# Exercise sheet 5 (Pipe flow)

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1 Exercise sheet 5 (Pipe flow) last edited June 4, 2018 These lecture notes are based on textbooks by White [13], Çengel & al.[16], and Munson & al.[18]. Except otherwise indicated, we assume that fluids are Newtonian, and that: ρ water = kg m 3 ; patm. = 1 bar; ρ atm. = 1,225 kg m 3 ; Tatm. = 11,3 C; µ atm. = 1, N s m 2 ; д = 9,81 m s 2. Air is modeled as a perfect gas (R air = 287 J K 1 kg 1 ; γ air = 1,4; cpair = J kg 1 K 1 ). In cylindrical pipe flow, we accept the flow is always laminar for [Re]D , and always turbulent for [Re]D & The Darcy friction factor f is defined as: f p L1 2 D 2 ρvav. (5/22) The loss coefficient KL is defined as: KL p ρvav. (5/23) Viscosities of various fluids are given in fig Pressure losses in cylindrical pipes can be calculated with the help of the Moody diagram presented in fig p.120. Figure 5.11 Viscosity of various fluids at a pressure of 1 bar (in practice viscosity is almost independent of pressure). Figure White 2008 [13] 119

2 120 Figure 5.12 A Moody diagram, which presents values for f measured experimentally, as a function of the diameter-based Reynolds number [Re] D, for different relative roughness values. Diagram CC-by-sa S Beck and R Collins, University of Sheffield

3 5.1 Revision questions non-examinable The Moody diagram (fig p.120) is simple to use, yet it takes practice to understand it fully... here are three questions to guide your exploration. They can perhaps be answered as you work through the other examples. 1. Why is there no zero on the diagram? 2. Why are the curves sloped downwards should friction losses not instead increase with increasing Reynolds number? 3. Why can the pressure losses p be calculated given the volume flow V, but not the other way around? 5.2 Air flow in a small pipe Munson & al. [18] E8.5 A machine designed to assemble micro-components uses an air jet. This air is driven through a 10 cm-long cylindrical pipe with a 4 mm diameter, roughness 0,0025 mm, at an average speed of 50 m s 1. The inlet air pressure and temperature are 1,2 bar and 20 C; the air viscosity is quantified in fig p What is the pressure loss caused by the flow through the pipe? 2. Although that is not possible in practice, what would the loss in the case where laminar flow could be maintained throughout the pipe? 5.3 Couette flow CC-0 o.c. We consider the laminar flow of a fluid between two parallel plates (named Couette flow), as shown in fig Figure 5.13 Two-dimensional laminar flow between two plates, also called Couette flow. Figure CC-0 o.c. 1. Starting from the Navier-Stokes equations for incompressible, two-dimensional flow, [ u ρ t + u u x + v u ] = ρд x p [ 2 ] y x + µ u ( x) u ( y) 2 (4/41) [ v ρ t + u v x + v v ] = ρд y p [ 2 ] y y + µ v ( x) v ( y) 2 (4/42) 121

4 show that the velocity profile in a horizontal, laminar, steady, fully-developed flow between two horizontal plates separated by a gap of height 2H is: u = 1 ( ) p (y 2 H 2 ) (5/9) 2µ x 2. Why would this equation fail to describe turbulent flow? 5.4 Kugel fountain Derived from Munson & al. [18] 6.91 A Kugel fountain is erected to entertain students of fluid mechanics (figs and 5.15). A granite block (2 700 kg m 3 ) is sculpted into a very smooth sphere of diameter 1,8 m. The sphere is laid on a cylindrical concrete stand of internal diameter 1,2 m, whose edges are also smoothed out. The stand is filled with water, and a pump creates a flow which lifts the sphere to make a decorative fountain. Water flows between the stand and the sphere along a length of 10 cm, with a thickness of 0,13 mm. Entrance effects are negligible, and because the Reynolds number is low, the flow is entirely laminar. 1. What is the pressure exerted by the sphere in the central part of the fountain? 2. The velocity distribution for flow in between two flat plates of width Z separated by an interval 2H, known as Couette flow, is modeled as a function of the stream-wise pressure gradient p/ x with the equation: u = 1 2µ ( p x ) (y 2 H 2 ) (5/9) Starting from this equation 5/9, show that the pressure gradient can be quantified as a function of the volume flow V: p x = 3 µ 2 ZH 3 V (5/10) 3. What is the volume flow between the sphere and the concrete foundation? 4. What is the pumping power required for the fountain to work? Figure 5.14 Schematic layout for a Kugel fountain. A pump (bottom left) increases the water pressure to a value high enough that it can support the weight of the sphere. A thin strip of water flows between the concrete foundation and the sphere. Figure CC-0 o.c. 122

5 5. How would this power change if the diameter of the concrete support (but not of the sphere) was increased? (briefly justify your answer, e.g. in 30 words or less) Figure 5.15 A Kugel fountain, featuring questionable aesthetics but providing the opportunity for amusing interaction. A granite sphere is supported by the flow of water between the sphere and a solid strip of concrete. Photo CC-by-sa by Commons User:Atamari 5.5 Water piping CC-0 o.c. A long pipe is installed to carry water from one large reservoir to another (fig. 5.16). The total length of the pipe is 10 km, its diameter is 0,5 m, and its roughness is ϵ = 0,5 mm. It must climb over a hill, so that the altitude changes along with distance. Figure 5.16 Layout of the water pipe. For clarity, the vertical scale is greatly exaggerated. The diameter of the pipe is also exaggerated. Figure CC-0 o.c. The pump must be powerful enough to push 1 m 3 s 1 of water at 20 C. Figure 5.11 p.119 quantifies the viscosity of various fluids, and fig p.120 quantifies losses in cylindrical pipes. 1. Will the flow in the water pipe be turbulent? 2. What is the pressure drop generated by the water flow? 3. What is the pumping power required to meet the design requirements? 4. What would be the power required for the same volume flow if the pipe diameter was doubled? 123

6 5.6 Pipeline CC-0 o.c. The Trans-Alaska Pipeline System is a cylindrical smooth steel duct with 1,22 m diameter, average roughness ϵ = 0,15 mm, and length km. Approximately 700 thousand barrels of oil ( m 3 ) transit through the pipe each day. The density of crude oil is approximately 900 kg m 3 and its viscosity is quantified in fig p.119. The average temperature of the oil during the transit is 60 C. In industrial pumps, oil starts to cavitate (change state, a very undesirable behavior) when its pressure falls below 0,7 bar. The pipeline is designed to withstand ground deformations due to seismic movements in several key zones, and crosses a mountain range with a total altitude variation of m. 1. How much time does an oil particle need to travel across the line? 2. Propose a pumping station arrangement, and calculate the power required for each pump. 3. How would the pumping power change if the speed was increased? 5.7 Pump with pipe expansion A pump is used to carry a volume flow of 200 L s 1 from one large water reservoir to another (fig. 5.17). The altitude of the water surface in both reservoirs is the same. Figure 5.17 Layout of the water pipe. For clarity, the diameter of the pipe and the vertical scale are exaggerated. Figure CC-0 o.c. The pipe connecting the reservoirs is made of concrete (ϵ = 0,25 mm); it has a diameter of 50 cm on the first half, and 100 cm on the second half. In the middle, the conical expansion element induces a loss coefficient of 0,8. At the outlet (at point D), the pressure is approximately equal to the corresponding hydrostatic pressure in the outlet tank. The inlet is 14 m below the surface. The total pipe length is 400 m; the altitude change between inlet and outlet is 12 m. 1. Represent qualitatively (that is to say, showing the main trends, but without displaying accurate values) the water pressure as a function of pipe distance, when the pump is turned off. 2. On the same graph, represent qualitatively the water pressure when the pump is switched on What is the water pressure at points A, B, C and D?

7 5.8 A more complex ducted flow non-examinable. From institute archives A more complicated laminar flow case can be studied with the following case. In the middle of a vertical container of width 2h filled with oil (fig. 5.18), a plate of width b (perpendicular to the plane) and length L with negligible thickness is sinking at constant velocity v p. Under the assumption that the flow is laminar everywhere, steady and fully developed, give an analytical expression for the friction force applied on the plate. Figure 5.18 The hypothetical case of a plate sinking vertically in an oil reservoir. 5.9 Wind tunnel non-examinable Describe the main characteristics of a wind tunnel that could be installed and operated in the room you are standing in. In order to do this: Start by proposing key characteristics for the test section; From these dimensions, draw approximately an air circuit to feed the test section (while attempting to minimize the size of the fan, whose cost increases exponentially with diameter). Quantify the static and stagnation pressures along the air circuit, by estimating the losses generated by wall shear and in the bends (you may use data from fig. 5.19); Quantify the minimum power required to generate your chosen test section flow characteristics. 125

8 Filter screen: η = 0,05 Figure 5.19 Loss coefficients K L (here noted η) generated by the use of various components within wind tunnel ducts. Figure Barlow, Rae & Pope 1999 [9] 5.10 Politically incorrect fluid mechanics non-examinable In spite of the advice of their instructor, a group of students attempts to apply fluid mechanics to incommendable activities. Their objective is to construct a drinking straw piping system that can mix a drink of vodka and tonic water in the correct proportions (fig. 5.20). They use a Strawz kit of connected drinking straws, two bottles, and a glass full of ice and liquid water to cool the mix. For simplicity, the following information is assumed about the setup: Vodka is modeled as 40 % pure alcohol (ethanol) with 60 % water by volume; Ethanol density is 0,8 kg m 3 ; Tonic water is modeled as pure water; The viscosity of water and alcohol mixes is described in table 5.1 (use the nearest relevant value); The pipe bends induce a loss coefficient factor K Lbend = 0,5 each; The pipe T-junction induces a loss coefficient factor K L = 0,3 in the line direction and 1 in the branching flow; 126 Figure 5.20 Conceptual sketch of a student experiment. Figure CC-0 o.c.

9 Percentage of alcohol by weight Viscosity in centipoise 0 1, , , , , , , , , , ,2 Table 5.1 Viscosity of a mix of ethanol and water at 20 C. data from Bingham, Trans. Chem. Soc, 1902, 81, 179. The pipe has inner diameter D = 3 mm and roughness η = 0,0025 mm. The students wish to obtain the correct mix: one quarter vodka, three quarters tonic water. For given levels of liquid in the bottles, is there a straw pipe network configuration that will yield the correct mix, and if so, what is it? 127

10 Answers 5.2 1) Calculating inlet density with the perfect gas model, [Re] D = (turbulent), a Moody diagram read gives f 0,029, so p friction = 1 292,6 Pa = 0,0129 bar. 2) If the flow were (magically) kept laminar, with equation 5/24, p friction = 200 Pa. 5.3 The structure is given in the derivation of equation 5/9 p. 111, and more details about the math are given in the derivation of the (very similar) equation 5/19 p p = F W sphere A basin 35,3 W. = 71,51 kpa. Then V = 0,494 L s 1 which enables us to obtain W pump = 5.5 p alt. = ρд( ) = 1,57 bar and p friction = 51,87 bar : W pump = 5,345 MW. 5.6 The total pumping power is W = 10 MW (!). Be careful not to cavitate the oil in the ascending sections, and not to burst the pipe in the descending sections! 5.7 Pressure losses to friction are Pa in the first half, +71 Pa in the throat, 117 Pa in the second half. Add hydrostatic pressure to obtain final result. 5.8 Calculate the velocity distribution in the same way as for equation 5/9 p.111. Evaluate the pressure gradient p y using mass conservation (total cross-section mass flow is zero). With the full velocity distribution, derivate v with respect to x to obtain F τ = ρsдh + 2µS h v p. 5.9 This is a fun (and quick) exercise, but the results strongly depend on your proposed design! Get help during office hours The author cannot remember which exercise you are referring to. 128

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