
Isentropic Flow Equation Derivations

Glenn
Research
Center

As a gas is forced through a tube, the gas molecules are deflected
by the walls of the tube. If the speed of the gas is much less than
the speed of sound of the gas, the
density
of the gas remains
constant. However, as the speed of the flow approaches the
speed of sound
we must consider
compressibility effects
on the gas. The density of the gas varies from
one location to the next.
If the flow is very gradually compressed (area decreases) and then
gradually expanded (area increases), the flow conditions return to their
original values. We say that such a process is reversible.
From a consideration of the
second law
of thermodynamics,
a reversible flow maintains a constant value of
entropy.
Engineers call this type of flow an isentropic flow;
a combination of the Greek word "iso" (same) and entropy.
On this page we will derive some of the equations which are
important for isentropic flows.
We begin with the definitions of the
specific heat coefficients:
Eq. 1:
gamma = cp / cv
Eq. 1a:
cp  cv = R
where cp is the specific heat coefficient at constant pressure,
cv is the the specific heat coefficient at constant volume,
gamma is the ratio of specific heats, and R is the
gas constant from the
equation of state.
If the specific heat capacity is a constant value, the gas is said
to be calorically perfect and if the specific heat capacity
changes, the gas is said to be calorically imperfect.
At subsonic and low supersonic
Mach numbers, air is calorically perfect.
But under low hypersonic conditions, air is
calorically imperfect.
Derived flow variables, like the
speed of sound and the
isentropic flow relations
are slightly different for a calorically imperfect gas
than the conditions predicted for a calorically perfect gas
because some of the energy of the flow excites the vibrational
modes of the diatomic molecules of nitrogen and oxygen in the air.
Returning to our derivation, divide Eq 1a by cp:
Eq. 2:
1  1 / gamma = R / cp
Regroup the terms:
Eq. 3:
cp / R = gamma / (gamma  1)
Now, the equation of state is:
Eq. 4:
p = r * R * T
where p is the
pressure, r is the
density, and T is the
temperature. The
entropy of a gas is given by:
Eq. 5:
ds = cp * dT / T  R dp / p
where ds is the differential change in entropy, dT the
differential change in temperature, and dp the differential
change in pressure. For an isentropic process:
Eq. 6:
ds = 0
Eq. 6a:
cp * dT / T = R dp / p
Substitute from Eq. 4 into Eq. 6a
Eq. 7:
cp * dT = dp / r
Eq. 7a:
(cp / R) d(p / r) = dp / r
Differentiate Eq. 7a
Eq. 8:
(cp / R) * (dp / r  p * dr / r^2) = dp / r
Eq. 8a:
((cp / R)  1) dp / p = (cp / R) dr / r
Substitute Eq. 3 into Eq. 8a:
Eq. 9:
(1 / (gamma  1)) * dp / p = (gamma / (gamma  1)) * dr /r
which simplifies to:
Eq. 10:
dp / p = gamma * dr /r
For the calorically perfect gas, we can integrate this equation because gamma
is a constant value.
For the calorically imperfect case, we can not perform the simple integration
of Eq. 10. We will develop equations for the calorically imperfect case
beginning at Equation 23 at the bottom.
Continuing with our derivation for the calorically perfect case, integrate Eq. 10 to obtain:
Eq. 11:
p / r ^ gamma = constant
We evaluate the constant as being the total pressure and density
that occur when the flow is brought to rest isentropically:
Eq. 12:
p / r ^ gamma = pt / rt ^ gamma
Eq. 12a:
p / pt = ( r / rt) ^ gamma
where pt is the total pressure, and rt is the total density.
Usng Eq. 4 we can likewise define the total temperature Tt:
Eq. 13:
(r * T) / (rt * Tt) = ( r / rt) ^ gamma
Eq. 13a:
T / Tt = ( r / rt) ^ (gamma  1)
Combining Eq. 13a and Eq. 12a:
Eq. 14:
p / pt = ( T / Tt) ^ (gamma / (gamma  1))
Let us now derive the relation between the static and total
variables in terms of the Mach number.
From the definition of the
Mach number:
Eq. 15:
V = M * a
where V is the flow velocity, M is the Mach number,
and a is the speed of sound:
Eq. 16:
a^2 = gamma * R * T
The enthalpy h of a gas is given by:
Eq. 17:
h = cp * T
Then the conservation of energy equation
can then be expressed as:
Eq. 18:
ht = h + (V^2) / 2
Substitute Eqs. 15 and 17 into Eq. 18:
Eq. 19:
cp * Tt = cp * T + (M^2 * a^2) / 2
Now substitute Eq. 16 into Eq. 19:
Eq. 20:
cp * Tt = cp * T + (M^2 * gamma * R * T) / 2
Divide Eq. 20 by cp:
Eq. 21:
Tt = T + (M^2 * gamma * R * T) / (2 * cp)
Eq. 21a:
Tt / T = 1 + (M^2 * gamma * R ) / (2 * cp)
Finally, substitute Eq. 3 into Eq. 21a:
Eq. 22:
Tt / T = 1 + ((gamma  1) / 2) * M^2
Eqs. 14 and 13 can be used with Eq. 22 to obtain the relations between
the static and total pressure and static and total density in terms of the
Mach number. These equations are summarized on the
isentropic flow page.
We now return to the integration of the Eq. 10 for the calorically imperfect case.
Mathematical models
based on a simple harmonic vibrator have been developed for the calorically
imperfect case..
The details of the analysis were given by Eggars in
NACA Report 959.
A synopsis of the report is included in
NACA Report 1135.
The relation for the total temperature is
given as:
M^2 = (2 (Tt/T) / gam) * [(gamma/(gamma1) * (1  T/Tt) + (theta/Tt) * (1/(e^theta/Tt 1)  1/(e^theta/T 1)]
where gamma is the ratio of specific heats for a perfect gas, theta is
a thermal constant equal to 5500 degrees Rankine, and gam is the ratio of
specific heats including a correction for the vibrational
modes:
gam = 1 + (gamma  1) / ( 1 + (gamma1) * [(theta/T)^2 * e^(theta/T) /(e^(theta/T) 1)^2])
The equation relating Mach number and total temperature must be solved iteratively
to obtain a value for the total temperature. Having the total temperature ratio,
the relations between density and total temperature is:
rho/rhot = [(e^(theta/Tt)  1)/(e^(theta/T)  1)] * [(T/Tt)^(1/(gamma1))] * exp[(theta/T) *
(e^(theta/T)/(e^(theta/T) 1))  (theta/Tt) * (e^(theta/Tt)/(e^(theta/Tt) 1)) ]
where rho/rhot is the ratio of the static density to the total density. There is a similar relation
for the ratio of the static and total pressure p/pt:
p/pt = [(e^(theta/Tt)  1)/(e^(theta/T)  1)] * [(T/Tt)^(gamma/(gamma1))] * exp[(theta/T) *
(e^(theta/T)/(e^(theta/T) 1))  (theta/Tt) * (e^(theta/Tt)/(e^(theta/Tt) 1)) ]
The equation for the dynamic pressure coefficient q/p is given by:
q/p = (gamma/(gamma1)) * (Tt/T  1) + (theta/T)*[1/(e^(theta/Tt) 1)  1/(e^(theta/T) 1) ]
Activities:
Guided Tours
Navigation ..
 Beginner's Guide Home Page
