Introductory Transport Phenomena
Introductory Transport Phenomena CBE 320
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This 1 page Class Notes was uploaded by Cordie Balistreri on Thursday September 17, 2015. The Class Notes belongs to CBE 320 at University of Wisconsin - Madison taught by Daniel Klingenberg in Fall. Since its upload, it has received 46 views. For similar materials see /class/205198/cbe-320-university-of-wisconsin-madison in Chemical Engineering at University of Wisconsin - Madison.
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Date Created: 09/17/15
The Shell Momentum Balance StepbyStep Method CBE 320 DJK Fall 2006 Step 1 Draw a crude picture and list assumptions 0 What velocity components are nonzero o What spatial variables does the velocity depend on 0 Choose coordinate system 0 Start a list of assumptions Step 2 Select system 0 Draw a second picture showing the shell in detail 0 Shell 0 a box with sides or I to the velocity v 0 box faces should lie on coordinate surfaces Qlou choose your coordinate system to guarantee this 0 Make shell thin in the direction the velocity is varying 0 You will develop intuition about how to do this as you work prob lems Step 3 Apply conservation of momentum compo nent of interest Rale al Wmch Rate al whtch momentum ts momentum ts Force ot grath Iransporled Iransponed 361th on O tnto out ot the system the system the system 7 add other bad tl unsleady Ihen l u P 5V 13 PVtV mes he y the RHStsIheltme re eg eteotnoat magneltc rale ot change ot elc momentum Add arrows to second picture indicating Where ux components transport momentum in and out Write out the conservation of momentum equation momentum balance for the system Step 4 Simplifyeliminate terms if possible 0 Insert appropriate de nitions for the lj components ie 4w 10511 W WW 0 Use assumptions to eliminate terms that are zero or terms that cancel 0 Do as much simplifying here as possible but you will have more opportunities to simplify later Step 5 Let Shell thickness a 0 0 Let the small dimension of the box A 0 o This will give a differential equation for the momentum ux the relevant component of m or 4 depending how much simplify ing you did in the previous step 0 If you haven t already make sure you have substituted for all components of ns and simpli ed as much as possible Step 6 Integrate once if possible 0 This will give a constant of integration 0 If you have a boundary condition for the momentum ux compo nent 739 apply the boundary and solve for the constant 0 You will not evaluate the constant here if both of your boundary conditions are for the velocity Step 7 Insert Newton s Law of Viscosity 0 Use Appendix Bl pp 8437844 to replace the components of W with the appropriate expressions in terms of velocity compo nents o Simplify as much as possible 0 Appendix BI is general some velocity components and derivatives Will be zero 0 Make use of your assumptions and add to the list of assump tions if new ones occur to you This will lead to a differential equation for the desired velocity component First order if you completed step 6 second order if you skipped step 6 Step 8 Solve the differential equation 0 This will produce another unknown constant 0 Use a second boundary condition to get this constant 0 The result of this step Will be the velocity pro le for example 1 7 Step 9 Use the velocity pro le to engineer 0 Using the velocity or stress pro le you can calculate a variety of quantities of interest eg 0 force on a surface 0 torque on an obj ect o volumetric ow rate 0 maximum ow rate 0 etc