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midterm_2 - Problem 2 A large beaker with a shallow pool of...

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Problem 2 A large beaker with a shallow pool of liquid Heptane of depth l , is under an array of mercury lamps which provide an energy flux of q 00 s , as shown in the figure. From experiments it was determined that the energy flux inside the heptane pool decays as q 00 s = q 00 0 e - γx , where q 00 0 and γ are positive constants. The temperature of the room is T , the heat transfer coefficient at the liquid surface is h and the thermal conductivity of liquid heptane is k . The bottom of the beaker is made of a transparent material that is perfectly insulating. At steady state conditions, assuming that the atmosphere is perfectly transparent to radiation and that there is no mass loss to the atmosphere: a. Show that the differential equation for variation of temperature in the hep- tane pool is d 2 T d x 2 + γq 00 0 k e - γx = 0 b. Setup the boundary conditions at the upper and lower boundaries. c. Solve the equation to get the temperature profile in the heptane pool. (Hint: Integrate the equation twice and apply the two boundary conditions) 1
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