Chapter 3 Unidirectional transport - Chemical Engineering ...

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40 CHAPTER 3. UNIDIRECTIONAL TRANSPORTq=0T 0TBT 0TAq=0Figure 3.11: Conduction from a cube.8. A rotating cylinder geometry consists of a cylinder of radius R andheight H, filled with fluid, with two end caps. The cylinder rotates withan angular velocity Ω, while the end caps are stationary. Determinethe fluid velocity field using separation of variables as follows.(a) Choose a coordinate system for the problem. Clearly, the onlynon-zero component of the velocity is u φ . Determine the boundaryconditions for this component of the velocity.(b) Write down the mass balance condition for an incompressible fluid.For a uni-directional flow in which the density is a constant, whatdoes this reduce to?(c) Use a shell balance to determine the conservation equation for thevelocity. Can you eliminate the pressure term using a result fromthe mass balance condition?(d) Solve the conservation equation at steady state using the methodof separation of variables. Frame the orthogonality conditionswhich would be required to solve the problem.Data:
3.3. EFFECT OF PRESSURE ON MOMENTUM TRANSPORT 41(a) Bessel equation:Solution:x 2 d2 ydx 2 + xdy dx + (x2 − n 2 )y = 0y = A 1 J n (x) + A 2 Y n (x)where J n (x) is bounded for x → 0, and Y n (x) is bounded forx → ∞. tem Modified Bessel equation:Solution:x 2 d2 ydx 2 + xdy dx − (x2 + n 2 )y = 0y = A 1 I n (x) + A 2 K n (x)where I n (x) is bounded for x → 0, and K n (x) is bounded forx → ∞.
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