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Chapter 8: Problem 33

How is the friction factor for flow in a tube related to the pressure drop? How is the pressure drop related to the pumping power requirement for a given mass flow rate?

Short Answer

Expert verified
Question: Explain the relationship between the friction factor for flow in a tube, the pressure drop, and the pumping power requirement for a given mass flow rate. Answer: The friction factor is a dimensionless number that represents the resistance to fluid flow due to friction between fluid particles and the tube walls. The pressure drop refers to the difference in pressure between the entrance and exit of the tube. These two quantities are directly related through the Darcy-Weisbach equation. The pumping power requirement is the mechanical work needed to move a given mass flow rate of fluid through a tube. It is directly proportional to the pressure drop, and therefore, also related to the friction factor. A higher friction factor results in a higher pressure drop, which in turn requires a greater amount of mechanical power to move the fluid through the tube.

Step by step solution

01

Definition of friction factor and pressure drop

The friction factor, denoted as 'f', is a dimensionless number that characterizes the resistance of fluid flow due to the friction between fluid particles and the walls of the tube. The pressure drop, denoted as 'ΔP', is the difference in pressure between the entrance and exit of the tube.
02

Relationship between friction factor and pressure drop

The friction factor is related to the pressure drop through the Darcy-Weisbach equation, which is given as follows: ΔP = f * (ρ * L * v² / (2 * D)) where ΔP = pressure drop (Pa) f = friction factor (dimensionless) ρ = fluid density (kg/m³) L = length of the tube (m) v = flow velocity (m/s) D = diameter of the tube (m). This equation shows that the pressure drop is directly proportional to the friction factor. The higher the friction factor, the greater the resistance to fluid flow, and thus, the greater the pressure drop.
03

Definition of pumping power requirement

Pumping power requirement, denoted as 'P', is the mechanical work required by a pump to move a given mass flow rate (ṁ) of fluid through a tube. It is usually expressed in units of Watts (W) or horsepower (hp).
04

Relationship between pressure drop and pumping power requirement

The pumping power requirement for a given mass flow rate can be calculated using the following equation: P = ΔP * ṁ / η where P = pumping power requirement (W) ΔP = pressure drop (Pa) ṁ = mass flow rate (kg/s) η = pump efficiency (dimensionless, usually in decimal form) This equation shows that the pumping power requirement is directly proportional to the pressure drop. The higher the pressure drop, the greater the amount of mechanical power needed to move the fluid through the tube.

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Most popular questions from this chapter

Inside a condenser, there is a bank of seven copper tubes with cooling water flowing in them. Steam condenses at a rate of \(0.6 \mathrm{~kg} / \mathrm{s}\) on the outer surfaces of the tubes that are at a constant temperature of \(68^{\circ} \mathrm{C}\). Each copper tube is \(5-\mathrm{m}\) long and has an inner diameter of \(25 \mathrm{~mm}\). Cooling water enters each tube at \(5^{\circ} \mathrm{C}\) and exits at \(60^{\circ} \mathrm{C}\). Determine the average heat transfer coefficient of the cooling water flowing inside each tube and the cooling water mean velocity needed to achieve the indicated heat transfer rate in the condenser.

What is the physical significance of the number of transfer units NTU \(=h A_{s} / \dot{m} c_{p}\) ? What do small and large NTU values tell about a heat transfer system?

Hot water at \(90^{\circ} \mathrm{C}\) enters a \(15-\mathrm{m}\) section of a cast iron pipe \((k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose inner and outer diameters are 4 and \(4.6 \mathrm{~cm}\), respectively, at an average velocity of \(1.2 \mathrm{~m} / \mathrm{s}\). The outer surface of the pipe, whose emissivity is \(0.7\), is exposed to the cold air at \(10^{\circ} \mathrm{C}\) in a basement, with a convection heat transfer coefficient of \(12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the walls of the basement to be at \(10^{\circ} \mathrm{C}\) also, determine \((a)\) the rate of heat loss from the water and \((b)\) the temperature at which the water leaves the basement.

What is the generally accepted value of the Reynolds number above which the flow in smooth pipes is turbulent?

Water enters a 5-mm-diameter and 13-m-long tube at \(45^{\circ} \mathrm{C}\) with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\). The tube is maintained at a constant temperature of \(8^{\circ} \mathrm{C}\). The exit temperature of water is (a) \(4.4^{\circ} \mathrm{C}\) (b) \(8.9^{\circ} \mathrm{C}\) (c) \(10.6^{\circ} \mathrm{C}\) (d) \(12.0^{\circ} \mathrm{C}\) (e) \(14.1^{\circ} \mathrm{C}\) (For water, use \(k=0.607 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=6.14, v=0.894 \times\) \(10^{-6} \mathrm{~m}^{2} / \mathrm{s}, c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \rho=997 \mathrm{~kg} / \mathrm{m}^{3}\) )
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