Chapter 6.6, Problem 16P

A)

Interpretation Introduction

Interpretation:

The exit temperature and velocity of the ideal gas has to be determined.

Concept introduction:

Use the entropy balance to determine the exit temperature of the gas and then similarly use the enthalpy balance for a nozzle to determine the exit velocity of the nozzle.

Write the expression for the change in molar entropy for an ideal gas.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=C_P^{*}\ln \left(\frac{T_2}{T_1}\right)+R\ln \left(\frac{P_1}{P_2}\right)$

Here, constant pressure heat capacity of an ideal gas is $C_P^{*}$, inlet and exit temperature and pressure is $T_1,T_2,P_1,\text{and}{P}_2$, and gas constant is R.

Write the expression for enthalpy balance for a nozzle.

$\frac{v_2^2}{2}={\widehat{H}}_1-{\widehat{H}}_2$

$\frac{v_2^2}{2}=C_P^{*}\left(T_1-T_2\right)$

Here, specific enthalpy for inlet and exit state is ${\widehat{H}}_1$ and ${\widehat{H}}_2$, and exit velocity is $v_2$.

B)

Interpretation Introduction

Interpretation:

The exit temperature and velocity of the gas has to be determined.

Concept introduction:

Write the expression for the change in molar entropy for an ideal gas.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=C_P^{*}\ln \left(\frac{T_2}{T_1}\right)+R\ln \left(\frac{P_1}{P_2}\right)$

Here, constant pressure heat capacity of an ideal gas is $C_P^{*}$, inlet and exit temperature and pressure is $T_1,T_2,P_1,\text{and}{P}_2$, and gas constant is R.

Write the expression for enthalpy balance for a nozzle.

$\frac{v_2^2}{2}={\widehat{H}}_1-{\widehat{H}}_2$

$\frac{v_2^2}{2}=C_P^{*}\left(T_1-T_2\right)$

Here, specific enthalpy for inlet and exit state is ${\widehat{H}}_1$ and ${\widehat{H}}_2$, and exit velocity is $v_2$.

C)

Interpretation Introduction

Interpretation:

The exit temperature and velocity of the gas has to be determined.

Concept introduction:

Write the entropy balance for an irreversible nozzle, but still adiabatic and steady state.

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1=\dot{n}\left({\underset{\_}{S}}_1-{\underset{\_}{S}}_2\right)+{\dot{S}}_{gen}$

Here, rate of entropy of generation is ${\dot{S}}_{gen}$, and molar flow rate is $\dot{n}$.

Write the expression for enthalpy balance for a nozzle.

$\frac{v_2^2}{2}={\widehat{H}}_1-{\widehat{H}}_2$

$\frac{v_2^2}{2}=C_P^{*}\left(T_1-T_2\right)$

Here, specific enthalpy for inlet and exit state is ${\widehat{H}}_1$ and ${\widehat{H}}_2$, and exit velocity is $v_2$.