CSU MECH 344 - HW7 - Solutions (8 pages)

PROBLEM 7 17 KNOWN Temperature pressure and Reynolds number for air flow over a flat plate of uniform surface temperature FIND a Rate of heat transfer from the plate b Rate of heat transfer if air velocity is doubled and pressure is increased to 10 atm SCHEMATIC ASSUMPTIONS 1 Steady state conditions 2 Uniform surface temperature 3 Negligible radiation 4 Re x 5 105 c PROPERTIES Table A 4 Air Tf 348K 1 atm k 0 0299 W m K Pr 0 70 ANALYSIS a The heat rate is q h L w L Ts T 4 Since the flow is laminar over the entire plate for ReL 4 10 it follows that h L 1 2 1 3 Nu L L 0 664 Re1 2 Pr1 3 0 664 40 000 0 70 117 9 L k k 0 0299 W m K Hence h L 117 9 117 9 17 6 W m2 K L 0 2m and q 17 6 W m2 K 0 1m 0 2m 100 50 o C 17 6 W b With p2 10 p1 it follows that 2 10 1 and 2 1 10 Hence u L u L ReL 2 2 10 20 ReL 1 8 105 2 1 and mixed boundary layer conditions exist on the plate Hence h L 1 3 0 037 8 105 4 5 871 0 70 1 3 Nu L L 0 037 Re4 5 871 Pr L k Nu L 961 Hence h L 961 q 143 6 W m2 K 0 0299 W m K 143 6 W m2 K 0 2m 0 1m 0 2m 100 50 o C 143 6 W COMMENTS Note that in calculating ReL 2 ideal gas behavior has been assumed It has also been assumed that k and Pr are independent of pressure over the range considered PROBLEM 7 24 KNOWN Plate dimensions and initial temperature Velocity and temperature of air in parallel flow over plates FIND Initial rate of heat transfer from plate Rate of change of plate temperature SCHEMATIC ASSUMPTIONS 1 Negligible radiation 2 Negligible effect of conveyor velocity on boundary layer development 3 Plates are isothermal 4 Negligible heat transfer from sides of plate 5 5 Re x c 5 10 6 Constant properties PROPERTIES Table A 1 AISI 1010 steel 573K kp 49 2 W m K c 549 J kg K 7832 3 6 2 kg m Table A 4 Air p 1 atm Tf 433K 30 4 10 m s k 0 0361 W m K Pr 0 688 ANALYSIS The initial rate of heat transfer from a plate is q 2 h As Ti T 2 h L2 Ti T With Re L u L 10 m s 1m 30 4 10 6 m 2 s 3 29 105 flow is laminar over the entire surface and 2 1 3 0 664 3 29 105 Nu L 0 664 Re1 L Pr 1 2 0 688 1 3 336 h k L Nu L 0 0361W m K 1m 336 12 1W m 2 K Hence q 2 12 1W m 2 K 1m 2 300 20 C 6780 W Performing an energy balance at an instant of time for a control surface about the plate E out E st we obtain L2c dT h 2L2 Ti T dt i 2 12 1W m 2 K 300 20 C dT 0 26 C s dt i 7832 kg m3 0 006m 549 J kg K COMMENTS 1 With Bi h 2 k p 7 4 10 4 use of the lumped capacitance method is appropriate 2 Despite the large plate temperature and the small convection coefficient if adjoining plates are in close proximity radiation exchange with the surroundings will be small and the assumption of negligible radiation is justifiable PROBLEM 7 92 KNOWN Geometry surface temperature and air flow conditions associated with a tube bank FIND Rate of heat transfer per unit length SCHEMATIC ASSUMPTIONS 1 Steady state conditions 2 Negligible radiation effects and incompressible flow 3 Gas properties are approximately those of air PROPERTIES Table A 4 Air 300K 1 atm Pr 0 707 Table A 4 Air 700K 1 atm 68 1 6 2 3 10 m s k 0 0524 W m K Pr 0 695 0 498 kg m cp 1075 J kg K ANALYSIS The rate of heat transfer per unit length of tubes is q hN D Tlm hN D With Vmax ST ST D V 20 Ts Ti Ts To ln Ts Ti Ts To 5 m s 10 m s Re D max 10 Vmax D 10 m s 0 01 m 68 1 10 6 2 1468 m s Tables 7 5 and 7 6 give C1 0 27 m 0 63 and C2 0 97 Hence from the Zukauskas correlation 0 36 Nu D C1C2 Re m Pr Prs 0 26 1468 0 695 0 695 0 707 D max Pr k Nu D 22 4 h Nu D 0 0524 W m K 22 4 0 01 m 117 W m 2 K D 1 4 0 63 0 36 1 4 Hence 400K exp VN TST c p Ts To Ts Ti exp DNh 0 01 m 500 117 W m 2 K 0 498 kg m3 5 m s 50 0 02 m 1075J kg K Ts To 201 3K and the heat rate is q 117 W m 2 K 500 0 01 m 400 201 3 K 532 kW m ln 400 201 3 COMMENTS 1 There is a significant decrease in the gas temperature as it passes through the tube bank Hence the heat rate would have been substantially overestimated 768 kW if the inlet temperature difference had been used in lieu of the log mean temperature difference 2 The negative sign implies heat transfer to the water 3 If the temperature of the water increases substantially the assumption of uniform Ts becomes poor The extent to which the water temperature increases depends on the water flow rate PROBLEM 7 105 KNOWN Exit diameter of plasma generator and radius of jet impingement surface Temperature and velocity of plasma jet Temperature of impingement surface Droplet deposition rate FIND Rate of heat transfer to substrate due to convection and release of latent heat SCHEMATIC ASSUMPTIONS 1 Steady state conditions 2 Negligible radiation 3 Negligible sensible energy change due to cooling of droplets to Ts ANALYSIS The total heat rate to the substrate is due to convection from the jet and release of the latent heat of fusion due to solidification q qconv qlat With Re VeD 400 m s 0 01 m 5 6 10 3 m2 s 714 Ar D2 4r2 0 04 and H D 10 F1 2Re1 2 1 0 005 Re0 55 1 2 58 2 and G 2 1 2 1 2 2A1 r 1 2 2A r 1 0 2 H D 6 A r 0 193 the correlation for a single round nozzle Section 7 7 yields Nu GF1 Pr 0 42 0 193 58 2 0 600 42 9 07 h Nu k D 9 07 0 671W …