The Impulse Function in Isothermal Nozzle

Previously Impulse function was developed in the isentropic adiabatic flow. The same is done here for the isothermal nozzle flow model. As previously, the definition of the Impulse function is reused. The ratio of the impulse function for two points on the nozzle is

$${F_2 \over F_1} = { P_2A_2 + \rho_2 {U_2}^2 A_2 \over P_1 A_1 + \rho_1 {U_1}^2 A_1 }$$ (4.112)

Utilizing the ideal gas model for density and some rearrangement results in

$${F_2 \over F_1} = {P_2 A_2 \over P_1 A_1} { 1 + {{U_2}^2 \over RT} \over 1 + {{U_1}^2 \over RT }}$$ (4.113)

Since $U^2 / RT = kM^2$ and the ratio of equation (4.86) transformed equation into (4.113)

$${F_2 \over F_1} = {M_1 \over M_2} { 1 + k {M_2}^2 \over 1 + k {M_1}^2}$$ (4.114)

At the star condition ( M=1 ) (not the minimum point) results in

$${F_2 \over F^{*}} = {1 \over M_2} { 1 + k {M_2}^2 \over 1 + k }$$ (4.115)