PowerPoint PresentationTemperature DistributionTemperature Distribution on the PlateHeat Transfer to the TubeTemperature Distribution in Flow DirectionTemperature Distribution in Flow Direction (2)Temperature distribution in flow direction (3)Solar Collector PlateCollector plate1 m wide, W=1 mWater tubesxinsulationRadiation heat transfer inTQuestion: what will be the temperature distribution of the plate, T(x), if there is a constant radiation heat flux q” coming in. For simplicity, assume the temperature immediately below the water tube is constant, T(x=L)=TO.x=L=0.1(m)WaterflowTemperature DistributionModel the solar heat flux as continuous generation inside the plate:averge solar flux around noon time: q"= 700 W / mAssume it uniformly distributes inside the plate with thickness = 1 mmq = q / = 700000 W / m separation between tubes 2L = 2 cm, L = 10 cmHeat diffusion equation: T +qk= 0, k = 240 W / m K for Aluminumwith boundary conditions: dTdx(x = 0) = 0, T(x = L) = T integrate twice to get T(x) = -q2kApply boundary conditions: A = 0, B = 40 +q2k232O,,,( )( ) 404024022222 2Cd Tdxqkx Ax BLT xqkL x 1458 3 0 012. ( . ) xTemperature Distribution on the PlateT xqkL x x( )( ) . ( . ) 40240 1458 3 0 012 2 20 0.025 0.05 0.075 0.140455055Distance from centerlineTemperature T( )xxxTqcondqcondq kdTdxqxcond" qcondHeat Transfer to the Tubeq kdTdxqxq x L Wcond",)()(( ) ( ) Total heat transfer q(x) = q" A = q" (W)( ) = q(W)( x)q(x = 0) = 0, no heat transfer at the middle between two tubesq(x = L) = q(W)( L) = 700000(1)(0.001)(0.1) = 70(W)The total heat transfer into the tube should be twice of this amountqcond condtube 2 140Tube bondat temperature of 40°Cqtube = hAs(T-T)As=RWh=140/[(RW)(T- T)] =111.5(W/m2K)for a 2-cm radius pipeand an incoming water temperature of 20°CWater, 20°CTemperature Distribution in Flow DirectionzE mc Tz p z E EdEdzdz mc T mcdTdzdzz dz zzp z pz Assume constant heat flux q” into the tube000 q q E Eq dzRdEdzdz q dzR mc dTdTdzq Rmcz Tq Rmczin out z z dzzp zzpzp "( )"( )",( ) ( )", integrate from z = 0 to z = WT the temperature increases linearlydue to the heat transfer from the collector platezTemperature Distribution in Flow Direction (2)Tz( ) ( )",z Tq Rmczzp 0q”=h(TO-T)=2230 (W/m2), R=0.02 (m), dm/dt=0.05 (kg/s), cp=4200(J/kg K)Tz(z)=Tz(0)+0.212z=20+0.212z0 25 50 75 100202530354045distance (m)temperature (deg. C)T( )zz• Temperature increases linearly. It increases 0.212°C for every 1 m increase in length. The temperature increase in water can be significant if the tube length is long.Temperature distribution in flow direction (3)Assume h(convection heat transfer coefficient) is constant; q” is not a constant)0040 20 0 0106 q q E Eq dzRdEdzdzh dzR T T mc dTdTT ThRdzmcz T T ThRmczzin out zz dzzO z p zzO z pO Op "( )( )( )( ),( ) ( ) exp()exp( . ) integrate to getTz0 25 50 75 100202530354045distancetemp.T( )zTT( )zzAs the water temperature increases, both the heat transfer and the rate of temperature change decrease
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