Thermodynamic Head Loss in a Channel with Combined Radiation and Convection Heat Transfer

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1 Jounal of Poe and Enegy Engineeing, 04,, Published Online Septembe 04 in SciRes. hemodynamic Head Loss in a Channel ith Combined Radiation and Convection Heat ansfe Deodat Makhanlall, Peixue Jiang Depatment of hemal Engineeing, singhua Univesity, Beijing, China deodat@mail.tsinghua.edu.cn Received July 04 Abstact Losses in channel flos ae usually detemined using a fictional head loss paamete. Fluid fiction is hoeve not the only souce of loss in channel flos ith heat tansfe. Fo such flo poblems, themal enegy degadation, in addition to mechanical enegy degadation, add to the total loss in themodynamic head. o assess the total loss in a channel ith combined convection and adiation heat tansfe, the conventional fictional head loss paamete is extended in this study. he analysis is applied to a 3D tubulent channel flo and identifies the citical locations in the flo domain hee the losses ae concentated. he influence of Boltzmann numbe is discussed, and the best channel geomety fo flos ith combined heat tansfe modes is also detemined. Keyods Radiation-Convection Heat ansfe, hemodynamic Head Loss, Entopy Geneation. Intoduction hemodynamic ievesibility educes the iciency of all pactical momentum and heat tansfe pocesses. he measuable ects of themodynamic ievesibility in a flo though a channel ae the dop in pessue and tempeatue. he non-measuable ects involve entopy geneation. A loss in total head is thus elated to entopy geneation. his elationship can be itten as []: hf = S gen, f d m () and shos that the fiction head loss tem in the enegy equation is a measue of the themodynamic ievesibility associated ith momentum tansfe, i.e. mechanical enegy degadation. In flos involving heat tansfe, themal enegy degadation also occus in addition to the mechanical enegy degadation. he objective of this ok is to use an extended fom of Equation () to study both mechanical and themal enegy degadation in a 3D channel ith combined adiation and convection heat tansfe. Such flo poblems ae impotant in high-tempeatue applications, such as sola enegy collectos, nuclea eactos, and Ho to cite this pape: Makhanlall, D. and Jiang, P.X. (04) hemodynamic Head Loss in a Channel ith Combined Radiation and Convection Heat ansfe. Jounal of Poe and Enegy Engineeing,,

2 electonic equipments. he FLUEN 6.3 CFD code is employed to solve the govening equations. he poblem schematic is shon in Figue and consists of a 0 m long 3D channel ith aspect atio HW= 0.5. ubulent flo is consideed.. heoy.. Govening Equations he equations fo mass, momentum and enegy tansfe in the channel can be itten in Catesian tenso fom as folloing, espectively: ui = 0 x i ( ρuu ) i j u u i j u k p = µ + δij ρkδij xj xj xj xi 3 x k 3 xi ρu i ui ρ c pd + x 0 i c µ u u u = x x x x x p t j i k km ui µ µ δij q j P t j i j 3 k hee, u is a velocity component, ρ is the density, k m is themal conductivity, c p is specific heat, P t is the tubulent Pandtl numbe, and µ t and µ ae the tubulent and ective tubulent viscosities, espectively. hey ae computed as folloing: k = + = (5) ε µ µ µ t, µ t ρc µ Hee, C µ = is a model constant, k is the tubulent kinetic enegy, and ε is the tubulence dissipation ate. k and ε ae detemined by solving to additional tanspot equations []. Evaluation of the enegy equation equies knoledge of the adiative intensity field. his is obtained by solving the adiative tansfe equation (RE), ith its bounday condition fo opaque sufaces [3]: κ s s I ( s, ) = κaib ( ) ( κa+ κs) I ( s, ) + I (, ) (, ) d 4 4π Φ Ω π s s s (6) () (3) (4) Figue. Mesh vie of the computational domain. 58

3 ε I, s = I + I, s n s dω (7) ( ) ε ( ) ( ) b π ns Hee, κ a and κ s ae the adiative absoption and scatteing coicients, espectively, is avelength, I is spectal intensity, Ib is blackbody spectal intensity, Φ is the scatteing phase function, Ω is solid angle, n is a unit suface nomal vecto, is a unit position vecto, s is a unit diection vecto, and ε is the all emissivity. Once the intensity distibution is knon, the adiative souce tem in the enegy equation is computed as follos Solution paametes ae pesented in able... hemodynamic Head Loss ( I I ) q = κ 4π dω d (8) 0 a b 4π Fo channel flos in themal equilibium ith the efeence state at tempeatue 0 heat tansfe ects can be neglected, and viscous dissipation is the only ievesible pocess (electic and magnetic fields, chemical eactions, and othe souces of ievesibility ae not consideed in this study). he fictional head loss in Equation () then gives an exact measue of themodynamic ievesibility. Hoeve, fo flos involving heat tansfe, the fiction head loss is only a measue of mechanical enegy degadation. When momentum and heat tansfe (including adiation and convection) occu, a themodynamic head loss can be intoduced that include both mechanical and themal enegy degadations as folloing: ht = hf + hcc + h (9) Hee, h f, is the conventional head loss due to viscous dissipation, hcc is the head loss due to heat conduction and convection, and h is the head loss due to themal adiation. hese head loss tems can be computed as folloing hf = S 0 gen, f d m (0) hcc = S d 0 gen, cc m h aes = S gen, d + S gen, d m A m () A he adiation tem has to pats: the fist pat accounts fo losses due to absoption, emission and scatteing of adiation heat in the medium, and the second pat ae the losses that occu nea solid sufaces due to all adiation able. Model paametes. Paamete Symbol alue Unit () Density ρ 0.08 Specific heat c p hemal conductivity k iscosity µ kg m 3 J kg K W m K kg m s ubulent pandtl numbe P t 0.85 Absoption coicient Scatteing coicient κ a 0.5 m κ s 0.0 m Scatteing phase function Φ +Ω Ω Wall emissivity Inlet tempeatue Inlet velocity ε 0.9 in 000 K u in m s 59

4 absoption-emission pocesses..3. Computation of Local Entopy Geneation Rates he local entopy geneation ates in Equations (3)-(6) ae computed folloing pocedues discussed in pevious studies: ) Fo viscous dissipation in tubulent flo [4] = ( µ + µ ) Φ (3) S gen, f t hee, Φ is the viscous dissipation function. ) Fo heat conduction and convection in tubulent flo [4] and c pµ t S gen, cc = km + (4) Pt 3) Fo adiation absoption, emission, and scatteing in semitanspaent medium [5] I (,s) Ib (,s) ( ) 0 4π 0 4 (,s) π (,s) κ I s b ( ) κa ( ) 0 4π 4 4π 0 4 (,s π ) ( ) S = κ + κ dω d+ κ dωd aes, gen a s a,s + Φ s,s d Ω d Ω d + I (,s ) Ib ( ) d Ω d 4) Fo adiation absoption-emission pocesses at the solid alls [6] Hee, (5) I ( ) S,s, gen = L ( ) d d 0 4π,s n s Ω ( ) (6) L {( ) ( ) } 4 (, ) kc b ln ln s = Γ+ Γ+ Γ Γ (7) ε Γ= exp ( ) [ hc k ] hee, k b is Boltzmann s constant, h is Planck s constant, and c is the speed of light. he solves in FLUEN ae assigned a UDF (use defined function) fo solving Equation (3) and Equation (4). Equation (5) and Equation (6) ae solved ith a FORRAN 77 pogam. he solution methodology is discussed in [6]. 3. Results and Discussion Gid independence study as caied out to ensue that essential physics ee not stongly dependent on gid size. It as assumed that maximum eos occued in high-tempeatue-gadient and high-velocity gadient egions. he gid in the high-gadient egions as efined until the change in entopy geneation ates ee less than %. he final gid consisted of 00,000 cells ith nea-all cell clusteing. he mesh vie is shon in Figue. Figue shos coss-section slices and iso-sufaces of the computed velocity and tempeatue fields. he velocity field quickly develops into a small nea-all zone ith stong gadients, and a much lage channel cente zone ith an almost unifom velocity. his is because of the lo viscosity and no-slip bounday condition employed. he tempeatue field also quickly establishes a high tempeatue zone at the channel cente, and a high-tempeatue gadient nea-all zone at the entance egion of the channel. he fictional and heat tansfe head losses ae shon in Figue 3. he velocity and tempeatue fields detemine ho these losses ae distibuted in the channel. Losses due to heat conduction and convection ae dependent on the local tempeatue gadients (see Equation (4)). he conduction and convection head losses ae high b (8) 60

5 Figue. he computed velocity and tempeatue fields. Figue 3. he computed local head losses. in locations ith stong tempeatue gadients. Fictional head losses ae significant in egions ith high velocity gadients. he high tempeatue and high velocity gadient egions ae identified in Figue. hese citical egions ae located at the nea-all zone, mainly in the fist half of the channel. In accodance, the head losses due to heat conduction and convection, and viscous dissipation ae significant nea the solid alls in the beginning half of the channel. While heat conduction and convection, and viscous dissipation ae shot-ange phenomena, themal adiation is a long-ange phenomenon. Hence, the themal adiation head losses ae detemined by the entie tempeatue field. In geneal, the adiation entopy geneation is most impotant in high-tempeatue egions. hus the adiation head losses ae also elatively vey stong at the channel cente in the fist half of the channel, hee the fluid is hottest. 6

6 Oveall, the total head loss is fo 50% due to heat conduction and convection. Radiation absoption, emission and scatteing pocesses in the medium account fo 30% of the total head loss, hile all adiation pocesses account fo 0%. he total adiation ect is thus as impotant as that of heat conduction and convection in the high-tempeatue flo. otal loss in head due to viscous dissipation can be neglected. iscous ects usually become impotant only hen heat tansfe ects can be neglected. It is impotant to notice that most of the head losses occu in the nea-all zone. Hence, themodynamic optimization of the channel flo should involve educing the influence of this citical egion. his can be achieved by educing the all suface aea of the channel. he esults, in hich the channel aspect atio is vaied at constant channel volume, ae shon in Figue 4. It can be seen that the total losses in themodynamic head ae minimized in the channel ith aspect atio (squae channel). he squae channel has the smallest suface aea fo a given channel volume, and thus the smallest citical nea-all zone. In the squae channel, 60% of the total loss in themodynamic head is due to themal adiation. he adiation ects in the medium in this case ae about 5% lage than that at the alls. A method to educe the influence of adiation is discussed belo. In combined adiation and convection heat tansfe poblems, Boltzmann numbe appoximately epesents the atio of convection to adiation heat tansfe. Boltzmann numbe can be defined as: ρcu p in Bo = (9) 3 σ hee, σ is the Stefan-Boltzmann constant. Since themal adiation is the main ievesible pocess in the squae channel, inceasing Boltzmann numbe may educe the oveall losses in themodynamic head. he paametic study shoing the ects of Boltzmann numbe is pesented in Figue 5. Both, the pecentage adiation head losses and the total themodynamic head losses, educes ith inceasing Boltzmann numbe. hus, in combined adiation and convection heat tansfe, the themodynamic losses ae loe hen the faction of heat tanspoted by adiation is educed. Figue 4. Effects of channel aspect atio. Figue 5. Effects of Boltzmann numbe. 6

7 4. Conclusions he conventional fictional head loss paamete is extended in this ok to account fo both mechanical and themal enegy degadation in flos involving heat tansfe. he ne head loss paamete is applied in the analysis of a 3D channel flo ith combined adiation and convection heat tansfe. he esults sho that: ) hemodynamic head losses in high-tempeatue flo occu mainly at the channel nea-all zone. ) he size and influence of this zone is minimized in squae channels. 3) Radiation head losses can be vey stong in channels flos ith combined adiation and convection heat tansfe. 4) Lage Boltzmann numbe should be selected fo channel flos ith combined adiation and convection heat tansfe in ode to educe the total themodynamic head losses. Acknoledgements his ok is financially suppoted by singhua Univesity. Refeences [] Adeyinka, O.B. and Natee, G.F. (005) Entopy-Based Metic fo Component-Level Enegy Management: Application to Diffuse Pefomance. Intenational Jounal of Enegy Reseach, 9, [] (006) FLUEN 6.3 Use s Guide. [3] Modest, M.F. (003) Radiative Heat ansfe. nd Edition, Academic Pess, San Diego. [4] Yapici, H., Kayatas, N., Albayak, B. and Bastuk, G. (005) Numeical Calculation of Local Entopy Geneation in a Methane-Ai Bune. Enegy Convesion and Management, 46, [5] Caldas, M. and Semiao,. (005) Entopy Geneation though Radiative ansfe in Paticipating Media: Analysis and Numeical Computation. Jounal of Quantitative Spectoscopy and Radiative ansfe, 96, [6] Liu, L.H. and Chu, S.X. (007) eification of Numeical Simulation Method fo Entopy Geneation of Radiation Heat ansfe in Semitanspaent Medium. Jounal of Quantitative Spectoscopy and Radiative ansfe, 03,