Chapter 6.6, Problem 20P

A)

Interpretation Introduction

Interpretation:

The rate of work done in the pump has to be determined.

Concept introduction:

Expression for the rate of work done in the pump as given as follows

$\begin{array}{c}{\dot{W}}_{S,pump}={\displaystyle \int \dot{V}dP}\\ =\dot{V}{\displaystyle \int dP}\\ =\underset{\_}{V}\dot{n}\left[P_{out}-P_{in}\right]\end{array}$

Here, volumetric flow rate is $\dot{V}$, molar volume is $\underset{\_}{V}$, change in pressure is dP, molar flow rate is $\dot{n}$, inlet and exit pressure of liquid is $P_{in}$ and $P_{out}$ respectively.

B)

Interpretation Introduction

Interpretation:

The temperature of the liquid leaving the pump has to be determined.

Concept introduction:

Write the energy balance around the pump.

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

Here, time is t, total number of moles of the system is N, molar internal energy of the system is $\underset{\_}{U}$, velocity of the system is v, height of the system is h, acceleration due to gravity is g, molar flow rate for inlet and outlet streams is ${\dot{n}}_{j,in}$ and ${\dot{n}}_{k,out}$, molar enthalpies of streams inlet and outlet is ${\underset{\_}{H}}_j$ and ${\underset{\_}{H}}_k$, 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 $\dot{Q}$.

C)

Interpretation Introduction

Interpretation:

The final molar volume has to be determined.

Concept introduction:

Write the total derivative of $d\underset{\_}{V}$.

$d\underset{\_}{V}={\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_{P}dT+{\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_{T}dP$        (4)

Here, change in molar volume with respect to change in temperature at constant pressure is ${\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P$ and change in molar volume with respect to change in pressure at constant temperature is ${\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$.

Write the coefficient of thermal expansion.

$\begin{array}{c}{\alpha }_V=\frac{1}{\underset{\_}{V}}{\left(\frac{\partial \underset{\_}{V}}{\partial T}\right)}_P\\ =0.00237\frac{1}{\text{K}}\end{array}$

Here, molar volume is $\underset{\_}{V}$.