Isentropic nozzle flow describes the movement of a gas or fluid through a narrowing opening without an increase or decrease in entropy.
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 approaches 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 and adiabatic 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 The isentropic relations are no longer valid and the flow is governed by the oblique or normal shock relations.
Set of Equations
Below are nine equations commonly used when evaluating isentropic flow conditions. These assume the gas is calorically perfect; i.e. the ratio of specific heats is a constant across the temperature range. In typical cases the actual variation is only slight.
Properties without a subscript are evaluated at the point of interest (this point may be chosen anywhere along the length of the nozzle, but once chosen, all properties in a calculation must be evaluated at the same point) Subscript denotes a property at total/stagnation conditions. In a rocket or jet engine, this means the conditions inside the combustion chamber. For example, is total pressure/stagnation pressure/chamber pressure (all equivalent). is the local Mach number of the gas is the speed of the gas (m/s) is the local speed of sound through the gas (m/s) is the ratio of specific heats of the gas is the pressure of the gas (Pa) is the density of the gas (kg/m3) is the temperature of the gas (K) is the cross sectional area of the nozzle at the point of interest (m2) is the cross sectional area of the nozzle at the sonic point, or the point where gas velocity is Mach 1 (m2). Ideally this will occur at the nozzle throat.
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.