# 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 conditions and appear as shown below when the flow is developing COMMENTS 1 Note that cp should be evaluated at Tm 27 44 C 2 309 K 2 Why did we show Ts 0 Tm 0 for both types of history when the flow was developing 3 Why must Tm x be linear with distance in the case of uniform wall flux heating PROBLEM 8 25 KNOWN Inlet temperature and flowrate of oil flowing through a tube of prescribed surface temperature and geometry FIND a Oil outlet temperature and total heat transfer rate and b Effect of flowrate SCHEMATIC ASSUMPTIONS 1 Negligible temperature drop across tube wall 2 Incompressible liquid with negligible viscous dissipation PROPERTIES Table A 5 Engine oil assume Tm o 140 C hence Tm 80 C 353 K 852 kg m3 37 5 10 6 m2 s k 138 10 3 W m K Pr 490 0 032 kg m s cp 2131 J kg K ANALYSIS a For constant surface temperature the oil outlet temperature may be obtained from Eq 8 41b Hence DL Tm o Ts Ts Tm i exp h mc p To determine h first calculate ReD from Eq 8 6 ReD 4 0 5 kg s 4m 398 D 0 05m 0 032 kg m s Hence the flow is laminar Moreover from Eq 8 23 the thermal entry length is x fd t 0 05D ReD Pr 0 05 0 05 m 398 490 486 m Since L 25 m the flow is far from being thermally fully developed Since Pr 5 h may be determined from Eq 8 57 Nu D 3 66 0 0668Gz D 1 0 04Gz D2 3 With GzD D L ReDPr 0 05 25 398 490 390 it follows that 26 11 95 1 2 14 k 0 138 W m K Hence h Nu D 11 95 33 W m 2 K and it follows that D 0 05 m Nu D 3 66 Continued PROBLEM 8 25 Cont 0 05 m 25 m 33 W m 2 K Tm o 150o C 150o C 20o C exp 0 5 kg s 2131J kg K Tm o 35 C From the overall energy balance Eq 8 34 it follows that p Tm o Tm i 0 5 kg s 2131J kg K 35 20 o C q mc q 15 980 W The value of Tm o has been grossly overestimated in evaluating the properties The properties should be re evaluated at T 20 35 2 27 C and the calculations repeated Iteration should continue until satisfactory convergence is achieved between the calculated and assumed values of Tm o Following such a procedure one would obtain Tm o 36 4 C ReD 27 8 h 32 8 W m2 K and q 15 660 W The small effect of reevaluating the properties is attributed to the compensating effects on ReD a large decrease and Pr a large increase b The effect of flowrate on Tm o and q was determined by using the appropriate IHT Correlations and Properties Toolpads 30000 36 Heat rate q W Outlet temperature Tmo C 40 32 28 25000 20000 24 20 15000 0 5 1 1 5 2 0 5 1 Mass flowrate mdot kg s 1 5 2 Mass flowrate mdot kg s due to the corresponding increase in ReD and hence h The heat rate increases with increasing m causing Tm o Tm i q mc p and hence Tm o to However the increase is not proportional to m 0 20 The maximum heat rate corresponds to the maximum flowrate m decrease with increasing m kg s COMMENTS Note that significant error would be introduced by assuming fully developed thermal conditions and Nu D 3 66 The flow remains well within the laminar region over the entire range of m PROBLEM 8 27 KNOWN Inlet and outlet temperatures and velocity of fluid flow in tube Tube diameter and length FIND Surface heat flux and temperatures at x 0 5 and 10 m SCHEMATIC ASSUMPTIONS 1 Steady state conditions 2 Constant properties 3 Negligible heat loss to surroundings 4 Incompressible liquid with negligible viscous dissipation 5 Negligible axial conduction PROPERTIES Pharmaceutical given 1000 kg m3 cp 4000 J kg K 2 10 3 kg s m k 0 80 W m K Pr 10 ANALYSIS With VA 1000 kg m3 0 2 m s 0 0127 m 2 4 0 0253 kg s m Eq 8 34 yields c p Tm o Tm i 0 0253 kg s 4000 J kg K 50 K 5060 W q m The required heat flux …

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