Chapter 4.8, Problem 26P

Interpretation Introduction

Interpretation:

The rate at which the entropy of the universe is increased by the boiler and condenser

Concept Introduction:

Write the steady state energy balance equation for valve.

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

Here, time is t, total mass of the system is M, specific internal energy of the system is $\widehat{U}$, velocity of the system is v, height of the system is h, acceleration due to gravity is g, mass flow rate for inlet and outlet streams is ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific enthalpies of streams inlet and outlet is ${\widehat{H}}_j$ and ${\widehat{H}}_k$, heights at which streams enters and leave the system is $h_j$ and $h_k$, rate at which work is added to the system through expansion or contraction of the system is ${\dot{W}}_{EC}$, rate at which shaft work is added to the system is ${\dot{W}}_S$, and the rate at which heat is added to the system is $\dot{Q}$.

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, 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 ${\dot{S}}_{gen}$.