Class12: Energy losses in pipe flow When a fluid is flowing through a pipe, the fluid experiences some resistance due to which some of the energy of the fluid is lost. Energy losses Major energy loss (due to friction) Minor energy losses a. Sudden expansion b. Sudden contraction c. Bend in pipe d. Pipe fitting e. An obstruction in pipe Entry length Near the beginning of the pipe, the flow is not fully developed. If we have uniform velocity profile at the pipe entrance, how far down the pipe must we go before entrance effects are negligible? Entry length Laminar flow: Turbulent flow: L Re ≈ D 16 L ≈ 10 − 50 D Fully developed flow: The flow no longer changes in the flow direction. Frictional losses in pipe flows • The viscosity causes loss of energy in flows which is known as frictional loss. Expression for loss of head: 1 p1 A 2 p2 A Consider a horizontal pipe, having steady flow as shown above. Let L = length of the pipe between sections 1 and 2. d = diameter of the pipe f = friction factor hf = loss of head due to friction. p1 = pressure at section 1 v1 = velocity at section 1 p2, v2 are the corresponding values at section 2. Applying Bernoulli’s equations for real fluid at sections 1 and 2, we get 2 2 p1 v1 p v + + z1 = 2 + 2 + z2 + h f ρ g 2g ρ g 2g But z1 = z2 , and V1 = V2 , as the pipe is horizontal and the diameter of the pipe is same in both sections. hf = p1 p − 2 ρg ρg (1) Darcy friction factor is defined as , dp f ρV 2 ⇒− = dx 2 Dh f ρV 2 L ⇒ p1 − p2 = 2 Dh dp Dh f = − dx 1 ρV 2 2 Hydraulic diameter, Dh = 4( Area) Perimeter Using Eq. (1), we obtain L V2 hf = f Dh 2 g Darcy-Weishbach equation Loss of energy due to friction Darcy-Weishbach equation: Head loss due to friction, L V2 LV2 hf = f = 4C f d 2g d 2g L: length of the pipe V: mean velocity of the flow d: diameter of the pipe f is the friction factor for fully developed laminar flow: f = 64 ( for Re < 2000) Re ρ uavg d Re = µ C f is the skin friction coefficient or Fanning’s friction factor. 1 2 16 For Hagen-Poiseuille flow: C f = τ wall / ρ uavg = 2 Re For turbulent flow: ε 1 18.7 = 1.74 − 2.0 log10 p + f R Re f Moody’s Diagram R: radius of the pipe ε p : degree of roughness (for smooth pipe,ε p = 0) Re → ∞ : completely rough pipe Minor Energy head loss: Sudden expansion A2p2 A1p1 Let A1 = Area at section 1. p1 = pressure at section 1 v1 = velocity at section 1 A2, p2, v2 are the corresponding values at section 2. 1 2 Applying Bernoulli’s equations for real fluid at sections 1 and 2, we get 2 2 p1 v1 p2 v2 + + z1 = + + z2 + he ρ g 2g ρ g 2g 2 p1 − p2 v1 − v22 ⇒ he = + 2g ρg (1) But z1 = z2 as the pipe is horizontal. Consider a control volume of liquid between sections 1 and 2. Resolving the forces acting on the liquid inside the control volume, we get Fx = p1 A1 − p2 A2 + p '( A2 − A1 ) where p’ is pressure of the liquid eddies in the area (A2-A1). Experimentally it is known that p ' = p1 , hence Fx = ( p1 − p2 ) A2 ρ A1v12 2 Momentum of liquid/sec at section 2= ρ A2 v2 Change of momentum of liquid/sec = ρ A2 v22 − ρ A1v12 = ρ A2 ( v22 − v1v2 ) = Fx Momentum of liquid/sec at section 1= (Using the continuity equation) p1 − p2 v22 − v1v2 = ρg g Thus from (1), we obtain he v1 − v2 ) ( = 2g 2 Minor energy head loss: Sudden contraction A1p1 A2p2 Let A1 = Area at section 1. p1 = pressure at section 1 v1 = velocity at section 1 A2, p2, v2 and Ac, pc, vc are the corresponding values at section 2 and c, respectively. 1 c 2 Continuity equation: Ac vc = A2 v2 ⇒ Head loss due to expansion from section c to 2 hc (v − v ) = c 2 2g 2 2 2 2 v vc v 1 v22 = − 1 = k − 1 = 2 g v2 2 g Cc 2g 2 2 2 2 1 − 1 . The value of k is 0.5 to 0.7. where, k ≡ Cc vc A2 1 = = v2 Ac Cc v2 • Head loss at the entrance of the pipe: hi = 0.5 , 2g where v is the velocity of the liquid in the pipe. v2 • Head loss at the exit of the pipe: h0 = , 2g where v is the velocity of the liquid at the outlet of the pipe. kv 2 • Head loss due to bend in pipe: hb = , 2g where v is the velocity of the flow, k is the coefficient of the bend which depends on the angle of the bend, radius of curvature of the bend and diameter of pipe. • Head loss due to pipe fittings: kv 2 hf = , 2g where v is the velocity of the flow, k is the coefficient of pipe fitting.