Chapter 3.9, Problem 12P

(A)

Interpretation Introduction

Interpretation:

Determine the velocity of the water at that point that it hits the surface of the pool.

Concept Introduction:

Draw the waterfall A and pool B is shown in Figure (1).

The energy balance on the waterfall, “System A” by considering Figure (1),

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

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 rate of heat addition to the system is $\dot{Q}$.

(B)

Interpretation Introduction

Interpretation:

Determine the temperature of water in the pool.

Concept Introduction:

The energy balance on the pool, “System B” by considering Figure (1).

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

The expression to calculate the change in specific enthalpy.

${\widehat{H}}_{in}-{\widehat{H}}_{out}=C_P\left(T_1-T_2\right)$

Here, heat capacity of liquid water is $C_P$, temperature of water in a creek is $T_{in}$, and temperature of water in the pool is $T_{out}$.