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pipe-flow-example

pipe-flow-example - Pipe Flow Example Water flows steadily...

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Pipe Flow Example Water flows steadily into the circular pipe with a uniform inlet velocity profile as shown. Due to the presence of viscosity, the velocity immediately adjacent to the inner pipe wall becomes zero and this phenomenon is called the no-slip boundary condition. It is found out that the velocity distribution reaches a parabolic profile at a distance downstream of the entrance and can be represented as V 1 (r)=V max [1-(r/R) 2 ], where V max is the velocity at the center of the pipe and r is the radial distance measured away from the center axis. Use the mass conservation equation, determine V max. V O P O R: radius of pipe V 1 (r )=V max [1-(r/R) 2 ] P 1
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Mass Conseration CS IN OUT 1 1 OUT r=R 2 1 1 r=0 2 2 max 2 Mass conservation: V dA 0 V dA V dA 0 V dA 0, since V & dA are in the same direction V (r)(2 rdr), since dA =d( r ) 2 r ( ) V 1- (2 R CV O O O O O d t V A V A rdr V R ρ ρ ρ ρ ρ ρ π π π π π 2200 + = + = - + = = = = r r r r r r r r r=R 3 max 2 r=0 0 2 2 max
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