Chapter 3.9, Problem 25P

(A)

Interpretation Introduction

Interpretation:

Determine the work added $\left({\dot{W}}_S\right)$ in the compressor and heat removed by the heat exchanger

Concept Introduction:

The steady state energy equation for the compressor.

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{V_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{V_{out}^2}{2}+g{h}_{out}\right)\\ +{\dot{W}}_{\text{S}}+W_{\text{EC}}+\dot{Q}\end{array}\right]$

Here, total mass is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, time taken is $t$, initial mass flow rate is ${\dot{m}}_{in}$, initial specific enthalpy is ${\widehat{H}}_{in}$, initial velocity is $V_{in}$, initial height of the gas is $h_{in}$, final mass flow rate is ${\dot{m}}_{out}$, final height of the gas is $h_{out}$, rate at which shaft work is added to the system is ${\dot{W}}_{\text{S}}$, rate at which work is added to the system through expansion or contraction of the system is ${\dot{W}}_{\text{EC}}$, and rate at which heat is added to the system is $\dot{Q}$.

The formula to calculate the constant pressure heat capacity for ideal gas $\left(C_P^{\ast }\right)$.

$\begin{array}{l}\frac{C_P^{\ast }}{R}=A+BT+C{T}^2+D{T}^3+E{T}^4\\ C_P^{\ast }=R\left(A+BT+C{T}^2+D{T}^3+E{T}^4\right)\end{array}$

Here, gas constant is $R$, temperature is $T$, and constant are $A,B,C,D\text{and}E$ respectively.

Derive the formula to calculate the change in enthalpy for an ideal gas.

$d\underset{\_}{H}=C_P^{\ast }dT$

Here, derivative of molar enthalpy is $d\underset{\_}{H}$, constant pressure heat capacity for ideal gas is $C_P^{\ast }$, and change in temperature is $dT$.

(B)

Interpretation Introduction

Interpretation:

Determine the work added $\left({\dot{W}}_S\right)$ in the compressor and heat removed by the heat exchanger

Concept Introduction:

The steady state energy equation for the compressor.

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{V_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{V_{out}^2}{2}+g{h}_{out}\right)\\ +{\dot{W}}_{\text{S}}+W_{\text{EC}}+\dot{Q}\end{array}\right]$

Here, total mass is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, time taken is $t$, initial mass flow rate is ${\dot{m}}_{in}$, initial specific enthalpy is ${\widehat{H}}_{in}$, initial velocity is $V_{in}$, initial height of the gas is $h_{in}$, final mass flow rate is ${\dot{m}}_{out}$, final height of the gas is $h_{out}$, rate at which shaft work is added to the system is ${\dot{W}}_{\text{S}}$, rate at which work is added to the system through expansion or contraction of the system is ${\dot{W}}_{\text{EC}}$, and rate at which heat is added to the system is $\dot{Q}$.