Chapter 5, Problem 68P

A fluid of density $\rho$ and viscosity $\mu$ flows through a section of horizontal converging-diverging duct. The duct cross-sectional areas $A_{\text{inlet}}$and $A_{\text{outlet}}$ are known at the inlet, throat (minimum area), and outlet, respectively. Average pressure $P_{\text{outlet}}$ is measured at the outlet, and average velocity $V_{\text{intel}}$ is measured at the inlet. (a) Neglecting any irreversibilities such as friction, generate expressions for the average velocity and average pressure at the inlet and the throat in terms of the given variables. (b) In a real flow (with irreversibilities). do you expect the actual pressure at the inlet to be higher or lower than the prediction? Explain.

To determine

(a)

The expression for the average pressure at the inlet and the throat in terms of the given variables without any irreversibility.

Answer to Problem 68P

The average pressure at inlet is $P_{inlet}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{out}}\right)}^2-1\right)+P_{out}$.

The average pressure at throat is $P_{throat}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{out}}\times V_{inlet}\right)}^2-{\left(\frac{A_{inlet}}{A_{throat}}\times V_{inlet}\right)}^2\right)+P_{out}$

Explanation of Solution

Given information:

The cross sectional area at inlet is $A_{inlet}$, the cross sectional area at throat is $A_{throat}$, the outlet area of nozzle is $A_{outlet}$, the average pressure at outlet is $P_{out}$, the average pressure at inlet is $P_{inlet}$, the average velocity at inlet is $V_{inlet}$, the density of the fluid is $\rho$, the acceleration due to gravity is $g$and the viscosity of fluid is $\mu$.

Write the expression for conservation of mass equation for section (1) and section (2).

$m_{in}=m_{throat}$...... (I)

Here, the mass flow rate at inlet is $m_{in}$, the mass flow rate at throat is $m_{throat}$.

Write the equation for mass flow rate through inlet

$m_{in}=\rho A_{inlet}{V}_{inlet}$

Write the equation for mass flow rate through throat.

$m_{throat}=\rho A_{throat}{V}_{throat}$

Write the expression for conservation of mass for section (1) and section (3).

$m_{in}=m_{outlet}$...... (II)

Here, the mass flow rate at the outlet is $m_{out}$and velocity at the outlet is $V_{outlet}$.

Write the equation for mass flow rate through throat

$m_{outlet}=\rho A_{outlet}{V}_{outlet}$

Calculation:

Substitute $\rho A_{inlet}{V}_{inlet}$for $m_{in}$and $\rho A_{throat}{V}_{throat}$for $m_{throat}$in Equation (I).

$\begin{array}{l}\rho \times A_{inlet}\times V_{inlet}=\rho \times A_{throat}\times V_{throat}\\ A_{inlet}\times V_{inlet}=A_{throat}\times V_{throat}\\ V_{throat}=\frac{A_{inlet}}{A_{throat}}\times V_{inlet}\end{array}$

Substitute $\rho A_{inlet}{V}_{inlet}$for $m_{in}$and $\rho A_{outlet}{V}_{outlet}$for $m_{outlet}$in Equation (II)

$\begin{array}{l}\rho \times A_{inlet}\times V_{inlet}=\rho \times A_{outlet}\times V_{outlet}\\ A_{inlet}\times V_{inlet}=A_{outlet}\times V_{outlet}\\ V_{outlet}=\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\end{array}$

Write the Bernoulli's equation for section (1) and (3).

$\begin{array}{l}\frac{P_{inlet}}{\rho g}+\frac{V_{inlet}^2}{2g}=\frac{P_{outlet}}{\rho g}+\frac{V_{outlet}^2}{2g}\\ \frac{P_{inlet}-P_{outlet}}{\rho g}=\frac{V_{outlet}^2-V_{inlet}^2}{2g}\\ \frac{P_{inlet}-P_{outlet}}{\rho }=\frac{V_{outlet}^2-V_{inlet}^2}{2}\end{array}$...... (III)

Substitute $\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}$for $V_{outlet}$in Equation (III).

$$$\begin{array}{l}\frac{P_{inlet}-P_{outlet}}{\rho }=\frac{{\left(\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\right)}^2-V_{inlet}^2}{2}\\ \frac{P_{inlet}-P_{outlet}}{\rho V_{inlet}^2}=\frac{{\left(\frac{A_{inlet}}{A_{outlet}}\right)}^2-1}{2}\\ P_{inlet}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{outlet}}\right)}^2-1\right)+P_{outlet}\end{array}$

Write the expression for Bernoulli's equation for section (2) and (3).

$\begin{array}{l}\frac{P_{throat}}{\rho g}+\frac{V_{throat}^2}{2g}+z_{throat}=\frac{P_{outlet}}{\rho g}+\frac{V_{outlet}^2}{2g}+z_{outlet}\\ \frac{P_{throat}}{\rho g}+\frac{V_{throat}^2}{2g}=\frac{P_{outlet}}{\rho g}+\frac{V_{outlet}^2}{2g}\\ \frac{P_{throat}-P_{outlet}}{\rho g}=\frac{V_{outlet}^2-V_{throat}^2}{2g}\end{array}$..... (IV).

Substitute $\frac{A_{inlet}}{A_{throat}}\times V_{inlet}$for $V_{throat}$and $\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}$for $V_{outlet}$in Equation (IV).

$\begin{array}{l}\frac{P_{throat}-P_{outlet}}{\rho g}=\frac{V_{outlet}^2-V_{throat}^2}{2g}\\ \frac{P_{throat}-P_{outlet}}{\rho g}=\frac{{\left(\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\right)}^2-{\left(\frac{A_{inlet}}{A_{throat}}\times V_{inlet}\right)}^2}{2g}\\ P_{throat}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\right)}^2-{\left(\frac{A_{inlet}}{A_{throat}}\times V_{inlet}\right)}^2\right)+P_{outlet}\end{array}$

Conclusion:

The average pressure at the throat is $P_{throat}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\right)}^2-{\left(\frac{A_{inlet}}{A_{throat}}\times V_{inlet}\right)}^2\right)+P_{outlet}$.

To determine

(b)

Whether the actual pressure at the inlet is higher or lower than the prediction.

Answer to Problem 68P

The pressure at the inlet in real flow is higher than the predicted value.

Explanation of Solution

Write the expression for the Bernoulli's equation for section (1) and section (3) including frictional losses.

$\begin{array}{l}\frac{P_{inlet}}{\rho g}+\frac{V_{inlet}^2}{2g}+z_{inlet}=\frac{P_{out}}{\rho g}+\frac{V_{out}^2}{2g}+z_{outlet}+h_f\\ \frac{P_{inlet}}{\rho g}+\frac{V_{inlet}^2}{2g}=\frac{P_{outlet}}{\rho g}+\frac{V_{outlet}^2}{2g}+h_f\\ \frac{P_{inlet}-P_{outlet}}{\rho g}=\frac{V_{outlet}^2-V_{inlet}^2}{2g}+h_f\end{array}$......(V)

Here, the frictional loss is $h_f$.

Substitute $\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}$for $V_{outlet}$in Equation (V).

$\begin{array}{l}\frac{P_{inlet}-P_{outlet}}{\rho }=\frac{{\left(\frac{A_{inlet}}{A_{outlet}}\times V_{inlet}\right)}^2-V_{inlet}^2}{2}+g{h}_f\\ \frac{P_{inlet}-P_{outlet}}{\rho V_{inlet}^2}=\frac{{\left(\frac{A_{inlet}}{A_{outlet}}\right)}^2-1}{2}+g{h}_f\\ P_{inlet}=\frac{\rho \times V_{inlet}^2}{2}\left({\left(\frac{A_{inlet}}{A_{outlet}}\right)}^2-1\right)+P_{outlet}+\rho g{h}_f\end{array}$

Conclusion:

The actual pressure at the inlet is higher than the predicted value of the pressure at inlet.