
Isentropic Flow Equation Derivations


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
importatnt 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.
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
Subsitute 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
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.
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