Compressible Flow Tables for Engineers: With Appropriate Computer Programs, for Estimating Property Changes Caused by Friction Heat Transfer and/or Shock Waves

95 Pages·1987·5.08 MB·English

by  James Palmer

Compressible Flow Tables for Engineers
with appropriate computer programs, for estimating property
changes caused by friction heat transfer and/or shock waves
James Palmer, Kenneth Ramsden and Eric Goodger
Senior Lecturers
The School of Mechanical Engineering
Cranfield Institute of Technology
M
MACMILLAN
EDUCATION
©J. R. Palmer, K. W. Ramsden and E. M. Goodger 1987
All rights reserved. No reproduction, copy or transmission
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No paragraph of this publication may be reproduced, copied
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with the provisions of the Copyright Act 1956 (as amended).
Any person who does any unauthorised act in relation to
this publication may be liable to criminal prosecution and
civil claims for damages.
First published 1987
Published by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
ISBN 978-0-333-44764-2 ISBN 978-1-349-09724-1 (eBook)
DOI 10.1007/978-1-349-09724-1
Macmillan title of related interest
E. M. Goodger Principles of Engineering Thermodynamics (second edition)
Contents
Introduction
Notation 2
Unit Conversions 3
Typical Values of Specific Heat Ratio and Gas Constant 4
1. Isentropic Flow: Introduction 5
Program 6
Tables 7
2. Isothermal Flow: Introduction 31
Program 32
Tables 33
3. Rayleigh Flow: Introduction 45
Program 46
Tables 47
4. Fanno Flow: Introduction 65
Program 66
Tables 67
5. Plane Shock Wave: Introduction 79
Program 81
Tables 84
6. Prandtl-Meyer Expansion: Introduction 89
Program 90
Tables 91
Introduction
Large numbers of publications exist, both in the open literature and within companies, presenting fundamental compressible
flow data important to the engineer. The data are presented in a variety of both graphical and tabulated forms. In the
authors' experience, each style of presentation has, in general, limitations in terms either of the wide increments of Mach
Number, or of the small size of graphical presentation. These make interpolation less accurate than is generally desirable.
Furthermore in many cases, especially in the open literature, the data are restricted to one value of specific heat ratio, and
apply to air only.
This particular presentation covers increments of Mach Number that are sufficiently small to avoid the need for inter
polation, and to promote the accuracy of the end result by reducing rounding errors.
The data are presented for several important types of compressible flow, namely Isentropic, Isothermal, Rayleigh, Fanno,
Plane Shock and Prandtl-Meyer Expansion. For the first four of these, three values of specific heat ratio are chosen, namely
1.400, 1.333 and 1.250, to represent typical values within the temperature range 270-2000 K. The given values allow the
tables to be used for air, gaseous hydrocarbon fuels and their combustion products. A correction procedure is provided to
allow their use for values of the Gas Constant other than 287 J/kg K. In each case, the range of Mach Number chosen is
appropriate to the particular type of flow, as indicated in the explanatory notes.
In the cases of the Plane Shock and Prandtl-Meyer Expansion, only the single value of 1.400 for specific heat ratio is
given.
To enable the reader to generate values of the various functions for values of specific heat ratio and/or Mach Number
other than those tabulated, the computer programs from which the tables were computed are also presented. These are
written in FORTRAN 77, since this is thought likely to be the language most widely understood by readers. However,
since the programs are all fairly simple, and are sufficiently annotated, conversion to another language would not be
difficult.
Explanatory notes are given at the beginning of each set of tables. The various formulae used in calculating the data
are stated but, in the interests of brevity, are not derived. Their derivations are, however, well documented in most standard
texts on gas dynamics.
1
Notation
A Area,m2
A* Area at 'star' point, m2
a Sonic velocity, m/s
a* Sonic velocity at 'star' point, m/s
Cc Friction coefficient
Cp Specific heat at constant pressure, J/kg K
Cv Specific heat at constant volume, J/kg K
d Diameter of duct, m
F/A Fuel/air mass ratio
L Length of duct, m
M Mach Number= flow velocity/sonic velocity
M* 'Star' Mach Number= flow velocity/'star' sonic velocity
Mn Component ofMnormal to shock
P Stagnation pressure, Pa
p Static pressure, Pa
Q Mass Flow Parameter (stagnation)= W..j(T)/AP
q Mass Flow Parameter (static)= W..j(T)/Ap
R Gas constant, J/kg K
T Stagnation absolute (thermodynamic) temperature, K
t Static absolute (thermodynamic) temperature, K
u1 Flow velocity component at shock inlet, parallel to inlet velocity, m/s
u2 Flow velocity component at shock outlet, parallel to inlet velocity, m/s
uf, etc Abbreviation for uda*, etc.
V Absolute velocity, m/s
v Flow velocity component at shock outlet, normal to inlet velocity, m/s
W Mass flow rate, kg/s
'Y Ratio of specific heat capacities Cp/Cv
6 Flow deflection, degrees
JJ. Wave angle, degrees
p Density, kg/m3
For Isentropic, Rayleigh and Fanno flows, the superscript '*' refers to the (possibly notional) point in the flow where the
local Mach Number is unity.
For Isothermal flow, the subscript 't' refers to the (possibly notional} point in the flow where the local 'Isothermal Mach
Number' is unity- that is, where the Mach Number as normally defmed is 1/V'Y·
2
Unit Conversions
All the quantities listed in these tables are non-dimensional, with the exception of Q, q and V/vT in the Isentropic Flow
tables, which are tabulated in SI units.
To convert to Imperial units, the following factors may be used:
(I) Q and q are tabulated in kg y(K)/N s
To convert to lb y(K}/lbf s multiply by 9.806 65
To convert to lb y("R}/lbf s multiply by 13.157 0
(2} V/vT is tabulated in m/s VK
To convert to ft/s.../K multiply by 3.28084
To convert to ft/s vR multiply by 2.445 39
3
Typical Values of Specific Heat Ratio and Gas
Constant
Kerosine*-Air Combustion
Products
Temp. (K) Air F/A =0.015 F/A = 0.068** Methane
300 1.400 1.396 1.372 1.303
400 1.395 1.387 1.360 1.257
500 1.387 1.377 1.348 1.218
600 1.376 1.366 1.336 1.188
700 1.364 1.354 1.324 1.167
800 1.354 1.344 1.313 1.151
900 1.344 1.334 1.303 1.138
1000 1.336 1.326 1.295 1.129
1100 1.329 1.318 1.287 1.121
1200 1.323 1.313 1.281 1.115
1300 1.319 1.308 1.274 1.110
1400 1.314 1.304 1.271 1.106
1500 1.311 1.300 1.267 1.103
1600 1.308 1.296 1.264 1.100
1700 1.305 1.293 1.261 1.098
1800 1.302 1.291 1.258 1.096
1900 1.300 1.289 1.256 1.094
2000 1.298 1.287 1.254 1.092
R (J/kgK) 287.0 287.0 287.0 518.2
*Standard Kerosine (see D. Fielding and J. E. C. Topps, Thermodynamic
Data for the Calculation of Gas Turbine Performance, R & M No. 3099,
H.M.S.O., 1959), combustion products of which have the same molar mass
as Air, namely 28.969 g/mol.
**Stoichiometric.
4
1. Isentropic Flow
(Frictionless adiabatic flow with area change)
Introduction
The assumptions of one-dimensional isentropic flow have two important applications:
Firstly, calculation of local conditions at any point in any flow.
Secondly, calculation of the changing flow conditions from point to point in frictionless adiabatic flow in ducts or
stream tubes.
'Y-
Tft = 1 + 1M2 (1.1)
2
_l
'Y;
Pfp = (1 + 1M2) -y-1 (1.2)
VfyT = Mv['YR/~ 'Y; 1 (1.3)
+ M1J
~
2(')'-1)
Q = Wy(T)/AP = Mv(~) ( 1 + 'Y; 1M2) (1.4)
q = Wy(T)/Ap = My[~ (1 + 'Y; 1M2)] (1.5)
1 + 1
{-2-
2(')'- 1)
A/A* = _!_ ~ + 'Y - 1 M2)} (1.6)
M '"f+1 2
The Isentropic flow tables, on pages 7 to 30 inclusive, cover the following ranges and increments of Mach Number:
for O<M< 1.5 increment = 0.005
<M< =
for 1.5 5 increment 0.05
'Y
for = 1.400, 1.333 and 1.250.
Note
The values of the quantities Q and q vary inversely as the square root of the Gas Constant R - that is, directly as the
square root of the molar mass, while the value of V/yT varies directly as the square root of R -that is, inversely as the
square root of the molar mass. In these tables R is taken as 287 J/ kg K, corresponding to the molar mass of air
(28.969 g/mol). The user may readily adapt the tabulated data to other values of R and molar mass.
5
6 COMPRESSIBLE FLOW TABLES FOR ENGINEERS
FORTRAN Program for Isentropic Flow
PROORAM !SENT
1 WRITE(6,2)
2 FORMAT(/' Enter value of gamna (or "0" to exit)'/)
READ(S,*)GAMMA
IF(GAMMA.EQ.O.)STOP
C REQUIRED FUNCTIONS OF GAMMA CALCULATED
A= (GAMMA-!. ) /2.
B= (GAMMA+!. )/2.
C=B/A
D=SQRT( GAMMA* ( 1./B) **C/287. )
E=l./(GAMMA-1.)
F=GAMMA*E
WRITE(6,3)
3 FORMAT(/' Enter min. M, increment of M & max. M' /)
READ( 5, * )AMl ,DAM,AM2
C SET LINE COUNTER "N" TO INCLUDE THE CASE M=O AS AN EXTRA LINE,
C IF PRESENT
IF(AMl.NE.O. )THEN
N=O
ELSE
N=-1
END IF
C MAIN COMPUTING LOOP OVER REQUIRED RANGE OF MACH NUMBER
CO 7 AM=AM1 ,AM2, DAM
C CALCULATION OF GENERAL CASE ( M NON-ZERO)
IF(AM.NE.O.)THEN
TR=1. +A*AM*AM !(1.1)
PR='l'R**F !(1.2)
VT-AM*SQRT(287.*GAMMA/TR) !(1.3)
Q=VT*TR/287. !(1.5)
QQ=QIPR !(1.4)
AR=D/QQ ! (1.6)
END IF
C INSERT NEW HEADING EVERY 50 LINES
IF(AM.EQ.O •• OR.(AM.NE.DAM.AND.N.EQ.(50*(N/50))))WRITE(2,4)GAMMA
4 FORMAT(///24X,'ISENTROPIC FLOW'//24X,'GAMMA = ',F5.3///
14X, 'M' ,SX, 'T/t' ,8X, 'P/p' ,6X, 'V//T' ,4X,
2' 1000Q' , SX, '1000q' , 6X, 'A/A*'/)
N=N+1
C OUTPUT OF GENERAL CASE (M NON-ZERO)
IF(AM.NE.O.)THEN
WRITE(2,5)AM,TR,PR,VT,1000.*QQ,1000.*Q,AR
5 FORMAT(X,FS.3,F9.S,F11.5,F9.5,3F10.5)
ELSE
C OUTPUT OF SPECIAL CASE ( M. .O )
WRITE(2,6)
6 FORMAT(' 0.000 1.00000 1.00000 0.00000 0.00000 0.00000',
1' Infinity' )
ENDIF
7 CONTINUE
C JUMP BACK FOR NEW GAMMA VALUE OR TO EXIT
GOTO 1
END