Topic 2.2 Hydraulic Radius and non circular flow.pptx

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Topic 2.2 – Flow in Closed Conduits (Non-Circular) Official (Open) 2.2 Flow in Closed Conduits (Non- Circular) Learning Objectives Explain what is laminar, transition and turbulent flow. Relate types of flow to the Reynolds number....

Topic 2.2 – Flow in Closed Conduits (Non-Circular) Official (Open) 2.2 Flow in Closed Conduits (Non- Circular) Learning Objectives Explain what is laminar, transition and turbulent flow. Relate types of flow to the Reynolds number. Compute Reynolds number for the flow of fluids in round pipes and tubes. Introduce the terms, hydraulic radius and Reynolds number for non-circular conduits. Explain the role of friction loss in pipe flow, friction factor and Darcy’s equation. Determine friction factor in turbulent flow using Moody’s diagram. Explain various types of energy losses in pipe flow and the resistance coefficient. Page Official (Open) 2.1.3 Hydraulic Radius & Re for Non-Circular Closed Conduits Re for full flow in circular pipe characteristic dimension for flow path = inside diameter, d 𝜌𝑣 d Internal 𝑑 𝑅𝑒 = diamet er Re for flow in non-circular cross-sections characteristic dimension for flow path = Hydraulic Radius, R 𝜌𝑣𝟒 d= d=? 𝑅𝑒 𝑹 Hydraul ? ic = Radius Page 3 Official (Open) 2.1.3 Hydraulic Radius & Re for Non-Circular Closed Conduits Examples of typical closed, non-circular cross sections d D S d S H B S S e.g. shell-and- e.g. air distribution ducts, rainwater or e.g. flow path tube sewage drains inside a heat machine exchanger Characteristic dimension of non-circular cross sections is the hydraulic radius, R Page 4 Official (Open) 2.1.3 Hydraulic Radius & Re for Non-Circular Closed Conduits Examples of typical closed, non-circular cross sections d D S d S H B S S 𝜋 2 A =4 𝐷2 A= A = BH A = 𝑆2 4 𝜋𝑑 − 𝑑2 S2 WP = 2B − WP =  (𝐷 + WP = 4S 𝑑) WP = + 2H +𝑑 4S Characteristic dimension of non-circular cross sections is the hydraulic radius, R A area R  WP wetted perimeter Page 5 Official (Open) 2.1.3 Hydraulic Radius & Re for Non-Circular Closed Conduits Hydraulic radius, R A area R  WP wetted perimeter Area = net cross-sectional area of flow stream Wetted perimeter = sum of length of boundaries, of sections in contact with (i.e. wetted by) the fluid The unit for R is meter (SI unit system) or feet (U.S. Customary System) Page 6 Official (Open) 2.1.3 Hydraulic Radius & Re for Non-Circular Closed 2.2 Example Conduits Determine the hydraulic radius of the section d S shown in Figure. If the inside dimension of each side of the square is 250 mm and the outside diameter of the tube is 150 mm. S 2 Solution: the square Theand net the flow areaarea is the of the difference circle: A = 𝑆2 𝜋𝑑 4 − between A = S2 -  the=area d2/4 (250)of 2 -  (150)2/4 = 44,829 mm2 WP = 4S +𝑑 The wetted perimeter is the sum of the four sides of the square and the circumference of the circle: WP = 4S +  d = 4(250) + (150) = 1,471 mm A 44,829 Then, the hydraulic radius R  2 30.5 mm 0.0305 mm mm m is R WP 1,471 mm Page 7 Official (Open) 2.1.4 Reynolds Number for Non-Circular Closed Conduits From continuity equation, we know v = Q/A This equation can be used when: fluid completely fills the available cross- sectional area fluid is under pressure Note : the Area is the same as that used to compute the Hydraulic Radius. The Reynolds number for flow in non-circular sections is computed Page 8 Official (Open) 2.1.4 Reynolds Number for Non-Circular Closed Conduits The only alteration is the replacement of the diameter d with 4R, i.e., four times the hydraulic radius. Re   vd   v(4R) v(4R)     The validity of this substitution can be demonstrated by calculating the hydraulic radius for a circular pipe. (Tutorial 2, Q7) This approach will give reasonable results if the cross section has an aspect ratio (ratio of the width of the section to its height) not much different from that of the circular cross Page 9 End of Topic 2.2 Flow in Closed Conduits (Non- Circular) Try Tutorial 2 (Questions 6 – 8) Page 10

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