A DIGITAL COMPUTER SOLUTION FOR LAMINAR FLOW
HEAT TRANSFER IN CIRCULAR TUBES
by
MAR 13
Perry Goldberg
53
L I rf R A-R
SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREES OF
BACHELOR OF SCIENCE AND MASTER OF SCIENCE
IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February, 1958
Signature redacted
-
of the Author
Signature
~~0~~~~~~
Department of N9chanical Engir
C
'a.
Certified by
e ring, January 20,
1958
3ignature redacted
Thesis Supervisor
Accepted by
..
Signature redacted
.a
e
l./Ga
a
S
n
Chairman, Departmental Commit7 e on Graduate Students
0,
A DIGITAL COMPUTER SOLUTION FOR LAMINAR
FLOW HEAT TRANSFER IN CIRCULAR TUBES
by
Perry Goldberg
Submitted to the Department of Mechanical Engineering
on January 20, 1958, in partial fulfillment of the requirements for the degrees of Bachelor of Science and Master
of Science in Mechanical Engineering.
ABSTRACT
A need exists for determining solutions to the laminar flow heat transfer
problem for circular tubes in which the velocity and temperature profiles
are developing simultaneously. The equations describing the heat transfer
can be derived rather easily and must be solved using numerical or other
approximate techniques. Programs for the IBM 650 and 704 computers are
written to obtain heat transfer solutions using finite difference equations.
These programs and the solution methods are discussed in the present thesis.
Also presented in the thesis are numerical results obtained for various
Prandtl numbers and two different boundary conditions. The computer solutions are in agreement with the solutions calculated by W. M. Kays and
verified with experimental data.
Thesis Supervisor:
Title:
Warren M. Rohsenow
Professor of Mechanical Engineering
ii
ACKNOWLEDGMENT
The author is indebted to Professor W. M. Rohsenow, who supervised
the thesis, for his valuable comments, advice, and encouragement.
The
writer also wishes to thank Mr. B. Thurston of the MIT Computation
Center for his valuable assistance in running programs on the 704 electronic
data-processing machine.
This work was done in part at the MIT Computation Center,
Massachusetts.
Cambridge,
111
TABLE OF CONTENTS
Page
-Section
ABSTRACT . .
. . . . . . . . . . . . . . . . . .. . . .
i
. ..
ii
ACKNOWLEDGMENT
. . . . . . . . . . . . . .
LIST OF FIGURES . . . . . . . . . . . . . . . .
LIST OF TABLES
1.
INTRODUCTION
vii
. . . . . . . . . . . . . . . . . . . .
I
1. 1
Available Solutions
. . . . . . . . . . . . . . . . .
1
1. 2
Derivation of Physical Equations . . . . . . . . . .
5
. . . . . . . .
8
. .
9
NUMERICAL METHODS . . . . . . . . . . . . . . . . .
12
. . . . . . . . . . . .
12
. . . . . . . . . . . . . . . . . .
17
2. 21
Langhaar Velocity Profiles . . . . . . . . . .
17
2. 22
Parabolic Velocity
. . . . . . . . . . . . . .
19
. . . . . . . . . . . . . . . .
20
. . . . . . . . .
21
. . . . . . . . . . . . . . . .
23
. . . . . . .
24
. . . . . . . . . . . . . .
24
1.4 Objectives of the Report
..
2. 1
Finite-Difference Equations
2. 2
Velocity Profiles
2. 3 Boundary Conditions
3.
v
. . . . . . . . . . . . . . . . .. . .
1. 3 Boundary Conditions Considered . .
2.
. . .
..
. . ..
2.4
Local and Mean Nusselt Numbers
2. 5
Computing Machines
MECHANICS OF COMPUTER SOLUTIONS
3. 1
Logical Block Diagrams
. ...
3. 11
Velocity Block Diagrams
. . . . . . . . . . .
24
3. 12
Diagrams for 704 Solutions . . . . . . . . . .
25
3. 2
650 Program for Langhaar Velocity Profiles
. . .
29
3.3
704 Programs for Heat Transfer Solutions . . . . .
33
3.4
650 Test Solutions . . . . . . . . . . . . . . . . . .
37
iv
Page
Section
3. 5
4.
5.
704 Test Solutions . . . . . . . . . . . . . . . . . .
RESULTS OF STUDY......
39
... 42
...................
4. 1
Velocity Results . . . . . . . . . . . . . . . . . . .
42
4. 2
Heat Transfer Results
. . . . . . . . . . . . . . .
46
4. 3
Discussion of Computer Results
. . . . . . . . . .
63
CONCLUSIONS.
....
-68
..............
. . . . . . . . . . . . . ..
APPENDIX A.
GLOSSARY
APPENDIX B.
LANGHAARtS VELOCITY PROFILES
APPENDIX C.
BIBLIOGRAPHY . . . .. .
. . . . . . .
69
71
75
y
LIST OF FIGURES
F page No.
Caption
3
1
Available Constant Wall Temperature Solutions.....
2
Fluid and Surface Temperature Variations, Constant
4
Finite-Difference Approximation
5
Logical Block Diagram for Velocity Calculation
6
Block 00 for Velocity Calculation
. . . . . . . . . . .
7
Block 10 for Velocity Calculation
. . . . ..
8
Logical Block Diagram for 704 Solution . . . . . . . .
9
Block 00 for 704 Temperature Calculations
. . . . . . . . . . .
10
16
.
26
..
. .
. . . . . .
.
..
.
Temperature Difference . . . . . . . . . . . . . . .
.
Constant
27
.
Fluid and Surface Temperature Variations,
28
.
3
10
.
. . . . .
Wall Temperature
30
.
Fig. No.
31
Block 10 for 704 Mean Fluid Temperature Calculation
32
11
Generalized Velocity Profiles, Langhaar
45
12
Local Nusselt Numbers for Constant Wall Temperature,
Numerical Solutions
13
. . . . . . .
.
10
57
. . . . . . . . . . . . . . ...
Local Nusselt Numbers for Constant Temperature
Numerical Solutions . . . . . . . . . . . . . . . . .
Numerical Solutions . . . . . . . . . . . . . . . . .
Constant Wall Temperature Solutions,
Numbers
15a
59
Mean Nusselt
. . . . . . . .. . .
.. . .
..
.
14b
58
Mean Nusselt Numbers for Constant Wall Temperature,
.
14a
.
Difference and Langhaar Velocity Profiles,
60
Mean Nusselt Numbers for Constant Temperature
Difference and Langhaar Velocity Profiles, Numerical Solutions.
. . . . . . . . . . . . . . . . . . . . .
61
vi
Fig. No.
15b
Page No.
Caption
Semi-Log Plot of Mean Nusselt Numbers for
16
.
Solutions . . . . . . . . . . . . . . . . . . . . .
62
.
Constant Temperature Difference Numerical
66
Solutions for Parabolic Velocity Profiles and
Constant Wall Temperature . . . . . . . . . . .
vii
LIST OF TABLES
.
.
11
.
.
38
.
Numerical Solutions Presented in This Report for
.
40
.
1
Page No.
Caption
Table No.
.
41
5
Values of Langhaar Velocity Profiles Obtained on 650 .
42
6
Summary of Numerical Solutions Constant Wall
Langhaar Velocity Profiles . . . . . . . . . . . .
2
Comparison of 650 Computer Solution and Test
Solution for
3
Y =
2. 0
. . . . . . . . . . . . . . .
Test Solution and Computer Solution for Pr
= 1. 0,
Langhaar Velocity Profiles, and Constant Wall
Temperature
4
. . . . . . . . . . . . . . . . . . .
Kays' Solution and 704 Solution for Pr
= 0. 7,
.
.
Langhaar Velocity,
Pr = 1. 0 . . .
Langhaar Velocity, Pr
Langhaar Velocity,
49
= 2. 0 . . .
50
Pr = 5. 0 . . .
51
Langhaar Velocity,
Pr = 0. 5 . .
.
Summary of Numerical Solutions Constant Temperature Difference,
12
48
Summary of Numerical Solutions Constant Wall
Temperature,
11
Langhaar Velocity, Pr = 0. 7 . . .
Summary of Numerical Solutions Constant Wall
Temperature,
10
47
Summary of Numerical Solutions Constant Wall
Temperature,
9
= 0. 5 . . .
Summary of Numerical Solutions Constant Wall
Temperature,
8
Pr
52
Summary of Numerical Solutions Constant Temperature Difference,
Langhaar Velocity,
Pr = 0. 7 . .
.
7
Langhaar Velocity,
.
Temperature,
. . . . . . . . . . . . . . . . . . .
.
Temperature
.
Langhaar Velocity Profiles, and Constant Wall
53
viii
ture Difference,
55
Pr
Local Nusselt Numbers Parabolic Velocity,
= 5. 0 . .
.
Summary of Numerical Solutions Constant Tempera-
56
Different
. . . . . . . . . . . . . .
L Increments
17
Pr = Z. 0 . .
.
Langhaar Velocity,
ture Difference, Langhaar Velocity,
16
54
Summary of Numerical Solutions Constant Temperature Difference,
15
1. 0 . .
.
14
Langhaar Velocity, Pr
.
,Summary of Numerical Solutions Constant Tempera-
64
Mean Nusselt Numbers for Parabolic Velocity, Constant Wall Temperature,
A L
. . . . . .
...
and Varing Increments
. . . . . . . . . . . . . .
.
13
Page No.
Caption
Table No.
65
1.
INTRODUCTION
With the advent of the gas -flow heat exchanger, the solution of the
laminar-flow heat transfer problem for low density fluids in small diameter
tubes received added importance.
this problem exists.
However,
At present, no closed form solution to
solutions have been obtained for a fluid
with Prandtl number equal to 0.7 *and
for very high and very low Prandtl
number fluids where the flow can be idealized as fully developed or uniThese solutions are limited in their usefulness because they were
form.
obtained for special cases.
Interest in the past was directed toward the solution of the problems
associated with oil-flow heat exchangers.
When oils are the primary
fluid satisfactory solutions have been obtained because oils are high Prandtl
number fluids; the velocity profile develops much faster than the temperature profile, hence the idealization of fully developed flow is adequate.
For heat exchangers utilizing liquid metals the idealization of uniform flow is adequate because liquid metals have low Prandtl numbers.
Therefore, the temperature profile developes much faster than the velocity
profile.
However, when fluids with intermediate Prandtl numbers are
used, no such flow idealizations yield an adequate discription of the velocity
distribution that exists within a circular tube.
-Hence, a more accurate
discription of the flow must be used along with numerical or approximate
techniques to obtain useful solutions.
1. 1.
Available Solutions.
The majority of available solutions have been obtained for constant
tube wall temperatures.
However,
this does not restrict the use of these
solutions because Klein and Tribus2 have demonstrated that by using
-*These numbers refer to references in Appendix C.
step functions the constant wall temperature solutions can be manipulated
to yield the solutions for any wall-temperature distribution.
The available constant wall-temperature solutions are plotted in
Fig.
1.
The ordinate is the mean Nusselt number, the local Nusselt
numbers averaged over the tube length, and the abscissa is the inverse
of the independent nondimensional length variable that appears in the
energy equation, sometimes called the Graetz number.
All of these
solutions are based upon the idealization that the velocity and temperature
profiles are uniform at the tube entrance. , The following
is a list of the
available solutions that are plotted in Fig. 1:
1.
Two constant wall temperature solutions have been
proposed by Graetz, 3 one for uniform-velocity flow
and one for parabolic-velocity flow.
The uniform
velocity solution can be obtained for all Graetz
numbers.
However, the parabolic velocity solution
can be obtained only for Graetz number less than
100.
This solution involves an infinite series of
which only three terms are known.
For Graetz
numbers less than 100, the series converges rapidly
and three terms are sufficient.
But, for larger
Graetz numbers the convergence is slower and three
terms do not yield adequate results.
2.
Leveque3 developed a solution for parabolic-velocity
flow and high Graetz numbers by using a flat plate
solution as an asymptotic approximation.
This
solution is good only near the tube entrance, Graetz
numbers greater than 1000.
/00
-0*
80
60
-10
*10
GRAETZO
ELOCITY
-IivFORM
POHLHAU5EN
__
_
'EU
MODIFIED FOR
TUBE,
PPROXIMATION
-
PR= 0.7
-____
/0
__POHLH
AUSEN,
PLATE,
______FLAT
6
-
NORRIS AND STREID
-----
NTERPOLATION
NuMERICAL
LANGHAA
Pr=07-
-
--
-
20
BY WM.ifqy6,
VE LOCITY PROFILE, Pr= 0.7
SOLUTION
~
3
PARABOLIC VELOCITY
.20
.30 .40
.60
.so J.o
2.O
3.0
X/IL
FiG.I. AVAILABLE
W.0
I.0
aoi /o
20
Re Pr x /0-2
/
.10
CONVSTPSVT WALL T-EvPERATURE SOLUTIONS
30 40
60
S0
100
3.
Norris and Streid
have proposed that on a log log
plot, such as that in Fig. 1, a straight-line interpolation be made between the Graetz and Leveque
parabolic -velocity solutions.
4.
For any tube cross-section, when the velocity and
temperature are uniform at the tube entrance,
the
behavior near the entrance is approximated by the
laminar boundary layer on a flat plate.
Therefore,
at high Graetz numbers the Pohlhausen 3 flat plate
solution is a good approximation to the true solution.
For 0. 5
< Pr < 15 the Pohlhausen solution is
RePr
Nu
x/D
m
ln
(1)
4
4Nu.
1-mi
RePr
(x/D)
where the mean Nusselt number based on the initial
temperature difference Nu mi is
Nu
mi
0.664
Pr0 . 167
RePr 0.5
(x/D)J
The Pohlhausen flat plate approximation for circular
tubes can be improved by correcting the flow cross
Symbols are defined in Appendix A.
(2)
sectional area with the boundary layer displacement
thickness from the Blasius
3'
laminar boundary layer
solution.
5.
A numerical solution has been obtained by W. M. Kays
employing the Langhaar5 velocity profiles for a
fluid with Pr = 0. 7.
This solution obtained by
"hand". methods agrees well with experimental data.
A solution has been obtained by Sparrow using integral methods.
How-
ever, the accuracy of this solution is poor and, therefore, it is not plotted
in Fig. 1.
Solutions "2" and "4" are adequate for Graetz numbers
)
1000, but
inadequate for the real area of interest in heat exchanger design, Graetz
numbers 4 100.
All of the available solutions, with the exception of the
solution by Kays, are satisfactory only for very long or very short tubes
and for high or low Prandtl number fluids.
Therefore, to obtain adequate
design data in the laminar-flow region for gas-flow heat exchangers with
both temperature and velocity profiles approximately uniform at the tube
entrances,
a direct solution of the energy-balance differential equation
is necessary.
Numerical solutions of the energy equation for various Prandtl numbers
and for two different boundary conditions are presented in the present
report.
These solutions were obtained on a high speed digital computer
using numerical methods similar to those used to obtain the solution for
Pr= 0.7 byW.
1. 2.
M. Kays.
Derivation of Physical Equations.
When the only heat transfer mechanism in the tube is conduction,
when fluid properties are constant, and when conversion of mechanical to
thermal energy is neglected, the following differential equation is obtained
by applying the energy equation to an annular control surface in a space
bounded by cylindrical surfaces dx in length and of radii r and r + dr:
=Vc --+vrc -- (3)
kr -r ar rx Jr
+ k
r x2
4X
r
The terms to the left of the equal sign represent the radial and axial
conduction and the terms to the right represent the radial and axial
enthalpy flux.
This equation is reduced to
1 14
t
r Jr
r)
=
c
k
(4)
ax
or
.-Jr2
+
-r ar
k
(L)
2jx
if the following assumptions are made:
1.
The axial conduction terms are small compared
to the other transfer terms and can be neglected.
2.
The presence of the radial enthalpy term greatly
complicates the solution of the reduced equation
and because this term is small except at large
Graetz numbers the radial enthalpy flux can be
neglected without introducing very large errors.
The reduced equation, Eq. (5),
nondimensional variables
can be normalized by defining the
Ar
Tt 1
7 tI
V A
,
T
v
and
kx
LA
2
vmecr0
4(x/D)
RePr
Substituting, Eq. (5) becomes
.--.
++21-..-. JT
T
P2
R JR
V ...-TT
JL
(6)
An expression for the local Nusselt number Nu
is obtained by
equating the heat transfer rate per unit of length q as determined by the
difference between the wall temperature and the mean fluid temperature
with q as determined by the temperature gradient at the wall.
The
former expression for q is
q = hZlr0 At = hZrr0t
- Tm(7)
where h is the unit conductance for convective heat transfer.
The
latter expression for q is
= k21Tr
q = k2nr0 ~
r
wlr
wall
~~
0
JR
.
wall
(8)
The local Nusselt number is defined as
2hr 0
A hD
Nu
=.-.
Therefore,
by equating Eqs.
9
k
k
(7) and (8) and by solving for Nu
the
following result is obtained,
2
Nu
=
lJ R Iwall
(10)
TW - T m
The mean Nusselt number Nu
m
is the local Nusselt number
Hence,
averaged over the tube length.
L
Nu
Nu dL
=.
(11)
m L_0
After the temperature distribution T is determined at any cross
section L from Eq. (6), the mean fluid temperature is found by performing the following integration,
T
=2
VTRdR.
0
Finally, the local and mean Nusselt numbers are evaluated with
Eqs. (12) and (13).
1. 3.
Boundary Conditions Considered.
For the solution to the temperature equation the following two
boundary conditions are considered:
1.
Constant wall temperature.
2.
Constant wall-to-fluid temperature difference.
(12)
These two boundary conditions represent the extreems generally encountered in two-fluid heat exchangers.
The fluid and surface temperature
variations for both conditions are plotted in Figs. 2 and 3, with L the
normalized length variable as the abcissa.
Figure 2 shows that for
Condition I the difference between the wall and the fluid temperatures
Figure 3
is large at the tube entrance, but decreases as L increases.
shows that for Condition 2, both the wall and fluid temperatures increase
as L increases, but that the difference between the two remains constant.
Condition 1 is a good approximation of the wall temperature distribution
that occurs in condensers, evaporators,
and often in parallel-flow heat
exchangers when the two fluid capacity rates and heat transfer resistances
are nearly equal.
For many fluid to fluid heat exchangers,
such as
counter-flow heat exchangers utilizing two fluids with equal capacity rates
and heat transfer resistances,
Condition 2 has application.
When either
boundary condition is used, the primary information of interest is the
mean Nusselt number.
With Nu m, the fluid exit temperature and the over-
all heat exchanger performance can be predicted.
Therefore, the local
Nusselt number is of secondary importance.
1'. 4.
Objectives of the Report.
The objectives of this report are:
1.
To present numerical solutions for laminar flow
heat transfer in circular tubes.
2.
To describe the digital computer solutions for
obtaining these solutions.
The numerical solutions should help simplify the trial-and-error
heat-exchanger design proceedure.
Table 1 is a summary of the numerical
solutions presented in this report for fluid flow described by the
Langhaar5 velocity profiles.
i0
5URFACE
K
i.e-
T
PLUID
I .6 -I
E
/,0
7.0
.1
0.z 0.3 .4
0.5
0.6
Q7
0.8
4CX/)
-
ReP
Fi.2.
FLUID
AND SURFACC
TEMP1RATURE
CONSTA/IT
V..A7AT/O1N.S
WAL-
T-MPERAT-iUqE
6
5-1
T
-LVI
4
/
35
0.0
0./
o.3
02
o.4
0.5 0.6
0.7
0.8
(xID)
4
Re P-
Fi&.3.
FLU/Da
/qND
SURFAC.E
TMPERATURE-
CONSTANr
VAR/A r/OM.1,
rEMPERqATURE
ThFFERG'SCE
TABLE 1.
NUMERICAL SOLUTIONS PRESENTED IN THIS
REPORT FOR LANGHAAR VELOCITY PROFILES
(Temperature and Velocity Assumed Uniform at Tube Entrance)
Prandtl Numbers
Boundary Condition
1
0.7
0.5
1.0
2.0
5.0
2
Solutions have been obtained also for fully developed flow (parabolic
velocity profiles) and these solutions are compared with the Graetz analytic
solution to help determine the accuracy of the numerical methods.
Solutions have been obtained for Pr = 10 and Pr = 50.
However, when
Pr = 10 or Pr = 50, the velocity profile developes much faster than the
temperature profile and the idealization of fully developed flow yields
adequate solutions for Graetz numbers 4
100.
Therefore, these solutions
are not included in the present report.
The following programs are discussed in the report:
1.
A program for the 650 magnetic drum data-processing
machine for calculating numerical values of the
Langhaar velocity profiles.
2.
A program for the 704 electronic data-processing
machine for obtaining the constant wall temperature
solutions.
3.
A 704 program for obtaining the constant temperature
difference solutions.
4.
A 704 program for determining the parabolic velocity
solutions.
2.
NUMERICAL METHODS
When the velocity distribution is other than a constant or some
simple function of R and L, the normalized temperature equation, Eq. (6),
is difficult to solve analytically.
Therefore,
alternate solution techniques
are a necessity when the fluid flow is described by the Langhaar velocity
profiles.
The proposed solution method discussed in the present section
involves approximating the continuous system with an "equivalent"
lumped-parameter system.
For this finite difference method of solution,
the basic approximation is obtained by replacing a continuous domain
with a pattern of discrete points within the domain.
Instead of obtaining
the continuous solution for T within the domain, approximations to T are
obtained only at the isolated points and when necessary,
values,
derivatives,
intermediate
and integrals are obtained from the discrete solution
by interpolation techniques.
The remainder of Sec. 2 presents (1) the finite difference approximations used to reduce the normalized temperature equation,
(2) the
resultant finite-difference equations for computing the temperature T,
(3) a discussion of the Langhaar and parabolic velocity profiles,
methods used to evaluate the mean fluid temperature,
number,
(4) the
the local Nusselt
and the mean Nusselt number, and finally (5) a brief discussion
of the computing machines used.
2. 1.
Finite -Difference Equations.
After the continuous formulation of the physical system is deter-
mined, the discrete formulation is obtained by replacing derivatives with
finite-difference approximations.
For the present study the following
commonly used approximations are utilized:
1
S T
=------TR-AR, L + TR+AR, L
JR
ZA R
= --
Equations (13),
-\
+ O AR
(TR-AR, L - 2TR,L+ T R+AR,L
JT
JT-
- 1 T R,L+AL - TRL
JL
AL
+
(13)
(A-)
+(ALI
(14)
(15)
\
(14), and (15) are termed explicit finite-difference
formulae because the only unknown appearing in the derivative expressions
is the desired quantity, in this instance TR, L+AL, for which the calculation is being performed and once the relationship between derivatives
is defined the unknown quantity can be obtained easily.
The derivative approximations are derived from Taylor series
expansions and the last term in Eqs. (13) through (15) expresses the
order of the Taylor series remainder.
When the intervals AR and AL
are small enough the remainder behaves essentially like a constant times
--2
-2
AR or AL.
/-/Hence, the terms 0 (ZAR
and 0 (AL
are indications
of the size of the approximation errors.
When Eqs. (13),
(14), and (15) are substituted into Eq. (6),
22T
.- -+.1.-. =1V._._. 2T (6) vl
R2
R J)R
L
the resultant expression can be solved for TR, L+AL to obtain the relationship
TR, L+AL
f1 TR+AR, L + f2 TR, L + f 3 TRR-AR, L.
(16)
14
The coefficients f , f 2 , and f 3 are defined as
A
L
iAR
2R
VJ1R
1
1
f
2
Z AL
-2
1
AR
and
-
f3
3
lil
V
AL
AR
I1
AIR
2R
An examination of the expressions for f 1 and f 3 indicates that as R goes
to zero the two coefficients and, hence, the centerline temperature go to
infinity.
in any real situation the centerline temperature can
However,
never go to infinity.
Eq. (16) cannot be used for finding the
Therefore,
centerline temperature T
Furthermore,
because the temperature
distribution is radially symmetric, when R is zero
TO+AR, L =
(17)
O-AR,L
Hence, to determine the centerline temperature f
and f3 can be added,
thus eliminating the 1/2R terms and yielding the equation
TO, L+AL
4 T AR, L +2
TO, L
(18)
where f 4 is defined as
)(ZAL)
V
AR
The finite-difference equations, Eqs. (16) and (18),
imply that when
the temperature and boundary conditions are known at the cross-section
115
AB of Fig. 4, the solution for all points within the shaded triangle ABC
How-
can be obtained if the velocity values are known within the triangle.
ever, the continuous system is parabolic6 and has a single characteristic
perpendicular to the tube walls across which discontinuities can occur.
Therefore, 'before the solution can be determined at points such as C in
Fig. 4, the boundary conditions at cross-section C must be known.
This
result indicates that implicit finite-difference formulae, which behave
more like the continuous system than explicit formulae, would yield more
accurate results than explicit finite-difference formulae.
the implicit formulae are used,
However, if
1/AR simultaneous equations must be
solved at each AL. - Therefore, this type of solution is more complex
than the explicit method and is much harder to handle on a digital computer.
The explicit formulae are used to solve Eq. (6) and the increment in AL
is chosen small enough such that the errors introduced by the finitedifference approximations are small.
The solution to the finite-difference equation, Eq. (16), will converge as long as all the coefficients f 1 , f.,
and f 3 are of the same sign.
Hence, the values for AL and AR cannot be chosen independently.
For
all solutions, the radius R is divided into 10 equal parts and AR is 0. 1.
Therefore, the limiting value of AL is chosen so that the coefficient f 2 is
positive throughout the calculation.
Because the smallest value of V used
in the calculation is 0. 384, the limiting value of AL is
AL
=0.0019.
(19)
For all but one of the solutions AL is less than AL
The solution for
Pr = 0. 5 is obtained by choosing AL greater than AL
and letting the
calculation proceed until f2 becomes negative,
at which time the
16
rse
WAWL
5~u
Fi G. 4.
FINITE -PDiFEREfcE
APPRoxlMATION
ac We A/L L
17
calculation is ended.
The convergence constraint was not violated for any
of the solutions presented in this report.
2. 2.
Velocity Profiles.
Numerical solutions for laminar-flow heat transfer in circular tubes
are obtained for two different velocity distributions, the Langhaar velocity
profiles and the parabolic velocity profiles.
The solutions for parabolic
velocity profiles are used to estimate the accuracy of the numerical
solutions obtained for the Langhaar velocity profiles.
2. 21.
Langhaar Velocity Profiles.
The equation proposed by Langhaar for the velocity distribution of
a laminar flowing fluid in a circular tube is
V(Y,
where the parameter
R)=
I('R)
-
(20)
is some function of a nondimensional length
variable (4 x/D)/Re that is independent of the Prandtl number,
(21)
\
Re
/
4 x/D
y. =4
The I's appearing in Eq. (20) are modified Bessel functions of the
first kind and are defined as
2k+ p
I (W)=
.
k=0
k!(k+p)!
Therefore, Eq. (20) can be rewritten
(22)
V(
(L
Z~)
Zk
(k!) 2
k= o
(k!
= -k= 0
)
,R)
$0
2k
00
(23)
r2
k=0
k! (k + 2)!
or removing the k=0 term from the summation and adding parens
R2
k=1I
V( 2, R) =
k! j
.
k
2
'2
(2
k-- I
. k!
(24)
2
(k + 2)(k + 1)
Equation (24) is in a convenient form to program for a digital computer.
The velocity V( 3, R) is calculated from this equation on a 650 magnetic
drum data-processing machine.
The two parameters that affect the nondimensional-velocity distribution are the tube radius R and the parameter (4x/D)/Re which
represents the length from the tube entrance to some cross section x.
The radius R was varied from its value at the tube center line to its
value at the tube wall in increments of 0. 1.
0
and
Therefore,
R .1
A R = 0. 1 for all solutions.
The length variable L of Eq. (6) and the parameter (4x/D)/Re are
related by the Prandtl number,
19
4x
L
I
Pr
(25)
D
Re
Hence, to make the values of V as calculated from Eq.
(24) directly
applicable to a finite-difference solution of Eq. (6) without interpolation,
the increment in (4x/D)/Re is chosen to be 0. 001, and for any Prandtl
number the increment
. 001/Pr.
AL for a finite-difference solution is chosen as
Also. Langhaar's data indicates that the velocity profile is
fully developed at (4x/D)/Re equal to 0. 300.
Therefore
when the
stability conditions discussed in Sec. 2. 1 permit, the solutions are
calculated for values of (4x/D)/Re less than or equal to 0. 250, at which
point the temperature profile is sufficiently developed such that interpolation to the fully developed temperature solution yields adequate
results.
The actual values of J used on the digital computer are obtained
by first plotting the values tabulated in Langhaar's paper 5 then
selecting values of
'
for
(4x
0 .
=
0. 250
Re
at increments of (4x/D)/Re equal to 0.001.
2. 22.
Parabolic Velocity.
When the flow in a circular tube is fully developed the velocity
profile is parabolic and constant at each cross section.
The velocity
is a function of radius only
V = 2(1 - R )
(26)
and the coefficients fl, f 2 ' f 3 , and f4 can be expressed in terms of R,
and
AL.
By substituting Eq.
(26) for V in the definitions of fy
f'
A R,
f3'
and f 4 the coefficients for fully developed flow are obtained;
f
11
(29)
AL (
1
,.
AR
A--R2
2. 3.
Z7
R2)
AL
2A L
.
2R
1(28)
--
AR
f
R2
2 AL
1
f
(Z(l
1
2(1 - R2
A R
2R
(30)
1
2(l - R2
Boundary Conditions.
The numerical solutions presented in this report are obtained
assuming the velocity and temperature constant at the tube entrance.
The
nondimensional fluid temperature at the entrance is assumed always to be
one and for the constant wall temperature solutions the wall temperature
is assumed to be two.
For the constant temperature difference solutions
the difference between the wall temperature and the mean fluid temperature
is taken as one along the length of the tube.
When computing the constant
temperature difference solutions the wall temperature at any cross section
is obtained by adding one to the mean fluid temperature at that particular
cross section.
Z. 4.
Local and Mean Nusselt Numbers.
Once the temperature distribution is obtained at some cross-section L,
the local Nusselt number Nu
is determined from Eq. (10).
However, before
this calculation can proceed the mean fluid temperature and the temperature
The mean fluid temperature is
gradient at the wall must be evaluated.
R
T
But, R
is one.
m
=
R
max
2
(31)
RVTdR.
0
max
Hence,
Tm = 2
(32)
RVTdR.
0
The integral in Eq.
(32) is evaluated with Simpson's rule.
The resultant
approximation to the mean fluid temperature is
R= 0. 9
Tm = ZAR
m
R=0. 1
[(RVT)R- AR + 4 (RVT)R + (RVT)R+ AR1,
ARRR1R
(33)
R = 0. 1, 0. 1 + 2 AR, 0. 1.+ 4 AR, -.. 0.9,
where the subscripts refer to the radius at which the values of R, V, and T
are obtained to form the product (RVT).
To determine the mean fluid
temperature at any cross section L, the wall temperature at L is not needed
because the velocity is zero at the wall and, therefore, the term (RVT)R+
AR
is always zero when R = 0. 9.
Hence, for the constant temperature difference
solutions the mean fluid temperature is obtained at section L although the
wall temperature at L is unknown.
To evaluate the temperature gradient at the wall, the temperature
profile at any cross section L is approximated as the parabola
T =Twall + a(1
-
R) + b(
-
R)
(34)
and the gradient is obtained by first differentiating Eq. (34) with respect
to R, then letting R equal one.
The gradient is
(R
= - a.
(35)
R=
The constant a is found by evaluating Eq. (34) at two different radii,
R = 0. 9 and R = 0. 8, and solving the two resultant equations for a in
terms of T o8,'
T 0.9, and Twall;
a = 5 (4T 0 . 9 - T 0 . - 3 Twall)
8
(36)
and the gradient at the wall is
T
(R/R=I
With Eqs. (10),
= - 5(4T 0 .9
- T0.
8
- 3 Twall).
(37)
(33), and (37) the local Nusselt number at any
section L can be determined when the temperature profile at L is known.
The mean Nusselt number given by Eq. (11)
L
Nu
mL
is evaluated also with Simpson's rule.
Nu dL
J
X
Hence, Num is approximated as
(11)
L- A L
Nu
AL
m
(Nu
L- AL
3
+ 4Nu + Nu+
),
L+ AL)
L
(38)
A L
L =
A L,
3 AL, 5 AL,
L - A L,
where the quantities being summed are local Nusselt numbers evaluated
at the cross sections indicated by the subscripts.
2. 5.
Computing Machines.
The solutions to the temperature equation, Eq. (6),
and the velocity
equation, Eq. (24), are both obtained with the aid of IBM digital computers.
The temperature solutions are determined with the 704 electronic dataprocessing machine and the velocity calculations are performed with the
650 magnetic drum data-processing machine.
The former computer has
magnetic-core storage and over eight thousand storage registers with an
access time of 12 microseconds.
This machine stores information and
performes all of its arithmetic operations in the binary number system.
The 650 computer is a decimal machine, all information is stored
and all calculations are done in the decimal number system.
This machine
has magnetic drum storage with a capacity of 10, 000 or 20, 000 digits
and an access time that varies with the position of the read out heads with
respect to the position of the desired information on the drum.
3.
MECHANICS OF COMPUTER SOLUTIONS
This section presents a discussion of the programs used for determining (1) velocity values within a circular tube for laminar flow, (2) the
solution to the temperature equation, and,
(3) the values of local and
mean Nusselt numbers for a laminar-flowing fluid in a circular tube.
Because two different computers are used for these calculations, this
section can be divided into two distinct parts, one for the discussion of
the 650 program and the other for a description of the 704 programs.
3. 1.
Logical Block Diagrams.
To facilitate the programming of equations and, hence, the solution
of problems involving equations,
logical block diagrams are constructed.
These diagrams are obtained by placing arithmetic and logical operations
in boxes and connecting these boxes with vectorsthat indicate the direction
in which the calculation is proceeding.
The block diagram has two
purposes:
1.
It helps keep the over-all requirements of the problem
well in mind.
2.
It reduces the large problem that is difficult to visualize
and comprehend into a series of many smaller problems,
each of which may be readily visualized, understood,
and programmed.
The diagrams can be drawn at many levels.
Depending upon the complexity
of the problem being solved, a single box might represent one arithmetic
operation or a multitude of arithmetic operations.
3. 11.
Velocity Block Diagrams.
Because the calculation of the velocity from Langhaar's equation,
25
2
)ok
1 - R2k
.2
V(6, R)=
k=
Z
2
C12
+
2
, k! J
Wo
0
)2
. ...
,
(39)
(k + 2) (k + 1)
.k!
k= 1
2
2
involves determining values of two infinite series,
some criterion for
terminating both series must be established before any computation or
programming can proceed.
In Appendix B it is shown that all terms of
both series decrease monotonically for k greater than or equal to two.
Therefore,
because k is always two or greater, both series are terminated
by assuming that all fractional terms less than some quantity
neglected, where
A can be
A is defined as
A
A
-f
5x10.
The exponent f in the definition is one plus the number of significant
digits desired in the numerator and denominator of V.
Also, because the
number of significant digits in a quotient is equal to the number of
significant digits in either numerator or denominator whichever is smaller,
f plus one is the number of significant digits desired in the velocity V.
Figure 5 is the complete logical block diagram for the velocity
calculations.
The blocks labeled 00 and 10 in Fig. 5 are expanded and
redrawn in Figs. 6 and 7 respectively.
3. 12.
Diagrams for 704 Solutions.
The logical block diagram for the complete 704 solution, Fig. 8, is
composed of five major blocks.
ture solution,
These blocks represent (1) the tempera-
(2) the evaluation of the mean fluid temperature,
(3) the
26
READ
AN H 5ET
5ELECr y
AND
SET
'R=0
SET
|N v V = 0
SET
M=I
BLOCK 00
'YES
-R
VNK= -
-7
K.K
REPLACE'S
K
YES
SLOC< 10
xf2
VD - K!
(Kf 2}X+ 1)
R + 0.1
R EPLA CES
,q
DK
PCH
R$ V4 L
R EPLA C S
{YES
Lv
Fic. 5.
14K<
A
VIN
No
Lo&lciAL BLocK
DIAGFM
ro'
VELOCTrY CAqLCULArION
K 2
K!
RK R -RK- 1
FiG. 6.
B1LOCK
00
FOR VELOCITY
C A LCU L ATION
(i
Fi. 7.
(T K
2
ja)
(K+-
l)
BLOCr< 10 FOR
CA LCULATION
VELOCITY
determination of the temperature gradient at the wall, (4) the calculation29
of the local Nusselt number, and (5) the calculation of the mean Nusselt
number.
Each of these blocks can be subdivided.
However, the third
and fourth blocks represent straightforward algebraic manipulations and
the fifth block is very similar to the second block.
first and second blocks are expanded.
Therefore,
only the
Figures 9 and 10 represent the
temperature solution and the evaluation of the mean fluid temperature
respectively.
3. 2.
650 Program for Langhaar Velocity Profiles.
With the aid of the logical block diagrams Eq.
for the 650 magnetic drum data-processing machine.
(24) was programmed
An optimized
program was assured by subscribing to the rules of coding for the Symbolic
Optimum Assembly Program (S. O. A. P. ).
Because the parameter
X is generally a mixed number, the floating
point subroutines of the MIT Selective System (MITSS) were used.
Thus,
the need for scale factoring and for shifting decimal points either before
or after performing fixed point multiplication or division was eliminated.
In this manner, the time ordinarily needed for coding the problem was
reduced and the program was simplified.
However, no real time advantage
was gained because the floating point routines require more machine time
than the fixed point routines.
The assembled program is outlined here:
1.
Initially, seven values of X are read into the machine and
placed in appropriate storage registers.
2.
The first of the seven values of
Y
is selected and R is set
equal to zero.
3.
The selected value of X is operated upon to obtain the value
0
AND INITIAL
INFORMATION
T-EMPERATUR E DISTRIUTION
READ IN VELOCirY
L=O
SET
BLOCK
00
1
RL+AL
T,L+AL
REPLACE
BLOCK
7
'fl TR+AR,L + fa TR,L t+liTR-AR,L
4 TAR.,L +
T
R
0
2 'O,L
WITH TRL+4L
10
Tr
VT) R+AR+ 4(AVT)
+ (RV T R-AR]
R 0.I
BLOC/
20
-- 3
R
BLOCK
30
U/JT%
-'~Rl
flu
\
7- - .- rY-
1BLOCK
40
L-ATL
Nu,= 3
PRINT
AL
OUT
N
+ NVL.
T
"I
AND
L+2 AL
+
Nu4_
, +, L 7-y7 4aL ,
Nux,
iVlm
R EPLACES
L+AL
F~io...
LIiCAL fBLoerw
DIAGRAM
FoR
704
SOLUTION
31
PcMO
G.
~
OAL*L
4
Z 4L
7ARL
REPLACES
41 V
A R +Z
I7--
-A)
YK()(A)(?
RSL+AL
IR+,AR,L
R =.
OYE
Fi.
1BLOCK
00
o
R-ARL
TRL
704
NO
T EMP ERA TURre
C ALCULATIONS
32
S ET
R 0.1
(RvT)
R +2?R
R EFLACES
EIIR
-4(RVT)
(-vr)
T
=(RVT)RA2
- q(RvT)
+ (Rvr)
NO
1 qo.q
Yes
I7
r =PA
K
Fi. 10.
BLOCK 10 rom 704 MEAm FLu
Tr-eMPATURE
CA LCUL AT 1 ION
I
33
of V at the radius R and the result of this calculation plus
some identification is punched on an IBM card.
4.
The parameter R is indexed and control is transferred
back to (3).
After R has been indexed nine times, control
is transferred to the next part of the program step (5).
5.
The machine selects the next value of Y , sets R back to
zero, and transferres control back to (3).
This procedure
is repeated six times, thus using all seven values of 6 and
the program then continues with step (6).
6.
A new card is read into the machine and seven new values
Control is then
of X are placed on the magnetic drum.
transferred back to (2).
This procedure is repeated until
the read card hopper is empty, at which time a read light
is energized and the program is stopped.
All machine results obtained in this study were for
(40)
A= 5(10 4).
Hence, the calculated values of velocity V used for the temperature solutions.
are good to three decimal places.
3. 3.
704 Programs for Heat Transfer Solutions.
With the aid of block diagrams, Figs. (8),
(9), and (10), the heat
transfer problem was coded for the 704 electronic data-processing machine.
The program was written in the standard symbolic coding language agreed
upon by members of the Share organization so that the Share Assembly
Program (SAP) could be utilized.
All arithmetic operations are performed
on floating-point numbers with floating-point instructions and index
registers are used to count and regulate cyclic processes.
34
The 704 programs are composed of five major parts, the calculation
at any cross section L for (1) the temperature distribution,
fluid temperature,
(3) the temperature gradient at the wall,
Nusselt number, and (5) the mean Nusselt number.
(2) the mean
(4) the local
The assembled program
for the constant wall temperature solutions is outlined here.
Initially:
1.
The velocity data and all other starting information are
read into the machine.
Temperature Distribution:
2.
The coefficients f2 and f 4 are determined for R equal to
zero and the center-line temperature is evaluated.
3.
The radius R is indexed and the coefficients f , f , and f
2
3
are calculated.
4.
The temperature at the indexed radius is obtained and
control is transferred back to (3).
This procedure is
continued, always using the proper velocity values, until
the entire temperature distribution at section L is determined.
5.
The program then continues with step (5).
The temperature distribution at L replaces the distribution
at L - A L.
Hence, the first time through the program
the temperature distribution at A L replaces the starting
temperature distribution.
Mean Fluid Temperature:
6.
After step (5) is completed, the mean fluid temperature
is computed with Simpson's
rule by averaging the product
VTR over the tube cross section.
35
Temperature Gradient at the Wall:
7.
The gradient at the wall is evaluated by assuming that the
temperature distribution between R equal to 0. 8 and the
wall is a parabola.
Hence, the gradient is the derivative
of the temperatuire evaluated at R equal to 1. 0 and is
obtained from Eq.
(37).
Local Nusselt Number:
8.
The local Nusselt number at section L is determined by
dividing twice the temperature gradient at the wall by the
wall temperature minus the mean fluid temperature.
All
values of local Nusselt number are preserved in the
machine s-o that the mean Nusselt number can be determined at any section L.
Mean Nusselt Number:
9.
The mean Nusselt number at section L is obtained with
Simpson's rule by averaging the local Nusselt numbers
over the tube length between section L and the tube
entrance.
For the computer solutions, the local Nusselt
number at the tube entrance is assumed to be 30.
This
figure is obtained by evaluating Eqs. (10) and (37) at the
tube entrance.
However, this assumption appears to be
rather poor for many Prandtl numbers.
Hence, mean
Nusselt numbers are also evaluated on a desk calculator
with the values of local Nusselt numbers obtained on the
704 and with a starting mean Nusselt number at L = 0. 004
as determined from the Pohlhausen flat plate solution,
Eqs. (1) and (2).
Finally:
10.
After the calculations at section L are completed, the
values of L, T
, ( 4 T/a R)R = 1, Nu , and Nu
mx
written on magnetic tape.
m
are
Also, at some fixed interval
of L other than A L the temperature distribution TR is
placed on tape.
The length L is incremented and control
is transferred back to (2).
This cyclic process is re-
peated until f 2 becomes negative or until the desired
number of length increments
A L have been considered.
When the program stops, the magnetic tape that contains
the results is taken to special off-line equipment and a
list out of the results is obtained.
The constant temperature difference solutions proceed as the constant
wall temperature solutions with the following exceptions:
1.
After the mean fluid temperature is found, the wall temperature at L is obtained by adding one to the mean fluid
temperature.
This value of wall temperature then re-
places the value at L - A L.
2.
Because the difference between the wall temperature
and the mean fluid temperature is always one, the local
Nusselt number is obtained by multiplying the temperature
gradient at the wall by two.
The 704 program for the parabolic-velocity,
constant wall-temperature
solutions is much simpler than the two programs discussed in the preceding
paragraphs.
of R alone.
For the parabolic-velocity solutions, the velocity is a function
Hence, the f. functions do not need to be calculated at each
1
station along the tube length but can be determined at the first station, stored
in the machine, and used at all stations.
This simplification plus the fact
that a very small amount of velocity data need be stored in the machine
greatly reduces the complexity of the program for parabolic-velocity solutions.
3. 4.
650 Test Solutions.
To check-the assembled program and to determine whether or not
production running can be started, a test solution for
=
2, that was cal-
culated by hand, is compared with a trace of the 650 program.
The test
solution values and the values obtained from the trace are tabulated in
Table 2.
A comparison of these values reveals that the machine numbers
are in agreement with the test solution.
Therefore, it is assumed that no
errors were made in the final assembly of the program and production
running is started.
After completing the production runs a comparison of machine results
is made with some values of center-line velocity that are listed in
Langhaar's paper.
5
The following values are compared:
Langhaar
650 Results
(4x/D)/Re
1.9800
1.9799860
.227
1.9434
1.9434035
.156
1.8573
1.8572992
.095
1.3514
1.3514404
.010
When the 650 results are rounded off to four places, the machine values and
those of Langhaar are in exact agreement.
Hence, it is assumed that the
program accomplished what it was designed to agcomplish, namely determine the velocity V at various radii R and lengths L.
38
Table 2.
Comparison of 650 Computer Solution
and Test Solution for W= 2.0.
Program Position
End of 1st Loop
Computer Solution
Test Solution
-1. 6616667x10~
1
-1. 6616667x10
1
2nd Loop
-2. 0333333x10-2
-2.0333333x10-2
3rd Loop
-8. 8888900x10~ 4
-8.8888900x10
4th Loop
4. 4212963x10~4
4. 4212963x10 44
1.8573035
1.8572992
Final Value
4
704 Test Solutions.
3. 5.
Two test solutions are used for the 704 computations.
solution was obtained with a desk calculator for Pr
tube wall temperature.
of
=
The first test
1. 0 and a constant
Calculations were made for the first two increments
A L, that is, values were obtained for L =
A L and L = 2 A L.
This
solution is used to check the assembled program and after a satisfactory
machine solution is obtained production running is started.
The test
solution and machine results appear in Table 3.
The second test solution was obtained by W. M. Kays at Stanford
University and is used to verify the 704 program.
Kays computed a solution
for Pr = 0. 7 and a constant tube wall temperature with numerical techniques similar to those used in the present study.
Table 4 is a tabulation
of Kays' results and the 704 results obtained for Pr = 0. 7 and a constant
tube wall temperature.
From this table it is apparent that the machine
results agree very well with the test solution.
are not in exact agreement.
However, the two solutions
A large initial discrepancy is present
because the incremental length A L used for both solutions was not the
same.
For Kays' solution A L = 0. 001 was employed for L = 0 to L = 0. 010,
and A L = 0. 002 for L > 0. 010, whereas for the 704 solutions A L = 0. 00143
was employed for all L.
Kays verified his solution with experimental data.
Therefore, the foregoing discussion indicates that the 704 program is
correct and can be used to compute within the region of interest heat
transfer solutions for laminar flow in circular tubes.
40
Table 3.
Test Solution and Computer Solution
for Pr = 1. 0, Langhaar Velocity
Profiles, and Constant Wall Temperature.
L= .002
L= .001
.Test Solution
Radius
Computer Solution
Test Solution
Computer Solution
.Temperature
1. 00000
1 .00000000
1 .00000000
1 .00000
1 .00000
1.0&0000000
.99999985
0.2
1. 00000
.99999992
.99999
.99999992
0.3
1. 00000
1 .00000000
.99999992
0.4
1. 00000
.99999992
.99999
.99999
0.5
1. 00000
.99999992
1 .00000
.99999985
0.6
1. 00000
.99999992
1 .00008
.99999992
0.7
1. 00099
1 .00000000
1 .00080
1.00000000
0. 8
99999
.99999992
1 . 01124
1.01124864
0.9
1. 11430
1 .11431486
1 .21116
1.21118170
Q
.1. 00000
0.1
.99999985
Mean Fluid Temperature
1.01993
1.01992723
1.04155
1.04156030
Gradient at Wall
12.7140
12.7137023
10.8330
10.8326090
'Local Nusselt Number
25. 9451
25. 9444049
22. 6052
22. 6046752
.Mean Nusselt Number
16. o642
16.0637156
41
Kays' Solution and 704 Solution
0.7, Langhaar Velocity
for Pr
Profiles, and Constant Wall Temperature.
Table 4.
Kays
704
.004
18.46
17.45
.010
11.31
10.87
.020
7.90
.030
6.53
6.52
. 040
5.82
5.80
.050
5.34
5.34
.060
5.02
5.01
.070
4.75
4.7 6
.080
4.57
4.57
.090
4.42
4.41
.100
4.29
4.29
.110
4.18
4.19
. 120
4.09
4.11
. 130
4.03
4.04
.140
3.97
3.98
.150
3.91
3.93
.160
3.88
3.89
.170
3.85
3.85
.180
3.82
3.82
.190
3.79
3.78
.200
3.77
3.76
.
RePr/(x/D)
Nu
Nu
(4x/D)/RePr
Kays
704'
1000
17.44
17.58
400
200
7.77
11.28
11.25
133.
100.
9.12
9.09
80.
66.
7.95
7.93
57.
50.
7. 19
7.18
44.
40.
6.67
6.66
36.
33.
6.27
6.26
30.
28.
5.96
5.96
26.
25.
5.72
5.72
23.
22.
5.52
5.52
21.
20.
RESULTS OF STUDY
4.
In this section the results obtained for particular Prandtl-numbers
are presented.
Only those results useful in heat exchanger design have
been included, other machine results have been deleted.
Also, because
the results are useless without some indication of their accuracy, a discussion of the accuracy of the numerical methods is included in this
section.
4. 1.
Velocity Results.
The velocity calculation was made for twenty-five hundred different
positions within a circular tube on the 650 magnetic drum data-processing
machine.
The calculations for cross sections close to the tube entrance
used the most machine time, twenty-nine seconds per cross section.
The
shortest solutions used four and a half seconds for cross sections where
the velocity profile was very nearly developed.
The average solution
time was six and a third seconds and the total 650 time used for production
running was slightly over two hundred minutes.
A summary of the resultant velocity solutions appears in Table 5.
The values in this Table are good to at least three figures to the right of
the decimal point.
profiles.
Also, Fig. 11 is a plot of the generalized velocity
In this Figure,
V is plotted against the tube radius R with the
expression LPr as a parameter.
These profiles are invariant with
changes in Prandtl number.
The velocity values obtained from the 650 were used with the 704 to
compute heat transfer solutions for various Prandtl numbers and boundary
conditions.
Table 5.
Values of Langhaar Velocity
Profiles Obtained on 650.
(rounded off to three places)
4(x/D)
Re
RADIUS
0
0.1
0.2
0.3
0.4
0.5
.004
1.253
1.253
1.253
1.251
1.247
1.237
.010
1.351
1.350
1.347
1,340
1.326
1.300
.020
1.483
1.480
1.469
1.450
1.416
1.362
.030
1.577
1.571
1.554
1.522
1.472
1.396
.040
1.647
1.640
1.616
1.575
1.511
1.418
.050
1.703
1.693
1.665
1.615
1.540
1.434
.060
1.751
1.740
1.707
1.649
1.564
1.446
.070
1.790
1.778
1.741
1.677
1.584
1.456
.080
1.821
1.808
1.767
1.699
1.598
1.463
.090
1.846
1.832
1.789
1.716
1.610
1.469
.100
1.868
1.853
1.808
1.731
1.621
1.473
.110
1.887
1.871
1.824
1.744
1.629
1.477
.120
1.903
1.886
1.837
1.755
1.636
1.481
.130
1.916
1.900
1.849
1.764
1.643
1.484
.140
1.928
1.911
1.859
1.772
1.648
1.486
.150
1.938
1.920
1.867
1.779
1.653
1.488
. 160
1.947
1.929
1.875
1.785
1.657
1.490
.170
1.954
1.936
1.881
1.790
1.660
1.491
.180
1.960
1.942
1.886
1.794
1.663
1.492
.190
1.965
1.947
1.891
1.797
1.665
1.493
.200
1.970
1.951
1.895
1.800
1.667
1.494
.210
1.974
1.955
1.898
1.803
1.669
1.495
.220
1.978
1.959
1.902
1.806
1.671
1.496
.230
1.981
1.962
1.904
1.807
1.672
1.496
.240
1.983
1.964
1.906
1.809
1.673
1.497
.250
1.986
1.966
1.908
1.810
1.674
1.497
Table 5.
(continued)
0.6
0.7
0.8
0.9
1.215
1.162
1.036
.733
1.251
1.159
.983
.646
1.275
1. 138
.918
.565
1.284
1. 118
.876
.521
1.287
1.103
.847
.493
1.288
1.090
.826
.472
1.288
1.079
.807
.455
1.287
1.070
.793
.442
1.287
1.063
.782
.432
1.286
1.057
.773
.424
1.285
1.051
.765
.418
1.285
1.047
.758
.412
1.284
1.043
.753
.407
1.284
1.040
.748
.403
1.283
1.037
.744
.400
1.283
1.035
.741
.397
1.283
1.033
.738
.395
1.282
1.031
.735
.393
1.282
1.030
.733
.391
1.282
1.028
.731
.389
1.281
1.027
.730
.388
1.281
1.026
.729
.387
1.281
1.025
.727
.386
1.281
1.025
.726
.385
1.281
1.024
.726
.385
1.281
1.023
.725
.384
2.o
Re
.8.
.250
/.64
'.4
4/D-
K
V
v-rn
.8
L Pp=
Re
=.
004
.6-
.4-
2
/o
.6
FiG. /I.
.6
-4
GENERALIZED
.Z
6
.I
VS-LOCITY
.4
.6
.
1.o
PRqoF-1LEr, L-ANrH ARR
4. 2.
Heat Transfer Results.
The 704 electronic data-processing machine solutions are summarized
in Tables 6 through 15.
The constant wall temperature solutions appear in
Tables 6 through 10 and the constant temperature difference solutions appear
in Tables 11 through 15.
Solutions were obtained for Prandtl numbers
0. 5, 0. 7, 1. 0, 2. 0, 5. 0,
10. 0, and 50. 0.
However, the solutions for
Pr > 5. 0 are not presented in this report.
For Pr > 5. 0 the velocity pro-
file develops much faster than the temperature profile and the idealization
of fully developed or parabolic flow yields adequate results within the region
of interest in heat exchanger design.
Two columns of each of these Tables are devoted to mean Nusselt
numbers.
The values obtained from the 704, "704 Num", were obtained by
making the simplifying assumption that for all Prandtl numbers the local
Nusselt number at the tube entrance is 30.
This particular value was
chosen because it is consistent with Eqs. (10),
(12), and (37).
However,
because errors are propagated downstream and because near the tube
entrance finite difference errors are large, the mean fluid temperature
changes rapidly,
and the radial enthalpy flux, neglected in the reduced
temperature equation, may be of some importance, this approach does not
yield accurate results.
A more accurate method is to employ an approximate
solution in the vicinity of the tube entrance and then to use Eq. (12) and
the local Nusselt numbers to obtain the downstream mean Nusselt numbers.
The values appearing in the columns headed "Nu m" were obtained on a
desk calculator using the Pohlhausen flat plate solution to obtain a starting
mean Nusselt number at RePr/(x/D) = 1000.
The computer results are also summarized in Figs. 12, 13, 14, and
15.
The constant wall temperature results are plotted in Figs. 12 and 14,
417
Table 6.
Summary of Numerical Solutions
Constant Wall Temperature,
Langhaar Velocity, Pr = 0. 5.
4 (x/D)/(Re Pr)
Nu
.004
17.02
22. 40
.007
13.11
20. 35
.010
10.91
17.09
.020
7.91
13.05
.030
6.65
11.12
.040
5.93
-9.89
.050
5.46
9.05
.060
5. 12
8.42
.070
4.87
7.93
.080
4.67
7.53
.090
4.52
7.21
.100
4.39
6.93
.110
4.28
6.69
.120
4.20
6.49
.130
4.12
6.31
.140
4.06
6. 15
. 150
4.01
6.01
. 160
3.96
5.89
.170
3.93
5.77
.180
3.89
5.67
. 190
3.86
5.57
. 200
3.84
5.49
. 210
3.82
5.41
.244
3.76
5.18
. 278
3. 72
4.99
704 Nu
m
Nu
m
24.5
RePr/(x/D)
1000
571.4
17.84
400
200
11.41
133.3
100
9.24
80
66.7
8.06
57.2
50
7.31
44.4
40.0
6.78
36.4
33.3
6.38
30.8
28.6
6.07
26. 7
25.0
5.82
23.5
22.2
5.62
21. 1
20. 0
5.45
19.0
16.4
5.04
14. 4
Table 7.
4(x/D)/RePr
Summary of Numerical Solutions
Constant Wall Temperature,
Langhaar Velocity, Pr = 0. 7.
.Nux
704 Nu
m
. 004
17.45
22. 79
.007
13.13
19.75
.010
10.87
17.26
.020
7.77
13. 11
.030
6.52
11.10
.040
5.80
9.86
.050
5.34
9.00
.060
5.01
8.36
.070
4.76
7.86
.080
4.57
7.46
.090
4.41
7. 13
.100
4.29
6.85
.110
4.19
6.61
.120
4.11
6.41
.130
4.04
6.24
.140
3.98
6.07
.150
3.93
5.93
.160
3.89
5.80
.170
3.85
5.69
.180
3.82
5.57
.190
3.80
5.49
.200
3.78
5.41
.210
3.76
5.33
.280
3.68
4. 92
.350
3.65
4.67
.Nu
m
-23. 8
(RePr )/(x/D)
-1000
571.4
17.58
400
zoo
11.25
133.3
100
9.09
80
66.7
7.93
57.2
50
7. 18
44. 4
40.0
6.66
36.4
33.3
6.26
30.8
28.6
5.96
26.7
25. 0
5.72
23. 5
22. 2
5.52
21.1
20.0
5.35
19.0
14. 3
4.69
11.4
49
Table 8.
4(x/D)/(RePr)
Summary of Numerical Solutions
Constant Wall Temperature,
Langhaar Velocity, Pr = 1. 0.
.Nu
704 Nu m
..004
-17.47
.22. 96
.007
13. 04
19. 61
.010
10..72
17. 22
.020
7.60
13.04
.030
6.36
10.99
.040
5.66
9.74
.050
5.21
8.88
.060
4.89
8.24
.070
4.65
7.74
.080
4.46
7.34
.090
4.32
7.01
.100
4.20
6.74
.110
4.11
6.50
.120
4.03
6.30
.130
3.97
6.12
.140
3.91
5.97
.150
3.87
5.83
.160
3.83
5.71
.170
3.80
5.59
.180
3.77
5.49
.190
3.75
5.40
.200
3.73
5.32
.210
3.71
5.24
.230
3.69
5.11
.250
3.67
4.99
Nu
21.5
RePr/(x/D)
1000
571. 4
16. 64
400
200
10.82
133.3
100
8.77
80
66.7
7.67
57.2
50
6.96
44.4
40.0
6.46
36.4
33.3
6.08
30.8
28.6
5.79
26.7
25.0
5.56
23. 5
22. 2
5.37
21. 1
20. 0
5.22
19. 0
17. 4
4.97
16. 0
5c0
Summary of Numerical Solutions
Constant Wall Temperature,
Langhaar Velocity, Pr = 2. 0.
Table 9.
4(x/D)/RePr
Nu
..004
17. 15
22. 88
. 007
12. 57
19. 33
.010
10.24
16. 92
. 020
7.23
12. 67
.030
6.05
10. 64
.040
5.40
9. 40
.050
4.98
8. 56
.060
4.69
7. 94
.070
4.47
7. 45
.080
4.31
7. 07
. 090
4.18
* 100
4.08.
.110
4.00
.120
3.94
6.
6.
6.
6.
x
704 Nu
m
76
Nu
m
19.20
08
1000
571.4
15.47
400
200
10.18
133.3
100
8.28
80
66.7
7.26
57.2
50
6.61
44.4
40.0
49
27
(RePr)/(x/D)
6.15
36.4
33.3
Table 10.
4(x/D)/RePr
.Summary of Numerical Solutions
Constant Wall Temperature,
.Langhaar Velocity, Pr = 5. 0.
Nu
704 Nu
m
. 004
.16. 16
22. 33
.007
11.62
18.59
. 010
9.46
16.14
.020
6.74
11.98
.030
5.71
10.04
.040
5.15
8.88
.050
4.79
8.10
Nu
m
16. 90
RePr/(x/D)
.1000
571.4
14.01
400
200
9.35
133.3
100
7.68
80
52
Table 11.
Summary of Numerical Solutions
Constant Temperature Difference,
Langhaar Velocity, Pr = 0. 5.
4(x/D)/RePr
Nu
.004
17.65
22. 75
.007
13.81
20. 79
.010
11.62
17.64
. 020
8.61
13. 69
.030
7.37
11.78
. 040
6.67
10. 57
.050
6.21
9.75
.060
5.88
9. 13
. 070
5.64
8.65
.080
5.45
8.26
.090
5.30
7.94
. 100
5. 18
7. 67
.110
5.08
7.44
.120
5.00
7.23
.130
4.92
7.05
. 140
4.87
6.90
.150
4.82
6.77
. 160
4.78
6.64
.170
4.74
6.53
.180
4.71
6.43
190
4.68
6.34
.200
4.66
6.26
.210
4.64
6. 18
x
704 Num
Nu
24.50
RePr/(x/D)
1000
571.4
18.25
400
200
12.02
133.3
100
9.90
80
66. 7
8.75
57. 2
50
8.02
44.4
40. 0
7.50
36.4
33.3
7.12
30.8
28. 6
6.82
26.7
25. 0
6.58
23.5
22.2
6.38
21.1
20.0
6.22
19. 0
Table 12.
4(x/D)/RePr
Nu
-
x
Summary of Numerical Solutions
Constant Temperature Difference,
Langhaar Velocity, Pr = 0.7.
704 Nu
M
Nu
23.8
004
17.97
23. 23
007
13.75
19. 97
010
11.52
17. 72
020
8.43
13. 67
030
7.19
11. 70
040
6.49
10. 47
050
6.04
9. 68
060
5.72
9. 00
070
5.48
8. 52
080
5.29
8. 12
090
5.15
7. 80
10.0
5.04
7. 53
110
4.94
7. 30
120
4.87
7. 10
130
4.81
6.95
140
4.76
6. 93
6. 77
150
4.71
6. 63
6.66
160
4.67
170
4.64
6. 52
6. 41
180
4.62
6. 31
190
4.59
6. 22
200
4.57
6. 14
210
4.56
6. 06
280
4.49
5. 68
350
4.46
5. 43
RePr/(x/D)
1000
571.4
17.95
400
200
11.81
133.3
100
9.70
80
66. 7
8.56
57.2
50
7.84
44.4
40. 0
7.33
36. 4
33.3
30. 8
28.6
26.7
25.0
6.43
23.5
22.2
6.24
21.1
20.0
6.08
19.0
14.3
5.44
11.4
54
Table 13.
Summary of Numerical Solutions
Constant Temperature Difference,
Langhaar Velocity, Pr = 1. 0.
Nu
704 Num
Num
. 004
17.89
23. 17
21.5
.007
13.60
19. 94
.010
11.30
17. 63
.020
8.21
13. 54
.030
6.99
11. 53
.040
6.31
10. 31
.050
5.87
9.34
.060
5.56
9. 46
8. 83
.070
5.33
8. 35
8.26
.080
5.16
7. 96
.090
5.03
7. 64
.100
4.92
7. 37
.110
4.84
7. 15
.120
4.77
6. 95
.130
4.71
6. 78
.140
4.67
6. 63
.150
4.63
6. 50
.160
4.60
6. 38
. 170
4.57
6. 28
.180
4.55
6. 18
.190
4.53
6. 09
.200
4.52
6. 02
.210
4.51
5. 94
.230
4.49
5. 82
.250
4.47
5. 71
4(x/D)/RePr
-RePr/(x/D)
1000
571.4
16. 96
400
200
11.33
133. 3
100
80
66.7
57.2
50
7.58
44.4
40.0
7.09
36.4
33.3
6.74
30. 8
28.6
6.46
26.7
25.0
6.24
23. 5
22.2
6.06
21.1
20.0
5.92
19.0
17.4
5.69
16.0
55
Table 14.
-Summary of Numerical Solutions
Constant Temperature Difference,
Langhaar Velocity,. Pr = 2. 0.
4(x/D)/RePr
Nu
704 Nu m
.004
17.50
23. 04
.007
13.03
19. 60
.010
10.74
17. 25
.020
7.75
13.
.030
6.60
11. 10
.040
5.97
.050
5.58
9. 89
9. 06
.060
5.31
8. 46
.070
5.11
7. 99
.080
4.97
7.63
.090
4.86
7. 32
100
4.78
7.07
4.72
6.86
.110
Nu
.m
19.20
(RePr)/(x/D)
1000
571.4
15.74
09
400
200
10.62
133.3
100
8.71
80
66.7
7.74
57.2
50
7.13
44.4
40.0
6.70
36.4
56
Table 15.
Summary of Numerical Solutions
Constant Temperature Difference,
Langhaar Velocity, Pr = 5. 0.
704 Nu
4(x/D)/RePr
Nu
.004
16.45
22. 47
.007
1Z. 00
18. 82
.010
9.87
16.42
.020
7.21
12. 34
.030
6.22
10.44
.040
5.70
9.32
.050
5.38
8.56
rn
Nu
.m
16. 90
(RePr)/(x/D)
1000
571.4
14.23
400
200
9.74
133.3
100
8. 13
80
6.0
_r_ 0.5-
PPr=
PrPr
.--
iR ASOL C
-
-
-
-
-
0.7
5.7
'1.0
o
/0
?0
30
40
60
s0
80
70
90
/00
Re PvCx! D)
F/c. /2. LOCAL
NU55GLT
UMBERS
FOk
NUMEAICRL
CONSTAt
WALL
SOLUTIONS
TEropeRruRe,
7.0
--
-
Pr=
-
Pr =0.7A
-Pr
6.0
-
Pr
o
.o
Nux
4.0
0
10
20
30
40
50
60
70
80
'?0
/0
RePr
(x/o)
Fl&. 13.
LOCAL NUSSELT
DIFFERENCE
NYUMBEAS
AND
LAN&qHAAR
FOR CONsrnNr TcMPERA TUR E
VELocirr
NUMERICAL
PROFILES,
SOLTrIONS
= 0.7
.____Pr
/B
j
0
ooo
-
8
7
40
~
P= 2.0
PARAqOLlC VELOCITr
5
/
_______
____0_
000-
0
-- r--
-
10
l0
30
+0
50
60
70
80
90
/00
//0
IZ0
130
RePr
(xID)
Fi&. Ma. MEAto NUSSELT NJMBERS
FOR CONSTANT
NUMERICAL
SoLUrToNS
WALL TMPE,,qrURE
1,0
60
40
30
GRAETZ UNIFORM VELOCITY
Na7"n
(ANALYTIC)
p
Prz 0.7
Pr=z.o
/0
Pe= 6:otcVtcy
PiA.BOLIC
VELOCITY
(NVmeRiCAL)
- TuBe
ENTRANCE
0
100
/000
i0
Re Pr
(X/D)
Fi.. 14b. CONSTRNT WALL
TErIPERRTRE $0LUTIONS
Nu5seL-r
NumssPs
3
61
/
----
-
-
Pr0. 7
/h
1.0
-
-04
9
pi
C
Pe2.
4
oo-
0
10
20
30
40
50
60
70
80
90
/00
/10
10
RePr
(x/o)
Fic. 16a. MEAN NusseLr
DIFFERENCE
NUmscR65 FOR CONSTANT Th-MPEaA'UR E
ANo LANGHARR VELOCITY PROFIL ES,
NuMERICA.. SOLUrIONS
/30
140
62
q0
30
Pr=o.s
e0
-
Pr 0.
Pr= 1.0
/0
Tu,6E EnTRAtC E
0
1000
to
100
-
.Re P
(X/D)
Fic.Ib.
SEmi-Lo& PLOT OF MEN NUSSELT NuMBER5
CONSTANT TE mPS7RATURE DIFFFRENcE
SOLUTIONS
FOR
NuMERICAL
63
the constant temperature difference results in Figs. 13 and 15.
vs. RePr/(x/D) and Figs. 14 and 15 are plots of
and 13 are plots of Nu
Nu
Figures 12
Figures 14(b) and 15(b) are semilog plots.
vs. RePr/(x/D).
Only the
more accurate mean Nusselt numbers have been plotted.
The total 704 time used for production running was 14. 9 minutes.
The average solution time was 43 seconds.
4. 3.
Discussion of Computer Results.
By comparing numerical solutions with the analytic solution of the
same problem a qualitative statement of the accuracy of the numerical
Therefore,
methods can be made.
computer solutions were obtained for
parabolic velocity profiles and a constant tube wall temperature with the
same numerical methods used to obtain the results presented in Sec. 4. 2.
Solutions were computed for each length increment . 001/Pr used previously.
These computer solutions are summarized in Tables 16 and 17.
The mean Nusselt numbers were obtained with the second method discussed in Sec. 4. 2 using the Leveque solution in place of the Poh-lhausen
solution.
Figure 16 is a plot of the analytic solution of Graetz and the computed
numerical solutions.
An investigation of these curves indicates that (1)
as the length increment
A L increases,
the difference between the
analytic solution and the numerical solutions increases,
(2) the numerical
solution values are always less than the analytic values, and (3) the
maximum difference between the computer solutions and the analytic
solution occurs for A L =
002, the largest increment plotted, and is less
than 4 per cent for RePr/(x/D) 4
120.
Therefore, the numerical
methods yield good results within the region of interest and heat exchangers
designed with the results of Sec. 4. 2 should always be more than adequate
for the application under consideration.
Table 16.
Local Nusselt Numbers
Parabolic Velocity,
Different L Increments
Nu
4 (x/D)/RePr
.002
,00143
..004
11.27
. 007
x
64
for A L=
.. 001
.0005
0002
10. 51
10. 90
11.54
.11. 93
8.39
8.68
8.83
9.05
9.21
571.4
.010
7.51
7.66
7.76
7.89
7.97
400
. 020
6.03
6.08
6. 12
6. 16
6.19
200
.030
5.33
5.36
5.39
5.41
5.43
133.3
.040
4.92
4.94
4.95
4.97
4.98
100
. 050
.4. 64
4.65
4.66
4.68
4.68
80
. 060
4.43
4.45
4.45
4.64
66. 7
.070
4.28
4.29
4.30
4.31
57.2
.080
4.16
4.17
4.17
4.18
50
.090
4.07
4.07
4.08
4.08
44.4
.100
.3. 99
4.00
4.00
4.01
40.0
.110
3.93
3.93
3.94
3.94
36.4
. 120
3.88
3.88
3.88
3.89
33. 3
.130
3.83
3.84
3.84
30.8
. 140
3.80
3.80
3.81
28. 6
.150
3.77
3.77
3.78
26.7
. 160
3.75
3.75
3.75
25. 0
.170
3.73
3.73
3.73
23.5
.180
3.71
3.71
3.71
22. 2
.190
3.69
3.70
3.70
21.1
.200
3.68
3.68
3.69
20.0
.210
3.67
3.67
3.68
19.0
RePr/(x/D)
1000
65
Table 17.
Mean Nusselt Numbers for
Parabolic Velocity, Constant
Wall Temperature, and
Varing Increments A L.
AL
4(x/D)/RePr
. 002
.00143
.001
.0005
.0002
RePr/(x/D)
..004
16. 10
.16. 10
16. 10
.16. 10
16. 10
1000
.010
11.67
11.73
12. 05
12. 00
12. 14
400
.030
8.00
8.06
8. 20
8. 22
8. 29
.050
6.78
6.82
6.11
6.14
6. 93
6.23
6.97
.070
6. 91
6. 21
.090
5.68
5.71
5. 76
5. 77
44.4
.110
5.37
5.40
5. 44
5. 45
36. 4
.130
5.14
5.16
5. 20
30.8
.150
4.96
4.98
5. 01
26.7
.170
4.82
4.84
4. 86
23.5
.190
4.70
4.72
4. 74
21.1
.210
4.60
4.62
4. 64
19.0
4. 49
16
.250
.350
.500
4.23
4.04
133,3
80
57.2
11.4
8
9.0
AL=.0005'
GRAETZ
8.0
_
L =.OO/43
7.0
____
.0014
___L
AL-..O02.
Nam
6.0
s.o
4.0
.0 >
20
q0
60
80
/00
/20
140
160
lao
2O
Re P(X/D)
Fic.16.
SOLUTIONS FOR
PARABOLIC VELOCiTY
CONPiTANT
PROFILE,5
WALL TEMPERATURE
AND
67
An estimate of the rate of convergence of the numerical methods can
be obtained from Table 16.
In general, the numerical solution converges
to the true solution as the working increments become infinitely small or
as the number of increments taken become infinitely large.
Therefore,
at any tube cross section, the local Nusselt numbers computed with the
smaller length
increments approach the true value faster than the numbers
computed with the larger increments.
for
At 4(x/D)/RePr = 0. 050,
the values
A L = 0. 0005 and A L = 0. 0002 have apparently converged to the true
value rounded off to two decimal places and the value for the largest
increment
A L = 0. 002 is in error by less than 1 per cent.
For the
A L = 0. 002 solution 25 increments must be taken to have 4(x/D)/RePr
0. 050.
Hence, the parabolic-velocity solutions all converge to within
1 per cent of the true solution after 25 increments have been taken along
the tube length.
Because the numbers used to compute the Langhaar-velocity solutions
differ from those used for the parabolic-velocity solutions, the preceding
discussion can be applied only qualitatively to the Langhaar-velocity
solutions.
The local Nusselt numbers obtained on the 704 computer for
the Langhaar velocity profiles are adequate for most practical applications.
The solutions are very good at 4(x/D)RePr -
0. 050 or RePr/(x/D) !._ 80.
8
5.
CONCLUSIONS
The results presented in this report are by no means exact.
Aside
from using approximate methods of solution, the problem had to be
idealized before any solution could be attempted.
of these approximations,
However, being cognizant
the results can be used to good advantage to
simplify the trial and error design of circular-tube gas-flow heat exchangers.
Further studies of the laminar flow heat transfer problem in circular
tubes should include:
1.
A study of the errors involved in neglecting the radial
enthalpy flux.
2.
A more precise investigation of the accuracy of numerical
methods of solution.
APPENDIX A.
Unit
Definition
Symbol
a, b
GLOSSARY.
Constants appearing in approximation of temperature
profile at a. tube cross section.
ft 2
A
Area
c
Specific heat.
D
Tube diameter.
f
One plus the number of significant digits desired in V.
f.
Coefficients for evaluating TR, L+ A
L
Btu/ (lb OF)
ft
i= 1, 2, 3, 4
h
Unit conductance for convection heat transfer.
Ii
Modified Bessel function of the first kind i
k
Unit thermal conductivity
k
Summation variable.
L
Nondimensional length variable,
p
Any positive integer.
q
Heat transfer rate per unit length
r
Radial distance measured from tube centerline
ft
r
Total tube radius
ft
R
Nondimensional radial distance variable,
t
Temperature.
0F
tl
Initial fluid temperature.
0F
T
Nondimensional temperature variable,
Tm
Nondimensional mean fluid temperature.
T
Nondimensional wall temperature.
v
w
Axial fluid velocity.
Btu/hr ft2o F,
= 0, 2.
Btu/hr ft2oF/ft
4(x/D)/RePr.
1
.
t/t
r/r
.
Btu/ft hr
ft/sec
Unit
Definition
Symbol
vm
Mean fluid velocity.
ft/sec
v
Radial component of fluid velocity.
ft/sec
V
Nondimensional velocity,
VD
Denominator of velocity expression.
VN
Numerator of velocity expression.
Vt
Sample series.
x
Axial distance from tube entrance.
v /vm'
ft
Parameter in Langhaarts velocity expression.
A
Number used for terminating series.
AL
Increment in L.
A Lx
Limiting value of
A R
Increment in R.
a L.
lbs/ft 3
Fluid density.
Fluid viscosity.
Nu.
mi
lbs/(hr ft)
Mean Nusselt number based on initial temperature
difference.
Mean Nusselt number.
Nu
Nu
x
P
r
Re
Local Nusselt number,
Prandtl number,
hD/k.
,c/k.
Reynolds number based on tube diameter,
vmD
/g
Subscripts usually refer to a position within a circular tube where
the velocity or temperature is evaluated.
APPENDIX B.
LANGHAAR'S VELOCITY PROFILES
The equation for Langhaar's velocity profiles
V('X
R)= k=0
k =O0
involves two infinite series.
(.2
(k!)
O0
k
00R
2k
2
00
k=O
(
(k!)
2
(B-i)
\2k+2
k! (k +2)
The purpose of this Appendix is to show that
the fractional terms of both the numerator and denominator series decrease
The following series is used to illustrate this point:
)
monotonically.
V:A =
0
(A )22k
k=0
(k!)Z
.(B
Three cases are of interest:
1.
For A less than one the numerator of Eq. (B-2) is a
fraction raised to a positive power greater than one and
as the exponent increases the numerator decreases.
Also, the denominator increases as k increases.
There-
fore, the fractional terms for this case, 0 4 A 4. 1,
decrease monotonically.
(o
2.
For A equal to one Eq. (B-2) is simply
V' (1) =
7
k= 0 (k
2 .(B
The terms of this series obviously decrease smoothly.
-3)
3,
For A greater than one the initial term or terms of the
However, when k becomes
series are greater than one.
greater than A by some amount the k'th term becomes a
fraction and the terms following this'fractional term decrease monotonically.
To illustrate this result a general
term of the series is written out as
Vk-
(A
A)(A
(k -k)
A)x-
(k -1)(k
- x(A
-1)
A)
x: ---
(B-4)
x(l
- 1)
or
V
A"- A
k
A
k - k
x
A - A
(k-
1)(k - 1)
x
A - A
-x
(B-5)
1 - 1
When k is less than A, all the terms in the product of
Eq.
V
(B-5) are greater than one.
is greater than one.
Hence, the entire term
However, when k becomes larger
than A the first term of the product becomes less than one.
As k gets even larger, more and more of the terms in the
product become less than one.
Finally, enough of the
terms become sufficiently less than one so that their product
also becomes less than one.
After this occurs, the next
term of the series, the k + 1 term, becomes
AA
xx -- x.
A
A
.. (B-6)
(k + 1)(k + 1)
(k)(k)
A
A
1
1
__+_=_
or
V
k+1
=
A - A
(k + 1)(k + 1)
k yt
k
(B-7)
where k is greater than A and Vt
fore, V+
is less than one.
There-
is less than V1 and all succeeding terms decrease
monotonically.
The denominator of Eq. (B-1) can be written
o
2k
k=0
(k!.) 2
2
A~ A(B-8)
VD
(k + 2)(k+ 1)
or
A2
(k + 2)(k + 1)
Vj(
VD=C
k=- 0
(B-9)
The terms A 2 /(k + Z)(k + 1) decrease smoothly as k increases.
because V
decreases monotonically, the fractional terms of V
Hence,
also
decrease monotonically.
To complete the proof it is necessary to prove that the fractional
terms of the numerator series decrease monotonically.
The mathematical
statement of this problem is
A k+I
(k + 1)!)
2
2
2
2
L Ak
(AR)k+
(k + 1)!]
(AR)k
k!
k!
(
0
for
2
Ak
k!
2
(AR)
--k!
If Eq. (B-10) can be proved for
(k)
A.. k 2 <(B-12)
k!
)2
1.
(B-11)
74
then Eq. (B-10) will certainly be true for Eq. (B-11).
Also, because the
maximum value of R used in Eq. (B-1) is 0. 9, when Eq. (B-10) is true
for R
then it must be true for R less than R
This follows because
.
maxma
as R becomes smaller, the terms containing R become less significant and
the proof of Eq. (B-10) approaches the proof for Eq. (B-3).
Equation (B-10) can be rewritten
A
A
A2
1 - RkR2
1 - R2k.
(B-13)
(k!)2
(k! ) 2(k + 1)2
Cancelling the common term and separating terms involving A and R,
Eq. (B-13) becomes
1 - R 2kR2
-
2k
Equation (B-12) implies that A <
k.
1
proved for A
(k + 1)2
.(B-14)
A2
Therefore,
if Eq. (B-14) can be
= k, then when A -4 k the proof is still true.
Substituting
k for A in Eq. (B-14)
1
(k + 1)2
RZkR 2
1 -
This expression is true for k
-15)
.(
k2
2k
2 when R
-
0. 9.
Hence, for k =
2
the fractional terms of the numerator of V,
00 AZk
Vn
k
k= 0
(k! )
2
j
k=0
(AR)
2k+ 2
(B-16)
(k!)
decrease monotonically.
The fractional terms of both numerator and denominator series
decrease monotonically and both series converge very rapidly.
75
APPENDIX C.
1.
BIBLIOGRAPHY.
"Numerical Solutions for Laminar Flow Heat Transfer in Circular
Tubes", by W. M. Kays, Tech. Report No. 20, Navy Contract
N60NR-251 T.O.6, Stanford University, California, October 15,
2.
"Forced Convection from Nonisothermal Surfaces", by John Klein
and Myron Tribus, ASME Paper No. 53-SA-46,
Semiannual Meeting, Los Angeles, California,
3.
1953.
1949.
Trans. ASME, August,
1940.
"Steady Flow in the Transition Length of a Straight Tube", by
H. L. Langhaar, Journal of Applied Mechanics,
6.
June,
"Laminar Flow Heat Transfer Coefficients for Ducts", by R. H. Norris
and D. D. Streid.
5.
presented at ASME
"Heat Transfer", Vol. 1, by Max Jakob; John Wiley and Sons,
New York, N.Y.,
4.
1953.
June,
1942.
"Engineering Analysis", by Stephen H. Crandall, McGraw-Hill Book
Co.., Inc., New York, N.Y.,
1956.