Chapter 6 Fundamentals of Convection 6-37 Parallel flow of oil between two plates is considered. The velocity and temperature distributions, the maximum temperature, and the heat flux are to be determined. Assumptions 1 Steady operating conditions exist. 2 Oil is an incompressible substance with constant properties. 3 Body forces such as gravity are negligible. 4 The plates are large so that there is no variation in z direction. Properties The properties of oil at the average temperature of (40+15)/2 = 27.5 ° C are (Table A-13): k = 0.145 W/m-K and μ = 0.580 kg/m-s = 0.580 N-s/m 2 Analysis ( a ) We take the x-axis to be the flow direction, and y to be the normal direction. This is parallel flow between two plates, and thus v = 0. Then the continuity equation reduces to Continuity : 0= ∂ ∂ + ∂ ∂ y v x u ⎯→0 = ∂ ∂ x u ⎯→ u = u ( y ) 12 m/s Oil L =0.7 mm T 2 =40 ° C T 1 =25 ° C Therefore, the x-component of velocity does not change in the flow direction (i.e., the velocity profile remains unchanged). Noting that u = u ( y ), v = 0, and 0/ = ∂ ∂ x P (flow is maintained by the motion of the upper plate rather than the pressure gradient), the x-momentum equation (Eq. 6-28) reduces to x-momentum : x P y u y u v x u u ∂ ∂ − ∂ ∂ μ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
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