THERMAL BOUNDARY LAYER DEVELOPMENT
IN DISPERSED FLOW FILM BOILING
Lawrence M. Hull
Warren M. Rohsenow
.
Report No. 85694-104
Contract No. CME-76-82564 A02
I.-
Heat Transfer Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
June, 1982
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THERMAL BOUNDARY LAYER DEVELOPMENT
IN DISPERSED FLOW FILM BOILING
Lawrence M. Hull
Warren M. Rohsenow
Report No. 85694-104
Contract No. CME-76-82564 A02
Heat Transfer Laboratory
Department of Mechanical Engineering
Massachusetts Institute of Technology
Cambridge, Massachusetts
02139
June, 1982
Heat Transfer
Laboratory
THERMAL BOUNDARY LAYER DEVELOPMENT
IN
DISPERSED FLOW FILM BOILING
by
LAWRENCE M. HULL
Submitted to the Department of Mechanical Engineering
on May 11, 1982 in partial fulfillment of the
requirements for the Degree of Doctor of Philosophy in
Mechanical Engineering
ABSTRACT
Dispersed flow film boiling consists of a dispersion of droplets which
are carried over a very hot surface by their vapor. This process occurs in
cryogenic equipment and wet steam turbines. It is also of interest in the
analysis of a nuclear reactor loss of coolant accident and many other applications.
The integral approach to boundary layer theory is first used to
analyze heat transfer to the turbulent flow of single phase vapor in the
entrance region of a circular tube. The single phase predictions compare
well with published correlations and data. The integral analysis is then
extended to analyze the heat transfer to turbulent dispersed flow. A numerical solution and a simplified explicit solution for the dispersed flow case
are presented. These analyses are verified by comparisons to experimental
data.
An experiment was designed which allowed control over the key parameters
appearing in the analysis: vapor Reynolds number, quality, and drop diameter.
The heat transfer tests were carried out at vapor Reynolds numbers of 2x10 4
and 4x10. The quality varied between 10% and 50%. A photographic study of
the dispersion provided the estimates of the drop sizes. For each pair of
Reynolds number and quality, a test with relatively large drops and one with
relatively small drops was carried out. This provided an experimental parametric study in addition to the data required for the model verification.
Thesis Supervisor:
Title:
Warren M. Rohsenow
Professor of Mechanical Engineering
-4-
ACKNOWLEDGEMENTS
I sincerely thank Professor Warren M. Rohsenow for all his helpful
guidance and support throughout my stay at M.I.T.
I wish to express
thanks to Professors Peter Griffith, Borivoje Mikic and Ain Sonin for
their careful suggestions and review of this work.
I greatly appreciate the daily assistance and support of Wayne
Hill and the other members of the M.I.T. Heat Transfer Laboratory.
Technical assistance was provided by W. Finley, F. Johnson and J. Caloggero.
Thanks to Dr. Harold Edgerton for his help with the photography.
Special
thanks to Sandy Tepper for all her cheerful help and guidance.
This work was funded by a grant from the National Science Foundation.
Thanks to Ms. Gisela Rinner for her aid with contract business.
-5-
TABLE OF CONTENTS
PAGE
ABSTRACT
2
ACKNOWLEDGEMENTS
4
LIST OF FIGURES
8
11
NOMENCLATURE
CHAPTER 1 -
1.1
1.2
1.3
1.4
CHAPTER 2 -
INTRODUCTION
Dispersed Flow Film Boiling
Thermal Boundary Layer Development
Review of Related Work
Objectives
ANALYSIS OF SINGLE PHASE FLOW
2.1 Introduction
2.2 Integral Energy Equation
2.3 Dimensionless Variables
2.4 Single Phase Velocity and Temperature
Distributions
2.5 Single Phase Boundary Layer Development
2.6 Average Temperature
2.7 Wall Friction and the Effect of Developing Velocity Profiles
2.8 Comparisons to Other Investigations
CHAPTER 3 - ANALYSIS OF DISPERSED FLOW FILM BOILING
3.1 Introduction
3.2 Vapor Energy Balance
3.3 Radiation
3.4 Drop-Wall Interactions
3.5 Temperature Profile for Dispersed Flow
3.6 Evaluation of the Loading Parameter B
16
16
17
19
22
23
23
25
26
27
31
33
35
39
47
47
48
51
52
53
56
-6-
PAGE
3.7
General Solution Procedure
3.8
Simplified Solution
CHAPTER 4 -
EXPERIMENTAL APPARATUS
4.1
Introduction
4.2
4.3
Steam Delivery System
Liquid Delivery System
4.4
4.5
Inlet Section
Liquid Film Removal System
84
88
94
4.6
Set Points
97
4.7
Free Spray Test Section
97
4.8
Heated Test Section
99
CHAPTER 5 -
EXPERIMENTAL RESULTS AND MODEL PREDICTIONS
5.1
Introduction
5.2
Drop Size Measurement
5.3
Single Phase Tests and Heat Loss Calibration
5.4
5.5
Parametric Behavior
Dispersed Flow Data and Model Predictions
CHAPTER 6 6.1
6.2
CONCLUSIONS AND RECOMMENDATIONS
Conclusions
Recommendations
80
82
105
105
105
124
127
133
150
150
151
153
REFERENCES
APPENDIX A -
AXIAL CONDUCTION NEAR DRYOUT
156
APPENDIX B -
INTEGRAL APPROXIMATION
160
APPENDIX C -
INSTRUMENT SPECIFICATION
164
APPENDIX D -
VENTURI CALIBRATION
165
APPENDIX E -
ROTAMETER CALIBRATION
173
-7-
PAGE
APPENDIX F -
SET POINTS
176
APPENDIX G -
WALL TEMPERATURE ERROR ESTIMATE
192
SINGLE PHASE DATA
195
TWO PHASE DATA
204
APPENDIX H
-
APPENDIX I -
-8-
LIST OF FIGURES
NUMBER
TITLE
PAGE
1.1
Dryout in Annular Flow
18
2.1
Single Phase Boundary Layer
24
2.2
Single Phase Temperature Profile
32
2.3
Single Phase Friction Factor
38
2.4
Comparison to McAdams Equation
41
2.5
Comparison to Heinmann Equation
42
2.6
Comparison to Deissler's and Sparrows' Analyses
44
2.7
Comparison to the Data of Mills
45
2.8
Effect of Developing Velocity Profile
46
3.1
Dispersed Flow Boundary Layer
49
3.2
Differential Element for Temperature Profile
54
3.3
Temperature Profile for Dispersed Flow
57
3.4
Force Balance on a Drop
59
3.5
Variation of the Loading Parameter
63
3.6 a,b
Comparison of the Simple Model to the
Local Conditions Solution
3.7
Distribution Function $l
74
3.8
Distribution Function $2
75
3.9
Parametric Behavior of the Simple Model
79
4.1
Conceptual Experimental Design
81
4.2
Flow Schematic
83
4.3
Steam Supply System
85
4.4
Steam Measurement System
86
71 - 72
-9-
TITLE
NUMBER
PAGE
4.5
Liquid Delivery System
87
4.6
Schematic of Inlet Section
89
4.7
Photographs of Nozzle and Inlet Section
90
4.8
Film Removal System
95
4.9
Pressure-Flowrate Characteristic for the
Film Removal System
96
4.10
Spray Photography Set-Up
98
4.11
Heated Test Section
100
4.12
Thermocouple Attachment
101
4.13
Inlet of Heated Test Section
104
5.1
Photograph of Spray
106
5.2
Histogram for Drop Sizes
108
Drop Size Distribution
111
5.15
Correlation for Volume Mean Diameter
123
5.16
Loss Coefficient
126
5.17
Single Phase Test
128
5.18
Single Phase Test
129
5.19
Parametric Behavior of Dispersed Flow
130
5.20
Parametric Behavior of Dispersed Flow
131
5.21
Parametric Behavior of Dispersed Flow
132
5.22 - 5.33
Dispersed Flow Test
134
5.34
Dispersed Flow: Effect of Properties
147
A.1
Axial Conduction at Dryout
157
B.1
Effect of Laminar Sub-layer on Integral
161
B.2
Effect of Laminar Sub-layer on Wall Temperature
163
5.3
-
5.14
-10-
NUMBER
TITLE
PAGE
D.1
Venturi Calibration
166
D.2
Venturi Calibration
168
D.3
Velocity Coefficient
169
D.4
Venturi Calibration
171
D.5
Velocity Coefficient
172
E.1
Film Flow Rotameter Calibration
174
E.2
Liquid Supply Rotameter Calibration
175
G.1
Resistance Network for Temperature Error
Estimates
193
- M111.MInMM0INUMIMMnINI
Ifi .u1W
-11-
NOMENCLATURE
English
A
Drop size distribution parameter
a
Acceleration
B
Droplet loading parameter
c
Drop size distribution parameter
CD
Drag coefficient on a drop
cp
Vapor specific heat
d
Drop diameter
DT
Tube diameter
f
Friction factor
F
Force
g
Acceleration of gravity
G
Mass Flux
h
Heat transfer coefficient
hfg
Latent heat of vaporization
i
Local specific enthalpy of vapor
k
Thermal conductivity of vapor
kd
Deposition velocity
K
Non-equilibrium parameter
m
Mass
N
Total number of drops
n
Droplet number density or number frequency
P
Pressure
-12-
English (Cont'd.)
q
Local vapor heat flux
q"
Wall heat flux
r
Radial position from tube centerline
S
Slip ratio
t
Time or thickness
T
Temperature
Uloss
Loss coefficient
u
Local vapor velocity
U*
Friction velocity
V
Average vapor velocity
w
Mass flow rate
We
Weber number
x
Quality
y
Distance from wall
z
Axial position from dryout
Greek
Void fraction or thermal diffusivity
U
Emissivity
Thermal boundary layer thickness
6m
Momentum boundary layer thickness
Drop-wall effectiveness
n
Surface tension
-13-
Greek (Cont'd.)
Non-equilibrium parameter
i
Vapor viscosity
x
Kinematic vapor viscosity
Distribution parameter
p
Density (of vapor unless subscripted)
Stephan-Boltzman constant
Shear stress
$1,2
Distribution functions
Subscripts
a
Atomizing
av
Average
b
Dryout or burnout
bl
Average over boundary layer
6m
Edge of momentum boundary layer
d
Drop
dw
Drop-wall
e
Equilibrium or entrained
f
Film properties
g
Saturated vapor
li
Liquid inlet
1
Liquid
p
Patch point
-14-
Subscripts (Cont'd.)
rad
Radiation
s
Saturation or sink
T
Tube or turbulent
tot
Total
vd
Vapor to drop
v
Bulk vapor or "to vapor"
w
Wall
wd
Wall to drop
Fully developed
0
Wall or dryout
Count mean
2
Area mean
3
Volume mean
Dimensionless Variables
Nu
Nusselt number
Pr
Prandtl number
Re
Reynolds number (of vapor unless subscripted)
DT+
Tube diameter, DTu*/v
I+
Vapor superheat parameter, Eq. (2.25c)
r+
Radius, ru*/v
T+
Local vapor temperature, (T-Ts)/(q"DT/k)
u,
Friction velocity,
u
Local vapor velocity, u/u*
/T 0 /P
ljlim
W11111,1,14
111d
1.1
mu. miII ig~
-15-
Dimensionless Variables (Cont'd.)
y+
Distance from wall, yu*/v
y
Distance from wall, y/DT
z
Axial position from dryout, z/DT
Boundary layer thickness, 6u*/v
6
Boundary layer thickness, 6/DT
-16-
CHAPTER 1
INTRODUCTION
1.1
Dispersed Flow Film Boiling
Boiling heat transfer encompasses all heat transfer processes in
which the liquid to vapor change of phase is involved.
Throughout this
work it is assumed that only one component is present; pure water, pure
nitrogen, etc.
When the heated surface temperature is high enough, no
liquid may wet the wall and film boiling conditions are said to exist.
(The term "wetting" is used here to indicate intimate contact between
the liquid and the heated surface, rather than the phenomenon associated
with surface tension.)
Heat transfer rates are low as compared with
wetted wall boiling processes and wall temperatures may be high enough
to cause material damage or failure.
Film boiling can exist with any imaginable two-phase flow regime;
i.e., the wall is dry but the core may be in slug flow, churn flow,
bubbly flow, etc.
The flow regime considered in this work is the dis-
persed flow regime.
In this regime the liquid is present in the form
of droplets which are entrained and carried over the heated surface by
their own vapor.
The heat transfer is mostly by vapor convection, but
this is augmented by the presence of the liquid via three mechanisms:
1) Vapor to drop. Since the vapor may become
superheated and the liquid remains at the saturation
temperature, heat may be transferred from the vapor
to the drops.
-17-
2) Wall to drops. As a drop comes close to the
wall, there is vapor generation from the facing surface
of the drop which provides a cushion of vapor that prevents liquid-wall contact. This is termed the drop wall
interaction and is modeled as a heat transfer mechanism.
3) Radiation. If the number density of droplets
in the dispersion is such that there is a significant
view factor and the wall temperatures are high enough,
radiation from the wall to the drops may be important.
It is of interest to predict wall temperatures in dispersed flow
film boiling in such applications as the analysis of the reflood phase
of a nuclear reactor loss of coolant accident.
Other areas of applica-
tion include wet steam turbines and once through steam generators.
1.2
Thermal Boundary Layer Development
A common situation which produces dispersed flow film boiling is
shown in Figure 1.1.
Liquid is fed to the bottom of a vertical heated
tube. When the heat flux is moderate, nucleate boiling conditions will
prevail in the lower portion of the tube. At some point, called the
dryout point, enough liquid has been evaporated so that the rate of
deposition of liquid entrained in the core is not sufficient to prevent
the liquid film on the wall from drying out.
Downstream of this point,
the wall temperature increases rapidly and film boiling conditions
exist.
Since the wall is wet upstream of the dryout point, the vapor
at the dryout point is essentially at the saturation temperature. Since
the liquid does not wet the wall downstream of the dryout point, the
-18-
o
a
6
I
DRYOUT
POINT
6
-/O
FIGURE 1.1
Dryout in Annular Flow
TEMPERATURE
-19-
vapor superheats and the mechanisms described in the previous section
come into play.
The region immediately downstream of the dryout point
may therefore be modeled as the development of a thermal boundary layer.
The investigation of this region, both experimentally and theoretically,
is the subject of this work. Flow regimes in which the liquid is not
in the form of a dispersion at and downstream of the dryout point are
not considered.
1.3
Review of Related Work
Dispersed flow film boiling has been the subject of many investi-
gations. A very thorough review of the available literature has been
performed by Groeneveld [1].
A few of the investigations which are of
interest in this study are discussed below. The available experimental
data are considered first, and then the prediction methods.
There is a fairly large data base for dispersed flow film boiling
which includes many heating methods and geometries:
flat plate, cir-
cular tube, heated rods, etc. The circular tube is considered in this
investigation and the following selections from the data base are for
this geometry and a constant wall heat flux:
Bennett [2] et al., and
Era [3] et al., have published data for water, Groeneveld [4] for freon
12, Forslund [5] and Hynek [6] for nitrogen and Koizumi [7] et al. for
freon 113.
Relatively few experimental investigations of dispersed flow
film boiling have included measurements of drop sizes.
Recently, ex-
perimental data for two-component dispersed flow film boiling have been
published in which drop diameters were measured.
However, to the
knowledge of the author, only the work of Koizumi [7] et al., reports
measured drop diameters in single component dispersed flow film boiling in a circular tube.
Unfortunately, the dryout point occurred well
downstream of the point at which the drop sizes were measured, and
therefore allowed the entrainment and deposition processes to further
affect the drop size distribution before dryout.
Another important parameter which is difficult to measure is the
vapor superheat.
Forslund [5] used a helium tracer technique to measure
actual quality and more recently Chen [8] has developed a technique of
measuring the vapor temperature directly.
No experiments known to the author have paid particular attention
to the region in which the thermal boundary layer would be developing.
Very often there is only one wall temperature measurement in this region.
There are a large number of correlations for predicting dispersed
flow film boiling, and the majority are based on the data mentioned.
The
most accurate include the effect of thermal non-equilibrium; they account
for the vapor superheat in some fashion.
Of these, the Chen [9] or the
Groeneveld and Delorme [10] correlation currently appear to be the choice
selections.
Since these are based on the data mentioned above, and use
fully developed single phase correlations to estimate the vapor convection, they cannot be expected to do well in the "entrance region" just
beyond the dryout point.
This is well represented in the model compari-
sons presented by Hill [11].
It has been suggested that a single phase
-21-
entrance length correction be used, but it is not clear that this is the
correct procedure.
Many workers have developed phenomenological models of dispersed
flow film boiling. These analyses model the individual mechanisms mentioned in Section 1.1, combine them with differential mass, energy and
momentum balances, and integrate stepwise downstream from the dryout
point.
Recently, Yoder [12] has presented a simple graphical solution
of a one-dimensional phenomenological model.
This model, which does not
require a computer, is termed the Local Conditions Solution and applies
to points well downstream of dryout.
This work uses the integral approach of boundary layer theory to
analyze the region just beyond dryout.
It is directly linked to the
Local.Conditions Solution of Yoder [12]; there is a simple transition
from the entrance length calculation to the fully developed flow calculation.
Both the Local Conditions Solution and the Droplet Laden
Boundary Layer model require that the conditions, including the characteristic drop diameter, be known at the dryout point. An analysis of
the annular flow regime by Hill [11] has resulted in a method for calculating the proper drop diameter for dispersed flow film boiling.
To
the knowledge of the author, the only other analysis of thermal boundary
layer development in dispersed flow film boiling has been performed by
Yao [13].
He used a finite difference technique to solve the two-
dimensional energy equation. While the analysis method used by Yao [13]
is probably the most accurate, the simplified model presented in Chapter
3 is easy to use and yields reasonable results.
-22-
1.4 Objectives
As mentioned in the previous section, the experimental and theoretical investigations of dispersed flow film boiling have not concentrated on the "entrance region" just downstream of the dryout point.
This investigation adds to the information, both experimental and
theoretical, available about this region.
The following table lists
the objectives of this work.
1) Perform a detailed analysis of thermal boundary
layer development in dispersed flow film boiling.
2) Present a simple method of estimating the heat
transfer in this region, and indicate the link to the
Local Conditions Solution.
3) Collect data specific to the entrance region.
In particular, obtain experimental evidence of the parametric behavior by controlling independantly and systematically varying the variables deemed to be of importance by the
analysis.
-23-
CHAPTER 2
ANALYSIS OF SINGLE PHASE FLOW
2.1
Introduction
The heat transfer to a single phase vapor in the entrance region
of a circular tube is analyzed using the integral approach of boundary
layer theory. The flow is assumed to be turbulent and is divided into
two zones:
a wall zone and a turbulent zone. Velocity and temperature
distributions are postulated for each zone and then patched at a point
near the center of the buffer layer. The case of a fully developed
initial velocity profile is worked out in detail and then a method for
including the effect of a developing velocity profile is presented.
Figure 2.1 indicates diagramatically the transport quantities to
be considered in the single phase analysis.
The following assumptions
are used repeatedly:
1) The wall heat flux is constant. This assumption
neglects conduction through the tube wall material for the
case of a constant internal heat generation. However, it is
shown in Appendix A that this is a good assumption for many
such cases.
2) The properties of the vapor are not a function of
radial position but may vary with axial position. It is
shown that properties evaluated at the average temperature
of the boundary layer yields adequate results even when there
are severe radial temperature gradients.
3) Axial conduction of heat through the fluid is
negligible.
WeJ
I
FIGURE 2.1
Single Phase Boundary Layer
M06IMNAft
IWI
-25-
In the subsequent sections the single phase analysis is presented
according to the following outline. The integral energy equation is
derived from first principles. Then the dimensionless variables normally used for the analysis of turbulent flow are introduced.
Next, velocity
profiles applicable to turbulent flow are used to derive temperature distributions. These distributions are then substituted into the integral
energy equation and the result solved to find the variation of the
Nusselt number with axial position.
The method for estimating the effect
of developing velocity profiles is then given and all of the results compared to data and analyses published by other authors.
2.2
Integral Energy Equation
From Figure 2.1 the following balances are written
Energy:
dI
Iz + I
+ q" DTdz
I
=
+
(2.1)
dz
Mass:
dw
wz +w
ze
or
w
= wz
+
z
dw
=
The flow entrained, we
(2.2)
z dz
dz
dz
,
is at temperature Ts
so that
dw
w is =
dzi
(2.3)
-26-
Making use of the formulas
r
w
rz
0
p u 2Trrdr
(2.4)
(> u iz-ardr
(2.5)
r00
z
r0-6
Equations (2.1) to (2.3) are combined to give the integral energy
equation
r
r0
2.3
pu cp(T-Ts) 21lrdr
= q"TDT
(2.6)
Dimensionless Variables
The dimensionless variables used in this analysis are
Velocity:
u
=
u/u*
(a)
where
u* = /1 O/p
(b)
Temperature:
T+=
(T-T)/ k
(c)
(2.7)
-27-
Radial Position:
r
=
ru*(d)
or
y+ yV=
etc.
(e)
Axial Position:
z
= z/D T
(f
Using the above definitions the integral energy equation
becomes
dz
0
d u+T+(r
DT
-y+)dy+
=
1
2.4 Single Phase Velocity and Temperature Distributions
The following assumptions, suggested by Prandtl for turbulent
flow over a flat plate, are used to derive temperature distributions
from velocity distributions known to fit experimental data.
1) The turbulent Prandtl number is unity. This
should be a reasonable assumption, especially for the
region near the wall. See Schlichting [14], 7th edition,
pp. 706-712.
2) In the laminar sublayer the eddy diffusivities
are negligible; only the molecular transport properties
are important.
(2.8)
-28-
3) In the turbulent zone the molecular transport
properties can be neglected.
4) The ratio, q/T is constant over the boundary
layer and may be evaluated at the wall.
For the turbulent zone the velocity distribution
1/7
(2.9)
= 8.74 y
is based on the Blasius friction factor, which is used since it
yields simple and familiar results.
For a derivation of Eq. (2.9) see
Schlichting [14], 7th Edition, p. 600.
Equation (2.9) is used for de-
veloping velocity profiles where it must be remembered that
+I
-I
u = u6m for y
4
+4
The shear stress and heat flux are expressed, for the turbulent zone,
by
du
=
( .10)
(2
q = -pc( T
(2.11)
Dividing Eq. (2.11) by Eq. (2.10) and invoking assumptions 1 and 4
results in
=
-
c d/dy
(2.12)
-29-
Substituting Eq. (2.9) into (2.12), integrating with the boundary condition
T = Ts when
y = 6 and writing the result in terms of the
dimensionless variables gives
T
6
L
8.74
-
4Pr
T
T
1/7
1/7
(2.13)
T
For the laminar zone the linear velocity distribution is used
u+
+
(2.14)
Equations (2.14) and (2.9) cross at y' = 12.54 , determining the patch
point for both the temperature and the velocity distributions. The heat
flux and shear stress for this case are
du
T
=
(2.15)
pdyv
q = -p ca
d
(2.16)
Invoking assumption 4 again yields
"=
T0
-
cp
Pr
dT/dy
du/dy
Substituting Eq. (2.14) into (2.17) and integrating with the boundary
condition T = Tw at y = 0 results in
(2.17)
-30-
w-T
q" DT
=
k
y
(2.18)
DT
Equations (2.13) and (2.18) are now patched at the point y+ = 12.54
If 6+ is less than 12.54 then only the laminar equation need be considered.
At the patch point, Eq. (2.18) gives
12.54
D+
T
Tw -T p =q kDT
(2.19)
and Eq. (2.13) gives
qk DT
Tp - Ts
Equating
T
874
D+6/7 Pr
P
DT
_
T
1/7
12.54
1/7
(2.20)
[
in these equations results in the expression for the wall
temperature
8=74
DT Pr
6 1/7
T
+
(2.21)
12.54 (Pr - 1)
DT+ Pr
Using Eq. (2.18) in (2.21) yields the temperature distribution for the
laminar sublayer
+
74
DT
Pr
T
6 1/7
DT
12.54
6
1/7
12.54 - y
DT
T
(2.22)
-31-
Figure 2.2 is a plot of the distribution provided by equations (2.2) and
(2.13) for typical values of the parameters
2.5
D+ , Pr , and
6/D
Single Phase Boundary Layer Development
Substitution of the velocity and temperature distributions into
the integral energy equation and integration in the radial and axial
directions results in an expression for the thermal boundary layer
Use of Eq. (2.21) gives the
thickness as a function of axial position.
corresponding wall temperature.
It is shown in Appendix B that in the evaluation of the integral
energy equation the turbulent zone temperature distribution, Eq. (2.13),
may be extrapolated to the wall with negligible effect on the predicted
This approximation, which results in a considerable
Nusselt numbers.
algebraic simplification, is used in the following presentation and all
of the single phase predictions.
Appendix B also shows that it is
using the extrapolated
possible to make a significant error in T+
w
profile. Therefore, Eq. (2.21) is used to estimate Tw
When Eq. (2.13) is substituted into Eq. (2.8) and the axial and
radial integrations carried out, the relation between boundary layer
thickness and axial position is obtained. Thus
z
DT
=
7.43 D+2/
7
(
T
(2.23)
-
T
Nusselt numbers for single phase flow are usually based on bulk
temperature. This definition and the dimensionless variables give
1.0
Dr
.8
=
1500
Pr =
S/Dr
-
.6-
a
+
6
8
10
1+
T+ 2
FIGURE 2.2
Single Phase Temperature Profile
Sp3
-33-
Nu
Tw
=
b
(a)
(2.24)
Nu
2.6
=
(b)
+
+
T - Tb
Average Temperature
Two average temperatures are presented in this section:
temperature and the bulk boundary layer temperature.
the bulk
The bulk tempera-
ture is obtained by averaging over the whole tube and is used in the
definition of the single phase Nusselt number. The bulk boundary layer
temperature is obtained by averaging over only the boundary layer and
is the temperature at which the properties are evaluated.
For the case of developing flows the bulk temperature can be
represented as an increase of average temperature over the inlet
temperature:
pcuT-T )2irrdr
Tb - T
s
=
0
r0
p
(2.24)
s
Jpcpu 2ffrdr
0
Since the fluid properties are assumed constant, the denominator of
Eq. (2.24) is just the flowrate through the tube. Additionally, the
numerator of Eq. (2.24) is zero for y > 6
variables, Eq. (2.24) can be rewritten as
.
Using the dimensionless
-34-
6+2Pr +
+
+
kD T
)dy+
y
Tf u+T+(r0
c wT
pv 0
DT
kI
T
(a)
(b)
cw
(2.25)
where
+ =
u+T+(r
0
- y+)dy+
(c)
T
Integrating Eq. (2.8) in the axial direction and using Eq. (2.25c)
results in I+ = z/DT
.
Substituting this result into Eq. (2.25b) gives
kD
(a)
c w z/D T
pv
+
4 z/DT
b
Re Pr
(2.26)
(b)
For single phase flow Eq. (2.26) can be obtained from a one-dimensional
energy balance
(w c (Tb-Ts)
= 7DTzq").
However, Eqs. (2.25) apply
equally well to single phase or dispersed flow.
The bulk boundary layer temperature is defined as
-35-
Tbl
-
O pcpu(T-Ts)2rdr
(2.27)
~~=
Ts
pc pu27nrdr
S
In dimensionless form and using Eq. (2.25c)
T
Tbl =
{
I
(2.28)
u+(r+- y+)dy+
~
DT
Assuming that Eq. (2.9) may be extrapolated to the wall in the evaluation of the integral in Eq. (2.28) gives
1+
bl
2.7
6 8/7
T --T )(6)
8/7
7.65 PrDT+
16(2.29)
15
DT
Wall Friction and the Effect of Developing Velocity Profiles
The wall shear stress is intrinsic in the preceeding equations
through the dimensionless variables.
For a fully developed velocity profile,
the wall shear stress is estimated from the Blasius friction factor
correlation.
The velocity and temperature distributions used for the
fully developed flow field case are assumed to apply to developing
flow field cases if the variation of wall shear stress with distance
is accounted for.
-36-
The definition of the friction factor is
f
TO
Using the definition of
u0=
(2.30)
PV2
u* gives
(2.31)
V/f77 /7~
10'* < Re < 105
where the factor
is used to correct the fully developed friction
/Tf
factor for the entrance region. The Blasius correlation is used for
f:
.0791/Re 1
4
Combining Eqs. (2.31) and (2.32) and the definition of
=
(2.32)
gives
.199 Re7/ 8 /fT7T-
(2.33a)
For fully developed flow f =fCf
and Eq. (2.33a) reduces to
f
D
=.
199 Re7/ 8
(2.33b)
MONNIM111111,1411111111W
I
-37-
is estimated from the
For a developing velocity profile, f/fC
results of Deissler [15], Figure 20 of reference [15].
f/fm
=
491
1
1 -1.53(z/DT)
(2.34)
Equation (2.34) essentially replaces the solution of the momentum equation.
Figure 2.3 shows the points used to correlate Deissler's analysis.
The correlation is for a Prandtl number near 1.
The velocity distribution may be obtained by combining Eqs. (2.9),
(2.31), and (2.32):
u
1.38 (
)
-
1 /7
T
(2.35)
where u = u6m for y > 6m '
The momentum boundary layer growth is found by requiring the
velocity profile to satisfy the mass balance:
pfrV
Since
u = u6m
nr 2
=
0
for
r0
O
(2.36)
.pu2rdr
y > 6m
r0
r0 6
u
2 rdr +
r0-6
2ardr
2Tr0
r
(2.37)
-38-
S oo
1.5
A -
ANALYSIS OF DEISSLER.
-EQUATION
2.34
L>
.0
-
2.
4
6
8
FIGURE 2.3 Single Phase Friction Factor
10
Imwil
WIN
,,,
-39-
Substituting Eq. (2.35) into (2.37), integrating and solving for u /V
results in
u6m
-
(1 --
1
f
26
2 [1D- .2 2 (7)
-)
7
4/7
{g(
8/7
-)
-T
(2.38)
T
14m14 6m)15/7
T
Equation (2.35) can be applied at y = 6m
u6m4/7
=
.38(
)
-
(6 M1/7
(
(.9
(2.39)
T
Equations (2.34) and (2.38) can be solved for the variation of 6m/DT
with axial position Z/DT . For the case of single phase vapor, the
Prandtl number is near 1 and so
6m
6
Therefore, the only added
complication in the calculation of the heat transfer for developing
velocity profiles arises in the evaluation of
D+ ; Eqs. (2.33a) and
(2.34) should be used rather than just Eq. (2.33b).
2.8
Comparisons to Other Investigations
The general calculation procedure is to solve Eq. (2.21) to find
wall temperature and Eq. (2.23) to determine the corresponding axial
position.
ted at Tbl
The calculation is iterative since the properties are evalua,
Eq. (2.29).
The bulk temperature and Nusselt number are
'. , ,- iM00----
-
-40-
then calculated from Eqs. (2.26) and (2.24).
In the following compari-
sons, the properties were assumed constant and the velocity profile
fully developed, resulting in an explicit calculation for Nusselt
number versus axial position.
At the point where the thermal boundary layer reaches the center
of the tube, Ts
becomes the centerline temperature, and the analysis
provides a prediction for fully developed flow.
Figure 2.4 compares
this analysis to the widely accepted McAdams [16] correlation.
Nub
.023 Re
8
Prj 4
(2.40)
The Heinmann [17] correlation
'04
D
Nuf
=
.0157 Re 84
3(
)
(2.41)
6 < z/DT < 60
is for steam and applies over a portion of the developing region.
A comparison for a specific case is given in Figure 2.5.
Deissler [15] has carried out an analysis of the entrance region
heat transfer, also using the integral approach.
This analysis differs
from his in that he calculated velocity and temperature profiles from
eddy viscosity distributions and allowed for temperature dependent fluid
properties.
Sparrow [18] analyzed the entrance region heat transfer, for
a fully developed flow field, using a differential formulation along with
-41-
2.5
Pr =.73
z
.73
20 -
dj.j
--
4.5
FIGURE 2.4 Comparison to McAdams Equation
-
--
-Mc
AAms
-42-
1500 -
LL
0
HULL
-
1000
HEINMANN
Soo
2,
FIGURE 2.5
4
8
Comparison to Heinmann Equation
c
Z , nc.h
-43-
Deissler's eddy viscosity distributions.
Figure 2.6 compares this work
with that of Deissler and Sparrow.
Mills [19] reported heat transfer coefficients for air, drawn in
from his laboratory, for a variety of initial conditions.
Figure 2.7
compares this analysis to the data of Mills where a value of .015 Btu/hr/
ft/F has been assumed for thermal conductivity in the evaluation of his
data.
All of the above comparisons have been for the case of a fully
developed velocity profile.
The predictions of this model for a develop-
ing velocity profile are compared to that for a fully developed velocity
profile in Figure 2.8.
It was assumed that
case of developing velocity profiles.
6m
= 6 in calculating the
Nu
Nu..
1.4
Re = 10 5
1.3
PEISSLEA
1.0
HULL
l.0
1.0
2.0
3.0
4.0
I
I
5.0
6.0
7.0
I
I
I
8,0
9.0
10.0
P1FIGURE 2.6
Comparison to Deissler's and Sparrows' Analyses
300
)ATA
OF MILLS
o
10.Z6 x 104
8.150 x 104
C)
4.860
A
Re
o
3.012
x
x
104
104
1.667 x 10+
v
zoo
0
0
0
200
O
1.0
2.0
FIGURE 2.7
3.0
o
4.0
5.0
6.0
Comparison to the Data of Mills
7.0
8.0
9.0
t0.0
1.6
N1
Re = )0 5
1.5
Pr = .73
14 1DEVELOPING
VELOCITY
PROFILE
1.3
FULLY
DEVELOPEP
VELOCITY
PROFILE
1.1
1.0
Z/PT
FIGURE 2.8
Effect of Developing Velocity Profile
-47-
CHAPTER 3
ANALYSIS OF DISPERSED FLOW FILM BOILING
3.1
Introduction
In this chapter the concept of a volumetric heat sink is used to
analyze boundary layer development in dispersed flow film boiling. The
heat sink is due to the presence of the liquid, which remans at the
saturation temperature.
The model presented in this chapter reduces to
that presented in Chapter 2 when the liquid loading goes to zero.
It is assumed that it is valid to superpose heat transfer mechanisms
with,a wall heat balance:
q1o
The terms q ad
and
qna
(3.1)
+ q1w + q"
q w are estimated as if they were totally indepen-
dant of each other and the vapor velocity and temperature fields.
The following assumptions, in addition to those stated in Section
2.1 are used in this chapter.
1) The liquid is in the form of a mono-dispersion
of spherical droplets.
2) The void fraction is near 1.
3) The vapor velocity profile is unaffected by the
presence of the liquid.
In the subsequent sections the analysis for dispersed flow film
boiling is presented according to the following outline.
The integral
-48-
energy equation is modified to account for the heat transfer from the
vapor to the liquid and the result written in terms of the dimensionless
variables of Chapter 2. The methods used for evaluating drop-wall interactions and radiation from the wall to the drops are presented.
Next,
the velocity profiles used in Chapter 2 along with the heat sink formulation of this chapter are used to obtain temperature profiles for dispersed
flow. Then I method for evaluating the sink strength, using accepted
correlations, is presented. The general solution procedure, a simplified
solution of the equations, and the connection to the fully developed flow
model are given. The comparisons of the model to experimental data are
saved for Chapter 5.
3.2
Vapor Energy Balance
The integral energy equation must be modified to account for the
heat transferred from the vapor to the drops.
With reference to Figure
3.1, the vapor energy balance is
dz
j
r0-6
uc
(T-Ts) 2frdr +
r0 nrd2hvd (T-Ts) 2ffrdr = q"DT
r0-6
(3.2)
where assumptions 1 and 2 of Section 3.1 have been employed.
Some assumption
must be made about the radial distribution of
liquid in the flow.
Since film boiling conditions exist, there is no
liquid on the wall.
One distribution function which satisfies this con-
dition is
-
av
1/7
n~
= 1. 2 5 ( y )
0
(3.3)
0
o
0CG o
00
62
0
0
0
|
40
a
IIf'
FIGURE 3.1
Dispersed Flow Boundary Layer
G
-50-
Equation (3.3) is a convenient choice since it has the same form as the
velocity distribution.
Using Eqs. (3.3), (2.9) and the dimensionless
variables of Section 2.3 in Eq. (3.2) yields
d [q"
u+T+ (r - y+ y+]
dz
0
D+0
T
158 q
+
(3.4a)
B
0
2 Pru+T+(r+
- y+)dy+
+8/7 v
T
T
Pr
Introducing the definition of
I+
,
Eq. (2.25c),
d (q" I ) +
~
dz v
.158 B (
I+)
8/7 Pr(qI
B
2
navd
av Nu dd T
(3.4b)
q"
where
(3.5)
The droplet loading parameter, B, can be thought of as a sink strength.
When
B=O
the analysis reduces to that of Chapter 2.
The integral appearing in Eq. (3.4) was defined as
I+ by Eq.
(2.25c) and is termed the vapor superheat parameter. This parameter is
related to the actual and equilibrium qualities.
A one-dimensional
energy balance, written for the total flow, leads to
ill@|lllilllt||ululillalu
lll I il
a
-51-
x -x
x
i hfg
(3.6)
iv - i , is related to
The bulk vapor superheat,
I+ and actual quality
through their definitions:
4q"
=GI+
i -i
(3.7)
Substituting this into Eq. (3.6) gives the relation between
I+ and the
equilibrium and actual qualities:
4q"
x
3.3
=
x
e
--
Gh fg
(3.8)
I+
Radiation
The formulation used here is the same as used by Yoder [12].
The
radiant flux from the wall to the drops is
q1a rad= aF wd (T'- T)
S
where a is the Stefan-Boltzman constant.
Fwd
=
1
1
Yd
Yw
(3.9)
From the analysis of Sun [20]
(3.10)
where Yd and yw are the emissivities of the drops and wall, respectively.
These are evaluated from
-52-
Yd
and
yw = .76
DT
-1.1(-a
= 1-ed(3.11)
for inconel.
3.4 Drop-Wall Interactions
The total heat flux due to drop-wall interactions is formulated in
terms of the droplet heat transfer effectiveness for one interaction.
For each impact of a drop
Qd
d
P
h
e(3.12)
The product of deposition velocity,
density,
kd
,
and average droplet number
nav , determines the number of impacts per unit time and area.
Thus, the total drop-wall heat flux is
w
dw
d3
~~- h
hfg Ana
Kdlv
The results of Liu and Ilori [21] show that
(3.13)
kd
should be linear
with the friction velocity for the conditions of interest in dispersed
flow.
Iloeje [22] suggests
Kd
=
.15 u*
=
.15 --DT D+
T
(3.14)
Both the data of Watchers [23] and the analysis of Kendall [24] indicate
that the effectiveness is nearly constant and of the order 10-3 for wall
superheats above about 200*F.
Even for the developing flows considered
-53-
in this work wall superheats attain values well above 200*F a very short
Therefore, a value of c = 0.0025
distance downstream of dryout.
was
chosen from the data of Watchers.
3.5 Temperature Profile for Dispersed Flow
The assumptions of Section 2.4, except number 4, are used in the
following presentation.
For the case of dispersed flow, it is assumed
that the liquid does not affect the velocity distribution, but it is expected that the presence of the heat sink will affect the shape of the
temperature profile. The approach used here is to use the velocity distribution to derive an eddy diffusivity for use in the determination of
the temperature profile.
Deissler found that the radial distribution of shear stress and heat
flux had little effect on velocity and temperature profiles in turbulent
flow.
The eddy viscosity for the turbulent zone is found from Eqs. (2.9)
and (2.10) and the assumption of a constant shear flow:
6/7
D+6/7 (Y - )
8.74 T
DT
7
vT
v
Assuming
aT
vT
(3.15)
results in
-
=
T 7
8 .74
+6/7
Pr DT
6/7
- )
T
Considering Figure 3.2, the following energy balance is written:
(3.16)
-54-
tL
III
)
S
lyf-dy
I
FIGURE 3.2
.< .dy
Differential Element for Temperature Profile
-55-
a
q"" +
+
3 (pui
(3.17)
0
q"'. , is formulated in the same way as in
The volumetric heat sink,
Eq. (3.2), and the z-gradients are neglected in comparison to the y-gradients.
Thus,
Bk (T - T
d
dy
D2
T
n
(3.18)
s nav
Notice that if B=O and Eqs. (3.16) and (2.11) are used, Eq. (3.18)
reproduces the single phase turbulent zone temperature distribution, Eq.
(2.13).
Combining Eqs. (2.11), (3.3), (3.16) and (3.18) gives the equation
for the turbulent zone:
d
dy
7
8.74
D6/7
Pr
DT
.38 B y1/7
6/7 dT +
9
(3.19)
dy
where
D
DT
For the laminar zone, Eqs. (2.16), (3.3) and (3.18) give
d2T+
2
dy
= 1.38 B y
117 +
T
(3.20)
-56-
For specified values of
B, D
Pr
,
and
3
,
the temperature distribu-
tion is obtained by numerical integration of Eqs. (3.20) and (3.21), subject to the boundary conditions
T
9
at
= 0
= 6
and
(3.21)
y = 0
at
-1
dy
The two zones are patched at the point y+ = 12.54; the heat flux and
p
temperature are continuous at y+
p
A plot of the distribution for typical values of the parameters
is shown in Figure 3.3.
increasing B .
As expected, the temperature is suppressed for
The shooting technique was used to solve the equations
of this section (see Hildebrand [25], pp. 290-297).
3.6
Evaluation of the Loading Parameter B
In this section a method for estimating the loading parameter,
B = nidNu d T ,2from accepted correlations is presented.
The droplet Nusselt number is calculated from the hard sphere
correlation suggested by McAdams [16] for gases
Nudf
= .7
Re
for
17 < Redf < 70,000
(3.22a)
\.0
DT =1000
S/DT
-
Pr
.6
=
.5
=1
B =2000
13 = 0
L,
8
2+
FIGURE 3.3
10
1z
14
Temperature Profile for Dispersed Flow
16
zo
18
T+ x 103
and the result for pure conduction
Nudf
= 2
Redf
<
(3.22b)
for
17
The droplet Reynolds number is based on the relative velocity,
Redf
Pvf(VPp -
d
vf
V, - V
(3.23)
The relative velocity is estimated with the help of a force balance on a
drop. Considering Figure 3.4,
FD -
mdg
=
(3.24)
mdad
The drag force
FD D~ 1 P
'v W dV
d2(V
-
V
)2C
V~)D
(3.25)
is estimated using a hard sphere correlation:
CD
=
Red (1 + .142 Re 698)
for
Red
<
2000
(3.26a)
Mlliftfl
Will
10,
1'.,
-59-
I VA
v
t
FIGURE 3.4
j1tt)Ilt
Force Balance on a Drop
v
-60-
with a lower limit of
=
.45
for
(3.26b)
Red
2000
>
The maximum acceleration that the liquid could experience occurs
for an equilibrium flow (x = xe) and the minimum for a complete nonequilibrium flow (x = xb).
The final results are not overly sensitive
to liquid acceleration and the choice of the maximum acceleration has
been shown to yield reasonable results for the case of fully developed
flow by Yoder [12].
Therefore, the maximum acceleration case (x = xe)
is used here.
The one-dimensional liquid velocity is written in terms of the slip
ratio
v
S
Gx
V
e
(3.27)
(3.28)
l
The liquid acceleration is therefore
dVV
d
=
d
Gx
V d ( PvS
V2Qdx
x
dz
(3.29)
where it was assumed that
1 dx
Saj dz
d 1/Sa
N
x'
dz
(3.30)
-61-
and that
G/p
is constant to obtain the last result.
The assumption, Eq. (3.29), is valid since
S and a
are both
asymptotic to 1 and nearly equal to 1 for all cases encountered here and
was also shown to be valid for other data sets by Yoder [12].
Thus, the
maximum acceleration is
dV
V2 dx
dxdz
(3.31)
Substituting Eqs. (3.25) and (3.31) into (3.24) results in an expression
for the slip ratio
2
S
1+
1/2
()]
(3.32)
The average number density is related to the void fraction through
nav
=
(3.33)
1 -ca
The void fraction is related to quality and the slip ratio
a
1
+ LX
(3.34)
p29
S
and the equilibrium quality is given by
Xe
=
Xb
+
4q"I
tot z/D
fg
(3.35)
-62-
The preceeding equations are solved simultaneously according to the
following procedure.
Given:
d, G, X, DT
,
and
qtot
1) Assume an S
2) Use Eqs. (3.34), (2.38), (3.23), (3.26) and (3.35), in
that order, to calculate values of a , V
and
,
Red, CD
dX /dZ .
3) Use Eq. (3.32) to calculate
S
.
If this value of S
does not agree with the assumed value, then return to
Step 1.
4) When the equations have balanced out, calculate
from Eq. (3.22),
nav
from
Nud
Eq. (3.33) and B from
Eq. (3.5)
B
=
nay 'dNuD
2
(3.5)
Figure 3.5 shows the result of this calculation for
typical values of X , Re, d, DT , and fluid properties.
In the general solution procedure, the calculation of the loading
parameter is updated at each axial position.
Since evaporation is taking
place, the drops will decrease in size as they move downstream. The drop
diameter can be related to actual quality and the drop diameter at dryout.
The liquid flux is
oinil~mloililllilliilll1
illi
-63X=.I
los
X=.3
X =.5
10+
Re =2
x104
Re= 4- x 10
io2
-
10
FIGURE 3.5
100
Variation of the Loading Parameter
1000
-64-
G(l-x)
=
navV p
(3.36)
If no coalescence or breakup occurs, the number of drops is conserved.
This may be expressed by
= constant
navV
(3.37)
Equation (3.36) may be applied at the dryout point
Ad3
G(1-xb)
=
navV kpk
(3.38)
6
Dividing Eq. (3.38) into (3.36) and using (3.37) results in
d
=
db(1 -
x
b
1/3
)
Equations (2.33) and (2.34) are used to calculate
(3.39)
D+
The
Reynolds number is based on the vapor velocity
Gx DT
Re
(3.40)
=
v
3.7
General Solution Procedure
In this section the method by which the equations of the previous
sections may be used to predict wall temperature as a function of axial
position is presented.
-65-
The solution procedure, which involves numerical integration of
Eqs. (3.4), (3.19), (3.20) and (2.25c), begins at the dryout point. The
boundary layer thickness is zero at this point and therefore the heat
transfer coefficient is infinite.
Since the vapor is at the saturation
temperature and the heat transfer coefficient is infinite, the wall will
also start at Ts . Since there is no temperature difference between the
wall and the drops, the drop-wall flux and the radiant flux are zero and
The calculations are iterative because of the temperature
q" = qo.
dependance of the fluid properties and the coupling of the wall temperature to the radiant heat flux. The following steps outline the procedure.
Given:
G, Xb, db, DT, and
qtot
1) The loading parameter, B , is calculated for the
dryout conditions according to the method described in
Section 3.6.
2) The Reynolds number is calculated from Eq. (3.40)
and then
D
from Eq. (2.33).
3) Equation (3.4b) is integrated by one increment in
the axial direction. This results in a value of q" I+
corresponding to an axial position Z/DT
4) Guess a wall temperature, Tw *
5) The radiant flux is calculated as described in
Section 3.3 using the current estimate of wall temperature,
void fraction, and drop diameter.
-66-
6) The drop-wall flux is calculated as described in
Section 3.4, again using the current values of the required
parameters.
7) Equation (3.1) is used to get q"
and then
V
I+ can
be calculated from the current value of q" I+
8) In this step the temperature distribution is found
and then integrated to find
I+ .
I+ calculated here matches that
is adjusted until the value of
found in step 7).
The boundary layer thickness
This is described in the following sub-
procedure.
Given:
I+, B, DT , and Pr
a) Guess a value of
~
b) Find the temperature distribution by numercial
integration of Eqs. (3.19) and (3.20) with the boundary
conditions (3.21) and the patch at
y+ = 12.54.
p
c) Use this distribution, Eq. (2.9), and the
assumed value of
(2.25c) to find
6 , and numerically integrate Eq.
I+
d) Check the value of
against the required value.
return to step a).
I+ calculated in step c)
If they do not agree,
If they doagree the iteration stops.
The above sub-procedure yields, in particular, the dimensionless wall
temperature, Tw , and the boundary layer thickness,
.
111MININ
-67-
9) Tbl
is calculated from Eq. (2.29) for the value of
S
found in step 8) and the current values of the other parameters.
10) The properties at Tbl
are found.
The properties
found in this step are used throughout the procedure.
11)
Calculate the wall temperature from the results of step 8),
the current properties and q" . If it agrees with the value assumed
in step 4), qo to step 17).
12)
If they do not then qo to step 12).
Calculate the quality from Eqs. (3.8) and (3.35) and
then the drop diameter from Eq. (3.39)
13)
Recalculate the loading parameter, using the current
d, G, X, and fluid properties.
14) Recalculate the Reynolds number from Eq. (3.40) and
DT from Eqs. (2.33) and (2.34).
15) Reintegrate Eq. (3.4b) to obtain a new estimate of
q" I+ but at the same axial position Z/DT
16) Return to step 4).
17)
The correct value of wall temperature corresponding
to an axial position has been found.
The calculation returns
to step 3) and repeats until the boundary layer reaches the
center of the tube.
The results of this calculation are compared to data in Chapter 5.
3.8 Simplified Solution
It is desirable to have a solution which does not require numerical
integration.
The following assumptions result in an explicit solution
-68-
and have been found to be reasonable by examination of the results of
the general solution procedure.
a) The heat flux to the vapor is uniform with distance.
b) The temperature profile is unaffected
by the drops.
c) The group B/RePr is constant.
d) The velocity profile is fully developed.
Using assumption a) and Eqs. (2.25c) and (2.33), Eq. (3.4) becomes,
for a fully developed velocity profile,
-I
+
I+
(3.41)
= 1
dz
With the help of assumption c), this is integratable
I+
- B Z/D
T)
eRePr
RePr
(3.42)
Use of the single phase temperature profile in Eq. (2.25c) results in
I+
9/7
= 7.4373.D+2/7
T
The functional dependance of
(3.43)
T
T
I
on
6/DT
in Eq. (3.43) may be
approximated by
9/7
(
)
[
- ]
10DT
~ .645(
+)
DT
(3.44)
IMMINININIII'l
-69-
Substituting this into Eq. (3.43) and the resulting expression for
I+
into (3.42) gives
B Z/D
=
.366 Re-. 277
-
.908
(3.45)
B/RePr
Neglecting the second term in Eq. (2.21) and using (3.45) to eliminate
6/D T results in
.13
B -Z/D
T+
30.23
Re. 78 Pr
w
(3.46)
1 - e
B/RePr
The Nusselt number is related to T+
w through their definitions
Nu
=
T
(T
w
s
(3.47)
1
TW
Therefore, from Eq. (3.46)
1 .13
B
Nu
=
.0331 Re. 78 Pr
RePr
(3.48)
RePr /DT
Equation (3.48) relates the wall superheat,
Tw - Ts
,
to the vapor heat
flux. Equation (3.1) is used to relate this to the total flux in cases
where the radiative and drop wall interactions are not negligible.
The group B/RePr is calculated at the dryout point and remains
constant. However, the Reynolds number dependance in the leading factor
-70-
of Eq. (3.48) results from the Reynolds number dependance of the temperature and velocity profiles and is not constant with axial position. The
Reynolds number may increase because of the increase in actual quality
or decrease because of increasing viscosity.
A simple expression for actual quality as a function of equilibrium
quality is obtained by substitution of Eq. (3.42) into (3.8) and using
Eq. (3.35) to write the result in terms of quality
- (xe -x
_"/q_
X
=
-
Xe
tot[
1
- e
b
(3.49a)
where
A
=
B Gh f
RePr 4q
34b
(3.49b)
tot
The parameter
A is closely related to the non-equilibrium parameter,
K, derived by Yoder [12] for use in the Local Conditions Solution
for fully developed flow. A comparison of their definitions leads to
(1-xb)7/12
A =)3/ 2
K x3/4
b
(3.50)
The Local Conditions Solution for fully developed flow is therefore
directly linked to the developing flow solution through Eq. (3.50).
A
comparison of Eq. (3.49) to the Local Conditions Solution is provided in
Figure 3.6.
From the figure it is seen that Eq. (3.49) may be used for
a change in equilibrium quality of about 10%.
This provides a simple
.7=.
.52
T
.3
2.-
.1
~ .2-
FIGURE 3.6a
.3
.q
.
7
3
Xe
Solution
Comparison of the Simple Model to the Local Conditions
-9
1.0
.0
Hull
Yoder
Hull
.5.
.E
.7
.9
Xe
FIGURE 3.6b
Comparison of the Simple Model to the Local Conditions Solution
-73-
criteria for switching from the developing flow model to the Local
Conditions Solution:
The developing flow model is used until the
boundary layer reaches the center of the tube or a change in equilibrium
quality of 10%, whichever occurs first. The two analyses have an overlap region.
The fluid properties are evaluated at the average temperature of
the boundary layer, Tbl
.
may be ob-
A simplified expression for Tbl
tained by using Eqs. (3.42), (3.43), (3.44), (3.45), (2.29) and an additional approximation
852
8/7
(
DT
[1 -1
)
(3.51)
.471 ( ' )
DT
15 DT
Thus,
.23
ReB
T+
bl
Re14.13
.07Pr
Z/DT
RePr
1 - B/RePr
2
(3.52)
T
When annular flow precedes dryout, the analysis of Hill [11] may be
used to calculate conditions at dryout. The necessary results are taken
directly from [11] and presented here for completeness.
The non-
equilibrium parameter, K , is found from
$1$2
Go
)
5/6
v x + 1)5/2
Pr2/3 Re4b 3
(
K = .0013 ~p
5/4_b5/12
7/4
1/12
)
(3.53)
-74-
557
4.0
!Th
{$1i bt
I+4A
I
6<t
10
10
4
T
"
5<i
44
4
I
-
;
4x10
4
-~.
-
44
u
--
14
7+4
l
tt
3
*
6x1
I
-ti
-
3.04
W2x10
'
Vt
--
- +1
--
t
3
2.0
2x1 0T
3
106XO
1j
II
:jf3t
Fu
tl;i
- Qi
+
250
iT-h}
4,
tit4
3
.44+
f ~ ~ r
,
I--
t
7ftI'
i
--
r
t~
+
--
-17-
-
--
-
-- ,-+
---..-.--
1.0
0.3
0.5 .
Figure 3.7
1.0
2.0
3.0
5.0
Distribution Function$
10
20
30
-75-
2.0
-
1.0
1.9
1.8
1.7
1.6
.90
1 1.5
1.4
.80
1.3
.70
1.2
.60
.50
.40
.30
1.1
1.0
10 2
Figure
3.8
14
We
Distribution Function $2
105
106
xb
-76-
where
2 are read from the graphs in Figures 3.7 and 3.8 and
I and
We
=
G2 x2D
bT
(3.54)
py i
.0338 xb
Rel1/8
b
(3.55)
to
Ghfg
The slip ratio is taken as unity and the void fraction calculated
from Eq. (3.34).
K
=.554(
In cases where the drop diameter is known
d( 5/4
T
q1 3/4
tot )
Pr2/3 Re12
Rb
Gh
(
5/12 12
(1-X b)
T
(3.56)
Xb1
The following is a summary of the calculation procedure.
It is
assumed that annular flow precedes dryout and that G, Xb, DT , and
q1ot
3.8.
are known.
Calculate
K using Eqs. (3.53), (3.54), (3.55) and Figures 3.7 and
Calculate
A from Eq. (3.50) and then B/RePr
from Eq. (3.49b).
Calculate the drop diameter from Eq. (3.56), and the void fraction from
(3.34) with
S = 1 .
The radiation shape factor,
Fwd , is now found
from Eqs. (3.10) and (3.11).
The drop-wall interaction may be written in terms of the void fraction and Reynolds number by combining Eqs. (3.13), (2.33b), and (3.33):
INUMUNIMINIIIINNI,
-77hfg
1Iw
pv
DT
-
.03
=
Pv
E(1-at)Re
7/
(3.57)
Equation (3.57) and the following equations are solved simultaneousTw, Tbl, X, X , Re, q",
ly to obtain
,
and qa
as a function of
Z/DT '
=
Re
-v
(3.40)
T
Re. 78
.0331
=
a
~ RePr Z/DT
+
qII
'1I
ot
Xe
+
=
1
qad
4q"I
+ Ghtot
fg
Xb
q"/q"_
X
=X
v
tot [1-
-A(X -xbI
e
e
~ RePr
s
RePr
(3.48)
(3.35)
Z/DT
B
bl
K
(3.1)
q1I
+
JT
1.13(Tw-T)
B/RePr
B
Pr Pr
i
B/RePr
(3.49)
b]
.23
Z/D
T
q" D
v T
KV
(3.52)
-78-
qrad
=
Fwd(T4
-
(3.9)
T )
The properties are evaluated at Tbl
.
Figure 3.9 shows the predicted
Nusselt number variation for a few specific cases.
-79-
160
140
B/RePr= 1.0
I2o
B/Re Pr = .1
100
80
G0
40
Pr
Ldw
=
c11~c
4
Figure 3.9
8
I
-1 0
~
0
1a
16
Parametric Behavior of the Simple Model
-80-
CHAPTER 4
EXPERIMENTAL APPARATUS
4.1
Introduction
This chapter contains descriptions of the apparatus and the pro-
cedures used to acquire the necessary data.
The objective of the experi-
mental work is to obtain data in the entrance region for dispersed flow
film boiling. Additionally, it is of interest to test the sensitivity of
the heat transfer in this region to the parameters appearing in the analysis:
vapor Reynolds number, drop diameter and quality. Figure 4.1 in-
dicates the conceptual design of the experiment.
To obtain some control over the drop diameter present at dryout,
it was decided to use an atomizing nozzle to produce the dispersion.
The
ratio of water to steam supplied to the nozzle determines the drop size
produced. To control the overall vapor flowrate, and thus the vapor
Reynolds number, the nozzle sprays the dispersion into a pipe carrying
an independently controllable flow of vapor.
To obtain a data set which
fulfills the objectives in mind, five sets of experiments were required:
1) Calibration of the flow measurement equipment.
2) Determination of set points giving flows at thermal
equilibrium at the dryout point.
3) Measurement of drop sizes produced by the nozzle.
4) Calibration for heat losses from the test section.
5) Single and two-phase heat transfer tests.
11=01101hiiii
,..U
-81
-
4ea4+e4
Nozzle
Atornizing
S1eamP"
Water
Primxxr
P
Figure 4.1
9em
Conceptual Experimental Design
-82-
As can be seen from Fig, 4.1, the experiments require two steam
flows and a water supply.
Figure 4.2 is a schematic of the system used
to deliver the required flows.
The assumptions used in the analysis of
dispersed flow film boiling, both that presented here and that of other
workers, become more questionable as the liquid loading increases.
As
a practicality, if the quality is much less than 10% in a "naturally
occurring" dryout, the liquid will probably be in the form of slugs or
ligaments rather than a dispersion of droplets.
It was therefore de-
cided to acquire data spanning a quality range of 10% to 50%.
The analy-
sis of Chapter 3 predicts that, at constant quality and drop diameter,
the augmentation of the heat transfer due to the liquid decreases with
increasing Reynolds number.
It was therefore decided to test Reynolds
numbers of 2x10 4 and 4x10 4 . These are on the low side for turbulent
flow, yet the factor of 2 is enough to adequately test the Reynolds number sensitivity of the analysis.
Pue to the practical reasons mentioned
in Section 4.4, a 7/8 inch tube diameter was chosen.
The above require-
ments allowed the maximum steam and water flowrates to be estimated to
aid in the sizing of the supply systems.
The following sections describe
the steam and water supply systems, the test sections, and procedures
used in the experiments.
The specifications of the instruments mentioned
in the text are listed in Appendix C.
4.2
Steam Delivery System
The steam available in the laboratory is at 60 psi with a maximum
demand of 3200 pph.
The flowrates needed by this experiment are compara-
VAC-(ckU?4 T
Figure 4.2
FLOW
SCHEMATIC,
-84-
tively small and 60 psi is enough to operate a commercial atomizing
nozzle.
Figure 4.3 shows the steam supply system. The wet steam is fed to
the separation tank on the left. The velocity in this tank is very low,
allowing the condensate to be separated. Also, the majority of the crud
is knocked down along with the condensate. All pipes and fittings carrying the steam used in the experiment are made of brass to reduce rust
problems.
Vapor is drawn from the top of the separation tank and split into
three lines.
One line is used for the outer side of the double pipe heat
exchangers, the other two for the two flows required by the experiment.
The heat exchangers and electric heaters allow the steam to be superheated slightly before being fed to the venturis for measurement.
It is difficult to prevent the steam in the venturi tap lines
from condensing and the differential pressure gauges must not be overheated, so the system shown in Fig. 4.4 was used.
Each tap line must be
level to prevent errors due to the condensate slugs which form. The
liquid head on each side of the differential pressure gauge is maintained
constant by the overflows.
The upstream pressure is measured at the
corresponding condensate pot, and the temperature and pressure are measured
downstream of the venturis.
Guard heaters prevent heat losses between the
venturis and the nozzle. The calibration and venturi system checks are
presented in Appendix D.
4.3 Liquid Delivery System
The liquid delivery system, Fig. 4.5, supplies a measured flow of
GUARP
HEATERS
--4
SEPARATION
TANK
DOUBLE PIPE
HEAT EXCRANGERS
STEAM
PRAIN
PRAIN
Figure 4.3
Steam Supply System
TO VENTURIS
PRESSURE GAUGE
-CONDENSING
POTS-S
OVEAFLOW
OVERFLOW
RE.SERVOl K
RESERVOIA
PIFF ER.E NTIAL
P-
P RE5URE
~RM
SUPERHEATERS
VENTU
SECTION
l
Figure 4.4
Steam Measurement System
GAUGE
TO NOZZLE
N ITROGEN
THERMOCOUPLE
ROTAMETER
VLNT
Q
)4
FiLTER
FILL
I00
ELECTRIC
HEATER
ELECT RIC
HEATER
Figure 4.5
Liquid Delivery System
PRAIN
-88-
water at a specified temperature and pressure.
All of the piping and
fittings are made of brass.
The electric heaters submerged in the tank are used to set the
temperature of the water in the tank just below the desired delivery
temperature. The heater between the filter and the rotameter is used
to make any necessary fine adjustments.
Regulated nitrogen is used to pressurize the tank.
The tank is
made from a 2 foot length of 30-inch diameter steel pipe welded to 1/2inch steel plates to close the ends.
It was designed to safely operate
at pressures up to 120 psi, but 60 psi was found to be a convenient
pressure to run at.
A rotameter was used to measure the flowrate and its calibration
is in Appendix E. The water temperature is measured at the delivery
point with a chromel-alumel thermocouple and the Omega trendicator.
4.4
Inlet Section
The atomizing nozzle chosen is a Spraying Systems Company 1/4
J-SU42.
This is intended to be fed with 1/4 NPT pipes and mounted in a
thick wall with a 3/4" NPT bushing.
The inlet section, Fig. 4.6, consists of four main components: a
mount for the nozzle, a mixing chamber, a bellmouth contraction, and a
porous tube.
Photographs of these components are shown in Fig. 4.7.
All
of the components are made of brass except the porous tube, which is
made of bronze.
The mixing chamber is made from a short length of 2-inch brass pipe.
-89-
1~~
I
SHAPE
2" brass pipe
MIXI NG
CHAMBER
I.]LI
NOZZLE
MOUNT
Figure 4.6
PRIMAKY
STEAM
HEAPER
Schematic
of
Inlet Section
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missing.
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is missing.
-93-
The bottom is threaded to fit into a 2-inch brass cross.
The primary
steam is fed to the bottom of the cross and the atomizing steam and
water are fed to the nozzle with 1/4-inch brass pipes that enter the
cross through the cross-flow legs.
The purpose of the mixing chamber
is to allow the spray produced by the nozzle to mix, and attain thermal
equilibrium with, the primary steam flow.
A disk serves as both a header plate for the primary steam and
a mount for the atomizing nozzle. There are eight 1/8-inch holes
drilled around the periphery through which the primary steam flows.
The center is drilled and tapped for the nozzle mount, 3/4-NPT. The disk
was press fitted into the bottom of the mixing chamber and then silver
soldered in place.
The bellmouth contraction was constructed according to the ASME
flow nozzle shape. The piece that the porous tube is mounted in was
machined to fit with the bellmouth contraction. These were silver soldered
together and then pressed into the mixing chamber and soldered in place.
It was decided to use a 7/8-inch test section diameter to avoid large
amounts of liquid deposition in the contraction.
The purpose of the contraction is to provide a uniform velocity profile at the entrance of the heated length.
The uniform velocity profile
was chosen as the entrance condition because a long velocity development
section would allow large amounts of deposition. This is undesirable for
two reasons:
The entrainment/deposition process affects the drop size
distribution and the presence of a liquid film on the wall will either
-94-
force the dryout quality to be very high or the required dryout flux to
be very high.
Thus, control over two of the major variables of interest
would be lost.
The purpose of the porous tube is to remove the liquid film just
before the start of the heated length.
The reasons for doing this are
much the same as stated above for the contraction.
The film would affect
the drop size and the dryout flux, and probably cause a quench of the
test section.
4.5
Liquid Film Removal System
As mentioned in Section 4.4, a porous tube is used to remove the
liquid film from the wall just before the flow enters the heated section.
A pressure drop must be applied across the porous tube to cause the liquid
to flow through.
Further, no vapor should be withdrawn and the liquid flow
must be metered. The system shown in Fig. 4.8 fulfills these requirements.
The flow from the porous tube is first passed through a sight glass
for visual observation.
flashing.
Next, a heat exchanger cools the water to prevent
The flow is measured with a rotameter (see Appendix E for the
calibration).
The water is then collected in the cold trap.
A suction
is applied to the cold trap with a vacuum pump. The pressure in the cold
trap is maintained at about 8 inches Hg vacuum by allowing the pump to draw
some air through the vent valve.
The method used to determine the point at which all of the liquid
film is being removed, and none of the vapor, can be understood with the
help of the pressure drop-flowrate characteristic of the film removal system, Fig. 4.9.
As the pressure drop is increased, bubbles are first ob-
VENT
SIGHT
GLASS
HEAT
EY4CHlANGER
ROTAMETER
-*
SUCTION
COLP
WATER
FROM
POROUS TUBE.
GOLD TRAP --
Figure 4.8
Film Removal System
-96-
STEAM BUBBLES
A
0
o
n0
An
AO
Qf
Cc/s
STEAM BUBBLE S
6-
0
OA
20
40
Figure 4.9
60
80
to
20
Pressure Drop-Flowrate Characteristic
for the Film Removal System
1+0
160
A P Cm HzO
-97-
served in the sight glass at the knee and this is associated with the desired operating point. Therefore, all that is necessary is to adjust the
pressure drop to the point at which the bubbles are first observed.
4.6
Set Points
To determine the best way to operate the apparatus, the following
facts were taken under consideration.
The test section pressure was always very nearly atmospheric.
steam in the nozzle is at a relatively high pressure.
The
Both the primary
and atomizing steam are slightly superheated to prevent condensation in
the venturis.
Flashing of the liquid will have a strong effect on the
drop size distribution of the spray. However, the performance of the
nozzle is unstable when cold water is used. Therefore, the water subcooling is determined such that neither condensation or flashing should
occur. The performance of the nozzle is steady in a fairly large region
around this point. Use of this set point should also ensure that the
flow is as close as possible to thermal equilibrium at the entrance of
the test section.
The set points, which cover the range of conditions mentioned in
Section 4.1, are tabulated in Appendix F. All of the experiments were
carried out with the apparatus operating at one of these set points.
4.7
Free Spray Test Section
The set point determination tests and the photographic study were
carried out using a free spray.
for the photographic tests.
Figure 4.10 shows the arrangement used
The spray issues out of the top of the inlet
ACTUATING
STEAM
4.
STEAM
EJECTOR
I4
I,,,
I
-
SPRAY
CAMERA
MICKOFLASH
PIFFUSE.R
Figure 4.10
Spray Photography Set-up
UNIT
-99-
section, travels across a gap of about 1-inch as a free spray, and is then
collected by the suction of the steam ejector. This allows optically undistorted visual observation.
The steam ejector was constructed from
standard pipe and pipe fittings.
The photographs were taken with diffuse back lighting.
was a Canon Fl and the lens a 135 mm macro.
flash duration of about .3 microseconds.
The camera
The microflash unit has a
Kodak plus-X film was used since
it has a very fine grain and a fairly slow speed. The photographs were
taken by setting the camera on bulb, opening the shutter and then firing
the strobe by hand. This ensured that the shutter was fully open when
the light pulse reached it. Clouding of the film was prevented by dimming
the lights in the laboratory during the photographic sessions.
The film
was slow enough that total darkness was not required.
4.8 Heated Test Section
The heated test section is a 14.
inch length of 7/8 OD, .015 to
.018 wall thickness inconel 600 tubing, Fig. 4.11.
Inconel was chosen as
the test section material since its electrical resistivity varies little
with temperature. This allows the heat input to be assumed constant, even
when there are severe axial temperature gradients present.
The heat is applied by imposing a voltage across the electrical
connectors.
The electrical connectors are heavy pieces of copper flat
stock silver soldered to the outside of the tube. The power source was
the 15 kw DC motor-generator set available in the laboratory. The electrical power input was found by measuring the voltage between the electrical
connectors on the test section and the voltage drop across a shunt resistor
-100-
F
LI
ELE..TACAL
CONNECTOR.
DISTA NCE
T HERMOCOLUPLE
FROM BOTTOM
NUMBiER
LLZ
1:3.0
25
I 2.5
20
11.5
19
10.5
18
17
9.5
9.25
24
8.5
16
8.0
15
7.5
14.5 "
7.0
13
6.0
if
.5.5
5.25
5.0
i0
23
9
4.5
8
4.0
7
3.5
6
3.0
5
2.5
4
2.0
3
I.5
2
22
0-
).0
I ''
\
Figure 4-11
ELECTRIc-AL CONNECTOR
Heated Test Section
1
1
wlllalihw
-101-
SANTOCE L
THERMOC.OUPLE -
-1/4 INCH
ASBESTOS
1116 INCH
INCONEL -
ASDESTOS
MICA
STRINCt
GLASS TAPE
Figure 4.12
Thermocouple Attachment
ROPE
-102-
in the line. These voltages were recorded on the Perkin-Elmer minicomputer available in the laboratory.
The arrangement used to attach the chromel-alumel thermocouples to
the tube wall is shown in Fig. 4.12.
The thermocouple is electrically
insulated from the tube by a thin slice of mica, about .001 inch thickness, and then tied onto the tube with asbestos string.
A 1/4-inch dia-
meter asbestos rope was wound over the string to give the assembly more
strength.
This prevented the thermocouples from coming loose while the
test section was being installed into the apparatus.
The thermocouple
output was recorded on the minicomputer. Thermocouples 22, 23, 24, and
25 are connected to the Omega Trendicator so that the wall temperature may
be observed during the experiments.
A wooden box with asbestos end pieces was used to hold the powdered
(Santocel) insulation. This provided a minimum thickness of 1-inch of
santocel.
The expected temperature measurement error is a maximum of
about 50*F.
The method used to arrive at this estimate is discussed in
Appendix G.
To prevent warping of the test section during the experiments, the
tube was put into tension.
This was done by first weighing the complete
assembly: tube, connectors, box and insulation, and then applying a force
equal to the weight plus about 5 pounds symmetrically to the top electrical
connector with a spring scale. The top pipe fitting was designed to provide electrical isolation from the drain pipe and allow travel for the
thermal expansion of the test section.
-103-
An insert, Fig. 4.13, made from 3/4-inch OD type 304 stainless
steel, was used to prevent conduction controlled quenching of the test
section. This allowed stable film boiling to exist at heat fluxes considerably less than the critical heat flux.
This was necessary since the
top of the test section would melt at the higher fluxes.
The test procedure for the film boiling tests is as follows. With
no power to the test section, the flows are set at the desired set point.
The power is then set at the desired point. The liquid flow is turned off
momentarily. When the wall temperature near the inlet increases above
about 350*F, the liquid flow is turned on. The final set point adjustments can then be made and when the wall temperatures reach steady values,
about 15 minutes, the computer is started and the data recorded.
-104-
7/g"
OD 0.0I8 -0.O5ISWALL
INCONEL
GOO
TU BING
1/,/ v x
VI
VI:
O-RINCS
CoPPER
Jc
t
E ECTRI CA L
CON N ECTOR
"00
30/
O.00
STAINLES5
IN.S ERT
Figure 4.13
Inlet of Heated Test Section
WALL
STEEL
-105-
CHAPTER 5
EXPERIMENTAL RESULTS AND MODEL PREDICTIONS
5.1
Introduction
This chapter contains the results of the drop size measurements and
the single and two-phase heat transfer experiments.
The methods used to
extract the drop size information from the photographs is first discussed
and the results presented in graphical form.
The single phase heat trans-
fer data and the corresponding predictions of Chapter 2 are presented
along with the heat loss correction technique. Section 5.4 discusses the
sensitivity of
dispersed flow heat transfer to the major parameters:
Vapor Reynolds number, quality and the drop diameter. The comparisons of
the analyses of Chapter 3 and the data are presented and the differences
between the simplified model and the numerical solution discussed.
5.2
Drop Size Measurement
The accurate measurement of drop size distributions is a difficult
task. Average drop sizes obtained from measured size distributions, for
the same flow conditions, scatter considerably.
For example, the data of
Nukiyama et al. [26] show a factor of at least 2 scatter in the Sauter mean
diameter.
The purpose of the experiments performed for this study was to
show that the atomizing nozzle controls the drop size and to provide an
order of magnitude estimate of average drop sizes present.
Photographs of the dispersion at each set point were taken according
to the procedure outlined in Section 4.7; Figure 5.1 is an example.
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Page 106 is missing.
-107-
black line across the corner is a .032-inch diameter wire which provided
the scale factor (about 14x) and an easy object to focus on.
The focal
point (the wire) was set at a point about 1/4-inch from the front of the
spray. The depth of field of the lens was about 1/2-inch so this arrangement prevented any out of focus drops from obstructing the view of those
that are in focus.
The number frequency distribution was obtained by counting the number
of drops between selected size pairs and fitting an assumed distribution
function to these data, Fig. 5.2.
During the counting process, only those
drops which were objectively in focus were counted.
It was generally
possible to discern a difference of 1/32-inch in the drop diameters seen
in the pictures.
Therefore, at least four data points could usually be
obtained. The exceptions are the cases in which very small drops were produced by the nozzle. When the drops are too small, it is easy to confuse
them with the photographic grain.
The distribution function chosen is a specialization of the function
used by Nukiyama et al. [26] to fit their air atomized sprays.
This function
was also used by Cumo [27] to fit drop size distributions found in mist flows.
Thus,
n
dN
=
Ad e-cd
(5.1)
A least squares curve fit of the data to this equation yields the constants
A and c.
-108-
Figure 5.2
Histogram for Drop Sizes
IMNIIIN
MINNININ
liilmlmlliilllmllii
611MINIIINININIMIMI
Wili
-109-
The average drop sizes are obtained by integration of Eq. (5.1).
Since the minimum diameter was not measured and since there is no
assurance that the largest drop was seen in any given picture, the limits
of integration were assumed to be
0 and
co
. Thus, the total number of
drops is
ndd
N =
=
A/c2
(5.2)
0
The number frequency distribution is, from Eqs. (5.1) and (5.2),
c2 d e-cd
n
(5.3)
The count mean diameter, or most probable drop diameter, is the maximum
of Eq. (5.3),
d
=
1/c
(5.4)
The area mean diameter is found from
N7Td
=
f0nlTd2dd
(5.5)
Thu0
Thus
d2
=
(5.6)
(56c)
-110-
The volume mean diameter is found from
3
N Id
6 3
n
J6
=
(5.7)
7Td3dd
0
Thus
d
=
r 4/c
(5.8)
Figures 5.3 to 5.14 show the frequency distributions and average sizes
obtained for flow conditions spanning the range considered in this work.
Nukiyama et al. [26] showed that the ratio of flowrates fed to an
atomizing nozzle has a major effect on the average drop size produced.
The same was found to be true in this study. The volume mean diameter is
plotted as a function of the ratio of atomizing steam mass flow to water
input mass flow in Fig. 5.15. The data shown cover all of the conditions
tested and are seen to be independent of the vapor Reynolds number. Thus,
the data are reasonably correlated as a function of
d3
=
468( wa
x 100)- 7 .24
W1 iI
wa/Wli
,
only.
(5.9)
Figure 5.15 and Eq. (5.9) indicate that the nozzle controls the average
drop diameter present. The experiments where "the drop diameter was held
constant" were accomplished by holding the flow ratio wa/wli
constant.
The magnitudes predicted by Eq. (5.9) should not be taken too seriously.
As can be seen in Fig. 5.15, there is probably a factor of 3 scatter in
d3 for the same flow ratio.
-111-
0
x
=
Qg;
=
dla
da
=62.8""
= 74.0,7
=-.30
X
40
80
Figure 5.3
x10 4
A
Re
12.0
)20 Ibm/hr
160
Drop Size Distribution
ZOO
-112-
Re
=
2x104
\Ali = I2O Ibm/hr
'C)
dl 2
=
ds3
= 59,,A
.34-
A4
14
50o
12?
10
8
6
4
Z5
50
Figure 5.4
75
100
Drop Size Distribution
125
150
-113-
8
x
Re
4
WIL
120
d13
138K
X
.46
x
104
lbm/hr
117 o
6
5
4
3
50
100
Figure 5.5
150
200
Drop Size Distribution
250
-114-
0
x
Re
= 4-x-10+
120 Ibm/hr.
o
W I-=
d 2
= 39,g
20
d3
=46-5,-
X
=.45
10
20
40
Figure 5.6
60
80
Drop Size Distribution
100
Re. = Zxi10
60-
w'
q=29
so-
/7'A
10o
Z9'fp
lbw/Ar
K.30-
2.0-
/0-
A
0
oo
200
300
A100
Soo
d94
Figure 5.7
Drop Size Distribution
600
700
Re
=x
2.
xI
9o0-
2'0 b /r
dz =92.5
so
6/0
0
zo
0
-
00
160
zoo
Z'10
do)
Figure 5.8
Drop Si zeDistribution
280
.320
A-17
Re ='xlo"
bs/
= 2'O l0
rZi
'x = .47
diZ
= zo,I
d,3
=
27
0
x
100
200
300
'q00
500
d(t)
Figure 5.9 Drop Size Distribution
600
700
e ='xI 1
wa=290 lbmw/A~r
.27
60
C
da=/0.5
d3 =
s10
IZo
160
oo
d (P)
Figure 5.10
Drop Size Distribution
23
ZV0O
2PO
320
-119-
80
Re =2-x104
vr,=z360Ib/Mr
d: 23qp
d =276r
x
5:lz
20 -
10.
I0
00
200
Figure 5.11
300
'00
Soo
Drop Size Distribution
600
-120-
0
Re =
2 x 104
W; =
360 lbv/hr
X( = .Iz
d.
130A
d 3 =15 3
70
60
50
40
30
20
10
12D
Figure 5.12
160
z0
Drop Size Distribution
2.40
Z80
320
-121-
90
Re = 4fx1o
= 360 lbnV/r
X= .20
d.= 1 15 r
10
d
= ZO6 r
60-
50
>Cs
40-
30
20
i0
zoo
0
Fgr
()
Figure 5.13
Drop Size Distribution
:OO
/00-1
-122-
Re
t
360
X
lbv-n/hr
.2.0
70 j
9 .
da
60I
>04
L10
304\
7r
-5
I0
0IGO
Figure 5.14
Zoo
Drop Size Distribution
0
o
300 y
100)
A
0
0
O
'7
V
(00
Symbol
v- , (pp)
3r0
240
120
0
Creai x 100
Figure 5.15
Correlation for Volume Mean Diameter
As can be seen in Figs. 5.3 to 5.14, there are no data points to
verify the low end of the distribution. To estimate the magnitude of
error possible due to the lack of low end data, the data were fitted to
a second distribution function:
n =ae-cd
(5.10)
The estimates of d3 using Eq. (5.10) average 14% lower than those previously given. The individual decreases range from 1 to 25%, the largest
decreases occuring at the smallest average drop diameters.
This indicates
that the average drop size estimates for the cases with small drops may be
too large but the estimates for the cases with large drops are fairly good.
5.3 Single Phase Tests and Heat Loss Calibration
The single phase tests were used for comparison to the analysis
of chapter 2 and to provide a calibration of the heat losses from the test
section. This calibration was then used to correct for the heat losses in
the two-phase tests.
The electrical heat input was known from the voltage and current
measurements recorded by the computer. The actual heat input to the steam
was found by measuring the steam inlet and outlet temperatures and the
flowrate.
Thus,
nDTL q Tctalp = w c (Tout - T.n)
in
l oss
input
actual
(5.11)
(5.12)
11011MIN
Ij
-125-
Both of the temperature measurements were made well away from the test
section. The piping between the measurement points and the test section
was insulated and guard heated.
The temperature distribution in the steam
should therefore be flat and the measurement a good estimate of the bulk
temperature of the steam.
Due to the insulation arrangement, pipe fittings, and electrical
connectors on the test section, the heat loss would be expected to be uneven.
Regardless, the losses should be roughly proportional to the differ-
ence between the average temperature of the test section and the room temperature:
Uloss Tavewall
oss
-
Troom)
(5.13)
The loss coefficient is also expected to be a (slight) function of the
temperature difference. The loss coefficient found from the single phase
tests are plotted in Fig. 5.16. A least square curve fit to these points
gives
U
where Tavewall
=
.0085(Tave,wall - 80) - 2.65
is in F and
Uloss
(5.14)
is in Btu/hr/ft 2/OF. The loss
coefficient is correlated with a scatter of about 12%. The heat losses
were a maximum of 25% of the total heat input in the two-phase tests.
Therefore, the actual heat input for the two-phase runs is known to within
about ±3%.
-126-
"p4)
6
~
.4
0
5
V
700
800
800 900
P000
900
1000
TC
Figure 5.16
1100
-80
Loss Coefficient
F)
1200
j300
-127-
Figures 5.17 and 5.18 show comparisons of the prediction of Chapter
2 to the single phase data. The heat input to the flow was calculated from
the temperature rise of the steam (not the loss coefficient).
5.4
Parametric Behavior
The dispersed flow heat transfer tests were run according to the
procedure outlined in Section 4.8.
The tests resulted in wall temperature
as a function of axial position for two vapor Reynolds numbers, qualities
between 50 and 10% and a case with large drops and one with small drops
for each.
It was not possible to predetermine the heat flux.
The sensitivity of the heat transfer to changes in quality, drop
diameter and vapor Reynolds number can be seen by plotting Nusselt numbers
as a function of axial position.
The Nusselt number chosen for the compari-
sons is defined as
Nutot
where
qt
tot
T
(5.15)
is the total heat flux to the flow (corrected for the heat
losses).
Figure 5.1.9 shows that the sensitivity to changes in quality at
constant drop diameter and vapor Reynolds number is small.
Figure 5.20
shows that the sensitivity to changes in drop diameter is somewhat larger.
Figure 5.21 shows the sensitivity to changes in vapor Reynolds number is
by far the largest.
Therefore, to obtain accurate estimates of the heat transfer to dispersed flow film boiling, it is most important to know the vapor Reynolds
1000
Tw *F
900
800
7,00
-DT
G = 1.0
x
104 )bm/ti+2k
0=9.3
x
103
Btu/-Ft-Inhr
= 7/8 inch
Ti= ZBO
.
4
Figure 5.17
6
Single Phase Test
8
10
12
14
ollilgAiiiii1wilil1w
WAIIIiI
I,
-129-
14 00r-
1300
Tw*F
M0o0
1100
1000 |-
104 pttz
1.5
900 -
,
Dr
Ti
=2. 1
=
10
4
7/8 inch
281 *F
800
700
Figure 5.18
x
Single Phase Test
-130-
200
Re =
XO
q
Ai
L --)(=.5
q.L
Q
I-o*
100
D
EFFECT OF QUALITY
AT
CONSTANT DROP/ DIAMETER
AND VAPOR REYNOLDS NUMBER
2
96
la
810
Z) INCH
Figure 5.19
Parametric
Behavior of Dispersed Flow
-131-
200
Re
a X /o
)(- ./0
S-
J,
=660
F -d,=130,A
EFFECT OF PROP PIAMETER
AT CONSTANT QUALITY AND
VAPOR REYNOLDS NUMBER
100
0 El
L
6
00
0
8
10
12-
2, INCH
Figure 5.20
Parametric Behavior of Dispersed Flow
-132-
0
RUN TPI6
Re = 4 x10 4
=
40,
=.45S
200
6
RUN TP20
Re
=
- -104
di= 40/(
>
.31
EFFECT OF VAPOR REYNOLDS
NUMBF-R AT CONSTANT DROP
PAMETER AND QUALITY
oo
A
D
OE[
1O f-1
El
r-1L
AAAA
Z
A
A
2, inc.h
Figure 5.21
Parametric Behavior of Dispersed Flow
-133-
accurately.
If only the total mass flux is known then this implies that
the dryout quality must be known accurately. The second most important
parameter is the drop diameter.
In any application, the drop diameter
will not be known accurately and will probably be a function of the upstream flow history.
In higher pressure systems, the sensitivity to quality may be greater.
This is because of an expected sensitivity to void fraction. The radiation
view factor and the coverage area for the drop wall interaction increases
with decreasing void fraction. These tests were all at atmospheric pressure
where the density ratio is about 1600. Because of this large density ratio,
the void fraction changes little over the tested quality changes.
5.5
Dispersed Flow Data and Model Predictions
In this section the dispersed flow film boiling data are compared
to the models of Chapters 2 and 3. The wall superheat, as calculated by
the three models, (single phase, complete two-phase, and the simplified
two-phase) are compared to the data in Figs. 5.22 to 5.33.
A short vertical line at the end of the curve marks the point at
which the thermal boundary layer is fully developed,
6/DT
-
1
In most
cases the single phase and simple two-phase models did not predict fully
developed flow in the distance considered.
The single phase model was applied by assuming that the Reynolds
number was constant and equal to the vapor Reynolds number at dryout.
Further, the radiation and drop wall interactions were neglected.
The complete model was applied according to the procedure outlined
in Section 3.7.
The velocity profile was assumed to be fully developed and
-1342000
PHASE
LL
0
NUMERICAL
AND
SIMPLE MODELS
PAOPERTIES ATTu,
1500
RUN TPii
1000
c, -q.q x i0'
3. 0 X10''
Re
Ibm/f/ hr
Btu/h)/hr
2.xl0Y
.08q
43
375<,,
z.31
500
Z,INCH
Figure 5.22
Dispersed Flow Test
-135-
2.000
U-
SINGLE PHASE
NUMERICAL AND
SIMPLE MODELS
PRO PERTIES AT T
|5.00
SIMPLE, cs
(000
SI MPLL c,
RUN TPlO
G-=g.'9 X 10''
)bm/ fi/f
'=2.3 X o
6lu/fe'/hr
E00
Re= 2-X 10'
d 1 5 8,a
3 =Ibst,
Figure 5.23
K, =.013
K3 -. 10
Dispersed Flow Test
-1362000
PHASE
NUMERICAL AND
SIMPLE MODELS
PROPERTIES AT Ti
LL
I500
SIMPLE,
RUN TP Iq
1000
G =
5g,(1 oN&i
I he /f t'hr
I
10 Bfv/iP/hr
7t,= 2.. VX
Xb
.16
Re = 2-XIDO
/03m
a3
=
3057
K,=./0
K3 =.y2
500
2.
6
"Z,I NLIH
Figure 5.24
Dispersed Flow Test
-137-
2000
SINGLE PHASE
NUMERICAL AND
SIMPLE MODELS
PROPERTIES AT Tb
1500
SIMPLE,
100D
K
AALP ~L~
L
RUN TPH1
G = 5.3 X10"
I6m/ft"/hr
q;+m-Z.7 X 10
Etu/ft/hr
Xb
Re
19
-..
1.11 10
dS7=
C3, =(7e'ob
*6
)
K,
.027
K3 =.Z22
500
z
4
7,INC H
Figure 5.25
Dispersed Flow Test
-1382000
SINGLE
PHASE
6-
1500
.fSMPLE , da
'n,'n,'n
RUN TP20
G =2.7 x 0'O
1000
y .-25 XJO"
X6 =.31
Re =2X O
d,I Y2.
, K,
d3 = IZ.0M
,
.032.
k3 =2.
PROPERT1ES AT TI
0 0o
2
4
6
10
g
Z mch
Figure 5.26
Dispersed Flow Test
-139-
Z000
SINGLE PHASE
L0
NUMERICAL AND
SIMPLE MODELS
PROPERTIES AT T
1500 -
d3
SIMPLE) d
1000 -
RUN TP21
G = -. 5-S 10 4 lbmn//
= 2.4 x 104 BtLkt
500 -
/hr
2 /hr
x = 33
Re = 2 x 104
d I = a22,
7 K , = .0090
= .077
d1 = 63
, K 3
2.
+
ZE, inch
Figure 5.27
Dispersed Flow Test
-140NUMEICAL AND
SIMPLE MODE.LS
PROPERTIES AT Tb
1500
SINGLE PHASE,,,
IL
0
1000
Annnz
RUN
500
C
G,
TP)3
=2.9 x 104
=
4 x 104
= 70^ , K.056
=
dl 3
2.
Figure 5.28
i6-//e/hr
-19
Xb
Re
c
8.7 x 104
Btu/tl/hr
68
Dispersed Flow Test
K= .2.8
2O,
to
2
inch
-1412000
SINGLE PHASE
LL
NUMERICAL AND
SIMPLE MODELS
PROPERTIES AT Ti,
1500
SIMPLE
1000
RUN TP3
G= 5.5 X 10 I bm/iL'/kr
'=2..5 X1o0 8t A/hr/ff'
SO00
Re= 2X10
d,
3, =
K, =.007g
3
5q
K3 = .07/
2-.
Z,INCH
Figure 5.29
Dispersed Flow Test
-14?-
5 I NGLE PH4ASE
I500
LL
M PLE , d3
SI MPLE ,,A
1000 -,
NUMERICAL AND SIMPLE MODELS
PRO PERT IES AT T
RUlN TPI8
G=6.q X ty10 |bmfL1/hr
-:3.4 X10' Bru/W/hr
4
2.Y
Re =X iO
00-
d,
=
"7
c)3 = 13i
2
9
)
,
6
Z, INCH
Figure 5.30
Dispersed Flow Test
=
.33
- .0q0
111mm
NOW
114111111111
Wild
-143-
Z000
NUMERICAL ANP SIMPLE MODELS
PROPE-RTIES AT Tbi
TP6
RUN
F-
CG
=
-,
1500
X=
3.6 x 104 1bm/ftz/hr
). 9 x 104 B/hr/ft2
.45
4 x 104
Re
di = 22^
K, = .012da = 64A ) K 3 = .J0
1000
SINGLE. PHASE.
500
ZA nL Z
2
Figure 5.31
4
6
8
Dispersed Flow Test
10
-144-
2000
NUMERICAL ANP SIMPLE MODELS
PROPERTIES AT Tb1
tL
IF-
1500
SINGE PHASE
1000
n
6
I
RUN TP12,
500
Q
8.2.
x04
2.7
x
)04
x
10
Ibm/ftZ/hr
Btak/ftz/'hr
Xb = .21
Re
li
d3
a
4.3
-
48 ,
=- 138,
K,= .027
K
3
= ..
4
12
2,
Figure 5.32
Dispersed Flow Test
INCH
kW
,
-145-
2ooo
NUMERICAL ANP SIMPLE MODELS
PROPERATIES AT Tbl
U-
SINGLE PHASE.
1500
SIMPLE) d3
1000
n ins
4n,
n
n
RUN TP9
500
6.2.x 10" 1bst/W/hr
G
2. 8 x 104
9Jt
X6
2.
4
6
Re
.27
4 x 104
d3
40, ,
d(3
) 1 6 /Ak
8
8tu/
t
Dispersed Flow Test
hr
K=.02.4
. K3 = . ?-3
10
~ , inc.h
Figure 5.33
/
-146-
the calculation was carried out for both the count mean and the volume mean
drop diameters.
These drop sizes were chosen since they represent a span
of the size distribution.
The pairs of runs in which the drop diameter was
held constant were generally fit equally well by choosing the same drop size
for both. The cases with large drops were fit best with the volume mean
diameter.
However, use of the count mean diameter most often gave the best
fit to the data.
It is not clear that this has any physical implications,
and is discussed further in the conclusions, Chapter 6.
The simplified model was then used to predict the data. To provide a
consistent comparison, the same drop diameter which was used in the complete
model was used in the simplified version. The complete model always predicts lower wall superheats than the simple model, but the differences are
not very great.
The major difference between the simple solution and the complete
solution is that the complete model predicts shorter development lengths.
This is because of the suppression of the vapor temperature profile, Fig. 3.3.
The heat sink term in the energy equation tends to suppress the growth of
the boundary layer.
However, the suppression of the temperature distribu-
tion tends to increase boundary layer growth and this effect dominates.
Figure 5.34 shows the sensitivity of the predictions to the temperature
at which the fluid properties are evaluated.
In each case, all of the vapor
properties were evaluated at the indicated temperature.
Four temperatures
were chosen for comparison: saturation temperature, wall temperature, film
temperature (Tw + Ts )/2), and the average temperature of the boundary layer.
WN1011011111IN11W
Wmill
WiNifti.
-147-
S500
SINGLE
PHASE
d3 , PROPERTIES
I-
IDOD
&OL
RUN TPib
G = 3.7 X10
b6,/UL/hr
Z-= 3.0X10'
Btu/ML/hr
Re = .I Xl10
d., = 106 ,
K3 S=.50
goo I.
NUMERICAL MODEL
Z INCH
Figure 5.34a
Dispersed Flow:
Effect of Properties
-148-
PROPERTIES
EVALUATED
AT
SIMPLE MODEL
1500
TN#
Twa
66L
1000F
RUN TP11
B4/RJhr
s
.0(3
1' /
Re= 2. X /0
d =c
=375
,<K=.'6I
500
Z, rNC H
Figure 5.34b
Dispersed Flow:
Effect of Properties
-149-
The average boundary layer temperature was used in the previous figures
since it worked well for the single phase predictions and has some
theoretical basis. However, the two-phase models appear to be the most
successful when the film temperature is used.
that the film temperature be used.
It is therefore recommended
-150CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
1) The simplified solution yields reasonable wall temperature
estimates, but maybe in error in development length estimates.
2) The calculation of the loading parameter is the most questionable part of the analysis. Use of the count mean diameter usually resulted in the best prediction of the dispersed flow data.
There are
several possibilities of why it was necessary to choose such a small
drop:
a) It is possible that the droplet Nusselt number
should be increased due to the turbulent fluctuations in
the flow field.
b) It may be important to include the complete
distribution of drop sizes.
c) Different radial liquid flux distributions may
have been produced at different set points.
At high
liquid flowrates it is possible for the liquid to be
concentrated in the center of the pipe.
d) The drop size data may be slanted toward the
larger sizes, especially for the small drop cases.
e) A radial distribution of drop sizes may exist;
i.e. the small drops may collect near the wall.
NOUNHOW
-151-
3) Due to the suppression of the temperature distribution in
dispersed flow, it may be reasonable to evaluate the fluid properties
at a temperature higher than the average temperature of the boundary
layer. The chosen temperature would most likely be a function of the
liquid loading.
6.2 Recommendations
Additional experiments could be used to clear up the questions
raised in Conclusion 2.
1) There are techniques for producing mono-dispersions.
Therefore, it should be possible to compare the heat transfer
for a mono-dispersion to that for a wide distribution of sizes.
2) Run experiments in which the radial liquid flux distribution is controllable.
3) Make measurements of radial drop size distributions
to determine if the drops have a preferred location in the
flow field in dispersed flow.
4) Develop a technique to measure the radial vapor
temperature distribution.
5) Develop a technique for measuring the radial
velocity profile in dispersed flow.
If this analysis is to be used to estimate the heat transfer just
downstream of a quench, it may be necessary to include the effect of conduction in the tube wall material.
Additionally, it may be desirable to
eliminate the infinite heat transfer coefficient at the dryout point. This
could be done by extrapolating a transition boiling heat transfer
coefficient until it intersects with that predicted by this analysis.
mw
mw
NMIW111941
-153-
REFERENCES
[1]
Groeneveld, D.C., Gardiner, S.R.M., "Post-CHF Heat Transfer
Under Forced Convective Conditions", AECL-5883.
[2]
Bennett,A.W., Hewitt, G.F., Kearsey, H.A., Keeys, R.F.K.,
"Heat Transfer to Steam-Water Mixtures in Uniformly
Heated Tubes in Which the Critical Heat Flux has been
Exceeded", AERE-R 5373.
[3]
Era, A., Gaspari, G.P., Hassid, A., Milani, A., Zaveltarelli, R.,
"Heat Transfer Data in the Liquid Defficient Region for SteamWater Mixtures at 70 Kg/cm 2 Flowing in Tubular and Annular
Conduits", CISE-R-184, 1966.
[4]
Groeneveld, D.C., "The Thermal Behavior of a Heated Surface at
and Beyond Dryout", AECL-4309, 1972.
[5]
Forslund, R.P., "Thermal Non-Equilibrium in Dispersed Flow Film
Boiling in a Vertical Tube", Ph.D. Thesis, Massachusetts
Institute of Technology, December, 1966.
[6]
Hynek, S.J., "Forced Convection Dispersed Flow Film Boiling",
Ph.D. Thesis, Massachusetts Institute of Technology, 1969.
[7]
Koizumi, Y., Ueda, T., Tanaka, H., "Post Dryout Heat Transfer
to R-113 Upward Flow in a Vertical Tube", Int. J. of Heat
and Mass Transfer, Vol. 22, pp. 669-678, 1979.
[8]
Nijhawan, S.M., Chen, J.C., Sundaram, R.K., "Parametric Effects
on Vapor Non-Equilibrium in Post-Dryout Heat Transfer", ASME
80-WA/HT-50.
[9]
Chen, T.C., Ozkaynak, F.T., Sundaram, R.K., "Vapor Heat Transfer
in Post-CHF Region Including the Effect of Thermodynamic
Non-Equilibrium", Nuc. Eng. Des. 51 (1979), 143-155.
[10]
Groeneveld, D.C., Delorme, G.G.J., "Prediction of Thermal NonEquilibrium in the Post-Dryout Regime", Nuc. Eng. Des. 36
(1976), pp. 17-26.
[11]
Hill, W.S., "Dryout Droplet Distribution and Dispersed Flow Film
Boiling", Ph.D. Thesis, Massachusetts Institute of Technology,
May 1982.
[12]
Yoder, G.L., "Dispersed Flow Film Boiling", Ph.D. Thesis,
Massachusetts Institute of Technology, March 1980.
-154[13]
Yao, S., Rane, A., "Numerical Study of Turbulent Droplet Flow
Heat Transfer", ASME Paper WAM 1980.
[14]
Schlichting, H., "Boundary Layer Theory", 7th ed. McGraw-Hill
1979.
[15]
Deissler, R.G., "Turbulent Heat Transfer and Friction in the
Entrance Regions of Smooth Passages", Trans. ASME, November
1955.
[16]
McAdams, W.H., "Heat Transmissions", 3rd ed., McGraw-Hill, New
York, 1954.
[17]
Heinmann, J.B., "An Experimental Investigation of Heat Transfer
to Superheated Steam in Round and Rectangular Tubes", ANL6213, 1960.
[18]
Sparrow, E.M., Hallman, T.M., Siegel, R., "Turbulent Heat Transfer
in the Thermal Entrance Region of a Pipe with Uniform Heat
Flux", Appl. Sci. Res. Section A, Vol. 7.
[19]
Mills, A.F., "Experimental Investigation of Turbulent Heat
Transfer in the Entrance Region of a Circular Conduit", J.
Mech. Eng. Sci., Vol. 4, No. 1, 1962.
[20]
Sun, K.H., Gonzalez, J.M., Tien, C.L., "Calculations of Combined
Radiation and Convection Heat Transfer in Rod Bundles Under
Emergency Cooling Conditions", ASME Paper, 75-HT-64, 1975.
[21]
Liu, Y.H., Ilori, J.A., "Aerosol Deposition in Turbulent Pipe
Flow", Environmental Science and Technology, Vol. 8, No. 4,
April 1974, pp. 351-356.
[22]
Iloeje, 0.C., "A Study of Wall Rewet and Heat Transfer in Dispersed Vertical Flow", Ph.D. Thesis, Department of Mechanical
Engineering, Massachusetts Institute of Technology, February
1975.
[23]
Watchers, L.H.J., Westerling, N.A.J., "The Heat Transfer from a
Hot Wall to Impinging Water Drops in the Spheroidal State",
Chemical Engineering Science 21, pp. 1047-1056, (1966).
[24]
Kendall, G.E., Rohsenow, W.M., "Heat Transfer to Impacting
Drops and Post Critical Heat Flux Dispersed Flow", M.I.T. Heat
Transfer Laboratory Report No. 85694-100, March 1978.
[25]
Hildebrand, F.B., "Introduction to Numerical Analysis", 2nd ed.,
pp. 290-297, McGraw-Hill, 1974.
UMM1
111&1111
-155-
[26]
Nukiyama, Tanasawa, "Experiments on the Atomization of Liquids
in an Air Stream", Trans. Soc. Mech. Eng. (Japan), vol. 5,
18, 1939.
[27]
Cumo, M., Farello, G.E., Ferrari, G., Palazzi, G., "On Two-Phase
Highly Dispersed Flows", Trans. ASME, Nov. 1974, pp. 496-500.
-156-
APPENDIX A
AXIAL CONDUCTION NEAR DRYOUT
The most severe wall temperature gradients occuring in dispersed
flow film boiling result from the rapid change in heat transfer coefficient
near the dryout point.
The simple model depicted in Figure A.1 is used
to obtain an estimate of the distance beyond the dryout point over which
axial conduction is an important heat transfer mechanism.
The dryout point is the point
z = 0 .
For
heat transfer coefficient is assumed infinite.
z less than zero the
For z greater than zero
the heat transfer coefficient is relatively small.
The vapor temperature
is assumed constant and there is constant volumetric heat generation within
The applicable energy equation is
the tube wall material.
h
d2
dz
where,
w
(A.1)
= T-T
with boundary conditions
e
6 =
Ts - T
=
The solution of Eq. (A.1)
=
e
z = 0, 0 finite at
z =o
is
-/h/ktiz
0
at
. +
wt
-/h/kt z
- - (1 - e
)
(A.2)
*--
TV
h = oo
TS
h
Tt
wi
+-
Figure A.1
Axial Conduction at Dryout
iNSULATION
-158-
An estimate of the length over which conduction is important is therefore
Lcond
(A.3)
=
The order of magnitude of
h can be estimated from the McAdams
equation
Nu
=
.023 Re. 8 Pr'4
(A.4)
For steam flowing in a one inch tube with a Reynolds number of 104 at
500*F and atmospheric pressure
h
~ 10 Btu/hr/ft 2 /OF
The tube used in the experiments has a wall thickness of .015 - .018
inches and the conductivity of Inconel 600 is 9 Btu/hr/ft/*F. Therefore
Lcond
~
.4 inch
Since the development length is usually much greater than .4 inches,
it is reasonable to neglect axial conduction.
In addition, many applica-
tions involve flows where the Reynolds number is much larger than 104
and
will therefore exhibit negligible axial conduction effects over the majority
of the thermal boundary layer development length.
However, the effect of
-159-
conduction should be checked if the wall material has a high thermal
conductivity or a large. effective thickness.
-160-
APPENDIX B
INTEGRAL APPROXIMATION
In this appendix it is shown that the extrapolation of the turbulent
zone temperature profile to the wall results in negligible error in the
Pr / 1, the
evaluation of the integral Equation (2.8) but that, for
laminar zone profile should be used in the evaluation of the wall temperature.
Substituting the turbulent and laminar temperature and velocity profiles, Eqs. (2.9), (2.13), (2.14) and (2.22), into the integral energy
equation, Eq. (2.8), and carrying out the integrations results in:
z/DT
= 7.43 D+217
6 9/7
T
66
1-lOT
1203
77
1/7
T
13.4
DT
T
+ 1535
DT
+ 986
DT
21650
+
D+2
D+6/
DTT
DT
(Pr - 1)[I
687
-
16.7
DT
-) [1
( 61/7
T
657 Pr
DT
This result is compared to Eq. (2.22) in Fig. B.l.
6.7
DI
(B.1)
DT+
18
DT
Equations (2.21),
(2.26b), and (2.24b) were used to calculate the Nusselt number. The
approximation is seen to be within about 5 percent of the exact integral.
110
Nu.
Re
=
Pr
=.73
.667 x
)04
90
70
APPROXIMATE
JNTEGRAL
zEXACT
S34
INTEGRAL
5
6
7
8
9
Z/
Figure B.1
Effect of Laminar Sub-layer on Integral
T
-162-
The error made by extrapolating the turbulent zone profile to the
wall to obtain wall temperature may be more serious. The extrapolated
turbulent zone profile and the patched distributions predict the same
wall temperature when Pr = 1. The laminar zone and the extrapolated
turbulent zone distributions are compared for the case of a fully developed
boundary layer,
6/DT
I
, and
Pr = .71
in Fig. B.2.
In this case the
extrapolated turbulent profile overestimates the wall temperature by about
18 percent.
Therefore, the laminar zone distribution was used to calculate
the wall temperature in all of the predictions shown at the end of Chapter
2,
.014
.o2
.010
.008
.006
.004
.002
4
6
8
10
03x
Figure B.3
Effect of Laminar Sub-layer on Wall Temperature
-164-
APPENDIX C
INSTRUMENT SPECIFICATION
Perkin-Elmer minicomputer
12-bit accuracy
provides temperatures measurements to
within ± 1F
Omega Trendicator, Model 403A
1VF resolution
-165-
APPENDIX D
VENTURI CALIBRATION
The system shown in Fig. D.1 was used to calibrate the venturis.
The steam was condensed completely and the flowrate found by measuring
the volume of water collected and the corresponding time.
For each flow-
rate, the upstream pressure was measured at the corresponding condensate
pot.
The temperature and pressure of the steam were recorded at a position
several diameters downstream of the venturi and the upstream temperature
was assumed to be equal to the downstream temperature. The isentropic
flow equations, along with a velocity coefficient, were used to correlate
the pressure drop-flowrate data. These were cast in terms of the isentropic
expansion factor and the density at the inlet of the venturi, station 1:
w = cYA2
fl
(D.1)
4
1 - 8
k-l
P
Y
P2
2/k
2/k
=
-2/
k
P
4](
k
3 is the diameter ratio of the venturi and
heats for steam:
1/2
4
I2
(D.2)
k is the ratio of specific
k = 1.33.
The atomizing steam flow was measured with a 1/2-192 Barco venturi.
The expansion factor for this venturi is closely approximated by
PRESSURE GAUGE
-O
CONPENSING PoTS-,
OVEU\FLOW
RF-SERVOI K
vORFLOW
RESERVOI A
'
PIFFERENTIAL
PRE5$URE- CA UGE
on
IJ
VENTUKI
,,-CONPENSEPs
Figure D.1
Venturi Calibration
-167-
Y
=
(D.3)
.3898 + .6121 ( P2
over the range of operating conditions. A curve fit to the calibration
data, Fig. D.2, gives:
wa
= 5.0648 Y /p-AP
- 1.9942
The density is in lbm/ft 3 and AP
(D.4)
in inches of water to get mass flow
Cv , can be calculated from
v
Eqs. (D.1), (D.3), and (D.4) if it is assumed that Cv is a function of
rate in lbm/hr. The velocity coefficient,
Re
only and that
Cv
=
y is constant:
975
1
-
41
+
]
(D.5)
This is plotted if Fig. D.3 and is seen to have the same general shape
specified for ASME venturis.
Since the magnitude is near 1, it is concluded
that the venturi is reliably calibrated.
The same procedure of calibration and check of velocity coefficient
was done for the 3/4-390 Barco venturi used to measure the primary steam
flowrate. The relevant equations are
Y, = .3759 + .6257(
)
P
P1
wp
34.95 Y/p1AP
-
4.489
(D.6)
(D. 7)
-168-
28-
i/2
- 192
2.4-
20
d
8-
4
I
I
1
2
|
3
YNd"A,
Figure D.2
4
V
5
m9'mhH0
Venturi Calibration
-169-
1.0
.q7 5
Cv
W/v - M2
0.9
0.8 I
0.7
InRe
Figure D.3
Velocity Coefficient
-
-
-170-
cV
=
.936[1
3]
-
4.9x10
Re + 1
and the results are plotted in Figs. D.4 and D.5.
(D.8)
-171-
803/4 -390
70
50
-
40 -
30
-
20 -
10 -
3
2
Y-/
Figure D.4
Venturi Calibration
,
3 inch HZO
lbrn/f+
16
-172-
1.0
3/19 - 390
0.9
0.2
S9
nRe
Figure D.5
Velocity Coefficient
-173-
APPENDIX E
ROTAMETER CALIBRATION
Rotameters were used to measure the liquid input flowrate and the
film removal flowrate.
Both were calibrated by measuring the volume
collected over a known period of time. Also, they were calibrated with
water near the operating temperature. As expected, the rotameters show
a very linear characteristic. The film flow rotameter calibration is
shown in Fig. E.1 and is correlated by
Q[cc/s]
where
= .1606 X +
X is the reading.
.5657
(E.1)
The liquid input rotameter calibration data is
shown in Fig. E.2 and is correlated by
Q[cc/s]
where
= .5257 X - .07114
(E.2)
X is the reading. The point on the float at which the reading was
taken is shown on the corresponding graphs.
FILM
FLOW
ROTAMETER
TUBE: FP 1/2 - 17 - (A - 10/55
CUSVT 4OT60
FLOAT:-/Z
EO
K
READI NG
H E.RE
gi
U
10 1-
20
80
30
90
X (Reading)
Figure E.1
Film Flow Rotameter Calibration
100
35
(
-
ROTAMETER
LIQUID
SUPPLY
TUBE:
FLOAT:
FP-1/2.-2.7-GA-I0/55
T6-603A943
30 -
READING
L
I
HF-RE.
20
15-
10
10
20
30
Figure E.2
40
so
60
70
Liquid Supply Rotameter Calibration
80
90
100
-176-
APPENDIX F
SET POINTS
The set points used are tabulated in the following tables.
-1774 x 104
120 ibm/hr
198 OF
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
3
116 yi
Atomizing Steam Flow
Primary Steam Flow
P,, PSIG
d P mnH2.0
P,, PSIG
260
1.0452
97
3.4
4.4
T *F
P,, 16/ff
r
-60.5
Liquid Film Flow
.
READI NG
'T, *F
27
50
WoS l/r
38.9
Test Section Conditions
Wy Ibm/hr
68.7
W
XRe
81.1h
.459
4.02xl04
W,. 16,/hf
-178Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
:
Nominal Drop Size, d3
4 x 10'
120 ibm/hr
197 OF
62 y
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
READING
26
IT
*F
50
Wf Ibw/hr
37.6
Test Section Conditions
-179Nominal
Liquid
Liquid
Nominal
Reynolds Number: 2 x
120
Input Flowrate:
Input Temperature: 204
: 120
Drop Size, d3
101
lbm/hr
OF
yp
Atomizing Steam.Flow
Primary Steam Flow
Liquid Film Flow
READING
T *F
W; \bmVhr
29.5
50
41.8
Test Section Conditions
-180Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
3
2 x 10'
120 ibm/hr
204 0F
120 i
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
Test Section Conditions
-181-
2
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
.
Nominal Drop Size, d3
x
10'
120 ibm/hr
203 'F
62 p
Atomizing Steam Flow
Primary Steam Flow
P,, FSIG
6l/fI
AP, nH,0 1,
P,.PSIG
-
.7
Liquid Film Flow
READING
T
37.5
Wf Ib/hr
50
51.9
Test Section Conditions
Wy lb/hr
34.5
W Ibm/hr
68.1
X
.336
I
I W IL6/hrl
-I
260
.0364
8.7
Y,
T *OF
Re
2.02x10'
15.0
-1824 x 10'
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
3
240 ibm/hr
205 OF
189 yi
Atomizing Steam.Flow
Primary Steam Flow
P,, PSIG
4.3
P,F,. Plo
3.3
APm H,0
fee I M
.0451
95
T*F
-59.8
260
Liquid Film Flow
READING
40
I4
*F
Y
Wf Ibwvhr
54.4
50
Test Section Conditions
We ib./hr
WNMEMIIIM
110IM1111,
1011111MEM11111111
i'11111
-183-
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
4 x 10'
240 lbm/hr
205 OF
189 yi
3
Atomizing Steam Flow
Primary Steam Flow
P, PSIG
4.3
P,. PSIG
3.3
A P,MH1
95
P,
I6./* '
.0451
T, 'F
Y,
M
260
-
Liquid Film Flow
Test Section Conditions
W.
I&../hrI
I 59.8
-184Nominal
Liquid
Liquid
Nominal
Reynolds Number:
Input Flowrate:
Input Temperature:
:
Drop Size, d
3
4 x 10
240 lbm/hr
204 "F
108 y
Atomizing Steam Flow
Primary Steam Flow
P,, PsLG
3.2
P.. PSI G A inH'0
68
2.4
f,, I6./fi
.0422
T F
262
Liquid Film Flow
READING
| T50F
W4 \b"/hr
40
50
55.1
Test Section Conditions
Y,
-49.8
W,
l,,/hr
-185-
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
:
Nominal Drop Size, d
3
2 x 10'
240 ibm/hr
209 "F
297 y
Atomizing Steam Flow
Primary Steam Flow
P,, PSIG
1.6
P,. PIGI AP., mH,0
1.3
Pp, I6/ft'
26.5
Yi,
.965
T *F
263
.0383
Liquid Film Flow
READING
Wf Ibw/he
| 'TF
67.7
50
50
Test Section Conditions
Wy lbnm/hr
34.0
W
171.9
/
Re
.165
I
1 . 99x104
W, b../hrI
29.5
-186Nominal
Liquid
Liquid
Nominal
2x
Reynolds Number:
Input Flowrate:
240
Input Temperature: 208
- 98
Drop Size, d3
104
1bm/hr
F
V
Atomizing Steam Flow
Primary Steam Flow
P,, PSIG
.8
PF,
PSIGow
APmHzO
fe I
-7.6
1p'F
.0367
258
Liquid Film Flow
READING
50
IT
5
0
50
Wf \bm/hr
1
67.7
Test Section Conditions
Y
.988
VIe 160/hr
13.8
-187-
4 x 104
Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
360 ibm/hr
207 0F
203 y
3
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
READING
*F:
Wf I /hr
71
50
94.3
Test Section Conditions
-188Reynolds Number: 4 x 10'
360 ibm/hr
Liquid Input Flowrate:
Liquid Input Temperature 207 OF
Nominal
Nomival
Drop Size, d
3
140 y
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
READIN,
62
T *F
50
Wf
hm/hr
82.9
Test Section Conditions
-189Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nomipal Drop Size, d
2 x 104
360 ibm/h r
210 *F
364 y
3
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
READING
T0*F
76
50
Wf 1b/hr
101
Test Section Conditions
-190Nominal Reynolds Number:
Liquid Input Flowrate:
Liquid Input Temperature:
Nominal Drop Size, d
2
x
10'
360 lbm/hr
210 *F
364 y
3
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
Test Section Conditions
IMM111
lit
11
-191-
Nominal Reynolds Number: 2 X 104
360 ibm/hr
Liquid Input Flowrate:
Liquid Input Temperature: 210 OF
: 164
Nominal Drop Size, d3
Atomizing Steam Flow
Primary Steam Flow
Liquid Film Flow
|READING
7T,"F
69
50
Wf \bm/hr
92.2
Test Section Conditions
-192-
APPENDIX G
WALL TEMPERATURE ERROR ESTIMATE
The error resulting from the method of thermocouple attachment shown
in Fig. 4.12 can be estimated from the handbook values of thermal conductivity for the Santocel, the asbestos and the wood.
The thermal contact
resistances should fall within the range 10<hc<1000 Btu/ft 2/hr/*F. The
effective thermal conductivity of Santocel is .01 Btu/hr/ft/0 F when the
intersticial gas is nitrogen and is at an average temperature of 350*R.
This corresponds very closely to the conductivity of the nitrogen alone.
Therefore, the effective conductivity used for the Santocel in these
estimates was .03 Btu/hr/ft/*F. This corresponds to air at 800*F.
From Fig. B.1, the heat flux through the insulation is
Ul(Tmeasured - T )
ins
(G.1)
where
U
=
t
()+
asbestos
(z )
Santocel
+ (
(G.2)
)
wood
+
0
The temperature error is then estimated from
ATerror
qins/U2
where
U2
t
(G.3)
)-
mica
+ h
contact
MICA
I/hc
+/k
ASBESTOS
t/k
SANTOCEL
WOOP
t/k
t/k
ThT
I/ho
To
Figure G.1
Resistance Network for Temperature Error Estimates
-194-
Using these equations and the typical ranges of the values of the
properties discussed results in estimates of wall temperature error between
0 < ATerror < 50'F .
-195-
APPENDIX H
SINGLE PHASE DATA
The single phase data are tabulated in the following tables.
-196RUN SP 4
Mass Flux: 2.9xl03
Inlet Temperature:
Outlet Temperature:
Room Temperature:
Power Input:
I bm/ft 2 /hr
254
390
75
.42
*F
0F
*F
*F
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, *F
1.5
638
2
629
2.5
652
3
704
3.5
734
4
4.5
763
780
5
~
824
856
5.5
6
6.5
875
7
7.5
885
880
8
931
8.5
9.5
942
960
10.5
985
11.5
994
970
12.5
INA11111,1101
61
-197-
RUN SP 5
Mass Flux: 7.8xlO
Inlet Temperature: 275
Outlet Temperature: 45
80
Room Temperature:
1.2
Power Input:
1bm/ft 2 /hr
0F
0F
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
-198RUN SP 6
Mass Flux: 1.1x10
Inlet Temperature:
4
280
Outlet Temperature: 489
Room Temperature:
8Q
Power Input:
2.0
1bm/ft 2/hr
*F
0
F
*F
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, *F
1
--
1.5
903
2
966
2.5
1021
1098
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
1135
1174
1196
--
1252
1294
1318
1328
1315
1385
1397
9.5
1417
10.5
1448
11.5
1464
12.5
1450
-199RUN SP 7
1.4xlO1
Mass Flux:
273
Inlet Temperature:
Outlet Temperature: 421
Room Temperature:
'80
.6
Power Input:
l bm/ft'/hr
kF
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature,
1.5
633
2
653
2.5
3
683
736
3.5
762
4
789
4.5
802
5
5.5
6
6.5
7
7.5
8
--
843
873
889
896
888
--
8.5
948
9.5
10.5
964
11.5
996
12.5
979
989
"F
-200-.
RUN SP9
1.OX104
Mass Flux:
Inlet Temperature:
Outlet Temperature:
Room Temperature:
Power Input:
lbm/ft 2 /hr
0F
280
oF
398
.80 kF
kw
.92
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
Wall Temperature, *F
--
1.5
612
2
646
2.5
676
3
3.5
722
744
4
766
4.5
777
5
5.5
6
--
810
6.5
837
849
7
7.5
855
846
8
--
8.5
901
9.5
10.5
913
935
11.5
943
12.5
935
-201RUN SP 10
1.5x104
Mass Flux:
Inlet Temperature: 281
Outlet Temperature: 474
Room Temperature: .8Q
2.35
Power Input:
lbm/ft 2 /hr
OF
*F
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, *F
1.5
869
2
932
985
1061
2.5
3
4
1093
1131
4.5
1149
3.5
5
5.5
6
6.5
7
7.5
8
--
1202
1242
1263
1272
1253
--
8.5
1337
9.5
10.5
1359
1391
11.5
1406
1394
12.5
-202RUN SP 11
6.4x103'
Mass Flux:
250
Inlet Temperature:
Outlet Temperature: - 439
".8.0
Room Temperature:
1.17
Power Input:
lbm/ft'/hr
*F
kF
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1.5
2
2.5
3
3.5
4
4.5
Wall Temperature, *F
753
799
846
922
958
996
1015
5
5.5
1066
6
1107
6.5
7
7.5
8
8.5
9.5
10.5
11.5
12.5
1126
1134
1118
1186
1196
1219
1245
1257
1237
-203RUN SP 12
8.2x1O
Mass Flux:
Inlet Temperature:
Outlet Temperature:
Room Temperature:
Power Input:
1bm/ft 2 /hr
' 252
418
,..80
1.2
0F
*F
*F
kw
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1.5
2
2.5
3
3.5
Wall Temperature,
715
759
802
872
905
4
4.5
940
952
5
1002
5.5
6
6.5
7
1039
1056
1063
7.5
8
1048
1111
8.5
1121
9.5
1138
10.5
1165
11.5
1174
1160
12.5
"F
-204-
APPENDIX I
TWO-PHASE DATA
The two-phase data are tabulated in the following tables.
-205RUN TP2
FLOW RATES: ltm/hr
14.8
Primary Steam:
19.4
Atomizing Steam:
118
Liquid Input:
37.6
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.5
2.4x10'
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
1.5
2
2.5
3
Wall Temperature, F
1037
896
937
3.5
961
996
1012
4
1036
4.5
1038
927
1081
1108
5
5.5
6
6.5
1128
7
7.5
8
8.5
1137
1135
1186
1198
9.5
10.5
1217
1245
11.5
1262
12.5
1249
-206RUN TP3
FLOW RATES: ibm/hr
13.4
Primary Steam:
20.8
Steam:
Atomizing
240
Liquid Input:
42.7
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.6
2.5x10
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1081
1.5
929
2
2.5
1024
1071
3
3.5
1119
1129
4
4.5
1148
1144
5
5.5
1018
1165
1195
6
6.5
7
7.5
8
8.5
9.5
1203
1204
1192
1247
1253
1266
10.5
1293
11.5
1307
12.5
1292
Nkmlilfili
11111111111
hikuiflu
-207RUN TP4
FLOW RATES: lbm/hr
60.0
Primary Steam:
8.2
Steam:
Atomizing
120
Liquid Input:
33.8
Liquid Film:
HEAT INPUT
Electrical Power Input: 1.8
1.9x10
Corrected Heat Flux:
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
Wall Temperature, F
1.5
727
578
2
638
2.5
3
672
713
3.5
730
746
4
4.5
749
6
678
770
792
6.5
798
7
7.5
8
8.5
800
791
827
830
9.5
837
10.5
855
11.5
858
12.5
848
5
5.5
-208RUN TP6
FLOW RATES: lbii/hr
49.2
Primary Steam:
Atomizing Steam: 19.7
120
Liquid Input:
37.0
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
kw
2
Btu/hr/ft
1.8
2.0x10'
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
Wall Temperature, F
762
1.5
~584
2
609
2.5
622
3
642
3.5
651
4
4.5
666
668
5
5.5
609
690
6
6.5
707
7
716
719
7.5
715
8
8.5
742
739
9.5
713
10.5
251
11.5
229
12.5
224
-209RUN TP8
FLOW RATES: lbm/hr
51.6
Primary Steam:
Atomizing Steam: 18.7
240
Liquid Input:
52.9
Liquid Film:
HEAT INPUT
Electrical Power Input: 1.8 4
2.0x10
Corrected Heat Flux:
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
1.5
2
Wall Temperature, F
763
588
6
6.5
7
7.5
8
8.5
658
700
741
758
777
778
707
798
816
822
823
815
847
849
9.5
857
10.5
868
11.5
871
12.5
855
2.5
3
3.5
4
4.5
5
5.5
-210RUN
TP9
FLOW RATES: lbm/hr
51.6
Primary Steam:
Atomizing Steam: 18.7
240
Liquid Input:
52.9
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.7
2.8x10
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1017
1.5
2
764
854
2.5
907
3
3.5
963
983
4
4.5
1003
5
5.5
6
6.5
903
7
7.5
8
8.5
1003
1026
1048
1055
1055
1044
1083
1086
9.5
1091
10.5
1107
11.5
1110
12.5
1091
501101111441
-211RUN TP10
FLOW RATES: lbn/hr
Primary Steam:
Atomizing Steam:
Liquid Input:
Liquid Film:
19.3
15.3
360
44.0
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.5
2.4xl0'
kw
Btu/hr/ft
2
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1007
1.5
2
848
2.5
3
979
1033
3.5
1052
4
4.5
1077
5
5.5
954
1111
6
1141
6.5
1155
1158
7
7.5
8
8.5
933
1078
1149
1201
1209
9.5
1222
10.5
1245
11.5
1254
12.5
1236
-212RUN
TP1l
FLOW RATES: ibm/hr
Primary Steam:
29.7
Atomizing Steam: 5.03
Liquid Input:
360
Liquid Film:
44.6
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
3.5
3.1x104
kw
Btu/hr/ft2
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1236
1071
1.5
2
1151
1191
2.5
3
3.5
1242
1262
4
4.5
1294
1306
5
5.5
1158
1352
6
1401
1423
6.5
7
7.5
8
8.5
9.5
1435
1426
1499
1514
_1537
10.5
1570
11.5
1584
12.5
1562
110111101,10.11,11
1,
-213RUN TP12
FLOW RATES: ibm/hr
52.7
Primary Steam:
Atomizing Steam: 19.7
360
Liquid Input:
91
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.5
2.7x10"
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
951
1.5
2
705
2.5
3
787
837
895
3.5
917
4
4.5
941
945
5
849
5.5
975
6
6.5
995
7
7.5
8
8.5
1009
1011
1002
1041
1049
9.5
1057
10.5
1060
11.5
1056
12.5
1027
-214RUN TP13
FLOW RATES: Ibm/hr
59.07
Primary Steam:
Atomizing Steam: 11.56
360
Liquid Input:
65.6
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.7
3.0x10'
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
997
1.5
2
762
826
2.5
859
3
3.5
906
927
4
953
4.5
5
5.5
962
862
997
6
1031
6.5
7
1045
1052
7.5
1041
8
1095
8.5
1105
9.5
1118
10.5
1135
11.5
1133
12.5
1109
-215RUN TP14
FLOW RATES:lbm/hr
30.0
Primary Steam:
Atomizing Steam: 4.3
240
Liquid Input:
58.0
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
3.4
2.8x10"
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
Wall Temperature, F
1.5
1264
1117
2
1205
2.5
3
1255
3.5
1320
1345
4
4.5
1380
5
5.5
1226
1450
1500
6
6.5
1394
1527
7
7.5
8
8.5
1538
1522
1608
9.5
1651
10.5
1685
11.5
1712
12.5
1683
1625
-216RUN TPl5
FLOW RATES: ibm/hr
60.6
Primary Steam:
8.6
Atomizing Steam:
240
Liquid Input:
58.0
Liquid Film:
HEAT INPUT
Electrical Power Input: 3.5
3.5xl0
Corrected Heat Flux:
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
1.5
2
2.5
3
3.5
4
4.5
Wall Temperature,
1192
909
999
1049
1115
1138
1166
1177
5
5.5
1041
1216
6
1256
6.5
1273
7
7.5
8
8.5
1279
1262
1328
9.5
1356
10.5
1374
11.5
1378
12.5
1349
1339
F
-217RUN TP16
FLOW RATES: ibm/hr
59.07
Primary Steam:
Atomizing Steam: 9.31
120
Liquid Input:
35.1
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
3.03
3.1x10'
kw
Btu/hr/ft 2
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9.5
10.5
Wall Temperature, F
1094
844
923
968
1020
1035
1054
1059
946
1087
1115
1127
1130
1114
1167
1174
1186
11.5
1197
1199
12.5
1176
-218RUN TPI7
FLOW RATES: lbm/hr
24.3
Primary Steam:
Atomizing Steam: 10.6
120
Liquid Input:
37.6
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.7
kw
2
2.6xl04 Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1.5
2
1123
988
1052
2.5
3
3.5
4
4.5
1081
1129
1136
1157
1157
5
5.5
6
1013
1196
1229
6.5
1249
7
7.5
8
8.5
9.5
10.5
,1254
1238
1307
1319
1338
11.5
1356
1367
12.5
1340
-219RUN TP18
FLOW RATES:
55.2
Primary Steam:
Atomizing Steam: 13.6
240
Liquid Input:
42.7
Liquid Film:
ibm/hr
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
3.3
3.4x10'-
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1157
1.5
2
882
2.5
3
1024
1084
3.5
4
1101
1125
4.5
1130
5
1001
5.5
1158
6
6.5
1190
7
7.5
8
8.5
1204
1183
1245
1251
9.5
1262
10.5
1273
11.5
1265
12.5
1236
975
1203
-220RUN TP19
FLOW RATES: ibm/hr
Primary Steam:
Atomizing Steam:
Liquid Input:
Liquid Film:
23.3
10.5
240
51.0
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.9
2.7xl 0'
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1
1146
1.5
2
996
1074
2.5
3
1113
1168
3.5
4
1182
1207
4.5
1218
5
5.5
1084
6
19
1300
6.5
1319
7
1326
7.5
1311
8
1381
8.5
1394
9.5
1416
10.5
11.5
1440
1453
12.5
1428
-221RUN TP20
FLOW RATES: Ibm/hr
26.6
Primary Steam:
Atomizing Steam: 7.9
120
Liquid Input:
43.3
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.4
2.5x10'
kw
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
1
Wall Temperature, F
-
1.5
914
2
994
2.5
3
3.5
1039
1101
1116
4
4.5
1140
1145
5
5.5
-1185
6
6.5
1221
7
7.5
8
8.5
1243
1221
-1302
9.5
1321
10.5
1343
11.5
1348
12.5
1314
1238
RUN TP21
FLOW RATES: ibm/hr
14.81
Primary Steam:
Atomizing Steam: 19.5
120
Liquid Input:
50.3
Liquid Film:
HEAT INPUT
Electrical Power Input:
Corrected Heat Flux:
2.5
2.7x10
kw
2
Btu/hr/ft
AXIAL WALL TEMPERATURE DISTRIBUTION
Axial Position, inches
Wall Temperature, F
1.5
910
2
932
940
976
989
1012
2.5
3
3.5
4
4.5
1017
5
5.5
6
1064
1094
6.5
1118
7
1127
7.5
1118
8
--
8.5
1193
9.5
1218
10.5
1236
11.5
1252
12.5
1229