file: Fluid Flow Calculator equations 14.pdf fro: Mark van Dijk revision: DEC 01 LAMINAR FLOW (Reynolds < 30, parabolic velocity profile) Nae sybol forula unit gravity g 9. 81 pipe length L elevation change h inner diaeter Di distance to pipe centre r pipe crosstional area A A = 0. 5 π Di average velocity v v = flow / A Reynolds Re ρ v Di Re = η equivalent length Leq L L Leq = Di fro tables Di ( ) Di friction factor F 64 / Re relative roughness surface k/di not applicable for lainar flow friction of pipe Kw(pipe) Kw( pipe) = F L / Di friction of appendages Kw(app) Kw( app) = F Leq / Di pipe volue V V = A L litre average residence tie in pipe, transportation lag, t t = L / v = V / flow distance/velocity lag power loss in fluid P P = p flow pressure drop pipe p = g h +??? p ρ bar dynaic viscosity η Also called Newtonian shear viscosity. average shear rate γ η = τ / γ σ v v = = η x Di 4 γ (see Figure 1) N = Watt Pa.s = centipoise (estiated shear rate in round pipes) shear stress σ or τ σ = F / A N / velocity profile, v(r) V position Vposition = v (1 ( r /(0.5 Di)) ) = velocity @ position r residence tie profile, t(r) = residence tie @ position r t position t position = L / Vposition shear rate profile, γ (r) γ position 4 v r γ position = 1 = shear rate @ position r (0.5 Di) 1 1
Figure 1: Lainar shear field due to applied shear stress. Figure : Velocity profile and shear rate profile in lainar flow. The highest shear rate is where the velocity gradient is the highest: at the wall of the pipe.
revision: JAN 006 LAMINAR FLOW POWERLAW FLUIDS (Reynolds < 30, parabolic velocity profile) nae sybol forula unit Gravity g 9. 81 pipe length L elevation change h inner diaeter Di distance to pipe centre r pipe crosstional area A A = 0. 5 π Di average velocity v v = flow / A Reynolds powerlaw Re n n 1 ρ v Di 4n Di Re = * * K 3n + 1 8v equivalent length Leq L L Leq = Di fro tables Di ( ) Di friction factor F 64 / Re relative roughness surface k/di not applicable for lainar flow friction of pipe Kw(pipe) Kw( pipe) = F L / Di friction of appendages Kw(app) Kw( app) = F Leq / Di pipe volue V V = A L litre t t = L / v = V / flow average residence tie in pipe, transportation lag, distance/velocity lag power loss in fluid P P = p flow pressure drop pipe p ρ ( ( ) ( )) 1 p = g h + Kw pipe + Kw app ρ v dynaic viscosity η Also called Newtonian shear viscosity. average shear rate γ η = τ / γ σ v v = = η x Di 4 γ (see Figure 1) bar N = Watt Pa.s = centipoise (estiated shear rate in round pipes) shear stress σ or τ σ = F / A N / velocity profile, v(r) V position ( n+ 1) / n = velocity @ position r 3n + 1 r Vpos = v 1 n + 1 0.5Di residence tie profile, t(r) = residence tie @ position r t position t position = L / Vposition shear rate profile, γ (r) γ position 4 v r γ position = = shear rate @ position r (0.5 Di) 1 NB. The highest shear rate is where the velocity gradient is the highest: at the wall of the pipe. 3
TURBULENT FLOW (Reynolds >= 30, flat velocity profile) nae sybol forula unit gravity g 9. 81 pipe length L elevation change h surface roughness k inner diaeter Di distance to pipe centre r pipe crosstional area A A = 0. 5 π Di average velocity v v = flow / A Reynolds Re ρ v Di Re = η Equivalent length Leq L L Leq = Di fro tables Di ( ) Di friction factor F = f = λ (iterative algorith of k. 5 λ = log + ColebrookWhite) 37. Di Re λ relative roughness surface k/di k / Di friction of pipe Kw(pipe) Kw( pipe) = F L / Di friction of appendages Kw(app) Kw( app) = F Leq / Di pipe volue V V = A L litre average residence tie in pipe, transportation lag, distance/velocity lag t t = L / v = V / flow power loss in fluid P N P = p flow = Watt pressure drop pipe p 1 p = ρ g h + ( Kw( pipe) + Kw( app) ) ρ v bar velocity profile, v(r) V position 1/ n Vposition = v (1 ( r /(0.5 Di)) = velocity @ position r n is a function of the Reynolds nuber. Varies fro about 6 to 10. Specific values: n=7 for Re=10^5, n=9 for Re=10^6 residence tie profile, t(r) = residence tie @ position r t position t position = t = L / v = V / flow The residence tie at position r is the sae as the average residence tie. The assuption ade is that the fluid is ideally ixed due to the turbulence. 4
shear rate profile, γ (r) = shear rate @ position r r 0.5 Di γ position = v ( ) 6 The integral over V position. In reality the shear rate will be higher due to turbulence. γ position 5/ 6 1 average shear rate γ σ v 4 v γ = = (see Figure 1) η x Di The assuption ade is that the turbulent fluid iniu shear rate is the sae as the average shear rate for lainar flow. In reality the shear rate will be higher due to turbulence. 1 In fact it is the integral over γ position Friction factor: The chart above shows the relationship between Reynold s nuber and pipe friction. Calculation of friction factors is dependant on the type of flow that will be encountered. For Re nubers <30 the fluid flow is lainar, when Re nuber is >= 30 the fluid flow is turbulent. Lainar flow (Re < 30) : f = 64/Re Turbulent flow (Re > 30) k.5 : f = λ = log + 3.7 Di Re λ In case of turbulent flow, the inner roughness of the pipe work can have a significant effect on the friction factor. See also: http://www.pipeflow.co.uk/theory.htl 5
Figure 3: Flow disturbance of a fluid passing a cylindrical obstacle. The flow changes fro lainar to turbulent. 6
Tables EQUIVALENT PIPE LENGTH OF FITTINGS Bends / Tees 180 bend, R=5D [ 8 L/Di ] 90 bend, R=5D [ 16 L/Di ] 90 bend (square, R=1.5D) [ 0 L/Di ] 45 bend (square, R=1.5D) [ 16 L/Di ] Tee (flow straight through) [ 0 L/Di ] Tee (flow through side outlet) [ 65 L/Di ] Valves Gate valve open [ 13 L/Di ] ¼closed [ 39 L/Di ] ½closed [ 195 L/Di ] ¾closed [ 300 L/Di ] Mebrane valve [ 00 L/Di ] Ball valve (spherical plug valve) [ 18 L/Di ] Needle valve [ 1000 L/Di ] Butterfly valve (larger then 6 inch) [ 0 L/Di ] Globe valve [ 300 L/Di ] Nozzle (suction nozzle on vessel) [ 3 L/Di ] Check valve (inline ball type) [ 150 L/Di ] Check valve (swing type) [ 135 L/Di ] Filter (Ytype and bucket type) [ 50 L/Di ] Surface roughness value [E] Various aterials Glass Lead Copper Brass Concrete tube Steel Pipe New After longer use Slightly rusted Very rusted Galvanised Cast iron New With bituen layer Slightly rusted Very rusted 0.015 [] 0.015 [] 0.015 [] 0.015 [].0 [] 0.04 [] 0. [] 0.4 [] 3.35 [] 0.15 [] 0.5 [] 0. [] 1.5 [] 3.0 [] MvD 7