Chapter 3.9, Problem 28P

(A)

Interpretation Introduction

Interpretation:

The heat added to the system.

Concept Introduction:

The steady state energy balance equation for piston cylinder device.

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

Here, mass flow rate for inlet and outlet is ${\dot{m}}_{in}$ and ${\dot{m}}_{out}$, specific enthalpies of inlet and outlet is ${\widehat{H}}_i$ and ${\widehat{H}}_{out}$, heights at which streams enters and leave the system is $h_{in}$ 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 heat addition to the system is $Q$.

(B)

Interpretation Introduction

Interpretation:

The heat added to the system.

Concept Introduction:

Here, the change in enthalpy is determined by adding the amount of heat needed to heat the ice, amount of heat needed to change the phase, and the amount of heat needed to heat the new fluid up to $125°\text{F}$.

Then the Equation (3) becomes.

$\begin{array}{l}\frac{Q}{M}=\Delta \widehat{H}\\ Q=M\left(\Delta \widehat{H}\right)\\ Q=M{\displaystyle \underset{T_1}{\overset{T_2}{\int }}{C}_{P,solid}}dT+\Delta {\widehat{H}}^{fus}+{\displaystyle \underset{T_2}{\overset{T_3}{\int }}{C}_{P,liquid}}dT\end{array}$

(C)

Interpretation Introduction

Interpretation:

The heat added to the system.

Concept Introduction:

Here, all the solids need to dissolve and be raised to allocate the proper temperature, and only a quarter of the aggregate fluid is changed over into vapor, the ̇ condition is broken into the fluid and the vapor segments.

Then the Equation (3) becomes.

$\begin{array}{l}\frac{Q}{M}=\Delta \widehat{H}\\ Q=M\left(\Delta \widehat{H}\right)\\ Q=0.75M\left(\begin{array}{l}{\displaystyle \underset{T_1}{\overset{T_2}{\int }}{C}_{P,solid}}dT+\Delta {\widehat{H}}^{fus}\\ +{\displaystyle \underset{T_2}{\overset{T_3}{\int }}{C}_{P,liquid}}dT\end{array}\right)+0.25M\left(\begin{array}{l}{\displaystyle \underset{T_1}{\overset{T_2}{\int }}{C}_{P,solid}}dT+\Delta {\widehat{H}}^{fus}\\ +{\displaystyle \underset{T_2}{\overset{T_3}{\int }}{C}_{P,liquid}}dT+\Delta {\widehat{H}}^{vap}\end{array}\right)\end{array}$

(D)

Interpretation Introduction

Interpretation:

The heat added to the system.

Concept Introduction:

Here, all of the solid turns into a vapor before the process to end, then the procedure is demonstrated in five stages: heat added to the solid, melting, heating the liquid, boiling and heating vapor.

Then the Equation (3) becomes.

$\begin{array}{l}\frac{Q}{M}=\Delta \widehat{H}\\ Q=M\left(\Delta \widehat{H}\right)\\ Q=M\left({\displaystyle \underset{T_1}{\overset{T_2}{\int }}{C}_{P,solid}}dT+\Delta {\widehat{H}}^{fus}+{\displaystyle \underset{T_2}{\overset{T_3}{\int }}{C}_{P,liquid}}dT+\Delta {\widehat{H}}^{vap}+{\displaystyle \underset{T_3}{\overset{T_4}{\int }}{C}_P}dT\right)\end{array}$