# CSU MECH 344 - HW8 - Solutions (12 pages)

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# HW8 - Solutions

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## HW8 - Solutions

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Mech 344 - Heat and Mass Transfer
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PROBLEM 8 12 KNOWN Internal flow with prescribed wall heat flux as a function of distance FIND a Beginning with a properly defined differential control volume the temperature distribution Tm x b Outlet temperature Tm o c Sketch Tm x and Ts x for fully developed and developing flow conditions and d Value of uniform wall flux q s instead of q s ax providing same outlet temperature as found in part a sketch Tm x and Ts x for this heating condition SCHEMATIC ASSUMPTIONS 1 Steady state conditions 2 Constant properties 3 Incompressible liquid with negligible viscous dissipation PROPERTIES Table A 6 Water 300 K cp 4 179 kJ kg K ANALYSIS a Applying energy conservation to the control volume above p dTm dq conv mc 1 where Tm x is the mean temperature at any cross section and dqconv q dx Hence p ax mc dTm dx 2 Separating and integrating with proper limits gives x T x p m dTm xdx mc x 0 Tm i a Tm x Tm i ax 2 p 2mc 3 4 b To find the outlet temperature let x L then p Tm L Tm o Tm i aL2 2mc Solving for Tm o we find Tm o 27o C 5 20 W m 2 30 m 2 2 450 kg h 3600s h 4179 J kg K 27o C 17 2o C 44 2o C c For linear wall heating q s ax the fluid temperature distribution along the length of the tube is quadratic as prescribed by Eq 4 From the convection rate equation q s h x D Ts x Tm x 6 For fully developed flow conditions h x h is a constant hence Ts x Tm x increases linearly with x For developing conditions h x will decrease with increasing distance along the tube eventually achieving the fully developed value Continued PROBLEM 8 12 Cont d For uniform wall heat flux heating the overall energy balance on the tube yields p Tm o Tm i q q s DL mc Requiring that Tm o 44 2 C from part a find q s 450 3600 kg s 4179 J kg K 44 2 27 K D 30 m 95 3 D W m 2 where D is the diameter m of the tube which when specified would permit determining the required heat flux q s For uniform heating Section 8 3 2 we know that Tm x will be linear with distance Ts x will also be linear for fully developed

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