computer program for liquid metal condensing heat transfer
TRANSCRIPT
MANUAL
COMPUTER PROGRAM FOR LIQUID METAL CONDENSING HEAT TRANSFER COEFFICIENTS INSIDE TUBES
H., L. Ornstein
prepared for
C ontrac t NAS3- 2 335 National Aeronautics and Space Administration
E A S T H A R T F O R D 0 C O N N E C T I C U T
8 .
MANUAL
COMPUTER PROGRAM FOR LIQUID METAL CONDENSING HEAT TRANSFER COEFFICIENTS INSIDE TUBES
7
/'I NASA CR-54350 PWA- 25 5 6
Prepared for
C ontract NAS 3- 2 33 5 National Aeronautics and Space Administration
March 1965
Written by 6 4 x 4 H. L. Or n s tein , Sr. Anal. Engr. - Approved by S. S. Wyde, Program Manager
W. J. Lueckel, Chief, I \ I Space Power Systems
- PRATT h WHITNIY AIRCRAFT PWA-2556
FOREWORD
This report i s a manual of computer programming for the calculation of liquid-metal condensing-heat- transfer coefficients inside tubes. It was prepared by the P ra t t & Whitney Aircraft Division of United Aircraft Corporation for the National Aeronautics and Space Administration, Lewis Research Center, under Contract NAS3-2335, Experimental Investiga- tion of Heat Rejection Problems in Nuclear Space Powerplants.
PAGE NO. ii
I * PRATT e WHITNCY AIRCRAFT
8 8 8 1 t I 1 1c 1 1 d 8 8 8 1 II 1 IJ
Foreword
Table of Contents
I. Introduction
11. Program Logic
111. Input Instructions
IV. Printout Format
V. E r r o r Printouts
PWA - 2 556
TABLE OF CONTENTS
Page
ii
iii
1
2
9
1 2
13
Appendix A - Simpson's Rule Integration
Appendix B - Method of False Position
Appendix C - Tables
Appendix D - Lis t of Computer Statements
Appendix E - Computer Flow Diagrams
PAGE NO. iii
14
18
23
32
47
PWA-2556
I. INTRODUCTION 3364 Report PWA-2530, NASA CR-54352, Analytical Study of Liquid Metal Condensing Inside Tubes, presents a theoretical analysis that can be used to determine the liquid film thickness and condensing-heat-transfer coefficient a t an axial location inside a tube where condensing is occurring. The method also enables determination of the liquid-layer shear- s t r e s s distribution, liquid velocity profile, eddy diffusivity for momentum, rat io of eddy diffusivities, and liquid temperature profile. The analysis assumes annular two-phase flow inside circular tubes with no liquid entrainment in the gas core. were programmed in For t ran I1 f o r the IBM 7094 digital computer.
The final equations derived in that report
This manual presents the following i tems of that computer program in
Program logic, Input format and operational instructions, Printout for mat, Listing of program statements, and
PAQL NO. 1
I PRAlT-L WHITNCY AIRCRAFT
1
11.
PWA-2556
PROGRAM LOGIC
F o r a given set of input conditions the program assumes a liquid film thickness and calculates the shear s t r e s s distribution and velocity pro- file within the liquid layer. profile from the wa l l to the edge of the film to calculate the liquid flow rate . An iteration routine i s used until a film thickness which'results in the cor rec t liquid flow ra te i s found. flow ra t e is calculated in this manner, the program calculates the liquid temperature distribution and the condensing-heat-transfer co- efficient.
The program then integrates the velocity
When the co r rec t liquid
The determination of the cor rec t liquid flow ra te is an i terative process which depends upon the value of liquid film thickness and hence upon the liquid fraction R,*. In order to s ta r t the iteration process, the program calculates the liquid fraction RL based on the Lockhart- Martinelli correlation. the following equation a s the initial guess to calculate a corresponding value of the dimensionless l iquid film thickness **
It then uses this value of liquid fraction in
The liquid film is then divided into 2N increments whose end points have the following radial locations
2N i t , . . . . , , - 3 6 + t d- 2 b + y = o , - 9
2N 2N 2N 2N
*Nomenclature is defined in Table 1 , Appendix C
**Most of the equations presented in this section a r e derived in Report PWA-2530, NASA CR-54352, Analytical Study of Liquid Metal Con- densing Inside Tubes, by H. R. Hunz.
PAOL NO. 2
8 . I PRATT e WHITNLY AIRCRAFT P W A - 2 5 5 6
The shear s t r e s s is evaluated a t each of the 2N t 1 radial locations from the following equation *
r
0 2 T = -
friction
and**
static head t d P
friction ‘1 ’momentum
is a program input item. (5) The frictional pressure gradient friction
The momentum pressure gradient is
X - - - - 2 --[ *T qO 2 ( . 1 -x momentum 7r gC r 3 0 x P C R L Pg (-J
*If r / i s negative a t any radial location, the program will proceed with i ts normal calculations, but an e r r o r printout will occur. Section V for a discussion of e r r o r printouts.
See
**All of the pressure gradient terms in this equation a r e actual pressure gradients except for the frictional pressure gradient which is the negative of the actual gradient. printout a r e the negative of the actual gradients in a l l cases.
The pressure gradient t e r m s in the
PAOLNO. 3
I 1 1 1 8 1 3 1
P R A l T (L WHITNLY AIRCRAFT PWA-2556
dRL where
upon Lockhart and Martinelli's liquid fraction correlation.
i s calculated using an empirical expression based dx
The p res su re gradient due .to static head is
and (5) a r e dependent momentum static head
The values of
upon R, , therefore the iteration procedure is necessary.
eM t - i s evaluated a t each of the selected y locations from the following VL
equation
( 3) EM T 2 t 2 - VL ='[ 2 - 1 t i l t 4 - T K y (1-e
0
t where K = 0 .4 and A = 26.0
Generally, the values obtained for 2 f rom Equation (3) increase
with increasing values of y from a value of zero a t the
wal l ( yt = 0).
mum within the film, and then decreases.
realist ic, the program offers two options for the values of 2 used
in s uc c e e ding c alc ulations :
'L t
However , under certain conditions - reaches a maxi- "L
Since this might not be
VL
1M L a) Option 1 - The program uses the values of 7 obtained
from Equation ( 3),
t M b) Option 2 = The program uses the values of - VL obtained EM
f rom Equation (3) until Beyond that point this maximum value of
reaches a maximum . is used.
L
P A Q C N O . 4
'I . P R A T T a WHlTNlY AIRCRAFT PWA-2556
1 8
Q M
V L Once the shear s t r e s s distribution and - a r e obtained, the program
solves for the velocity at each radial location by numerical integration of the following equation
Q M
V L 1 t-
0
The final solution i s obtained when the value of liquid flow ra te calculated in this manner is equal to the input liquid flow ra te (1-x) WT within a specified tolerance. Thus, the final solution i s obtained when the follow- ing condition occur s
Integration of Equation (4) and all subsequent ,-itegrations a r e performed with the aid of Simpson's rule. in Appendix A.
A discussion of Simpson's rule appears
Under certain conditions, integration of Equation (4) can resu l t in negative values of u , If this should occur, the program will continue to solve for the desired flow using the absolute value of ut. In such a case, the program will provide an e r r o r printout in addition to i ts normal printout, since the resul ts of such a case a r e not valid. fo r a discussion on e r r o r printouts.
t
See Section V
Once the liquid velocities a r e known a t each radial location, the program calculates a liquid flow ra t e by performing the following integration
t ( 5)
2 A P , v , 2 /- t t - u bo - Y f dY W L - V*
whereTolerance 1 i s a program input item.
PAGLNO. 5
I P R A T T h WHITNgY A I R C R A f T P W A - 2 5 56
However, i f the calculated liquid flow ra t e f rom this f i r s t t r y i s not within the specified tolerance, the program wil l r e s o r t to the use of a false-position subroutine in order to rapidly zero in on a value of liquid fraction which matches the input liquid flow r a t e (1-x) WT . The false-position subroutine requires two values of liquid fraction which bracket the root. That is , one value of RL must, when used in Equations (1) through (51, yield a value of liquid flow ra t e which is l e s s than the input liquid flow rate. Another value of R L must provide a liquid flow ra te greater than the input value. The initial guess of R provides one of these values. If the liquid flow ra t e obtained using the initial guess of liquid fraction R L i i s greater than the input liquid flow ra te , the program chooses a new value of liquid fraction equal to one-
R ~ i i s l e s s half the initial guess. If the liquid flow ra te obtained using
than the input liquid flow ra te , the brackets needed for the false-position subroutine a r e obtained. i s greater than the input liquid flow ra te , the program cal-
culates liquid flow ra te using - S ---s - , etc., until a value
of liquid fraction for which the calculated liquid flow ra t e is l e s s than (1-x) WT is obtained.
2
If the liquid flow ra t e obtained using R L i -2-
RLi R L i R t i 4 8 - 16
If the liquid flow ra te obtained using the initial value of liquid fraction i s l e s s than the input liquid flow rate , the program chooses a new value of liquid fraction equal to twice the initial value if R L i i s l e s s than 0. 3 3 3 3 ,
or if R L i i s greater than o r equal to 0. 3333. If this second
value of R L fails to provide the upper bracket, then another value of R, i s t r ied by a similar routine until the two bracketing values of liquid fraction a r e obtained.
1.0 t R L i 2
The logic of the false-position subroutine used to find the value of liquid friction that matches the input liquid flow ra t e i s described in Appendix B.
After convergence of liquid flow r a t e i s achieved, the program cal- culates the rat io of eddy diffusivities radial locations, using the following equation
01 at each of the selected
1 -Alpha Constant 1 Alpha Constant 2
u = exp
PAGE NO. 6
11 . I PRATT e WHITNLY AIRCRAFT P W A - 2 5 56
Recommended values for the constants in Equation (7) a r e
Alpha Constant 1 = 2.0 Alpha Constant 2 = 0. 5
After evaluating the rat io of eddy diffusivities the program calculates the temperature profile from the following equation
t Y
where
t t t Once t is evaluated a t 6 , the condensing-heat-transfer coefficient i s calculated from the following equation
1 = -
0 film T - T h
V t t t t a t y = 6
This coefficient hfilm involves only the temperature difference due to the thermal resistance of the liquid film. If liquid-vapor interfacial resistance is present, the program calculates an interfacial res i s t - ance coefficient and an overall condensing-heat-transfer coefficient, using the following equations
PAGE NO. 7
8 - P R A T T h W H I T N C Y AIRCRAFT PWA-2556
1 (11)
t 1 h =
hfilrn h interface
If a value of zero is input for sigma ( u ) the program bypasses the liquid-vapor interfacial resistance calculation. A s a resul t h = hfilm.
PAGE NO. 8
1. PRATT a W H I T N L Y AIRCRAFT
III. INPUT INSTRUCTIONS
A . Instructions for F i r s t Case
PWA-2556
Enter the cards in the following order:
1. Title Card
Enter title in Colurrins 2 through 72
2. Control Card
The outline below shows the field locations of the various i tems used to control the program
All i t ems must be entered a s fixed-point right-adjusted numbers.
Symbo 1 Column Instruction
N 1 + 3 Enter the number of double intervals used in the calculations (maximum number for N = 499)
ICHANG 4 and 5 Enter ze ro (0). This instructs the machine that a mas ter case is being loaded.
IEMNU Enter one ( 1 ) in Column 7 to use Option 1. Enter two (2) in Column 7 to use Option 2. See Page 4 for description of Options.
6 and 7
3 . Data Cards
Five data cards a r e entered with the format shown below. (Defini- tions and units a r e listed in Table 2 , Appendix C). A l l data input
PAGE NO. 9
iTT h WHITNIY A I R C R A F T PWA-2556
must be entered in floating-point mode within the field widths indicated.
Field Width 1- 14 15-28 29-42 43-56 57-70
Card No. 1 T'Sat P'Sat Tube Radius Total Flow Quality
Card No. 2 Heat DP/DL G Field Cosine Viscosity
Card No. 3 Viscosity Specific Latent Thermal Density
Card No. 4 Density Mole Sigma Alpha Alpha
Card No. 5 Tolerance 1 Tolerance 2
Flux Friction Theta ( L)
(VI Heat (L) Heat Cond. (L) (L)
(VI Weight ' Constant 1 Constant 2
B. Instructions for Each Succeeding Case
The following procedure should be used if it is desired to run more than one case in a loading when only a few input i tems a r e to be changed from the preceding case.
Load cards immediately behind the preceding case in the following order :
1. Title Card
Enter 1 in Column 1 Enter title in Columns 2 through 72
2. Control Card
N
ICHANC
IEMNU
Enter one-half the number of double intervals used in the calculations (maximum number for N = 499).
Enter one (1) in Column 5. machine that one o r more input i tems from the previous case will be changed.
This instructs the
Enter one (1) in Column 7 to use Option 1. Enter two (2) in Column 7 to use Option 2. for description of options.
See Page 4
I P R A R L WHITNLY A I R C R A F T P W A - 2 5 5 6
II 8 8 8 I I 8 8 8 I 4
3. Input Change Cards
F o r each input i tem on data cards to be changed from the preced- ing case , enter a card with the input a s follows:
a. changed (see Table 2). a fixed-point right-adjusted number.
Columns (1 and 2) - Identification number of variable to be Identification number must be entered as
b. point mode.
Columns (3 through 16) - New value of input item in floating-
4. Blankca rd
C. Instructions to End Deck
After the las t case input, inser t two blank cards .
PAQC NO. 1'1
II I PW A - 2 5 56
I
P R A T T & WHlTNtY A I R C R A F T
IV. PRINTOUT FORMAT
A sample of the printout format of the program is shown in Table 3. The information contained in this format is arranged in four blocks which appear in the following order:
Block 1
Block 1 is a tabulation of the input data. Symbols, definitions, and units for the input data appear in Table 2.
Block 2
Block 2 is a tabulation of values of the radially-independent output items. Symbols, definitions, and units of these i tems appear in Table 4.
Block 3
Block 3 is a tabulation of values of the radially-dependent output i tems. Symbols, definitions, and units of these items appear in Table 5 .
Bloak 4
Block 4 is a tabulation of the values of liquid flow rate and liquid fraction R,, used in the i teration routine to obtain the final solution.
If a set of conditions a r e entered so that an unrealistic answer resul ts , o r convergence upon an answer is impossible, an e r r o r printout will result. See Section V for an explanation of e r r o r printouts.
P R A n & WHlTNlLY AIRCRAFT PW A - 2 5 5 6
V. ERROR PRINTOUTS
In the event that unusual flow conditions exist o r the computer cannot find a liquid flow rate that will satisfy the program equations, a n e r r o r printout w i l l occur.
E r r o r printouts- occur if:
1) The program cannot converge on the proper liquid flow rate Wliquid calculated f (1-X)w~
2) Negative liquid velocities occur
3) Negative shear s t r e s ses occur
I I I 1 I I I I I I I
4) The calculated value of the Lockhart-Martinelli liquid fraction is g rea t e r than o r equal to one.
5 ) The calculated value of the Lockhart-Martinelli liquid fraction is less than o r equal to zero.
If Item 1 is the reason for the e r r o r printout, the following statement w i l l be printed. answers that w i l l be printed a r e f rom the last pas s through the itera- tion and a r e not valid.
"The method of false position failed to iterate". The
If the reason for the e r r o r printout i s Item 2, the following statement w i l l be printed. velocities occur".
"Valid solution not obtained -- negative liquid
If the reason for the e r r o r printout is Item 3, the following statement w i l l be printed. "Note: Negative shear s t r e s s e s occurred".
In the event that Item 4 o r 5 is the reason for the e r r o r printout, the program wi l l terminate without returning any answers other than list- ing the input (Block 1). These e r r o r s a r e possible only if an e r r o r is made in the program input.
PAGLNO. 13
1 . P R A R & WHITNLY AIRCRAFT
I
I
I I
PWA-2556
APPENDIX A
Simpson's Rule Integration
PRATT WHITNLY AIRCRAfT PWA-2556
APPENDIX A
Simpson's Rule Integration
A l l integration within the program is performed with the aid of Simpson's rule. ing functions with a high degree of accuracy. section of a function to be integrated can be approximated by a parabola through its ends and midpoint.
Simpson's rule provides a rapid method for integrat- It assumes that each
The sketch below shows three points of function with a parabolic a r c drawn through them. Thus
Z = f(y) = Ay2 t By t C
Figure A 1
A t the midpoint of the double interval sh z = z2
wn in the ketch y = 0 and
Substituting these values into Equation ( A l ) and solving for C
c = 2.2 (A2)
thus
Z = Ay2 t By t 2 2 (A31
I P R A l T (L WHITNCY A l R C R A f T PWA -2556
A t the left end point y = -h and Z = Z1
Substituting these values into Equation (A3) gives
Z1 = Ah2 - Bh t Z2
A t the right end point y = t h and 2 = Z3
Substituting these values into Equation (A3) gives
Z3 = Ah2 t Bh t Z2
Adding Equations (A4) and (A6) and solving fo r Ah2
Z1-k z 3 - z2 Ah2 =
2
Subtracting Equation (A4) from Equation (A5) yields
Bh = 2 3 - z1 2
(A4 1
(A7 1
Integrating Equation ( A l ) between limits of y1 = -h and y2 = 0 gives
Area of Y 2 = 0
-. 2
Increment I = f f ( Y ) dY = (Figure A 1)
y1 = -h
Similarly
A r e a of Y3 = h Ah (A101
2 Increment I1 = (Figure A l )
Y2 = 0
PAQL NO. 16
? . P R A T T & WHITNEY AIRCRAFT P W A - 2 5 5 6
Substituting Equations (A2), (A7) and (A8) into Equations (A9) and (A10)
0
0
These las t two equations provide the integrals of two increments of a function in t e rms of values of the function at the increment end points.
Equations ( A 11) and (A12) a r e the forms of Simpson's rule used through- out the program.
The smal le r the width of the increment, the grea te r w i l l be the accuracy of the approximation. vals ( N = 499) and hence 998 increments. w i l l it be necessary to use that many increments.
The program can accept up to 499 double inter- However, in few instances
P R A T T h WHITNLY AIRCRAFT I
E
~~
I
PWA-2556
APPENDIX B
Method of Fa lse Position
PAQC NO. 18
PRATT h WHlTNEY AIRCRAFT
APPENDIX B
PW A - 2 5 56
Method of Fa lse Position
The principle of false position provides a rapid method for solving i t e r a - t ive problems. to obtain a value of liquid fraction RL corresponding to the input value of liquid flow rate (4-x) W, .
The computer program uses a false-position subroutine
The false-position subroutine requires
1) 2)
the function to be evaluated be a monotonic function, and values of liquid fraction with corresponding liquid flow rates that bracket the desired root.
The method of false position can be explained with the aid of Figures B1 through B4. liquid flow rate and liquid fraction.
Assume that Figure B1 shows the t rue relationship between
W L
. -X)!w,
R,l
Figure B1 Figure B2
In Figure B2, R known to bracket the cor rec t answer (see page 6 for a discussion of the method used to bracket the correct answer). The program determines where a straight line through Points 1 and 2 intersects the horizontal l ine WL = (1-x) WT . R L (namely R L ~ ) .
and R L 2 correspond to the values of liquid fraction
This point corresponds to a new guassed value for
P A O L N O . 19
I t . I P R A T T & WHlTNEY A I R C R A F T PW A - 2 5 5 6
Using Equations (1) through (5) of Section IV, the program calculates the value of liquid flow rate corresponding to R L3, which is W L on Straight lines a r e then determined between Points 1 and 4 and Points 2 and 4, intersecting W L = (1-x) WT at Points 5 and 6, respectively. R L and R L 6 bracket the solution.
(Point 4) Figure B3.
(1-
L 4
WT
=f(R
Figure B3
The program then selects the two values of RL which provide bracketing liquid flows closest to the t rue liquid flow rate, and uses them as the new brackets (in this ca se , R
After each process , the program checks to see whether o r not the dif- ference between the RL brackets is within a specified tolerance.
and R L 5) to continue the process.
is less than, greater than, o r equal to /RL5 - RL1/
/RL5 + RL1/ It checks if
Tolerance 2 where Tolerance 2 is a program input item.
/ R L 5 - R L 1 /
/ R L 5 -t RL1/ a ) If I Tolerance 2, the program picks a new value
PAGE NO. 20
P R A T T & W H I T N I Y A I R C R A F T PWA -2556
and evaluates a corresponding of liquid fraction R L ~ - - RL5 -t R q
2 liquid flow rate W L ~ on Figure B4.
WL I
L~~ solution
Figure B4
The program then checks whether o r not W L 7 matches (1- x ) WT within a specified tolerance.
It checks i f
Tolerance 1.
/ W L 7 - ( l - x ) wT is l e s s than, greater than, o r equal to
( l - X ) W ,
where Tolerance 1 is a program input item.
If I Tolerance 1, /wL7 - ( l - x ) WT /
( l - X ) w T
convergence on liquid flow rate is attained, and the velocity profile and fi lm thickness corresponding to WL and RL,7 are the final answers.
the program decreases Tolerance 2 by a factor of 10 and repeats the i teration process using R L respectively.
and R L 5 a s the left and right brackets
PAQL NO. 21
a - P R A T T h WHITNEY AIRCRAFT P W A - 2 556
/ R L 5 - RL1/
R L 5 + RL1 b) If > Tolerance 2,
the program repeats the iteration process using R L 1 and R L 5 a s the left and right brackets. If there is no convergence upon liquid fraction and liquid flow rate after 30 iterations, the program prints out the las t calculated values of liquid fraction and liquid f low ra te along with an e r r o r printout stating that the method of false position failed to i terate. If this happens, it i s probably due to an input e r ro r .
PAQE NO. 22
P R A T T h WHITNEY A I R C R A F T
APPENDIX C
Tables
PAGE NO. 23
P W A - 2 5 56
I P R A T T h WHITNEY A I R C R A F T
TABLE 1
PWA - 25 56
Svmbol
Alpha Constant 1 Alpha Constant 2 CP dp/ dJ? g gc h
h f i l m
h interface
J
M N
P s at Pr
R
r0 R L
r0 +
TO Tv tv t+ Tolerance 1
Tolerance 2
U+
V* WL WT
Nomenclature
Definition
constant defined in Equation (7) constant defined in Equation (7) fluid specific heat static pressure gradient local gravitational acceleration Newton's conversion factor local condensing-heat-transfer coef-
ficient heat-transfer coefficient due to liquid
film thermal resistance heat-transfer coefficient due to liquid
vapor interfacial r e s is tanc e thermal conversion factor
(778 ft lbf/Btu) molecular weight number of double intervals used in
calculations vapor saturation p res su re Prandtl number heat flux universal gas constant liquid fraction tube inner radius dimensionless tube inner radius wall temperature fluid saturation temperature fluid saturation temperature dimensionless fluid temperature tolerance on liquid flow rate (see
tolerance on liquid fraction (see
dimensionless liquid velocity friction velocity dTm0 gc/pL liquid flow rate total flow rate (liquid t vapor)
Appendix B)
Appendix B)
PAGE NO. 24
Units
Btu / lbm O F
lbf/ ft3 f t /hr2 lbrn ft/lbf hr2
Btu / h r ft2 O F
Btu/hr f t 2 OF
Btu/hr f t2 OF
-- lbm/lb-mole
-- lbf/ f t2
Btu/hr f t2 ft lbf/lb-mcle OR
ft
O F
"F OR
_-
-- - -
f t /h r lbm/hr lbm/hr
I PRATT e WHITNEY AIRCRAFT
X
X
Y Y+
a
6 +
C M
8
Svmbol
g i L
0
TABLE 1 (Cont'd.)
Lockhart -Ma rtinelli two -phas e flow parameter
W fluid quality =a distance measured
WT f rom wall
dimensionless distance measured
ratio of eddy diffusivity of heat to
dimensionless thickness of liquid
eddy diffusivity of momentum angle of tube orientation measured
f r o m vertically upward latent heat of vaporization kinematic viscosity density condensation coefficient, see
shear s t r e s s
f rom wa l l
eddy diffusivity of momentum
film
Equation ( 10)
Subs c r ipt s
Definition
r e fe r s to vapor re fers to initial guess re fers to liquid refers to conditions at the wall
PAGE NO. 2 5
PWA - 2556
P R A T T L W H I T N E Y A I R C R A F T
Variable Ide ntificati on Number
8 9
10
11
12
13 14
15 16 17 18
19
20
21
22
Symbol
PWA -2556
TABLE 2
Input Data
T'Sat P'Sat Tube Radius Total Flow Quality Heat Flux
Friction G Field Cosine Theta
DP/DL
Viscosity (L)
Viscosity ( V )
Specific Heat
Latent Heat (L)
Definition
fluid saturation temperature fluid saturation p res su re tube inner radius total flow rate fluid quality, x heat flux absolute value of frictional pressure gradient
gravity field (g /gc) cosine0 (dmeasured f rom vertically upwa-r d axis)
liquid dynamic viscosity, P L
vapor dynamic viscosity,
liquid specific heat %
latent heat of vaporization Thermal Cond liquid thermal conductivity ( L) Density (L) Density (V) Mole Weight Sigma
Alpha Con- stant 1 Alpha Con- stant 2 Tolerance 1
Tolerance 2
liquid density vapor density fluid molecular weight condensation coefficient
constant appearing in
constant appearing in
tolerance on liquid flow ( see
tolerance on liquid fraction
appearing in Equation( 10)
Equation (7)
Equation (7)
Appendix B)
(see Appendix B)
PAGE NO. 26
Units
'F lbf / f t2
ft l bm/h r
.-
- - Btu/hr ft2
lb,/hr ft
lbm/hr ft Btu/lbm 'F
Btu/lb, Btu/hr ft 'F
lbm / f t lbm/ f t lbm/lb-mole
- -
P W A - 2 5 56 P R A T T (L WHITNEY A I R C R A F T
- m P O a W L U
E m O N I N - N
1 - 0 0 2 U U
U
1 9 3 0 o u
n r - U L U d o w o ~m i & I N
e o - * - 1 .
0 0 \ I
0 a
I .
A 0
V 4 4 u
4 - 0
00 Z l O W v o
0 4 L n a m
UJO I
x m a *
- x m - 1 0
N 0
0 0
- I S L Y - . n > w e o
-0 A N 4 d 3 m 0 . 0
r L Y s - o r - I N
- - 1 0 - 0 -
- r - 2 - 0 . \ o P n
N
0
~m a .
I-
4 4 C O w X U C O 0
LUO Z O - 4 m . a 0 U I
c m W I U . o e 0 Z * w m I -@
2 0
a 0
a *
NN 0
W l V U Z O
Q O W O A 0 0 .
a 0
N - 0 V I d r y 4 m V N - 9
X O 0 a w L O
N
I o w I o I 4
I 0
v) *
0
4 a Ani 4 9 c u a * e o
* - t - .
0
u i
a . A m 0 c.4
- w V
LL M W c z
a w 0 0
-1
X
a
I
a - e
I
-1
I-d
W X U 0
u o -0 u o u - w o m
- a 0
1.4
a
W d -0
> w C O - 0 VI+ O N u m m . -0 w
I
40 -0
> L u C O -0 v)O O N V 9 m . -0 >
m-l 30 - 1 O U a m
N UN
3. C O
atu
cad
u 0
c 4 w v)O - 0 0 9
0, N
0 .
m 0
c Q U I IAO - 0 e m N N
0
.4 0
a w LUO -0 u o
0 a d 0
n
*
z a m -0 c U U -0 d 9 u u
m A d
\ O
D
n e
n
0 x o 2 d U u . 0 0
clc
W d x . 0
a m
N C O X O W -0 UJO X O
m L u d -1. 00 x
- 4 > O - 1
w > O t-0 I N m d z m w * n o
- N -10
w > O e 0 - * w e ZVI W . n o
-JN 0
U J I UUJ Z O a 0 a 0 LUO A 0 0 . C d
N
C O 20 c w v ) O 20 00 uo
VI 4 1 I O a a
a
4
d
crl 20 CUI m 0 20 00 v o N
a . IO a .J 4
a
- 0 0 0 a w r u I + - m
* - 0 a t - u v ) *
a w
- 4
z 0 C N
a
21 L w
I
a
a
U c4 2 0
X a
2
v) c z S w u z
LUO -1R1
3 0 D
u. 0
e W
I: 3 z
a
w
m
m
4 -
\ m
I
n m a . n o
m > N *
0
- a 4 w m - 0
- z o m - 0
* 0
s w - 1 9 - 9 u . 4 - m
0
- x m .
v ) N 3 0 c
v u - 0 a 9 L L U - . m - I d 0 . \ o a n
4 Q U I - m u e * 4 - w . P O
o w z * N
I > Y (z
- 3
- 1 m - 0
o w z m
9 m d
A m 0 . 2 0 > w
I
n u
a
.? - 0 -1 4 w d 9 a 9
- N d o c w 0 9 e m - 4
a
n m w . 4 w m o w I O - . \ .
a 0 n i
u 0 J
PAGE NO. 27
R A T T L WHITNEY A I R C R A F T PWA - 2 5 56
......................................... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
-1 lb
Q M N d N d d d d -00000000 3
. . . > . . . . . . . . . . * . . . . . . . . ~ * . . * . . . . . . , . * . . . ~ * ~ . . O O Q O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
>
PAGE NO. 28
I . P R A R & WHlTNEY A I R C R A F T
~~ ~ ~
PWA-2556
Symbol
TABLE 4
Radially - Inde pendent Printout Items
Calc . Liquid Flow
DELTA (Plus) DP/DL (Friction) D P / D L (Head)
D P / D L (Mom)
D P / DL (Mom' Liq)
DP/DL (Mom'Vap)
DP/DL (Total
Definition Units
liquid flow rate obtained using final value of liquid fraction in Equations ( 1 ) to (5) l b m / h r
dimensionless film thickness - - ': f r i c t i onal p r e s s u r e g r a di e n t lbf / f t3 Apressure gradient due to static
*static pressure gradient due to liquid
*static pressure gradient due to liquid
*static pressure gradient due to vapor
*static pressure gradient due to friction,
head lbf / f t3
lbf / ft 3 and vapor momentum
momentum lbf/ft3
momentum lbf / f t
static head,and mome ntum lbf/ft3
Film Thickness film thickne s s ft H (Film)
layer resistance Btu/hr ft2 OF H (Overall) overall heat t ransfer coefficient, see
Equation ( 1 1) Btu/hr ft2 O F
H (Liq'Vap Interface) heat transfer coefficient due to inter- facial resistance, see Equation( 10) Btu/hr ft2 OF
Reynolds No . (Liq)
Reynolds N o . (Vap)
heat transfer coefficient based on liquid
- - 2WL full bore liquid Reynolds number - = ropL 2xw full bore vapor Reynolds number -1
r r o p g - -
R L (Calc) final value of liquid fraction - -
R L (L'M) liquid fraction obtained f rom Lockhart- - - Martine I l i cor r e lation
~ ~
',t dp/dQ used here is the negative of the true pressure gradient and therefore a negative static pressure gradient indicates a pressure r i s e in the direction of flow
PAOENO. 29
P R A T T & WHITNEY A I R C R A F T PW A - 2 5 5 6
TABLE 4 (Cont'd.)
T ' INTERFACE *liquid temperature a t liquid-vapor
TI WALL wall temperature in presence of
TI WALL (H = H Fi lm) wall temperature when interfacial
V (STAR) friction velocity =
interface
interfacial resistance
resistance is neglected
" F
"F
'F f t / h r
t When no liquid-vapor interfacial resistance is considered, the temperature a t the liquid vapor interface is the saturation tempera- tu re
PAQE NO. 30
PRATT & W H l T N t Y AIRCRAFT
Symbol
Alpha
Epsilon M/NU
Tau/Tauo
I 1 I I 8 I I 8 8 8 I
T (Plus)
Y u (Plus)
Y (P lus)
Y / R O
PW A - 2 5 56
TABLE 5
Radially- Dependent Printout Items
Units .- Definition
ratio of eddy diffusivity of heat to eddy diffusivity of momentum, - - (4%)
ratio of eddy diffusivity of momentum
ratio of local shear s t r e s s to shear
dimensionless temperature , t dimensionless velocity, u
dimensionless distance measured from
ratio of distance measured f rom wall to
to liquid kinematic viscosity, ( t M / v L ) - -
s t r e s s a t wall, ( T / zo) - -
distance from wall, y ft
- - t - - +
- - t wall, y
tube inner radius, Y / r o - -
PAOE NO. 3 1
I . PRATT e WHITNEY AIRCRAFT
APPENDIX D
List of Computer Statements
PAQENO. 32
~
PWA- 2556
I - ' P R A T T h WHITNEY A I R C R A F T
I I I
o o o o o o o o o o o o o o o c 0 o o o o O O O O O o o o o o o o o o o o O o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
Y Y Y V Y Y Y J J J J J J J m D u a .n a 'n
J U roz W
W z Id
u U Z Z Z W W 0
e
Z Z
d n a t-
I- 3 I-
I I 3
r. a
a - y a o x w u
PAGE NO. 33
P R A T T & WHITNLY A I R C R A F T PWA - 25 56
V Y Y - -JJ
z - - - m a
0 Ln
PAGE NO. 3 4
P R A T T & WHITNEY A I R C R A F T
I> 3 3 Q
' 3 0
c
4
c
0
-1- CY0
-1J 0 3 -I
3 LL I * 3
- I l L > a * n x -
P A Q E NO. 35
P R A T T h WHITNEY A I R C R A F T P W A - 25 56
I I D 1 I I I I I I- I
+ . m
E + -J u -
N - Lo J
+-E n
KJ rl 0 0 4
PAGE NO. 36
PRATT h WHITNEY AIRCRAFT I PWA -2556
J Q 3
I- c
I
iy 0 0 4
c n N 6 .cU m a c u - 6 -
- z -1z !l 3 '11 r a > Q U l I -
c
0 3 4
* c
z z
a I-
c a r h r
P W A - 2 5 5 6
1 E R
n 0
3 z 5 W
w c3 z a I U -
1J J 3 3 3 3
LL 3
3 1 0 rd
t-
W I1
3 I-
IL
c3
z 0
t- a 'J I -
-1 > n I - \
X 3 _I h.
3
id -1 0 2
> - > I- - rn Z ;I 3
J - 2- I-
Id - * \ 0
IL LO Z
Ly t-
I-
91 I 0 Z
U1 z JJ
Z 0 V
I- lL Q Ly 3 [Y
4
t
a
- n
-
-1- a - n I -0 W 3 I- a Q C -1 \ 3 3 U fl J I- U U
(I1 3 -13
- \ > x a z
Z c)
.. 0 a W W I- 4 -1 > 3 U J Q U G r
a -I
- a Lrl
PAGLNO. 3 8
* P R A T T L WHITNCY A I R C R A F T P W A - 2 5 56
E P
E 1
(r 0 - PJ r) 3 Ul - r4 N nl w ry nl N N nl CJ C 1 N nl 0 0 0 0 0 0 0
- - u J - > \ \
3 - 3 3 - J
.4 - l J - : l J ! i J t- t- t- - t-
c c c -
P A G E NO. 3 9
I . PWA - 2 556 P R A V h WHITNLY A I R C R A F T
I
!> Ly J
0 u * l-
4
Y
I t u l
a (Y 0
z J G
-I 0 I m -I
(Y I 0
2. * 0
-1 _ I O
T * t
o n I O
\ cr U c 4
Y
LL I-
+ Lrl
m > o \
- > \
J Y Z - J 4 3 ~ - n
Ly 0 ul - *
I
[Y LL
1
0 . 4
Y * cn 3
LL 0 I-3 a a
ul
;3 > * * - u
I
II
J Q 3 Iy
D
-I n
Lrl 3 II
a I- W
_I
3 (Y
a m I I
CAGE NO. 4 0
PW A - 25 56
i- * 0
Q .
- a a -)
J -
> - a 3 L - l
o > c n * L L a *
0 + . ;o
x + 3-1 7
a. SI
J
> c3
a
P R A n & WHlTNtY AIRCRAFT 1 . PW A - 2 5 56
I I 1 I It I
PAQC NO. 42
1 P R A n h WHlTNEY A I R C R A F T P W A - 2 556
* c 8 . -
* * * * * * * * . * * * * - - - - . - - - - - - I - - I - - -
+ - I - - - - - - - - - - - - - - - Cd a a. - I
n
u
PAGE NO. 43
~ ~~~
PWA-2556 1 PRATT L WHITWEY AIRCRAFT
0
N \ X \ X \ In
* * X E ld z 0
.
8
n
-A
J LY
1
0 8 d - \ X * 0
cu +
8
O U U - - A - A - A - - - - 0 0 0 0 0 - - -000cu0 - - - A 0 0 0 0
J ---I...-
I y o + A M p * 4
n z z _I
.-(
8 * * X z w Z 0 \ I- U [Y > -
m E 3 0 * In
* t
i n + .=l
* [ Y * > x *
. X A uI1: o\
X \ E
I l l J Z n o x *
* N
n o .
-1
u a Y
A a u Y
-I
u a Y
0 Iy -1 a
* X 0
IY * * o\ m
.4 V \ d N N . II
CI
0 0 to .-4
I
A . a it
I
rJ
8
. Y
I I
m r 3 a
\ In Q
.* * X \ 0
I
II
X
8
I- Q \ I- O t- 3 II
J X cn I- 0
- u > \ \ L9
ln 0
I 3 n
a 0 A
Q 3
J U 0
a > N
I
0 8 d
II
x n 0 cu A 0 u x
. a,
1 4
II
> a * * F4
0 .-(
I
+ 0 0 - II
I
> II
II I- d
c(
Q
II
c 0 I- a
II I I I I II r-
0 I-
O 0
c 0 t-
r 0 I-
ll II
N H 3 Q
J W [Y
-1 w LY
X u X E W Z 0
X
X n
X
X u n
n
X
-I U a
n
I - a a 0 U I > a
n
n X u
w
LL - Y
LL - Y
c I
0 0
0 0
V n 0
N r) 4 cu P) Q cu 0 d In 9 r al
PW A - 2 5 5 6
I
> > -1
(Y
w r X
. . . .
.
L? 0
n - 0 -4 0 - 0 c > o o I I F 0 (v C U I . LL I L - I Y Y I L " \ \ - - 0 - \ A
0 Z W
CAQL NO. 4 5
?
I P R A T T h WHITNCY AIRCRAFT
In 4
d
t
0 m d
w 0 m 4
0
a W
I
c.
LL I
[Y
I a 4
a X +
XLL * 0 .u a a - a
PWA-2556
CAOC NO. 46
1 . P R A T T & WHITNEY A I R C R A F T
~
PWA-2556
APPENDIX E .
Computer Flow Diagrams
I
I ' ' P R A T T (L WHlTNEY A I R C R A F T P W A - 2 5 56
F low Diagram for Main Program
I
I= I I
PAQLNO. 48
P R A T T 8 NHITNEY A I R C R A F T PWA - 2 5 56
Flow Diagrams for Program Subroutines
Subroutine FWFLOL
Subroutine SDP DL M
Subroutine SRLLM
CAQE NO. 49