SolutionHW_C7 - Homework Set#7 Due date by 11:00 am Problem...

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Homework Set #7: Due date: 11/10/06, by 11:00 am. Problem # 1: A crude oil rich in wax must be heated to enable its transportation in an annular space of a production well. If the clearance between the two coaxial casings forming the annulus is significantly smaller than the radius of the inner casing, one can treat flow in this annulus as that between two flat surfaces. In this sense, the crude oil with temperature-independent physical properties is in fully developed laminar flow between two flat surfaces placed a distance 2B apart, as shown in the below figure. For z < 0 the fluid is uniform at T = T 1 . For z > 0 heat is added at a constant, uniform flux q 0 at both walls. It is assumed that heat conduction in the flow direction is negligible compared to energy convection, and that the viscous works is negligible. a. State necessary assumptions. b. Derive a partial differential equation describing temperature distribution in the crude oil. You do NOT need to solve this equation. c. State boundary conditions. Answer: 1. Assumptions a. The heat conduction in the z-direction is negligible compared to energy convection b. Works done by viscous stresses are negligible. c. The oil is incompressible. d. Steady state heating 2. Derive a partial differential equation for temperature Since the temperature depends on both x and z, we make an energy balance over a region of volume W x z , in which W is the dimension of the slit in the y direction. The various contributions of the energy balance are:
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Total rate of energy in at x: ( ) x x W z e Total rate of energy out at x x + ∆ : ( ) x x x W z e +∆ Total rate energy out at z : ( ) z z W x e Total rate energy out at z z + ∆ : ( ) z z z W x e +∆
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