Thermofluids Data Book

School of Engineering,
University of Southampton, United Kingdom.

2022 Edition
 CONTENTS
Preface 3
Nomenclature 3
General data 4
Non-dimensional groups 5
Thermodynamic definitions 6
Ideal gas relationships 7
Perfect gas relationships 7
Mixtures of perfect gases 7
Equations for systems 8
Equations for control volumes with steady flow 9
Equations for streamlines 10
Incompressible viscous flow 10
Equations of motion in differential form for a viscous steady flow of a fluid 10
Perfect gas relationships for compressible flow 11
One-dimensional flow of a perfect gas 12
Heat transfer 13
Combustion 14
Thermodynamics efficiencies 14
Rocket propulsion 17
Table 1 Calorific values 18
2 Perfect gases – Miscellaneous data 18
3 Semi-perfect gases – Molar enthalpies at low pressures 19

THERMOCHEMICAL TABLES
4 Enthalpies of reaction 20
5 Equilibrium constants 21

GAS FLOW TABLES

19 Gas flow tables (γ=1.400): Subsonic flow 22
20 Gas flow tables (γ=1.400): Supersonic flow 24
21 Gas flow tables (γ=1.333): Subsonic flow 28
22 Gas flow tables (γ=1.333): Supersonic flow 30

INTERNATIONAL STANDARD ATMOSPHERE

23 Properties gases at sea level conditions 32
24 Properties of the International Standard Atmosphere at selected altitudes 32
25 Properties of the International Standard Atmosphere at altitude 33
Appendix A: Principal sources of data 34

2
 PREFACE
These tables have been produced in order to provide students of Thermodynamics and Fluid Mechanics
with an up-to-date and consistent set of property tables and charts. The lay-out of these tables has been
modelled on the Thermodynamic Tables in SI (metric) Units, Third Edition, by R.W. Haywood.
Throughout these tables all pressures are absolute pressures. Exact numerical values are printed in bold
type.

NOMENCLATURE
Nomenclature for key quantities is given below. Other symbols are introduced where they appear in the
subsequent pages.
𝐴 Area
𝑎 Speed of sound
𝐵, (𝑏) Steady-flow specific availability function
𝑐 Speed (𝑐 ≡ |𝑽|)
c𝑝 Constant pressure specific heat capacity
c𝑣 Constant volume specific heat capacity
CV Calorific value
D Drag force
d Length scale
𝐸, (𝑒) Energy, (specific)
𝐻, (ℎ) Enthalpy, (specific)
ℎ𝑐𝑜𝑛𝑣. Convective heat transfer coefficient
L Lift
𝑀 Mach number
𝑚 Mass
𝑛 Number
P Power
𝑝 Pressure
Q, (𝑞) Heat, (specific)
𝑅, (𝑅) Gas constant, (Universal)
𝑆, (𝑠) Entropy, (specific)
𝑇 Temperature
𝑇𝑛𝑒𝑡 Net thrust
𝑈, (𝑢) Internal energy, (specific)
𝑽, (𝑉𝑥 ) Velocity vector, (velocity component in x-direction)
𝑉, (𝑣) Volume, (specific)
W, (𝑤) Work, (specific)
W𝑥 , (w𝑥 ) Shaft work, (specific)
z Vertical displacement

α Thermal diffusivity
β Coefficient of volume expansion
λ Thermal conductivity
μ Dynamic viscosity
𝜈 Kinematic viscosity
𝜈𝑃𝑀 Prandtl-Meyer angle
ρ Density
𝜏𝑤 Wall shear stress
𝜓 Steady-flow specific exergy

3
 GENERAL DATA
Temperature: At a temperature (thermodynamic) of T kelvins (namely T K), the corresponding
truncated thermodynamic temperature called the Celsius temperature is t Celsius, written symbolically
as t °C, where:
t = T – 273.15
Note: Thermodynamic temperatures expressed in kelvins are commonly described as absolute
temperatures.
Pressure:
1 bar = 105 N/m2.
1 atm = 1.01325 bar = 0.101325 MN/m2.
Standard temperature and pressure (s.t.p.): 0 °C and 1 bar.
1
Unified atomic mass unit (12 of an atom of the nuclide 12C): 1.66054 × 10-27 kg.

Atomic weights: Hydrogen 1*, Helium 4, Carbon 12, Nitrogen 14, Oxygen 16, Sulphur 32, Argon 40.
Mole: A mole (mol) is a unit of quantity of particles of specified kind.
Molar mass: The molar mass is the mass in grams (kilograms) of a mole (kilomole) of the specified
substance.
Molar number (Avogadro constant), 𝑵𝑨 : 6.022 × 1023 particles/mol (≡ 6.022 × 1026 particles/kmol).
Stefan-Boltzmann constant, σ: 5.67 × 10-8 W/m2 K4.
Boltzmann’s constant, k: 1.381 × 10-23 J/K.
Acceleration due to gravity, g: 9.80665m/s2.
*
A more exact value is 1.008.

4
 NON-DIMENSIONAL GROUPS
𝜌|𝑽|𝑑 |𝑽|𝑑
Reynolds Number 𝑅𝑒 = =
𝜇 𝜈
|𝑽|
Mach Number 𝑀=
𝑎
|𝑽|
Froude Number 𝐹𝑟 =
√𝑔𝑧
𝜇𝑐𝑝 𝜈
Prandlt Number 𝑃𝑟 = =
𝜆 𝛼
𝐷
Drag Coefficient 𝐶𝐷 =
1 2
2 𝜌|𝑽| 𝐴
𝐿
Lift Coefficient 𝐶𝐿 =
1 2
2 𝜌|𝑽| 𝐴
𝜏𝑤
𝑐𝑓 =
Skin Friction Coefficient 1 2
2 𝜌|𝑽|

Friction Factor 𝑓 = 4𝑐𝑓

𝑚̇𝑎𝑐𝑡𝑢𝑎𝑙
Discharge Coefficient 𝐶𝑑 =
𝑚̇𝑖𝑑𝑒𝑎𝑙

ℎ𝑐𝑜𝑛𝑣. 𝑑
Nusselt Number 𝑁𝑢 =
𝜆
𝑔𝑑3 𝛽∆𝑇
Grashof Number 𝐺𝑟 =
𝜈2
𝑁𝑢 ℎ𝑐𝑜𝑛𝑣.
Stanton Number 𝑆𝑡 = =
𝑅𝑒𝑃𝑟 𝜌|𝑽|𝑐𝑝

5
 THERMODYNAMIC DEFINITIONS

Specific enthalpy ℎ ≡ 𝑢 + 𝑝𝑣

𝜕𝑢
Specific heat capacity at constant volume 𝑐𝑣 ≡ ( )
𝜕𝑇 𝑣
𝜕ℎ
Specific heat capacity at constant pressure 𝑐𝑝 ≡ ( )
𝜕𝑇 𝑝
𝑐𝑝
Ratio of specific heat capacities 𝛾≡
𝑐𝑣
1 𝜕𝑣
Coefficient of volume expansion 𝛽≡ ( )
𝑣 𝜕𝑇 𝑝
1 𝜕𝑣
Coefficient of compressibility 𝜅≡− ( )
𝑣 𝜕𝑝 𝑇

For a simple compressible substance, in the absence

𝑇𝑑𝑠 = 𝑑𝑢 + 𝑝𝑑𝑣 = 𝑑ℎ − 𝑣𝑑𝑝
of capillarity, electric and magnetic fields

Steady flow specific availability function, for a

𝑏 = ℎ − 𝑇0 𝑠
dead-state indicated by subscript 0

Steady flow specific exergy, for a dead-state

𝜓 = 𝑏 − 𝑏0
indicated by subscript 0

6
 IDEAL GAS RELATIONSHIPS
𝑝𝑉 = 𝑛𝑅̅ 𝑇
𝑝𝑉 = 𝑚𝑅𝑇
Equation of state {
𝑝𝑣 = 𝑅𝑇
𝑝 = 𝜌𝑅𝑇

Relationship between 𝑐𝑝 , 𝑐𝑣 and 𝑅 𝑐𝑝 − 𝑐𝑣 = 𝑅

Speed of sound 𝑎 = √𝛾𝑅𝑇

PERFECT GAS RELATIONSHIPS

Change in specific internal energy 𝑢2 − 𝑢1 = 𝑐𝑣 (𝑇2 − 𝑇1 )

Change in specific enthalpy ℎ2 − ℎ1 = 𝑐𝑝 (𝑇2 − 𝑇1 )

𝑇2 𝑣2
𝑐𝑣 ln ( ) + 𝑅 ln ( )
𝑇1 𝑣1
𝑇 𝑝
Change in specific entropy 𝑠2 − 𝑠1 = 𝑐𝑝 ln ( 2 ) − 𝑅 ln ( 2 )
𝑇1 𝑝1
𝑝2 𝑣2
𝑐𝑣 ln ( ) + 𝑐𝑝 ln ( )
{ 𝑝1 𝑣1
𝑝𝑣 𝛾 = 𝑐𝑜𝑛𝑠𝑡.
For Isentropic changes 𝑇𝑣 𝛾−1 = 𝑐𝑜𝑛𝑠𝑡.
𝑇⁄𝑝(𝛾−1)⁄𝛾 = 𝑐𝑜𝑛𝑠𝑡.

MIXTURES OF PERFECT GASES

For a mixture of N perfect gases where, for component-i, 𝑚𝑖 = mass, 𝑝𝑖 = partial pressure, ℎ𝑖 =
ℎ𝑖 (𝑇) =partial specific enthalpy, 𝑠𝑖 = 𝑠𝑖 (𝑇, 𝑝𝑖 ) =partial specific entropy, 𝑛 =number of mols, and the
overbar signifies a partial molar quantity:
𝑖=𝑁
Pressure of the mixture (Dalton’s Law) 𝑝𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑝𝑖
𝑖=1
𝑖=𝑁 𝑖=𝑁
Enthalpy of the mixture 𝐻𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑚𝑖 ℎ𝑖 = ∑ 𝑛𝑖 ℎ̅𝑖
𝑖=1 𝑖=1
𝑖=𝑁 𝑖=𝑁
Entropy of the mixture 𝑆𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = ∑ 𝑚𝑖 𝑠𝑖 = ∑ 𝑛𝑖 𝑠̅𝑖
𝑖=1 𝑖=1

7
 EQUATIONS FOR SYSTEMS
Definition of a system
A system is a fixed quantity of matter.

Conservation of mass applied to a system

The mass of a system is constant, i.e.
𝑚 = 𝑐𝑜𝑛𝑠𝑡.

1st Law of Thermodynamics applied to a system

When heat is transferred to, and work done by, a system are defined as positive and the system is
undergoing a cyclic process

∮ 𝑑𝑄 = ∮ 𝑑𝑊

When heat transfer to, and work done by, a system of mass m undergoing a process between state 1 and
state 2 are defined as positive and in the absence of capillarity, electric and magnetic fields

𝑄 − 𝑊 = 𝐸2 − 𝐸1 = (𝑈2 + 1⁄2 𝑚𝑐2 2 + 𝑚𝑔𝑧2 ) − (𝑈1 + 1⁄2 𝑚𝑐1 2 + 𝑚𝑔𝑧1 )

2nd Law of Thermodynamics applied to a system

When heat is transferred to a system undergoing a cyclic process is defined as positive, the Clausius
inequality is
𝑑𝑄
∮ ≤0
𝑇
When heat transfer to a system of mass m undergoing a process is defined as positive
𝑑𝑄
𝑚𝑑𝑠 = + 𝑚𝑑𝑠𝑖𝑟𝑟𝑒𝑣 where 𝑑𝑠𝑖𝑟𝑟𝑒𝑣 ≥ 0
𝑇
where 𝑑𝑠𝑖𝑟𝑟𝑒𝑣 is the entropy created per unit mass by irreversibilities.

𝒑𝒅𝒗 work for a system

When the surface of a system undergoes displacement between state 1 and state 2 and the pressure
causing the displacement is known over the entire surface that is displaced, the displacement work done
by the system during this process is
2
𝑊 = ∫ 𝑝𝑑𝑉
1

8
 EQUATIONS FOR CONTROL VOLUMES WITH STEADY FLOW
Definition of a control volume with steady flow
A control volume is enclosed by a rigid control surface. For steady-flow, conditions within the control
volume are not, on average, changing so that the mass, momentum, energy and entropy within the
control volume remain constant.

Conservation of mass for a control volume with steady flow – the Continuity Equation
In general, the Continuity Equation may be written as

∑ 𝑚̇𝑖 = 0 or ∑ 𝑚̇𝑖𝑛 = ∑ 𝑚̇𝑜𝑢𝑡

𝑖

or, in vector form

∯ 𝜌𝑉 ∙ 𝑑𝐴 = 0
𝑐𝑠

where 𝑑𝐴 is positive out of the control volume.

Momentum equation for a control volume with steady flow – the Steady Flow Momentum Equation
(SFME)
In vector form, the SFME is

∯ 𝜌𝑉𝑉 ∙ 𝑑𝐴 = 𝐹 − ∯ 𝑝 ∙ 𝑑𝐴
𝑐𝑠 𝑐𝑠

where 𝑑𝐴 is positive out of the control volume and 𝐹 is the force on the flow due, for example, to
viscous forces and to body forces such as gravity.
In 𝑥 and 𝑦 coordinates, the SFME is

∑ 𝐹𝑥 + (∑ 𝑝𝐴) = ∑ 𝑚̇𝑜𝑢𝑡 𝑉𝑥,𝑜𝑢𝑡 − ∑ 𝑚̇𝑖𝑛 𝑉𝑥,𝑖𝑛

𝑥

∑ 𝐹𝑦 + (∑ 𝑝𝐴) = ∑ 𝑚̇𝑜𝑢𝑡 𝑉𝑦,𝑜𝑢𝑡 − ∑ 𝑚̇𝑖𝑛 𝑉𝑦,𝑖𝑛

𝑦

where care is needed with respect to the sign of the 𝑚̇ and 𝐴 terms.

1st Law of Thermodynamics applied to a control volume with steady flow – the Steady Flow Energy
Equation (SFEE)
When heat transfer to, and shaft work done by, the control volume are defined as positive, and in the
absence of capillarity, electric and magnetic fields, the SFEE is
1 2 1 2
𝑄̇ − 𝑊̇𝑋 = ∑ 𝑚̇𝑜𝑢𝑡 (ℎ𝑜𝑢𝑡 + 𝑐𝑜𝑢𝑡 + 𝑔𝑧𝑜𝑢𝑡 ) − ∑ 𝑚̇𝑖𝑛 (ℎ𝑖𝑛 + 𝑐𝑖𝑛 + 𝑔𝑧𝑖𝑛 )
2 2

2nd Law of Thermodynamics applied to a control volume with steady flow

When heat transfer to a control volume are defined as positive

𝑑𝑄̇
∫ ̇
+ 𝑆𝑖𝑟𝑟𝑒𝑣 = ∑ 𝑚̇𝑜𝑢𝑡 𝑠𝑜𝑢𝑡 − ∑ 𝑚̇𝑖𝑛 𝑠𝑖𝑛 𝑤ℎ𝑒𝑟𝑒 ̇
𝑆𝑖𝑟𝑟𝑒𝑣 ≥0
𝑇
𝐴

9
where 𝑑𝑄̇ is the rate of heat transfer to an area dA of the control surface at temperature T and the
̇
integration is over the whole area A of the control surface, and where 𝑆𝑖𝑟𝑟𝑒𝑣 is the rate of entropy
creation within the control volume due to irreversibilities.

EQUATIONS FOR STREAMLINES

Bernoulli’s Equation for incompressible inviscid steady flow along a streamline

1
𝑝 + 𝜌𝑉 2 + 𝜌𝑔𝑧 = 𝑐𝑜𝑛𝑠𝑡.
2

The pressure gradient normal to a streamline with a radius of curvature 𝒓

𝑑𝑝 𝜌𝑉 2
=
𝑑𝑛 𝑟

INCOMPRESSIBLE VISCOUS FLOW

The pressure drop along a pipe of constant diameter d and length L, with viscous flow
𝐿1 2 𝐿1 2
∆𝑝 = 4𝑐𝑓 𝜌𝑉 = 𝑓 𝜌𝑉
𝐷2 𝐷2

EQUATIONS OF MOTION IN DIFFERENTIAL FORM FOR

VISCOUS STEADY FLOW OF A FLUID
Mass conservation (continuity)
𝜕 𝜕
Intrinsic coordinates (𝑒𝑠 𝜕𝑠 + 𝑒𝑛 𝜕𝑛) . (𝜌𝑉𝑒𝑠 ) = 0

Vector notation ∇. (𝜌𝑉) = 0

Momentum
𝜕𝑉 𝑉2 𝜕𝑝 𝜕𝑝
Intrinsic coordinates 𝜌 (𝑉 𝑒 − 𝑒 ) =− 𝑒 − 𝑒 + 𝜌𝑔 + 𝑣𝑖𝑠𝑐𝑜𝑢𝑠
𝜕𝑠 𝑠 𝑟 𝑛 𝜕𝑠 𝑠 𝜕𝑛 𝑛

Vector notation 𝜌𝑉. ∇𝑉 = −∇𝑝 + 𝜌𝑔 + 𝑣𝑖𝑠𝑐𝑜𝑢𝑠

10
 PERFECT GAS RELATIONSHIPS FOR COMPRESSIBLE FLOW
Ratios of stagnation to static quantities
𝑇 𝛾 − 1 2 −1
= (1 + 𝑀 )
𝑇0 2
𝛾
𝑝 𝛾 − 1 2 −𝛾−1
= (1 + 𝑀 )
𝑝0 2
1
𝜌 𝛾 − 1 2 −𝛾−1
= (1 + 𝑀 )
𝜌0 2
Notes:
(1) 𝑇0 = 𝑐𝑜𝑛𝑠𝑡. in adiabatic flow with no shaft work
(2) If flow is isentropic, 𝑝0 = 𝑐𝑜𝑛𝑠𝑡. and 𝜌0 = 𝑐𝑜𝑛𝑠𝑡. when 𝑇0 = 𝑐𝑜𝑛𝑠𝑡.

Mach number relations (see tables)

1
𝑉 𝛾 − 1 2 −2
= √𝛾 − 1𝑀 (1 + 𝑀 )
√𝑐𝑝 𝑇0 2
1 𝛾+1
𝑚̇√𝑐𝑝 𝑇0 𝛾 𝛾 − 1 2 −2(𝛾−1)
= 𝑀 (1 + 𝑀 )
𝐴𝑝0 √𝛾 − 1 2
1
𝑚̇√𝑐𝑝 𝑇0 𝛾 𝛾−1 2 2
= 𝑀 (1 + 𝑀 )
𝐴𝑝 √𝛾 − 1 2
1
𝐹 √𝛾 − 1 1 + 𝛾𝑀2 𝛾 − 1 2 −2
= (1 + 𝑀 ) where 𝐹 = (𝑝 + 𝜌𝑉 2 )𝐴
𝑚̇√𝑐𝑝 𝑇0 𝛾 𝑀 2

1 2 −
𝛾
𝜌𝑉 1 𝛾 − 1 𝛾−1
2 = 𝛾𝑀2 (1 + 𝑀2 )
𝑝0 2 2
The Prandtl-Meyer angle 𝜈 through which the stream of perfect gas turns in expanding from 𝑀 = 1 to
a supersonic Mach number 𝑀 is

𝛾+1 𝛾−1 2
𝜈𝑃𝑀 = √ tan−1 √ (𝑀 − 1) − tan−1 √𝑀2 − 1
𝛾−1 𝛾+1

11
 ONE-DIMENSIONAL FLOW OF A PERFECT GAS

Superscript * refers to conditions at the throat of a choked convergent-divergent nozzle. Subscript ‘s’
refers to conditions immediately downstream of a normal shockwave.

Isentropic flow
1 𝛾+1
𝐴 1 2 𝛾 − 1 2 2(𝛾−1)
= { (1 + 𝑀 )}
𝐴∗ 𝑀 𝛾 + 1 2
1 𝛾−1 1/2
𝑚̇ 𝑝 𝛾 𝑝 𝛾
1/2
= 𝐴 ( ) [1 − ( ) ]
𝜌0 (2𝑐𝑝 𝑇0 ) 𝑝0 𝑝0

1
𝑚̇ ∗
2 𝛾−1 𝛾 − 1 1/2
1/2
=𝐴 ( ) ( )
𝜌0 (2𝑐𝑝 𝑇0 ) 𝛾+1 𝛾+1

Adiabatic constant area flow

4𝑐𝑓 𝐿𝑚𝑎𝑥 1 − 𝑀2 𝛾 + 1 (𝛾 + 1)𝑀2
= + ln ( )
𝐷 𝛾𝑀2 2𝛾 𝛾−1
2 (1 + 2 𝑀2 )

Normal shock waves in perfect gases

𝑉𝑉𝑠 = 𝑎∗2
1
𝛾−1 2
1 + 2 𝑀2
𝑀𝑠 = ( )
2 𝛾−1
𝛾𝑀 −
2
𝛾
𝛾+1 2 𝛾−1 1
𝑝0𝑠 2 𝑀 2𝛾 𝛾 − 1 1−𝛾
2
=( ) ( 𝑀 − )
𝑝0 𝛾−1 𝛾+1 𝛾+1
1 + 2 𝑀2

𝑝𝑠 2𝛾
=1+ (𝑀2 − 1)
𝑝 𝛾+1
𝑇0𝑠
=1
𝑇0
𝑇𝑠 𝛾−1 2 𝛾−1 2 2𝛾
= 2 2
(1 + 𝑀 )( 𝑀2 − 1)
𝑇 (𝛾 + 1) 𝑀 2 𝛾−1
𝜌𝑠 (𝛾 + 1)𝑀2
=
𝜌 2 (1 + 𝛾 − 1 𝑀2 )
2

12
 HEAT TRANSFER
Conduction, Convection and Radiation
Rate of heat transfer 𝑄̇ by convection from a body of surface area 𝐴

𝑄̇ = ℎ𝑐𝑜𝑛𝑣. 𝐴(𝑇𝑏𝑜𝑑𝑦 − 𝑇𝑠𝑢𝑟𝑟𝑜𝑢𝑛𝑑𝑖𝑛𝑔𝑠 )

Rate of heat transfer 𝑄̇ by conduction along a straight bar of cross-sectional area 𝐴

dT
𝑄̇ = −λ𝐴
dx
Rate of heat transfer 𝑄̇ by conduction radially in a straight circular bar of length 𝐿
dT
𝑄̇ = −λ2𝜋𝑟𝐿
dr
Rate of heat transfer 𝑄̇ by radiation1 from a grey body of surface area A and emissivity ε

𝑄̇ = ε𝑄̇𝑏𝑙𝑎𝑐𝑘 = εσAT 4

Net rate of heat transfer 𝑄̇ by radiation from a grey body of surface area A and emissivity ε1 at
temperature T1 exchanging heat with a black or large enclosure at temperature T2

𝑄̇ = ε1 σA(𝑇14 − 𝑇24 )

Logarithmic-mean Temperature Difference

∆𝑇1 − ∆𝑇2
∆𝑇𝑚 =
ln(∆𝑇1 ⁄∆𝑇2 )

Convective heat transfer for fully developed flow in circular pipes of diameter d
Overall heat transfer for laminar flow with constant wall temperature
𝑁𝑢𝑑 = 3.66 𝑅𝑒𝑑 < 2300 0.1 < 𝑃𝑟
Overall heat transfer for turbulent flow with constant wall temperature

𝑁𝑢𝑑 = 0.023𝑅𝑒𝑑0.8 𝑃𝑟 0.4 2300 < 𝑅𝑒𝑑 < 107 0.5 < 𝑃𝑟 < 120

Heat transfer due to natural convection from a vertical isothermal plate of height L
Overall heat transfer

𝑁𝑢𝐿 = 0.59(𝐺𝑟𝑃𝑟)0.25 Laminar 𝑃𝑟 ≈ 1

Reynolds Analogy
c𝑓
St =
2

1
Note that the net rate of heat transfer will be less due to incident radiation

13
 COMBUSTION
Air-fuel ratio
𝑚̇𝑎
𝐴𝐹𝑅 =
𝑚̇𝑓

Equivalence ratio
𝐴𝐹𝑅𝑠𝑡
𝜙=
𝐴𝐹𝑅
Steady flow combustion (inlet state 1, outlet state 2, reference state 0)

𝑄̇ − 𝑊𝑥̇ = ∑ 𝑚̇𝑖 (ℎ𝑖,0 − ℎ𝑖,1 ) − 𝑚̇𝑓 (𝐿𝐶𝑉) + ∑ 𝑚̇𝑗 (ℎ𝑗,2 − ℎ𝑗,0 )

𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠

THERMODYNAMIC EFFICIENCIES
Efficiency of a cyclic heat engine
𝑊̇𝑛𝑒𝑡
𝜂𝑐𝑦𝑐𝑙𝑒 ≡
𝑄̇𝐻
but

𝑊̇𝑛𝑒𝑡 = 𝑄̇𝐻 − 𝑄̇𝐶

so

𝑄̇𝐶
𝜂𝑐𝑦𝑐𝑙𝑒 = 1 −
𝑄̇𝐻

Coefficients of performance for refrigerators and heat pumps

Coefficient of performance for refrigeration

𝑄̇𝐶
𝐶𝑂𝑃𝑅 ≡
𝑊̇𝑛𝑒𝑡
Coefficient of performance for a heat pump

𝑄̇𝐻
𝐶𝑂𝑃𝐻𝑃 ≡
𝑊̇𝑛𝑒𝑡

14
Isentropic efficiencies of compressors and turbines

Compressor isentropic efficiency

Ideal Work Input ℎ2𝑠 − ℎ1
𝜂𝑐 ≡ =
Actual Work Input ℎ2 − ℎ1

Turbine isentropic efficiency

Actual Work Output ℎ3 − ℎ4
𝜂𝑡 ≡ =
Ideal Work Output ℎ3 − ℎ4𝑠

Efficiency of combustion power plant

Combustion efficiency

𝑄̇𝑐𝑜𝑚𝑏𝑢𝑠𝑡𝑖𝑜𝑛 Amount of heat released during combustion

𝜂𝑐𝑜𝑚𝑏𝑠𝑢𝑡𝑖𝑜𝑛 = =
𝑚̇𝑓𝑢𝑒𝑙 𝐶𝑉 Calorific value of the fuel burned

Boiler thermal efficiency, for a dead-state indicated by subscript 0

𝑄̇𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑟𝑒𝑑 thermal power transferred

𝜂𝑏 = =
𝑚̇𝑒𝑥ℎ𝑎𝑢𝑠𝑡 (ℎ𝑒𝑥ℎ𝑎𝑢𝑠𝑡,𝑖𝑛 − ℎ𝑒𝑥ℎ𝑎𝑢𝑠𝑡,0 ) sensible heat of incoming exhaust gas

Overall fuel efficiency

𝑊̇𝑥 power output

𝜂𝑜 = =
𝑚̇𝑓𝑢𝑒𝑙 𝐶𝑉 Calorific value of the fuel burned

Back work ratio, bwr

compression power input
bwr=
turbine power output

Efficiency of reciprocating internal combustion engines

The indicted power of an internal combustion engine with indicated mean effective pressure 𝑝𝑖 and
swept volume 𝑉𝑠 , operating at a rate of 𝑁𝑐 machine cycles per second is
𝑃𝑖 = 𝑝𝑖 𝑉𝑠 𝑁𝑐

The indicated efficiency is

indicated power output 𝑃𝑖
𝜂𝑖 = =
calorific value of the fuel burned 𝑚̇𝑓𝑢𝑒𝑙 𝐶𝑉

The mechanical efficiency is

brake power output 𝑃𝑏𝑟𝑎𝑘𝑒
𝜂𝑚𝑒𝑐ℎ = =
indicated power output 𝑃𝑖
The brake efficiency is

15
 brake power output
𝜂𝑏𝑟𝑎𝑘𝑒 = = 𝜂𝑚𝑒𝑐ℎ 𝜂𝑖
calorific value of the fuel burned
The volumetric efficiency is
mass of air inhaled per machine cycle 𝑚̇𝑎𝑖𝑟
𝜂𝑣𝑜𝑙 = =
mass of air to occupy swept volume at ambient 𝑝 and 𝑇 𝑉𝑠 𝜌𝑎𝑖𝑟,0

The specific fuel consumption sfc of a reciprocating internal combustion engine is

𝑚̇𝑓𝑢𝑒𝑙
sfc =
𝑃𝑏𝑟𝑎𝑘𝑒

Efficiency of aircraft engines

The propulsive efficiency of an aircraft engine illustrated is

power to aircraft 𝑉𝑥 [(𝑚̇𝑎𝑖𝑟 + 𝑚̇𝑓𝑢𝑒𝑙 )𝑉𝑥,𝑗 − 𝑚̇𝑎𝑖𝑟 𝑉𝑥 ]

𝜂𝑝 = =
power to jet 1 2 2
2 [(𝑚̇𝑎𝑖𝑟 + 𝑚̇𝑓𝑢𝑒𝑙 )𝑉𝑥,𝑗 − 𝑚̇𝑎𝑖𝑟 𝑉𝑥 ]
The fuel efficiency of the aircraft engine is given by
1
power to jet 2
2
[(𝑚̇𝑎𝑖𝑟 + 𝑚̇𝑓𝑢𝑒𝑙 )𝑉𝑥,𝑗 − 𝑚̇𝑎𝑖𝑟 𝑉𝑥 2 ]
𝜂𝑓 = =
calorific value of fuel burned 𝑚̇𝑓𝑢𝑒𝑙 𝐶𝑉

The overall efficiency of the aircraft engine is then

power to aircraft
𝜂𝑜 = = 𝜂𝑝 × 𝜂𝑓
calorific value of fuel burned
The specific fuel consumption sfc of an aircraft engine producing net thrust 𝑇𝑛𝑒𝑡 is
𝑚̇𝑓𝑢𝑒𝑙
sfc =
𝑇𝑛𝑒𝑡
Breguet range equation
𝐶𝐿 𝑉 𝑊1
𝑠= ln ( )
𝐶𝐷 𝑔 × sfc 𝑊2

16
 ROCKET PROPULSION
Specific impulse
𝐹
𝐼𝑠𝑝 =
𝑔 𝑚̇
Effective exhaust velocity
𝐹
𝑐𝑒 =
𝑚̇
Characteristic velocity
𝑝𝑐 𝐴𝑡
𝐶∗ =
𝑚̇
Nozzle thrust coefficient
𝐹
𝐶𝐹 =
𝑝𝑐 𝐴𝑡
Masses
Payload mass 𝑚𝑝𝑙
Propellant mass 𝑚𝑝
Deadweight mass 𝑚𝑑𝑤
Initial mass 𝑚0 = 𝑚𝑝𝑙 + 𝑚𝑝 + 𝑚𝑑𝑤
Burnout mass 𝑚𝑏𝑜 = 𝑚0 − 𝑚𝑝

Payload ratio
𝑚𝑝𝑙
𝜆=
𝑚0
Deadweight ratio
𝑚𝑑𝑤
𝛿=
𝑚0
Tsiolkovsky rocket equation
𝑚0 1
Δ𝑉𝑖𝑑𝑒𝑎𝑙 = 𝑐𝑒 ln ( ) = 𝑐𝑒 ln ( )
𝑚𝑏𝑜 𝜆+𝛿
Staging

(𝑚𝑝𝑙 )𝑖 = (𝑚0 )𝑖+1

Overall payload ratio

𝜆𝑜 = ∏ 𝜆𝑖

17
 TABLE 1. CALORIFIC VALUES
TABLE 1. CALORIFIC VALUES
In this T able, the calorific value is the enthalpy decrease on com bustion when the reactants and products
In this table, the calorific value is the enthalpy decrease on combustion when the reactants and products are at
are at 25°C. In the evaluation of the lower calorific value the steam is taken as being dry saturated.
25 °C. In the evaluation of the lower calorific value the steam is taken as being dry saturated.
Molar mass B.P. at 1 atm Calorific Value
Substance kg/kmol °C Phase kJ/kg
to CO 9190
C 12 sol.
to CO2 32760
CO 28 -191.5 gas 10100

Higher (gross) Lower (net)

(H2 O to water) (H2 O to steam)
H2 2* -252.9 gas 142000 120000
CH4 (methane) 16 -161.5 gas 55500 50010
C2 H6 (ethane) 30 -88.6 gas 51870 47470
C3 H6 (propylene) 42 -47.7 gas 48940 45800
C3 H8 (propane) 44 -42.1 gas 50360 46360
C4 H10 (butane) 58 -0.5 gas 49520 45730
C8 H18 (n-octane) gas 48270 44800
114 125.7
liq. 47900 44430
* A more exact value is 2.016.

PERFECT GASES
At normal atmospheric conditions, and over a limited range of temperature and pressure, the gases listed
in Table 2 may be assumed to behave as perfect gases. That is, they may be assumed to have the equation
of state pv=RT, and to have constant specific heat capacities.
Molar (universal) gas constant: 𝑅̅ =MR=8.3145 kJ/kmol K.
Molar volume of a perfect gas: 1 kmol of any perfect gas occupies a volume of approximately 22.4m3
at s.t.p (0°C and 1 bar).
TABLE 2
Molar mass Gas constant Specific heat capacity
Gas kg/kmol kJ/kg K kJ/kg K c p /c v

cp cv
Air 29 0.287 1.01 0.72 1.40
Atmospheric 28.15 0.295 1.03 0.74 1.40
nitrogen †
N2 28 0.297 1.04 0.74 1.40
O2 32 0.260 0.92 0.66 1.40
A 40 0.208 0.52 0.31 1.67
H2 2* 4.120 14.2 10.08 1.41
He 4 2.080 5.19 3.11 1.67
CO 28 0.297 1.04 0.74 1.40
CO2 44 0.189 0.82 0.63 1.31
SO2 64 0.130 0.61 0.48 1.26
CH4 16 0.520 2.23 1.71 1.31
C2 H6 30 0.277 1.75 1.47 1.19
C3 H6 42 0.198 1.52 1.32 1.15

*A more exact value is 2.016.

†Air contains 0.93% of argon (A) and traces of other gases; these and the nitrogen together are called
atmospheric nitrogen.
Real gases are not perfect gases, and the rounded values for R, cp, cv, and cp/cv listed above do not exactly
satisfy the relationships between these quantities that would obtain for perfect gases.

Air composition:
Volumetric (and molar): 21.0% O2, 79.0% atmospheric nitrogen.
Gravimetric: 23.2% O2, 76.8% atmospheric nitrogen.
18
 SEMI-PERFECT GASES
At low pressures, and over the temperature range quoted, the gases listed in this Table behave as semi-
perfect gases. That is, while having the molar equation of state p𝑣̅ =𝑅̅T, their specific heat capacities
are variable but are functions only of temperature.

TABLE 3. MOLAR ENTHALPIES AT LOW PRESSURES

Warning: T his table lists absolute temperatures.

Gas Air N2 O2 H2 CO CO2 H2 O

Molar mass 29 28 32 2* 28 44 18
kg/kmol
T emperature Molar enthalpy T emperature
K MJ/kmol K

200 5.79 5.81 5.79 5.69 5.81 5.96 6.62 200

298.15 8.64 8.67 8.66 8.46 8.67 9.37 9.90 298.15
300 8.70 8.72 8.71 8.52 8.72 9.44 9.96 300
400 11.62 11.64 11.68 11.42 11.64 13.37 13.35 400
500 14.57 14.58 14.74 14.34 14.60 17.67 16.82 500

600 17.59 17.56 17.90 17.27 17.61 22.27 20.39 600

700 20.66 20.61 21.16 20.21 20.69 27.12 24.09 700
800 23.81 23.72 24.50 23.16 23.85 32.18 27.90 800
900 27.03 26.89 27.90 26.13 27.07 37.41 31.83 900
1000 30.30 30.14 31.37 29.14 30.36 42.78 35.90 1000

1100 33.64 33.44 34.88 32.18 33.71 48.27 40.09 1100

1200 37.02 36.79 38.43 35.26 37.11 53.87 44.41 1200
1300 40.44 40.19 42.01 38.38 40.54 59.55 48.84 1300
1400 43.90 43.62 45.63 41.54 44.02 65.31 53.39 1400
1500 47.39 47.09 49.27 44.75 47.53 71.13 58.05 1500

1600 50.92 50.59 52.94 48.00 51.07 77.01 62.81 1600

1700 54.47 54.12 56.63 51.29 54.63 82.94 67.65 1700
1800 58.04 57.67 60.35 54.62 58.21 88.92 72.58 1800
1900 61.63 61.25 64.09 58.00 61.81 94.93 77.59 1900
2000 65.24 64.84 67.86 61.40 65.42 100.97 82.67 2000

2100 68.87 68.44 71.65 64.84 69.06 107.05 87.81 2100

2200 72.52 72.06 75.46 68.31 72.70 113.15 93.01 2200
2300 76.18 75.70 79.29 71.82 76.36 119.28 98.27 2300
2400 79.86 79.35 83.14 75.35 80.03 125.43 103.58 2400
2500 83.55 83.01 87.02 78.90 83.71 131.61 108.94 2500

2600 87.25 86.68 90.92 82.48 87.40 137.80 114.34 2600

2700 90.96 90.36 94.83 86.09 91.10 144.02 119.78 2700
2800 94.69 94.05 98.77 89.72 94.80 150.25 125.26 2800
2900 98.42 97.74 102.72 93.37 98.51 156.50 130.77 2900
3000 102.16 101.44 106.70 97.04 102.23 162.76 136.31 3000

*A more exact value is 2.016.

Note: (1) The molar enthalpies listed are those in the ideal gas state at zero pressure, but the values given are
also valid at and around atmospheric pressure.
(2) In this table, the arbitrary datum state for zero enthalpy is that of the substance in the ideal gas state
at zero pressure and zero absolute temperature. (Warning: In Tables 6-12, the arbitrary datum state for H2O is
that of the saturated liquid at the triple point, at which state the internal energy and entropy are taken to be zero.)
(3) The values for atmospheric nitrogen, N2*, may be taken to be the same as for N2.

19
 THERMOCHEMICAL TABLES
TABLE 4. ENTHALPIES OF REACTION
Reaction (kmol) - T he values of Δh̃ѳ relate to
the corresponding chemical equation with amounts of Δh̃ ѳ / (kJ/kmol)
substance in kilomoles at T =298.15 K (25 °C)
C(sol) + O2 → CO2 -393 520
CO + ½O2 → CO2 -282 990
H2 + ½O2 → H2 O(vap) -241 830
CH4 (vap) + 2O2 → CO2 + 2H2 O(vap) -802 310
C2 H4 (vap) + 3O2 → 2CO2 + 2H2 O(vap) -1 323 170
C2 H6 (vap) + 3½O2 → 2CO2 + 3H2 O(vap) -1 427 860
C6 H6 (vap) + 7½O2 → 6CO2 + 3H2 O(vap) -3 169 540
C8 H18 (vap) + 12½O2 → 8CO2 + 9H2 O(vap) -5 116 180
CO2 + H2 → CO + H2 O(vap) +41 160
½H2 + OH → H2 O(vap) -281 540
½N2 + ½O2 → NO +90 290
2H → H2 -435 980
2O → O2 -498 340
2N → N2 -945 300

At 298.15 K for H2 O Δh̃fg = 43 990 kJ/kmol of H2 O

for C6 H6 Δh̃fg = 33 800 kJ/kmol of C6 H6
for C8 H18 Δh̃fg = 41 500 kJ/kmol of C8 H18

The molar enthalpies of reaction, Δℎ̃ѳ, in Table 4 are for a reference temperature of T = 298.15 K and
are virtually independent of pressure. Corresponding values of Gibbs function of reaction, Δ𝑔̃ѳ, may be
found from values of equilibrium constant Kѳ using the relation

∆𝑔̃𝜃 (= 𝑔̃𝑃𝜃 − 𝑔̃𝑅𝜃 ) = −𝑅̅ 𝑇𝑙𝑛𝐾 𝜃 (Suffixes P and R refer to products and reactants).

20
 EQUILIBRIUM CONSTANTS
ln 𝐾 ѳ = ∑ ln(𝑝𝑖 ⁄𝑝ѳ )𝑣𝑖
𝑖

where pi is the partial pressure of the ith species in bars, and pѳ is the standard pressure ≡ 1 bar. vi are
the stoichiometric coefficients, those for the products and reactants being taken as positive and negative
respectively. The equilibrium constant so defined is dimensionless.

TABLE 5. EQUILIBRIUM CONSTANTS

Warning: T his table lists absolute temperatures.
ln Kѳ

Temp. (p H2O)(p ѳ )½ (p CO2 )(p ѳ )½ (p H2O)(p CO) (p H2O)(p ѳ ) ½ (p NO) (p H2 )(p ѳ ) (p O2 )(p ѳ ) (p N2 )(p ѳ )
½ ½ ½ 2 2
(K) (p H2 )(p O2 ) (p CO)(p O2 ) (p H2 )(p CO2 ) (p OH)(p H2 ) (p O2 )½(p N2 )½ (p H) (p O) (p N) 2
298.15 92.207 103.762 -11.554 106.329 -34.933 164.005 186.961 367.479
300 91.604 103.057 -11.453 105.627 -34.707 162.922 185.723 365.126
400 67.321 74.669 -7.348 77.360 -25.655 119.164 135.710 270.329
600 42.897 46.245 -3.348 48.956 -16.602 75.226 85.519 175.356
800 30.592 32.036 -1.444 34.670 -12.072 53.135 60.319 127.753
1000 23.162 23.528 -0.366 26.063 -9.353 39.808 45.145 99.128
1200 18.182 17.871 0.311 20.307 -7.541 30.878 35.005 80.011
1400 14.608 13.841 0.767 16.181 -6.245 24.468 27.742 66.329
1600 11.921 10.829 1.091 13.086 -5.273 19.637 22.285 56.055
1800 9.825 8.497 1.329 10.673 -4.518 15.865 18.030 48.051
2000 8.145 6.634 1.510 8.741 -3.912 12.840 14.622 41.645
2200 6.768 5.119 1.649 7.161 -3.417 10.358 11.827 36.391
2400 5.619 3.859 1.759 5.844 -3.005 8.281 9.497 32.011
2600 4.647 2.800 1.847 4.730 -2.657 6.517 7.521 28.304
2800 3.811 1.893 1.918 3.774 -2.360 5.002 5.826 25.117
3000 3.086 1.110 1.976 2.945 -2.102 3.689 4.357 22.359
3200 2.450 0.429 2.022 2.220 -1.877 2.538 3.072 19.936
3400 1.891 -0.170 2.061 1.582 -1.679 1.516 1.935 17.800
3600 1.391 -0.702 2.093 1.016 -1.504 0.609 0.926 15.898
3800 0.944 -1.176 2.121 0.507 -1.347 -0.202 0.019 14.198
4000 0.541 -1.600 2.141 0.051 -1.207 -0.934 -0.796 12.660
4500 -0.313 -2.491 2.178 -0.914 -0.914 -2.482 -2.514 9.414
5000 -0.997 -3.198 2.201 -1.683 -0.682 -3.725 -3.895 6.807
5500 -1.561 -3.771 2.210 -2.314 -0.493 -4.743 -5.024 4.666
6000 -2.033 -4.246 2.213 -2.839 -0.338 -5.590 -5.963 2.865

21
 TABLE 19. GAS FLOW TABLES (γ=1.400): SUBSONIC FLOW
1.4
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
0.010 1.0000 0.9999 1.0000 0.0063 0.0221 0.0221 45.1813 7134.405 0.0001
0.020 0.9999 0.9997 0.9998 0.0126 0.0443 0.0443 22.5994 1778.450 0.0003
0.030 0.9998 0.9994 0.9996 0.0190 0.0664 0.0664 15.0761 787.0814 0.0006
0.040 0.9997 0.9989 0.9992 0.0253 0.0885 0.0886 11.3173 440.3522 0.0011
0.050 0.9995 0.9983 0.9988 0.0316 0.1105 0.1107 9.0644 280.0203 0.0017

0.060 0.9993 0.9975 0.9982 0.0379 0.1325 0.1329 7.5645 193.0311 0.0025
0.070 0.9990 0.9966 0.9976 0.0443 0.1545 0.1550 6.4947 140.6550 0.0034
0.080 0.9987 0.9955 0.9968 0.0506 0.1764 0.1772 5.6939 106.7182 0.0045
0.090 0.9984 0.9944 0.9960 0.0569 0.1983 0.1994 5.0723 83.4961 0.0056
0.100 0.9980 0.9930 0.9950 0.0632 0.2200 0.2216 4.5762 66.9216 0.0070

0.110 0.9976 0.9916 0.9940 0.0695 0.2417 0.2438 4.1714 54.6879 0.0084
0.120 0.9971 0.9900 0.9928 0.0758 0.2633 0.2660 3.8350 45.4080 0.0100
0.130 0.9966 0.9883 0.9916 0.0821 0.2849 0.2883 3.5513 38.2070 0.0117
0.140 0.9961 0.9864 0.9903 0.0884 0.3063 0.3105 3.3089 32.5113 0.0135
0.150 0.9955 0.9844 0.9888 0.0947 0.3276 0.3328 3.0996 27.9320 0.0155

0.160 0.9949 0.9823 0.9873 0.1009 0.3488 0.3551 2.9172 24.1978 0.0176
0.170 0.9943 0.9800 0.9857 0.1072 0.3699 0.3774 2.7569 21.1152 0.0198
0.180 0.9936 0.9776 0.9840 0.1135 0.3908 0.3997 2.6151 18.5427 0.0222
0.190 0.9928 0.9751 0.9822 0.1197 0.4116 0.4221 2.4889 16.3752 0.0246
0.200 0.9921 0.9725 0.9803 0.1260 0.4323 0.4445 2.3758 14.5333 0.0272

0.210 0.9913 0.9697 0.9783 0.1322 0.4528 0.4669 2.2740 12.9560 0.0299
0.220 0.9904 0.9668 0.9762 0.1385 0.4731 0.4893 2.1820 11.5961 0.0328
0.230 0.9895 0.9638 0.9740 0.1447 0.4933 0.5118 2.0985 10.4161 0.0357
0.240 0.9886 0.9607 0.9718 0.1509 0.5133 0.5343 2.0225 9.3865 0.0387
0.250 0.9877 0.9575 0.9694 0.1571 0.5332 0.5568 1.9530 8.4834 0.0419

0.260 0.9867 0.9541 0.9670 0.1633 0.5528 0.5794 1.8892 7.6876 0.0451
0.270 0.9856 0.9506 0.9645 0.1695 0.5723 0.6020 1.8306 6.9832 0.0485
0.280 0.9846 0.9470 0.9619 0.1757 0.5915 0.6246 1.7766 6.3572 0.0520
0.290 0.9835 0.9433 0.9592 0.1819 0.6106 0.6473 1.7267 5.7989 0.0555
0.300 0.9823 0.9395 0.9564 0.1881 0.6295 0.6700 1.6805 5.2993 0.0592

0.310 0.9811 0.9355 0.9535 0.1942 0.6481 0.6928 1.6377 4.8507 0.0629
0.320 0.9799 0.9315 0.9506 0.2003 0.6666 0.7156 1.5978 4.4467 0.0668
0.330 0.9787 0.9274 0.9476 0.2065 0.6848 0.7384 1.5608 4.0821 0.0707
0.340 0.9774 0.9231 0.9445 0.2126 0.7027 0.7613 1.5262 3.7520 0.0747
0.350 0.9761 0.9188 0.9413 0.2187 0.7205 0.7842 1.4939 3.4525 0.0788

0.360 0.9747 0.9143 0.9380 0.2248 0.7380 0.8072 1.4637 3.1801 0.0829
0.370 0.9733 0.9098 0.9347 0.2309 0.7553 0.8302 1.4354 2.9320 0.0872
0.380 0.9719 0.9052 0.9313 0.2369 0.7723 0.8532 1.4090 2.7054 0.0915
0.390 0.9705 0.9004 0.9278 0.2430 0.7891 0.8763 1.3841 2.4983 0.0959
0.400 0.9690 0.8956 0.9243 0.2490 0.8056 0.8995 1.3608 2.3085 0.1003

0.410 0.9675 0.8907 0.9207 0.2551 0.8219 0.9227 1.3388 2.1344 0.1048
0.420 0.9659 0.8857 0.9170 0.2611 0.8379 0.9460 1.3182 1.9744 0.1094
0.430 0.9643 0.8807 0.9132 0.2671 0.8536 0.9693 1.2988 1.8272 0.1140
0.440 0.9627 0.8755 0.9094 0.2730 0.8691 0.9927 1.2804 1.6915 0.1186
0.450 0.9611 0.8703 0.9055 0.2790 0.8843 1.0161 1.2632 1.5664 0.1234

0.460 0.9594 0.8650 0.9016 0.2850 0.8992 1.0396 1.2469 1.4509 0.1281
0.470 0.9577 0.8596 0.8976 0.2909 0.9138 1.0631 1.2315 1.3441 0.1329
0.480 0.9559 0.8541 0.8935 0.2968 0.9282 1.0867 1.2170 1.2453 0.1378
0.490 0.9542 0.8486 0.8894 0.3027 0.9423 1.1104 1.2033 1.1539 0.1426
0.500 0.9524 0.8430 0.8852 0.3086 0.9561 1.1341 1.1903 1.0691 0.1475

22
 γ=1.400
1.4
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
0.510 0.9506 0.8374 0.8809 0.3145 0.9696 1.1579 1.1781 0.990 0.1525
0.520 0.9487 0.8317 0.8766 0.3203 0.9828 1.1818 1.1665 0.917 0.1574
0.530 0.9468 0.8259 0.8723 0.3262 0.9958 1.2057 1.1556 0.8496 0.1624
0.540 0.9449 0.8201 0.8679 0.3320 1.0084 1.2297 1.1452 0.7866 0.1674
0.550 0.9430 0.8142 0.8634 0.3378 1.0208 1.2538 1.1354 0.7281 0.1724

0.560 0.9410 0.8082 0.8589 0.3436 1.0328 1.2779 1.1261 0.6736 0.1774
0.570 0.9390 0.8022 0.8544 0.3493 1.0446 1.3021 1.1173 0.6229 0.1825
0.580 0.9370 0.7962 0.8498 0.3551 1.0561 1.3264 1.1090 0.5757 0.1875
0.590 0.9349 0.7901 0.8451 0.3608 1.0672 1.3507 1.1011 0.5317 0.1925
0.600 0.9328 0.7840 0.8405 0.3665 1.0781 1.3751 1.0937 0.4908 0.1976

0.610 0.9307 0.7778 0.8357 0.3722 1.0887 1.3996 1.0867 0.4527 0.2026
0.620 0.9286 0.7716 0.8310 0.3779 1.0990 1.4242 1.0800 0.4172 0.2076
0.630 0.9265 0.7654 0.8262 0.3835 1.1090 1.4489 1.0737 0.3841 0.2127
0.640 0.9243 0.7591 0.8213 0.3891 1.1186 1.4736 1.0678 0.3533 0.2177
0.650 0.9221 0.7528 0.8164 0.3948 1.1280 1.4984 1.0621 0.3246 0.2226

0.660 0.9199 0.7465 0.8115 0.4003 1.1371 1.5233 1.0568 0.2979 0.2276
0.670 0.9176 0.7401 0.8066 0.4059 1.1459 1.5483 1.0518 0.2730 0.2326
0.680 0.9153 0.7338 0.8016 0.4115 1.1544 1.5733 1.0471 0.2498 0.2375
0.690 0.9131 0.7274 0.7966 0.4170 1.1626 1.5984 1.0426 0.2282 0.2424
0.700 0.9107 0.7209 0.7916 0.4225 1.1705 1.6237 1.0384 0.2081 0.2473

0.710 0.9084 0.7145 0.7865 0.4280 1.1782 1.6490 1.0344 0.1895 0.2521
0.720 0.9061 0.7080 0.7814 0.4335 1.1855 1.6744 1.0307 0.1721 0.2569
0.730 0.9037 0.7016 0.7763 0.4389 1.1925 1.6999 1.0272 0.1561 0.2617
0.740 0.9013 0.6951 0.7712 0.4443 1.1993 1.7254 1.0239 0.1411 0.2664
0.750 0.8989 0.6886 0.7660 0.4497 1.2058 1.7511 1.0208 0.1273 0.2711

0.760 0.8964 0.6821 0.7609 0.4551 1.2119 1.7768 1.0179 0.1145 0.2758
0.770 0.8940 0.6756 0.7557 0.4605 1.2178 1.8027 1.0152 0.1026 0.2804
0.780 0.8915 0.6691 0.7505 0.4658 1.2234 1.8286 1.0126 0.0917 0.2849
0.790 0.8890 0.6625 0.7452 0.4711 1.2288 1.8547 1.0103 0.0816 0.2894
0.800 0.8865 0.6560 0.7400 0.4764 1.2338 1.8808 1.0081 0.0723 0.2939

0.810 0.8840 0.6495 0.7347 0.4817 1.2386 1.9070 1.0060 0.0638 0.2983
0.820 0.8815 0.6430 0.7295 0.4869 1.2431 1.9333 1.0041 0.0559 0.3026
0.830 0.8789 0.6365 0.7242 0.4921 1.2474 1.9598 1.0024 0.0488 0.3069
0.840 0.8763 0.6300 0.7189 0.4973 1.2514 1.9863 1.0008 0.0423 0.3112
0.850 0.8737 0.6235 0.7136 0.5025 1.2551 2.0129 0.9993 0.0363 0.3153

0.860 0.8711 0.6170 0.7083 0.5077 1.2585 2.0396 0.9979 0.0310 0.3195
0.870 0.8685 0.6106 0.7030 0.5128 1.2617 2.0665 0.9967 0.0261 0.3235
0.880 0.8659 0.6041 0.6977 0.5179 1.2646 2.0934 0.9956 0.0218 0.3275
0.890 0.8632 0.5977 0.6924 0.5230 1.2673 2.1204 0.9946 0.0179 0.3314
0.900 0.8606 0.5913 0.6870 0.5280 1.2698 2.1476 0.9937 0.0145 0.3352

0.910 0.8579 0.5849 0.6817 0.5331 1.2719 2.1748 0.9929 0.0115 0.3390
0.920 0.8552 0.5785 0.6764 0.5381 1.2739 2.2021 0.9922 0.0089 0.3427
0.930 0.8525 0.5721 0.6711 0.5431 1.2756 2.2296 0.9916 0.0067 0.3464
0.940 0.8498 0.5658 0.6658 0.5481 1.2770 2.2572 0.9911 0.0048 0.3499
0.950 0.8471 0.5595 0.6604 0.5530 1.2783 2.2848 0.9907 0.0033 0.3534

0.960 0.8444 0.5532 0.6551 0.5579 1.2793 2.3126 0.9903 0.0021 0.3569
0.970 0.8416 0.5469 0.6498 0.5628 1.2800 2.3405 0.9901 0.0011 0.3602
0.980 0.8389 0.5407 0.6445 0.5677 1.2806 2.3685 0.9899 0.0005 0.3635
0.990 0.8361 0.5345 0.6392 0.5725 1.2809 2.3966 0.9898 0.0001 0.3667
1.000 0.8333 0.5283 0.6339 0.5774 1.2810 2.4249 0.9897 0.0000 0.3698

23
 TABLE 20. GAS FLOW TABLES (γ=1.400): SUPERSONIC FLOW
1.4
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉 𝑝0𝑠 𝑝𝑠 𝑝0𝑠 𝑇𝑠
𝑀 2 𝑀𝑠 𝑃𝑀 𝑀
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐 𝑇
𝑝 0 𝐷 𝑝0 𝑝0 𝑝 𝑝 𝑇
1.010 0.8306 0.5221 0.6287 0.5821 1.2809 2.4532 0.9898 0.000 0.3728 0.9901 1.0000 1.0235 1.9152 1.0066 0.04 1.0100
1.020 0.8278 0.5160 0.6234 0.5869 1.2806 2.4817 0.9899 0.000 0.3758 0.9805 1.0000 1.0471 1.9379 1.0132 0.13 1.0200
1.030 0.8250 0.5099 0.6181 0.5917 1.2801 2.5103 0.9900 0.0010 0.3787 0.9712 1.0000 1.0711 1.9610 1.0198 0.23 1.0300
1.040 0.8222 0.5039 0.6129 0.5964 1.2793 2.5390 0.9903 0.0018 0.3815 0.9620 0.9999 1.0952 1.9844 1.0263 0.35 1.0400
1.050 0.8193 0.4979 0.6077 0.6011 1.2784 2.5678 0.9905 0.0027 0.3842 0.9531 0.9999 1.1196 2.0083 1.0328 0.49 1.0500

1.060 0.8165 0.4919 0.6024 0.6058 1.2773 2.5967 0.9909 0.0038 0.3869 0.9444 0.9998 1.1442 2.0325 1.0393 0.64 1.0600
1.070 0.8137 0.4860 0.5972 0.6104 1.2760 2.6258 0.9913 0.0051 0.3895 0.9360 0.9996 1.1691 2.0570 1.0458 0.80 1.0700
1.080 0.8108 0.4800 0.5920 0.6151 1.2745 2.6549 0.9917 0.0066 0.3919 0.9277 0.9994 1.1941 2.0819 1.0522 0.97 1.0800
1.090 0.8080 0.4742 0.5869 0.6197 1.2728 2.6842 0.9922 0.0082 0.3944 0.9196 0.9992 1.2195 2.1072 1.0586 1.15 1.0900
1.100 0.8052 0.4684 0.5817 0.6243 1.2709 2.7136 0.9928 0.0099 0.3967 0.9118 0.9989 1.2450 2.1328 1.0649 1.34 1.1000

1.110 0.8023 0.4626 0.5766 0.6288 1.2689 2.7432 0.9934 0.0118 0.3990 0.9041 0.9986 1.2708 2.1588 1.0713 1.53 1.1100
1.120 0.7994 0.4568 0.5714 0.6333 1.2667 2.7728 0.9940 0.0138 0.4011 0.8966 0.9982 1.2968 2.1851 1.0776 1.74 1.1200
1.130 0.7966 0.4511 0.5663 0.6379 1.2643 2.8026 0.9947 0.0159 0.4032 0.8892 0.9978 1.3231 2.2118 1.0840 1.94 1.1300
1.140 0.7937 0.4455 0.5612 0.6423 1.2618 2.8325 0.9954 0.0182 0.4052 0.8820 0.9973 1.3495 2.2388 1.0903 2.16 1.1400
1.150 0.7908 0.4398 0.5562 0.6468 1.2590 2.8626 0.9961 0.0205 0.4072 0.8750 0.9967 1.3763 2.2661 1.0966 2.38 1.1500

1.160 0.7879 0.4343 0.5511 0.6512 1.2562 2.8927 0.9969 0.0230 0.4090 0.8682 0.9961 1.4032 2.2937 1.1029 2.61 1.1600
1.170 0.7851 0.4287 0.5461 0.6556 1.2531 2.9230 0.9978 0.0255 0.4108 0.8615 0.9953 1.4304 2.3217 1.1092 2.84 1.1700
1.180 0.7822 0.4232 0.5411 0.6600 1.2500 2.9534 0.9986 0.0281 0.4125 0.8549 0.9946 1.4578 2.3500 1.1154 3.07 1.1800
1.190 0.7793 0.4178 0.5361 0.6644 1.2466 2.9840 0.9995 0.0309 0.4141 0.8485 0.9937 1.4855 2.3786 1.1217 3.31 1.1900
1.200 0.7764 0.4124 0.5311 0.6687 1.2432 3.0147 1.0004 0.0336 0.4157 0.8422 0.9928 1.5133 2.4075 1.1280 3.56 1.2000

1.210 0.7735 0.4070 0.5262 0.6730 1.2396 3.0455 1.0014 0.0365 0.4171 0.8360 0.9918 1.5415 2.4367 1.1343 3.81 1.2100
1.220 0.7706 0.4017 0.5213 0.6773 1.2358 3.0764 1.0024 0.0394 0.4185 0.8300 0.9907 1.5698 2.4663 1.1405 4.06 1.2200
1.230 0.7677 0.3964 0.5164 0.6816 1.2319 3.1075 1.0034 0.0424 0.4198 0.8241 0.9896 1.5984 2.4961 1.1468 4.31 1.2300
1.240 0.7648 0.3912 0.5115 0.6858 1.2279 3.1387 1.0045 0.0455 0.4211 0.8183 0.9884 1.6272 2.5263 1.1531 4.57 1.2400
1.250 0.7619 0.3861 0.5067 0.6901 1.2238 3.1700 1.0055 0.0486 0.4223 0.8126 0.9871 1.6563 2.5568 1.1594 4.83 1.2500

24
 γ=1.400
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉 𝑝0𝑠 𝑝𝑠 𝑝0𝑠 𝑇𝑠
𝑀 2 𝑀𝑠 𝑃𝑀 𝑀
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝𝑇0 𝐷 𝑝0 𝑝0 𝑝 𝑝 𝑇
1.260 0.7590 0.3809 0.5019 0.6943 1.2195 3.2015 1.0066 0.052 0.4233 0.8071 0.9857 1.6855 2.5875 1.1657 5.09 1.2600
1.270 0.7561 0.3759 0.4971 0.6984 1.2152 3.2331 1.0077 0.055 0.4244 0.8016 0.9842 1.7151 2.6186 1.1720 5.36 1.2700
1.280 0.7532 0.3708 0.4923 0.7026 1.2107 3.2648 1.0089 0.0582 0.4253 0.7963 0.9827 1.7448 2.6500 1.1783 5.63 1.2800
1.290 0.7503 0.3658 0.4876 0.7067 1.2061 3.2967 1.0100 0.0615 0.4262 0.7911 0.9811 1.7748 2.6816 1.1846 5.90 1.2900
1.300 0.7474 0.3609 0.4829 0.7108 1.2014 3.3287 1.0112 0.0648 0.4270 0.7860 0.9794 1.8050 2.7136 1.1909 6.17 1.3000

1.310 0.7445 0.3560 0.4782 0.7149 1.1965 3.3608 1.0124 0.0682 0.4277 0.7809 0.9776 1.8355 2.7459 1.1972 6.44 1.3100
1.320 0.7416 0.3512 0.4736 0.7189 1.1916 3.3931 1.0136 0.0716 0.4283 0.7760 0.9758 1.8661 2.7784 1.2035 6.72 1.3200
1.330 0.7387 0.3464 0.4690 0.7229 1.1866 3.4255 1.0149 0.0750 0.4289 0.7712 0.9738 1.8971 2.8112 1.2099 7.00 1.3300
1.340 0.7358 0.3417 0.4644 0.7270 1.1815 3.4581 1.0161 0.0785 0.4294 0.7664 0.9718 1.9282 2.8444 1.2162 7.28 1.3400
1.350 0.7329 0.3370 0.4598 0.7309 1.1763 3.4907 1.0174 0.0820 0.4299 0.7618 0.9697 1.9596 2.8778 1.2226 7.56 1.3500

1.360 0.7300 0.3323 0.4553 0.7349 1.1710 3.5236 1.0187 0.0855 0.4303 0.7572 0.9676 1.9912 2.9115 1.2290 7.84 1.3600
1.370 0.7271 0.3277 0.4508 0.7388 1.1656 3.5566 1.0200 0.0890 0.4306 0.7527 0.9653 2.0231 2.9455 1.2354 8.13 1.3700
1.380 0.7242 0.3232 0.4463 0.7427 1.1601 3.5897 1.0213 0.0926 0.4308 0.7483 0.9630 2.0551 2.9798 1.2418 8.41 1.3800
1.390 0.7213 0.3187 0.4418 0.7466 1.1546 3.6229 1.0226 0.0962 0.4310 0.7440 0.9607 2.0875 3.0144 1.2482 8.70 1.3900
1.400 0.7184 0.3142 0.4374 0.7505 1.1490 3.6563 1.0240 0.0997 0.4311 0.7397 0.9582 2.1200 3.0492 1.2547 8.99 1.4000

1.410 0.7155 0.3098 0.4330 0.7543 1.1433 3.6899 1.0253 0.1033 0.4312 0.7355 0.9557 2.1528 3.0844 1.2612 9.28 1.4100
1.420 0.7126 0.3055 0.4287 0.7581 1.1375 3.7236 1.0267 0.1069 0.4312 0.7314 0.9531 2.1858 3.1198 1.2676 9.57 1.4200
1.430 0.7097 0.3012 0.4244 0.7619 1.1317 3.7574 1.0281 0.1106 0.4311 0.7274 0.9504 2.2191 3.1555 1.2741 9.86 1.4300
1.440 0.7069 0.2969 0.4201 0.7657 1.1258 3.7914 1.0295 0.1142 0.4310 0.7235 0.9476 2.2525 3.1915 1.2807 10.15 1.4400
1.450 0.7040 0.2927 0.4158 0.7694 1.1198 3.8255 1.0308 0.1178 0.4308 0.7196 0.9448 2.2863 3.2278 1.2872 10.44 1.4500

1.460 0.7011 0.2886 0.4116 0.7732 1.1138 3.8598 1.0323 0.1215 0.4306 0.7157 0.9420 2.3202 3.2643 1.2938 10.73 1.4600
1.470 0.6982 0.2845 0.4074 0.7769 1.1077 3.8942 1.0337 0.1251 0.4303 0.7120 0.9390 2.3544 3.3011 1.3003 11.02 1.4700
1.480 0.6954 0.2804 0.4032 0.7805 1.1016 3.9287 1.0351 0.1288 0.4299 0.7083 0.9360 2.3888 3.3382 1.3069 11.32 1.4800
1.490 0.6925 0.2764 0.3991 0.7842 1.0954 3.9634 1.0365 0.1324 0.4295 0.7047 0.9329 2.4235 3.3756 1.3136 11.61 1.4900
1.500 0.6897 0.2724 0.3950 0.7878 1.0891 3.9983 1.0379 0.1361 0.4290 0.7011 0.9298 2.4583 3.4133 1.3202 11.91 1.5000

25
 γ=1.400
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉 𝑝0𝑠 𝑝𝑠 𝑝0𝑠 𝑇𝑠
𝑀 2 𝑀𝑠 𝑃𝑀 𝑀
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝𝑇0 𝐷 𝑝0 𝑝0 𝑝 𝑝 𝑇
1.510 0.6868 0.2685 0.3909 0.7914 1.0829 4.0333 1.0394 0.140 0.4285 0.6976 0.9266 2.4935 3.4512 1.3269 12.20 1.5100
1.520 0.6840 0.2646 0.3869 0.7950 1.0765 4.0684 1.0408 0.143 0.4279 0.6941 0.9233 2.5288 3.4894 1.3336 12.49 1.5200
1.530 0.6811 0.2608 0.3829 0.7986 1.0702 4.1037 1.0423 0.1470 0.4273 0.6907 0.9200 2.5644 3.5279 1.3403 12.79 1.5300
1.540 0.6783 0.2570 0.3789 0.8021 1.0638 4.1392 1.0437 0.1506 0.4266 0.6874 0.9166 2.6002 3.5667 1.3470 13.09 1.5400
1.550 0.6754 0.2533 0.3750 0.8057 1.0573 4.1748 1.0452 0.1543 0.4259 0.6841 0.9132 2.6363 3.6057 1.3538 13.38 1.5500

1.560 0.6726 0.2496 0.3710 0.8092 1.0508 4.2105 1.0467 0.1579 0.4252 0.6809 0.9097 2.6725 3.6450 1.3606 13.68 1.5600
1.570 0.6698 0.2459 0.3672 0.8126 1.0443 4.2464 1.0481 0.1615 0.4243 0.6777 0.9062 2.7091 3.6846 1.3674 13.97 1.5700
1.580 0.6670 0.2423 0.3633 0.8161 1.0378 4.2825 1.0496 0.1651 0.4235 0.6746 0.9026 2.7458 3.7244 1.3742 14.27 1.5800
1.590 0.6642 0.2388 0.3595 0.8195 1.0312 4.3187 1.0511 0.1688 0.4226 0.6715 0.8989 2.7828 3.7646 1.3811 14.56 1.5900
1.600 0.6614 0.2353 0.3557 0.8230 1.0246 4.3551 1.0526 0.1724 0.4216 0.6684 0.8952 2.8200 3.8050 1.3880 14.86 1.6000

1.610 0.6586 0.2318 0.3520 0.8263 1.0180 4.3916 1.0541 0.1760 0.4206 0.6655 0.8915 2.8575 3.8456 1.3949 15.16 1.6100
1.620 0.6558 0.2284 0.3483 0.8297 1.0114 4.4282 1.0555 0.1795 0.4196 0.6625 0.8877 2.8951 3.8866 1.4018 15.45 1.6200
1.630 0.6530 0.2250 0.3446 0.8331 1.0047 4.4651 1.0570 0.1831 0.4185 0.6596 0.8838 2.9331 3.9278 1.4088 15.75 1.6300
1.640 0.6502 0.2217 0.3409 0.8364 0.9980 4.5020 1.0585 0.1867 0.4174 0.6568 0.8799 2.9712 3.9693 1.4158 16.04 1.6400
1.650 0.6475 0.2184 0.3373 0.8397 0.9913 4.5392 1.0600 0.1902 0.4162 0.6540 0.8760 3.0096 4.0110 1.4228 16.34 1.6500

1.660 0.6447 0.2151 0.3337 0.8430 0.9846 4.5765 1.0615 0.1938 0.4150 0.6512 0.8720 3.0482 4.0531 1.4299 16.63 1.6600
1.670 0.6419 0.2119 0.3302 0.8462 0.9779 4.6139 1.0630 0.1973 0.4138 0.6485 0.8680 3.0871 4.0953 1.4369 16.93 1.6700
1.680 0.6392 0.2088 0.3266 0.8495 0.9712 4.6515 1.0645 0.2008 0.4125 0.6458 0.8639 3.1261 4.1379 1.4440 17.22 1.6800
1.690 0.6364 0.2057 0.3232 0.8527 0.9644 4.6892 1.0660 0.2043 0.4112 0.6431 0.8599 3.1655 4.1807 1.4512 17.52 1.6900
1.700 0.6337 0.2026 0.3197 0.8559 0.9577 4.7272 1.0674 0.2078 0.4098 0.6405 0.8557 3.2050 4.2238 1.4583 17.81 1.7000

1.710 0.6310 0.1996 0.3163 0.8591 0.9509 4.7652 1.0689 0.2113 0.4085 0.6380 0.8516 3.2448 4.2672 1.4655 18.10 1.7100
1.720 0.6283 0.1966 0.3129 0.8622 0.9442 4.8035 1.0704 0.2147 0.4071 0.6355 0.8474 3.2848 4.3108 1.4727 18.40 1.7200
1.730 0.6256 0.1936 0.3095 0.8654 0.9374 4.8418 1.0719 0.2182 0.4056 0.6330 0.8431 3.3251 4.3547 1.4800 18.69 1.7300
1.740 0.6229 0.1907 0.3062 0.8685 0.9307 4.8804 1.0734 0.2216 0.4041 0.6305 0.8389 3.3655 4.3989 1.4873 18.98 1.7400
1.750 0.6202 0.1878 0.3029 0.8716 0.9239 4.9191 1.0749 0.2250 0.4026 0.6281 0.8346 3.4063 4.4433 1.4946 19.27 1.7500

26
 γ=1.400
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉 𝑝0𝑠 𝑝𝑠 𝑝0𝑠 𝑇𝑠
𝑀 2 𝑀𝑠 𝑀
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝𝑇0 𝐷 𝑝0 𝑝0 𝑝 𝑝 𝑇 𝑃𝑀

1.760 0.6175 0.1850 0.2996 0.8747 0.9172 4.9580 1.0764 0.228 0.4011 0.6257 0.8302 3.4472 4.4880 1.5019 19.56 1.7600
1.770 0.6148 0.1822 0.2964 0.8777 0.9104 4.9970 1.0779 0.232 0.3996 0.6234 0.8259 3.4884 4.5330 1.5093 19.86 1.7700
1.780 0.6121 0.1794 0.2931 0.8808 0.9037 5.0362 1.0793 0.2352 0.3980 0.6210 0.8215 3.5298 4.5782 1.5167 20.15 1.7800
1.790 0.6095 0.1767 0.2900 0.8838 0.8970 5.0755 1.0808 0.2385 0.3964 0.6188 0.8171 3.5715 4.6237 1.5241 20.44 1.7900
1.800 0.6068 0.1740 0.2868 0.8868 0.8902 5.1150 1.0823 0.2419 0.3947 0.6165 0.8127 3.6133 4.6695 1.5316 20.73 1.8000

1.810 0.6041 0.1714 0.2837 0.8898 0.8835 5.1547 1.0838 0.2452 0.3931 0.6143 0.8082 3.6555 4.7155 1.5391 21.01 1.8100
1.820 0.6015 0.1688 0.2806 0.8927 0.8768 5.1945 1.0852 0.2485 0.3914 0.6121 0.8038 3.6978 4.7618 1.5466 21.30 1.8200
1.830 0.5989 0.1662 0.2776 0.8957 0.8701 5.2345 1.0867 0.2518 0.3897 0.6099 0.7993 3.7404 4.8084 1.5541 21.59 1.8300
1.840 0.5963 0.1637 0.2745 0.8986 0.8634 5.2747 1.0882 0.2551 0.3879 0.6078 0.7948 3.7832 4.8552 1.5617 21.88 1.8400
1.850 0.5936 0.1612 0.2715 0.9015 0.8568 5.3150 1.0896 0.2583 0.3862 0.6057 0.7902 3.8263 4.9023 1.5693 22.16 1.8500

1.860 0.5910 0.1587 0.2686 0.9044 0.8501 5.3555 1.0911 0.2616 0.3844 0.6036 0.7857 3.8695 4.9497 1.5770 22.45 1.8600
1.870 0.5884 0.1563 0.2656 0.9072 0.8435 5.3962 1.0926 0.2648 0.3826 0.6016 0.7811 3.9131 4.9973 1.5847 22.73 1.8700
1.880 0.5859 0.1539 0.2627 0.9101 0.8368 5.4370 1.0940 0.2680 0.3808 0.5996 0.7765 3.9568 5.0452 1.5924 23.02 1.8800
1.890 0.5833 0.1516 0.2598 0.9129 0.8302 5.4780 1.0955 0.2712 0.3790 0.5976 0.7720 4.0008 5.0934 1.6001 23.30 1.8900
1.900 0.5807 0.1492 0.2570 0.9157 0.8237 5.5191 1.0969 0.2743 0.3771 0.5956 0.7674 4.0450 5.1418 1.6079 23.59 1.9000

1.910 0.5782 0.1470 0.2542 0.9185 0.8171 5.5604 1.0984 0.2775 0.3753 0.5937 0.7627 4.0895 5.1905 1.6157 23.87 1.9100
1.920 0.5756 0.1447 0.2514 0.9213 0.8106 5.6019 1.0998 0.2806 0.3734 0.5918 0.7581 4.1341 5.2394 1.6236 24.15 1.9200
1.930 0.5731 0.1425 0.2486 0.9240 0.8041 5.6435 1.1012 0.2837 0.3715 0.5899 0.7535 4.1791 5.2886 1.6314 24.43 1.9300
1.940 0.5705 0.1403 0.2459 0.9268 0.7976 5.6853 1.1027 0.2868 0.3696 0.5880 0.7488 4.2242 5.3381 1.6394 24.71 1.9400
1.950 0.5680 0.1381 0.2432 0.9295 0.7911 5.7273 1.1041 0.2899 0.3677 0.5862 0.7442 4.2696 5.3878 1.6473 24.99 1.9500

1.960 0.5655 0.1360 0.2405 0.9322 0.7846 5.7695 1.1055 0.2929 0.3657 0.5844 0.7395 4.3152 5.4378 1.6553 25.27 1.9600
1.970 0.5630 0.1339 0.2378 0.9349 0.7782 5.8118 1.1069 0.2960 0.3638 0.5826 0.7349 4.3611 5.4881 1.6633 25.55 1.9700
1.980 0.5605 0.1318 0.2352 0.9375 0.7718 5.8542 1.1084 0.2990 0.3618 0.5808 0.7302 4.4071 5.5386 1.6713 25.83 1.9800
1.990 0.5580 0.1298 0.2326 0.9402 0.7655 5.8969 1.1098 0.3020 0.3598 0.5791 0.7255 4.4535 5.5894 1.6794 26.10 1.9900
2.000 0.5556 0.1278 0.2300 0.9428 0.7591 5.9397 1.1112 0.3050 0.3579 0.5774 0.7209 4.5000 5.6404 1.6875 26.38 2.0000

27
 TABLE 21. GAS FLOW TABLES (γ=1.333): SUBSONIC FLOW
1.3
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
0.010 1.0000 0.9999 1.0000 0.0058 0.0231 0.0231 43.2958 7493.200 0.0001
0.020 0.9999 0.9997 0.9998 0.0115 0.0462 0.0462 21.6560 1868.007 0.0003
0.030 0.9999 0.9994 0.9996 0.0173 0.0693 0.0693 14.4464 826.7890 0.0006
0.040 0.9997 0.9989 0.9992 0.0231 0.0923 0.0924 10.8442 462.6179 0.0011
0.050 0.9996 0.9983 0.9988 0.0288 0.1153 0.1155 8.6851 294.2161 0.0017

0.060 0.9994 0.9976 0.9982 0.0346 0.1383 0.1386 7.2475 202.8455 0.0024
0.070 0.9992 0.9967 0.9976 0.0404 0.1612 0.1618 6.2222 147.8292 0.0033
0.080 0.9989 0.9957 0.9968 0.0461 0.1841 0.1849 5.4546 112.1800 0.0042
0.090 0.9987 0.9946 0.9960 0.0519 0.2069 0.2080 4.8587 87.7848 0.0054
0.100 0.9983 0.9934 0.9950 0.0577 0.2297 0.2312 4.3831 70.3719 0.0066

0.110 0.9980 0.9920 0.9940 0.0634 0.2523 0.2544 3.9949 57.5186 0.0080
0.120 0.9976 0.9905 0.9928 0.0692 0.2749 0.2775 3.6724 47.7680 0.0095
0.130 0.9972 0.9888 0.9916 0.0749 0.2974 0.3007 3.4003 40.2012 0.0111
0.140 0.9967 0.9870 0.9903 0.0807 0.3197 0.3239 3.1678 34.2155 0.0129
0.150 0.9963 0.9851 0.9888 0.0864 0.3420 0.3471 2.9670 29.4027 0.0148

0.160 0.9958 0.9831 0.9873 0.0921 0.3641 0.3704 2.7920 25.4777 0.0168
0.170 0.9952 0.9810 0.9857 0.0979 0.3861 0.3936 2.6383 22.2372 0.0189
0.180 0.9946 0.9787 0.9840 0.1036 0.4080 0.4169 2.5022 19.5326 0.0211
0.190 0.9940 0.9763 0.9822 0.1093 0.4298 0.4402 2.3809 17.2536 0.0235
0.200 0.9934 0.9738 0.9803 0.1150 0.4514 0.4635 2.2724 15.3166 0.0260

0.210 0.9927 0.9711 0.9783 0.1207 0.4728 0.4869 2.1747 13.6578 0.0285
0.220 0.9920 0.9684 0.9762 0.1264 0.4941 0.5102 2.0863 12.2273 0.0312
0.230 0.9913 0.9655 0.9740 0.1321 0.5152 0.5336 2.0061 10.9859 0.0340
0.240 0.9905 0.9625 0.9717 0.1378 0.5362 0.5570 1.9330 9.9026 0.0370
0.250 0.9897 0.9594 0.9694 0.1435 0.5569 0.5805 1.8662 8.9522 0.0400

0.260 0.9889 0.9562 0.9669 0.1492 0.5775 0.6040 1.8049 8.1146 0.0431
0.270 0.9880 0.9529 0.9644 0.1549 0.5979 0.6275 1.7486 7.3731 0.0463
0.280 0.9871 0.9494 0.9618 0.1605 0.6181 0.6510 1.6966 6.7140 0.0496
0.290 0.9862 0.9459 0.9591 0.1662 0.6380 0.6746 1.6486 6.1261 0.0530
0.300 0.9852 0.9422 0.9563 0.1718 0.6578 0.6982 1.6042 5.5998 0.0565

0.310 0.9843 0.9384 0.9534 0.1775 0.6774 0.7218 1.5629 5.1272 0.0601
0.320 0.9832 0.9346 0.9505 0.1831 0.6967 0.7455 1.5245 4.7016 0.0638
0.330 0.9822 0.9306 0.9475 0.1887 0.7158 0.7692 1.4888 4.3173 0.0675
0.340 0.9811 0.9265 0.9444 0.1943 0.7347 0.7929 1.4555 3.9693 0.0714
0.350 0.9800 0.9224 0.9412 0.1999 0.7533 0.8167 1.4244 3.6535 0.0753

0.360 0.9789 0.9181 0.9379 0.2055 0.7717 0.8405 1.3953 3.3663 0.0793
0.370 0.9777 0.9137 0.9346 0.2111 0.7898 0.8644 1.3680 3.1046 0.0834
0.380 0.9765 0.9093 0.9311 0.2167 0.8077 0.8883 1.3425 2.8655 0.0875
0.390 0.9753 0.9047 0.9276 0.2223 0.8253 0.9122 1.3185 2.6469 0.0917
0.400 0.9741 0.9001 0.9241 0.2278 0.8427 0.9362 1.2959 2.4466 0.0960

0.410 0.9728 0.8954 0.9204 0.2334 0.8598 0.9603 1.2747 2.2627 0.1003
0.420 0.9715 0.8906 0.9167 0.2389 0.8766 0.9843 1.2548 2.0937 0.1047
0.430 0.9701 0.8857 0.9130 0.2444 0.8932 1.0085 1.2360 1.9382 0.1091
0.440 0.9688 0.8807 0.9091 0.2499 0.9095 1.0326 1.2183 1.7949 0.1136
0.450 0.9674 0.8757 0.9052 0.2554 0.9255 1.0569 1.2016 1.6627 0.1182

0.460 0.9660 0.8706 0.9012 0.2609 0.9412 1.0811 1.1858 1.5405 0.1228
0.470 0.9645 0.8654 0.8972 0.2664 0.9567 1.1055 1.1710 1.4276 0.1274
0.480 0.9631 0.8601 0.8931 0.2718 0.9718 1.1299 1.1569 1.3231 0.1321
0.490 0.9616 0.8548 0.8890 0.2773 0.9867 1.1543 1.1436 1.2263 0.1368
0.500 0.9600 0.8494 0.8847 0.2827 1.0012 1.1788 1.1310 1.1365 0.1415

28
 γ=1.333
1.3
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
0.510 0.9585 0.8439 0.8805 0.2881 1.0155 1.2033 1.1192 1.053 0.1463
0.520 0.9569 0.8384 0.8761 0.2935 1.0295 1.2279 1.1079 0.976 0.1511
0.530 0.9553 0.8328 0.8717 0.2989 1.0431 1.2526 1.0973 0.9041 0.1559
0.540 0.9537 0.8271 0.8673 0.3043 1.0565 1.2773 1.0872 0.8373 0.1608
0.550 0.9520 0.8214 0.8628 0.3097 1.0696 1.3021 1.0777 0.7752 0.1656

0.560 0.9504 0.8157 0.8583 0.3150 1.0823 1.3269 1.0687 0.7174 0.1705
0.570 0.9487 0.8099 0.8537 0.3204 1.0948 1.3518 1.0601 0.6636 0.1754
0.580 0.9470 0.8040 0.8490 0.3257 1.1069 1.3768 1.0520 0.6136 0.1803
0.590 0.9452 0.7981 0.8443 0.3310 1.1188 1.4018 1.0444 0.5669 0.1852
0.600 0.9434 0.7921 0.8396 0.3363 1.1303 1.4269 1.0371 0.5235 0.1901

0.610 0.9417 0.7861 0.8348 0.3416 1.1415 1.4521 1.0303 0.4830 0.1950
0.620 0.9398 0.7801 0.8300 0.3469 1.1524 1.4773 1.0238 0.4452 0.1999
0.630 0.9380 0.7740 0.8252 0.3521 1.1630 1.5026 1.0176 0.4101 0.2048
0.640 0.9362 0.7679 0.8203 0.3573 1.1733 1.5280 1.0118 0.3773 0.2096
0.650 0.9343 0.7618 0.8153 0.3626 1.1833 1.5534 1.0063 0.3467 0.2145

0.660 0.9324 0.7556 0.8104 0.3678 1.1930 1.5789 1.0011 0.3183 0.2194
0.670 0.9305 0.7494 0.8054 0.3729 1.2023 1.6045 0.9962 0.2918 0.2242
0.680 0.9285 0.7431 0.8003 0.3781 1.2114 1.6301 0.9916 0.2671 0.2290
0.690 0.9266 0.7368 0.7953 0.3833 1.2201 1.6559 0.9872 0.2441 0.2338
0.700 0.9246 0.7306 0.7902 0.3884 1.2285 1.6817 0.9831 0.2227 0.2386

0.710 0.9226 0.7242 0.7850 0.3935 1.2367 1.7075 0.9792 0.2028 0.2433
0.720 0.9205 0.7179 0.7799 0.3986 1.2445 1.7335 0.9755 0.1843 0.2480
0.730 0.9185 0.7116 0.7747 0.4037 1.2520 1.7595 0.9721 0.1671 0.2527
0.740 0.9164 0.7052 0.7695 0.4088 1.2592 1.7856 0.9688 0.1512 0.2574
0.750 0.9144 0.6988 0.7643 0.4139 1.2661 1.8118 0.9658 0.1364 0.2620

0.760 0.9123 0.6924 0.7590 0.4189 1.2727 1.8381 0.9629 0.1227 0.2666
0.770 0.9102 0.6860 0.7537 0.4239 1.2790 1.8644 0.9603 0.1100 0.2711
0.780 0.9080 0.6796 0.7484 0.4289 1.2850 1.8908 0.9578 0.0983 0.2756
0.790 0.9059 0.6732 0.7431 0.4339 1.2907 1.9174 0.9554 0.0875 0.2800
0.800 0.9037 0.6668 0.7378 0.4389 1.2961 1.9440 0.9533 0.0776 0.2844

0.810 0.9015 0.6603 0.7325 0.4438 1.3013 1.9706 0.9513 0.0685 0.2888
0.820 0.8993 0.6539 0.7271 0.4487 1.3061 1.9974 0.9494 0.0601 0.2930
0.830 0.8971 0.6475 0.7217 0.4536 1.3107 2.0243 0.9477 0.0524 0.2973
0.840 0.8949 0.6411 0.7164 0.4585 1.3149 2.0512 0.9461 0.0454 0.3015
0.850 0.8926 0.6346 0.7110 0.4634 1.3189 2.0782 0.9446 0.0391 0.3056

0.860 0.8904 0.6282 0.7056 0.4683 1.3226 2.1053 0.9433 0.0333 0.3097
0.870 0.8881 0.6218 0.7002 0.4731 1.3260 2.1326 0.9420 0.0281 0.3137
0.880 0.8858 0.6154 0.6948 0.4779 1.3292 2.1599 0.9409 0.0235 0.3176
0.890 0.8835 0.6090 0.6893 0.4827 1.3321 2.1873 0.9399 0.0193 0.3215
0.900 0.8812 0.6026 0.6839 0.4875 1.3347 2.2147 0.9390 0.0156 0.3253

0.910 0.8788 0.5963 0.6785 0.4923 1.3370 2.2423 0.9383 0.0124 0.3291
0.920 0.8765 0.5899 0.6731 0.4970 1.3391 2.2700 0.9376 0.0096 0.3328
0.930 0.8741 0.5836 0.6676 0.5018 1.3410 2.2978 0.9370 0.0072 0.3364
0.940 0.8717 0.5773 0.6622 0.5065 1.3425 2.3256 0.9365 0.0052 0.3400
0.950 0.8694 0.5710 0.6568 0.5111 1.3439 2.3536 0.9360 0.0035 0.3435

0.960 0.8670 0.5647 0.6514 0.5158 1.3449 2.3817 0.9357 0.0022 0.3469
0.970 0.8646 0.5585 0.6459 0.5205 1.3458 2.4098 0.9354 0.0012 0.3502
0.980 0.8621 0.5522 0.6405 0.5251 1.3464 2.4381 0.9353 0.0005 0.3535
0.990 0.8597 0.5460 0.6351 0.5297 1.3467 2.4664 0.9351 0.0001 0.3567
1.000 0.8573 0.5398 0.6297 0.5343 1.3468 2.4949 0.9351 0.0000 0.3598

29
 TABLE 22. GAS FLOW TABLES (γ=1.333): SUBSONIC FLOW
1.3
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
1.010 0.8548 0.5337 0.6243 0.5389 1.3467 2.5234 0.9351 0.000 0.3628
1.020 0.8524 0.5276 0.6189 0.5434 1.3464 2.5521 0.9352 0.000 0.3658
1.030 0.8499 0.5215 0.6136 0.5479 1.3458 2.5809 0.9354 0.0011 0.3687
1.040 0.8474 0.5154 0.6082 0.5525 1.3450 2.6097 0.9356 0.0019 0.3715
1.050 0.8449 0.5093 0.6028 0.5569 1.3440 2.6387 0.9359 0.0029 0.3743

1.060 0.8424 0.5033 0.5975 0.5614 1.3428 2.6678 0.9363 0.0042 0.3769
1.070 0.8399 0.4974 0.5922 0.5659 1.3414 2.6970 0.9367 0.0056 0.3795
1.080 0.8374 0.4914 0.5869 0.5703 1.3397 2.7263 0.9371 0.0071 0.3820
1.090 0.8349 0.4855 0.5816 0.5747 1.3379 2.7557 0.9376 0.0089 0.3845
1.100 0.8323 0.4796 0.5763 0.5791 1.3359 2.7852 0.9381 0.0108 0.3868

1.110 0.8298 0.4738 0.5710 0.5835 1.3337 2.8148 0.9387 0.0128 0.3891
1.120 0.8272 0.4680 0.5658 0.5878 1.3313 2.8446 0.9394 0.0150 0.3913
1.130 0.8247 0.4622 0.5605 0.5922 1.3287 2.8744 0.9401 0.0173 0.3934
1.140 0.8221 0.4565 0.5553 0.5965 1.3259 2.9043 0.9408 0.0197 0.3954
1.150 0.8195 0.4508 0.5501 0.6008 1.3229 2.9344 0.9415 0.0223 0.3974

1.160 0.8170 0.4452 0.5449 0.6050 1.3198 2.9646 0.9424 0.0250 0.3993
1.170 0.8144 0.4396 0.5398 0.6093 1.3165 2.9949 0.9432 0.0277 0.4011
1.180 0.8118 0.4340 0.5347 0.6135 1.3131 3.0253 0.9441 0.0306 0.4028
1.190 0.8092 0.4285 0.5295 0.6177 1.3094 3.0558 0.9450 0.0335 0.4044
1.200 0.8066 0.4230 0.5245 0.6219 1.3057 3.0864 0.9459 0.0366 0.4060

1.210 0.8040 0.4176 0.5194 0.6261 1.3017 3.1172 0.9469 0.0397 0.4075
1.220 0.8014 0.4122 0.5143 0.6302 1.2976 3.1481 0.9479 0.0429 0.4089
1.230 0.7988 0.4068 0.5093 0.6344 1.2934 3.1791 0.9489 0.0462 0.4102
1.240 0.7962 0.4015 0.5043 0.6385 1.2890 3.2102 0.9500 0.0495 0.4115
1.250 0.7936 0.3963 0.4994 0.6426 1.2845 3.2414 0.9511 0.0529 0.4127

1.260 0.7909 0.3911 0.4944 0.6466 1.2798 3.2727 0.9522 0.0564 0.4138
1.270 0.7883 0.3859 0.4895 0.6507 1.2751 3.3042 0.9533 0.0599 0.4148
1.280 0.7857 0.3808 0.4846 0.6547 1.2701 3.3358 0.9545 0.0634 0.4158
1.290 0.7830 0.3757 0.4798 0.6587 1.2651 3.3675 0.9557 0.0670 0.4167
1.300 0.7804 0.3706 0.4749 0.6627 1.2599 3.3993 0.9569 0.0707 0.4175

1.310 0.7778 0.3657 0.4701 0.6667 1.2547 3.4313 0.9581 0.0744 0.4182
1.320 0.7751 0.3607 0.4654 0.6706 1.2493 3.4633 0.9594 0.0781 0.4189
1.330 0.7725 0.3558 0.4606 0.6746 1.2438 3.4955 0.9606 0.0819 0.4195
1.340 0.7698 0.3510 0.4559 0.6785 1.2382 3.5279 0.9619 0.0857 0.4200
1.350 0.7672 0.3462 0.4512 0.6824 1.2325 3.5603 0.9632 0.0895 0.4205

1.360 0.7646 0.3414 0.4465 0.6862 1.2266 3.5929 0.9645 0.0934 0.4209
1.370 0.7619 0.3367 0.4419 0.6901 1.2207 3.6256 0.9659 0.0973 0.4212
1.380 0.7593 0.3320 0.4373 0.6939 1.2147 3.6584 0.9672 0.1012 0.4215
1.390 0.7566 0.3274 0.4328 0.6977 1.2086 3.6914 0.9686 0.1051 0.4216
1.400 0.7540 0.3229 0.4282 0.7015 1.2025 3.7245 0.9700 0.1091 0.4218

1.410 0.7513 0.3183 0.4237 0.7053 1.1962 3.7577 0.9714 0.1130 0.4218
1.420 0.7487 0.3139 0.4192 0.7090 1.1899 3.7910 0.9728 0.1170 0.4218
1.430 0.7460 0.3094 0.4148 0.7127 1.1835 3.8245 0.9742 0.1210 0.4217
1.440 0.7434 0.3051 0.4104 0.7164 1.1770 3.8581 0.9756 0.1250 0.4216
1.450 0.7407 0.3007 0.4060 0.7201 1.1704 3.8918 0.9771 0.1290 0.4214

1.460 0.7381 0.2965 0.4017 0.7238 1.1638 3.9257 0.9785 0.1331 0.4212
1.470 0.7354 0.2922 0.3974 0.7275 1.1571 3.9597 0.9800 0.1371 0.4209
1.480 0.7328 0.2880 0.3931 0.7311 1.1504 3.9938 0.9815 0.1411 0.4205
1.490 0.7301 0.2839 0.3888 0.7347 1.1435 4.0281 0.9829 0.1452 0.4201
1.500 0.7275 0.2798 0.3846 0.7383 1.1367 4.0625 0.9844 0.1492 0.4196

30
 γ=1.333
1.3
𝑉 𝐹 4𝑐𝑓 𝐿𝑚𝑎𝑥 1 2
𝑇 𝑝 𝜌 𝑚̇ 𝑐𝑝 𝑇0 𝑚̇ 𝑐𝑝𝑇0 𝜌𝑉
𝑀 2
𝑇0 𝑝0 𝜌0 𝑐𝑝𝑇0 𝐴𝑝0 𝐴𝑝 𝑚̇ 𝑐𝑝 𝑇0 𝐷 𝑝0
1.510 0.7248 0.2758 0.3804 0.7419 1.1298 4.0970 0.9859 0.153 0.4191
1.520 0.7222 0.2718 0.3763 0.7454 1.1228 4.1317 0.9874 0.157 0.4185
1.530 0.7195 0.2678 0.3722 0.7489 1.1158 4.1665 0.9889 0.1613 0.4178
1.540 0.7169 0.2639 0.3681 0.7524 1.1087 4.2014 0.9905 0.1654 0.4171
1.550 0.7143 0.2600 0.3641 0.7559 1.1016 4.2365 0.9920 0.1694 0.4164

1.560 0.7116 0.2562 0.3600 0.7594 1.0945 4.2717 0.9935 0.1734 0.4156
1.570 0.7090 0.2524 0.3561 0.7629 1.0873 4.3070 0.9950 0.1775 0.4147
1.580 0.7064 0.2487 0.3521 0.7663 1.0801 4.3425 0.9966 0.1815 0.4138
1.590 0.7038 0.2450 0.3482 0.7697 1.0729 4.3782 0.9981 0.1855 0.4129
1.600 0.7011 0.2414 0.3443 0.7731 1.0656 4.4139 0.9997 0.1895 0.4119

1.610 0.6985 0.2378 0.3405 0.7765 1.0583 4.4498 1.0012 0.1935 0.4109
1.620 0.6959 0.2343 0.3367 0.7799 1.0510 4.4859 1.0028 0.1975 0.4098
1.630 0.6933 0.2308 0.3329 0.7832 1.0436 4.5220 1.0043 0.2015 0.4087
1.640 0.6907 0.2273 0.3291 0.7865 1.0363 4.5584 1.0059 0.2055 0.4075
1.650 0.6881 0.2239 0.3254 0.7898 1.0289 4.5948 1.0075 0.2094 0.4063

1.660 0.6855 0.2206 0.3217 0.7931 1.0215 4.6314 1.0090 0.2134 0.4051
1.670 0.6829 0.2172 0.3181 0.7964 1.0141 4.6682 1.0106 0.2173 0.4038
1.680 0.6803 0.2139 0.3145 0.7996 1.0066 4.7051 1.0122 0.2213 0.4025
1.690 0.6777 0.2107 0.3109 0.8028 0.9992 4.7421 1.0137 0.2252 0.4011
1.700 0.6751 0.2075 0.3074 0.8061 0.9918 4.7793 1.0153 0.2291 0.3997

1.710 0.6726 0.2044 0.3039 0.8093 0.9843 4.8166 1.0169 0.2330 0.3983
1.720 0.6700 0.2012 0.3004 0.8124 0.9769 4.8541 1.0184 0.2369 0.3968
1.730 0.6674 0.1982 0.2969 0.8156 0.9694 4.8917 1.0200 0.2407 0.3953
1.740 0.6649 0.1951 0.2935 0.8187 0.9620 4.9294 1.0216 0.2446 0.3938
1.750 0.6623 0.1922 0.2901 0.8218 0.9545 4.9673 1.0232 0.2484 0.3922

1.760 0.6597 0.1892 0.2868 0.8249 0.9471 5.0054 1.0247 0.2522 0.3906
1.770 0.6572 0.1863 0.2835 0.8280 0.9396 5.0435 1.0263 0.2560 0.3890
1.780 0.6546 0.1834 0.2802 0.8311 0.9322 5.0819 1.0279 0.2598 0.3874
1.790 0.6521 0.1806 0.2770 0.8341 0.9248 5.1204 1.0294 0.2636 0.3857
1.800 0.6496 0.1778 0.2737 0.8372 0.9173 5.1590 1.0310 0.2673 0.3840

1.810 0.6471 0.1751 0.2706 0.8402 0.9099 5.1978 1.0326 0.2711 0.3822
1.820 0.6445 0.1723 0.2674 0.8432 0.9025 5.2367 1.0341 0.2748 0.3805
1.830 0.6420 0.1697 0.2643 0.8461 0.8951 5.2758 1.0357 0.2785 0.3787
1.840 0.6395 0.1670 0.2612 0.8491 0.8878 5.3150 1.0373 0.2822 0.3769
1.850 0.6370 0.1644 0.2581 0.8521 0.8804 5.3544 1.0388 0.2858 0.3751

1.860 0.6345 0.1619 0.2551 0.8550 0.8731 5.3939 1.0404 0.2895 0.3732
1.870 0.6320 0.1593 0.2521 0.8579 0.8658 5.4336 1.0419 0.2931 0.3714
1.880 0.6295 0.1568 0.2491 0.8608 0.8585 5.4734 1.0435 0.2967 0.3695
1.890 0.6271 0.1544 0.2462 0.8636 0.8512 5.5134 1.0450 0.3003 0.3676
1.900 0.6246 0.1520 0.2433 0.8665 0.8439 5.5535 1.0466 0.3039 0.3656

1.910 0.6221 0.1496 0.2404 0.8693 0.8367 5.5938 1.0481 0.3074 0.3637
1.920 0.6197 0.1472 0.2376 0.8722 0.8295 5.6342 1.0497 0.3110 0.3617
1.930 0.6172 0.1449 0.2348 0.8750 0.8223 5.6748 1.0512 0.3145 0.3598
1.940 0.6148 0.1426 0.2320 0.8778 0.8152 5.7155 1.0527 0.3180 0.3578
1.950 0.6123 0.1404 0.2292 0.8805 0.8081 5.7564 1.0543 0.3215 0.3558

1.960 0.6099 0.1382 0.2265 0.8833 0.8010 5.7974 1.0558 0.3249 0.3537
1.970 0.6075 0.1360 0.2238 0.8860 0.7939 5.8386 1.0573 0.3284 0.3517
1.980 0.6051 0.1338 0.2212 0.8888 0.7869 5.8800 1.0588 0.3318 0.3497
1.990 0.6026 0.1317 0.2185 0.8915 0.7799 5.9215 1.0603 0.3352 0.3476
2.000 0.6002 0.1296 0.2159 0.8942 0.7729 5.9631 1.0619 0.3386 0.3455

31
 TABLE 23. PROPERTIES OF GASES AT SEA LEVEL CONDITIONS
The following data are at p =101.325 kN/m2 and T sl =288.15 K.

Air CO 2 H2 He
Density p kg/m 3 1.225 1.86 0.0852 0.1693
Viscosity μ kg/m s 17.9x10 6 14.4 x10 -6 8.9x10 4 19.7 x10 -6
Kinematic viscosity ν m 2 /s 14.64x10 -6 7.7x10 -6 104x10 6 116x10 6
Speed of sound m/s 340 264 1292 988
T hermal conductivity λ W/m K 0.0252 0.0153 0.18 0.15

THE INTERNATIONAL STANDARD ATMOSPHERE

The sea-level condition for the International Standard Atmosphere is given by

𝑝𝑠𝑙 = 101.325 kN/m2 = 760 mm Hg

𝑇𝑠𝑙 = 15 °C = 288.15 K
for which

𝜌𝑠𝑙 = 1.225 kg/m3

𝜇𝑠𝑙
𝜈𝑠𝑙 = = 16.64 × 10−6 m2 /s
𝜌𝑠𝑙

𝑎𝑠𝑙 = √𝛾𝑅𝑇𝑠𝑙 = 340 m/s

The speed of sound at altitude is given by

𝑎 𝑇
=√
𝑎𝑠𝑙 𝑇𝑠𝑙

The viscosity at altitude is given by

1.458 × 10−6 𝑇 3/2

𝜇= kg m-1 s-1
𝑇 + 110.4

TABLE 24. PROPERTIES OF THE INTERNATIONAL

STANDARD ATMOSPHERE AT SELECTED ALTITUDES
Altitude T emperature Pressure Density
feet km K bar kg/m 3
0 0.00 288.15 1.013 1.225
31,000 9.45 226.73 0.287 0.442
33,000 10.06 222.77 0.262 0.410
35,000 10.67 218.81 0.238 0.380
37,000 11.28 216.65 0.217 0.348
39,000 11.89 216.65 0.197 0.316
41,000 12.50 216.65 0.179 0.287
51,000 15.54 216.65 0.111 0.178

32
TABLE 25. PROPERTIES OF THE INTERNATIONAL
STANDARD ATMOSPHERE AT ALTITUDE
Height 𝑇 𝑝 𝜌
m 𝑇𝑠𝑙 𝑝𝑠𝑙 𝜌𝑠𝑙 𝑠𝑙
0 1.0000 1.0000 1.0000 1.0000
1000 0.9774 0.8870 0.9075 1.0826
2000 0.9549 0.7846 0.8217 1.1739
3000 0.9324 0.6920 0.7422 1.2753
4000 0.9098 0.6085 0.6689 1.3881
5000 0.8873 0.5334 0.6012 1.5138
6000 0.8648 0.4660 0.5389 1.6543
7000 0.8423 0.4057 0.4816 1.8117
8000 0.8198 0.3519 0.4292 1.9887
9000 0.7973 0.3040 0.3813 2.1881
10000 0.7748 0.2615 0.3376 2.4137
11000 0.7523 0.2240 0.2978 2.6697
12000 0.7519 0.1915 0.2546 3.1206
13000 0.7519 0.1636 0.2176 3.6514
14000 0.7519 0.1398 0.1860 4.2722
15000 0.7519 0.1195 0.1590 4.9983
16000 0.7519 0.1022 0.1359 5.8476
17000 0.7519 0.0873 0.1162 6.8408
18000 0.7519 0.0747 0.0993 8.0023
19000 0.7519 0.0638 0.0849 9.3606
20000 0.7519 0.0546 0.0726 10.9488
21000 0.7551 0.0467 0.0618 12.9031
22000 0.7585 0.0399 0.0527 15.2021
23000 0.7620 0.0342 0.0449 17.8964
24000 0.7654 0.0293 0.0383 21.0515
25000 0.7689 0.0252 0.0327 24.7434
26000 0.7723 0.0216 0.0280 29.0603
27000 0.7758 0.0186 0.0239 34.1039
28000 0.7792 0.0160 0.0205 39.9926
29000 0.7826 0.0137 0.0175 46.8626
30000 0.7861 0.0118 0.0150 54.8714

33
 APPENDIX A

PRINCIPAL SOURCES OF DATA

Wagner, W. and Pruss, A., "The IAPWS Formulation 1995 for the Thermodynamic Properties of
Ordinary Water Substance for General and Scientific Use," J. Phys. Chem. Ref. Data, 31(2):387-535,
2002.
Tillner-Roth, R. and Baehr, H.D., "An international standard formulation of the thermodynamic
properties of 1,1,1,2-tetrafluoroethane (HFC-134a) for temperatures from 170 K to 455 K at pressures
up to 70 MPa," J. Phys. Chem. Ref. Data, 23:657-729, 1994.
Lemmon, E.W., Huber, M.L., McLinden, M.O. NIST Standard Reference Database 23: Reference
Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of
Standards and Technology, Standard Reference Data Program, Gaithersburg, 2013.
Stull, D.R., and Prophet, H. (eds). “Janaf Thermochemical Tables,” The Dow Chemical Company,
Midland, Michigan (U.S. G.P.O., 1971, Supplements 1974, 1975, 1978).
US Standard Atmosphere, NOAA-S/T 76-1562, 1976.

34