1K......K..4 K K
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4 M M H M 4
ILLIL
4 N M K
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M H M M 4 M
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LLLLL L. LL
LL LLL LL L LILL LL LLLLL LLLL L.
K44 4H4H
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JJJJJJJJJJJJJJJJJJJJ
MMI44MMHM4HMKMMIJM4MMHMM
NNNNN N. NNNNNNNNNNNNNNNNN
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.
Figure 3.1.1.2.
Pressure field, Case No. 27.

SCALE.
DELTA X= .190CE+00
DELTA Y= .1000E4.00
CASE NO.
49
.-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G ------------ H-H-H-H-H-H-H
- - - ------- -G-G-G-G-G-G-G-G-G-G-G-G-G --------- -
---------------- - - - - - -G-G-G-G-G-G-G-G-G- - - -
.-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F - r --------- G-G-G-G-G-G-G-G
- ------------- F-F-F-F-F-F-F-F-F-F ----- - -G-G-G
.-E-E-E-E-E-E-E-E-E -------------- F-F-F-F-F-F ------
.-E-E-E-E-E-E-E-E-E-E-E-E-E-E-E-E-E-E-E - - - - - - - - F-F-F-F-F-F-
- ------ - - -
- - - - -E-E-E-E-E-E-E ------ F-F-F-F-F
- - -E-E-E-E-E ------ F-F
. -0 -0 -0 -0- 0- D - -0
.-0 ---------- 0-0-0-1-C-0-0-0-0-0-0 ----- E-E-E-E -----
------------- - - - - - - - -0-1-0-1- - - - -E-E-E- - -
.-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-G-C-C - - - - - -0-0-0-0- - -E-E-E-E-
- - - - - - -
- - ----- C-C-C-C-C-C- - - -0-0- - - -E-E-E
- - -9-B-8-8-8-8-8-8-9-9-3-E-8 ----- - -C-C-C- - -0-0- - -E-E
.-9-9-9-8-9-8-8-8-4-8-8- -8-8-9-9-8-8-8-9-8- - - -C-C-C- -0-0-E-E
9 8 8 C C
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0
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.-H-H-H-H-H-H-H-H-H-H-H-H-H
.-H-H-H-H-H-H-H-H-H- - - -
---------
.-G-G-G-G-G-G-G-G-G-G-G-G-G
.-G-G-G-G-G-G-G-G-G-G-G-G-G
- -
------ - - -
---------- F-F-F
.-F-F-F-F-F-F-F-F-F-F-F-F-F
.-F-F-F-F-E-F-F-F-F- - - -
.-F-F-F-F --------- E
. --------- E-E-E-E-E
. ------ E-E-E-E-E-E-
- - -E-E-E-E-E- - - - -
.-E-E-E-E-E-E ----- 0-0-0
.-E-E-E-E-E- - - -0-0-0-
I
I
I
I
I
I
I
I
I
J
J
J
JJ JJ
H
H
H
H
H
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- -
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A A A A .
.
.
G G
H
H H
H H
F
H
H
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F
H
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C
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D 0
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E E
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F
F F F
F F
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.
H H H
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I I
I I .
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.
Figure 3.1.1. 3.
Pressure field, Case No. 49.

the bubble boundary.
The plotting symbols used in these figures are listed in Table 3. 1. 1. 1.
33
3. 1. 2 Average Gas Flow Inside an Attached Two-
Dimensional Bubble
The solution for the average gas velocity described above does not apply within the boundary of the bubbles.
Inside the bubbles, the motion of the gas is governed by the Navier -Stokes equations and is estimated here by calculating the average mass flow across rays which intersect the cylinder surface and outer bubble boundaries.
The average mass flow across such a ray must be balanced by the average mass flow through the bubble boundary (see Fig. 3. 1. 2. 1) and this is estimated by using the results of Sec. 3. 1. 1 for the average interstitial velocity.
So based on continuity, the average velocity is u
0
(1,0) b
.+ s (A) so
E (Q g ri
B
)ds
(3. 1. 2. 2)
The above integral is evaluated using the trapezoidal rule as follows.

SYMBOL
A
B
C
I
3
E
F
G
H
K
L
N
0
P
Q
S
U
W
X
Table
3.1.1.1.
POTENTIAL PRESSURE IN VICINITY OF A TUBE
IN A FLUIDIZED SEC WITH BUBBLES
0.
.1500E+00
.3ocoE+on
.4500E+00
.6000E+00
.7500E+40
.9000E+00
.1050E+01
.1200E+01
.1350E+01
.1500E+01
. 1650E+01
.1800E+01
. 1950E+01
.2100E+01
.2250E+01
.24013E+01
.2550E+01
.2700E+01
.2850E+01
.3000E+D1
.3150E+01
.3300E+01
.3450E+01
.3600E+01
.3750E+01
.3900E+01
.4050E+01
.4200E+01
.4350E+01
.4500E+01
.4650E+01
.4800E+01
.4950E+01
.5100E+01
.5250E+01
.5400E+01
. 5550E+01
.5700E+01
.5850E+01
.6000E+01
. 6150E+01
.6300E+01
.6450E+01
.6600E+01
.6750E+01
.6900E+01
.7050E+01
.7200E+01
.7350E+01
.7500E+01
PLOTTING SYM9CLS
MAG. DELTA P/TCPOYO*A)
MIN.
MAX.
.1500E .00
.3000E+00
.4500E+10
.6000E+00
.7500E+00
.9000E+00
.1050E+01
.1200E+01
.1350E+01
.1500E+01
.1650E+01
.1800E+01
.1950E+01
.2100E+01
.2250E+01
.2400E+01
.2550E+01
.2700E+01
.2850E+01
.3000E+01
.3150E+01
.3300E+01
.3450E+01
.3600E+01
.3750E+01
.3900E+01
.4050E+01
.4200E+01
.4350E+01
.4500E+01
.4650E+01
.4800E+01
.4950E+01
.5108E+01
.5250E+01
.5400E+01
.5550E+01
.5700E+01
.5950E+01
.6000E+01
.6150E+31
.6300E+01
.6450E+01
.6600E+01
.6750E+01
.6900E+01
.7050E+01
.7200E+01
.7350E+01
.7500E+01
.7650E+01
34

QgX
QgY
Figure 3.1. 2. 1.
Average gas velocity inside bubble.
35

36
(3. 1. 2. 3)
Q
+Q+ sin AO ±
Qi-gY cos Pe so that for two points on the bubble surface, and j+1: mj
+1
EPS.
+1
2
{±(Q+ +Q+ gXj gXj+1
)s in
.
Aej+1
±(Q
+ g+Q+ gYj+1 kos AO j+1
} but, from the geometry shown in Fig. 3. 1. 2. 1
(3. 1. 2. 4)
(3.
1.
2. 5) sin Aej+1 -
As.
j4-1 cos A0i-1-1 =
IAX.+11
As j+1 thus
Note that
11+1 = 1j
+
2
1
-(Q gXj
+Q+
Xj+1
)( y+
+
1 j )
+(Q+ Q+ gYj+1
)
X j+1
-X j
)} u mf
00 g co co
( -K (
) dY
00
)
SO
(3.
1.
2. 6)
(3.
1.
2. 7)
(3.
1.
2. 8)

E g
E umf
Vp cip_.
dY +
37
(3.1.2.9) for constant E and Ko
,
(3. 1.2. 10)
And Eq. 's 3. 1. 1. 12-14 can be used directly to calculate the average velocity inside the bubble.
The net velocity of the gas inside the bubble relative to the cylinder surface is obtained by adding to the results of Eq. 3. 1.2. 10 the tangential component of the bubble velocity.
assumed that the vertical component of the bubble velocity is constant and given by (41, 22) uB =
0 umf + K
B
(gR
B
)1/2 (3. 1. 2. 11) or where uB =
0-1)
+
B
B
KB umf
F
2
2
(3. 1. 2.12) and the value for KB recommended by Ref.41 is approximately 0. 7.
Now, the net gas velocity inside the attached bubble is u0net u
Orel
+ u8 cos 0 (3. 1. 2.13)

38
3.2 Analysis of the Two-Dimensional Boundary Layers
Heat transfer from the gas to the cylinder wall is determined by the magnitude of the normal derivative of gas temperature at the wall.
This parameter is established through analysis of the boundary layers which form along the wall.
By neglecting diffusion of heat and momentum parallel to the average gas flow, the two dimensional form of the boundary layer equations can be used.
It is also assumed that viscous dissipation effects are negligible (valid at the low gas velocities involved).
Furthermore, it has been shown by Levy (44) and discussed by Kays (36) that gas property variations have little effect upon laminar heat transfer when free stream values are used.
However, the presence of small amounts of turbulence have a strong effect on the heat transfer, even at Reynolds numbers for which the boundary layer is normally laminar.
Accordingly, an eddy diffusivity term will be included but all gas properties will be evaluated at bed temperature and pressure.
With these assumptions, the appropriate time averaged boundary layer equations are (36, 50, 52) av au
8y az p v g au p u au sip a
-r ay g az dz ay
E
,
) au
(3.2. 1)
(3. 2. 2)

39 p v g ae
8y
+ p u g ae
= a az ay
(
g T 8® ay
2o-a
2- (2(19+Tw)
4 4 4
-(TB+Tw)) (3. 2. 3) where e = T Tw (3. 2. 4)
E and optically thin radiation has been included (see Ref. 's 52 and 62), is the eddy diffusivity and molecular and turbulent Prandtl numbers are assumed to be equal.
The boundary conditions are a) y = 0 (cylinder wall) u = 0 (3. 2. 5)
(3. 2. 6) = 0 a ay au)
T ay a ay 1l1-LTIDE
, ae,
T ay 0 dz
(3. 2. 7)
(3. 2. 7a)
(3.2.7b)
E
= 0 b) y = 5 (edge of boundary layer)
U = U e = TB TW u.
duo dz
dz
(3. 2. 8)
(3.2.9)
(3. 2. 10)

= ay
2 ay
ay ay2
40
(3.2.10a)
(3.2.10b)
(3.2.10c)
E T E T°°
Next dimensionless parameters are introduced yielding the following dimensionless equations:
+ v + u ay
+ au
+ az+ ay+
+ au+ ay az
= 0 (3.2.11) sta._
dz
+
1
ReD a ay+
(0.+E )) (3.2.12)
T + ay v+ ae+
T ay
U
+ az
1 a + a ®+
+ ((l+E T)
)
D ay ay
-2( p a p o-e
3
BD
) (2(® + +T+ )
4 +4
-(Tco
+4
+TW )) (3.2.13) where
E
T
= p
E g T
/p.
.
reduced to
The coefficient of the radiation term can be a p
3D o- A
B
1
ReD r ( a
P o-D
283
B
K g
)
(3. 2. 14) and, for CO2 emission (expected to be the dominant contributor of

41 gas radiation) the order of magnitude of this term is 1/Re
D the radiative term will be neglected.
so
Equations 3.2.12 and 3.2.13 are complicated by the presence of the eddy diffusivity in the higher derivatives.
This difficulty is partially eliminated by stretching the normal coordinate,
Y through the integral transformation
(3. 2. 15)
This transformation does not completely eliminate the parameter from the equations, but the resulting form is more amenable to an integral solution.
Here, it is noted that similarity solutions have been obtained for flow past a horizontal cylinder by Smith and Kuethe (54) and Traci and Wilcox (61), but with more complexity than reported here. As will be established below, the above transformation provides a compact way of including the effect of small levels of turbulence in an integral solution of the equations.
The transformed equations are
0.+E
1 av
+ n
+
+ au
+ az
+ o v+
(1+E
T
) au+ u+ au+ az
+ dp+ dz+
1
2 +
Re
D
(1+E+) n2
T
(3.2.16)
(3. 2. 17)

v+
(1+E
+
) ae
Ot ae+
+ u+
8z+
1
829+
Re DPr(l+e+) at
2
42
(3. 2. 18)
Integration of the above equations from the wall to the edge of the boundary layers gives dz
+
0
V
+ + + u (u 00-u )(1+E
+
)clt +
Sby + +
(u00 -u+ )(1+E+)dt duo° dz
+
1
Re
D
8u+
(0) (3. 2. 19)
SFl u+(1-13+)(1++)d dz+ 0
Re
1
D
Pr
88-
(0) (3. 2. 20)
These equations are further reduced by introducing the thickness
6
1
=
6
V
(1- --L-1-u
)(1+E+)dt (3. 2. 21)
6
62 =
(1- )(1+E+)dt uoo
00
0 e2 = S
0 bH
u co
+)0.+E+)cit
(3. 2. 22)
(3. 2. 23) so that the integral equations are

43 d +26
{u dz+ 2
}
+ u 61 du oo dz
+ d +
{u
+ dz
A2}
1
89
8g
(0)
Re
1
(0)
3g
(3.
2.
24)
(3.
2.
25)
Furthermore, defining
2
M = Re
D62
2
W = Re DPrb
2
(3.
2.
26)
(3.
2.
27) the above equations become where d(u00M) dz du
2{H2vf1(0)
-(-1-+H12)1\4 dz d(u00W)
- 2 {G g' (0) -
2H 0 dz+
2 duo() dz
+
62
H
2V 6V
67
H12 62
(3. 2. 28)
(3. 2. 29)
(3. 2. 30)
(3. 2. 31)
(3. 2. 32)
G
ZH 6H

f = u uoo
44
(3. 2. 33) and ry
5 e
= e
(3. 2. 34)
+ + +
Next, profiles for u
, E , and le are assumed so that the integral parameters can be determined.
Following Pohlhausenis analyses, the following profiles are assumed oo
2 3 4 f a 111V + a271V a3rIV a4TIV e+
(3. 2. 35)
(3. 2. 36) also, the dimensionless eddy diffusivity is assumed to be of the form
E
T
= p
T V
=
TX
1V
(3. 2. 37) where
TIV
'V'
"H' and PT will be established by comparing calculated and experimental results for stagnation point flow with varying levels of interstitial turbulence.
Application of the boundary conditions result in the following values for the constants
(see Ref. 50)

al = 2 +
0
V
6 a
2 2
V a3 = -2 +
2
V a
4
= 1 -
S2
V
6
(3.2.39)
(3. 2. 39)
(3. 2. 40)
(3. 2. 41) b
1
= 2 b2 = 0 b
3
= -2 b
4
= 1
45
(3. 2. 42)
(3. 2.43)
(3. 2.44)
(3. 2. 45) where 0 v factor.
= Re
D
2
5V du
00 dz
+ is the traditional velocity profile shape
Note that these results parallel Pohlhausen's except that
(his
X) is based upon a stretched thickness parameter. Now the integrals are
V
51
5v HIV r
0
(l-f)(1+(3TTIV)driV
= (0. 3-
2
V
120 T 15
1
V
360
)
(3. 2. 46)
(3. 2.47)
5
2
= H2v =
° V 0
1 f(1-f)(1+P v
V
= (0. 11746 -0. 0010580v- 0. 0001102352v)
(3. 2. 48)
+ (3T(0. 03968- 0.00105852V0.0000330752V) (3. 2. 49)

(3. 2. 50) 2
5H
= G2H =
0 f(1-g)(1-1-1,11V)d ri H
4 m=1 a m
Nm
HV
1
(.m+1)
NHV
+13
T (m+2)
1
NHV
4
1 = 1
1
N
HV bi m +1 +1 +1jT m+1 +2 for NHV < 1
4
=1 a
(-
P
1
T
+ m m+1 m+2
N/1
1=1 HV
1 m+1 +1 +
PT m+1
(1+13T)
(
1-
4
1 b/
(/+1)N1
HV
+ (1+13T)
(
1b1
,e
+ 1) for NHV > 1 (3. 2. 51)
Thus the integral parameters are found to be functions of and NHV,
V' f3T where NHV is the boundary layer thickness ratio,
46
N
5
H
=
HV 5V
(3. 2. 52)
The nonlinear differential equations presented above are solved numerically for the bubble region using the IMSL library routine
DVOGER (35).
The parameters
C2v and NHV are obtained from
method (16) as follows.
First note that

thus
M -"7 Re D62
2
Re
D
6
2 2
H
V 2V
2 duco dz duo°
M dz
+
- E2
V
2
H2V
= 0
47
(3. 2. 53)
(3. 2. 54)
(3.
2. 55)
Now define then du
FS-2 = M dz
+
2
C2VH2V
(2V )i +1 (2v)
O'm
F2
8F2
(3.
2. 56)
(3.
2. 57) where i and i+1 denote the ith and (i+l)th iterations and aF v
-H
2V
(2E2V dH2V dC2v
+ H2V) dH
2V dE2-
V
(-0.
0010582-0. 00022046C2v)
+
001058-0. 00006614C2v)
(3.
2.
58)
(3 2.
59)
Next NHV is obtained as follows:

W = Re
D
PrA
2
2
= Re
D
Pr6
2 2
N
V HV
CI
2
2H
48
(3. 2. 60)
(3. Z. 61)
SO defining
W -
C2 Pr
V du
+
(N
HV
G
2H
)
2
= 0 dz+
G
HV
= W -
S2 Pr
V
+ duoo dz
+
(NHV G2H)2
(3. 2. 62)
(3. 2. 63) then
(N
HV
)i+1 = (N
HV
)i -
GHV i a G
Fiv
8N
HV
(3. 2. 64) where i and i+1 denote the ith and (i+l)th iterations respectively and with a G
HV
8N
HV
N
HV
G
2H
8G
2H
(G
2H+ aNHV
)
S2 Pr du: dz
+
(3. 2. 65)

NHV aG
2H
8N
HV
-
1 4 m=1 a mNHm rn
V m+1 ( m+1
T HV m+2
4
b ( m+i +1
) m+1
+ (3TNHV( m+1 +2
1=1
))
< 1
49
N.' m=1
(
1 m m +1 m+
PT
/=1
13/
+1)
NHV
+1
PT m+/+2'
(14-(37,)( i=1
NHV
NHV > 1
(3. 2. 66) u 00W
The initial conditions for the dependent variables uooM and at the stagnation point are t d(u+ co
M) dz+ stag.
pt.
= NI stag.
pt.
co} dz+ stag.
pt.
stag: pt:
W stag pt.
(du«) dz+ stag.
pt.
(3. Z. 67)
(3.
2. 68)
Furthermore, the stagnation point values of M and W are obtained by noting that Eq. 's 3.2.28 and 29 reduce to the following at

50 the stagnation point du a
1
H2V - (H 12+2)M dz
+
= 0 duco
G
2H b
1
-W dz+
= 4
(3.2.69)
(3.2.70)
These equations can be alternately written al - (2H
ZV
+H
)S-2
1V V
= 0 b
1
- N
HV
PrC2
V
G214 = 0
(3.2.71)
(3. 2. 72) and are nonlinear algebraic equations for
12 and NHV at a stagnation point with PT
solved in the same manner described above using Newton's iterative method.
For
and over a broad range of values of PT,
NHV has been found to lie between 1.5 and 1. 6 and the variation in
Clv is as shown in Fig. 3. Z. 1 (note that the value for PT = 0 agrees with previous results (50)).
and N
HV have been established, the two dimensional Nusselt number is easily obtained. From the definition
NuD
W D
(TB -TW) Kg
(3. 2. 73)

4
0
0 2
1
4
1
6
Turbulence parameter, PT
8
Figure 3. 2. 1.
Shape factor at stagnation point.
51

52
Nu
D
K aT D g ay K
TB -TW with 5H obtained from a
+
(o)
(o)=
2 sH
5H = NHV 5V
(3. 2. 74)
(3.2.75)
(3. 2. 76)
= NHV du
Re
D + dz
(3. 2. 77)
Spaulding and Pun (55) have demonstrated the validity of integral methods for analysis of heat transfer due to laminar, constant property flows.
Using the technique to analyze flows with small levels of
"imposed turbulence" requires coupling with experimental data to establish the form of the eddy diffusivity parameter, pT.
This parameter is
R
Ttx
00
= KT vco
A
(3. 2. 78)
(3. 2. 79) where KT is assumed to be a universal constant, /
00 is the length

scale of the turbulence and ulu is the R. M. S. fluctuation of the velocity.
Following Traci and Wilcox (61), the length scale is assumed to be of the form
53 lc°
TR7e Loo
(3. 2. 80) where L is an appropriate characteristic length.
Now, the diffusivity ratio is p
T
= K
T
L
(3. 2. 81)
The constant parameter, KT' is established by matching experimental and analytical results for flow past a cylinder in an air stream. These results are shown in Fig. 3. 2. 2 with KT = 0. 13 and
L = D.
In the case of a fluidized bed, the appropriate characteristic length is taken as the particle diameter and the characteristic velocity is the maximum interstitial velocity.
These parameters were selected on the basis of experimental work by Galloway and Sage (28).
Galloway and Sage inferred interstitial turbulence levels in packed and fluidized beds by measuring heat transfer to an instrumented sphere in a bed of equal size spheres and correlating the results with similar measurements in an air stream. Their correlations were based upon particle Reynolds number (hence particle diameter as the characteris tic length) evaluated at the maximum interstitial velocity.

54
2.0
KT = 0. 13
0
1.5
I.Exp. from
Ref. 54
.111
1.0
10 20 30
Free stream turbulence
(uINIFTeD)
40
Effect of free stream turbulence on Nu
D point for flow past cylinder.
at stagnation
With the above assumptions
(3T = 0. 13u'
D
Nc ip+ Nr2(71+ (3. 2. 82) with 2Q+ being the maximum interstitial velocity.
bubble region

55
(3.2.83)
(3TB = O. 13uI TR;
Dq dp co
3.3 Analysis of Interstitial Corner Regions
The Stokes flow assumption for the cusped interstitial corners reduces the analysis of heat transfer in these regions to solution of the steady conduction equation for the region shown in. Fig. 3. 3. 1.
The edge of this region along the cylinder wall is established by requiring
{Nu
D
}Stokes =7 {Nu
D
}2D at xs
(3.3.1) with {Nu
D
}
2D obtained from the two-dimensional solution for the central region of the channel. Then, the inner boundaries of the region are established by assuming they are lines of constant radius and angle appropriate for the conformal mapping of a unit circle onto the interior of the channel cross section boundary. This mapping is assumed to produce a natural coordinate system for the flow.
thermore the region is assumed to form a curvilinear rectangle on the unit circle with a width in the radial direction equal to half the length in the angular direction.
This assumption is derived from experimental observations of flow along a right angle corner (see e. g. Zamir and Young (73)).
Experimental results indicate that the lateral extent of the corner influence is approximately twice the boundary layer thickness at the corner.

Physical plane Unit circle plane
Stokes 2-D flow B. L.
Stokes flow
Figure 3. 3. 1. Stokes flow regions.
1 -R
=
1
4:'s
0
2

57
3. 3.1 Validation of Stokes Flow Model
The approach used in the analysis of the cusped corner region has been tested against flow along a right angle corner.
Since the pressure gradient in the direction of flow is zero for this case, the solutions for the skin friction and heat transfer coefficients are similar.
Therefore, skin friction coefficients computed on the basis of the model will be compared with available analytical and experimental results obtained by previous investigators.
For constant density flow, the equations governing the axial velocity in the Stokes region (see Fig. 3. 3. 1. 1) is
V2u = 0 (3. 3. 1. 1) with boundary conditions as shown in Fig. 3. 3. 1. 1.
The natural coordinate system selected for this case corresponds to the conformal mapping of an upper half plane onto the cross flow plane as shown in Fig. 3. 3. 1. 1.
This mapping is produced by the transformation z = zh
1/2
(3. 3. 1. 2)
Equation 3. 3. 1. 1 is easily solved in the transform plane and for the stated boundary conditions, the solution is

u 00
Cross flow plane
U=U
00
Transform plane x2
Stokes region x
-D
B. L.
x au
8x1
0
Figure 3.3. 1. 1. Flow along a right angle corner.
58

u = u
( -)
00 xz6
(3.3. 1. 3)
59
Now, the skin friction coefficient is c f
1
2
TO p u
2
00
1
2
2 p u
00
1 h
0
8u
8x
2
(3.3.1.4)
(3. 3. 1. 5) ph
0 u where h
0 is the scale factor evaluated at transformation xz = 0.
For the stated ho =
I dz
I x =0 h 2
1
21 zhl
1/2
(3.3.
1.
6)
1
(3.
3.
1.
7) thus cf
4p.
00
47cl
Z6
(3.
3.
1.
8)
Along the lower wall (y = 0), 47ci = x, so Eq. 3. 3. 1. 8 becomes
4p.
x cf pu00 x26
(3. 3. 1. 9)
The two-dimensional skin friction coefficient corresponds to flow past a flat plate.
The solution obtained using the momentum integral

60 method is (50); c f2D
6 (
P-
)
2D pu
(3. 3. 1. 10)
Therefore, the ratio is c cf
(3. 3. 1. 11)
At the edge of the Stokes region, this ratio is required to be unity.
From the transformation and Eq. 3. 3. 1. 11 xs, the edge of the
Stokes region, is
(3.
3.
1.
12) x s
N
25
= 52D (3.
3.
1.
13) and the variation of skin friction with position is c, cf2D 82D
(3.
3.
1.
14)
This simple result is shown in Fig.
3. 3. 1. 2 where comparison is made with results of finite difference calculations by Rubin and
Grossman (48) as well as experimental data obtained by Zamir and
Young (73).
The simple procedure described above provides results which agree well with "more sophisticated calculations" of the skin friction.

1.2
l 0
0. 8
U
A
0.6 u
U
0. 4
0. 2 z
0-Calculated by
Rubin and
Grossman (48)
Experiments by
Zamir and Young
(73)
Simplified
Stokes model
52D
6 /4-17.ez assumed
0
1
2 3 x/6
2D
Figure 3.3. 1. 2. Skin friction coefficient for flow along a right angle corner.
61

62
However, the details of the flow field are not considered by the model.
Such refinement could be obtained by constructing an "overlap" domain between the inner Stokes region and the outer inertial and two dimensional regions.
This leads to considerable mathematical complication as exemplified by the work of Tokuda (59) and certainly cannot be justified on the basis of inaccuracies due to the simple approach described above.
3. 3. 2 Application of Stokes Flow Model to Cusped Corners
The remarkable agreement between experiment, finite difference calculations, and the simple Stokes flow model for flow along a right angle corner is sufficient justification for its use in analyzing flow along cusped corners.
In fact, the approach is expected to be more appropriate for the cusped corner because conditions producing
Stokes behavior are more pronounced.
The energy equation appropriate for the Stokes region is a ax
(K+
+ g ae+
) +
8x a ay+
8y
) = 0 (3. 3. 2. 1) with boundary conditions shown in Fig. 3.3. 2. 1.
Here it is assumed that the eddy diffusivity is negligible, but the conductivity is allowed to vary. Introduction of Kirchhoff's transformation (3),

a T al's1
T a T a`rs1
Temperature variation is approximately linear INS.
Power law temperature profile
/2
TW
INS.
Physical plane Circle plane
Figure 3. 3. 2. 1.
Stokes region spec ification and boundary conditions.

64
S =
.51
e+
+
(3.
3. 2. 2) reduces Eq. 3. 3. 2. 1 to v
2S
= 0 (3. 3.
2.
3)
The conductivity is assumed to vary linearly with temperature, but the Chapman-Rubes in constant (50) is introduced so that dimensionless conductivity is correct at the wall.
With these assumptions
S(e) = -
1
A
+ +
(e +TW) de+
T+ so
1
+
(0. 5 +T W)
{
2
+2
)+T
+ w
+
(3.
3. 2. 4)
(3.
3. 2.
5) where
T
-/ a
( TW )
/z
1 + 110K/TB
(
T
W
/TB + 110K/TB
(3.
3.
2.
6) and
S0 = S(0) = cR
T+
B
(0. 5+T w)
(3. 3.
2. 7)
Equation 3. 3.2. 3 is solved on the unit circle plane, then the heat
to the lower wall of the channel and the Nusselt number at the edge of the Stokes region are obtained through the mapping function.
The temperature variation along the curved portion of the corner is assumed to be such that a power law variation is produced on

65 the circle plane. With this assumption, the variation on the unit circle is
+
(co
) = 1-
24) s 1
(I's 0
T as
8r r, 0) = as
( r , 4
1
) = 0
(3. 3. 2. 8) and p.T
will be determined from the mapping function so that an approximately linear variation is obtained in the physical plane. For this variation in e, the boundary conditions on S
s( cos
S0
24's 01211T+T+
2
Sis0
1
(
2
[
+
+ T
)
W
1-
24's 1] 11T s0
(3. 3. 2. 9)
(l)s0 for 0 < c)s and 5(1' si's 1) SO fo r
S's0
2 5 `I's
1
< (1)s0
(3.
3.
2.
10)
S(R
1, cl)s 1) = 0 (3.
3.
2.
11)
The general solution to Eq. 3. 3. 2. 3 with boundary conditions
(Eq. is 3. 3. 2. 9 - 11) determined by the method of separation of variables (3, 33) is

66 co
S(r Os]) = c0 1n(Rr ) +
1 n=1 c n r
2
R
X
) n cos
(Xn4)s1)
(3. 3. 2. 12) where
Xn nir /cps .
The Fourier coefficients, Co, cn, are obtained by fitting the series to the outer boundary conditions.
The resulting expressions for the coefficients are
2
(1zth
T
81
)
2 p.
T cl)s0
+ T
+
W
(1-
2 c's 1 )4 T)
Os° uP
1
(3. 3. 2. 13)
( 1+2 p.T )
(1+11T)
1
+
2( +W T )(1+11T)(1+2p.T)
2
(3.3. 2. 14) and cn
S0
0
2)4.
(1-R n)
1
S
0 cl)
(1..
2
(1-
4) s 0 W
(1- s ) s0
II
T
X cos ( n"s1 s0
) cici)sl
1
+
+
)
0
2 W
4>s0 cos( n"
1 s°
)11051
(3. 3. 2. 15)
The above expression for cn contains integrals of the form

0
S1
(1-u
H)
H cos zHuH du
H
67
(3. 3. 2. 16) where zH = cyrr/2 and uH = 24s
1
0 as (24)
This integral is expressible
S
0
1
(1-uH) aH cos zH u
H du
H
2
1
1
SI (1
0
-uH)all(eizHull +e )duH
(3. 3. Z. 17)
1
112-Fa
H)
2 r(l+aH)
H
(1, 2+a
H ' iz ) +
H H
(1, 2+a
H
; -iz
)}
H where
(3. 3. 2. 18) r is the ordinary Gamma function and is the confluent hypergeometric function (see Ref.
1 or 24).
The parameter zH is generally large so that the asymptotic forms of the hypergeometric function will be used. As given by Ref.
24 these approximations are
'14(aH' cHI xH) r
(CH) r(cFraH)
N
X n=0 iTr E e xH aH
-cH+1)n
(-xli) n!
-n
HI
-a
H
-N-1
) +

68 r(cH)
F(aH) exH aH
KH
N
(cH n=0
HI a
H
-c
H n .
)n(1 )n
(3 3.2.19) where
= (a
H
)(a
H
+1)(a
H
+2)... (a +n)
(a
H
)
0
1
(3.3.2.20)
(3.3.2. 21) and EH = 1 for Im(xH) > 0 and -1 for Im(xH) < 0.
Using this approximation, together with the identity
e
--xH.T.
H
(1+a
H cr_r; xH) the asymptotic form of the integral (Eq. 3.3.2.16) is
(3.3.2.22)
1
Hcos(zHuH)duH
(1 +a )2 r(i+ce
H sin(zHa
TT
H
2
0 a
H
(1-a
H
)(1+a )2
2 zH
+0(
)
4 zH
(3.3.2.23)
Note that for the a = 0,
H

0 lcos(zHuH)du sin zH zH
(3.3. 2.24) and this is consistent with the limiting form of Eq. 3.3. 2. 16 for a
H
0.
Since a
H is indeed a small parameter, the approximation given above should be reasonable.
Using the above result to approximate the integrals appearing in Eq. 3.3. 2. 15, the Fourier coefficients become where cn
(1-R
1 kn
-S0
0
2
+
W
)
R t11T
1 s sin
2 cos nir
2 sin
2
1++ p
2s
1+111,
+ P3 s 1+4.1, n n n nn cos
2
P4s 1+21.i,
(3.3. 2. 25) n
2
P5s
R
1 s
T
W
(1+11
T
)2111+11T)
1+11T
( ) p2s
_
1-1, TIT
2 1 s
P3s
T
)
2 r(1+21.1.
T
)
1+2..irr
(3. 3. 2. 26)
(3.3.2.27)
(3.3. 2.28)
(3.3. 2. 29)
P4s -11T1TP3s
69

70
135s
2 f.I.T[(1-2p.T)(1+2 )2+T+
AAT(1+'1T) (1-i"Crn
Tr
2
( )
(3.3. 2.30)
Now the parameters required for determination of the heat transfer can be calculated.
First, the local Nusselt number is
Nu
D
K
B
(TB-TW)
KW
K
B
D h0
8T+ r
)r
= 1
_
KW h
+
0
(7
1
T+
)W as ar (11's1)
(3.3. 2.31)
(3.3.2.32)
(3.3. 2.33) and as+ aT
(3.3.2.34) so
{Nu
D
}Stokes = at x as h+ ar
0
(1,(I)s0) (3.3.2.35)
Also, the total heat transfer or integral of the Nusselt number is
(3.3. 2.36)

71
(1)so
4s0/2 as
(1 , it.
s 1 s1
(3.3.2.37)
Thus the two parameters required for analysis of the heat transfer in the channel corners are as
(1,4) sO and
Aso
4)s0 /2
( ,
)thpis
1
Differentiation and integration of Eq. 3.3.2.12
gives as ar
(1,4) so
= co
+
00 nn-
(i)s0
(1+R
1 cos nn (3.3.2.38) and
5,4s o c)s0/2 as ar s 1
c0(1)s0
00
2Xn
(1+R1
1
(3.3.2.39)
Using the approximations above for the Fourier coefficients and truncating the series, the above parameters become as ar
(1,(ISs0 ) co
TrS 0
1
+
(1)s0(-2 +TW)
X
1 co
1+R1 ls kn
(1-R1 n
11T sin 2 cos nn- +
P
2s cos nn-
2 cos nn +

and
72
P3s
142
T nrr s a cos mr +
2
P4s
112
T nrr cos ..., cos nrr
+
P5s cos nrr (3. 3. 2.40)
X
TrS
0
CO -
(13.
s0
(
1
+T+ )
W
2k
1+R1 n
1-R1 n nrr
2 cos nrr +
P2s
I-LT
CO s cos nrr
P3 s n
2I-LT rirr sin cos Tin
2
P4s
2
[1, nrr cos cos nrr +
2
P
5s cos nrr
2s oo nrr cos 2 co s nrr
1 sin nrr
2 cos nrr
2I1T
+
4s oo
1 nrr cos 2 cos nrr
2 nP-
T cos nrr n
(3. 3. 2.41)

73 s 0
/2
/a
8r (
1, c) sl kl.(1) c0(1)s0 sl
2
00
So
(
1
+
+ TW)
1
X
R
1 s
1+ p.
s in
2 nTr
2 n
R 2s
1+
11
T cos
2 n
P3 s
1+211T sin nn
2
+
134 s n1+211 T
R
5s n
2 sin
2 nrr
(3. 3. 2.42)
= cOs0
SO
+
2
(
1
+T
2
+
W
)
X
%ow
13
1 s
1+[1.1, n s
1+2 [IT s in
R
52 s
+ p
00
1 oo 00
2 nn
2
3s nT
1 n
1+211
(3. 3. 2. 43)
1

The infinite series appearing in Eq. 's 3. 3. 2. 42 and 43 are (see ref.
67):
1 nir s in 2 cos nrr naH
00
1
(2m-1)«H oo
1
(-1)m
(2m-1) a
H
Go naH
2
1
1
2 a
H
00
(-1) m -1 m a
H
(11
Zm
) a
H
{(1 -2
1-a
Hg(aH)}
=
/
(as)
(3. 3. Z. 44)
00
-1)m a
H
(2m) e
(3.3. 2.45)
1
74
00 co s nrr
11
-ln 2 (3. 3. 2.46) oo
1
1
/ oo
.
sin
2 nrr
2 n
1+a
H n
2
GE ' O. 91596
2
1 +a
H
(1-Fa
H
)
(3. 3. 2.47)
(3. 3. 2. 48)

75 where
(1-1-a,
H) using (see Ref. 23) is the Rieman zeta function and can be approximated
Ql+aH) = 0.435 +
0a 935
H
(3. 3. 2. 49)
So, the final forms of the required parameters are
8S
(1 sO
) CO
.0.0(
TrS
0
1
+
+Tw) x
1
2X.
[1+R n)
1 x
(1-R1n)
-] Pis P35 n
[
+
IIT n
211T sin nu
2 cos tyrr
[
132s
P4s nTr
+ cos cos tur
T n n
(Pls+(32s)I(117,) (1335+P45)(211T)
(3. 3. 2. 50) and

76
/2 as ar
(1, (ps c
0
(I)
1
2 s0
X
1
(-2 +T+
W)
(1+R
1
2X n)
(1-R 1n)
1 p3s
1+2111, sin
2 nTr
2 p5 s n
2 s in nTr
2
Pis
1+
[..L
T
1+p,
)
133s
2
1
1+2
11
T
(1+2[.1
T)
+ p5sGE
(3.3. 2.51)
The number of terms, N, that in the truncated series is selected such
1 <<
1
(3.3. 2. 52)
Calcluated values of
In R1 so as
8r
(1, (1)s0) were found to be within the range 1.3 to 1.5 for the following ranges of parameters

0.5 < R
1
< 0.99
1
<99.0
0.01 < p.T < 0.30
77
(3.3.2.53)
(3.3. Z. 54)
(3.3.2.55)
The parameter
1
o
S's0
8S
Or
(1,4) )4;4
was also found to be insensitive to variations in T+ and R1, but highly sensitive to p.T
for small values of p.T.
The variation of this parameter with p.T
results, the Nusselt number at the edge of the Stokes region is
13 s
SO
R 1)h0
0
(3.3.2.56) where
= -ln R1 as
(1' sO
)
S
0
= 1.4.
Also
(3. 3. 2. 57)
SO
R1
{Nur}x s
((l)c-4))
(3sso s
(x
) s
(3.3.2.58)

20
15
10
35
30
25
5
0
0
.2
Profile parameter, p.rr
.3
Stokes region total energy parameter.
78

with = -ho ln( -((l)c-0).
mapping function.
The parameter
3.3.3 Determination of the Mapping Function
79 is obtained from the
The parameter a function of xs derived in Sec. 3. 3. 2 is best obtained as s along the cylinder wall.
This is most easily obtained by determining the function which maps the channel cross section onto a unit circle.
During the initial stages of this investigation, the inverse mapping (unit circle onto cross section of channel) was developed for the purpose of defining a natural coordinate system for use in a three-dimensional integral analysis of the entire flow field.
This integral approach was abandoned because information regarding velocity and temperature profiles was inadequate.
2
However, the mapping technique was well established for the inverse mapping and will be used to obtain the function I's.
It is noted here that either mapping requires solution of an integral equation, but the equation is nonlinear in the case of the inverse mapping chosen (see
Ref.
17).
The cusped corners of the channel cross section initially resulted in convergence problems in the solution of the integral equation, so an intermediate step was introduced in which the unit
2
In any integral analysis of the boundary layer equations, it is necessary to assume the form of the solution in advance.
The boundary conditions involved for flow in the double cusped channel are not amenable to such an approach.

circle is mapped onto a unit semicircle then the unit semicircle is
80 mapped onto the channel cross section.
This mapping sequence is
The mapping of the unit circle onto the unit ,semicircle is obtained by combining the following (38) a) semi infinite strip onto unit semicircle zs = eiw
b) upper half plane onto strip w = Tr cos
-1U c) unit circle onto upper half plane
1-zc
U = i(
1+zc
(3. 3. 3. 3)
Combining Eq. 's 3.3.3.1-3 gives zs elm.
/2(1+z
)
(1-z )±47(1+z
2 1 2
Introducing polar coordinates z c
c i4 zs = p se iOs

zs
I
Figure 3. 3. 3. 1. Mapping sequence.
81

82 the mapping function gives where
0 s
Tr
2 + (01 -43
P s
Rs1
Rs3
(3. 3. 3. 7)
(3.3.3. 8)
Rs
1
= ((l+rc cos )2
+ r c sin 4)
)
2
)
1/2
1
= tan
-1
(
s in 4) c l+r cos 4) c
Rs3 = [(1+42Rs2 cos 4)2
)2
(3.3.3. 9)
(3. 3. 3. 10)
(3.3.3. 11)
+ (4-2Rs2 sin 4)2 - r sin (I) )21112
4)3
= tan
-1
(
NriR s2
1+42Rs2 sin 42 cos 4 -r
cos 4)
(3.3.3. 12)
42 = 2
1
-1
2 rc sin 24)
2 l+r cos 24) c
)
Rs2 = ((l+r
2 cos 24)
)
2
+rc
2 sin
2
24) )
1/4
(3.3.3. 13)
(3. 3. 3. 14)
The mapping of the unit semicircle on to the double cusped channel cross section is obtained by mapping a unit circle on to the region formed by reflecting the channel cross section about the axis as shown in Fig. 3. 3. 3. 1.
This function is obtained using

83
Theodorsen's method (Ref. 58) as outlined below.
First note that any function which is analytic inside and on a unit circle can be expressed as f (z ) = us + iv s s s
(3.3.3. 15)
1
2Tr
S'27 u s 1
(0
0
2
(1-p
)
1 -2p s cos(Os -4) +p
2 s dqi
+ i s(0) +
2Tr
51
0
2Tr us 1(4))
2ps sin(Os-qi)
1-4 s cos(0
-4J)-1-p s
2 s d4)
(3.3. 3. 16) where us 1(0s) is the value of us on the boundary of the unit circle.
Next, take fs = ln( z (z
)
P s zs
)
(3. 3. 3. 17) where z (z
) is the mapping function.
With this choice, Eq.
P s
3.3.3. 16 gives and
-y
- 0 s
1
=
2Tr y
0
2Tr l cos(O-0+p2 s s d r = ps exp
2ir
.51
0
2Tr
1 -2p
(1-13
2 s
) cos(0 s
(3.3.3. 18)
2 s d4 (3. 3.3. 19)

84 where rl(es) defines the boundary of the channel cross section.
Evaluation of Eq. 3.3.3.17 on the boundary of the unit circle gives the following integral equation for r 1(es): y(As) = As +
1 aff
0
271-
In r
1 cot(
O s
-LP
2
)4
(3. 3. 3. 20)
This equation actually provides y(es) and from r
1
(As ) = r
1
(y(0 s
)) rl(es) is determined
(3.3.3.21)
The integrand of the integral appearing in Eq. 3.3.3.20 is singular at
4i = A s so the principle value must be determined.
The solution is obtained numerically by iteration and the unit circle is divided into a finite number of arcs. Then the value of y for the kth arc and ith iteration is yk = Ask +-1-
2Tr
0
2-rr
In rL1 -i cot(
0
2
= Ask + 2n
0 e sk -1
ARG 4 + sk+1
ARG c14,
Ask-1
2-rr
Osk+1
ARG d)
(3.3.3.22)
(3.3.3.23)

85 where
ARG
i
-1 cot( q -e s z
)
The middle integral now contains the singularity and is approximated as follows
0 sk+1
In r cot(
1
0 sk-1
2
0 sk+1
In r
1 i-1
(Osk)+(tP-Osk)
0 sk-1 clan des
= 2(ln ri1
-1) k
In
0 sk+ 1-0 sk
0 sk-1-0 sk k
2 sk)
(3.3.3.24)
+ 2 d(ln ri-1 dOs k(esk+1
-0 sk-1)
(3.3.3.25)
The remaining parts of the integral are approximated as follows
0
0sk-1
ARG di + J
0 sk+1
ARG 4
2
jlk +1
[(l n r
).
1 i-
1
+(ln r 1
)ji+-111
1
0 sj esj-1 cot(LP
-O
2
(3.3.3.26)

86 jVk jVk+1
[(in r
).ji
1+(ln r di÷1]
In sin( es. -Osk
) sin( es.'
-1
-Osk
2
(3. 3. 3.27 ) so that the nonlinear system of equations at the ith iteration is yk = 0 sk
-
1
27r
2 (in In
A k -1
+
Lan r
1 dOs
N
X (A +A k -1
)1+
2 jVk jVIK4-1
[ (ln r
1
)i-1+(ln j r di-
_11
X In sin(
Os
2
0 sk
/
2
)1
(3. 3. 3. 28) where L.
= k eSk+1 esk
The derivative appearing in Eq. 3. 3.3. 28 is approximated using the Lagrange three point formula (16) as follows d ln r
1 des d ln r
1 dy dy dOs
(3. 3. 3. 29)
dy (0 sk
(0 sk+1
-0 sk)
-9 sk-1
)(0 sk+1
-0 sk-1
) Yk-1
1
(0 sk
-9 sk - 1
)
1
(0 sk+1
-Ask) Ilk

87
(0 sk
-0 sk-1
(0 sk+l-osk-1)(Ssk+1 -0 sk
) k+1
Finally, the boundary of the channel, geometry as follows (see Fig. 3. 3. 3. 2).
(3. 3. 3.30) rl(y) is related to the
Figure 3. 3. 3.2.
Channel geometry parameters.
r 1R(y) c r cos(V-Y c)-Lcos
2
(y-yc)-(1-(
R c
)
2
)j
1/2 for the circular arc portion, and r l(y) r for the linear portion. Also, the derivatives are
(3. 3. 3.31)
(3. 3. 3.32)

dr
1
1
= -s in(y-yc)
88 cos (y-yc) cos
2
(y-yc)-(1-(R--)"
(3. 3. 3. 33) and dr
1 dy
= cot y
(3.3.3.34) for the circular and linear sections respectively.
The computer program implementing this solution technique has been tested for the case of the mapping of the exterior of a unit circle onto the exterior of a square region. This same test was used by Theodorsen and will be used here, for convenience. The function fs
fs(zs) = zs
)
(3.3.3.35) so that the integral equation to be solved is y(0 s
1
) = 0 s 27r
0
2Tr Os -tts
In r1 cot( 2) dtP (3. 3. 3. 36)
The computer solution is shown in Fig. 3. 3.3.3 and compared with results originally reported by Theordorsen (58).

0.35
0.30
0.25
0.20
Fa
-r
0.10-t--
.
05
Theodorsen (58)
calculations
0
I
0.2
0.4
0.6
Circle plane coordinate, 4), radians
1T j4
in(ri(4))) for mapping exterior of unit circle onto exterior of square.
89

The parameters xs and '/s required in the heat transfer analysis are determined as follows.
First, note that along the cylinder wall portion of the boundary, y = 0, so xs = r and is given by Eq. 3.3.3.19. Thus, after ps is obtained using Eq. 's
3.3.3.8-13 and taking Os = 0, xs is
90 x s
= p s exp
0
21r
(1-p2)
1-2p s cos 4,+p2 s d4)}
= p s
1
0
(1-p
2)
1-2ps cos 4,+ps d
(3.3.3.37)
(3.3.3. 38) due to symmetry. Equation 3.3.3.38 is evaluated numerically using the approximation
ID ln rl
1-2ps cos tp+p2 dkp
J-1
(ln r ).+(ln r
1 j
)
1 j+1
2 j=1
2
1 -p
2
X
(
P
)tan
1(
(1+P
1
-ps ) tan -"-"))
2
(3.3.3. 39)
Thus

91 to x s p s exp
J-1
[ln r1).+ ln(rdi+11 j=1
X
l+p
1 -p s
)tan
1+p 1')
-tan .1((
2 1 -ps
(3.3.3.40)
Determination of the parameter §s involves evaluation of the scale factor h0.
Since rc = 1 and y = const, the scale factor reduces h
=
0 ax ac
= ax ap s
)8 =0( s a Ps act,
)r
=1
(3.3.3.41)
(3.3.3.42) with ax
' aps
)8 s
=0 x
GX
1 ---§+ ---s p s
Tr
J-1 j=1
).
)
1
1 3+1 tan Lpi+1/2
1 p )
2 2
+(l+p) tan s
LPj+1
2
J tan LiJi /2
2
2
41"
[(1-ps)++ps ) tan ---1]
(3.3.3.43) and aps
84 alts1 Rsi 8Rs3
1
Rs3 a4 Rs3
8(1)
(3.3.3.44) with

92 aRs
1
84) rc sin cl)
Rs
1
(3.3.3. 45) aRs3 acl)
1
Rs3
42(1-cos 4) cos (1)2 aRs2
-Rs2 s in
2
+ sin cl)
1
[
8R
-NE sin (1)2 acis: +Rs2 cos
2
+ 2R aR s2 act) s2
-I-1 2 Rs2(cos c1:12 sin (0-sin c1)2 cos cl))
(3. 3. 3. 46) aRs2 a(i)
_ sin 24)
Rs2
(3. 3. 3. 47)
Results of calculations for channel width parameter, s
, equal to r
, p
3. 3. 3. 5.
1.27 r
, and 1. 5 r are shown in Fig. 's 3. 3. 3.4 and p p
Figure 3. 3. 3. 4 is the integral equation solution for N(es) and Fig. 3. 3. 3.5 provides the matching parameter, cl's(xs).
For s p
= 1. 27r
, the integral equation solution did not converge completely in the vicinity of the upper corner, but I's appears to be little affected by this difficulty.
The parameter
[IT required for determination of the total energy transfer is estimated from the variation of (xc -x) /r as a function of
Assuming the mapping behaves like a straight sided corner onto a semi-infinite plane, the variation would be

x - x
I
(ci)c-(1))1
P.
93
(3. 3. 3. 48) so that
In (x -x) c
In Inc -14)1
(3. 3. 3. 49)
This approximation in indicated in Fig. 3. 3. 3. 6 and p.T
is seen to be approximately constant to within 0. 35r of the corner.
For the present analysis, p.T
will be assumed to be constant throughout the cusped corner region. In view of the variation of [IT
(decreasing toward the corner), this is equivalent to a temperature variation in the physical plane similar to that shown in Fig. 3. 3. 3.7.
0 0. 5 1.0
Os, radians
1.5 Tr/2
Figure 3. 3.3.4.
Integral equation solution for mapping function.

0. 5
0. 4
0. 1
0 0 . 2
0. 4
Figure 3. 3. 3. 5.
0. 6
S p
0.8
1. 0
Stokes region matching function.
1.2 2
1. 4

2.0
1.5
0.5
0
0
4 6 8
-1n14'c -41
10
function.
12 14 16

Figure 3. 3. 3. 7.
Temperature profile on cusped wall.
96

3. 4 Heat Transfer Due to Particle Radiation
The heat transfer due to particle radiation is approximated by assuming the bubble boundary and particle boundary are isothermal gray surfaces.
These boundaries produce the two enclosures shown in Fig. 3.4. 1 and the radiative exchange is analyzed using the net radiation method (52).
For the enclosure formed by the particle boundary and cylinder wall, the net transfer equations are
97 eW
1-e e p p
)q4
=
0_(7,4
W B
-(
1-e eW
)q1 q4 e
= -o-(T 4
W
-TB
) so that
(41 energy transferred to surface
1 e o-(T p
4
B W )
1
1+e
( p ew
-1)
(3.4. 1)
(3, 4. 2)
(3. 4. 3)
Similarly, for the enclosure formed by the bubble boundary and cylindar wall the net energies satisfy q2 eW
(1-eB)
4 4 q3 = o-(Tw-TB) eB
(3. 4. 4)

A2 (1-e
)
e
W q2+
1 eB
A
(1-e
2
-(1- A3) eB q3
A2
= o-
A-3 so that
-q2 r energy
Ito wall at
2
A2 e (T4
B B
A3
-T
4
W
)
1
+e B(
B e --w
A2
A3 with
A2
A3
=
(0
B2
-0
B1
)
-1 R 0
2
+R
B
2
-1
2RORB
(3. 4. 5)
(3. 4. 6)
(3.4. 7)
98
Figure 3.4. 1. Enclosures for radiative exchange calculation.

The above heat fluxes are converted to Nusselt numbers as follows.
99
NuD
WD
Kg(TB-TW)
(3. 4. 8) then
NuR
D
1
= Bi
R e p
+4
(TB -TW)
(3.4. 9)
Nu
D 2
= Bi
R
A2 e
B
+4 +4
(TB -TW)
1
+eB(
A
3 e
W
A2
A
3
(3. 4. 10) where
BiR o-D(T
B
-T
W
)3 g
(3.4. 11)
Since TB is used in Eq. 's 3.4.9-10 the emissivity of the particle surface adjacent to the wall is adjusted to account for cooling of particles there. Measurements of this effect have been performed by
Baskakov, et al. (4) and their results are used in the calculations reported in Chapter IV.
Note that the effective emissivity is reduced considerably by small changes in temperature at the surface, for example a 10% drop in temperature reduces the effective emissivity by about 35%.

100
3. 5 Composite Model
The total Nusselt number for the cylinder is obtained by integrating the contributions of flow within the attached bubble and interstitial channels and adding the particle radiation contribution.
The local values are computed by starting at the lower stagnation point and proceeding along the right side of the cylinder to the upper stagnation point and repeating the procedure for the left side of the cylinder (see Fig. 3. 1). The solution for the bubble region is obtained through numerical integration of Eq. is 3.2. 28 and 29.
When the lower stagnation point lies outside the bubble, the initial conditions for the bubble region are obtained by assuming the gas velocity accelerates from zero over a distance r to the average gas velocity inside the bubble.
The local velocity and gradient required for determination of the
Nusselt number are determined from the results of Sec.'s 3. 1. 1 and
2 as follows.
For the bubble region u+ ±(uOrel +uB cos 0) (3. 5. 1) and
duOrel
2( dO dz+ uB sin 0) (3. 5.2) where the upper sign is used for the right side and lower sign for the

left side.
For the channels, the velocity is assumed to vary linearly from zero to 2Q+ g so that the gradient is
101 dux+ 2Q+ dz+ Lc
(3. 5. 3)
This linear variation in velocity corresponds to stagnation point flow
(see Ref. 50) so values of v and NHV are obtained from the stagnation point solution. When the average gas flow stagnation point lies outside the bubble, the velocity gradient is obtained from differentiation of Eq. 3. 1. 1. 15 and is duco dz+
_ p_7 dY
00 a
2
KR ae
2
A ID a fdp dY+ with a2
802
Mak
Ap a ldp dY
/00R
=1
= -2 sin 0 +
N j=1
*2 2 sin 0
Bi
1 +R "2 -2R*
Oj
03 cos(e-e0j)
R =1
+ 4R0j[sin(20-0
O j
)-R
Oj sin 0
Oj cos(0-0
Oj
)]
*2
(1+R
Oj
*
-2R
Oj cos(0-0
))
Oj
2
(3. 5.4)

102
16R [sin
2
(0-0 )(-sin 0+R.
Oj Oj s n 0 )]
Oj
*
7' (1+R O2 -21R j
*
Oj
(3. 5. 5)
3)
The average Nusselt number for the channel is determined by using the two-dimensional value to locate the edge of the Stokes region then calculating the average from
(Nu
D
)
AVE
= s s p
51
0
P NuD(x)dx (3.5.6)
D2D
D _51 el
)s0 sP
(t)s0 /2 a as r (1Ps1)4s1 (3. 5. 7) with the integral parameter as shown in Fig. 3. 3. 2. 2. The parameters
'I' s
(x s
) and
1.1.7, required for the matching and evaluation of the integral are obtained from the mapping function for the particular channel geometry.
The primary input parameter for this analysis is the average particle spacing, Lc or s
.
By considering the region within one particle radius of the surface, the parameter s is related to surface voidage by
0. 724
for rectangular packing at the surface, and
(3. 5. 8)

sp
0.780
7177 rp
103
(3. 5.9) for hexagonal packing.
So the following approximation will be used sp
0.75
(3. 5. 10)
The local values of Nusselt number obtained using the analysis results described above are integrated numerically using the trapezoidal rule.
Then thermal radiation transferred to the cylinder wall from the bubble and particle region surfaces is computed and added to the convective contribution to obtain a total Nusselt number for the cylinder.

104
IV. RESULTS
The analytical model described in Chapter III for heat transfer to a horizontal cylinder immersed in a bubbling two-dimensional bed has been coded in Fortran compatible with the CDC CYBER 75 computing system.
Descriptions and listing of the programs developed can be found in the Appendix.
These programs have been used to perform calculations for cold-bed and hot-bed operation with particle diameters of 3 mm and 6 mm and cylinder diameter of 50.8 mm
(2 in. ).
The basic input parameters for these cases are listed in
Table 4.1, with Reynolds number computed using the correlation for particle Reynolds number at umf given by Kunii and Levenspiel
(41) as follows where
Re
=
Re
= rite
D
D d \
P'p / d
3g
33.72+0.0408
v2
B
(La
P
1/2_
(4. 1)
(4. 2)
The bed and wall temperatures for the hot-bed case are based upon recommendations by Ref. 57.
The baseline voidage distribution at the cylinder surface is shown in. Fig. 4.1.
This distribution is based in part on recent capacitance probe data taken by General Electric (56).
In all cases, the ambient voidage was assumed to be 0.5 and Prandtl number 0.7

105 and for the high temperature cases, the effective emissivity in the region of particle contact was assumed to be 0.45 (consistent with data given by Ref. 4 for the TB
bubble surface and wall emissivities were assumed to be 0.9. On the basis of data given by Galloway and Sage (28) the interstitial turbulence intensity was assumed to be 0.2.
Table 4. 1.
Baseline parameters D = 0. 0508 m, p = 1 atm.
T
B
(°K)
T
W
(°K) u mf d =3mm p
(m/s) d =6mm p
ReD d =3mm p d =6mm
P
310
1117
300
659
1.39
1.97
2. 11
3.66
4267
835
6492
1553
K /D g
2.
(W/m K)
0.531
1.44
BiR
0
3.93
Distributions of average velocity, Stokes region edge location and convective Nusselt number for the 3 mm hot-bed case without bubbles are shown in Fig. 's 4. 2-4.4. The sudden changes in Nusselt number and Stokes region edge location are directly linked to the voidage distribution through the average velocity distribution.
The relatively large Stokes region for this case suggests- that this represents a lower limit on the validity of the boundary layer model used in this thesis.
Figures 4. 5 and 4.6 are distributions of Stokes region edge location and Nusselt number for the 6 mm cold-bed parameters.
Comparison of Fig. 's 4. 3 and 4.4 and Fig. 's 4. 5 and 4. 6 indicate the

1.0
O
0 tt 0. 5
10. 65
10.438
1
Stack
3
Angle, radians
4
Figure 4. 1.
Surface voidage distribution.
5
I.--
6
2-n-

15
± b
..-1
0 i
10 "'"------.,.
73
-1-$
U)
F-4
.1) ai
>
1
1
V
I
3
I
Angle, radians
I
Figure 4. 2.
Average interstitical velocity at surface, no bubble.
I

1.0
..,
PA v)
0.75
a)
...
2
I
3
I
Angle, radians
4
I
5
I
6 2 Tr
Figure 4.3.
Location of edge of Stokes region, 3 mm hot parameters, no bubble.

200
Z 150
100
50
0
1
2 3
I
Angle, radians
4
5 6 2ir
Figure. 4. 4.
Convective Nusselt No. distribution, 3 mm hot parameters, no bubble.

1.0
cl)
Sa4
0.75
Cl)
0. 5
0. 25
0
Angle, radians
Figure 4. 5. Location of edge of Stokes region, 6 mm cold parameters, no bubble.

z
300
200 a) cn a)
100 a)
0
I
1
2
I I
3
Angle, radians
4
Figure 4. 6. Convective Nusselt No. distribution, 6 mm cold parameters, no bubble.

112 effect of Reynolds number on these parameters, that is a relatively smaller Stokes region and, hence, larger fluctuations in Nusselt number.
The effect of a single two-dimensional bubble with diameter equal to cylinder diameter on the distributions of average velocity and Nusselt number is demonstrated in Fig. 's 4. 7-4. 19 for the 3 mm hot-bed parameters.
The bubble centers for these cases are at 1.5
cylinder radii from the cylinder center and the angular locations are shown in Fig. 4.7. Figures 4. 8, 4. 13, 4. 14 and 4. 17 are print plots of the relative pressure fields for these cases (see Table 3. 1. 1. 1 for symbology).
Boundary layer separation occurred inside the bubble for configuration 42 so that results shown within the separated region are approximate.
A comparison of total heat transfer coefficients for cold-bed and hot-bed operation with 3 mm and 6 mm diameter particles is shown in
Table 4.2. Also indicated by Table 4.2 is the relative contribution of interstitial turbulence for these cases.
Note that the effect of inter stitital turbulence- is greater for cold-bed operation because the
Reynolds numbers are higher.
Also shown are the radiative contributions as well as the total convective heat transfer coefficients.
Generally, these data as well as the data reported below are within the range of experimental data reported in the literature
68).
However, detailed experimental validation of the model can only

39 40
Tr/6
41 42
Figure 4.7.
Bubble configurations.
113

SCALE.
DELTA X= .1000E+00
DELTA Y= .1000E+00
CASE NO.
39
. -G-G-G-G-G-G-G-5-G-G-G-G ------------
.--F-r-F-F-F-F-E-F-F-F-F ----------- - -
. ------------- G-G-G-G-G-G-G-G-G-G-G .
- - - - - - --G-G-G-G-G-G-G-G-G-G-G-G-G-G-G- -
.-F-r - - - - - - - -
-
- -G-G-G-G-G-G-G-G-G-G-G-G-G-G-G- - - - - -
.
- - - - -G-G-G-G-G-G-G-G-G .-G-G-G-G-G-G-G-G-G- - - - - --------- F
.
-------------- F-F-F-F-F-F-F-F-F-F .
. -------- F-F-F-F-F-F-F-F-F-F ------- -------F-F-F-F-F-F-F-F-F-F- -----
.-E-E ------------ F-F-F-F-F-F-F-F-F-F-F .-F-F-F-F-F-F-F-F-F-F-F ------------ E
.
.-E-E-E-E-E-E-E-E-E-E --------- F-F-F-F-F-F .-F-F-F-F-F-F --------- E-E-E-E-E-E-E-E-E .
- - - - - - -E-F-E-E-E-E-E - E - - - - - - - - -
---------- E-E-E-E-E-E-E-E- - - -
.
.-0-0-0-0 - - - - - - - - E-E-E-E-E-E - - - - - -
. ------- E-E-E-E-E-E- - - ------ 0-0-0 .
.-0-0-0-0-0-0-0-0-0-0 ------- E-E-E-E-E-E- - -E-E-E-E-E-E -------
.
- - - - - - - -0-0-0-C-0-0 - - - - - E-E-E-E-E-E .-E-E-E-E-E-E ----- 0-0-0-0-0-0- - - - - - -
.-G-C-C-C-C ------ - -0-0-0-0 ----- E-E-E-E .-E-E-E-E ----- 0-D-D-0- - ------ C-C-C-C
.
- -C-C-C-C-C-C-C-C-C ----- 0-0-0 ----- E-E .-E-E ----- 0-0-D ----- C-C-C-C-C-C-C-C-C-
.
. ----- - - - - -C-C-C-C-C- - - -0-0-0- - - -E .-E- - - -0-0-0- - -
- - - -C-C-C- - -0-0- - - -
-C-C-C-C-C- - - - - - - -
----- 0-0- - -C-C-C- - - - - -9-8-9-8-9-8-4 .
-0- - -C-C- - - -8-8-8-8-8-3 -----
.
. -A-A-A -------- -9-9-9- - -C-C -C-C- - -8-8-9 --------- A-A .
.
. -A-A-A-A-A-A-A-A-A-A- - - - -0-8-
A A A A A A A A A A- A- A -A -A- -
AAAAA-O-A
- -8-8- - - - -A-A-A-A-A-A-A-A-A .
-8- - -A-A-A-AAAAAAAAAA.
-A-A AAAAA
8 9 9 8 8 9 8 8
9 8 8 9 8 3 9
A
A
8E198888
8 8 8 8 3 8 9 .
.
. CCCCCCCCCCCCCCC CCCCCCCCCCCCCC.
.
.
.
.
.
.
.
.
0 0 0 0 9 0 0 0 0 0
0 0 D 0 0 0 0 0 0 .
.
.
o
EEEE
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.
.
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F F
FFFF
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FEFF
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.
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.
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3\4
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.
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0 .
0
0
4 1-
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LK
K J
1 I
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.
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III
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HHHHH.
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.
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.
Z I I I I .
K K J i J J 1 1 1 1
.
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I I I I
I I I
K K
J J A J
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JJJJJJ1\JJJJJ
J
J J J J J J
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. 1 1 1
J J
J
JJJJJJJ
1
J
\J
I J K K
K K K K K K K K K K K K K K
K K K K K K K K K K K K
N I .
J
1
I N
H G
I N
I
14.J
I
H
L L
.
L L
E-F .-F E
-C .-C
9
.
9
F
.
F
G
.
G
H
.
H
11.11
1 1
H
.1
I I
.
J
/ J
J J J
JJJJJ,JJJJJJJ
G N
J I/
H I
I
I
J,,,,
J/ J J
K K
KKKKKKKKKKKKK.
.
K K K K K K K K K K K.
.1,..1'
Figure 4.8. Pressure field, configuration No. 39.

15
b.0
10
5
0
1
2 3
Angle, radians
4
I
5
Figure 4. 9.
Average velocity distribution bubble configuration 39.
6 2Tr

200
0..
Z
Q..,.,,.
150
100
1
2 3
Angle, radians
Figure 4.10. Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 39.

.1000E+00
:ASE NO.
40 SCALE.
DELTA K.
.1000E+00 DELTA Y=
.
GGGGGGGGGGGGG
- - - - ------- G-G-G-G-G-G-G-G-G-G- - -
. --------------- G-G-G-G-G-G-G-G-G .
.-F-F-F-F-F-F-F ------- - - - -G-G-G-G-G-G-G .-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G-G- -
.
. ----- F-F-F-F-F-F-F-F-F - F -------- - -G -G-G-G-G-G-G-G-G-G-G- ------------ .
. - - - - - - - - - - -
-
- - - - -
. -E-E-E-E-E-E-E-E-E -------- F-r-F-F-F-F-F-
- - - - - - - --------- F-F-F-F-F-F-F .
-------- F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-
.
- - - - - -E-E-E-E-E-E-E-E ------- F-F-F-F .-F-F-F-F-F-F-F-F-F-F-F-F-F ----------
. -0-0-0 ------ - - - -E-E-E-E-E ------- F .-F-F-F-F-F-F-F ----------- E-E-E-E-E-E .
.-9-1-0-0-0-0-0-0-0-0 - - - - -E-E-E-E-E- - - . ------------ E-E-E-E-E-E-E-E-E-E-E-
.
. --------- D-0-0-0-0-9 ----- E-E-E-E- . ------- E-E-E-E-E-E-E-E --------- .
.
.-C-C-C-C-C-C-C- - ----- 1-C-1 ----- E-E-E-E .-E-E-E-E-E-E-E-E-E-E- - ------ 0-0-0-0-0-0
- - - -C-C-C-C-C-C-C-C- - - -0 -0-0- - - -E-E-E .-E-E-E-E-E-E-E- - - -0-0-0-0-9-0-0-0-0- -
.
. ----------- -C-C-C- - - -1-9-3- - - -E .-E-E-E-E ------ 0-0-0-0-0 ---------
- - - -C-C- - -0-0-0- - -
.-E-E-E ----- 0-0-0-0- ------ C-C-C-C-C-C .
- - ------- 9-8-0-8- - - -C-C- -0-0- - .-E-E- - - -D-O-D- - - - -C-C-C-C-C-C-C-C- - -
.
.-A-A-A-A-A-A-A-A ------ 0-8- - -C-C- -0-0- - - -C-C-C-C ---------
A A A A A A A A- A- A- A -A -A- - -9-B-
- - -C-C-C- - - - -8-8-8-8-8-3-8-3 .
.
.
A
0 0 0 0 9
4 A A A A A -A -A- -
.
9 A A-A
139 988088888889
A
. CCCCCC
C C C A
C C
0 0 0 0 0 0 0 9 r
E E
- - -0-8-8 ---------- A-A .
-8- - - - -A-A-A-A-A-A-A-A-A-A
-A-A-A-A-A AAAAAAAAA.
AAAAAA
8 9 8 8
.
80688888 9888813.
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Figure 4.11.
Pressure field, configuration No. 40.

20
15
5
0
0 1
2 3
Angle, radians
4
5
Figure 4. 12. Average velocity distribution, bubble configuration 40.
6 2Tr

250 z n 200
--
150
100
50
IIF
I
0 1
2 3
Angle, radians
4 5 6 21-
Figure 4.13. Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 40.

SCALE.
DELTA X= .1000E+00 DEL TA Y= .1000E+00
LASE NO.
41
.-G- G-G-G-G- G- G- G-G-G -G ---------- H- H- H-H . -H-H-H-H-H-H-H-H- H-H-H-H-H-H-H-H-H-H- H-H-H-H-H .
.
. - - - - - - - -G-G-G -G -G-G-G- C-G-G -------
F F
GGGGGGG
.
. - - -H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H ----- .
.
. -F-F-F-F-F -F-F-F-F-F -F -F ------- -G-G- G-G- G . -G-G-G-G-G ------ G-G-G- G-G-G-G-G-G-G-G-G-G .
. ----------- F -F-F -F-F-F-F ----- - -G . -G-G-G-G- G-G-G-G-G-G-G-G-G-G-CGG-G-G-G-G- -
.
. -E- E-E-E-E - E-E- E - - - - - - - - F-F- F-F-F-F - - -
. - - - - - - - - - - - - - - - - - - - - - - -
.
.
. - - - - - -E- E- E-E-E -E -E-E - E ------ F-F-F-F-F . ---------------- F-F-F-F -F-F-F-F .
. -------- - - - - -E-E-E-E ----- F-F-F . - F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F-F -F-F-F-
.
. -0-0-0-0-0-0-0-0-D-0-0 ----- -E- E-E-E- - - -
. -F-F-F-F-F-F-F-F-F-F-F-F ----------- .
0 0 0 0 C-E-E-E-E- -
. -C-C-C -C-C-C-C-C-1-C - - - - -C-0-0- - - -E-E-E-
. ------------------ E-E-E-.E-E-E .
. ----------- E-E-E-E-E-E-E-E-E-E-E-E-E .
- - - - -. - - - - -C -C -C-C-C- - - -B -0- - -E-E-E . ------- E-E-E-E-E-E-E-E- - - - - - - -
. - - -9-8-0-8-8-13-9- - - - -C-C-C- - -0-0- - -E-E . -E-E-E -E-E-E-E-E-E- - - ------ 0.-00-0-0-0 .
.
.
.
.
.
. -8-9-8-8- - - - -9-9-8-9-B- - - -C -C- -0-0- - -E . -E-E-E-E-E-E-E ------ 00-0-0-0-0-0-0-0-0-
.
. - - - - - - A- A- A-A - -
- - -B-2- - -C-C- -0- - -E . - E-E-E-E-E ----- 0-0-0-0-0- - - - - - - - -
.
. -A- A-A-A-A A A A A A - A -A-A -A- -8-9- -C-
A AA AA
A A A- A-A - -9
- - - -0-0-0-0 ------- C-C-C-C-C-C .
-0-0- - - - -C-C-C-C-C-C-C-C-C- -
.
B 8 B 8 8 8 B A A-A
9 8 8 8
- -C-C ------- 8-8-8-8-9-8-8 .
B
C C C
CC CC
0 0 0 0 0 0 3
3 9
C C
-8-8-8- - - - - ------
.
.
. C C
C I
0 0
D
0 0 0
E E . . . . . . -
E.,,,./FF
E E e..i. i-
I F F F F
G G G G G
- - - -A-A-A-A-A-A-A-A-A -A-A
.
-A-A-A-A-A AAA AA AA AA.
A A A A A A A A A A
.
E E/ F
EE,/ F
GHI
H
E E F G
E E/ F
G I
I
8 8 8 5 8 8 8 8 8 8 8 8 8 8
.
.
.
.
. E f
.
E i
F G I
G H J
KKK K
J
K L
L NCO
0 S
U Z V
L
K
K
E
CCCCCCCCCC
CCCCCCCC.
.
DODO
.
0 0 0 0 0 0 0 0 0 0 0 .
F F
F F 0 0 0 0 0 .
.
1
J S Z T
E
G
F F
E E E
EEE EEE
E
- -Z-
.
.
.
.
E E 9-C -J-C-J-C
EE
D C A- C- F -H -F-
0 E E
CO
E ir F
F
F F
E E C - - C -0-CC 0 E
ICE D C B A-A- -A A
CO E /F F
F F
E 0
C?
9 C E
/ F F
F F F \
E 0
CCCCC
0 E
FF
I I I 1 1
.
F F F
F F F F
E
F F F F F F F
E E E E
G
.
. GGGGGGGG
. GGGGGGGGGGGGGG
G G
F F F .
GGGGGGGGGGGGGG.
.
.
F F F .
E
E d'F F
.
.
G G
. GGGG
F F Nc.
G G G G
.
H H
. HHHHHH
E E ° °
F .F......
E E
0 ° E
F F F-T- F "F"-F F
4" F F
G G G G G G G G C G
. . ,
F F
G G G
G G G
G G G
.
G G G
G G G G
.
.
HHHHHHH
. HHHHHHHHHHHHHHHHHHHHHHH.
HH H. A HH
HHHHHH.
HHHHH
H H H H
.
.
IIIIIIIIIIIIIIIIII/ II.
.
HHHHHHHHHHFHH I. IIIIIIIIIIIIIIIIIIIIIII.
1 1 1
.
.
.
Figure 4. 14.
Pressure field, configuration No. 41

20 a on
15
10
0
2 3
Angle, radians
4
5
Figure 4. 15. Average velocity distribution, bubble configuration 41.
6 2n-

250 z n 200
150
100
50
Figure 4. 16.
0
0
1
2 3
Angle, radians
4
5 6 am
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 41.

SCALE.
DELTA X= .1000E+00 DELTA V= .1000E+00 CASE NO.
42
-----
HHHHHHH
.-G-G-G-G-G-G-G-G-G-G-G-G-G-G C
- - - - - ----- - - -G-G-G-G-G-G
- -F-F-F-F-F-r-F-F-F ----- - -
H H
.-F-F-F-F-F ----- F-F-f-F-F-F-F -- ---- G-G-G
---------------- F-F-F-F-F- - - -
.-E-E-E-E-E-E-E-E-E-E-E-F-E-E-E- - - -F-F-F-F-
- - - - - - - - - - - - -E-E-E-E- - -
-0-0-0-0-0-0-0-0-0-0-0-0-0- - - -E-E-E- - -F-F
.-0-0- - - -------- - -0-0- - - -E-E- - -
- - -C-C-C-C-C-C-C-C-C-C-C- - -0-0- - -E-E- -
.
- - -8-8-8-9 ------ 8-9- - -C-C- -0- - -E -E
.- -8-9-8- - -ArAA-411-44.0.1A-
. -B -R- - -4,0'8 A
-C-C- -B- -E-E
A AZOHNA- -B- -C-
.- - -A-AA
A -A B
R 01 8 8
CCCC a
C
0 C .-A1/14
-A
Al
8
.AA
: A
8
R C 0 0
C 0 E
E F
D J
F
H
K J
E
H
E
F
0
A A 0 G 0 C
.
. -----
.
H-H-H-H-H-M-H-H-H-H -----
------ - -
-
.-G-G ------------------- 6.06.4
-
------
.-F-F --------- F-F-F-F-F-F-F-F-FF-F-F-F .
.-F-F-F-F-F-F-FF-F-F-F-F-F-f-F:F - - - -
.
.-F-F-F-F-F-F-F-F-F --------------
- -F-f ---------- E-E- E-E-E-E- E -E-4-E
E E E E- E- E -E -E- - - - -
. ------ -E-E-E-E-E- - - - ------- -0
- -E-E-E ------ 0-0-00-0-00-0-0-0 .
- -0-0-0 C C .
-0-0 ----- C.CC-C-C-C-C-C-C-C .
-C- - 7 ------ 8-8-8-87.8
-
-
0-B- - - - - - --
- -A-A-A-A-A-A-A11-41+41-A-A
.-AVA-A- - - -0-6-0-2-0-G-0-C-
.-AVA ----- F- -
-8-
- - -9-
- -J-K-J-H-F- -C
.
A-A1,-8- -C-O-E- -F- -E- - -C-
A A-ht -8- -C-0-0- - -0- -C- -8
A A -Ar -8- -C-C-C-C-C- -8- -pA
.
9 9
A A=M- - -8-8-8-8-8-
A 14A-A- --
9 B B
8888
13
A
8
D or
A e
A
8
C c
C
C
C C 8 8 C C
CCCCCCCCCCC
0 o n
Inonop000p000
.EEEEEEEEFEEEEEE
E E
0
E E E
E E
F F
FFFF
.FFFFF
FFFrFFF
FFFFFFFFF
G
.GGGGGGGGGGGGGGG
G G
G G
G G G
H H H
0
0 E E
E
F F
F F F
F F
G G G
G G G
HHHH
H
Al AA AAAA AA AA
B
C
C
B
C
CCCC
CCCCCCCCCC
0 0
0 0 0
C C
E E
E E E
EEEEE
F F
FFFFFFFFF
0 0 0 0 0 0
0000000000
EEEEEEEE
EEEEEEEEEE
FFFFFFFFFFFFFFF
F F
FFFFF
F
FFFFFFFF
F
GGGGGG
GGGGGGGGG
G G
G G G
G G G
GGGGGGGGGGG
G G G
HHHHHHHH
HHHHHHHHH
H
HHHHHHHHHHHHHH
HHHHHHHHH
Figure 4.17. Pressure field, configuration No. 42.

20
5
0
1
3
Angle, radians
4
Figure 4. 18.
Average velocity distribution, bubble configuration 42.
6 2Tr

250
n 200
150
100
50
Figure 4.19.
i
4I
I
B. L. Sep.
I
I
I
0 1
2 3
Angle, radians
4
5 6 2Tr
Convective Nusselt No. distribution, 3 mm hot parameters, bubble configuration 42.

126 be accomplished through controlled experimentation and operation within the range of particle sizes for which the model is valid.
Table 4.2.
Total heat transfer for 3 mm and 6 mm hot and cold parameters, no bubble.
Case
Total Convective
Heat Transfer
Total Heat
Transfer
Total Convective
Nusselt No.
(W/m2°K)
2
(Wim °K) u' = 0.2
u' = 0.0
Li' = 0.2
u' = 0.0
u' = 0.2
u' = 0.0
3 mm hot 108.6
3 mm cold 244.2
6 mm hot 110.4
6 mm cold 262.6
98.04
156.4
194.7
129.7
82.45
159.0
173.3
139.4
141.2
103.4
118.7
92.0
231.9
129.7
234.5
139.4
216.7
103.4
194.3
92.0
The effect of bubble location on the total convective heat transfer to a horizontal cylinder for 3 mm hot-bed and 6 mm cold-bed parameters is shown in Table 4.3 and Fig. 4.20.
single bubble having a diameter equal to the cylinder diameter and center location as indicated in the table and on the figure (large dot adjacent to numbers).
Generally, these results indicate a relatively small effect of bubble presence on total heat transfer.
However, as discussed above, bubble location can have a strong effect upon local values. Also note the weak effect of Reynolds number on these results.
Figure 4.22 demonstrates the variation of convective heat transfer coefficient with time due to a single passing bubble having a diameter equal to the cylinder diameter and for 3 mm hot-bed

0
0
0
0
0
-1. 0
-1. 499
-1. 999
-1. 905
-2. 598
-3. 464
-2. 2
-3. 0
-4. 0
-1. 903
-2. 595
-3. 46
-1. 099
-1. 499
-1. 999
0.
0.
0.
0.
-0. 751
-1. 299
-1. 5
Bubble Location
X ia
0 YO
/a
O.
0.
0.
1. 104
1. 505
2. 007
1. 906
2. 599
3. 465
2. 0
3. 0
4. 0
-1. 5
-1. 298
-0.749
0.
-2. 0
-2. 5
-3. 0
-3. 5
-4. 0
-1. 732
-2. 598
-3. 465
-1. 10
-1. 5
-1. 999
Table 4. 3.
Effect of single bubble on convective Nusselt number, R
B
= D/2.
O. 971
1. 011
1. 0055
1. 0064
1. 0625
1. 0064
1. 028
1. 065
1. 035
1. 0211
O. 9291
0. 8619
O. 758
0. 752
1. 192
1. 1096
1. 071
1. 050
1. 039
1. 089
1. 043
1. 015
O. 928
O. 958
O. 9917
0. 9066
O. 948
(Nu
)
D cony
/(Nu
) D cony. no bubble
3 mm, Hot Case 6 mm, Cold Case
O. 9859
1. 0084
1. 0270
1. 1546
1.0659
1. 0377
1. 073
O. 8187
0. 6405
O. 6530
1. 229
1. 143
1. 097
1. 071
1. 054
1. 1466
1. 062
1. 0217
0. 8705
0. 9406
O. 9768
0. 8705
0. 9078
O. 9593
0. 9867
O. 9836
O. 9912
127

RB = D/2
1.021
1.038
4
1.028
1.027
Y /a
1.035
1.066
3
1.006
1.008
1.006
0.991
1.063
0.986
1.065
1.15
2
1.006
0.984
1.011
0.987
X /a
-4
0.971
0.959
-3
0.948
0.908
-2
1
0.907
0.752
0.871
0.653
0.958
0.941
0.928
0.871
0.758
0.640
1.09
1.15
0.861
-1
0.819
1 0.929
1.038
1.19
1.23
0.992
0.977
1.043
1.062
1.071
1.097
1.015
1.022
- 4
1.039
1.054
Figure 4.20. {(Nu
)
D cony.
/(Nu
) D cony. no bubble} for 3 mm hot parameters/6 mm cold parameters.
128

129 parameters. The bubble center is assumed to follow a potential flow streamline with a constant vertical velocity as given by Eq. 3. 1.2. 10.
The two trajectories considered are shown in Fig. 4.21.
The influence of the passing bubble on the convective heat transfer was approximately unaffected by the presence of a second bubble at position 39
Generally these results are consistent with single bubble influence as shown in Fig. 4.20.

Figure 4.21.
Bubble trajectories.
130

1.25
a) z
C
0 z
0
C
C
0
14" B. L. Sep.
0 Trajectory 2
0
1
A Bubble attached at configuration 39
2nd bubble trajectory 2
0
0
51
I
I
I
10 15
Elapsed time, Atumf /a
20
I
25 30
Figure 4.22. Convective heat transfer vs. time for 3 mm hot parameters.

132
V. CONCLUSIONS AND RECOMMENDATIONS
An analytical model of heat transfer to a horizontal cylinder immersed in a gas fluidized bed has been developed.
The operational equations obtained from the analysis have been coded in Fortran language so that results can be obtained for a range of operating conditions.
Based upon the results reported in this thesis, the following conclusions are noted:
1) The model was developed on the basis of an assumed dominance of gas convection as the heat transfer mechanism. This regime corresponds to operation with large particle sizes and the analysis approach based upon boundary layer theory further limits the applicability of the model to particle diameters greater than about 2-3 mm.
2) Most of the experimental data reported in the literature is based upon cold bed operation.
This is due to the difficulties involved in instrumenting a combustion temperature facility.
However, these cold bed data are used to estimate hot bed performance. Such cold-bed/hot-bed correlations must not only consider the effect of gas property variation but also differences in the relative importance of interstitial turbulence and radiative contributions to the heat transfer.

3) The total heat transfer to a cylinder is not strongly affected
133 by the presence of a single two-dimensional bubble with a diameter equal to the cylinder diameter.
This result is expected to hold qualitatively for the case of a single three dimensional bubble as well.
4) The total heat transfer coefficients obtained using the gas convection dominant model are within the range of experimental results obtained by previous investigators.
However, complete validation of the model can only be obtained as a result of controlled hot- and cold-bed experiments. The cold-bed experiments should attempt to correlate local values of pressure and heat transfer coefficient at the cylinder surface with bubble position and size.
Hot-bed experiments should at least correlate total heat transfer with bubble size and position, and if possible, obtain local data as well.
5) Gas velocity is a dominant factor in establishing the local heat transfer rate, so an important area for further analysis is more detailed modeling of the motion of the gas in the vicinity of a horizontal cylinder with bubbles present.
Also, further analysis should be done to investigate the unsteady effects caused by fast bubbles as well as three dimensional effects on the velocity distribution and heat transfer.

134
The physical model described in this thesis has been reduced to a series of mathematical problems for which initial analytical and numerical solutions have been obtained.
Once the model has been validated through detailed experimental measurements, any of the basic elements of the model should be developed further so that more complete understanding of the gas convection dominant heat transfer regime can be achieved.
Also, a suitable analytical model for the mid range between unsteady conduction dominant and gas convection dominant regime should be developed so that a complete picture of fluidized bed heat transfer is obtained.

135
BIBLIOGRAPHY
1.
Abramowitz, M. and Segun, I. A. , Handbook of Mathematical
Functions, Dover, New York (1965).
2. Andeen, B.R. and Glicksman, L. R. , "Heat Transfer to Horizontal
Tubes in Shallow Fluidized Beds", ASME Paper No. 76-HT-67
(1976).
3.
Arpaci, V. S. , Conduction Heat Transfer, Addison-Wesley,
Reading, Mass. (1966).
4.
Baskakov, A. P. , Berg, B. V. , et al. , "Heat Transfer to Objects
Immersed in Fluidized beds", Powder Technology, Vol. 8, pp. 273-282 (1973).
5.
Baskakov, A. P. and Suprum, V. M. , "Determination of the Convective Component of the Heat Transfer Coefficient to a Gas in a
Fluidized Bed", Int. Chem. Eng'g. , Vol. 12, pp. 324-326 (1972).
6.
Basu, P. , "Bed to Wall Heat Transfer in a Fluidized Bed Coal
Combustor", Presented at AIChE Mtg. Nov. 28-Dec. 2, 1976.
7.
Batchelor, G. K. , An Introduction to Fluid Dynamics , Cambridge
University Press (1967).
8.
Bieberbach, L. , Conformal Mapping, Trans. by F. Steinhardt,
Chelsea, New York (1953).
9.
Botterill, J. S. M. ,
New York (1975).
Fluid-Bed Heat Transfer, Academic Press,
10.
Botter ill, J. S. M. and Desai, M. , "Surface to Fluidized Bed Heat
Transfer", Chem. Eng'g. Sci. , Vol. 25, p. 749 (1970).
11.
,
"Limiting Factors in Gas-Fluidized Bed Heat
Transfer", Powder Technology, Vol. 6, pp. 231-8 (1972).
12.
Botter ill, J. S. M. and Denloye, A. O. O. , "Gas Convective Heat
Transfer to Packed and Fluidized Beds -
1.
A Theoretical
Model", Presented at AIChE Mtg. Nov. 28-Dec. 2, 1976.

13.
Botterill, J. S. M. and Sealy, C. J. , "Radiative heat transfer between a gas fluidized bed and an exchange surface", Brit.
Chem. Eng'g. , Vol. 15, pp. 1167-1168 (1970).
14.
Botterill, J. S. M. and Williams, J. R. , "The mechanism of heat
Vol. 41, pp. 217-30 (1963).
136
15.
Broughton, J. and Kubie, J. , "A model of heat transfer in gas fluidized beds", Int.
pp. 289-99, (1975).
J. Heat and Mass Transfer, Vol. 18,
16.
Carnahan, B. , Luther, H. A. , and Wilkes, J.O. , Applied
Numerical Methods, Wiley, New York (1969).
17.
Carrier, G. F. , Krook, M. , and Pearson, C. E. , Functions of a
Complex Variable, McGraw-Hill, New York (1966).
18.
Chen, J. C. , "Heat Transfer to Tubes in Fluidized Beds", ASME
Paper No. 76-HT-75 (1976).
19.
Chung, B. T. , Fan, L. T. , and Hwang, C. L. , "A Model of Heat
Transfer in Fluidized Beds", J. of Heat Transfer, pp. 105-10
(1972).
20.
Churchill, R. V. , Introduction to Complex Variables and Applications, McGraw-Hill, New York (1948).
21.
Clift, R. and Grace, J. R. , "Bubble Interaction in Fluidized
Beds", Chem. Eng'g. Progr. Symp. Ser. , Vol. 66, No. 105, pp. 14-27 (1970).
22.
Davidson, J. F. and Harrison, D. , Fluidization, Academic Press,
New York (1971).
23.
Dwight, H. B.
, Mathematical Tables of Elementary and Some
Higher Mathematical Functions, Dover, New York (1958).
24.
Erdelyi, et al. , Higher Transcendental Functions, Vol. I,
McGraw-Hill, New York (1953).
25.
Fluidization and Its Applications, International Congress,
Toulouse, France, 1-5 Oct. 1973.

137
26.
Gabor, J. D. , "Wall to bed heat transfer in fluidized and packed
Vol. 66, No. 105, pp. 76-86 (1970).
27. Gabor, J. D. , "Wall-to-bed Heat Transfer in Fluidized Beds",
AIChE J. , Vol.
18, p. 249 (1972).
28. Galloway, T. R. and Sage, H. , "A model of the mechanism of transport in packed, distended, and fluidized beds", Chem.
Eng`g. Sci.
,
Vol. 25, pp. 495-516 (1970).
29.
Glass, D. H. and Harrison, D. , "Flow patterns near a solid obstacle in a fluidized bed", Chem. Eng'g. Sci. , Vol. 19, pp. 1001-1002 (1964).
30. Hager, W. R. and Thomson, W. J. , "Bubble Behavior Around
Immersed Tubes in a Fluidized Bed", Chem. Eng'g. Prog.
Symp. Ser. , Vol. 69, No. 128, pp. 68-77 (1973).
31. Hayes, J. K. , Kahaner, D. K. , and Kellner, R. G. , "A Numerical
Comparison of Integral Equations of the First and Second Kind for Conformal Mapping", Math. of Comp. , Vol. 29, pp. 512-21
(1975).
32.
,
"An Improved Method for Numerical Conformal
Mapping", Math. of Comp.
,
Vol. 26, No. 118 (1972).
33. Hildebrand, F. B. , Advanced Calculus for Applications, Prentice-
Hall, New Jersey (1962).
34. Howe, D. , "The Application of Numerical Methods to the Conformal Transformation of Polygonal Boundaries", J. Inst. Maths.
Applics. , Vol. 12, pp. 125-36 (1973).
35.
International Mathematical and Statistical Library, Library 3,
Edition 5, Fortran CDC 6000/7000, CYBER 70/170 Series (1975).
36. Kays, W. M. , Convective Heat and Mass Transfer, McGraw-Hill,
New York (1966).
37.
Keairns, D. L. (Ed. ), Fluidization Technology, McGraw-Hill,
New York (1976).
38. Kober, H. , Dictionary of Conformal Representations, Dover
(1952).

138
39.
Koppel, L. B. , Patel, R. D. , and Holmes, J. T. , "Statistical
Models for Surface Renewal in Heat and Mass Transfer Part IV:
Wall to Fluidized Bed Heat Transfer Coefficients", AIChE J. ,
Vol. 16, pp. 464-71 (1970).
40.
Korelev, V. N. and Syromyatnikov, N. I.
,
"The Fluid Mechanics and Structure of the Fluidized Bed in the Vicinity of a Plate Submerged in it", Heat Transfer -Soviet Research, Vol. 6, pp. 67-
70 (1974).
41.
Kunii, D. and Levenspiel, 0. , Fluidization Engineering, Wiley,
New York (1962).
42.
Lese, H. K. and Kermode, R. I.
,
"Heat Transfer from a Horizontal Tube to a Fluidized Bed in the Presence of Unheated
Tubes", Can. J. Chem. Eng'g. , Vol. 50, pp. 44-8 (1972).
43.
Levenspiel, 0. and Walton, J. S. , "Bed-wall heat transfer in fluidized systems", Chem. Eng'g. Prog. Symp. Ser. , Vol. 50, no. 9, pp. 1-13 (1954).
44.
Levy, S. , "Effect of Large Temperature Changes (Including
Viscous Heating) upon Laminar Boundary Layers with Variable
Free Stream Velocity", J. Aero. Sci. , pp. 459-73 (1954).
45.
Lockwood, D. N. , "An Investigation of the Effects of Immersed
Heat Exchanger Tube Spacing and Arrangement on the Quality of
Fluidization in a Cold Phase Two-Dimensional Fluidized Bed",
M.S. Thesis Oregon State University (1976).
46.
Mickley, H. S. and Fairbanks, D. F. , "Mechanism of heat transfer to fluidized beds", AIChE J. , Vol.
1, pp. 374-84 (1955).
47. Proceedings of the Fourth International Conference on Fluidized-
Bed Combustion, Dec. 9-11, 1975, MITRE Corp. , Virginia
(1976).
48. Rubin, S. G. and Grossman, B. , "Viscous Flow Along a Corner:
Numerical Integration of the Corner Layer Equations", Quart.
Appl. Math. , Vol. 29, pp. 169-86 (1971).
49. Schiedegger, A. E. , The Physics of Flow through Porous Media,
Third Ed. , U. of Toronto Press (1974).

50.
Schlichting, H. , Boundary Layer Theory, Sixth Ed.
,
McGraw-
Hill, New York (1968).
139
51. Schneider, P. J. , Temperature Response Charts, Wiley, New
York (1963).
52.
Siegal, R. and Howell, J. R. , Thermal Radiation Heat Transfer,
McGraw-Hill, New York (1972).
and Potter, 0. E. , "Solids Motion Caused by a Bubble in a Fluidized Bed", Powder Technology, Vol. 6, pp. 239-44 (1972).
54. Smith, M. C. and Kuethe, A. M. , "Effects of Turbulence on
Laminar Skin Friction and Heat Transfer", Phys. of Fluids,
Vol. 9, pp. 2337-44 (1966).
55.
Spalding, D.B. and Pun, W. M. , "A Review of Methods for Predicting Heat-Transfer Coefficients for Laminar Uniform-Property
Boundary Layer Flows", Int. J. Heat and Mass Trans. , Vol. 5, pp. 239-49 (1962).
56. Staub, F. W.
,
"Two-Phase Flow and Heat Transfer in Fluidized
Beds", Sixth Quarterly Report, GE Report SRD-77-011, Dec. 31,
1976.
57. Strom, S. S. , et al. , "Preliminary Evaluation of Atmospheric
Pressure Fluidized Bed Combustion Applied to Electric Utility
Large Steam Generators", EPRI Report No. RP 412-1 (1976).
"General Potential Theory of
Arbitrary Wing Sections", NACA Report No. 452 (1933).
59. Tokuda, N. , "Viscous flow near a corner in three dimensions",
J. of Fluid Mech. , Vol. 53, pp. 129-48 (1972).
60.
Touloukian, Y. S. , et al. , Thermophysical Properties of Matter,
Volume 2 Thermal Conductivity Nonmetallic Solids, Plenum,
New York (1970).
61. Traci, R. M. and Wilcox, D. C.
,
"Freestream Turbulence Effects on Stagnation Point Heat Transfer", AIAAJ. , Vol. 13, pp. 890-6
(1975).
62. Vincenti, W. G. and Kruger, C. H. Jr. , Physical Gas Dynamics,
Wiley, New York (1965).

140
63. Viskanta, R. , "Radiation Transfer and Interaction of Convection with Radiation Heat Transfer", Advances in Heat Transfer,
Vol. 3, pp. 176-248 (1966).
64. Vreedenberg, H. A. , "Heat transfer between a fluidized bed and a horizontal tube", Chem. Eng'g. Sci. , Vol. 9, pp. 52-60 (1958).
65. Wasan, D. T. and Ahluwalia, M. S. , "Consecutive film and surface renewal mechanism for heat or mass transfer from a wall", Chem. Eng'g. Sci. , Vol. 24, pp. 1535-42 (1969).
66. Weast, R. C.
,
Handbook of Chemistry and Physics, CRC Press
(1974).
67. Wheelon, A. D. , Tables of Summable Series and Integrals Involving Bessel Functions, Holden-Day, San Francisco (1968).
68. Wright, S. J.
, et al. , "Heat transfer in fluidized beds of wide size spectrum at elevated temperatures", Brit. Chem. Eng'g. ,
Vol. 15, pp. 1551-4 (1970).
69. Yoshida, H., Kunii, D. , and Levenspiel, 0. , "Heat transfer between a wall surface and a fluidized bed", Int.
Mass Transfer, Vol. 12, pp. 529-36 (1969).
J. Heat and
70.
Yoshida, H. , et al. , "Mechanism of Bed-Wall Heat Transfer in a
Fluidized Bed at High Temperatures", Chem. Eng'g. Sci. ,
Vol. 29, pp. 77-82 (1974).
71.
Zabrodsky, S. S. , Hydrodynamics and Heat Transfer in Fluidized
Beds, MIT Press (1966).
72.
Zabrodsky, S. S. , et al. , "Mechanism of heat transfer between a fluidized bed and the apparatus walls", Academia Nauk Beleruskai
BSSR, Minsk, Vetsi, Seryia fizikaenergetychnykh, Vol. 3, pp. 103-7 (1973).
73.
Zamir, M. and Young, A. D. , "Experimental Investigation of the
Boundary Layer in a Streamwise Corner'', Aero. Quart. J. , pp. 313-39 (1970).
74.
Ziegler, E. N. , et al. , "Effects of Solid Thermal Properties on
Heat Transfer to Gas Fluidized Beds", Ind. and Eng'g. Chem.
Fund. , Vol. 3, pp. 324-8 (1964).

APPENDICES

141
COMPUTER CODES
The analysis developed in Chapter III has been coded in Fortran compatable with the CDC CYBER 75 computer at OSU. The operational equations have been organized as seven separate programs, each containing one or more subroutines. This programming approach was selected so that maximum flexibility in the computation is possible with minimum duplication of lengthy calculations. Six of the seven programs provide input data for the seventh which actually performs the heat transfer calculation. All programs as well as typical output are listed on pages 145-179.
The final heat transfer computation is performed by MOMENGL
(see page 145
).
Input data for this program are listed and defined on page 145 and include fluid bed and bubble parameters as well as three tables. The first two input tables are the Stokes matching functions for the side and lee stack respectively (see Sec. 3.3. 3).
These tables are produced as punched output by TRFORM. The third table is the relative gas velocity and velocity gradient as a function of angular coordinate within the attached bubble and is produced as punched output by BUBBLE. The input data is organized so that several bubble configurations can be considered in a single run as determined by the parameter LBMAX.

142
The local Nusselt number calculations are performed as follows.
For the stagnation point and interstitical channels, the parameters and N
HV are established iteratively by subroutine STAGPT
V using Newton's iteration procedure.
Within the bubble region, these parameters are established through numerical integration using the
IMSL routine DVOGER with calls to DFUN for determination of the derivitives.
The average Nusselt number calculation within the channels is controlled by subroutine CHANNEL with calls to OUT for determination of the Stokes region contribution.
The local Nusselt number is numerically integrated using the trapezoidal rule so that the total convective Nusselt number is determined.
Finally, the radiative contributions of the bubble, sides and lee stack are computed so that an equivalent total Nusselt number is determined. Output data include listings of primary input parameters, tabulated values of local velocity, Stokes region edge location, Nusselt number and boundary layer thicknesses, total heat transfer contributions and an elapsed time parameter and a print plot of the convective Nusselt number as a function of polar angle in radians.
When computational difficulties arise due to poor convergence in the bubble region integration, warning statements are printed but the calculation is allowed to proceed.
If boundary layer separation occurs within the bubble, the velocity profile shape factor,
V' is fixed at -12.0 (consistent with zero wall shear) and if convergence problems

143 arise due to high velocity gradients, is fixed at the value which
V produces zero momentum thickness, 52
(see Eq. 3. Z. 49).
Bubble parameters are established by the two programs PRESS and BUBBLE. PRESS (pages 158 to 161) provides a field plot of dimensionless pressure in the vicinity of the cylinder and should be used to assure that the particular bubble configuration produces a realistic solution using the image method. The local pressure gradient at the cylinder surface is also tabulated and provides an estimate of the stagnation point location (when 0p/86 = 0).
BUBBLE performs the necessary integrations to determine the average velocity inside a two dimensional bubble contacting the cylinder surface.
The average velocity gradient is also calculated using finite differences.
Output from BUBBLE includes both tabulated values of the bubble boundary, average velocity and velocity gradient, as well as punched values of velocity and velocity gradient.
The angular location of the intersections of the bubble boundary with the cylinder
mined automatically based upon bubble diameter and center location or specified on input.
Programs CONFORM (pages 166-170) and TRFORM (pages 171 to
176) are used to obtain the Stokes region matching function.
CONFORM is used to obtain a solution to the integral equation defining the mapping function for the conformal mapping of a unit semicircle onto the channel cross section.
The solution is obtained by successive

144 substitution and is coordinated by subroutine GAMOPH with calls to function GEE (page 1.68) to establish the radial coordinate of the boundary. The calculation is performed in dimensionless form with particle radius as the length scale.
Output from CONFORM includes the solution details at each iteration as well as the converged solution.
The mapping function is produced as punched output for interfacing with TRFORM.
TRFORM performs the integrations necessary for determination of the scale factor and matching function tt'
(x s s
).
Input includes channel characteristics as well as punched output from CONFORM.
Results are both tabulated and punched and the punched output, l's(xs), interfaces directly with MOMENGL as described above.
The Stokes region heat transfer parameters are obtained using the program DSDR2 (pages 176-17$). Input parameters for this program are listed and defined on page 176. The calculations performed by
DSDR2 are described in Sec. 3.3.2 and involve evaluation of the truncated series necessary for determination of total energy transfer and temperature gradient at the edge of the Stokes region.
The results of these calculations are tabulated and typical output is shown on page 17 8.
Bubble trajectory coordinates were computed using TRAJCORD
(page 179).

pRnGRAm 10mENGL(INPUT,OUTPUT,TAPE 5=INPUT.TAPE 6=OUTpUT)
CALL mOMEOL
ENO
SUBROUTINE mOmEGL
MOMEGL CALCULATES THE GAS CONVECTIVE AND RADIATIVE HEAT TRANSFER
TO A CIRCULAR TUBE IN A GAS FLuIOIZED BED. APPROXIMATIONS ARE
VALID FOR LARGE ISOTHERMAL PARTICLE SYSTEMS AND ARE BASEJ UPON AN
INTEGRAL APPROXIMATION FOR THE B.L. EQUATIONS AND STOKES FLOW IN
INTERSTICIAL CORNER REGIONS.
RADIATION IS TREATE) AS UNCOUPLE).
ALL QUANTITIES APE DII:NSIONLESS, TEMPERATURES REL. TO TB-Tw,
LENGTHS REL. TO TU-IE DIA. OR PADIUS, VELOCITIES REL. To UHF.
7ATERNAL OFON
DIMENSION Y(9,2).WK(34).ERROw(2),YIAX(21,CASE(5),RB(10),(0(10),Y3(
1101.7MAX(21.7131(2).232(2).1ST(2),THETAP(1000),ENUF(iGGC)
OOMmON/d/PTC,INUP,RE3V2,0mAx
SOIMON/C/LTABS12),PRISS(10),2).X3RPS(100,2),ALPHAS(2),BS(2),SOOS12
11,RFSP(2),SORP
COMION/0/BFE.PHIDP.E0SI,C1.C2
COmmON/0/0mV,ENHVOUD7,PR,A(5).B(5),C9,U,DXUOZ,DwU07,BETAT,ITERP
COMMON/V/INDBU9,THETs1,THETB1,THET92,THETA,THLS1.THLS2,E(3,2),OPOY
II L'ABLE. THETAT ( 50 I ,UT (50 ) .OUDT (50) ,U4),INEIRIC,DP0Z commON/P/NPUB,THETA0(101,40(10),R02(10),RB2(10),ROS(10)
DATA CB,ENSUI.ENUS.ZSAVE,02UDZ2013/0.3,0.,0.,0..0..1.1
DATA 9/0.,2..0.,-2.,1./
INPUT DATA
LMAX= NUMBER OF ITERATIONS ALLOWED FOR SOLUTION AT STAGNATION
POINT
/TERP= NUMBER OF ITERATIONS ALLOWED AT AXIAL POSITIONS AWAY
FROM STAGNATION POINT
INN.= VELOCITY GRADIENT INDICATOR INDP=1 INDICATES 0007=0
LTABSC, LTABSS= NUMBER IF ELEMENTS IN STOKES REGION MATCHING
FUNCTION TABLE FOR SIDE AND STACK, RESP.
Ra. REYNOLDS NUMBER BASED ON TUBE DIAMETER ANO PROPERTIES AT
BCD TEMPERATURE AND PRESSUPE AND UHF (UmFo0=RHO/HU)
PR= PRAALITL NUMBER OF GAS
AIR= RADIATIVE HEAT TRANSFER PARAMETER (SIGmA0.(TB-Tw1=.3/KG)
UTI= INTENSITY OF INTERSTICIAL TURGULENCE
ATINF= CONSTANT IN SOUTHERLAND S VISCOSITY LAM DIVIDED BY BED
TEIPERATURL, (110 DEG K/TINF DEG K)
CMV= ESTIMATE) B.L. SHAPE FACTOR AT STAGNATION POINT
ENHJ= ESTIMATED B.L. THICKNESS RATIO (DELTAH/DELTAV) AT
STAGNATION POINT
T/NF=0ImENSIO4LESS 3E1 TEMPERATURE, TB /(T3 -TW)
TW=IIMENSION WALL TEMPERATURE. TW/(13-TW)
Sin. SPS= DIMENSIONLESS PARTICLE SPACING (HALF DISTANCE 3ETWEEN
CENTERS/TU3E DIA.) FOR SIDE AND STACK, FESP.
Co= PARTICLE DIA./TUBE DIA.
CCG.QCS= HEAT TRANSFER PARAMETER FOR STOKES CORNER FLOW REGION
FOR SITE AND STACK RESP.
D413= PARTICLE SoHERICITy (ApPF0x. 0.6 TO 0.7)
PHI9= RUBBLE VELOCITY PARAMETER
(1.34YEPSI40.31)=SOIRTIGY0 /2)/UIF
LO= SUPERFICIAL VELOCITY /UHF
EPSI= BED VOIDAGE FAR FROM TUBE
EPSC= SURFACE VOIDAGE, SIDE
EPSS= SURFACE VOIOAGE. STACK es(2). STOKES REGION MATCHING PARAMETERS. SIDE AND STACK
RASP.
EMISSP= AVE BED EMISSIVITY NEAP TUBE WALL
EMISSB= AVE BAD EMISSIVITY ON BUBBLE 90UNOARY
EMISSW= EMISSIVITY OF TUBE WALL
RAXIER. H, HIM HMAX, EPS ARE PARAMETERS USED BY THE IMSL
ROUTINE OVOGER
MAKDER= MAXIMUM ALLOWED ORDER OF THE METHOD OF SOLUTION
H, )(MIN. HMAX= INITIAL STEP SIZE, MINIMUM STEP SIZE.
MAXHIMUM STEP SIZE RESP.
EPS= ERROR PARAMETER
ELCCIELCS= DIMENSIONLESS CHANNEL LENGTH FOR SIDE AND STACK.
RASP.
PHISS, XSRPS= STOKES REGION MATCHING FUNCTION VALUE AND LATERAL
POSITION FROM CHANNEL CENTER (X/RP)
LAMAX= NUMBER OF BUBBLE CONFIGURATIONS TO BE CONSIDERED
CASE(I)= ALPHA NUMERIC CHARACTER STRING IDENTIFING CASE
LIABLE= NUMBER OF ELEMENTS IN BUBBLE GAS FLOW TABLE
HAUB= TOTAL NUMBER OF BUBBLES
LBUB=INOEX OF BUBBLE CONTACTING TUBE
THETSI. THETS2= ANGULAR LOCATIONS OF LOWER AND UPPER
STAGNATION POINTS. RASP.
THETT11,THET02= ANGULAR BOUNDS OF BUBBLE IN CONTACT WITH TUBE
THLS1, THL22= ANGULAR BOUNDS OF STACK REGION
RA= BUBBLE RADIUS/TUBE RADIUS
RO= RADIAL LOCATION OF BUBBLE CENTER/TUBE RADIUS
THETAO= ANGULAR LOCATION OF BUBBLE CENTER
THETAT= ANGULAR COORDINATE
UT= AVE VELOCITY INSIDE BUBBLE (U /UHF)
CUOT= AVE GAS VELOCITY GRADIENT INSIDE BUBBLE (OUDTHETA /UMF)
I
3
5
READ(5,3) LMAX.ITERP.INDP.LTABSC.LTABSS
REA0(5,5) RE.PROIR.UTI.ATINF.OMV,ENHVITINF.TW
REAC(5.4) SPC.SPSOP.00C.00S.PHIS,PHIA.U0
REAC(5,6) EPSI.APSC.EPSS.BS(1),BS(2).EMISSP.EMISSEIgEMISSW
REAC(5.71 MAXDAR.H.HNIN.HMAX.EPS.ELCC.ELCS
INPUT STOKES REGION MATCHING FUNCTION FOR SIDE
REAC(5,91 (PHISS(L.1).XSRPS(L.1)1.L=1.LTASSC)
INPUT STOKES REGION MATCHING FUNCTION FOR STACK
READ(5.9) IPHISS(L.2).XSRPS(L.2).L=1.LTABSSI
READ(5,1) LPMAX
TImE=0.
DO 1000 LBm=1.1.8mA
REAC(5,191 (CASE(I),I=1,5),LTABLE,NBUB.LBu9
REAC(5,6) IHETS1,THETS2,THETB1,THETB2.THLS1,THLS2
REA0(5,2) IRIBINI,R0(N),THETA0(N),N=1.NBUR)
REAC(5,8) (THETAT(LI.UT(L).DUDTIL).L=1,LTABLEI
FORMAT(I5)
FORmAT(3F10.0)
FORMAT(5I5)
FORmAT(8F10.0)
FORmAT(9F8.P)
UD

_
7
8
A
10
FORMAT(BF10.0)
POR.AT(I2IEF10.0)
FOR4AT(3E12.4)
POR.AT(2E12,4)
PORYAT15A2,315)
C
ENUS=0.
ZSAVE=0.
ENUSU4=0.
TWTINE=TW/TINR
OR=TMTINP..1.5.11.4ATINFMTWTINE.ATINFI
OUTPUT RUN FARAMETRS
C
C
;
C
C
C
50
WRITE(6,115)
MRITE(6,100)
WRITE16,101)
WRITE(6,114)
WR/TE16g119) 1CASE(I),I=1,5)
WRITE16,114)
WRITE(6.103) RE. PR
WRITE16,10.) BIR,UTI
WRITE(6.1051 TINF,TW
WRITE16,1061 SPC,SPS
WRITE(6,120) DP,PHIS
WRITE(6,121) OCC,OCS
WRITE(6,122) 3S(1),9S(2)
WRITE(6,123) UO,PHI9
WRITE(6,124) EPST.EPSC
WRITE(6,125) FPSS.EMISSP
WRITE(6,131) Emissa,Emissw
MR/1E16,12E) THETB1,THET92,TMLS1.THLS2
WRITE(6,127) ELCC,ELCS
CONVERT BUBBLE OATA
DO 53 N=1.NBUB
R021N)=RO(N).RO(N)
RE12(N)=R3(61.RB(N)
ROSIN)=R01WSIN1T4ETAO(N))
)(01NI=R01N).GOS( THETAO(N))
YO(N)=ROSIN)
IF(L3M.E).1) TS=V01LBUB)
C
OUTPUT BUBBLE PARAMETERS
WRITE16,114)
WRITE(6,210)
WRITE(6,2111
WPITE(6,2121
WRITE16,213)
WRITE15.2141 (L,RB(L),A3)1-1,TO1L),L=1,NBUR)
C
SET UP INITIAL VALUES AN) CONSTANT PARAMETERS
LTAPLE=LTAPLE-.1
LTARS(1)=LTA3SC...1
C
C
C
C
C
51
LTARS(21=LTA1SS...1
RP=OP/2,
PHICP=PHIS.OP
BRE=150./RE
CALL ORAG(EPSI,EPSI,E,1)
OPOTI=1C14C2/EPSI)/EPSI
CALL DRAG(EPSC.EPSC,E,1)
CALL ORAG(EPSS,EPSS,E.21
IP1°8(LBUB).GT,00 U9 =U0...1.4PHIB.SORT(RB(LBUBI)
S0 =C4.(.5+TW)/(1.+TM)
SORP=SO/RP s005(1)=sof0cc/sPc
SOOS(2)=SO*OCS/SPS
RPSP(1)=RP/SPC
RPSP12)=RP/SPS
00 51 L=1,2
ALPHASILI=ALOG1KSRPS11,0/XSRPS(2,L11/(PHISS11.LIPMISS12,L1)
ALPPA=0.13
BETO=ALPNA.UTI.SORT1RE.OP)
BETOB=BETO.SORT12,/EPSI)
BET0C=1.414+BET0
BETAT=BETOB
BTC=1.+BETAT/2.
F13=0.00011023+BETAT.0.00003307
E21=10,0010582+8ET4T+0.001058//(2..F131
P22=10.117464BETAT.0.03968)/F13
OMA0=(F214SORT(F21*F214F22)).0.95
ICOUNT=1
LOCATE BUBBLE AND STACK RELATIVE TO LONER STAGNATION POINT
29111)=0.
282(1)=0.
ZB1(2)=0.
ZB2121=0.
IF(TPET92eGT.TNETS11 282(1)=1THETB2.THETS11/2.
IF(THETBI.LT.THETSI) 29212)=( THETS1THETB1)/2.
IE(THETE12.1T.TMETSI) 281(2)=(THETSi...TNETB2)/2.
7ST11)=(6.2113+TMLS1THETS1)/2.
1ST(2)=(THETS1THLS2)/2.
ZMAX(21=1TMETSITHETS2)/2.
IMAX111=3.14159ZNAX(2)
5 =2
IFITNETB/oLT.TMETSI.ANO.THETB2eGT.THETS1) 18=1
INDBU9 =13
IF(T9.E0.11 3ETAT=BETOB
IF(T9.EQ.2) CALL UINE(UCI,OU020,ELCC1
IE(I3.E0.2) BETAT=BETOC.SORT(UCO)
CALL UINP(U,DUOZS,0.)
CALCULATE STAGNATION POINT PARAMETERS
DUOZ=BUOZS
CALL STAGPT(0.,LMAX,A,MOKUOZ2gUMUDZ2)
OMVS=04V
ENHVS=ENHV
IP=0

C r.
C
C
C
CALCULATE CONVECTIVE NEAT TRANSFER NUSSELT NUMBER
DO 62 I=1,2
INOBUB=Il
U1UO2.0.
INOR=I
01107=0UOZS
JSTaRT=0
2=0.
U=0.
THETA=THETS1
OUTPUT STAGNATION POINT VALUES
CALL OUTIO9VS,RE.OU0Z,ENNVS,ENUS,ZSAVE,Z,ENUSUM.ICOUNT.CASE.U)
IP=IP.1
THETANIR)=THETA
ENUP(IP1=ENUS
IF(INOBUB.E0.1) GO TO 76
PARTICLE REGION CALCULATION
21=].
22=ZST(I)
IF(I.EQ.2.AN7.281(21.NE.782(2)) 72=281(2)
CALL CHANNEL(RE,RETOC,ENUS.ZSAVE,ENUSUM,ICOUNT,CASE.THETAP.ENUF,21
1,22.ELGC.IR.LMAX,X,01.0XU022.0NUOZ2)
IFII.EQ.2.ANDZBI(2).NEa92(21) GO TO 79
21=72
22.7MAX(I)
CALL CHANNEL(RL.BETOC,ENUS,ZSAVEgENUSUM,ICOUNT,CASE,THETAR.ENUPal
1.72.ELCS,IP.LMAX.X.N.0XU922,3NU072)
GO TO 62
C
79
76
.0
al
77
BUBILE REGION CALCULATION
2=791(2)
UJU02=RP
GO TO BO
7=292(I1/10.
BETAT=1ETOE
CALL STAGPT(0.,LmAx.X,W.OXUD22.0AU022)
V(1,1)=OXLIOZAZ
Y(1,2)=0WUOZA7
GO TO il
BETAT=BETOB
CALL 3TAGPT(0.,LNAX,g,w,)XU022,014U022) r(1.1)=DAUC7AUOUIZ
Y(1,2)=OWUCZAu0U1Z rmAX(1)=1.0
vmAX(2)=1.
OmV=OmVS
ENHV=ENHVS
INDAUB=1
IF(Z.GE.Z82(I)) GO 73 79
CALL OJOSER(OFUN.Y.7.2.0.4AXOER,JSTART,H,HMIN,HMAX,EPS,NMAX,ERROR.
1WK,TER)
IF(7.GE.782(111 ;0 TO 78
JSTART=I
CALL OUT(ONV.RE.1U37.ENHV,ENUS,ZSAVE,2.ENUSUm,ICOUNT,CASE,U)
THETAP(IP)=THETA
ENUPIIP)=ENUS
GO TO 77
PARTICLE REGION CALCULATION
C
;
ON
62
INORUB=2
71=282tIl
72=ZST(I)
ELC=ELCC
Z2=ZMAX(I)
IF(Z2.E0.ZMAX(III ELC=ELCS
CALL CHANNEL(RE,BETOC,ENUS,ZSAVE,ENUSUM,ICOUNT.CASE.THLTAP.ENUP,21
1,22,ELC,IP.LMAx,X,W.TAUDZ2,0WU022)
Z1 =Z2
22=ZMAX(I)
IF(21.E0.22) GO TO 62
CALL CHANNELORE,RETOC,ENUS,ZSAVE,FNUSUm.ICOUNT,CASE.THETAP,ENUP,21
1,72,ELCS.IF,LmAX0(04.0XU022,0MU0221
CONTINUE
CALCULATE TOTAL CONVECTIVE, RADIATIVE, AND COMBINED NUSSELT NUMBER
ENUTC=ENUSUM/3.14159
ENUG8.0.
A2A3.0.
ENUR=BIR.(TI4F+.4,7N..9)
DTHET=THET02..THET01
IE(OTHET.NE.0.) A2A3.0THET/(2.*R3(LBUB).ACOS(1R021LBUBI.RB2(LBUB)..
11.)/(2..RB(LBUB).RO(LBU8))))
ENt.PR=ENUR.EMISSP/(1.+EMISSP.(1./EMISSN'1.))
IE(CTNET.NE.30 ENURB=ENUR.EMISS4/(A2A3+EMISSB.(1./EMISSW.A2A3))
OTNETR.DTMET/6.2932
ENUQ=ENURP.(1...0THETR)+ENURB.DTHETR
ENUTOT.ENUTC+ENUR
IF(LBM.GT.1.AND.UR.GT.0.) TIME=ARS(YO(LBUBI-YSI/UB wRITE(6,115) wPITE(6,11A) (CASE(I0),I0=1,6) wRITE(6,114)
WRITE(6,130) TIME
WP/TE(6,116) ENUTC
WRITE(6.132) ENUP
WRITE(6,1331 ENURB
WRITE(6,117) ENUR
WRITE(6,11A) ENUTOT
C PLOT NUSSELT NUMBER
IPAAX=IP
WRITE(6,I1E)
MRITE(6,119) (CASE(I3),I0=1,5)

WRITE(6.114)
WRITE(6.128)
NRTTE(6.129)
NRITE(6.I14)
CALL PLOTTER(IPMAX,THETAP.0..7..ENUP.0..1000..61.5.71,101
1000 CONTINUE
100
101
103
FORMAT(45X,28HGA5 CONVECTION HEAT TRANSFER)
FORMAT(33X.41HTO A CIRCULAR TUBE IN A GAS FLUIDI7ED BED)
FORMAT(5X,79HREYNOLOS NUMBER BASED ON TUBE DIA. GAS PROPS AT BED T
104
1J5
TEMP AND PRESSURE. AND UHF=.E12.4.5X,19HGAS PRANDTL NUMBER=.E12.41
FOPMAT(5X,39HRADIATIVE HEAT TRANSFER PARAMETER. BIR=oE12.4.5)(07HI
INTENSITY OF INTERSTICIAL TURBULENCE.,E12.4)
FORMATC5X,35HDIMENSIONLESS RED TEMP, TB/(T9TN)=.612.4,5X,36HDIMEN
/SIONLESS WALL TEMP. TW/(TBTW)=.E12.4)
FORMAT(5)(.37HPARTICLE HALF SPACING/TUBE DIA, SIDE=.E12.4.5)(.38HPAR
106
117
IT ICLE HALF SPACING/TUBE DIA. STACK=.E12.4)
114 FORMAT(1H0)
115 FORmAT(I.H11
116
FORMAT)5X.54HTOTAL GAS CONVECTION NUSSELT NO. (BASED ON TUBE UIA)=
1 .112.41
FORMATI5X.49HTOTAL RADIATIVE NUSSELT NO. (BASED ON TUBE OIA)= .112
118
119
120
1.4)
FORMAT(5)(09HTOTAL NUSSELT NO. (BASED ON TUBE DIA)= .112.4)
FORMAT(5)(.9HCASE NO. .502)
FORMAT(5X.22HPAR.T/CLE DIA/TUBE OIA=.E12.4.5X,20HPARTICLE SPHERICIT
121
122
123
IY=.E12.4)
FORMAT(5X.37HCORNER HEAT TRANSFER PARAMETER. SIDE=.E12.4.5X.38HCOR
INER HEAT TRANSFER PARAMETER. STACK=.E12.4)
FORMAT(5X,39HSTOKES REGION MATCHING PARAMETER. SIDE=.E12.4.5(.40HS
/TOXES REGICN MATCHING PARAMETER. STACK=,E12.41
FORMATI5X,29HSUPERFICIAL VELOCITY. UO/UMF=.E12.4.5X.26HBUBBLE VELD
124
/CITY PARANETER=.112.4)
FORMAT(5X.12H3LD VOIlAGE=1E12.4.5X,22HSUPFACE 0000001, SIOE=,E12.4
125
126
1)
FORMAT(5X.23HSURFACE VOIDAGE. STACK=.E12.4.5X.32HAVE BED FMESSIVIT
1.1.
AT TUBE hALL=.112.4)
FORMAT(5X,2IHBUBILE REGION, THETA=.E12.4,4H TO.E12.4,5X,20HSTACX
1RFGICN. THETA=.E12.4.4H
77,E12.4)
FORMAT(5X125HAVE CHANNEL LENGTH, SIDE=.E12.4,5X,26HAVE CHANNEL LEN
127
128
129 iGTH. STACK=.E12.4)
FORMAT(27X,34HNUSSELT NUMBER VS ANGULAR POSITION)
FORMAT(10X,30HVERTICAL SCALE= NUSSELT NUMBEROX.32HHORIZONTAL SCAL
13C
131
11= ANGLE, RADIANS)
FORMAT(5X.29HELAPSED TIME.UMF/TU3E RAOIUS=,E12.4)
FORMAT(5X,38HAVE RED EMISSIVITY ON BUBBLE BOUNDARY=.612.4,5X,24HEM
132
133
210
211
212
213
/ISSIVITY OF TUBE NALL=1112.4)
FORMAT(5X.39HPADIATIVE NUSSELT NO., PARTICLE REGION=.E12.41
FORMAT(5X,37HRADIATIVE NUSSELT NO., BUBBLE REGION=tE12.4)
FORmAT(28X,17HBUBBLE PARAMETERS)
FORMAT(43X,I1HCENTER COORDINATES)
FORMAT(5K.10HBUBBLE 40..5X.12HRADIUSIRB/A),9X14HX0/4,9X.RHYO/A)
FORMAT(5X,61H
214
FORMAT(9X1I2.9X,E12.4.5X.E12.4154.E12.41
END
C
SUBROUTINE CHANNEL(REOETOC.ENUS.ZSAVE,ENUSUM.ICOUNT.CASE.THETAP.E
INUP.21.22.ELC.I.LMAX,X.14.0hUD72,ONUOZ2)
CHANNEL CALCULATES THE AVE NUSSELT NO. IN THE CHANNELS
C
10
COMMON/0/0MV,ENHVOUOZ.PR,A(5).815)48.U.OXU02.0NUDZ,BETAT,ITERP
COMMON/P/NEU3ITHETA0(10),P0(10),R02(10).R82(10).ROS(10)
COMMON/V/IhDBUB.THETSI,THETBIITHET32.THETAITHLSI.THLS2.E(3.2).DPOY
II.LTABLE,THETATI501.UT(50),DUOT(50).UB,INDR.IC.DPDZ
OIMEhSION THETAP(10001,ENUP(1000)
01.(72-211/10.
Z3=72-1.5.02
7=21.5.32
2=2+01
CALL UINF(U.OUDZ.2)
11
EVALUATE STAGNATION POINT FLOW PARAMETERS
IF(U.E0.0.) CALL UINF(U.OUDZ.ELC)
BETAT=BET0CYSORT(U)
CALL STAGPT(O.,LMAX.X.W.DXU022,0NU072)
IF(U.EQ.0..OR.Z.LE.ELC) GO TO 11
DUDZ=2..U/ELC
CALL OUT(OMV.RE.DUDZ.ENHV.ENUS.ZSAVE.2.ENUSUN,ICOUNT,CASE.U)
THETAP(2)=THETA
ENUP(I)=ENUS
TFIZ.LT.231 GO TO 10
RETURN
END

C
C
C
C
C r
SUBROUTINE OFUN(Y.Z.10Y.PR,IND)
OFUN IS CALLED BY IMSL ROUTINE DVOGER AND RETURNS VALUES OF THE
DrRIVITIVES OF U.X AND U14 AT STATION 7
NEWTON S ITERATIVE RETHOO IS USED IN INTERMEDIATE CALCULATIONS TO
DETERMINE CMV AND NHV
T
C
C
COm40)4/8/BIC.INOR.REDV20MAX
ComMON/O/OPV.ENHVOUDZ,PR.A(5)0(51.C5.U,DXUOZOWUDZOETAT.ITERP
COMMON/P/NOUO.THETA0(10).R0(10).402(101.R82(10).ROS(10)
COMmON/V/INDBUI.TH:TSI,THET81,THET32,THETA.THLS1OHLS2.E13.2).OPOY
II,LTA3LE,THETAT(50).UT(50).DUOT(50).UO.INDR.ICOPD7
DIMENSION Y(3,2) .0Y(2),PR(M.1).04(2)
EPS=0.00001
CALL UINF(U.DU02,2)
TEST FOR STAGNATION POINT
IF(O.ED.O..OR.2.EQ.0.1 GO TO 20
X=Y(1.1)/U
W=Y(1.2)/U
IF(INDP.E0.1) GO TO 27
UPX.X.DUOZ
UPW=W+11107
ITERATIVE SOLUTION FOR DMA
C
C
C
C
C
11
100
In
27
28
I=0
ON(11=0MV
I=I+1
OMV=ON(1)
CALL HI2r(N2V.H1V.H2VP.H1VROMV.3ETAT)
ON(2)=ON(11-(0MV.H2V H2V-UPX)/(M2V.(H2V+2..0MV.H2VPII
TEST FOR CONVERGENCE
IF(ABSI(ON(2)-0N(1))/ON(2)).LT.EPS1 GO TO 10
ON(1)=0N(21
IF(I.LT.ITERP) GO TO 11
MR/TE(6.100) ITERP
FORMAI(5X,28HOMV FAILED TO CONVERGE AFTER.I3.11H ITERATIONS)
OMV=ON(2)
IF(OMV.LT.0..AND.0UW.GT.0.) GO TO 51
IF(OMV.GT.O.ANO.DUJZ.LT.0.) GO TO 52
IF(OMV.LT.-12.) GO TO 50
ARG=OPW/(PR.OMV)
IF15RG.LT.O.) GO TO 51
UPRO=SORT(ARG)
GO TO 28
REOV2=2/(U(0.02937..0092.3ETAT))
UPWC=SORT(4/(rR.REOV2))
CALL PROFLVIOMV.A,C41,C42)
C
C
ITERATIVE SOLUTION FOR NHV
I=0
ON(11=ENHV
12 I=I+1
ENHV=0)4(11
CALL G2HF(G2H,G2HP.A.B,CAI.C42.C3,ENHV.BETAT)
ON(2)=ON(1)-(ENHV.G2H-UPRO)/(G2H+G2HP)
C
C
C
TEST FOR CONVERGENCE
101
13
IF(ABS((ON(2)-ON(1))/ON(2)).LT.EPS) GO TO 13
ON(11=0N(2)
IF(I.LT.ITERP) G3 TO 12
WRITE(6,101) ITERP
FORMAT(5X,29HENHV FAILED TO CONVERGE AFTER.I3.11H ITERATIONS)
ENHV=ON(2)
H12=H1V/H2V
C
C
C
CALCULATION OF GRADIENTS
IF(INOP.EQ.1) GO TO 30
DY(1)=2..(H2V.A(2)-(1.54.H12)*X.OUDZ)
DY(21=2..(O2)4.13(21-0.5.M.OUDZ)
GO TO 21
30 0'1(1)=0.
20
DY(2)=4..G2H
GO TO 21
DY(11=0XU07
DY(2)=0MUD2
50
GO TO 21
OMV=-12.
WRITE(6.1021 OMV
102 FORmAT(5X01HBOUNDARY LAYER SEPARATION HAS OCCURED
1SOLUTION IS APPROXIMATE. OMV FIXED AT .E12.4)
DY(1)=0.
51
52
53
103
21
INSIDE BUBBLE,
DY(21=0.
GO TO 21
OMV=OMAX
GO TO 53
OMV=-12.
WRITE(6.103) OMV
FORBAT(5X.E6HVELOCITY GRADIENT TOO HIGH. SOLUTION IS APPROXIMATE.
10(10 FIXED AT .E12.41
DY(1)=0.
DY(2)=0.
RETURN
END

C
C
C
C
SUBROUTINE STAGPT(0211072,LMAX.X,W,DXUOZ2gDWUDZ2)
STAGFT PERFORMS THE FOLLOWING OPEFATIONS
1)
USES NEWTON S ITERATION PROCEDURE TO FIND OMV ANO NHV AT THE
STAGNATION POINT
2) CALCULATES THE INITIAL VALUES OF X ANO W ANO THEIR INITIAL
GRADIENTS
DIMENSION ON12).AP(5)
COMMON/0/0MV.ENHJOUOZ.PR.A(5),B(5),CB.U.OXU02.0WU07,3,TAT,ITEPP
COMMON/P/NOUB.THETA0(101.R0(1.0),R02(101,RB2(10),ROS(10)
COMMON/V/INDBUB,TMETS1,THETB1,THETB2,THETA,THLS/ITHLS2.E(3.2).DPOV
II,LTABLEsTHETAT(50),UT(50),DUOT(501,U3IINOR.ICOPD2
EPS=0.00001
ITERATIVE SOLUTION FOR OMV
C
C
12
ON(1)=OMV
L=0
L=L+1
ONV=ON(1)
CALL H/2F(H2VO4V.H2VP.H1V0.0MVOETAT)
H12V=2..)12114.141V
FP=1./6.(H12V+OMV.12..H2VP+HIVP))
F=2.+04V/6.DMVH12V
ON(2)=0N(1)F/FP
C
C
C
TEST FOR CONVERGENCE
102
13
IF(ABS(10N12)..ON(1))/04(2)).LT.EPS) GO TO 13
ON11)=ON(2)
IFIL.LT.LMAX1 GO TO 12
WRITE(6.102) LMAX
FoRmAT(5X,48HOmv FAILED TO CONVERGE AT STAGNATION POINT AFTER,I3,1
111.4 ITERATIONS)
OMV=ON(21
CALL PROFLvCONV.A.CAL,CA21
C
ITERATIVE SOLUTION FOR NHV
C
C
10
ON(/)=ENHV
G1P=2./(PR.OMV)
L=0
L=0.1
ENMV=ON(1)
CALL G2HF(C2H.G2HP.A.8.CA1.CA2tCB.ENMV.BETAT)
ON(2)=ON(1).-(ENHV.ENHV.G2H.-GOP)/(ENHV.(2..G2H+G2HF))
TEST FOR CONVERGENCE
103
11
IF(A3S110N(2).IN(111/0N(2)).LT.EPS1 GO TO It
ON(11=0N(2)
IF(L.LT.LMAX) GO TO 10
WRITE(6,103) LMAX
FORWAT(5X.49HENHV FAILED TO CONVERGE AT STAGNATION POINT AFTER.I3.
111M ITERATIONS)
ENHV=ON(2)
CALCULATION OF INITIAL GRADIENTS
0U072=DUOI.OUDZ
UPX=OMV.H2V.H2V
X=UPX/01112
UPW=PR.OMV*(ENMV.G2H)..2
W=UPW/OUDZ
FP=2.+H2VRFP
HOM=1./0411.2..H2VP/H2V
DX021=IFP/lHOMFP/UPX))/OU072
AP(2)=.166667
AP(3)=.-.5
AP14)=0.5
AP(5)=...1666667
CAP1=0.00833333
CAP2=0.0027777771
CALL G2HF(G2HAP,G2HAPP.AP.B.CAP1.CAP2gCBIENHV,BETAT)
G2H0=G2HAP/G2H
G2MN=G2HP/G2H
GPOM=...
2.*UPW.(1./OMV+G2H0)
GPNW.....2..UFW.(2.4.62HN)/ENHV
GN2=(1.4G2HN)/ENHV
GO2=1./OMV2..G2HO
OMP=OX021/X+1./DUOZ)/HOM
TGN2=2..GN2
OW021=(l TGN2.GPOM--G02.GPNH)*OMP/OUDI+GPMH/OUDI2)/(TGN2-.GPNH/UPW)
OX07=0XDZI*D2U022
OWDZ=OW011.02U022
OXUCZ=X.01102
EIWUOZ=W*OUO2
OXUOZ2=2..0X0I.DUOZ+X*02UDZ2
OWU012=2..0)102.0UOZ+W.02U022
RETURN
END

C
C
SUBROUTINE UINF(U,DU1Z,21
UINF CALCULATES THE LOCAL VELOCITY, VELOCITY GRADIENT. AND
PRESSURE GRAJIENT AT TUBE SURFACE
C
C
C
COMMONIP/NPUB,THETA01101,R0(10),R02(1011RB2(111,ROS(10/
COMMON/V/INDBUB,THETS1,THETB/vTHETB2ITHETAITHLS1,THLS2,1(3,2),OPDY iI,LTABLE,THETAT110),UT(501,DUDTISG),UB.INDR.IC.DP02
TEST FOR RIGHT OR LEFT SIDE OF TUBE
C
C
C
C
C
C
C
C
13
C
C
SGN=-1.
/FUNDR.EQ.1/ SGN=i.
THETA=TNETS1+SGN*2..2
IF1THETA.GT.6.28121 THETA=THETA5.2132
TEST FOR BUBBLE REGION
IF(THETElteEO.THETB2) GO TO 13
IF(INDBUB.E0.1) GO TO 10
PARTICLE REGION VELOCITY
IC=1
TEST FOR STACK REGION
IF(THETA.GT.THLSI.ANO.THETA.LT.THLS2) IC=2
IC=1 FOR CHANNEL REGION, IC=2 FOR STACK REGION
CALCULATE PRESSURE GRADIENT
C
CALL GRAOP(DPOT,02PDT2,THETA/
OPOZ=0PDYI.A9S(OPOT)
U=-0PD7/El1vIC)
DU07=-.(DPOYIKUPDT2)/E(1,IC)
DPOZ=...UFOUCZ
GO TO 11
RUBBLE REGION VELOCITY
11
12
11
IF(THETA.GE.THET02) THETA=THET82.0.999
IF(THETA.LE.THET111 THETA=THETB1.1.1001
DO 12 L=1,LTABLE
LI=L+1
IFITHETA.GT.THETAT(O.AND.THETA.LT.THETATILD) LINT=L
LINT1=LINT4.1
U=SGN.(UT(LINT)4.1THETA-THETAT(LINT)).(UTILINT1/-UT(LINT)1/(THETATI iLINTII-THETAT(LINT)1+UBFGOS(THETAI)
DU07=2.*(DUDF(LINT14.(THETA-THETAT(LINT)1.(OUDT1LINT1)-OUOT(LINT11/
11THETATILINTIA-THETAT(LINT1)-UBFSIN(THETA1/
IF(U.LT.0.) U=-U
DPOZ=-LI.JUCI
RETURN
ENO
SUBROUTINE FLOTTERIN.XONIN,KMAX,Y,YMIN,YHAX,IVH,IVD,IHH,IHO)
OINENSION A1125i.SY410L(31.X1W,YIN),IVI10001tIH11000)
DATA STMROL/K.S,K t,tK2/
WRITE 1,YWAX
FORMATI1X1E1.11
JELY=XNAX-XMIN
DELY=YMAX-.YMIN
VIV=FLOAT(IVH)..0.000101
1000
2000
KIH=FLOATIIHH/.-0.000001
00 1000 J=104
IVIJI=IYMAXYCJWYIV/3ELY4.1.
IH(J)=(X(J)AMIN).XIH/DELX+1.
DO 4000 I=11IVH
00 2000 K=1,INM
L=2
IF(MOD(K,IHD).EQ.1.0R04001I,IVD).E0.1)L=1
AIKI=SYMBOL(L)
DO 3000 J=1,N
IF(I.NE.IVIJI)G0 TO 3000
AIIM(J11=SYMBOL(3)
3000
IVIA=0
CONTINUE
4000 WRITE 10,(A(J),J=1,IHH)
10
FORMAT(10X,125A1)
30
WRITE 30,YMINIXHIN,XNAX
FORNAT(E44E8.1/* *,6X,E9.1.T78.E9.11
RETURN
END

C
C
C
C
C
SUBROUTINE OUTOMV.RE.OU17,ENHVIENUS.ZSAVE.Z.ENUSUM.ICOUNT.C4SE,U)
C
C
C
C
OUT COORDINATES OUTPUT OF LOCAL DATA INCLUDING CONSIOEFATION OF
STOKES REGION CONTRIBUTION
Al.
CONMON/C/LTAIS(2).PHISS4100.2),XSPPS(100.2).ALPHAS(2).BS(2).SOCIS(2
1).RRSP(2).SORP
CONMON/V/IND3UB.THETS1.THET31.THETB2.THETA.THLSI.THLS2.E(3.2).0PDY
1I.LTAILE.THETAT(5010UT(50).JUDT(501.U9.INDRoIC.OPOZ
DIMENSION CASE(5)
IF(ICOUNT.E0.55) ICOUNT=1
IF(ICOUNT.NE.1) GO TO Al.
WRITE(6.1/5)
WRITE(6.114)
WRITE(6,119) (CA5E(10).10=1.5)
WRITF(6.114)
WRITE(6.108)
WR/TE(6.109)
WRITE(6.107)
ICOUNT=ICOUNT.1
CALCULATE CONVECTIVE NUSSELT NUMBER
)(S./.
DELTAV=SCIRT(OMV/(RE.OUII))
DELTAH=DELTAVENHV
ENU=2./DELTAH
ENUI=ENU
IC=1
DETERMINE CORNER FLOW CONTRIBUTION
IFITHETA.GT.THLSteAND.THETA.LT.THLS2) IC=2
IFITHETA.LT.THET31.0R.THETA.GE.THET82) CALL CORNEP(FNU6.ENUO(SgIC)
IF(7.E0.0.) ZSAVE=4.
ENUSUM=ENUSUM(.(ENUSENUB)(77SAVE)/2.
ENUS=ENU3
ZSAVE=Z
C
C
OUTPUT LOCAL DATA
WRITE(6,113) THETA.Z.XS,ENU.ENUB.DELTAHIDELTAV.0
1
113 FORMAT(13X.E12.4.7K.E1Z.4.6X.E12.4.2X.E12.412X.E12.4.2K.E12.4.2XtE
112.4.2X.E12.4)
107 FORMAT(6)(0123)).-
108
1C9
119
114
115
2
FORMAT(16X.5HTHETA.15X.3HZ/0.9X.16HSTOKES EDGE X/SP.7X.4HNU20.80.
15HNUAVE.6)(03HDELTAH/3.6X.SHDELTA4/3.4X.12HAVE VELOCITY)
FORMAT(28)(116H(FROM STAG. PT.).27(.19)1(FROM CHANNEL CT4.1.2Kg12HIIA
1SEJ ON 0).7X.12H(BASED ON D).31)(1.5HU/UMF)
FORNAT(5X.9NCASE NO. .5A7)
FORIAT(IHO)
FORMAT(1N1)
RETURN
ENO
C
C
C
C
SUBROUTINE DRAGIEPS.EPSEF.EIL)
DRAG SETS UP CONSTANT RARAHETERS RELATING VELOCITY AND PRESSURE
GRADIENT
COMMON/0/06E.PHIOP.EPSI.C1X2
DIMENSION E(3.2)
A=11...EPS)/(EPSPHIOR)
C=EPSEF/EPS
C1=9REAAC
C2=1.75KAYCPC
E41.1.)=C1PC2IEPSI
RETURN
ENO

C
SUBROUTINE GRADP(OPOT.O2POT2,THET0)
GRAJP CALCULATES PRESSURE GRADIENT IN PARTICLE REGION AT TUBE
SURFACE BASED ON POTENTIAL SOLUTION
C f
10
COMMON/P/N9U9,THETA0(101,R0(10),R02(10),RB21101,ROS(101
TCS=2.*COS(THETA)
SN1=SIN(TMETA)
TSN=2..SN1
OPOT=TCS
02POT2=TSN
TTHETA=2..TMETA
DO 10 L=1,14BUB
0TH=THETATHETAO(L1
SN=SINOTH1
SN2=SN.S4
CS=COS(OTH1
RS=R0(11.SN
ONOM1=1.4R02(11-2..R0(L).C3
0110P2=0NOMI.ONOMI
014003=1NOMI.ONOM2
ROSS=ROS(L1SN1
OPOI=OPOT+R02(11+(TCS/ONON1-4..RS*ROSS/DNOM21
02PCT2=02POT2+RD2(Li.ITSN/ONOM1.4..R0(1) *(SIN(ITHETATHETA0l111R0
IS(O.CS1/0NOM2+16..402(L1+SN2*ROSS/ONOM31
RETURN
END
C
C
C
SUBROUTINE H12F(H24,HIV,H2VP,H1VP,OMV,BETAT1
H12F PROVICLS VALUES OF H1V AND H2V ANO GERIVITIVES
OMV2=0M11.0MV
HIV=0.3-0MV/120.4.9FTAT.(1./15.-01V/360.1
H1VP=-1./120.BETAT/360.
H2V=0.1174603-0.0010582.04V-0.000110229.0MV2+BETAT*(0.0346933-0.00
11050.0MV-0.000033069.0M02)
H20=-0.0010582-0.00105S.BETAT-2.'04V*(0.0001102291.0.000013069.BET
1AT1
RETURN
END
C
C
SUBROUTINE PROFLV(04V.A.CAI,CA21
PRJFLV CALCULATES THE VELOCITY PROFILE CONSTANTS
DIH7NSIOA 0(51
A(11=0,
A(,1=2.+0MV/5.
A(3)=-014V/2.
A(41=-20-0MV/2.
A(51=1.-0MV/6.
CA1=0.7+0MV/120.
CA2=0.43333+0MV/360.
Rc'TURN
END
C
C
SUBROUTINE G2HFIG2H,G2HP,A,B,CA1.CA2,CB.ENHV.BETAT1
G2HF PROVIOES VALUES OF G2H AND ITS PARTIAL DERIVITIVE WITH
RESPECT TO NHV
DIMENSION A(51.9(51
BETAT1=1.+BETAT
ENBT=ENHV*BETAT
SUM.J.
SUMP =O.
C
C
C
C
C
C
12
11
TEST ENHV
IFIENAV.GT.1.) GO TO 10
CALCULATE G2H ANO PARTIAL DERIVITIVE FOR ENHV.LT.1.
00 11 M=2,5
EM=FLOAT(M1
AN=A(M1ENHV..(M-11
SUM1=0.
SUM2=0.
00 12 L=2,5
EL=FLOATIM*1-1)
FL2=ENEIT/FLOAT(M+0
SUMI=SUM1+8(11.(1./FL+FL21
SUM2=SUM2.BILI.(FLOAT(M-11/FL+EM.E121
SUM=SUM+AN.11./EM+EN3T/FLOAT(M+11SUM11
SUMP=SUMP+AN.(FLOAT(M-11/EM+ENBT*EM/FLOAT(M411SUM21
G2H=SUM
G2HF=SUMP
GO TO 13
CALCULATE G2H AND PARTIAL DERIVITIVE FOR ENHV.GI.1.
10
15
14
13
SUMBP=0.
SUMB=0.
DO 14 M=2,5
SUM1=0.
5UM2=0.
DO 15 L=2,5
BN=P(11/ENHV..(L-11
BNM=BN.(1./FLOAT(M+L-11+BETAT/ELOAT(M+111
SOMI=SUM1+BN4
SU42=SUM2+ENI.FLOAT(L)
IF(M.NE.51 GO TO 15
SUm0=SUMB+BN/FLOAT(L1
SUmBP=SUNBF+BN
CONTINUE
SUM=SUM+A(M) +SUM1
SUMF=SUMP+A(MP.SUM2
G2H=CCA1+BETAT+CA2SUMBETATI.11.SUMB11/ENHV+BETATICB
G2HP=(CAIBETAT.CA2SUlPfBETAT1.(1.SUM9P11/ENHV
RETURN
ENO
Ln
Lo

C
C
C
SU9ROUT/NE CORNERIENUO.FNU.XSSP,IC)
COMMON/C/LTA9S(21.PHISS(100.2),XSRPS(100.21.ALPHAS(21.BS(2).SOOS(2
1),R0SP(2),SORP
CORNER LOCATES THE EDGE OF THE STOKES REGION VIA CALL TO FUNCTION
XSRPF AND CALCULATES THE AVE NUSSELT NO.
PHIS.BS(ICI*SORP/ENU
XSEP.XSRPFIPHIS,IC)*RPSP(IC)
ENU1=XSSP.ENU+SOOS(IC)
RETURN
ENO
FUNCTION XSRPF(PHIS,IC)
COHNON/C/LTA3S(21.PHISS(100.2).XSRPS(100,2).ALPHAS(21.BS(21.SOOS(2
11.RPSP(2),SORP
C
C
KSRPF LOCATES THE EO;E OF THE STOKES REGION
12
10
11
IF(PHIS.GT.PHISS(1.IC)) GO TO 10
LMAX=LTARS(IC)
DO 12 L=1,LMAX
11=14.1
IF(PHIS.LT.FHISS(L,IC).AND.PHIS.GT.PHISS(11,IC)) LINT.L
LINT1=LINT./
XSRPF.XSRPS(LINT./C).(PHIS-PHISS(LINT,IC)).(XSPPS(LINT1,IC)-XSRPS(
1LINT.TC))/(PHISS(LINT1.IC)-PHISS(LINT.ICI)
GO TO 11
XSRPF=KSRPS(1.11).EXPIALPHAS(IC).(PHIS-PHISS(1.IC)))
RETURN
END

GAS CONVECTION HEAT TRANSFER
TO A CIRCULAR TUBE IN A GAS FLUIDIZED RED
CASE NO.
2N
REYNOLDS NUMBER BASED ON TUBE DIA, GAS PROPS AT BED TEMP AND PRESSURE. AND UHF=
RADIATIVE HEAT TRANSFER PARAMETER, BIR= O.
.6492E+04
INTENSITY OF INTERSTICIAL TURBULENCE=
DIMENSIONLESS BED TEMP, TO/ITB-TW1= .3100E+02 DIMENSIONLESS WALL TEMP, TW/lTB-TW/=
GAS PRANOTL NUMBER=
.2000E+00
.3000E+02
.6000E-01 PARTICLE HALF SPACING/TUBE DIA, SIDE=
PARTICLE DIA/TUBE DIA= .1180E+00
CORNER HEAT TRANSFER PARAMETER. SIDE=
.7600E-01 PARTICLE HALF SPACING/TUBE DIA. STACK=
PARTICLE SPHERICITY= .6500E+00
.3000E+01 CORNER HEAT TRANSFER PARAMETER, STACK= .3900E+01
STOKES REGION MATCHING PARAMETER, SIDE=
SUPERFICIAL VELOCITY. UO/UMF= .1100E+01
BED VOIDAGE= .5000E+00
.1370E+01
BUBBLE VELOCITY PARAMETER=
SURFACE VOIDAGE. SIDE=
STOKES REGION MATCHING PARAMETER. STACK=
.6500E+00
.9530E-01
SURFACE VOIDAGE. STACK= .4330E+00
AVE BED EMISSIVITY ON BUBBLE BOLNDARY=
BUBBLE REGION. THETA= .4712E+01
AVE CHANNEL LENGTH. SIDE=
TO
.1520E+00
AVE BEO EMISSIVITY AT TUBE WALL=
.9000E+00
.4500E+00
EMISSIVITY OF TUBE WALL= .9000E+00
.4712E+01 STACK REGION. THETA=
AVE CHANNEL LENGTH. STACK=
.3490E+00
.1200E+00
TO
.1370E+01
.2793E+01
.7000E+00
BUBBLE NO.
1
BUBBLE PARAMETERS
CENTER COORDINATES
RADIUSTRB/A) 00/A TO/A
O.
.2204E-04 -.2000E+01

CASE NO.
2N
THETA
.1510E+01
.4712E+01
.4616E+01
.4424E+01
.42331+01
.4041E+01
.3849E+01
.3657E+01
.3465E+01
.3273E+01
.3081E+01
.2889E+01
.2732E+01
.2610E+01
.2487E+01
.2365E+01
.2243E+01
.2121E+01
.1999E+01
.1876E+01
.1754E+01
.4712E+01
.4808E+01
.5000E+01
.5192E+01
.5384E+01
.5576E+01
.57613E+01
.5960E+01
.6152E+01
.6086E-01
.2528E+00
.4099E+00
.5321E+00
.6643E+00
.7765E+00
.8987E+00
.1021E+01
.1143E+01
.1265E+01
.1357E+01
Z/3
(FROM STAG. PT.) o.
.4799E-01
.1440E+00
.2400E+00
.3359E+00
.4319E+00
.5279E+00
.6239E+00
.7198E+00
.8158E+00
.9118E+00
.9903E+00
.1051E+01
.1113E+01
.1174E+01
.1215E+01
.1296E+01
.1357E+01
.1418E+01
.1479E+01
.1540E+01
O.
.4799E-01
.1440E+00
.2399E+00
.3359E+00
.4319E+00
.5278E+00
.6238E+00
.7198E+00
.8157E+00
.9117E+00
.9903E+30
.1051E+01
.1112E+01
.1174E+01
.1235E+01
.1246E+01
.1357E+01
.1418E+01
.14/9E+11
STOKES EDGE X/SP
(FROM CHANNEL CTR.1
.1414E+00
.8078E-01
.2552E+00
.2297E+00
.2482E+00
.5558E+00
.6677E+00
.6874E+00
.6970E+00
.7022E+00
.7046E+00
.7050E+00
.7034E+00
.3842E+00
.3717E+00
.15531+00
.3348E+00
.1091E+00
.2774E+00
.2383E+00
.1922E+00
.1414E+00
.2552E+00
.2297E+00
.2482E+00
.5559E+00
.6678E+00
.6874E+00
.6970E+00
.7022E+00
.7046E+00
.7050E+00
.7034E+00
.3842E+00
.3717E+00
.3553E+00
.3348E+00
.3091E400
.2774E+00
.2383E+00
.1922E+00
NU20
(BASED ON 01
.1451E+03
.1312E+03.
.1413E+03
.3751E+03
.4516E+03
.5123E+03
.5581E+03
.5896E+03
.6066E+03
.6092E+03
.5975E+03
.3641E+03
.3506E+03
.3335E+03
.3127E+03
.2883E+03
.2598E+03
.2267E+03
.1882E+03
.1415E+03
.7746E+02
.1451E+03
.13/2E+03
.1413E+03
.3751E+03
.4516E+03
.5122E+03
.5581E+03
.5895E+03
.6066E+03
.6092E+03
.5975E+03
.3641E+03
.3506E+03
.3335E+03
.3128E+03
.2883E+03
.2598E+03
.2268E+03
.1882E+03
.1415E+03
.8500E+02
.7812E+02
.8305E+02
.2565E+03
.3496E+03
.4001E+03
.4370E+03
.4619E+03
.4754E+03
.4775E+03
.4682E+03
.2006E+03
.1911E+03
.1792E+03
.1655E+03
.1499E+03
.1326E+03
.1148E+03
.9693E+02
.8077E+02
.6702E+02
.8500E+02
.7812E+02
.8305E+02
.2564E+03
.3495E+03
.4001E+03
.4370E+03
.4619E+03
.4754E+03
.4775E+03
.4682E+03
.2007E+03
.1911E+03
.1793E+03
.1655E+03
.1499E+03
.1328E+03
.1146E+03
.9694E+02
.8077E+02
NUAJE
(BASED ON 01
OELTAH/D
.1378E-01
.1524E-01
.1415E-01
.5332E-02
.4428E-02
.3904E-02
.3583E-02
.3392E-02
.3297E-02
.3283E-02
.3348E-02
.5413E-02
.5705E-02
.5998E-02
.6395E-02
.6938E-02
.7699E-02
.8821E-02
.1063E-01
.1414E-01
.2582E-01
.1378E-01
.1524E-01
.1415E-01
.5332E-82
.4429E-02
.3904E-02
.3584E-02
.3342E-02
.3297E-02
.3283E-02
.3347E-02
.5493E-02
.5705E-02
.5997E-02
.6395E-02
.6938E-02
.7699E-02
.8820E-02
.1053E-01
.1414E-01
JELTAV/0
.9097E-02
.16771-01
.8766E-02
.9783E-02
.9004E-02
.3381E-02
.2803E-02
.2468E-02
.2263E-02
.2142E-02
.2081E-02
.2072E-02
.2113E-02
.3469E-02
.3625E-02
.3813E-02
.4069E-02
.4419E-02
.8766E-02
.9783E-02
.9004E-02
.3381E-02
.2802E-02
.2468E-02
.2263E-02
.2141E-02
.2061E-02
.2072E-02
.2113E-02
.3489E-02
.3625E-02
.3813E-02
.4069E-02
.4419E-02
.4911E-02
.5637E-02
.6811E-02
.4911E-02
.5636E-02
.6811E-02
.9097E-02
AVE VELOCITY
U/UMF
0.
.7396E+00
.2190E+01
.3560E+01
.4799E+01
.5863E+01
.6710E+01
.7312E+01
.7644E+01
.7696E+01
.7465E+01
.2796E+01
.2627E+01
.2419E+01
.2175E+01
.1898E+01
.1593E+01
0.
.1265E+01
.9171E+00
.5558E+00
.1862E+00
.7380E+00
.2189E+01
.3559E+01
.4799E101
.5862E+01
.6710E+01
.7311E+01
.7644E+01
.7696E+01
.7466E+01
.2797E+01
.2628E+01
.2420E+01
.2175E+01
.1899E+01
.1594E+01
.1265E+01
.9172E+00
.5559E+00

CASE NO.
2N
ELAPSED TIMEUMFFTUBE RADIUS= O.
TOTAL GAS CONVECTION NUSSELT NO. (BASED ON TUBE D/A)=
RADIATIVE NUSSELT NO.. PARTICLE REGION= O.
RADIATIVE NUSSELT NO.. BUBBLE REGION O.
TOTAL RADIATIVE NUSSELT NO. (BASED ON TUBE OIA1=
TOTAL NUSSELT NO. (BASED ON TUBE MAI= .2626E+03
O.
.2626E*03
CASE NO.
2N
NUSSELT NUMBER VS ANGULAR POSITION
VERTICAL SCALE= NUSSELT NUMBER
HORIZONTAL SCALE= ANGLE. RADIANS
.1E+04
157 x
.
X
.
.X X
A.
x x
.
x.
.
x
.
x x
.
X
Of..X
XX
X
.
X.
.
X x
X X
.
.
. X
X
X x.
X
X
XX
O.
X
X
X X
X .
.7E+11

PROGRAM PRESS(INPUTOUTPUTITAPE5=INPUTJAPE6=OUTPUT)
CALL PRES
END
SUBROUTINE PRES
C
C
C
C
C
C
C
C
C
C
C
C
C
PRESS PERFORMS THE FOLLOWING OPERATIONS
1) CALCULATES AND OUTPUTS THE PRESSURE GRAOIENT ON THE TUBE
AS A FUNCTION OF ANGLE
2) PRODUCES A PRINT PLOT OF PRESSURE LEVEL IN A TUBE CENTERED
COORDINATE SYSTEM
PRESSURE IS THE POTENTIAL SOLUTION OBTAINED BY PLACING IMAGES
OF AN ARBITARY NUMIER OF BOLES INSIDE THE TUBE SUCH THAT THE
CONDITION OF NO FLOW THROUGH THE TUBE IS SATISFIED.
C
C
C
C
C
C
C
C
C
C
C
C
INPUT DATA
NCASE= TOTAL NUMBER OF CASES TO BE CONSIDERED
CP= PRESSURE SCALE
OTHETA= INCREMENT IN ANGLE USED IN CALCULATION OF PRESSURE
GRADIENT ON TUBE
CASE(1). ALPHA NUMERIC IDENTIFIER OF CASE
MB= NUMBER OF BUBBLES PRESENT
XOR= DISTANCE FROM LEFT SIDE OF PAGE TO X.Y ORIGIN
YOR= DISTANCE FROM TOP OF PAGE TO XgY ORIGIN
CX.0f= SPATIAL SCALE IN X AND Y DIRECTIONS RESF.
'MAIO= BUBBLE RADIUS/TUBE RADIUS, R9/A
ROA(L), THETAO(L)= COORDINATES OF BUBBLE CENTER. ROA=RADIAL
COORO/TUBE RADIUS. THETA() IS IN RADIANS
DIMENSION SYMP(51),SYMN( 51),PSYM(70),R9A(10),R0A(10),THETA0(10),RB
1A2(101,R0A2110).TROA(10),ROAS(101.SNO(10),CS0(10).X0(1610Y0(10),CA
2SE(5)
I
DATA SYMP/3 A*,* 0,0 B*,* 0,0 Ct.* 0,0 00,0 2,7 17,0 0,0 Ffg2 t,t Gt.* 2,0 HO,* 0.0 It,* 0,0 i*,* 0,0 Ktot 0,7 L*.A
0,0
040,0 0.0 80.0
0,7 00,0 2,0 P0,0 0,0 00,0 0.2 R!,* 0.0 S*,*
30.0 TB,* 2.0 U2.!
0,0 00,0 0,0 WY.* 0.0 X*,* 0,0 Y2,2 0,0 00
4/
1
DATA SYMN/$11*,*.. 0,*...13*,*
0.0..G$10-. 1,0.)4*,*
*,2-0*,* 0.000.0
0.0E0,0- *.*Ft.*
0,012,2 0,2...10.0- !,*10),7 7,71.4.* *,*..
2010,0.. 0,32,t.. 0.0.<12.0.. t.*.P*.3. 0.0-00.0
0,0...120,0- 0,0...S0,0
3*,*14.0.. $,*-U0,* 0,0 -00.0- 2,0..Wf.*.. 0,010,0- $.$Y$,*' 0,$'7*
4/
DATA BLANX,007/0 2,0 .0/
C
C
C
100
REAC(5.1001 NCASE,1P.DTHETA
FoRmAT(15,2F10.01
OUTPUT TABLE OF PLOTTING SYMBOLS AND PRESSURE RANGES
WRITE(6.2001
WRITE(6,201)
WRITE16.202)
WRITE(6,203)
WRITE(6,204)
WRITE(6.205)
WRITE(6,20E)
WRITE(6,2071
WRITE(6,208) ISYMP(L),FLOAT(L1)=OP,FLOAT(L)=DP.L=1,51)
KMAX=IFIX(E.28316/0THETA1+1
DO 50 N=1,NCASE
C
C
101
102
READ(5,1011 ICASEIII.I=1,5),NBUB,X0R,YOR,DX.DY
REAC(5.102) (RBA(L),ROA(L).THETAD(L),L=1,NBUBI
FORMAT(5A2,I5,4F10.0)
FORMAT(3F10.0)
10
DO 10 L=1,NBUB
RBA2(L)=RBA(L).RBA(L)
ROA2(L)=ROA(L)mROA(L)
TROA(L)=2.+ROA(L)
SNOIL)=SIN(THETAO(L))
ROAS(L)=ROA(L)=SNOIL)
CSOILI=OOS(TMETAO(L))
XOIL)=ROA(LI.CSO(L)
YO/L)=ROA(LI=SNO(L)
C
C
C
OUTPUT TABLE OF BUBBLE PARAMETERS
WRITE16,200)
BR/7E16,209) (OASEII),I=1,5)
WRITE(6,203)
WR/TEI61210)
WRITE(6,211)
WRITE(6,212)
WRITS(6,213)
WRITE(6.214) (L.RBA(L),XO(L),Y0(LI.L=1,NBUB)
WRITE(6.203)
WRITE(6,203)
WRITE(6,215)
WRITE(6,21E1
WRITE(6,217)
THETA=0.
C
C
C
C
C
C
CALCULATE TUBE PRESSURE GRADIENT
12
11
DO 11 K=1,XMAX
CST=COS(THETA)
GRACP=2.=CST
SN=SIN(THETA)
DO 12 L=1,NBU8
CS=COS(THETATHETAO(L))
ONOM=1.+ROA2(L)..TROAIL).CS
GRAOP=GRADF+RBA2(L)=(2.=OST/ON04...4.=ROAILI=SINITHETATMETAO(L))=( iROAS(L)-.SN)/(0NOM.ONOM))
WaTE(6,2181 THETA,GRAOP
THETA=THETA+3THETA
OUTPUT TUBE PRESSURE GRADIENT
WRITE(6.200)
WRITE(6,219) (CASE(I),I=1,5),DX,DY
WRITE(6,203)
Y=YOR
C

C
CALCULATE PRESSURE AS FUNCTION OF 0,5
C
22
00 20 J=1,50
X=-XOR
Y2=vYY
DO 21 I=1,50
IF(J.EQ.1.0R.J.E0.50.0R.I.E0.1.0R.I.E0.50) GO TO 26
IFIABSIXI.LT.1.E-10.0R.A9S(Y).LT.1.E-101 GO TO 26
R2=0.X4.02
IF(R2.LT.1.) GO TO 25
R=SORT(R2)
SN=Y/R
OPUA=SNY(1.+1./R2)
CS1=X/R
DO 22 L=1,NBUB
CS=CSTCSO(L)6SNYSNO(L)
DPUA=DPUAYRBA2(LIY((-SNYROAS(L)/R)/(R2+ROA2(1.)-TROA(L)vRvCS).(-SNY
111YROAS(L11/(1.6R2YR0A2(L)-RvCSYTROAIL)))
OPUA=-DPUAvR
MATCH PRESSURE WITH PLOTTING SYMBOL
NSP=1+IFIX(ABS(0 PUA)/DP)
IF(NSP.GT.51) NSP=51
PSYm(I)=SYHF(NSP)
25
IF(DFUA.LT.0.) PSYN(/)=SYMN(NSP)
GO TO 21
PSYM(I1=BLANK
GO TO 21
26 PSYv(I)=00T
21 X =X +DX
C
C
C
PLOT PRESSURE
20
53
WR/TE(6.1000) (PSYM(II,I=1,501
V =Y -OY
CONTINUE
C
200 FORMAT(16111
201
FORMAT(11X,40HPOTENTIAL PRESSURE IN VICINITY OF A TUBE)
202 FORmAT(16X,31HIN A FLUIDIZED BED WITH BUBBLES)
1U3
204
FORmAT(1H
)
FORHAT(15)(.16HPLOTTING SYMBOLS)
005 FoRmAT(15X.22HNAG. DELTA P/(OPOYOvA))
206
207
208
FORmAT(5A,6MSYMBOL.5X,4HMIN.,13X,4HMAX.)
FORMAT(5X,38H
FoRmAT(7X,A2,7x,E12.4.5X,E12.4)
209 FoRmAT(5X,9HCASE NO. ,5A2)
210 FORNAT(28X,17HBU3BLE PARAMETERS)
211 FORmAT(41X,18HCENTER COORDINATES)
212
211
FORPAT(5X,10HBUBBLE NO.,5X,12HRADIUS(RB/A1,9X,4HX0/A,9x,4HYO/A)
FORMAT(5x,61H
214
215
216
217
1
FORMAT(91,12,9X,E12.4.5X,E12.4,51,E12.4)
FORmAT(10X.22HTUBE PRESSURE GRADIENT)
FORmAT(8X,5NTHETA,iX,18HOPOTHETA/10P0y01A))
FORmAT(5X,35H
218
219
FoRmAT(5X,E12.4,1X,E12.41
FORMAT(5X,5HCASE NO. 1511205X.6NSCALE.OXOHDELTA X= ,E12.4,5X,9N3E
1LTA 1= ,012.4)
1000 FORmATI5X,50A21
RETURN
END

SYMBOL
C
A
8
0
F
H
I
K
L
4
N
0
P
R
S
U
V
X
2
POTENTIAL RRESSURE IN VICINITY OF A TUBE
IN A FLUID/2EI 8E0 NITH BUBBLES
.3450E+01
.3600E+01
.3750E+01
.3900E+01
.4050E+01
.4200E+01
.4350E+01
.4500E+01
.4650E+01
.40000.01
.4950F+01
.5100E+01
.5250E+01
.5400E+01
.5550E+01
.5700E+01
.5050E+01
.6000E+01
.6150E+01
.63001+01
.6450E101
.6600E+01
.6750E+01
.6910E+01
.7050E191
.7200E+01
.7350E+00
0.
.1500E+00
.3000E+00
.4500E+00
.6000E+00
.7500E+00
.9100E+00
.1050E+01
.1200E+01
.1350E+01
.1500E+01
.1650E+01
.1000E+01
.1950E+01
.2100E+01
.2250E+01
.2400E+01
.2550E101
.2700E+01
.2150E+01
.3000E+01
.3150E+01
.3300E+01
.7500E+01
PLOTTING SYMBOLS
MAG. OFLTA Pf9ORTWA1
YIN.
MAX.
.1500E+00
.3000E+00
.4500E+00
.E000E+00
.7500E+00
.9000E+00
.1050E+01
.1200E+01
.1350E+01
.1500E+01
.1650E+01
.1100E+01
.1350E+01
.2100E01
.2250E+01
.2400E+01
.25500.01
.2700E+31
.2950E +01
.3000E+01
.3150E+01
.3300E+01
.3450E+01
.3600E+01
.3750E+01
.3900E+01
.4050E+01
.4200E+01
.4350E+01
.4500E+01
.4650E+01
.4130E+01
.4950E+01
.5100E+01
.5250E+01
.5400E+91
.5550E+01
.5700E+01
.5350E+01
.6000E+01
.6150E+01
.5300E+01
.6450E+01
.6600E+01
.1750E+01
.6910E+01
.7050E+01
.7200E+01
.7350E+01
.7500E+01
.7550E+01
CASE NO.
79
BUBBLE NO.
9LBILE PARAMETERS
CENTER
RADIUS(RBiAl X0fA
COORDINATES
YO/A
.1000E+01 -.1439E+01 .4005E+00
O.
.3142E+00
.6283E+00
.9425E+00
.1257E+01
.1571E+01
.1885E+01
.2199E+01
.2513E+01
.2827E+01
.3142E+01
.3456E+81
.3770E+01
.4004E+11
.4398E+01
.4712E+01
.5927E+01
.5341E+01
.5655E+01
.5969E+01
TUBE PRESSURE GRADIENT
THETA
OPOTHETAMPOYO*A
.1691E+01
.1510E+01
.1267E+01
.7767E+00
.1445E+00
-.58090+07
-.1317E+01
-.1737E+01
.1450E+02
.1127E+01
-.1599E+01
-.3593E+01
-.2001E+01
-.2047E+01
-.1252E+01
-.4931E+01
.2111E+0E
.0213E+00
.1294E+01
.1593E+01

SCALE, DELTA A= .1100E+00 DELTA Y.
.1000E+00
CASE NO.
79
- - ------ H-H-+-F-P-H-F-Y-H-H-H-4-H-H-H-4-H-H-H-H ----- -
.-H-H-H-H-H- - - - ----- G-C-G-G-G-G-G-G-G-G-G-G- - - - -
- - ----- G-G-G-G-G-G-G-C- - - - - - - - -0-0-0-0-0- - - - -
.-I-I-I-I-I-I-I-I-I-I-I-I-I
.-H-H ----------- .
.-G-G-G-G-G- - ----- F-F-F-F-F-F- - - - - -F-F-F-F-F- - - - -G-G-G . ------- - - - - - -
.
- - - - - - - F-F-F-F-F- - - - - -E-E-E-E-E- - - -
-F-F-F ----- G .-G-G-G-G -------- G-G .
.
.-F- - ------ E-E-E- - - -C-1- - - - - - - -0-0-0- -E-E- - - -F-F
. ----- E-E-E-E-F- -
- -0-0- - -C-C-C- - -C-C-C- -0-0- - -E-E- - -F .-F -------F-F -F -F -F
.
.-F-F-F-F-F-F-F-F-F-F-F-F-
.-E-E-E-E- - - - - - 0-0- - -C-C-
- ------ 0-0-0- - -C-C- -9-
A
-A-A-A-4- -9- -C-C- -0- - -E-
. ----- 0-0-0-0-0- - -C- -EA
.-0-1-0-0-0-0-0-0- - - -C-C- -B -A
C C B A- -9- -C- -0
0 E E 0 C -A- -9-
.-2-0-0-0-0- ------C -C- -E-A
---------C-C- - -
A
4 H
9 N
0 8-A- -8
0
-A-9
.-F-F-F-F-F-F ------ E-E .
- - -0-0-0 -
C
.-C-C-t-C-C-C-C-C-C ----- - - -0-0-E ----- 0-0
.-C-C-C-C-C-C-C-C-C-C- - - -0-C-E-F-H-H-7-Z- -I-G-F
- -C-C-C
-C- - -
- -8-8-8 .
. -------- C-C-C-C- -0- -E-F-G-I-K ----- I-
.
A A A A
.
A .
.
....11-A ------ 9-9-8- - -C- -0- -E-E- -F-F-F-F- -
.
.
.
.
.
.
.-A-A-A-A-A ----- 9-E- - -C- - -D- - -E-E-E-E- -D
A A A-A-A-A-A- - - -e-o-B- -C -C- - 0- 0- 0 -0 -0- - -C
A A A A A A- A- A -A -A- - -8-9- - - -C-C-C-C-C-C- - -9
9 9
A
A A A A- A -A -A- - - -9-8-9- - - - -8-8- -A-A
A
A A-A-4-4-A- -------- A-A A
1
. 999 B B B
.
.
AAA A- A- A- A- A -A -A -A A A A
C C
9 9 0 9 9
8E989 e
A A A
A A A a o 3 A
0 0 0 0
9 8 9 9 9
C C C C C C C C
C C C C C C C C
CCCCCCC
C C C C C
0 0 0 0 0 0 7 0 0 0 0
00001E0 01110
E E E F E E
FFFFFFFFFFFFFF
1 0 0 0 0
EEEEEEEEFEEEEEEEEEEFE
F F F F F F F F F F F F
8 9
C C
8
C C
0 0 0
E E E E
F F F F F F F F F F
C C
FFFFF
3 C I
0 0
CC
0
0 0
EEEE
E E E
E E
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.
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.
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.
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.
FFFF .
FFFF
.
.
.
.
.
.
.
0 0 0 0.
E E
EEEE 0.
EEEEEE
EEEEEE.
F E .
FFFF
.FFFFFFFFFFFF
FF.FFFFFFFFFFFFF.
.00000GGGGGGGG
0 6 0 .
0 0 0 0 0 G
GGGGGGG.
GGGGGGG
G GGGGGGGGGCGGGCGGGGGGGGG
.
0
8 9
C C
C C C C
0 0 0
.
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.
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.
HHHHHHHHHH
HH.HHHHHHHHHHHHH
.
.
.
.

PROGRAM BUBBLE(INPUTOUT.UT,FuNCH.TAPE5=INPUT.TAFE6=OUTPUT)
CALL RUBLE
ENO
C
C
C
C
C
SUBROUTINE BUBLE
BUBBLE CALCULATES THE AVERAGE GAS VELOCITY INSIDE A GAS BUBBLE IN
A FLUIDIZED BED RASED UPON CONSERVATION OF MASS. INTEGRATIONS ARE
PEPFCRMED NUMERICALLY USING THE TRAPESOIOAL RULE. GAS VELOCITY
AT THE BUBBLE BOUNDARY IS OBTAINED VIA POTENTIAL SOLUTION FOR
ARBITRARY NUMBER OF BUBBLES IN VICINITY OF A TUBE.
C
C
C
INPUT DATA
C
C
C
NCMAX= TOTAL NUMBER OF CASES TO BE CONSIDERED
MAX= NUMBER OF SECTIONS AT WHICH VELOCITY IS TO BE CALCULATED
LBUB= INDEX OF BUBBLE IN WHICH GAS VELOCITY IS CALCULATED
N91/1= NUMBER OF BUBBLES NEAR TUBE
C
C f.
C
C
C
C
MASS FLOW RATES
CASE(Il= ALPHA NUMERIC IOENTIFIER OF CASE
THETAD=ANGULAR POSITION OF CENTER OF BUBBLE CONTACTING TUBE
(INDEX NO. LBUB)
THETB1= INITIAL ANGULAR BOUND OF BUBBLE CONTACTING TUBE
IF THETBI IS SET EQUAL TO ZERO ON INPUT, THET31 WILL
BE COMPUTED FOR A CIRCULAR BUBBLE OF RADIUS R9ILBU91
AND CENTER AT ROITHETAO
FNALP= NUMBER OF ANGULAR INCREMENTS USED IN CALCULATION OF MASS
FLOW THROUGH BUBBLE ARC AT THETA LT. THETBI
R9= DIMENSIONLESS BUBBLE RADIUS (REL. TO TUBE RADIUS)
XO.Y0= DIMENSIONLESS COORDINATES OF BUBBLE CENTER
C
C
DIMENSION XB1300),Y3(30111.THETA(3001.3(300),U1300).CASE(5).RB(10),
10UOT(300)
COHMON/E/OX,IY.NMAX.ENT,XI.YI
COMMON/P/X0(101,Y0(0),R02(10).RB2(10),OPDX.OPOY,OPH,NBUB
C
110
101
REA0(5,102) NCHAX
DO 1000 NC=1,NCMAX
REA9(5,100) MAX,L3UB,NBU9.(CASE(II.I=1.51,THEYAO.THETB1.ENALP
REAC(51101) (R0(L),X0(LI,Y0IL).L=1,N9U9)
FORMAT(3I5,5A2.3F10.0)
FoRwAT(3E10.01
102 FORMATION)
C
C
COMPUTE BUBFLE PARAMETERS r
IT
30
00 12 L=1,NBUB
RBT(LI=RB(L)YRB(11
P0211.1=X0(1)*X0(L)+YO(L).Y0(L)
RO=SORT(R02(LBUB))
TEST THETBI INPUT. IF TRETBi=0. COMPUTE TRET81
IF(THETB1.E0.0.1 GO TO NT
BFTMAX=THETAO-THET41
RBP2=1.4.202(LBUBI-2..RO.COS(BETMAX1
GO TO 51
R9P1=RB2(LBUB)
RETYAW.AC:OS1(1.+R02(LBU3)-R3F21/(2..R011
C
51
30
TwET31=THETAO-BETmAX
THETA(/)=THETBI
OTHETA=2.*BETMAX/FLOAT(MAX-.1)
MAX2=MAX4.1
XB(1)=COS(THETBil
YB(11=SINITHETB1/
DO 30 m.2,rax2
THETA(M)=THETA(1.1.1THETAYFLOATIM-2)
CSB.COS(THETA(M)-THETA01
SN221.-CSBYCSB
RBO=ROCSB.SORT(RBF2-R02(L3U01.SN21
XE1(MI.RBEOCOSITHETA(M1)
YR(141=RBB.SIN(THETA(4)1
OUTPUT BUBBLE PARAMETERS
C
C
C
WRITE(6.200)
WRITE(6.201)
WRITE(6,202)
WRITE(6,2031
WRITE(6.204) (CASE(I1,I=1.51
WRITE(6,2031
WRITE(6,2101
WRITE(6,2111
WRITE(6,2121
WRITE(6,213)
WRITE(6.214) (L.RB(L),X0(L),Y0(L).L=1,NBUB)
WRITE(6,203)
WRITE(6,205) LBUB
WRITE(6.203)
WRITE(6.217)
WR/TE(6.218)
WRITE(6.219)
WR/TE(6.2201
(X9(M).YEI(M),THETA(M)01=2.MAX2)
WRITE(6,200)
WRITE(6,204) (CASE(I),I=1,5)
WRITE(6,206)
WRITE(6.203)
WRITE(6,207)
WRITEI6,201)
CALCULATE GAS VELOCITY AT BUBBLE ENO POINTS
X=X9(1)
Y=Y0(11
CALL GRAOPSIA,Y)
UXS=OPDX
UYS=OPIY
U(21=UXS.SIN(THETA(1))+UYSYCOS(THETA(111 x=xe(mAx2)
Y=TB(NAX2)
CALL GRAOP9(X,Y)
U(40(2)=-DPDX.SIN(THETA(MAX2))+DPDYCOSITHETA(MAX2))
DEFORM MASS FLOW INTEGRATIONS TO OBTAIN GAS VELOCITY INSIDE RUB

C
C
20
C
77
COMPUTE MASS FLOW THROUGH RUBBLE ARC FOR THETA LT. THETB2
C
C
C
52
40
ENDOT=0.
B(2)=SORTIXE)(20X8(2).Y9(2)Y13(2)0-1.
IF(B(2).LT.1.E.-02) GO TO 52
XS=X9(11
YS =YB(1)
RBP=SORT(RBP2)
ALPMAX=2.ACOS(ROYSIN(BETMAX)/RBP)
GAMA=1.57081BETMAX+ALPMAX/2.0
DALF=ALPMAX/ENALF
ALPHA=0.
R1 =R02(LBUE)+RBP2
R2=2.R0
R3=R2RBP
R4=R02(LBUB)-RBP2
NALP=1,IFIX(ENALP)
00 40 N=2,NALP
ALPHA=ALPNA.DALP
R=SORTIR1R3YCOS(GA4A+ALPHA))
THETI=THETAOACOSUR44R.01/(R2.R))
X=RCOSITHETI)
Y=RSIN(THETI)
CALL GRADP2(X.Y)
UX=DPDX
UY=OPOY
EM00T=EMOOT.0.5.((UX+UXS).(YS-Y)+(UY.UYS).(X..XS))
XS=X
YS=Y
UXS=UX
UYS=UY
U(2)=EMOOT/B(2)
CALCULATE MASS FLOW THROUGH BUBBLE ARC AT THETA GT. THEM.
C
C
10
DO 10 M=3,MAX2
M1=M-1
X=X13(M)
Y=Y9(4)
CALL GRAOPB(XtY)
UX=OPDX
UY=ORDY
EMOCT=EMOOT+.5.((UX.UTS).(YB(M1)-VB(M)).(UY4UYS/.1)(9(MI-X9(041)0)
UXS=UX
UYS=UY
B(M1=SORT(XEI(M).X8(41+1.8(M)YB(4))1.
IF(B(4).GT.1.E-.04) U(M)=EMOOT/B(M)
CALCULATE AVE VELOCITY GRADIENTS VIA DIFFERENCE AFPROXIMATIONS
DO 20 M=3.MAX
DUOTIM)=1U1M+10U(M..1))/(THETAIM(.10-THETA(M-1)0
OUDT(2)=(U(3)-U(21)/(THETA(3)..THETA(2)1
DUDT(MAX2).(U(MAX2) ..U(MAX1)/(THETA(MAX2)..THETA(MAX))
DO 37 4=1.MAX
IF(THETA(M).GT.6.2831 THET1(M)=THETA(M) -6.283
;
OUTPUT AVE VELOCITY AND GRADIENT
C
WRITE(692160 (THETA(M).B(01),U(M).DUDT(01),M=2,MAX2)
PUNCH 301. NC
300
301
PUNCH 300, (THETA(M),U(4).OUIT(M).M=2.MAX2)
FORMAT(3E12.4)
FORMATII2)
1000 CONTINUE
200
FORMAT(1H1)
201
FORMAT(331(.34HAVERAGE GAS VELOCITY INSIDE BUBBLE)
202 FORMAT(32)(136HCONTACTING A TUBE IN A FLUIDIZED BED)
203 FORMAT(TH
1
204 FORMAT(5X.9HCASE NO. .5/12)
205
206
FORMAT(5X,23HFLOW INSIDE RUBBLE NO. ,12.12H IS COMPUTED)
FORMAT(5)(01HAVE. GAS VELOCITY IN BUBBLE REGION AS A FUNCTION OF P
IOLAR ANGLE IN TUBE CENTERED COORDINATE)
207
FORMATT5X.13HANGLE (THETA).5X,19NSECTION WIDTH 18/A0.5)(.20HAVE VEL
IC/CITY (U/UMF),5X.36HAVE VELOCITY GRADIENT (OUDTHETA/UMF))
208 FORMAT(5)(.105H
1
210 FORMAT(28X.17HBUBBLE PARAMETERS)
211 FORMAT(43X,18HCENTER COORDINATES)
212 FORM4T(5X,10HBUBBLE NO.,5)(912HRAOIUS(RB/A)00(.4HX0/A.9X1,4HYO/A)
213 FORMAT(50(.61H
214 FORMAT(9)(tI219X.E12.4.5X.E12.4.5X.E12.4)
216 FORMAT(5)(9E12.41,9X.E12.41;13X.E12.4021X.E12.4)
217 FORMAT(15X.27HBUBBLE BOUNDARY COORDINATES)
218 FORMAT(10X0MX/A.148,3HY/A.12X,5HTHETA)
219 FORMAT(5X1,46H
220 FORMAT(5X0E12.4.5X1E12.4.5XsE12.4)
RETURN
ENO
C
C
SUBROUTINE GRACIPB(X.Y)
GRAOP CALCULATES THE PRESSURE GRADIENT AS FUNCTION OF A,Y IN
PRESENCE OF A TUBE ANO AN ARBITRARY NUMBER OF BUBBLES
10
COMMON/P/A0(101,Y0(1.00,R02(10),RB2(10).0PDX,OPOY,DPHTNBUR
R2=YX+YY
R4=R2R2
OPDX=-.2.XY/R4
OPOY=1..1./R2-.2.YY/R4
DO 10 L=1,NBUB
YY0=YO(L)..0
YY02=YY0YY0
P1=2.*(X0(1).X.Y0(LIYY)
ONOM1=R02(100.R2P1
ONOM12=0NOMIONOM1
DNOM2=1.+R02(L)42-P1
ONOM22=0N0M20N0M2
OFOT=OPDX+RB2(l).(2..YY0Y(X0(L)/0/3NOM12+2..X.YO(L)/DNOM2-.2..(Y0
11L0R2...Y1(R02(L).X..)(0(L))/ONOM220
DFDY=OPOYYRB2(0.(1./ONOM142.*YY02/DN01124(2..Y0(L).Y-1.),DNOM2+2
1.(Y-221.0(L)0(R02(L)Y..Y0(L)//INOM220
OPHrSORT(SORT(OPDX1.0X+OPOYDPOY))
RETURN
END

AVERAGE GAS VELOCITY INSIDE BUBBLE
CONTACTING A TUBE IN A FLUIDIZED aEo
CASE NO.
79
BUBBLE NO.
BUBBLE PARAMETERS
CENTER COORDINATES
PADIUSIRB/A1 X0/8 10/A
.4089E+00 1 .1000E+01 -.1439E101
Flow INSIOE BUBBLE NO.
1 IS COMPUTED
-.6664E+00
-.8052E+00
-.9138E100
-.1014E+01
-.1109E+01
-.1202E0.01
-.1292E101
-.1380E+01
-.1466E+01
-.1549E,01
-.1630E+01
-.1708E401
-.1784E+01
-.1856E+01
-.1925E+01
-.1991E+01
-.2053E+01
-.2111E+01
-.2164E+01
-.2214E+01
-.2259E+01
-.2299E+01
-.2134E401
-.2365E+01
-.2390E+01
-.2411E401
-.2425E+01
-.2435E+01
-.2439E+01
-.2437E4.01
-.2430E+01
-.2418E+01
-.2400E+01
-.2376E+01
-.2347E+01
-.2312E+01
-.2271E+01
-.2226E+01
-.2174E+01
-.2117E401
-.2055E101
-.1987E+01
-.1913E+01
-.1834E101
-.1748E+01
-.1655E+01
-.1553E+01
-.1439E+01
-.1306E401
-.1115E01
.9814E+00
.9190E100
.8539E+00
.7864E+00
.7170E+00
.6459E+00
.5734E+00
.5000E+00
.4260E4.00
.3517E+00
.2775E+00
.2037E+00
.1307E+00
.5892E-01
-.1134E-01
-.7969E-01
-.1457E.00
-.2092E+00
-.2695E+00
-.3264E+00
-.3794E00
-.4281E+00
-.4719E+00
-.5103E+00
-.5427Es.00
-.568CE+00
-.5150E+00
-.5915E+00
-.5826E+00
-.5379E+00
.1043E+01
.1182E+01
.1259E+01
.1314E+01
.1350E+01
.1380E.01
.1391E+01
.1407E+01
.1405E+01
.1402E01
.1390E+01
.1372E+01
.1347E101
.1317E+01
.1282E+01
.1243E+01
.1190E+01
.1150E+01
.1097E+01
.1041E+01
X/A
BUBBLE 9OUNOARY COORDINATES
V/A THETA
.2732E+01
.2761E+01
.2791E+01
.2821E+01
.2850E+01
.2510E+01
.2909E+01
.2939E+01
.2969E+01
.2990E+01
.3029E+01
.3058E+01
.3087E+01
.3117E +01
.3146E+01
.3176E+01
.3206E+01
.3235E+01
.3265E+01
.3295E+01
.3324E+01
.3354E+01
.3383E+01
.3413E+01
.3443E+01
.3472E+01
.3502E+01
.3532E+01
.3561E+01
.3591E+01
.2139E+01
.2169E+01
.2198E+01
.2228E+01
.2258E+01
.2287E+01
.2317E+01
.2347E+01
.2376E+01
.2406E+01
.2435E+01
.2465E+01
.2495E+01
.2524E+01
.2554E+01
.2584E+01
.2613E+01
.2643E+01
.2672E+01
.2702E+01
164

CASE 40.
79
AVE. GAS VELOCITY IN 5UEOLE REGION AS
ANGLE (THETA, SECTION WIDTH (9/A1
A FUNCTION OF POLAR ANGLE IN TUBE CENTERED COORDINATE
AVE VELOCITY (U/UMF) AVE VELOCITY GRADIENT (DUOTHETA/UMF)
.2139E+01
.2169E+01
.21980+01
.2228E+01
.2258E+01
.2287E101
.2317E+01
.2147E+01
.2376E+01
.2406E+01
.2435E+01
.2465E+01
.2495E+01
.2524E+01
.2554E+01
.2584E+01
.2E13E+01
.2643E+01
.2672E+01
.2702E+01
.2732E+01
.2761E+01
.2791E+01
.2421E+01
.2850E+01
.2880E+01
.2909E+01
.1939E+01
.2969E+01
.2998E+01
.0028E+01
.3054E+01
.3987E+01
.3117E+01
.3146E+01
.1176E+01
.3206E+01
.3235E+01
.3265E+01
.3295E+01
.3324E+01
.3354E+01
.3383E+01
.5413c+01
.3443E+01
.3472E+01
.3502E+01
.1532E+01
.3561E+01
.3591E+01
.2379E+00
.4302E+00
.5560E+00
.6593E+00
.7493E+00
.8300E+00
.9034E+00
.9706E+00
.1033E+01
.10900+01
.1142E+01
.1191E+01
.1215E+01
.1276E+01
.1313E+01
.1347E+01
.1377E+01
.1403E+01
.1426E4.01
.1446E+01
.1463E+01
.1476E+01
.1446E+01
.1492E+01
.1496E+01
.149E0+01
.1492E+01
.1446E+01
.1476E+01
.1463E+01
.1446E+01
.1426E+01
.1403E+01
.1377E+01
.1347E+01
.1313E+01
.1276E+01
.1235E+01
.1191E+71
.1142E+01
.1090E+01
.1033E+01
.9706E+00
.9034E+00
.9300E+00
.7433E+00
.6593E+00
.5560E+00
.4302E+00
.2379E+00
-.1624E+01
-.1649E+01
-.1693E+01
-.1742E+01
-.1792E+01
-.1842E+01
-.1892E+01
-.1941E+01
-.1988E+01
-.2035E+01
-.20800+01
-.2124E101
-.2166E+01
-.2208E+01
-.2247E+01
-.2286E+01
-.2322E+01
-.2358E+01
-.2392E+01
-.2424E+01
-.2454E+01
-.2484E+01
-.25110.01
-.2537E+01
-.2562E+01
-.2585E+01
-.2606E+01
-.2626E+01
-.2644E+01
-.2661E+01
-.2677E+01
-.2691E+01
-.2703E+01
-.2715E+01
-.2725E+01
-.2734E+01
-.2742E+01
-.2748E+01
-.27540+01
-.2760E+01
-.2765E+01
-.2770E+01
-.2776E+01
-.2782E+11
-.2791E+01
-.2802E+01
-.2820E+01
-.2848E+01
-.2899E+01
-.3026E+01
-.8363E+00
-.1171E+01
-.1579E+01
-.1671E+01
-.1691E+01
-.1682E+01
- .1656E +01
-.1626E+01
-.1589E+01
-.1549E+01
-.1505E+01
-.1460E+01
-.1414E+01
-.1366E+01
-.1317E+01
-.1267E+01
-.1217E+01
-.1166E+01
-.1114E+01
-.1062E+01
-.1010E+01
-.9576E+00
-.9054E+00
...85300+00
-.00070+00
-.7486E+00
-.6969E+00
-.6458E+00
-.5953E+00
-.5458E+00
-.4973E+00
-.4502E+00
-.4047E+00
-.3613E+00
-.3203E+00
-.2824E400
-.2481E+00
-.2186E+00
-.1948E+00
-.1785E+00
-.1718E+00
-.1781E+00
-.2023E+00
-.2523E+00
-.3413E+00
-.4947E+00
-.7673E+00
-.1314E+01
-.3003E+01
-.4321E+01

C
PROGRAM CONFORM(INPUT.OUTPUT.PUNCH.TAPE 5=INPUT.TAPE 6=OUTPUT)
CALL CNFORM
ENO
SUBROUTINE CNFORM
CONFORM PRCVIDES AN ITERATIVE SOLUTION TO THE INTEGRAL EQUATION
FOR THE FUNCTION WHICH MAPS THt INTERIOR OF AN ARPITRARY BOUNDARY
ONTC A UNIT CIRCLE.
SPECIFIC APPLICATION IS TO A REGION BOUNDED
BY CIRCULAR ARCS AND LINEAR SECTIONS.
C
C
C
COMMON/A/GANA2.GAMAC.ARC.RC.RF
COMMON/0 ,N.ITER.EPS.GAMA(2401.PHI1240).U0(240).X(60)0(601
COMNON/P/PI.PIL.PI2.RI3
PI=3.141592654
P/1=PI,2.
P12=2.RPI
PI3=3.*PI
C
C
C
INPUT DATA
C
C
C
C
C
C
C
N= NUMBER OF ANGULAR STEPS ON CIRCLE PLANE
ITER= NUMBER OF ITERAIONS ALLOWED FOR SOLUTION OF INTEGRAL
EQUATION
SP= DIMENSIONLESS PARTICLE HALF SPACING (IN PARTICLE RADII)
EPS= ERROR PARAMETER USE/ IN ITERATIVE SOLUTION OF INTEGRAL
EQUATION
C
READ(5.100) N.ITER.SP.EPS
100 FORMAT(2I5.2F10.0)
RP=1.
CALCULATE ARC PARAMETERS FOR CHANNEL CROSS SECTION
GAMA1=0.
GAMA2=PI1
IF(SP.NE.RP) GANA2=ATAN(RP/(SP-RP)1
GAMAC=ATAN(FP/SP)
RC=SORT(RP.FP +SP.SP)
ARC=1.-(RP /RC) ..
2
L=1
WRITE(6.5001
WRITE(6,522) wRITE(6.523)
WRITE(6,5241
WRITE(6.501)
WRITE (6.52'1
WRITE(6.501)
TSP=2.*SP
WRITE(6.5271 TSP
WR/TE(6.528) RP
WPITE(6,501)
WRITE(6,525)
NPITE(6,5291
WRITE(6.5301
WRITE(6.531)
WRITE(6,532)
500
WRITE(6.533) LgRC.GAMAC.RP.GAMAL,GAMA2
FORMAT(11411
501 FORMAT(P401
522 FORMAT(21X1146HCONFORMAL MAPPING OF INTERIOR OF A UNIT CIRCLE)
523 FORMAT126X.36HINT0 REGION BOUNDED BY CIRCULAR ARCS)
524 FORM4T(34X.19HANO LINEAR SECTIONS)
525 FORMAT(36X.14HARC PARAMETERS)
526 FORMAT(51(.40MALL DATA ARE DIMENSIONLESS OR IN RADIANS)
527 FORMAT(51(.37HPRIMARY ARC SPACING CENTER TO CENTER=.E12.4)
528 FORM8T(5X.19HPR/MARY ARC RADIUS=.E12.4)
529 FORMAT(5X.7HARC 40..7X.17HPOLAR C00RDINATES.7X.10HARC RAO/US.10X.1
14HANGULAR BOUNDS)
530
531
FORMAT(21X.13HOF ARC CENTER.32X.6HOF ARC)
FORMAT(17)(.6HRADIAL.8)(.7HANGULAR.18X,13H/NITIAL ANGLEOX.11HFINAL
JANGLE)
532 FoRPAT(5X,78N
C
533 FokmAT(IX.I3,4X,E12.4,2X.E12.4.2X.E12.4.2X,E12.4,3X,E12.41
SOLVE INTEGRAL EQUATION FOR MAPPING FUNCTION ON HALF CIRCLE
CALL GAMOPH
PUNCH ARC PARAMETERS AND MAPPING FUNCTION
300
PUNCH 300, PC.GA4AC.RP.GAMAL.GAMA2
PUNCH 300, (PHI(L).GAMA(L).U0(1).X(L).Y(L).L=1,141
FORmAT(5F14.61
WRITE(6.5001
WRITE(6.507)
WRITE16.508)
WRITE(6.509)
WRITE161501)
WRITE(6.514) ITER
WRITE(6.501)
WRITE(6.5101
WRITE(61511)
WRITE16.5121
507 FORWAT(36X,22HFIRST YAPPING FUNCTION)
508 FoRmAT(27X.41NINTEGRAL EQUATION SOLUTION FOR MAPPING OF)
509 FORMAT(26X.40HUPPER HALF UNIT CIRCLE INTO GIVEN REGION)
510 FORMAT(10X.13HANGLE IN HALF.2%./6101APPING FUNCTION.15X,15HBOUNDARY
/ COOR0.1
511 FORMAT(11)(.12HC/RCLE PLANE.8X,6HU(PHI).10)(.7HANGULAR.10X./11X.13X11
512
1HY)
FORmAT(11X,73M
1 1
513 FORMAT(11X.E12.*,5X.E12.40.X.E12.4.2X.E12.4,2X.E12.4)
514 FORNAT110X,21HNUMBER OF ITERATIONS=.I3)
WRITE(6.5131 (PHI( L).U0(L).GAMA(L).X(L).Y(L).L=1.N)
RETURN
END
(a fT

CC
C
SUBROUTINE GAMOPH
GAMCPH SOLVES THE NONLINEAR INTEGRAL EQUATION FOR THE MAPPING
FUNCTION BY SUCCESSIVE SUBSTITUTION
COMMONFA/GAMA2vGANAC,ARC,RC,RP
COSMON/B/NtITER.EPS.GAMA(240),PNI(2401,U0(240),X160)tY(601
COMMON/P/PI.PIL,PI2,PI3
DIMENSION DELTA(6010(INT1(59,2371,GAMAB(60),ERROR(601
NU2=2.111
NU3=3.N...2
NU4=4.44..3
PHI (11=0.
PHI(NU2)=PI
PHI(NU3)=3..PI1
PHI(NU4)=PI2
C
C
CALCULATION OF ANGULAR STEP SIZE
CC
10
20
DELTAI=PII/FLOAT(11.4)
00 10 LK=1,11
DELTAILK)=DELTA1
GAMA11)=PHI(1)
NU=N-1
DO 20 L=2,11
Li=L/
PHIlL)=PHI(L1)+DELTA(11)
GAMA(L)=PHI(L)
NL1=2.W.-
NL2=4.N...2
NL3=2.Nel-2
PHI(A1.11=FIPHI(L)
PHI41412)=P/2PHI(L)
PHI(NL3)=FI+RHI(L)
CONTINUE
GAMAINI=PH/(N)
CALCULATE ANO STORE KERNAL rUNCTION
C
C
C
30
31
00 31 L1=2,NU
ERROR(L11=1.
L1M=L1,1
10 31 L2=2,NU4
IF(L1.EQ.12) GO TO 30
L2M=L21
PHII2M=(PHI(L1)PHI(L2M11/2.
PHI12.(PHI(L1)PHI(L2))/2.
K/NT1(11,1.2)=ALOG(43S(SIN(PHI121/S/N(PHI1201)))
GO TO 31
XINT1(1.1.L1)=ALOG(DELTA(L1)/DELTA(L/01
CONTINUE
ITFRATIVE SOLUTION FOR UO(PH/)
ERRCR(1)=0.
ERROR(N)=0.
1=0
C
22
27
1=141
00 27 L=1,N
IF(GAMA(L).GT.PI1) GAMA(L)=PI1
IF(GAMA(L).LT.0.1 GAMA(L)=0.
GAMAB(L)=GAMA(L)
GAMAP=GAMA(L)
UOIL)=ALOG(GEEIGANAP,1))
NL1 =2.NL
NL2=4,14.2
NL3=2.11+1...2
O0(NLII=U0(1)
00(14L2)=UOIL/
UO(NL31=UO(L)
NRITE(6,500)
NRITE(6.515)
NRITE(6,516)
WRITE (6,501)
IN=/1
NRITE(6.517) IM
WR/TE(6,5011
NR/TE(6,518)
WRITE(6,519)
WRITE /6,520)
MRITE(6.521) (PHI(IL1.GAMAIILI,ERROR(IL),IL=1,11)
500 FORMAT(1H1)
501 FORMAT41H0)
515
FORMAT(14X,37HOETAILS OF INTEGRAL EQUATION SOLUTION)
516
FORMATI23X,26HFOR FIRST MAPPING FUNCTION)
517 FORMAT(21X.13HITERATION NO.,I3)
5111
FORMAT(10X0HANGLE IN,SX.OHANGLE IN)
519 FORMAT(13X.17HHALF CIRCLE PLANEg2X,124GIVEN REGIONs7X.5HERROR1
520 FORMAT(12X,4014
521 FORMAT(16X,E12.4.4X,E12.4.4X.E12.4)
DO 21 K=2,NU
SUM =O.
1
NPH/=K
NPH/M=1(..1
NPNIP=K+1
NUMERICAL INTEGRATION
C
C
37
DO 37 KI=2,14U4
K2=K1-1
IF(1(1,E0.14FHI.OR.KI.EQ.NPHIPI GO TO 37
SUN=SUM.(UO(K1)+00(K2)1mXINT1(K,K1)
CONTINUE
ESTIMATE DER/VITIVE FOR USE IN CALCULATING PRINCIPLE VALUE OF
INTEGRAL
GAMAP =GAMAY(K)
OUVAM=GEE(GAMAP,2)
OG4MDP=DELTA(KI.GAIAB(NPHIM)/(DELTAINPHIM)*(DELTA
1(K)41ELTAINFHIH)11+(1./DELTA(NPHIM)1./DELTA(K)).
2GAMA9(K)+DELTAINPHIM).GANAB(NPHIP)/(OELTA(K).(DELTA
3(K)+DELTA(NPHIM))/
OUDP=DUOGAM.OGAMOP

21
ENT=-(SU44.2..(XINTI(K0(1=UOIK)+10ELTA(K)+DELTA(NPHIM)).DUOP))/PI2
EN1=ENT/PI2
N1=IFIX(EN1)
EN2=FLOAT(N1)
ENT=(EN1-EN21.PI2
GANA(K)=PHI(K) i.INT
IF(GAMA(K).LT.0.) G4IA(K)=GAMA(K)+PI2
IF(GAMAIK).GT.PI2) GAMAIKI=GAMA(K) -PI2
ERROR(K)=ABSUGAMA(KI-GAMA3(K))/GAMAIK))
CONTINUE
C
C
C
C
C
C
TEST FOR CONVERGENCE
INOE=1
DO 24 L=2.NU
24 IF(ERROR(11.GT.EPS) INOE=2
IF(INDE.E0.1) GO TO 23
522
IF1I.LT.ITER) GO TO 22
WRITE(6.522) ITER
FORMAT(5)(.34HCONVERGENCE WAS NOT OBTAINED AFTER.I3.11H ITERATIONS)
FINAL CALCULATION OF MAPPING FUNCTION AND BOUNDARY COORDINATES
23 ITER=I
00 25 IK=1.N
GAMAP=GAMAIIK)
UO(IKI=ALOG(GEE(GAMAP.1))
RI=EXPIUO(IKI)
25
X(IK)=R1=CCS(GAMAP)
YlIKI=R1SIN(GANAPI
RETURN
END
C
C
C
FUNCTION GEE(GANAP.INDI
COMMON/A/GAMA2.GAMAC.ARCORC.RP
GEE CALCULATES THE ARC RADIUS AND DERIVITIVE. WHEN INO.NE.1 THE
OERIVITIVE IS CALCULATED.
10
11
IF(GAMAP.GT.GAMA2) GO TO 10
CSG=COS(GAMAP-GAMACI
ARG=CSG.CSG-ARC
IF(ARG.LT.0.) ARG=0.
SOC=SORT(ARGI
IFM2C.EQ.O.4NO.IND.AE.1) CALL EXIT
SEE=RC=4CSG-SOC)
IF(INO.E0.11 GO TO 11
H=SIN(GAMAP-GAMACI.(1.-CSG/SOCI
GEE=-RC.H/GEE
GO TO 11
SNG=SIN(GAMAP)
GEE=RP/SNG
IF(INO.E0.1) GO TO 11
GEE =-COS(GAHAP)/SNG
RETURN
ENO

169
CONFORMAL NAPPING OF INTERIOR OF A UNIT CIRCLE
INTC REGION 90UNOE0 SY CIRCULAR ARCS
AN1 LINEAR SECTIONS
ALL DATA ARE DIMENSIONLESS OR IN RADIANS
PRIMARY ARC SPACING CENTER TO CENTER=
PRIMARY ARC RADIUS.
.1400E+01
.3000E+01
ARC NO.
POLAR COORCINATES
OF ARC CENTER
RADIAL
ARC PARAMETERS
ARC RADIUS
ANGULAR
.1803E+01 .5890E+00 .1001E+01
ANGULAR ROUNDS
OF ARC
INITIAL ANGLE FINAL ANGLE
O.
.1107E+01
DETAILS OF INTEGRAL EQUATION SOLUTION
FOR FIRST 'OFFING FUNCTION
ITERAT/CN NO.
C
ANGLE IN ANGLE IN
HALF CIRCLE PLANE GIVEN REGION
O.
.2708E-01
.54171-11
.0125E-01
.1083E+00
.1354E+10
.16251.50
.1896E+10
.2167E+00
.2437E+10
.2708E+00
.8125E+00
.8396E+00
.8666E+00
.9937E+00
.5208E+00
.9479E+00
.5750E+00
.11021+01
.1029E+01
.1056E+01
.1043E+01
.1110E+01
.1117E+01
.11651+01
.1192E+01
.1219E+01
.1246E+81
.1273E+01
.1310E+0i
.1327E+01
.2979E+00
.7250E100
.3521E+00
.3792E+00
.4062E+00
.4333E+00
.4604E+00
.4875E+10
.5146E+00
.5417E+00
.5687E+00
.5958E+00
.0229E+00
.6500E+00
.6771E+00
.7042E+00
.7312E+00
.7553E+00
.7854E+00
.1354E+01
.1341E+01
.1408E+91
.1435E+01
.1462E+01
.1490E+01
.1517E+01
.1544E+01
.1571E+01
O.
.270181-01
.5417E-01
.8125E-01
.1183E+00
.1354E+00
.1625E+00
.1096E0.00
.2167E900
.2437E+00
.2700E+00
.2979E+00
.3250E+00
.3521E+00
.3792E+00
.4062E+10
.4333E+00
.4604E+00
.4975E+00
.5146E+01
.5417E+00
.56187E+00
.5958E+00
.6229E+10
.6500E+00
.6771E+00
.7042E+00
.7312E+00
.7583E8.00
.7454E+00
.8125E+00
.8396E+00
.8666E+00
89371+00
.9208E+00
.9479E+00
.9750E+00
.1002E+01
.10291+,11
.1056E+01
.1083E+01
.1110E+01
.1137E+01
.1165E+01
.1192E+01
.1219E+01
.1246E+01
.1273E+01
.1100E+01
.1327E+01
.1354E+01
.1391E+01
.1409E8.01
.1435E+01
.1462E+01
.1490E+01
.1517E+01
.1544E+01
.1571E+01
ERROR
O.
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1400E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+11
.1000E+01
.1000E8.01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.10006+01
.1000E+01
.1000E+01
.1000E+01
.1000E8.01
.1000E+01
.1000E+81
.1000E+01
.100014.01
.1000E+01
.1000E+01
.1000E+01
.1800E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1800E+01
.1000E+01
.1000E+01
.1000E+01
0.
DETAILS OF INTEGRAL EQUATION SOLUTION
FOR FIRST NAFFING FUNCTION
ITERATION NO. 31
ANGLE IN ANGLE IN
HALF CIRCLE PLANE GIVEN REGICN
1.
.2708E-01
.5417E-01
.6125E-01
.1083E+00
.1354E+00
.1625E+00
.1896E+00
.2167E+00
.2437E+00
.2709E+00
.2379E+10
.3250E+00
.3521E+00
.3792E+00
.4062E+00
.4333E+00
.4604E+00
.4875E+00
.5146E+00
.5417E+00
.5687E+00
.5958E+00
.6229E+00
.6500E+00
.6771E+00
.7042E+00
.7312E+00
.7583E+00
.7854E100
.9125E+00
.9196E+00
.8666E+00
.8937E+00
.9209E+00
.9479E+00
.9750E+00
.1002E+01
.132qE+51
.1056E+01
.1083E+01
.1110E+01
.1137E+01
.1165E+81
.11921+01
.1219E801
.1246E+01
.1273E+01
.1300E801
.1127E+01
.1354E+01
.1381E+01
.1409E+01
.1435E8.01
.1462E+01
.1490E+01
.1517E+01
.1544E+01
.1571E+01
O.
.1628E+00
.1036E+00
.1982E+00
.2730E+00
.2462E+00
.2646E000
.2904E+00
.3115E+00
.1322E+00
.3524E+00
.3723E+00
.7919E+00
.4113E000
.4304E+00
.4493E+00
.4681E+10
.4868E+10
.5053E+00
.52371+10
.5421E+00
.5604E+08
.5716E+00
.5968E+00
.6150E+00
.6332E+00
.6514E+00
.6696E+00
.6878E+00
.7061E+00
.7245E+00
.7429E+00
.7614E+00
.7801E+00
.7999E+00
.8179E+00
.0371E+00
.8567E1.00
.8767E+00
.8973E+00
.5199E +00
.5417E+10
.5657E+80
.5895E+00
.0010E+01
.10421+01
.1043E+01
.1181E+01
.1223E+01
.1275E+01
.1315t+01
.1352E+01
.1316E+01
.1419E+01
.1450E+01
.1481E+01
.1511E+01
.1541E+01
.1571E+01
ERROR
.5294E-03
.5508E-03
.5732E-03
.5966E-03
.6214E-03
.6478E-03
.6761E-03
.7065E-03
.7395E-03
.7754E-03
.8144E-03
.8567E-03
.50161-03
.9471E-03
.58581 -03
.5954E-03
.5090E-03
.5462E-03
.5211E-03
.3020E-02
.7417E-02
.1266E-11
0.
.3137E-03
.5994E-03
.6349E-03
.3607E-03
.1042E-03
.5699E-04
.14711.03
.7015E-03
.2401E-03
.2715E-03
.29896-13
.3238E-73
.3468E-03
.3696E-03
.3894E-03
.4096E-03
.4294E-03
.4491E-03
.4687E-03
.4885E-03
.5047E-03
.1398E-01
.6319E-02
.2444E-02
.2360E-02
.2801E-02
.1500E-02
.8621E-03
.5261E-03
.33421-03
.2168E-03
.1410E-03
.89576-04
.5271E-04
.2426E-04
0.

FIRST MAPPING FUNCTION
INTEGRAL EOUATION SOLUTION FOR MAPPING OF
UPPER HALF UNIT CIRCLE INTO GIVEN REGION
NUm9ER OF ITERATIONS= 32
ANGLE IN HALF
CIRCLE PLANE
RAPPING FUNCTION
UIPHI)
-.2191E+00
-.2179E+00
-.2161E+00
-.2136E+00
-.7118E+00
-.2072E+00
-.2029E+00
-.1940E+00
-.1923E+00
-.1060E+00
-.1748E+00
-.1/09E+00
-.1620E+00
-.1522E+00
-.1413E+00
-.1292E+00
-.1157E+00
-.1000E+00
-.0102E-01
-.5664E-01
-.2740E-01
.2127E-01
.2064E-01
.7643E-01
.6072E-01
.4386E-01
.3251E-01
.2361E-01
.1692E-01
.1146E-01
.7197E-02
.3992E-02
.1758E-02
.4369E-03
O.
.4055E+00
-.2665E-01
-.4067E-61
-.6275E-01
-0470E-01
-.1133E+00
-.1195E+00
-.1337E+00
-.1463E+00
-.1574E+00
-.1673E+00
-.1761E+00
-.1839E+00
-.1907E+00
-.1960E+00
-.2020E+00
-.2065E+00
-.2103E+00
-.2134E+00
-.2159E+00
-.2177E+00
-.2190E+00
-.2196E+00
-.2196E+00
.1002E+01
.1029E+01
.1056E+01
.1003E+01
.1110E+01
.1137E+01
.1165E+01
.1192E+01
.1219E+01
.1246E+01
.1273E+01
.1300E+01
.1327E+01
.1354E+01
.1361E+01
.1408E+01
.1435E+01
.1462E+01
.1490E+01
.1517E+01
.1544E+01
.0571E+01
O.
.2718E-01
.5417E-01
.8125E-01
.1063E+00
.1354E+00
.1625E+00
.1096E+00
.2167E+80
.2437E+00
.2746E+00
.2979E+00
.3250E+00
.3521E+10
.3792E+00
.4067E+00
.4333E+00
.4604E+00
.4475E+00
.5146E+00
.5417E+00
.5647E+00
.5954E+10
.6229E+00
.6500E+00
.6771E+10
.7042E+00
.7302E+00
.7503E+00
.7054E+00
.0125E+00
.0396E+00
.0666E+00
.8937E+00
.9206E+00
.9479E+00
.9750E+00
ANGULAR
O.
.1628E+00
.11136E+00
.1911E+00
.2226E+00
.2460E+00
.2614E+00
.2901E+10
.3113E+00
.3319E+00
.6670E+00
.7052E+00
.7235E+00
.7418E+00
.7602E+00
.77+7E+00
.7973E+00
.81E1E+00
.8350E+00
.4541E+00
.8734E+00
.6930E+00
.9131E+00
.9343E+00
.9576E+00
.9643E+00
.1013E+01
.1052E+01
.1052E+01
.1185E+01
.1226E+01
.1277E+01
.1317E+01
.1353E+01
.3522E+00
.3720E+00
.3916E+00
.4109E+00
.4301E+00
.4490E+00
.4677E+00
.4664E+00
.5049E+00
.5233E+00
.5416E+00
.5599E+00
.5781E+00
.59630+00
.6144E+00
.6326E+00
.6507E+00
.6610E+00
.1317E+01
.1420E+01
.1451E+01
.1411E+01
.1512E+01
.1541E+01
.1571E+01
BOUNOART COORO.
.1500E+01
.9E06E+00
.9365E+00
.9208E+00
.8960E+00
.8747E+00
0556E+00
.0383E+00
.1224E+00
.00771400
.7940E+00
.7812E+00
.7690E+00
.7575E+00
.7466E+00
.7361E+00
.7261E+00
.7164E+00
.7070E+00
.6980E+00
.6692E+00
.6607E+00
.67240+00
.6643E+00
.6564E+00
.6466E+00
.6410E+00
.E3360+00
.6262E+00
.6190E+00
.6119E+00
.6048E+00
.5979E+00
.5910E+00
.5042E+00
.5775E+00
.5706E400
.5E42E+00
.5576E+00
.5511E+00
.5445E+00
.5378E+00
.5307E+00
.52290+00
.51540+00
.5064E+00
.5065E+00
.4064E+00
.3593E+00
.3028E+00
.2592E+00
.22080+00
.10550+00
.1523E+00
.1204E+00
.19540-01
.5934E-01
.2957E-01
-.20510-09
.5269E+00
.5404E+00
.5543E+00
.5665E+00
.5832E+00
.5983E+00
.6141E+00
.6304E+00
.6475E+00
.6655E+00
.6445E+00
.7050E+00
.7276E+00
.7542E+09
.7970E+00
.1253E+00
.8071E+00
.8866E+00
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
O.
.1578E+00
.1739E+00
.1846E+00
.2030E+00
.7196E+00
.2353E+00
.2503E+00
.2646E+00
.2784E+00
.2918E+00
.3046E+00
.3176E+00
.3301E+00
.3425E+00
.3547E+00
.36E7E+00
.3704E+00
.3907E+00
.4027E+00
.4146E+00
.4266E+00
.4387E+00
.4508E+00
.4631E+00
.4755E+00
.4880E+00
.5007E+00
.5137E+00
1 70

PROGRAM TRFORM(INPUT.OUTPUT.PuNCH,TAPE5=INPUT.TAPE6=OUTPUT)
CALL TRFRM
ENO
SUBROUTINE TRFRM
C
C
C
C
TRFCRM USES THE YAPPING FUNCTION 01TAINE0 BY CONFORM TO EVALUATE
SCALE FACTORS AND TO DEVELOPE THE STOKES MATCHING FUNCTION USED BY
MOMEKGL.
COMmON/P/PI,PI1..I2.PI4.PI5,PI6
COMMON/S/RHO,PHI.THETAS.RHOSP.SQT2
COMHON/Z/RHOS.NU22.NU21.UB(2401.TN(240),TN2(240).XRHOS,X(2401
DIMENSION Y(2401.GAMA(240).RC(101.GAMAC(101,A1/01,GANA11/0).GANA2(
110).PHI1(240),U0(240)
C
C
C
C
C
C
C
C
C
C
INPUT DATA
N= NUMBER OF POINTS IN MAPPING FUNCTION TABLE
ITER= NUMBER OF ITERATIONS WHICH WERE USED BY CONFORM TO
SOLVE THE NONLINEAR INTEGRAL EQUATION
LMAX= NUMBER OF ANGULAR POSITIONS
SP= HALF DISTANCE BETWEEN PARTICLE CENTERS
END= PARAMETER USED TO DETERMINE EXPONENTIAL ANGULAR STEP
RCILI.GANAC(LIgAIL).GAMAI(L).SAMA2(L). TABLE OF ARC PARAMETERS
DEFINING CHANNEL CROSS SECTION ANO PUNCHED BY CONFORM
PH/1(L),GAMA(L).U3(L).X(L).Y(L). TABLE OF MAPPING FUNCTION ANO
EICUNDARY COORDINATES PUNCHED BY CONFORM
REA015.100) NgITER,LMAX.L1,SP.END
READ(5.300) IRC(L).GAMAC(1).A/LI.GANA1(1).GANA2(L).1=1.L1)
REA0(5.3001 (PHI)(L).GAMA(L).U0(L).X(1.),Y(L).L=1,N)
C
100
FORKAT(4I5,2F10.0)
300 FORmAT(5F14.61
C
RP=1.
WRITE(6.500)
WRITE16.522)
WRITE(6.523)
WRITE(6.5241
WRITE(6,501)
WRITE(6,526)
WRITE(6.501)
TSP=2..SP
WRITE16.527) TSP
WRITE(6.528) RP
WRITE(6.501)
WRITE(6.525)
WRITE(6.529)
HRITE(6.530)
WRITE(6.531)
WRITE(60532)
522
RR/1E(6.533) (LeRC(L).GANAC(L).A(11.GAMAi(L1gGAMA2(L).L=1,L1)
FORYAT(21X.46HCONFORMAL MAPPING OF INTERIOR OF A UNIT CIRCLE)
523
FORHAT(26X.36HINT0 REGION BOUNDED BY CIRCULAR ARCS)
524
FoRmAT(34X.19HAN0 LINEAR StCTIONS)
525 F0P4AT(36.,14HARC PARAMETERS)
526
FORKAT(5X.MOHALL DATA ARE DIMENSIONLESS OR IN RADIANS)
527 FORMATf5X.37HPRI4ARY ARC SPACING CENTER TO CENTER=.E12.41
528 FORmAT(5X.19HPR/MARY ARC RAD/US=gE12.4)
529 FORMAT(5X.7HARC NO..7X.17HPOLAR COORDINATES.7X.10HARC RAOIUS.10X.1
530
14HANGULAR BOUNDS)
FORMAT(21X.13HOF ARC CENTER.32X,6110F ARC)
531 FORMAT117X.6HRA0IALI8X.7HANGULAR.18X.13HINITIAL ANGLE.3X.11HFINAL
JANGLE)
532 FORMAT15X.78H
1
533 FORMAT(71,13.4X,E12...2X.E12.4,2X.E12.4.2X.E12.4.3X.E12.4)
WRITE(6.500)
MRITE(6,507)
MR/TE(6.508)
WRITE(6.509)
WRITE(6.5011
NR/TE16.514) ITER
WRITE(6.501)
WRITE(6,5101
WRITE(6.511)
WRITE(6.512)
507 FORNAT(36X,22HF/RST MAPPING FUNCTION)
508
FORMAT(27X.41HINTEGRAL EQUATION SOLUTION FOR MAPPING OF)
509
FORMAT(26X,40HUPPER HALF UNIT CIRCLE INTO GIVEN REGION)
510 FC71:70CJAX.13HANGLE IN HALF.2X.16HMAPPING FUNCTION.15X.15HBOUNDARY
511 FDRONIT(11X.12HCIRCLE PLANE.8X.6HUIPHI).10X.7HANGULAR.10X.IHX.13X,1
OH Y)
512 FORMAT(11X.73H
513 FORMAT(11X.E12.4.5X.E12.4.4X.E12.4.2X.E12.4.2X.E12.41
514 FORMAT(10X.21HNUMBER OF ITERATIONS=.I3)
WRITE(6, 513) (P4I1(1100(L).GAMA(L).X(L).Y(L).1=1.N)
SQT2=SCIRT(2.)
P1=3.141592654
PII=P//2.
PI2=3..PI1
RI4=PI.2.
PI5=911/2.
P/6=P11..999999
EXF1=EXP(...1./EMD)
EXP3=EXP1FLOAT(LMAX)/EMI)
EXP2=EXP1EXP3
EXP4=PI1EXPI/EXP2
GAMAIN)=PI1
RP=1.
00 201 L=1.N
NL1=2.N-L
UOINL1)=U0(1)
GAMAINL1)=FI-GAMA(L)
201 PHI1(NL1)=PI-PHI1IL)
NU2=2.4-1
NU21=NU2-1
NU22=NU2-2
DO 70 L=1gNU21
UB(L)=0.10(L)*U0(0.1))/2.
TN(1.1=TAN(FHI1(L)/2.)

C
C
70 TN2(L1=TN(L)*TN(L)
CALCULATE X AND Y COORDINATES IN PHYSICAL PLANE MAPPED BY CIRCLE
PLANE COORDINATES RHO AND PHI
C
WRITE(6,5001
WRITE(6,502)
WRITE(6,501)
WRITE(6.5031
WRITE(6.5041
WRITE(615051
RHO=1.
00 2 L=1,LMAX
PHI=PI.11.4.5.(EXP1 EXP(..FLOAT(L)/END))/EXP2)
C
G.
C
INTERMEDIATE MAPPING INTO SEMI - CIRCLE
50
IF(L.EO.LMAX) GO TO 50
CALL SEMCIRCIRHOS)
CALL ZEEO(L)
GO TO 73
RHOS=1.
X(L)=SP
C
C
CALCULATE MAGNIFICATION PARAMETERS
73
H=XRFOSRHOSP
OPC=EXP4.(EXPtFLOATILMAXL)/EHO)1.1
RHOS1=1..RHOS
HOLGP=1.E+16
ARGP=1.OPC
IFIARGP.LT.0.) GO TO 2
HOLGP=HALOG(ARGP1
IF(L.EO.LMAX) HOLGP60,
PUNCH 350. HOLGP,X(L)
350 FORMAT(2E12.61
2
WRITE(6,500) OPC.RHOSi.X(LIgH,HOLGP
500 FORNAT(1011)
501 FORMAT(1H0)
542 FORMATT64X,13HCONFORMAL MAP)
503
FORMAT(10)(019HCIRCLE PLANE COORO..5X,24HHALF CIRCLE PLANE COORO.,7
504
1)(1,18)(GIVEN PLANE COORD.,22X013HMAGNIFICATION)
FORMAT(5X.29HANGULAR COORDINATE (FHIC..PHI1,5X115HRADIAL (1-RHOS),2
11X.1.14)(127X,1HH,7X.21.H11.LOG(1....(PHIC-PHI)))
505 FORMAT15X,128H
2
506 FORMATI/31.E12.4,I4X.E12.4115X,E12.4.15X,E12.4.15X.E12.41
RETURN
END
C
C
C
C
SUBROUTINE ZEEO(L)
?EEO CALCULATES THE VALUES OF X ANO ITS DERIVITIVE WITH RESPECT
TO RHOS ALONG Y=0.
NECESSARY INTEGRATION IS PERFORMED BY PREINTEGRATING THE KERNAL
71
COMMON/I/RHOS,NU2204U21.UB(240),TNI2401,7N2(240),XRHOS,X(240)
COMMON/P/PI.PII,PI2,PI4,PI5IPI6
KRS=1.
RHOM=1...RHOS
RHOP=1.4.RHOS
RHOF=RHOP/RHOM
RHOM2=RHOMRHOM
RHOP2=RHOPRHOP
ENTP=O.
ENT's°.
DO it J=1,14022
ENTF=ENTP.U0(J).(IN( 1+1)/IRHOM2+RHOP2.TN2(J+1))-.TN(J)/(RHON24.RHOP2
1.TN2(J)))
ENT=ENT+UB(JI.IATAN(RHOFTNIJ.11)ATAN(RHOFTN(J)))
ENT=(ENTUB(NU21)(PI/..ATAN(RHOF.TN(NU2111)//PI1
ENTP=(ENTPUB( NU21).TN(NU21)/(RHON2.RHOP2.TN2(.1)))/PI5
X(L)=RHOSEXP(ENT)
IFIRHOS.GT.0.) XRS=X(L)/RHOS
XRHOS=XRS+X(L).ENTP
RETURN
ENO

C
C
C
C
C
C
SUBROUTINE SEMCIRC(RHOS)
SEMCIRC PERFORMS THE INTERMEDIATE MAPPING OF A FULL CIRCLE INTO A
SEMI-CIRCLE AND CALCULATES THE DERIVITIVE OF RHOS WITH RESPECT TO
PHI ALONG RHO=1.
cOMMON/S/RNO,PHI.THETAS.RHOSF.SOT2
PERFCRM MAPPING
RH02=RHO.RHO
Tp=2..PHI
RHOSN=RM0.5INIPHI)
RHOCS=RHOCOS(PHI)
SNTP= SIN(TP)
RH0252=RH02.SNTP
RN02C2=RH02.COS(TP1
X=1.4.RHOCS
PMI1=TINV(X.RHOSN)
RI=SORTIX.X*RHOSN.RMOSN)
X=1.4.RHO2C2
PHI2=TINVIX.RHO2S21 /2.
R2=SORT(SORTWX+RH0252.RHO2S2))
SO2R2=SOC2oR2
IF(PHT.GE.P/6.AND.PHI.LT.PI.AN0.RHJ.NE.0..OR.PHI.GE.FI2.ANO.PHI.LT
1.PI4.AND.RHO.NE.0.) SO2R2=-SO2R2
CSPT=COS(PNI2)
SNPT=SIN(PHI2) x=1.+SQ2R2.CSFT-RHOCS
V=S02R2.5NPT-RHOSN
PHI3=TINV(A,V)
R3=SORT(X.N.YmY)
THETAS=PI14FHI1-PHI3
IF(THETAS.LT.0.) THETAS=THETAS+P/N
IF(TNETAS.GT.PIm) THETAS=THETAS-PIm
RHOS=R1/R3
CALCULATE DERIVITIVE
DP2=.5
OR2=-SNTP/(R2mm3)
OR1=-RHOSN/P1
0123.((1.-RPOCSI.SQT2.(CSPT.DR2-R2.SNPT.OP21.2.*R2'0R2+RHOSN.(1.-SQ
1r2.(SNPT.DR24.R2' CSPT.OP2114.R2.5QT2.(RHOSN.CSPT-RHOCS.SNFT))/R3
RHOSP=l0R1-R1.0R3M)/R3
RETURN
END
C
C
10
11
FUNCTION TINV(X,Y)
COMMON/P/PI.PIl.PI2.PIm.PI5.PI6
TINY COORDINATES EVALUATION OF ARCTAN SO THAT PRINDIPLE VALUES
ARE OBTAINED
RESULTS ARE RETURNED TO SEMCIRC
IF(X.E0.0..ANO.Y.E0.4.) GO TO 10
TINV=ATAN2(Y,X)
TINV=TINV+PI
IF(X.GE.0..AND.Y.LT.0.1 TINV=TINV.I.PI4
GO TO 11
TINV=0.
CONTINUE
RETURN
END

CONFORMAL NAPPING OF INTERIOR OF A UNIT CIRCLE
INTO REGION SOUN0E0 BY CIRCULAR ARCS
ANO LINEAR SECTIONS
ALL DATA ARE OIMENSIONLESS OR IN RADIANS
PRIMARY ARC SPACING CENTER TO CENTER.
PRIMARY ARC RADIUS= .1000E+01
ARC NO.
.2540E+01
ARC PARAMETERS
POLAR COORCINATES
OF ARC CENTER
RADIAL ANGULAR
ARC RADIUS
.1616E+01 .6670E+00 .1100E+01
ANGULAR BOUNDS
OF ARC
INITIAL ANGLE FINAL ANGLE
O.
.1307E+01
FIRST MAPPING FUNCTION
INTEGRAL EQUATION SOLUTION FOR MAPPING OF
UPPER HALF UNIT CIRCLE INTO GIVEN REGION
NUMBER OF /TEPAT/ONS= 50
ANGLE IN MALT MAPNING FUNCTION
CIRCLE PLANE U(PHI)
.2390E+00
-.2561E+00
-.27/3E+00
-.2921E+00
-.3170E+00
-.3376E+00
-.3561E+00
-.3723E+00
-.3667E+00
-.3996E+00
-.4112E+00
-.4216E+00
-.4309E+00
-.4393E+00
-.4468E+00
-.4535E+00
-.4594E+00
-.4646E+00
-.4692E+00
-.4731E+00
-.4763E+00
-.4790E+00
-.4810E+00
-.4825E+00
-.4834E+00
-.4838E+00
-.4836E+00
-.4628E+00
-.4815E+00
-.4796E+00
-.4772E+00
-.4742E+00
-.4706E+00
-.4664E+00
-.4616E+00
-.4561E+00
-.4500E+00
-.4433E+00
-.4358E+00
-.4275E+00
-.4105E+00
-.4096E+00
-.3977E+00
-.3858E+00
-.3727E+00
-.3502E+00
-.3428E+00
-.3290E+00
-.3247E+00
-.3423E+00
-.3835E+00
-.4570E+00
-.2339E+00
-.2696E-01
.3431E-01
.1741E-01
.6629E-02
.4937E-02
0(
.8666E+00
.6937E+00
.9208E+00
.9479E+00
.9750E+00
.1002E+01
.1029E+01
.1056E+01
.1063E+01
.1110E+01
.1137E+01
.1165E+01
.1192E+01
.1219E+01
.1246E+01
.1273E+01
.1300E+01
.1327E+01
.1354E+01
.1381E+01
.1406E+01
.1435E+01
.1462E+01
.1490E+01
.1517E+01
.1544E+01
.1571E+01
O.
.2708E-01
.5417E-01
.6125E-01
.1063E+00
.1354E+00
.1625E+00
.1896E+00
.2167E+00
.2437E+00
.2708E+00
.2979E+00
.3250E+00
.3521E+00
.3792E+00
.4062E+00
.4333E+00
.4604E+00
.4675E+00
.5146E+00
.5417E+00
.5667E+00
.5956E+00
.6229E+00
.6500E+00
.6771E+00
.7042E+00
.7712E+00
.7563E+00
.7054E+00
.51250+00
.6796E+00
ANGULAR
BOUNDARY COORO.
X
.1270E+01
.7615E+00
.7427E+00
.7297E+00
.7075E+00
.6686E+00
.6716E+00
.6562E+00
.6421E+00
.6290E+00
.6164E+00
.6053E+00
.5945E+00
.5642E+00
.5743E+00
.5649E+00
.5559E+00
.5471E+00
.5387E+00
.5305E+00
.5225E+00
.5147E+00
.5071E+00
.49971+00
.4924E+00
.4852E+00
.4782E+00
.4713E+00
.4644E+00
.4577E+00
.4510E+00
.4443E+00
.4377E+00
.4312E+00
.4247E+00
.4182E+00
.4117E+00
.4052E0.00
.3987E+00
.3922E+00
.3856E+00
.3790E+00
.3724E+00
.3656E+00
.3588E+00
.3516E+00
.3449E+00
.3391E+00
.3374E+00
.34461 +00
.3644E+00
.4192E+00
.3073E+00
.2721E+00
.2665E+00
.1662E+00
-O.
.1155E+00
.9961E-01
O.
.5712E1.00
.5900E+00
.6097E+00
.6274E+00
.6460E+00
.6646E+00
.6632E+10
.70/7E+00
.7202E+00
.7387E+00
.7572E+00
.7758E+00
.7943E+00
.8129E+00
.8315E+00
.8502E+00
.8699E+00
.8877E+00
.9066E+00
.9255E+00
.94470+00
.9639E+00
.9633E+00
.1003E+01
.1023E+01
.1043E+01
.1063E+01
.1080E+01
.1085E+01
.1064E+01
.1007E+01
.8473E+00
.1805E+00
.1997E+00
.2140E+00
.2401E+00
.2644E+00
.2976E+00
.3105E+00
.3325E+00
.3540E+00
.3751E+00
.3957E+00
.4160E+00
.4361E+00
.4559E+00
.4755E+00
.4949E+00
.5142E+00
.5333E+00
.5523E+00
.1172E+01
.1267E+01
.1310E+01
.1305E+01
.1456E+01
.1472E+01
.1571E+01
O.
.1369E+00
.1503E+00
.1585E+00
.1732E+00
.1664E+00
.1986E+00
.2106E+00
.2217E+00
.2325E+00
.2426E+00
.2529E+00
.2627E+00
.4762E+00
.4669E+00
.4979E+00
.5093E+00
.5210E+00
.5332E+00
.5459E+00
.5592E+00
.5732E+00
.5861E+00
.6039E+00
.6203E+00
.6346E+00
.6392E+00
.6209E+00
.5759E+00
.4746E+00
.7293E+00
.9347E+00
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.1000E+01
.2722E+00
.2616E+00
.2909E+00
.3000E+00
.3090E+00
.3160E+00
.3269E+00
.3350E+00
.3446E+00
.3535E+00
.3623E+00
.3712E+00
.3802E+00
.3692E+00
.3983E+00
.4075E+00
.4166E+00
.4263E+00
.4356E+00
.4456E+00
.4556E+00
.4657E+00
174

CIRCLE PLANE CJORD.
ANGULAR COOPOINATE 1PRIC-PHI,
HALF CIRCLE PLANE COORC.
RADIAL 11 -RHOS)
.1571E+01
.1126E+01
.8065E+00
.5779E+00
.4141E+00
.2967E+00
.2126E+00
.1523E+00
.1091E+00
.7821E-01
.5604E-01
.4015E-01
.2677E-01
.2061E-01
.1477E-01
.1050E-01
.7514E-02
.5434E-02
.3893E-02
.2790E-02
.1999E-07
.1432E-02
.1026E-02
.7353E-03
.5268E-03
.3774E-03
.2704E-03
.1937E-03
.1388E-03
.9940E-04
.7119E-04
.5097E-04
.3649E-04
.2611E-04
.1867E-04
.1334E-04
.9525E-05
.6789E-05
.4828E-05
.3424E-05
.2417E-05
.1180E-05
.8092E-06
.5439E-06
.3538E-06
.2176E-06
O.
.1200E-06
.5011E-07
.10000+01
.8953E+00
.7902E+00
.7057E+00
.6285E+00
.5576E+00
.4924E+00
.4329E+20
.3704E+00
.3303E+00
.2965E+00
.2482E+10
.2142E+00
.1843E+00
.1583E+00
.1356E+00
.1160E+00
.9938E-01
.8451E-01
.7201E-01
.6129E-01
.5213E-01
.4430E-01
.3763E-01
.3194E-01
.2710E-11
.2299E-01
.1949E-01
.1652E-01
.1400E-01
.1186E-01
.1005E-01
.85060-02
.77000-02
.6092E-02
.5152E-02
.4355E-07
.36750-02
.3103E-02
.2613E-02
O.
.21960-02
.1840E-02
.1534E-02
.1271E-02
.1042E-02
.6402E-01
.656+1-03
.4586E-03
.3146E-03
CONFORMAL RAP
GIVEN PLANE COORO.
.7086E-10
.7917E-01
.1446E+00
.2024E+00
.2549E+00
.3033E+00
.3479E+00
.3692E+00
.4274E+00
.4628E+00
.4957E+00
.5262E+00
.5547E+00
.5814E+00
.6064E+00
.6380E+00
.6525E+00
.6739E+00
.6946E+00
.7146E+00
.7342E+00
.7536E+00
.7727E+00
.1916E+00
.8102E+00
.8285E+80
.8461E+00
.8629E+00
.8786E+00
.8931E+00
.9062E+00
.9180E+00
.9284E+00
.9375E+00
.9454E+00
.9522E+00
.9581E+00
.9632E+00
.9676E+00
.9713E+00
.9745E+00
.9772E+00
.9796E+00
.9816E+00
.9834E+00
.9649E+00
.9863E+00
.9877E+20
.9890E+00
.1270E+01
.1727E+00
.1884E+00
.2270E+00
.2052E+00
.3647E+00
.4694E+00
.6056E+00
.78210.00
.1011E+01
.13080+01
.1695E+01
.2203E+01
.2871E+01
.3756E+01
.4936E+01
.6521E+01
.8668E+01
.1160E+02
.1566E+02
.2129E+02
.29170.82
.4020E+02
.5555E+02
.7667E+02
.1052E+03
.1429E+03
.1914E+03
.2522E+03
.3268E+03
.4166E+03
.5233E+03
.6492E+03
.7971E+03
.9700E+03
.1175E+04
.1415E+04
.1698E+04
.2034E+04
.2433E+04
.2909E+04
.3480E+04
.4173E+04
.5022E+04
.6081E+04
.7437E+04
.9242E+04
.1181E+05
.1595E+05
.2481E+05
.2481E+05
MAGNIFICATION
-H.LOG(1.-(PHIC-PHI))
.1000E+17
.6000E+17
.3728E+00
.2460E+00
.1949E+00
.1652E+00
.1447E+00
.1292E+00
.1168E+00
.1065E+00
.9776E-01
.9027E-01
.0381E-01
.7824E-01
.7345E-01
.69300-01
.6599E-01
.6323E-01
.6186E-01
.5948E-01
.58370..01
.57610-01
.57030-01
.5640E-01
.5545E-01
.5395E-01
.51760-.01
.4066E-01
.4535E-41
.4141E-01
.3725E-01
.3309E-01
.2909E-01
.2535E-01
.2194E-01
.1808E-01
.1618E-01
.1301E-01
.1175E-01
.9958E-02
.8413E-02
.7079E-02
O.
.59240..02
.4921E...02
.4045E-02
.3270E-02
.2570002
.1914E-02
.1243E-02

C
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C.
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C
PROGRAM OSOR2(INPUT.OUTPUT.TAPE5=INPUT.TAPE6=OUTPUTI
CALL DSOR2S
END
SUBROUTINE OSOR2S
DIMENSION R1S1201.NS(20).US(201,G1S(201.G2S(201
STEADY STATE CONCUCTION IN ANNULAR SECTOR
EVALUATION OF OSOR(1.PHI0l AND THE INTEGRAL OF OSDR(1,P121
ON PHI0/2.LE.PHI.LE.PHIO
INPUT DATA
JmAX=NUMBER OF VALUES OF TW AND EXPONENT U TO BF CONSIDERED
KMAX=NUMBER OF VALUES OF INNER RADIUS TO BE CONSIDERED
LMAX=NUMBER OF VALUES OF TW TO 9E CONSIDERED
US=EXPONENT FOR POWER LAW PROFILE IN TRANSFORM PLANE
G1S=GAMA(14.US1
G2S=GAMA(14.2US1
R1S=INNER RADIUS OF ANNULAR SECTOR
NS=NUMBER OF TERMS IN FOURIER SERIES APPROXIMATION
TW=DIMENSTONLESS WALL TEMPERATURE TW/(T9-TWI
C
C
100
PI=3.141592654
PI2=PI/2.
G=0.693147
TLN=0.91596
RFA0(5.1001 JMAX.KMAX.LMAX
FORMAT(3I51
C
103
C
102
READ(5.103) (US(JI.G1S(J1.G2S(J).J=1.JMAX)
FORMAT(3F10.01
READ(5.1021 (RIS(KI.NS(K1.K=1.KMAX1
FORMAT(F10.0.I101
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101
DO 15 1=1.1MAX
READ(5.101) TW
FORNATIF10.01
00 15 J=1.JMAX
U=US(J1
01=01S(J1
G2=G2S(J)
WRITE(6.2001
WRITE(6.2011
WRITE(6,202)
WRITE(6.2031
WRITE(6.2041
WRITE(6.2051 U
WRITE(6.1061 TW
WRITE(6.2041
WRITE(6.2071
WRITE(6,200)
C
CALCULATE FOURIER SERIES AND ZETA FUNCTION PARAMETERS
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TWP=TW+.6
TU=2..0
OPU=1.4.LI
OPTU=1..TU
OPU2= OPU'OPU
OPTU2=OPTUOPTU
OMU=1.-U
OMTU=1.-TU
ZU=-.935/0M11+.435
ZTU=-.935/CMTU+.435
SEU=-(1.-2..YOMUlmZU/2....1
SOU=SEU
SOTU=-(1.-2. ..OHTUI=ZTU/2..*TU
SETU=SOTU
AOS01=OPU/OFTLI41.1/(2.=TWP=OPU.OPTU)
B1=T1I.G1=OPU2/PI2.=OPU
82=-U.P124'81
133=OPTU2.G2/(2..PI2.*OPTU1
84=-U4P1.133
135=U.((1.-TUI.OPTU2+TW.(1.-U).OPU21/(PI2=PI21
ZU=10.5844.1./U1*(1.-0.5.0PU1
ZTU=10.584e1./TU1.(1.-0.5.=OPTU1
DSOR1=B1=SOU,B2.SEU4.133=SOTU4.84.SETU-115TLN
ENTSP1=81.ZU+83.ZTU+95.G
DO 11 K=1.KMAX
R1=R1S(K1
NMAX=NS(K)
OMR1=1.-R1
ALGR1=ALOG(R1)
PIOPO=P12/0MR1
AOSO=AOS01/(2..ALGR11
CALCULATE DIFFERENCE FOURIER SERIES
SUM0=0.
SUMI=0.
SGE=-1..
SG0=-1.
DO 12 N=1,NMAX
EN=FLOAT(N1
ENU=ENY=U
ENTU=ENU+ENU
ALAM=EN.PIOPO
R11=41*.ALAM
COEF=(1.4211.R111/11.-R1L1-1.
B1ENU=91/ENU
B3ENU=B3/ENTU
TEST FOR OCO N
FN2=EN/2.
N2=IFIX(EN2)
AINOC=EN2-FLOAT (N21
IFIAINOC.NE.0.1 GO TO 13
N IS EVEN

C
S=0.
SS=0.
SC=0.
C=1.
CC=SGE
SGE=-SGE
GO TO 14
C
N IS ODD
C
13
14
12
SC=SGO
S=-SGO
SS=1.
C=-1.
CC=0.
SG0=-SGO
SUMO=SUMO+COEF(fSC81.00.321/ENLHASC93+CC84)/ENTU+C.B5/ENI
SU4 I=SUMI.COEF.4181ENU83ENU1SS/E1495*S/(ENEN))
C
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CALCULATE OSDR(1,PHI0)
CALCULATE INTEGRAL OF OSOR(1,PMI) ON PNI0/2.LE.PMI.LE.PHIO
OSIR=AOSO+PI0P0(SUND+OS3R11/TWP
ENTSP=A0S0.04P1-1SUMI+ENTSP1)/TWP
DSORN=OSORALGR1
11
15
WRITE(6,2091 R1,0SOR,OSORN,ENTSP
CONTINUE
200 FORMAT(1H11
201
FORNAT(26X,41HSTEADY STATE CONDUCTION IN ANNULAR SECTOR)
202
FORMAT/23X,46MFOR POWER LAW WALL B.C. ON 0.LE.PMI.LE.PHI0/2.)
203
FORmAT(26X.42MS IS NORMALISED -S/SO, AND PHI0=2.(1.-R1))
204 FORMAT(1H0)
205
FORmAT126X.24MPOWER LAW PARAMETER, U= ,F8.4)
206
FORMAT(26X,30HWALL TEMPERATURE, TW/(T8-TW1= ,F8.2)
207
FORMATI5X,17MINNER RADIUS, R1 .5X,12HOSOR(1.PHIO),5X,2CMJSOR(1,Plif
10).LOGIR1),5X,23HINTEGRAL OF OSOR(1.RHI))
208 FORMATI5X,93H
209 FORMATC8X,E12.4,7X,E12.4,9X,E12.4,15X.E12.41
RETURN
ENO

INNER RADIUS. RI
.9900E+00
.9808E+08
.8009E+10
.1000E00
.6000E+80
.5000E+00
.4880E+00
.3000E400
.2000E+00
.1050E+10
STEACY STATE CONDUCTION IN ANNULAR SECTOR
FOR POWER LAW WALL D.C. ON 0.LE.PMI.LE.PM/0/2.
S IS NORMALISED -3/31, AND PRI0=2.(1.-RI)
POWER LAW PARAMETER, Um .3000
WALL TEMPEPATURE. TW/174-TWI= .50
OSCR(1,PNI0)
-.1430E+03
-.1359E+02
-.6392E+01
-.3990E+01
-.2706E+01
-.2061E+01
-.1574E+01
-.1224E+01
-.9574E+00
-.7440E+00
D50R(1,PMI01LOGO111
.1437E+01
.1432E+01
.1426E+81
.1423E+01
.1423E+01
.1428E+01
.1443E+01
.1474E+01
.1541E+01
.1713E+01
INTEGRAL OF OSOR(1PRI)
-.2361E+01
-.2303E+01
-.2238E+01
-.2172E+01
-.2105E+01
-.2033E+01
-.1969E+01
-.1899E+01
-.1826E+01
-.1746E+01
178

C
C
PROGRAM TRJCRDIINPUT,OUTPUT,TAPE5=INPUT.TAPE6=OUTPUTI
TRAJCORO CALCULATES THE COOROINATES OF BUBBLE TRAJECTORIES WHICH
FOLLOW POTENTIAL FLOW STREAMLINES
C
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200
INPUT DATA
MAX= NUMBER OF TRAJECTORIES TO BE CONSIDERED
WAX= NUMBER OF POINTS ON EACH TRAJECTORY
XOMAX= MAXIMUM VALUE OF X TO 96 CONSIDERED
TRAJECTORIES WILL BE CALCULATED FOR 1.LE.X.LE.XOMAX
READ(5,200) MAX,NAX.X0HAX
FORMATUI5,F10.01
DX0=(X0MAX-1.1/FLOAT(MAX-1)
X00=1.-DX0+0.001
C
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10
20
C
100
101
102
103
DO 20 M=1,MAX
WRITE(6,1031
WRITE(6,101)
WRITE(6,102)
X00=000+000
C0=X00-1./X00
X=1.01CO
00=0000-X)/FLOAT(NAX-11
X=X-OX
DO 10 N=1,NAX
X=X+OX
Y=SORT(0/(X-001-X.X)
WRITE(6,1001 x.Y
CONTINUE
FORMAT(5X,E12.4,5X,E12.41
FORMAT(P1 .10X.1HX,15X.1HY1
FORMAT(5X,29H
FORmAT(1H1,11X,17H9UBBLE TRAJECTOFY1
END
.8045E+00
.5391E+00
.7751E+00
.9104E+00
.9456E+00
.9809E+00
.1016E+01
.1051E+01
.1087E+01
.1122E+01
.1157E+11
.1193E+01
.1228E+01
.12630+01
.1298E+01
.1334E+01
.1369E+01
.1404E+01
.1439E+01
.1475E+01
X
BUBBLE TR0JE7.702Y
.1002E+02
.4326E+01
.3222E+01
.2671E+01
.2335E+01
.208'1+01
.1896E+01
.171"E+01
.1601E+01
.14.0E+01
.1367E+11
.1261E+01
.11519+01
.1055E+01
.949,E+00
.8395E+00
.7091E+CO
.5126E+00
.4089E+00
.7053E-OE