Chapter 4.8, Problem 13P

Interpretation Introduction

Interpretation:

The feasibility of the inventor is possible or not.

Concept Introduction:

Thesteady state energy balance equation for the Nitrogen.

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\displaystyle \sum _{j=1}^{j=J}\left\{m_{j,in}\left({\widehat{H}}_j+\frac{V_j^2}{2}+g{h}_j\right)\right\}}\\ -{\displaystyle \sum _{k=1}^{k=K}\left\{m_{k,out}\left({\widehat{H}}_k+\frac{V_k^2}{2}+g{h}_k\right)\right\}}+{\dot{W}}_{EC}+{\dot{W}}_S+\dot{Q}\end{array}\right]$

Here, total mass is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, individual quantities of mass added to and removed from the process are $m_{j,in}$ and $m_{k,out}$ respectively, time taken is $t$, initial specific enthalpy is ${\widehat{H}}_j$, final specific enthalpy is ${\widehat{H}}_k$, initial velocity is $V_j$, final velocity is $V_k$, initial height of the gas is $h_j$, final height of the gas is $h_k$, rate of shaft work is added to the system is ${\dot{W}}_S$, rate of work done through expansion or contraction of the system is ${\dot{W}}_{EC}$, and rate of heat is added to the system is $\dot{Q}$.

The entropy balance equation for the machine.

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

Here, time is t, mass of the system is M, specific entropy of the system is $\widehat{S}$, mass flow rates of individual streams entering and leaving the system is ${\dot{m}}_{j,in}$, ${\dot{m}}_{k,out}$, specific entropies of streams entering and leaving the system is ${\widehat{S}}_j,{\widehat{S}}_k$, 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 labelled n occurs is $T_n$, and the rate at which entropy is generated within the boundaries of the system is ${\dot{S}}_{gen}$.

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

$C_V^{\ast }=\left(A-R\right)+BT+C{T}^2+D{T}^3+E{T}^4$

Here, ideal gas coefficients are $A,B,C,D\text{and}E$ respectively.