TAMU CHEN 304 - L8 (9 pages)

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L8



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L8

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

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CBE341 Lecture 8 9 28 Application of the macroscopic balances We have already discussed the integral forms of the mass conservation equation and the linear momentum balance These equations are also known as Macroscopic balance equations We will now go through some examples to illustrate the application of these equations The macroscopic mass balance applied to the control volume below is given by equation 42 see p 37 v n vs V t S t d dt V t r dV n r v v dS S 42 S t The corresponding momentum balance is given by see p 47 d rv dV S t rv v vs n dS V t rg dV S t Pn t dS Fother dt V t Let us now solve a few problems by using these equations 48 56 Example Force on a U bend An incompressible liquid density r is flowing through a stationary U shaped pipe with a Patm uniform ID D at an average velocity of U Pressures at the inlet and the outlet are P1 and P2 respectively Both arms of the bend are at the same elevation Assume that the flow is in the turbulent regime and that the velocity profile at the inlet and outlet can be approximated as plug flow i e the velocity is approximately independent of the radial location Neglect the viscous stresses over the inlet and outlet flow surface areas a What is the horizontal force exerted by the water on the U bend To do this problem consider a control volume enclosing just the liquid inside the tube b If the U tube is held in place with a clamp what is the horizontal force exerted by the clamp on the tube while the liquid is flowing through the pipe To do this problem consider a control volume that encloses the tube and the liquid inside it This control volume will cut the pipe wall at the liquid inlet and outlet neglect the traction exerted by pipe wall outside the control volume on the pipe wall inside the control volume at these inlet and outlet locations Solution a Since we seek the force exerted by the fluid on the tube wall it makes sense to consider a stationary control volume V around the fluid inside the U tube The surface S enclosing this control volume can be partitioned into three parts S1 tube cross sectional area at the inlet Let the area of this face be A1 S2 tube cross sectional area at the outlet Let this area be A2 S Curved area representing the pipe wall Clearly A1 A2 p D2 A say 4 58 We will take the inflow direction as the positive x direction Then at the inlet the velocity at various radial locations can be written as v v1e x so that v1 is positive everywhere and can in general be a function of the radial position At the exit where we know that the flow is pointing



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