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Chapter 8: Problem 37

Consider fully developed laminar flow in a circular pipe. If the viscosity of the fluid is reduced by half by heating while the flow rate is held constant, how will the pressure drop change?

Short Answer

Expert verified
Answer: The pressure drop will decrease by half.

Step by step solution

01

Understand the Hagen-Poiseuille equation

The Hagen-Poiseuille equation describes the pressure drop in laminar flow through a circular pipe. The equation is given by: \[\Delta P = \frac{8 \mu LQ}{\pi R^4}\] where: - ΔP: Pressure drop - µ: Fluid viscosity - L: Length of the pipe - Q: Flow rate - R: Radius of the pipe
02

Determine the initial pressure drop

Using the Hagen-Poiseuille equation, let's find the initial pressure drop with the given fluid viscosity. Keep the values as variables since the question asked is about how the pressure drop would change and not specific values. The initial pressure drop is: \[\Delta P_1 = \frac{8 \mu_1 LQ}{\pi R^4}\]
03

Determine the final pressure drop

Now, let's find the final pressure drop after the fluid viscosity is reduced by half while keeping the flow rate constant. The new viscosity is: \[\mu_2 = \frac{\mu_1}{2}\] Using the Hagen-Poiseuille equation with the reduced viscosity, the final pressure drop is: \[\Delta P_2 = \frac{8 \mu_2 LQ}{\pi R^4} = \frac{8 (\frac{\mu_1}{2}) LQ}{\pi R^4} = \frac{4 \mu_1 LQ}{\pi R^4}\]
04

Compare the initial and final pressure drops

To find the change in pressure drop, we can compare the initial and final pressure drops. Dividing the final pressure drop by the initial pressure drop gives: \[\frac{\Delta P_2}{\Delta P_1} = \frac{\frac{4 \mu_1 LQ}{\pi R^4}}{\frac{8 \mu_1 LQ}{\pi R^4}} = \frac{1}{2}\]
05

Conclusion

The pressure drop in the laminar flow through the circular pipe will decrease by half after reducing the fluid viscosity by half while keeping the flow rate constant.

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Most popular questions from this chapter

Water at \(15^{\circ} \mathrm{C}\) is flowing through a 5 -cm-diameter smooth tube with a length of \(200 \mathrm{~m}\). Determine the Darcy friction factor and pressure loss associated with the tube for (a) mass flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) and \((b)\) mass flow rate of \(0.5 \mathrm{~kg} / \mathrm{s}\).

The bulk or mixed temperature of a fluid flowing through a pipe or duct is defined as (a) \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} T d A_{c}\) (b) \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} T \rho V d A_{c}\) (c) \(T_{b}=\frac{1}{\dot{m}} \int_{A_{c}} h \rho V d A_{c}\) (d) \(T_{b}=\frac{1}{A_{c}} \int_{A_{c}} h d A_{c}\) (e) \(T_{b}=\frac{1}{\dot{V}} \int_{A_{c}} T \rho V d A_{c}\)

In the effort to find the best way to cool a smooth thin-walled copper tube, an engineer decided to flow air either through the tube or across the outer tube surface. The tube has a diameter of \(5 \mathrm{~cm}\), and the surface temperature is maintained constant. Determine \((a)\) the convection heat transfer coefficient when air is flowing through its inside at \(25 \mathrm{~m} / \mathrm{s}\) with bulk mean temperature of \(50^{\circ} \mathrm{C}\) and \((b)\) the convection heat transfer coefficient when air is flowing across its outer surface at \(25 \mathrm{~m} / \mathrm{s}\) with film temperature of \(50^{\circ} \mathrm{C}\).

A computer cooled by a fan contains eight printed circuit boards (PCBs), each dissipating \(10 \mathrm{~W}\) of power. The height of the PCBs is \(12 \mathrm{~cm}\) and the length is \(18 \mathrm{~cm}\). The clearance between the tips of the components on the \(P C B\) and the back surface of the adjacent \(P C B\) is \(0.3 \mathrm{~cm}\). The cooling air is supplied by a 10 -W fan mounted at the inlet. If the temperature rise of air as it flows through the case of the computer is not to exceed \(10^{\circ} \mathrm{C}\), determine (a) the flow rate of the air that the fan needs to deliver, \((b)\) the fraction of the temperature rise of air that is due to the heat generated by the fan and its motor, and ( \(c\) ) the highest allowable inlet air temperature if the surface temperature of the components is not to exceed \(70^{\circ} \mathrm{C}\) anywhere in the system. As a first approximation, assume flow is fully developed in the channel. Evaluate properties of air at a bulk mean temperature of \(25^{\circ} \mathrm{C}\). Is this a good assumption?

Water is to be heated from \(10^{\circ} \mathrm{C}\) to \(80^{\circ} \mathrm{C}\) as it flows through a 2 -cm-internal-diameter, 13 -m-long tube. The tube is equipped with an electric resistance heater, which provides uniform heating throughout the surface of the tube. The outer surface of the heater is well insulated, so that in steady operation all the heat generated in the heater is transferred to the water in the tube. If the system is to provide hot water at a rate of \(5 \mathrm{~L} / \mathrm{min}\), determine the power rating of the resistance heater. Also, estimate the inner surface temperature of the pipe at the exit.
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