Isentropic Flow

The fundamental equations for isentropic flows can be derived by considering a simplified model of a one-dimensional flow field, as follows.

Consider a streamtube differential in equilibrium in a one-dimensional flow field, as represented by the shaded area in Figure 9.1. p is the pressure acting at the left face of the streamtube and (p + ||ds) is the pressure at the right face. Therefore, the pressure force in positive s-direction, Fp, is given by:

dp dp

Fp = p dA — p + ds dA = — ds dA.

ds ds

For equilibrium, dm (dV/dt) = sum of all the forces acting on the streamtube differential, where dm is the mass of fluid in the streamtube element considered, and dV/dt is the substantial acceleration.

In the above equation for substantial acceleration, dV/dt is the local acceleration or acceleration at a point,

dV ds dV

that is, change of velocity at a fixed point in space with time. The convective acceleration——– = V —

ds dt ds

is the acceleration between two points in space, that is, change of velocity at a fixed time with space. It is present even in a steady flow.

The substantial derivative is expressed as:

dV dV dV

= + V.

dt dt ds

Therefore, the equilibrium equation becomes:

dp dV

— ds dA = dm. ds dt

But dm = p dA ds. Substituting this into the above equation, we get:

dV _ 1 dp

dt p ds

that is:

Equation (9.13) is applicable for both compressible and incompressible flows; the only difference comes in solution. For steady flow, Equation (9.13) becomes:

dV 1 dp
V— + – — = 0.

ds p ds

Integration of Equation (9.14) yields:

(9.15)

This equation is often called the compressible form of Bernoulli’s equation for inviscid flows. If p is expressible as a function of p only, that is, p = p(p), the second expression is integrable. Fluids having these characteristics, namely the density is a function of pressure only, are called barotropic fluids. For isentropic flow process:

where subscripts 1 and 2 refer to two different states. Therefore, integrating dp/p between pressure limits p1 and p2 , we get:

Using Equation (9.18), Bernoulli’s equation can be written as:

Equation (9.19) is a form of energy equation for isentropic flow process of gases.
For an adiabatic flow of perfect gases, the energy equation can be written as:

(9.20a)

y P2 v[ = y pi

Y – 1 P2 2 y – 1 Pi 2