UNIT II

Isentropic Flow with Variable Area

A steady one dimensional isentropic flow in variable area passages is called variable area flow. The following assumptions are made in this chapter.

1. The flow is assumed to be a steady flow

If the flow parameters do not vary with time is referred to as steady flow

2. The direction of flow is one—dimensional

If the flow parameters do not vary in directions normal to the flow direction is one—dimensional flow.

3. The flow is assumed as isentropic flow

It is assumed that the heat transfer is negligible and there are no other irreversibilities due to fluid friction etc.,

But the real systems deviate from the above assumed isentropic process. Therefore it is used as a standard reference for comparing the actual one.

2.1Comparison of Isentropic and Adiabatic processes

Fig 2.1 shows adiabatic and isentropic expansion of a perfect gas between two states I and 2.

By substituting different values of ‘M’ (subsonic to supersonic), it yields a graph asShown in Fig 2.5

Fig 25 Variation of Area ratio with Mach number

2.5 IMPULSE FUNCTION

For solving jet propulsion problems, it is sometimes convenient to employ a quantity called impulse function. It is defined as the sum of pressure force and impulse force.

One dimensional flow through a cQntrol surface is shown in Fig 2.6. The net thrust (or) side wall thrust produced by the stream is a result of changes in pressure and Mach number between sections 1 and 2 applying momentum equation between sections 1 and 2. The thrust exerted by the fluid

The above equation is convenient to find the thrust exerted by the flowing fluids using Mach number M. To obtain a relation between the non—dimensional impulse function and the Mach number the flow is assumed to be isentropic

We can also express the mass flow rate in four different non—dimensional forms and are(i) Mass flow rate interms of pressure ratio(ii) Mass flow rate interms of area ratio(iii Mass flow rate interms of Mach number and(iv) Numerical value of mass flow parameter

12.7 Flows through Nozzles As we have studied earlier in this chapter. convergent nozzles are used for sonic

and subsonic flows. They can also be used as flow regulating and flow measuring devices.

Convergent — divergent nozzles are used for super sonic flows. They are used in compressors and turbine blade rows etc.

12.7.1 Convergent nozzles The flow from an infinite reservoir to an exhaust chamber through a convergent

nozzle is shown is Fig 2.7. The stagnation conditions (Po,To etc) in the reservoir is kept constant, but the exhaust chamber pressure can be varied by using a valve.

Fig. 2. 7 Isentropic flow through a con vergetil nozzle

The pressure variation along the length of a nozzle and the exhaust chamber is shown in Fig 2.7. When there is no flow, the stagnation pressure (P is equal to the chamber pressure Pb (curve a).

When the chamber pressure is reduced by opening a valve, flow will takes place and the pressure decreases along the nozzle and up to the nozzle exit. But the nozzle exit pressure and the chamber pressure both are equal (curves b and c). The pressure

12.7.2 Convergent — divergent nozzles Fig 2.9 shows the flow from an infinite re through a convergent — divergent

nozzle to an exhaust chamber. The experiment is similar to the one above, except that a converging — diverging nozzle is to be used. In curves a, b and c the acceleration takes place in the convergent part and upto the throat. The diverging part acts as a diffuser through which the pressure rises to the chamber pressure Pb. Hence these cun’es acts as a “Venturi’ The pressure ratio at the throat is critical for the curve c’.

Fig. 2.9 Flow through a convergent— divergent nozzle Continuous acceleration takes place in curve ‘g’, hence it a design curve but

other curves are off—design curves. When back pressure is further lowered, the velocity is supersonic after the throat tO a point where discontinuity in the flow occurs. In curves a’ and e flow discontinuous and eddies are formed after the throat which results sudden increase in pressure (shock wave) so that back pressure is reached.

When the back pressure is lowered further, the shock wave moves downstream till it reaches the exit as in curve J. The back pressure increases suddenly through a shock wave and is above the nozzle exit pressure (Pe). When the back pressure is lowered further, the chamber pressure is reduced but the nozzle xit pressure is same as the design pressure is shown in Fig 2.9 and 2.10 (a) in curve h’.

When the back pressure is reduced, mass flow rate increases and it goes upto the critical state (c) and then there is no further increase in mass flow with decrease in back pressure. This condition of flow is chocked flow. The necessary conditions for this flow is (1) The pressure ratio must be equal to the critical pressure ratio and the corresponding mach number M = 1

2.7.3 Over expanding and under — expanding in nozzles If the back pressure is greater than the design pressure, the nozzle is said to be

over expanding and if it is less, the nozzle is said to be under expanding

12.7.4 Nozzle efficiency It is defined as the ratio of actual change or drop in enthalpy to the isentropic

change or drop in enthalpy.

It measures the fraction of available energy of expansion that is converted in to Kinetic form.

2.81 Flow through Diffusers Diffusion process is the deceleration of flow with rise in pressure. Fig. 2.1! shows

both reversible and irreversible diffusion of supersonic flow. The covergent part is supersonic and the divergent part is subsonic.

In an isentropic diffusion, continuous rise in static pressure takes place. In practical cases this kind of diffusion is not possible; hence a shock is introduced at the throat section which increases the pressure suddenly. The mach number after the shock is subsonic (M < 1)

Fig. 2.11 Reversible and irreversible flows in a diffuser