Chapter 5.7, Problem 14P

(A)

Interpretation Introduction

Interpretation:

The flow rate of liquid product leaving the flash chamber.

Concept Introduction:

The mass balance around the entire system is,

$M_{final}-M_{Initial}={\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,in}}-{\displaystyle \sum _{k=1}^{k=K}{\dot{m}}_{k,out}}$

Here, initial and final mass is $M_{initial}$ and $M_{final}$, mass flow rates of individual streams entering and leaving the entire system is ${\dot{m}}_{k,in}$ and ${\dot{m}}_{k,out}$ respectively.

The energy balance around the flash chamber is,

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

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 $\dot{Q}$.

(B)

Interpretation Introduction

Interpretation:

The specific enthalpy of the vapor by-product.

Concept Introduction:

The energy balance around the counter current heat exchanger is,

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

(C)

Interpretation Introduction

Interpretation:

The total heat removed by the heat exchangers.

Concept Introduction:

The energy balance around the compressors and heat exchangers is,

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

(D)

Interpretation Introduction

Interpretation:

The heat capacity of the compound.

Concept Introduction:

The equation of the ideal gas law is,

$d\widehat{H}=C_P^{*}dT$

Here, heat capacity at constant pressure for an ideal gas is $C_P^{*}$.