Chapter 7, Problem 67P

A liquid of density $\rho$ and viscosity $\mu$ is pumped at volume flow rate $\stackrel{\dot{}}{V}$ through a pump of diameter D. The blades of the pump rotate at angular velocity $\omega$ . The pump supplies a pressure rise $\Delta P$to the liquid Using dimensional analysis, generate a dimensionless relationship for $\Delta P$as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in you: result Hint: For consistency (and whenever possible), it is wise to choose a length, a density, and a velocity (or angular velocity) as repeating variables.

To determine

The dimensionless relationship for $\Delta p$.

Answer to Problem 67P

The dimensionless relationship for $\Delta p$ is $D^2\rho {\omega }^{2}f\left(\frac{\mu }{D^2\rho \omega },\frac{\dot{V}}{D^3\omega }\right)$.

Explanation of Solution

Given information:

The number of variable parameters is $6$.

Expression for number of pi terms,
$k=n-j$...... (I)
Here, number of variable parameters is $n$, number of primary dimensions is $j$.

Substitute $6$ for $n$ and $3$ for $j$ in Equation (I).

$\begin{array}{c}k=6-3\\ =3\end{array}$

Expression for first pi terms,
${\prod }_1=\Delta P{D}^{a_1}{\rho }^{b_1}{\omega }^{c_1}$...... (II)
Here, pressure difference is $\Delta P$, diameter is $x$, density is $\rho$, angular velocity is $\omega$.

Expression for second pi terms,
${\prod }_2=\mu D^{a_2}{\rho }^{b_2}{\omega }^{c_2}$...... (III)
Here, dynamic viscosity is $\mu$.

Expression for third pi terms,
${\prod }_3=\dot{V}{D}^{a_2}{\rho }^{b_2}{\omega }^{c_2}$...... (IV)
Here, volume flow rate is $\dot{V}$.

Expression for pressure,
$\Delta P=\frac{F}{A}$...... (V)
Here, force is $F$ and area is $A$

Substitute $\left[ML{T}^{-2}\right]$ for dimension of $F$ and $\left[L^2\right]$ for dimension of $A$ in Equation (V).

$\begin{array}{c}\Delta P=\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]}\\ =\left[M{L}^{-1}{T}^{-2}\right]\end{array}$...... (VI)
Expression for dimension diameter,
$D=\left[L\right]$...... (VII)

Expression for density,
$\rho =\frac{m}{v}$....... (VIII)
Here, mass is $m$, volume is $v$.

Substitute $\left[M\right]$ for dimension of $m$, $\left[L^3\right]$ for dimension of $v$ in Equation (V).

$\begin{array}{c}\rho =\frac{\left[M\right]}{\left[L^3\right]}\\ =\left[M{L}^{-3}\right]\end{array}$...... (IX)
Expression for angular velocity,
$\omega =\frac{\theta }{t}$...... (X)
Substitute $\left[M^0{L}^0{T}^0\right]$ for dimension of $\theta$ and $\left[T\right]$ for dimension of $t$.

$\begin{array}{c}\omega =\frac{\left[M^0{L}^0{T}^0\right]}{T}\\ =\left[T^{-1}\right]\end{array}$...... (XI)
Expression for dynamic viscosity,
$\mu =\frac{F}{A\frac{V}{y}}$...... (XII)
Substitute $\left[ML{T}^{-2}\right]$ for dimension of $F$, $\left[L^2\right]$ for dimension of $A$, $\left[L{T}^{-1}\right]$ for dimension of $V$ and $\left[L\right]$ for dimension of $y$ in Equation (IX).

$\begin{array}{c}\mu =\frac{\left[ML{T}^{-2}\right]}{\left[L^2\right]\frac{\left[L{T}^{-1}\right]}{\left[L\right]}}\\ =\left[M{L}^{-1}{T}^{-1}\right]\end{array}$...... (XIII)
Expression for dimension of volume flow rate,
$\dot{V}=\frac{v}{t}$...... (XIV)
Substitute $\left[L^3\right]$ for dimension of $v$ and $\left[T\right]$ for dimension of $t$ in Equation (XIV).

$\begin{array}{c}\dot{V}=\frac{\left[L^3\right]}{\left[T\right]}\\ =\left[L^3{T}^{-1}\right]\end{array}$...... (XV)
Expression for relation of ${\prod }_1$, ${\prod }_2$ and ${\prod }_3$,
${\prod }_1=f\left({\prod }_2,{\prod }_3\right)$...... (XVI)
Calculation:

Substitute $\left[M^0{L}^0{T}^0\right]$ for dimension of ${\prod }_1$, $\left[M{L}^{-1}{T}^{-2}\right]$ for dimension of $\Delta P$, $\left[L\right]$ for dimension of $D$, $\left[M{L}^{-3}\right]$ for dimension of $\rho$ and $\left[T^{-1}\right]$ for dimension of $\omega$ in Equation (II).

$\begin{array}{l}\left[M^0{L}^0{T}^0\right]=\left[M{L}^{-1}{T}^{-2}\right]{\left[L\right]}^{a_1}{\left[M{L}^{-3}\right]}^{b_1}{\left[T^{-1}\right]}^{c_1}\\ \left[M^0{L}^0{T}^0\right]=\left[M^{1+b_1}{L}^{-1+a_1-3b_1}{T}^{-2-c_1}\right]\end{array}$

Compare the exponent of $\left[M\right]$ to obtain value of $b_1$.

$\begin{array}{l}1+b_1=0\\ b_1=-1\end{array}$

Compare the exponent of $\left[T\right]$ to obtain value of $c_1$.

$\begin{array}{l}-2-c_1=0\\ c_1=-2\end{array}$

Compare the exponent of $\left[L\right]$ to obtain dimension of $a_1$.

$\begin{array}{l}-1+a_1-3b_1=0\\ -1+a_1+3=0\\ a_1=-2\end{array}$

Substitute the $-2$ for $a_1$, $-1$ for $b_1$, $-2$ for $c_1$ in Equation (II).

$\begin{array}{c}{\prod }_1=\Delta P{D}^{-2}{\rho }^{-1}{\omega }^{-2}\\ =\frac{\Delta P}{D^2\rho {\omega }^2}\end{array}$...... (XVII)
Substitute $\left[M^0{L}^0{T}^0\right]$ for dimension of ${\prod }_2$, $\left[M{L}^{-1}{T}^{-1}\right]$ for dimension of $\mu$, $\left[L\right]$ for dimension of $D$, $\left[M{L}^{-3}\right]$ for dimension of $\rho$ and $\left[T^{-1}\right]$ for dimension of $\omega$ in Equation (III).

$\begin{array}{l}\left[M^0{L}^0{T}^0\right]=\left[M{L}^{-1}{T}^{-1}\right]{\left[L\right]}^{a_2}{\left[M{L}^{-3}\right]}^{b_2}{\left[T^{-1}\right]}^{c_2}\\ \left[M^0{L}^0{T}^0\right]=\left[M^{1+b_2}{L}^{-1+a_2-3b_2}{T}^{-1-c_2}\right]\end{array}$

Compare the exponent of $\left[M\right]$ to obtain value of $b_2$.

$\begin{array}{l}1+b_2=0\\ b_2=-1\end{array}$

Compare the exponent of $\left[T\right]$ to obtain value of $c_2$.

$\begin{array}{l}-1-c_2=0\\ c_2=-1\end{array}$

Compare the exponent of $\left[L\right]$ to obtain dimension of $a_1$.

$\begin{array}{l}-1+a_2-3b_2=0\\ -1+a_2+3=0\\ a_2=-2\end{array}$

Substitute the $-2$ for $a_2$, $-1$ for $b_2$, $-1$ for $c_2$ in Equation (II).

$\begin{array}{c}{\prod }_2=\mu D^{-2}{\rho }^{-1}{\omega }^{-1}\\ =\frac{\mu }{D^2\rho \omega }\end{array}$...... (XVIII)
Substitute $\left[M^0{L}^0{T}^0\right]$ for dimension of ${\prod }_3$, $\left[L^3{T}^{-1}\right]$ for dimension of $\mu$, $\left[L\right]$ for dimension of $D$, $\left[M{L}^{-3}\right]$ for dimension of $\rho$ and $\left[T^{-1}\right]$ for dimension of $\omega$ in Equation (IV).

$\begin{array}{l}\left[M^0{L}^0{T}^0\right]=\left[L^3{T}^{-1}\right]{\left[L\right]}^{a_3}{\left[M{L}^{-3}\right]}^{b_3}{\left[T^{-1}\right]}^{c_3}\\ \left[M^0{L}^0{T}^0\right]=\left[M^{b_3}{L}^{3+a_3-3b_3}{T}^{-1-c_3}\right]\end{array}$

Compare the exponent of $\left[M\right]$ to obtain value of $b_2$.

$b_3=0$

Compare the exponent of $\left[T\right]$ to obtain value of $c_2$.

$\begin{array}{l}-1-c_3=0\\ c_3=-1\end{array}$

Compare the exponent of $\left[L\right]$ to obtain dimension of $a_1$.

$\begin{array}{l}3+a_3-3b_3=0\\ 3+a_3=0\\ a_3=-3\end{array}$

Substitute the $-3$ for $a_3$, $0$ for $b_3$, $-1$ for $c_3$ in Equation (IV).

$\begin{array}{c}{\prod }_3=\dot{V}{D}^{-3}{\rho }^0{\omega }^{-1}\\ =\frac{\dot{V}}{D^3\omega }\end{array}$...... (XVIII)
Substitute $\frac{\Delta P}{D^2\rho {\omega }^2}$ for ${\prod }_1$, $\frac{\mu }{D^2\rho \omega }$ for ${\prod }_2$ and $\frac{\dot{V}}{D^3\omega }$ for ${\prod }_3$ in Equation (XVI).

$\begin{array}{l}\frac{\Delta P}{D^2\rho {\omega }^2}=f\left(\frac{\mu }{D^2\rho \omega },\frac{\dot{V}}{D^3\omega }\right)\\ \Delta P=D^2\rho {\omega }^{2}f\left(\frac{\mu }{D^2\rho \omega },\frac{\dot{V}}{D^3\omega }\right)\end{array}$

Conclusion:

The dimensionless relationship for $\Delta p$ is $D^2\rho {\omega }^{2}f\left(\frac{\mu }{D^2\rho \omega },\frac{\dot{V}}{D^3\omega }\right)$.