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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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