Chapter 4, Problem 4.8P

(a)

Interpretation Introduction

Interpretation:

The pumping power at $\text{85}$ percent pump efficiency is to be calculated.

Concept Introduction :

The pump is used to provide mechanical energy to the fluid to maintain the flow of flowing fluid.

According to the Bernoulli’s theorem, the total mechanical energy of the flowing fluid remains constant.

The Bernoulli’s equation between two stations is written as follows:

  $\frac{P_1}{\rho }+\frac{V_1^2}{g}+z_1=\frac{P_2}{\rho }+\frac{V_2^2}{g}+z_2$

The Bernoulli’s equation with pump work is used to calculate the pump work.

The Bernoulli’s equation with pump work is written as follows:

  $\frac{P_1}{\rho }+\frac{V_1^2}{g}+z_1+\eta W_p=\frac{P_2}{\rho }+\frac{V_2^2}{g}+z_2+h_f$

The power of the pump is calculated by multiplying the pump work with mass flow rate of the fluid.

The formula for pump power is,

  $P=W_p\times \dot{m}$

(b)

Interpretation Introduction

Interpretation:

The power generated by the turbines using same total flow rate is to be calculated.

Concept Introduction :

In the case of turbines, the head loss due to friction is reduced in the Bernoulli’s equation to give the mechanical energy available to the flowing fluid since it is the energy generated, therefore, the Bernoulli’s equation is reduced to the following equation,

  $\frac{P_1}{\rho }+\frac{V_1^2}{g}+z_1+\frac{W_p}{\eta }=\frac{P_2}{\rho }+\frac{V_2^2}{g}+z_2-h_f$ ….. (1)

The power generated by the turbine is calculated as follows:

  $P=W_p\times \dot{m}$

Consider the efficiency of the turbines is $\text{0}\text{.85}$ .

(c)

Interpretation Introduction

Interpretation:

The overall efficiency of the system is to be calculated.

Concept Introduction :

The overall efficiency of the system is the ratio of the power generated to the power delivered.

  ${\eta }_{\text{overall}}=\frac{\text{Power}\text{generated}}{\text{Power}\text{delivered}}$