HomeworkSolutions_C9 - Problem#1 Rework question(a in problem 4(heated sphere submerged in the motionless fluid and problem#5(heat flow through an

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Unformatted text preview: Problem #1: Rework question (a) in problem # 4 (heated sphere submerged in the motionless fluid) and problem #5 (heat flow through an annular solid wall) in the homework set # 8, using the energy equation in the lecture note or tabulated equations in the textbook (appendix B in the 2 nd edition, or chapter 10 in the 1 st edition). Note that you DO NOT need to solve the differential equations of temperature. Only show how to simplify the general equation of energy in terms of temperature using the assumptions in the problems. Answers: Problem #4: The general equation of energy in terms of temperature written for the motionless liquid using spherical coordinates 2 2 2 2 2 2 2 1 1 1 ˆ sin sin sin p T T T T C k r t r r r r r δ δ δ δ δ S ρ θ ρ δ δ θ δ θ δ θ θ δ φ ⎡ ⎤ ∂ ⎛ ⎞ ⎛ ⎞ = + + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ∂ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ + There is no heat source within the liquid body (note that the sphere is not the source since the above equation describes the heat transfer within the liquid body which surrounds the sphere). This gives S ρ = It is assumed that the temperature at the sphere surface is constant, and that the heat flow is in steady state. These assumptions results in ˆ p T C t ρ ∂ = ∂ Furthermore, the symmetric nature of the heat flow leads to 2 2 2 1 1 sin sin sin T T r r 2 2 δ δ δ θ θ δθ δθ θ δθ ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ The final energy equation reduces to 2 2 T d d T r r r r dr dr δ δ δ δ ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Problem #5: The general equation of energy in terms of temperature written for the motionless liquid using cylindrical coordinates 2 2 2 2 2 1 1 ˆ p T T T T C k r t r r r r z δ δ δ δ S ρ ρ δ δ δ θ δ ⎡ ⎤ ∂ ⎛ ⎞ = + + ⎜ ⎟ ⎢ ⎥ ∂ ⎝ ⎠ ⎣ ⎦ + There is no heat source within the solid wall, resulting in S ρ = For steady state heat flow ˆ p T C t ρ ∂ = ∂ Furthermore, the symmetric nature of the heat flow leads to 2 2 2 1 T r δ δθ = The temperature is uniform in the z direction 2 2 T z δ δ = The final energy equation reduces to T d d T r r r r dr dr ∂ ∂ ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠ Problem #2: Rework the viscous heating problem discussed in Section 9-4 in the textbook, 1 st edition (or Section 10-4, 2 nd edition). In the current case oil is replaced by a polymer melt, whose viscosity can be adequately by the power law model 1 n z du m dx η − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ Show that the temperature distribution is ( ) 1 1 1 Br 1 2 Br b n b n b T T x x x T T b b b mu b k T T + − − ⎛ ⎞ = − + ⎜ ⎟ − ⎝ ⎠ = − c f d g d g d g e h Answer: Consider the idealized flow geometry shown in Fig. 10.4-2 (or Fig. 9.4-2 in the 1Consider the idealized flow geometry shown in Fig....
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This note was uploaded on 09/01/2011 for the course PGE 312 taught by Professor Peters during the Fall '08 term at University of Texas at Austin.

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HomeworkSolutions_C9 - Problem#1 Rework question(a in problem 4(heated sphere submerged in the motionless fluid and problem#5(heat flow through an

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