## L6

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## L6

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- Pages:
- 9
- School:
- Texas A&M University
- Course:
- Chen 304 - Chem Engr Fluid Ops

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CBE341 Lecture 6 9 25 Dimensional analysis of fully developed flow in a pipe Consider steady incompressible viscous flow of a fluid through a long straight pipe Let U be the average velocity the fluid viscosity D the pipe diameter and the fluid density We consider the possibility that the inner wall of the pipe is rough and characterize the wall roughness by an average roughness height k Let the axis of the pipe be inclined to the horizontal at an angle B A L U Intuitively one would expect that the flow patterns will change as a function of axial position along the length of the pipe both near the inlet where the flow changes from that in the upstream region to what is typical for the pipe under consideration and near the outlet where the flow field begins to adjust to conditions prevailing in the downstream region If the pipe is long then one can reasonably expect that there will be a region of pipe in the middle where the flow pattern is essentially fully developed where the flow behavior is for all practical purposes independent of the axial position Our goal in this example is to analyze this pressure drop per unit length in the fully developed region Consider the body of fluid inside a section of length L in the fully developed region as shown in the figure below Let us consider conservation of mass and linear momentum over the shaded region in the figure 56 Let A and B denote the two faces of the shaded region see figure The shaded region is thus bounded by the flow regions A B and the pipe wall Conservation of mass equation 38 r t dV n r t v r t dS t V t S t 38 Here the control volume is stationary and the fluid is specified to be incompressible so the left hand side is equal to zero The right hand side gets contributions from faces A and B across which flow occurs Face A contribution to the right hand side Face B contribution to the right hand side D2 UA 4 D2 UB 4 Here UA and UB denote the average velocities at faces A and B respectively These two contributions to the mass conservation equation clearly add up to zero if and only if U A U B U In other words according to mass conservation principle the average velocity U is independent of axial location Of course this is intuitively obvious Conservation of linear momentum equation 57 d v dV v v n dS g dV Pn t dS Fother dt V fixed S fixed V fixed S fixed Let us examine this term by term d v dV 0 as the flow is steady dt V fixed 57 57 v v n dS gets contributions from faces A and B across which flow occurs But S fixed if the flow is fully developed then the velocity profile over the tube

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