Chapter 4.8, Problem 22P

(A)

Interpretation Introduction

Interpretation:

The work added to the compressor.

Concept Introduction:

Write the general expression for an entropy balance equation.

$\frac{d\left(M\widehat{S}\right)}{dt}={\displaystyle \sum _{j=1}^{j=J}{\dot{m}}_{j,in}{\widehat{S}}_j}-{\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,out}{\widehat{S}}_k}+{\displaystyle \sum _{n=1}^{n=N}\frac{{\dot{Q}}_n}{T_n}}+{\dot{S}}_{gen}$

Here, mass of the system is M, specific entropy of the system is $\widehat{S}$, time taken is $t$, mass flow rates of individual streams entering and leaving the system are ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific entropies of streams entering and leaving the system are ${\widehat{S}}_j$ and ${\widehat{S}}_k$, actual rate at which heat is added to or removed from the system at one particular location is ${\dot{Q}}_n$, the temperature of the system at the boundary where the heat transfer labeled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$.

The steady state energy balance equation for compressor:

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{v^2}{2}+gh\right)\right\}={\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{v_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{v_{out}^2}{2}+g{h}_{out}\right)+{\dot{W}}_S+{\dot{W}}_{EC}+\dot{Q}$

Here, mass flow rate for inlet and outlet is ${\dot{m}}_{in}$ and ${\dot{m}}_{out}$, specific enthalpies of inlet and outlet are ${\widehat{H}}_i$ and ${\widehat{H}}_{out}$, heights at which streams enters and leave the system are $h_{in}$ and $h_{out}$, time taken is t, mass of the system is M, specific internal energy is $\widehat{U}$, velocity is v, height is h, acceleration due to gravity is g, rate at which work is added to the system is ${\dot{W}}_{EC}$, rate of shaft work is ${\dot{W}}_S$, and the rate of heat addition to the system is $\dot{Q}$.

(B)

Interpretation Introduction

Interpretation:

The work added to the compressor and the temperature of the exiting Freon.

Concept Introduction:

Write the compressor efficiency.

$\begin{array}{l}{\eta }_{compressor}=\frac{W_{S,reversible}}{W_{S,actual}}\\ W_{S,actual}=\frac{W_{S,reversible}}{{\eta }_{compressor}}\end{array}$

The steady state energy balance equation for compressor:

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{v^2}{2}+gh\right)\right\}={\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{v_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{v_{out}^2}{2}+g{h}_{out}\right)+{\dot{W}}_S+{\dot{W}}_{EC}+\dot{Q}$

Here, mass flow rate for inlet and outlet is ${\dot{m}}_{in}$ and ${\dot{m}}_{out}$, specific enthalpies of inlet and outlet are ${\widehat{H}}_i$ and ${\widehat{H}}_{out}$, heights at which streams enters and leave the system are $h_{in}$ and $h_{out}$, time taken is t, mass of the system is M, specific internal energy is $\widehat{U}$, velocity is v, height is h, acceleration due to gravity is g, rate at which work is added to the system is ${\dot{W}}_{EC}$, rate of shaft work is ${\dot{W}}_S$, and the rate of heat addition to the system is $\dot{Q}$.