Chapter 4.8, Problem 33P

Interpretation Introduction

Interpretation:

The temperature rise in the river

Concept Introduction:

Determine the efficiency of a Carnot heat engine $\left({\eta }_{CarnotH.E}\right)$.

${\eta }_{CarnotH.E}=1-\frac{T_C}{T_H}$

Here, temperature of a low temperature heat reservoir is $T_C$, and temperature of a high temperature heat reservoir is $T_H$.

The expression to calculate the actual efficiency $\left({\eta }_{actual}\right)$.

${\eta }_{actual}=\eta \cdot {\eta }_{CarnotH.E}$

Write the heat addition equation $\left({\dot{Q}}_{added}\right)$.

${\dot{Q}}_{added}=\frac{{\dot{W}}_{net}}{{\eta }_{actual}}$

Here, net work done added to the system is $W_{net}$.

The steady state energy equation for entire heat engine.

$\frac{d}{dt}\left\{M\left(\widehat{U}+\frac{V^2}{2}+gh\right)\right\}=\left[\begin{array}{l}{\dot{m}}_{in}\left({\widehat{H}}_{in}+\frac{V_{in}^2}{2}+g{h}_{in}\right)-{\dot{m}}_{out}\left({\widehat{H}}_{out}+\frac{V_{out}^2}{2}+g{h}_{out}\right)\\ +{\dot{W}}_S+{\dot{W}}_{EC}+\dot{Q}\end{array}\right]$

Here, time is $t$, total mass is $M$, specific internal energy is $\widehat{U}$, velocity is $V$, acceleration due to gravity is $g$, height is $h$, initial mass flow rate  is ${\dot{m}}_{in}$, initial specific enthalpy is ${\widehat{H}}_{in}$, rate of heat is added or removed from the system is $\dot{Q}$, initial velocity is $V_{in}$, initial height of the gas is $h_{in}$, final mass flow rate is ${\dot{m}}_{out}$, final height of the gas is $h_{out}$, rate of shaft work is added to the system is ${\dot{W}}_{\text{S}}$, and rate of work is added to the system through expansion or contraction of the system is ${\dot{W}}_{\text{EC}}$.