Flow Physics - Isentropic Flow
In order to easily understand flow through nozzles, some assumptions must be made to simplify the variables within the problem. The first assumption is that the flow through a nozzle is adiabatic. This means that during the course of the flow there is no heat gained or lost (J/s).
The second assumption follows the first and states that the flow in a nozzle is isentropic, or that there is no net change in entropy within the flow system. These systematic assumptions are easily made since within a nozzle, heat addition is normally conducted during the combustion phase and that the friction of the air flow through the nozzle is negligible.
From the second law of thermodynamics with a constant specific heat at steady pressure, the following equation can be created with the change in entropy (variable s) equal to zero. From this assumption, relationships between pressures, densities, and temperatures can be made with regard to the ratio of specific heats, k.
\[{s_2} - {s_1} = {c_p}\ln (\frac{{{T_2}}}{{{T_1}}}) - R\ln (\frac{{{P_2}}}{{{P_1}}}) = {c_v}\ln (\frac{{{T_2}}}{{{T_1}}}) - R\ln (\frac{{{\rho _2}}}{{{\rho _1}}})\]
The second assumption follows the first and states that the flow in a nozzle is isentropic, or that there is no net change in entropy within the flow system. These systematic assumptions are easily made since within a nozzle, heat addition is normally conducted during the combustion phase and that the friction of the air flow through the nozzle is negligible.
From the second law of thermodynamics with a constant specific heat at steady pressure, the following equation can be created with the change in entropy (variable s) equal to zero. From this assumption, relationships between pressures, densities, and temperatures can be made with regard to the ratio of specific heats, k.
\[{s_2} - {s_1} = {c_p}\ln (\frac{{{T_2}}}{{{T_1}}}) - R\ln (\frac{{{P_2}}}{{{P_1}}}) = {c_v}\ln (\frac{{{T_2}}}{{{T_1}}}) - R\ln (\frac{{{\rho _2}}}{{{\rho _1}}})\]
By setting the change in entropy equal to zero, the equation can be simplified to the following equation:
\[\frac{{{P_2}}}{{{P_1}}} = {(\frac{{{T_2}}}{{{T_1}}})^{k/(k - 1)}} = {(\frac{{{\rho _2}}}{{{\rho _1}}})^k}\]
Isentropic flow is flow that has no change in entropy, which includes adiabatic internally reversible conditions, with no friction or other energy losses. This assumption allows the creation of properties of flow called stagnation properties. These are the properties of the flow that would occur if the flow was allowed to come to rest adiabatically.
\[\frac{{{P_2}}}{{{P_1}}} = {(\frac{{{T_2}}}{{{T_1}}})^{k/(k - 1)}} = {(\frac{{{\rho _2}}}{{{\rho _1}}})^k}\]
Isentropic flow is flow that has no change in entropy, which includes adiabatic internally reversible conditions, with no friction or other energy losses. This assumption allows the creation of properties of flow called stagnation properties. These are the properties of the flow that would occur if the flow was allowed to come to rest adiabatically.
JL