Chapter 4.8, Problem 12P

Interpretation Introduction

Interpretation:

The change in entropy of the water, the surroundings, and universe for this process.

Concept Introduction:

Write the expression for steady state energy balance on the water.

$\Delta \left\{M\left(\widehat{U}+\frac{v^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{j,in}\left({\widehat{H}}_j+\frac{v_j^2}{2}+g{h}_j\right)-{\dot{m}}_{k,out}\left({\widehat{H}}_k+\frac{v_k^2}{2}+g{h}_k\right)+\\ W_{EC}+W_S+Q\end{array}\right]$        (1)

Here, 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, , time taken is t acceleration due to gravity is g, mass flow rate for inlet and outlet streams are ${\dot{m}}_{j,in}$ and ${\dot{m}}_{k,out}$, specific enthalpies of streams inlet and outlet are ${\widehat{H}}_j$ and ${\widehat{H}}_k$, heights at which streams enters and leave the system are $h_j$ and $h_k$, total expansion/contraction work added throughout the process is $W_{EC}$, total shaft work added throughout the process is $W_S$, and total heat added throughout the process is Q.

Write the expression for entropy.

$dS=\frac{d{Q}_{rev}}{T}$

Here, change in entropy is $dS$, temperature is T, and change in the reversible heat is $d{Q}_{rev}$.

Write the expression to calculate the change in entropy for solidifying state.

$\begin{array}{c}\Delta S_2=\frac{Q}{T}\\ =\frac{-\Delta H_{fus}}{T}\end{array}$

Here, change in enthalpy of fusion is $\Delta H_{fus}$.

Write the expression to calculate the change in entropy for surroundings.

$\begin{array}{c}\Delta S_{surr}=\frac{Q}{T}\\ =\frac{M\left(C_{V,liq}\right)\left(\Delta T_{liq}\right)+M\left(C_{V,sol}\right)\left(\Delta T_{sol}\right)+M\Delta {\widehat{U}}_{fus}}{T}\\ =M\left[\frac{\left(C_{V,liq}\right)\left(\Delta T_{liq}\right)+\left(C_{V,sol}\right)\left(\Delta T_{sol}\right)+\Delta {\widehat{U}}_{fus}}{T}\right]\end{array}$

Here, specific heat capacity of the liquid is $C_{V,liq}$, temperature of the liquid is $\Delta T_{liq}$, and fusion internal energy is $\Delta {\widehat{U}}_{fus}$.