Isentropic nozzle flow - Wikipedia [PDF]

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Idea Transcript


Isentropic nozzle flow Isentropic nozzle flow describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in entropy.

Contents Overview Stagnation properties Flow analysis Supersonic flow Converging nozzle See also References

Overview Whenever a gas is forced through a tube, the gaseous molecules are deflected by the tube’s walls. If the speed of the gas is much less than the speed of sound, the gas density will remain constant and the velocity of the flow will increase. However, as the speed of the flow approximates the speed of sound, compressibility effects on the gas are to be considered. The density of the gas becomes position dependent. While considering flow through a tube, if the flow is very gradually compressed (i.e. area decreases) and then gradually expanded (i.e. area increases), the flow conditions are restored (i.e. return to its initial position). So, such a process is a reversible process. According to the Second Law of Thermodynamics, whenever there is a reversible flow, constant value of entropy is maintained. Engineers classify this type of flow as an isentropic flow of fluids. Isentropic is the combination of the Greek word "iso" (which means - same) and entropy. When the change in flow variables is small and gradual, isentropic flows occur. The generation of sound waves is an isentropic process. A supersonic flow that is turned while there is an increase in flow area is also isentropic. Since there is an increase in area, therefore we call this an isentropic expansion. If a supersonic flow is turned abruptly and the flow area decreases, the flow is irreversible due to the generation of shock waves. The isentropic relations are no longer valid and the flow is governed by the oblique or normal shock relations.

Stagnation properties In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero. The isentropic stagnation state is the state a flowing fluid would attain if it underwent a reversible adiabatic deceleration to zero velocity. There are both actual and the isentropic stagnation states for a typical gas or vapor. Sometimes it is advantageous to make a distinction between the actual and the isentropic stagnation states. The actual stagnation state is the state achieved after an actual deceleration to zero velocity (as at the nose of a body placed in a fluid stream), and there may be irreversibility associated with the deceleration process. Therefore, the term "stagnation property" is sometimes reserved for the properties associated with the actual state, and the term total property is used for the isentropic stagnation state. The enthalpy is the same for both the actual and isentropic stagnation states (assuming that the actual process is adiabatic). Therefore, for an ideal gas, the actual stagnation temperature is the same as the isentropic stagnation temperature. However, the actual stagnation pressure may be less than the isentropic stagnation pressure. For this reason the term "total pressure" (meaning isentropic stagnation pressure) has particular meaning compared to the actual stagnation pressure. Enthalpy-Entropy diagram of stagnation state

Flow analysis The icentropic efficiency is h1-h2a/h1-h2. The variation of fluid density for compressible flows requires attention to density and other fluid property

relationships. The fluid equation of state, often unimportant for incompressible flows, is vital in the analysis of compressible flows. Also, temperature variations for compressible flows are usually significant and thus the energy equation is important. Curious phenomena can occur with compressible flows. For simplicity, the gas is assumed to be an ideal gas. The gas flow is isentropic. The gas flow is constant. The gas flow is along a straight line from gas inlet to exhaust gas exit. The gas flow behavior is compressible. There are numerous applications where a steady, uniform, isentropic flow is a good approximation to the flow in conduits. These include the flow through a jet engine, through the nozzle of a rocket, from a broken gas line, and past the blades of a turbine. Mach number = M Velocity = V Universal gas constant = R Pressure = p Specific heat ratio = k Temperature = T * = Sonic conditions Density = Area = A

Energy equation for the steady flow:

To model such situations, consider the control volume in the changing area of the conduit of Fig. The continuity equation between two sections an infinitesimal distance dx apart is

If only the first-order terms in a differential quantity are retained, continuity takes the form

The energy equation is:

This simplifies to, neglecting higher-order terms,:



Assuming an isentropic flow, the energy equation becomes:



Substitute from the continuity equation to obtain Steady, uniform, isentropic flow through a conduit

or, in terms of the Mach number:



This equation applies to a steady, uniform, isentropic flow. There are several observations that can be made from an analysis of Eq. (9.26). They are: For a subsonic flow in an expanding conduit (M 0), the flow is decelerating (dV 0), the flow is accelerating (dV >0). For a supersonic flow in a converging conduit (M >1 and dA

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