Question

The velocity profile in fully developed laminar flow of water at $40^{\circ} \mathrm{F}\( in a 140 -ft-long horizontal circular pipe, in \)\mathrm{ft} / \mathrm{s}\(, is given by \)u(r)=0.8\left(1-625 r^{2}\right)\( where \)r$ is the radial distance from the centerline of the pipe in \(\mathrm{ft}\). Determine (a) the volume flow rate of water through the pipe, \((b)\) the pressure drop across the pipe, and (c) the useful pumping power required to overcome this pressure drop.

Short answer

#Answer# (a) To find the volume flow rate \(Q\), integrate the given velocity profile: $$Q = \int_{0}^{R} 0.8\left(1-625 r^{2}\right)(2\pi r dr) $$ (b) To calculate the pressure drop across the pipe, use the Hagen-Poiseuille equation: $$\Delta P = \frac{32 \mu Q L}{\pi R^4}$$ (c) To find the useful pumping power needed to overcome the pressure drop, use the expression: $$P_{\text{pumping}} = Q \Delta P$$ Plug in the expressions for \(Q\) and \(\Delta P\) from Steps 1 and 2 to calculate the required pumping power.

Step-by-step solution

Calculate the volume flow rate \(Q\)

To find the volume flow rate, first, let's integrate the velocity profile \(u(r)\) over the pipe's cross-sectional area. Recall that for a circular cross-section with radius \(R\), the area element is given by \(dA = 2\pi r dr\). The volume flow rate is then the integral of the product of \(u(r)\) and \(dA\): $$Q = \int_{0}^{R} u(r) dA = \int_{0}^{R} u(r) (2\pi r dr) $$ Now we will integrate the given velocity profile: $$Q = \int_{0}^{R} 0.8\left(1-625 r^{2}\right)(2\pi r dr) $$

Find the pressure drop

For laminar flow in a pipe, the pressure drop can be calculated using the Hagen-Poiseuille equation: $$\Delta P = \frac{32 \mu Q L}{\pi R^4}$$ Where \(\Delta P\) is the pressure drop, \(\mu\) is the dynamic viscosity of water, \(Q\) is the volume flow rate, \(L\) is the pipe length, and \(R\) is the pipe radius. We will use the provided temperature (\(40^{\circ}\mathrm{F}\)) to find the dynamic viscosity of water.

Calculate the useful pumping power required

The useful pumping power needed to overcome the pressure drop is determined by the product of the volume flow rate and the pressure drop: $$P_{\text{pumping}} = Q \Delta P$$ Using the expression obtained in Step 1 for \(Q\) and the Hagen-Poiseuille equation obtained in Step 2 for \(\Delta P\), calculate the useful pumping power required.