Final Program EXPRES 2012 - Conferences

C. Energy balance equationMathematical model consists of equations ofenergy balance for all of the four components of theCPC modle, relations for determining heat transfercoefficient, relations for radiation absorbed by therelevant system components. In order to gain a unifiedsolution from the system of equations, initial andboundary conditions are defined. With apredetermined assumptions, equations of energybalance may be written as:(1) For the working fluidEnergy balance for an elementary fluid volumeof dz length in the axial direction, after sorting, maybe written as:T* * f c A *c A*wf p f f f p f f ztU T 2/ r T rr f f r , oTfzWhere ∗ , i w z represent density, specificheat, velocity in the axial direction of the elementary*fluid volume, respectively A 2 f r r , iis the area ofthe cross section of the elementary fluid particle. Thesecond term on the right hanside of the equation (2)represents heat gained from the outer side of thecollector pipeBoundary condition for this equation is:za z = 0,fin(2)T T . (3)(2) For the collector absorber pipeEnergy balance for elementary part of the collector ofdz length in the axial direction may be written as;* * TrTr* r cr Ar r Ar hc, r / ehr, r / et z zTr Te 2 rr , o-U r / fTr Tf2 rr , o qb, rqd , r2 rr , o (4)Where λ r , ∗ i c r are head conductioncoefficient, density and specific heat for elementary* 2 2pipe volume, respectively. Ar rr, o rr, i is thearea of the cross section of elementary part of thecollector pipe. The first term on the right hand side ofthe equation (4) represents heat transport byconduction in the axial direction of the collector, thesecond is the heat loss due to convention and radiationbetween the collector pipe and the surrounding layerof the collector pipe, the third term represents heatgiven to the working fluid, and the fourth representsheat gained via solar radiationBoundary conditions for this equation are:T za z 0, r 0 ; (5)zza z L,T r 0(6)z(3) For the transparent cover of the pipecollectorEnergy balance for the elementary part of thesurrounding layer of the collector pipe of dz length iswritten as:* * T eece Ae hc, r/ ehr, r/eTr Te 2 r,ot-hc, e/ chr, e/cTeTc2 re , o qb, eqd , e2 r(7)r,o∗Where i c e are density and specific heat og the* 2 2surrounding layer of the collector Ar re, o re, iis the area of the cross section of elementary part ofthe surraounding layer of the collector. The secondterm on the right hand side of the equation (7) is theheat lost due to radiation from the surrounding layerof the collector to the transparent cover(4) For the transparent coverEnergy balance for the elementary part of thetransparent cover of the concentrating collector, withdz length in the z direction and W width, may bewritten as:* * T cccc Ac hc, e/ chr, e/cTe Tct2 re , o ­h c, c/ aTc Ta W hr, c/sTc TsWq b , cq d , c2 r r , o(8)*A c W is the area of the cross section ofelementary part of the transparent cover of thecollector. The second term on the right hand side ofthe equation (8) is the heat lost due to convectionfrom the transparent cover to the environment, thethird term is the heat lost due to radiation between thetransparent cover and the sky.D. Initial conditionsIt was assumed for the initial conditions that inthe initial time point the temperature field in allcomponents is equal to the environment temperatureT a , while the fluid temperature at the inlet is constantin timeZa t 0, TfTr Te Tc Ta(9)Za z 0, T fT in(10)103