Question

Show that the power needed to drive a fluid through a pipe with uniform cross-section is equal to the volume rate of flow, \({\bf{Q}}\), times the pressure difference \({{\bf{P}}_{\bf{1}}}{\bf{ - }}{{\bf{P}}_{\bf{2}}}\), Ignore viscosity.

Short answer

The power needed to drive a fluid through a pipe is \(\left( {{P_1} - {P_2}} \right)Q\).

Step-by-step solution

Understanding Viscosity

The frictional force which acts between the adjacent layers of the fluid is termed as viscosity. The frictional force acts when the layers of fluid move with respect to each other.

Step 2: Calculating the power

The work done on the fluid can be given as,

\(W = \left( {{P_1} - {P_2}} \right)V\)

Here, \(W\) is the work done, \({P_1} - {P_2}\) is the pressure difference and \(V\)is the volume.

The power needed to drive the fluid can be given as,

\(P = \frac{W}{t}\)

Here, \(P\)is the power needed and \(t\)is the time taken.

Substitute the values in the above equation.

\(P = \frac{{\left( {{P_1} - {P_2}} \right)V}}{t}\)

The flow rate of the fluid can be given as,

\(Q = \frac{V}{t}\)

Here, \(Q\)is the flow rate.

Put the known value in equation of power,

\(P = \left( {{P_1} - {P_2}} \right)Q\)

Therefore, the power required to drive the fluid through pipe is \(\left( {{P_1} - {P_2}} \right)Q\).