Kun Yng School of Energy nd Power Engineering, Huzhong University of Science nd Technology, Wuhn 430074, PR Chin; Deprtment of Mechnicl Engineering, University of Cliforni, Riverside, Riverside, CA 95-045 Kmbiz Vfi Deprtment of Mechnicl Engineering, University of Cliforni, Riverside, Riverside, CA 95-045 e-mil: vfi@engr.ucr.edu Trnsient Aspects of Het Flux Bifurction in Porous Medi: An Exct Solution The trnsient therml response of pcked bed is investigted nlyticlly. A locl therml nonequilibrium model is used to represent the energy trnsport within the porous medium. The het flux bifurction phenomenon in porous medi is investigted for temporl conditiond two primry types of het flux bifurctions in porous medi re estblished. Exct solutions re derived for both the fluid nd solid temperture distributions for the constnt temperture boundry condition. The fluid, solid, nd totl Nusselt numbers during trnsient process re nlyzed. A het exchnge rtio is introduced to estimte the influence of interctions between the solid nd fluid phses through therml conduction t the wll within the het flux bifurction region. A region where the het trnsfer cn be described without considering the convection contribution in the fluid phse is found. The two-dimensionl therml behvior for the solid nd fluid phses is lso nlyzed. The temporl temperture differentil between the solid nd fluid is investigted to determine the domin over which the locl therml equilibrium model is vlid. In ddition, the chrcteristic time for reching stedy stte conditions is evluted. DOI: 0.5/.4003047 Keywords: porous medi, het flux bifurction, trnsient het trnsfer, locl therml nonequilibrium, nlyticl solution Introduction Porous medi re used to trnsport nd store energy in mny industril pplictions such s het pipe, solid mtrix het exchngers, electronic cooling, nd chemicl rectors. For solr collector with ir or wter s the working fluid, porous medium cn provide n effective mens for therml energy storge. During the period of chrging nd recovery, trnsient therml response spects of the process for the pcked bed re of mjor concerns. Locl therml equilibrium LTE nd locl therml nonequilibrium LTNE models re the two primry wys for representing het trnsfer in porous medium. Although LTE model is more convenient to use, more nd more studies hve suggested tht LTE model iot vlid for some problems such s storge of therml energy, or het trnsfer in porous medi with internl het genertion. In these cses, the LTNE model should be used for solid nd fluid phses in porous medi 3. Mny studies hve focused on the trnsient flow nd het trnsfer in porous medi. Schumnn 4 presented n erly nlyticl solution for trnsient temperture distribution of semi-infinite porous prism tht is initilly t uniform temperture nd the sides of the prism were dibtic. Using LTNE model, in which the diffusion terms in both the trnsverse nd xil directions were neglected, the fluid nd solid tempertures were found s function of the xil position nd time. Riz 5 investigted the trnsient response of pcked bed therml storge system, nd compred the nlyticl solutions obtined from simplified LTE nd LTNE models, in which Schumnn results were used nd the trnsient term in fluid phse ws ignored. It is obvious tht the trnsient term in fluid phse should be considered for mny types of pplictions. Spig nd Spig 6 nlyticlly investigted the dynmic response of porous medi nd pcked beds systems to n Corresponding uthor. Contributed by the Het Trnsfer Division of ASME for publiction in the JOUR- NAL OF HEAT TRANSFER. Mnuscript received September 7, 00; finl mnuscript received October 3, 00; published online Februry 4, 0. Assoc. Editor: Oronzio Mnc. rbitrry time vrying inlet temperture using LTNE model, in which the diffusion terms in both the trnsverse nd xil directions were neglected. The temperture responses for step, rmp, nd periodic vrying inlet tempertures were discussed. Using perturbtion technique, Kuznetsov 7 presented interesting nd importnt spects of the temperture difference between solid nd fluid phses in semi-infinite pcked bed bsed on LTNE model, in which the diffusion terms in trnsverse directions in both the fluid nd solid phses were neglected. Kuznetsov 7 estblished tht the temperture difference between the fluid nd solid phses forms therml wve loclized in spce. Using the sme technique, Kuznetsov 8 presented n nlyticl solution for pcked bed subject to constnt temperture condition t the wlls, in which the dimensionless solid phse temperture ws considered to differ from the fluid phse temperture by smll perturbtion. It ws shown tht the trnsient component of the temperture difference between the fluid nd solid phses describes wve propgting in the xil direction from the fluid inlet boundry. Hendl et l. 9 presented n nlyticl solution for the trnsient therml behvior of two dimensionl circulting porous bed bsed on LTE model. Their findings showed tht the temperture propgtes throughout the bed in wvelike form nd pproch stedy stte results for lrge vlues of time. Besley nd Clrk 0 developed numericl model to predict the trnsient response of pcked bed bsed on the LTNE model, in which the diffusion terms in both the trnsverse nd xil directions in the solid phse were neglected. Their numericl results compred fvorbly with the experimentl mesurement of temperture distribution in pcked bed of uniform spheres with ir s working fluid. Amiri nd Vfi 3 presented comprehensive investigtion of the trnsient response within pcked bed. The temporl impct of the non-drcin termd the therml dispersion effects on energy trnsport were investigted, nd the rnge of the vlidity for LTE condition ws estblished in detil. In the present work, the LTNE model is employed to represent the energy trnsport within porous medium. Two primry types of het flux bifurctions in porous medi re investigted for tem- Journl of Het Trnsfer Copyright 0 by ASME MAY 0, Vol. 33 / 0560-
where = xh i f c f u f =0 = s =0 = t = s c s /h i 4 = y ks,eff /h i, = H ks,eff /h i = h i H Bi where Bi = k s,eff Fig. Schemtic digrm for trnsport through chnnel filled with porous medium nd the corresponding coordinte system porl conditions. Het trnsfer performnces in terms of the fluid, solid, nd totl Nusselt number re presented. Qulittive nlyses of the effects of therml conduction t the wll on the totl het exchnge between the solid nd fluid phses within the het flux bifurction region re lso performed. Both the trnsient nd diffusion spects re considered in the solid nd fluid phses long with the convection nd the fluid-solid interction. The nlyticl solution for trnsient response of pcked bed subject to constnt temperture boundry condition is derived. The het flux bifurction phenomenon in porous medi is investigted for temporl conditions, nd the nlyticl two-dimensionl therml behvior nd the LTE model is exmined under trnsient conditions. Furthermore, the response time towrd stedy stte conditions is investigted. Modeling nd Formultion The schemtic digrm of the problem is shown in Fig.. Fluid flows through rectngulr chnnel filled with porous medium subject to constnt temperture boundry condition. The height of the chnnel is H nd the temperture t the wll is T w. The following ssumptions re invoked in the nlyzing this problem. The flow is incompressible nd represented by the Drcin flow model. Nturl convection nd rditive het trnsfer re negligible. 3 Axil het conduction in both the solid nd fluid phses re negligible. 4 Properties such s specific het, density, nd therml conductivity, s well s porosity re ssumed to be constnt.. Governing Eqution nd Boundry nd Initil Conditions. Bsed on these ssumptions, the following governing equtions re obtined from the work of Amiri nd Vfi 3 employing the locl therml nonequilibrium model. Fluid phse Solid phse Boundry conditions Initil conditions f + f = k f + s f s = s s f f = = s = =0 3 f = s =0 =0 =0 3b f =0 = in 3c = f c f s c s, k = k f,eff k s,eff, = T T w T 0 T w. Solution Methodology. The nondimensionl fluid nd solid temperture distributions, f,, nd s,, re represented s f,, = U f,v s,, = U s,v 7 Substituting Eqs. 6 nd 7 into Eqs. nd long with the boundry conditiond pplying the seprtion of vribled Lplce trnsformtion yield where f,, = U fn,cos s,, = U sn,cos = mw fn sin 5 6 8 9 n + 0.5, n = 0,,,... 0 + W fn W sn W fn = kw fn mw sn sin + W sn W fn = s n W sn where W sn nd W fn re the Lplce trnsformtions of U sn nd U fn, respectively, given by W sn =0 W fn =0 U sn e m d U fn e m d Solving Eqs. nd yields W fn = in m +m + s n + m + s n k ++m + s n sin exp m + s n k + +m + 3 4 +m + s n + sin + m + s n k ++m + s 5 n By utilizing inverse Lplce trnsform, U sn nd U fn re obtined s U sn = s + s + s3 + s4 + s5 + s6 sin 6 0560- / Vol. 33, MAY 0 Trnsctions of the ASME
where s4 = s5 = U fn = f + f + f3 + f4 + f5 sin s6 =+ s = in fp 0 p s = + s n + +fp p p p p p s3 = + s n + +fp p p p p + p p p p + p p p + +expp p p p + +expp p + +p + +exp + f = in gp 0 p f = + s n + +gp p p p p p f3 = + s n + +gp p p p p f4 = f5 = p p + s n + + p p p p expp + s n + + p p p p expp fp = exp ks n + + p I 0 t exp 0 + + pt Q t dt gp = exp k + + p 0 I t exp ++pt Q t dt + exp k + + p Q where Q is the unit step function,, 0 Q =0, 0 nd p 0 =0 p = s n + + ks n + + sn + ks n +4 7 p = s n + + ks n + sn + ks n +4 8 By substituting Eqs. 6 nd 7 in Eqs. 8 nd 9, the finl resulting solutions for Eqs.,, 3 3c, nd 4 re obtined s s = s + s + s3 + s4 + s5 + s6 sin cos 9 f = f + f + f3 + f4 + f5 sin cos 0 The verge temperture cn be clculted from s = 0 s d f = 0 f d Substituting Eqs. 9 nd 0 into Eqs. nd yields s = s + s + s3 + s4 + s5 + s6 f = f + f + f3 + f4 + f5 3 4.3 Stedy Stte Solution. The governing equtions for stedy stte conditions cn be obtined from Eqs. nd by deleting the trnsient term. This results in ss = in exp ks n + s n + sin s n + cos 5 fs = in exp ks n + s n + sin cos 6 nd the verge temperture under stedy stte conditions re obtined s ss = in exp k + s n +s n s n + fs = in exp ks n + s n + 7 8.4 Solution for the Cse Without the Convective Term in the Fluid Phse. The governing equtions for the cse without the convective contribution in the fluid phse cn be obtined from Eqs. nd. This results in snc = s4 + s5 + s6 sin cos 9 fnc = f4 + f5 sin cos 30 The verge tempertures for the cse without the convective contribution in the fluid phse re obtined s snc = s4 + s5 + s6 3 Journl of Het Trnsfer MAY 0, Vol. 33 / 0560-3
Fig. Dimensionless temperture distributions for fluid nd solid phses for k=0., =0.0, =5, =, nd in = 0.4: =0., b =.0, c =., d =3.0, e =5.0, nd f stedy stte fnc 3 Resultd Discussion = f4 + f5 3 The dimensionless temperture distributions for the fluid nd solid phses re shown in the Fig.. When is smll, the temperture distribution is minly dependent on the initil condition. However, when is lrge enough, the temperture distribution is primrily dependent on the inlet condition. Although the temperture difference between the fluid nd solid phses is reltively smll when stedy stte conditions re reched, it is reltively lrge compred with the fluid nd solid tempertures during the trnsient process. These results show tht the LTE model might be unsuitble to describe the trnsient het trnsfer process in porous medi. This figure lso discloses tht the therml boundry lyer grows s increses, which indictes substntil twodimensionl therml chrcteristic. It is importnt to note tht the direction of the temperture grdient for the fluid nd solid phses re different t the wll = in Figs. c nd d. This leds to het flux bifurction round these times. The concept of temperture grdient bifurction in the presence of internl het genertion in both the fluid nd solid phses hs been estblished in detil for the first time by 0560-4 / Vol. 33, MAY 0 Trnsctions of the ASME
Fig. 3 Bifurction region vritions s function of pertinent prmeters, k, nd in Yng nd Vfi. Utilizing the nlyticl solutions given in Eqs. 9 nd 0, the region over which het flux bifurction phenomenon occurs is estblished nd illustrted in Fig. 3. It is found tht this phenomenon occurs only over given xil region t given time frme. The bifurction region moves downstrem s increses, nd is dependent on the pertinent prmeters k,, nd in. When k,, nd in decrese, the bifurction region moves forwrd t fster rte. It should be noted tht bifurction phenomenon only occurs during the trnsient period. Bifurction phenomenon disppers when stedy stte conditions re reched. It should be noted tht the bifurction spects relted to phse chnge lyzed in Ref. hve not been investigted in this work. The dimensionless trnsverse verge temperture distributions for fluid nd solid phses for k=0., =0.0, =5, nd in = 0.4 re shown in Fig. 4. It is found tht the trnsverse verge tempertures pproch the cse with no convection in the fluid phse when the xil length is lrge enough. Bsed on Eqs. 9, 0, 9, nd 30, if stisfies the condition Journl of Het Trnsfer MAY 0, Vol. 33 / 0560-5
Fig. 4 Dimensionless trnsverse verge temperture distributions for fluid nd solid phses for k=0., =0.0, =5, nd in = 0.4 33 convection will hve n insignificnt impct on the temperture distributions for fluid nd solid phses. This is becuse tht the inlet condition effects do not propgte fr enough to influence tht time level. The difference between d ss presents the trnsient component of the verge solid temperture s, nd the difference between f nd fs presents the trnsient component of the verge fluid temperture f. These differences re shown in Fig. 5 for k=0., =0.0, =5, nd in = 0.4. It is found tht the pek positions for the trnsient components of the solid nd fluid phses moves downstrem with time, while the mgnitude of the pek decreses with time. The trnsverse verge temperture difference distributions between the solid nd fluid phses for k=0., =0.0, =0, nd in = 0.4 is shown in Fig. 6. It is found tht there is pek for the temperture difference t given, nd tht the pek moves downstrem s time progresses. For unstedy flow of gs through porous medium, Vfi nd Sozen 3 utilized the mximum difference between the solid nd fluid phse tempertures to estblish the vlidity of locl therml equilibrium ssumption. It ws found tht the locl therml equilibrium ssumption becomes more vible s both the Drcy nd prticle Reynoldumbers decrese. They hd shown tht decrese in the Drcy number trnsltes into decrese in the prticle dimeter, which results in n increse in the specific surfce re, thus incresing the fluid-to-solid het trnsfer interction by offering lrger surfce re. Furthermore, s the fluid velocity increses the time for the Fig. 6 Sptil nd temporl vritions of the verge temperture difference between the solid nd fluid phses for k=0., =0.0, =0, nd in = 0.4 solid-to-fluid het exchnge interction decreses, resulting in decrese in the efficiency of het exchnge between the solid nd fluid phses, thus incresing the devition from the locl therml equilibrium. Similrly here bsed on the definition of given in Eq. 5, n increse in the specific surfce re nd decrese in the fluid velocity cn be trnslted into n increse in. As such the temperture difference between the solid nd fluid phses becomes smller t lrger vlue of, s cn be seen in Fig. 6. The time s or f tht it tkes for either the solid or fluid phse to rech stedy stte condition is bsed on when the quntities defined by s, s ss ss = 0.0 34 f, f fs fs = 0.0 34b re chieved, respectively. The chrcteristic times for solid nd fluid phses to rech stedy stte re shown in Fig. 7. As cn be seen, the chrcteristic time for the solid is lwys lrger thn tht for the fluid phse. It cn lso be seen tht the chrcteristic times increse s k,,, or in increse. It is found tht the chrcteristic times remin lmost unchnged with k t ny given when k. This is due to the negligible influence of the fluid therml conduction. 4 Nusselt Number Results The Nusselt numbers for fluid nd solid phses cn be presented s Nu f = 4 f f = 35 Fig. 5 Vritions of the trnsient component of the verge temperture for fluid nd solid phses for k=0., =0.0, =5, nd in = 0.4 Nu s = 4 f k s = 36 The Nusselt numbers for fluid nd solid phses re presented long the xil coordinte in Fig. 8. It cn be seen tht the Nusselt numbers pproch infinity t specific xil loction t ny given time up to pproximtely when the stedy stte conditions re reched. It is lso found tht, fr enough downstrem of the entrnce, the Nusselt number becomes invrint with position. This phenomenon occurs when the dimensionl wll temperture vlue is within the rnge specified by the initil nd inlet temperture vlues. This is the reson why this phenomenon did not occur in the work of Amiri nd Vfi 3. In their work, the wll temperture ws lrger thn the entrnce nd the initil temperture. As 0560-6 / Vol. 33, MAY 0 Trnsctions of the ASME
Fig. 7 Chrcteristic time vritions of the solid nd fluid phses s function of pertinent prmeters k,,, nd in such in their work the dimensionless verge temperture did not pproch zero vlue. Furthermore, it should be noted this phenomenon is mnifesttion of nondimensionl temperture quntities. The fully developed temperture distributions for fluid nd solid phses under stedy stte conditions cn be derived from Eqs. 5 nd 6, fs_d = in exp ks 0 + s 0 + sins 0 coss 0 s 0 37 ss_d = in exp ks 0 + s 0 + sins 0 s 0 s 0 + coss 0 38 Furthermore, the verge fully developed temperture distributions for fluid nd solid under stedy stte conditions cn be obtined s fs_d = in exp ks 0 + s 0 + s 0 39 ss_d = in exp ks 0 + s 0 +s 0 s 40 0 + By utilizing Eqs. 37 40, the following equtions is obtined: ss_d ss_d = fs_d 4 fs_d T fs_d T ss_d T w As such the dimensionless fully developed temperture distri- = coss0 = Tfs_d T w = T ss_d T w T w butions, T fs_d T w /T fs_d T w nd T ss_d T w /T ss_d T w, become independent of the xil length when condition given by Eq. 39 is chieved. By utilizing Eqs. 37 39, the fully developed Nusselt numbers for fluid nd solid phses under stedy stte condition re obtined s Nu fs_d = 4 4 Nu ss_d = k +4 43 Defining totl Nusselt number, which is the sum of Nu f nd Nu s, we obtin 4 Nu ts_d =Nu fs_d +Nu ss_d = k +4 + 44 As cn be seen, the totl fully developed Nusselt number under stedy stte condition increses with, which is directly relted to the Biot number, nd decreses with the therml conductivity rtio, k. Journl of Het Trnsfer MAY 0, Vol. 33 / 0560-7
Fig. 8 = 0.4 Nusselt number distributions for fluid nd solid phses for k=0., =0.0, =5, nd in 5 Two Primry Types of Het Flux Bifurctions in Porous Medi In wht follows, we demonstrte the existence of two types of het flux bifurctions in porous medi. The first type is the sme s the one discussed by Yng nd Vfi. For the second type of het flux bifurction, we strt with representtion of the totl het flux t the wll s q w = k f,eff T f k y y=h s,eff T s 45 y y=h The dimensionless totl het flux t the wll is obtined from w = q w T 0 T w ks,eff h i = k f = s = 46 The dimensionless totl het flux t the wll for k=0., =0.0, =5, nd in = 0.4 is shown in Fig. 9. It is found tht the direction of totl het flux chnges long the chnnel. This leds to different type of het flux bifurction. This bifurction must Fig. 9 Dimensionless totl het flux t the wll for k=0., =0.0, =5, nd in = 0.4 0560-8 / Vol. 33, MAY 0 Trnsctions of the ASME
Fig. 0 An exmple of the requirement to chnge the imposed het flux direction t the wll, due to the bifurction effect, to obtin constnt temperture condition be tken into ccount for vrious pplictions, where there is need to mintin constnt temperture boundry condition. As shown in Fig. 0, the totl het flux bifurction region chnges with time, nd is dependent on the pertinent prmeters k,,, nd in. It should be noted tht this type of bifurction phenomenon only occurs during the trnsient process. The interfce line between the regions w 0 nd w 0 represents the loction for w =0, which moves downstrem with time. The speed, which the bifurction region moves downstrem, increses s either k,,, or in decreses. When q w =0, the het exchnge between the solid nd fluid phses through therml conduction t the wll is obtined from Q o = k f,eff T fy=h =k y s,eff T sy=h 47 y The integrted internl het exchnge between the solid nd fluid phses cn be clculted from H Q i = h i T s T f dy = h i HT 0 T w s 0 f The corresponding het exchnge rtio is defined s 48 = 49 Q i + Q o The het exchnge rtio vritions s function of prmeters, k, in, nd for q w =0 re shown in Fig.. It is found tht the het exchnge rtio is mostly dependent on nd k, wheres in nd hve little influence on the het exchnge rtio. The het Q o Journl of Het Trnsfer MAY 0, Vol. 33 / 0560-9
Fig. q w =0 Het exchnge rtio vritions s function of pertinent prmeters, k, in, nd for exchnge between solid nd fluid phses through the therml conduction t the wll is more prominent for smll nd lrge k. When = nd k=0, up to 68% of totl het exchnge between solid nd fluid phses within the bifurction region is through therml conduction t the wll. It should be noted tht the temporl vritions of the het exchnge rtio displys two distinct regimes. During the initil stge, the het exchnge rtio decreses shrply with time, while for the lter stge, the het exchnge rtio remins lmost unchnged. When q w 0, for the region where the first type of het flux bifurction occurs, the het exchnge between the solid nd fluid phses through the therml conduction t the wll cn be represented s Q o =min k f,eff T f k s,eff T s y y=h for y y=h, T fy=h T s 0 y y y=h 50 The corresponding het exchnge rtio for q w 0 is lso clculted using Eq. 49, nd shown in Fig.. The dshed line in Fig. represents the mxim loci of the het exchnge rtio. Compring Figs. 0, b, nd, it is found tht this dshed line is identicl to the corresponding curve for k=0. shown in Fig. b, which implies tht the het exchnge rtio for q w 0is lwys smller thn the corresponding one for q w =0. A exmple, the dimensionl chrcteristic time ws clculted for sndstone while the working fluid is ir. The following physicl dt were used: T in =300 K, T w =30 K, T o =335 K, H=0.05 m, d p =5 mm, nd =0.39; ir: f =.64 kg/m 3, c f =007 J/kg K, k f =0.063 W/m K, nd =.8460 5 kg/ m s; sndstone 3: s =00 kg/m 3, c s =70 J/kg K, k s =.83 W/ mk. The prticle Reynoldumber is defined s Re p = fud p The interstitil het trnsfer coefficient is expressed s 5 Fig. Het exchnge rtio for k=0., =0.0, =5, in = 0.4, nd q w Å0 0560-0 / Vol. 33, MAY 0 Trnsctions of the ASME
Fig. 3 Dimensionl chrcteristic time vritions of the solid nd fluid phses t different Re p for sndstone 0.6 5 /3 h i = k f +. Pr fud p d p The interfcil re per unit volume of the porous medium is clculted s = 6 53 d p The effective therml conductivity of the fluid nd solid phses of porous medi re represented by k f,eff = k f 54 k s,eff = k s 55 It cn be seen from Fig. 3 tht incresing Re p cn reduce the dimensionl chrcteristic time for both the fluid nd solid phses. However, the correltion between the dimensionl chrcteristic time nd Re p ionliner. 6 Conclusions Trnsient het trnsfer in pcked bed subject to constnt temperture boundry condition is investigted nlyticlly. A trnsient LTNE model, which incorportes diffusion in both the solid nd fluid phses, is employed to represent het trnsport. Exct solutions for trnsient solid nd fluid temperture distributions, s well s stedy solid nd fluid temperture distributions, re derived. Exct solutions of fluid, solid, nd totl Nusselt number for fully developed region under stedy stte condition re lso obtined. The results show substntil two-dimensionl therml behvior for the solid nd fluid phses, nd the LTE model is found to be unsuitble to describe the trnsient het trnsfer process in porous medi. The phenomenon of het flux bifurction for the solid nd fluid phses t the wll is found to occur over given xil region t given time frme. Het flux bifurction is lso found to occur long the chnnel. The bifurction region moves downstrem with time nd is dependent on the pertinent prmeters k,, nd in. The nondimensionl xil length scle,, introduced erlier cn be used to represent the indirect integrted influences of Drcy nd prticle Reynolds numbers on the temperture difference between the solid nd fluid phses. Therml conduction t the wll is found to ply n importnt role on the totl exchnge between the solid nd fluid phses within het flux bifurction region, especilly for smll nd lrge k. When /, it is found tht the het trnsfer cn be described using the LTNE model with no convection in the fluid phse energy eqution. A chrcteristic time is introduced to evlute the time tht it tkes for either the solid or fluid to rech stedy stte. This chrcteristic time is found to increse with n increse in k,,,or in. Nomenclture Bi Bi=h i H /k s,eff, Biot number c specific het J kg K d p prticle dimeter m h i interstitil het trnsfer coefficient W m K H hlf height of the chnnel m I 0 modified Bessel functions of the first kind of zero order I modified Bessel functions of the first kind of order k k=k f,eff /k s,eff, rtio of the fluid effective therml conductivity to tht of the solid, defined by Eq. 5 k f k f,eff therml conductivity of the fluid W m K effective therml conductivity of the fluid W m K k s therml conductivity of the solid W m K effective therml conductivity of the solid W m K m Lplce trnsform prmeter Nu Nusselt number q w Totl het flux t the wll W m k s,eff Q unit step function defined by Eq. 8 Q i integrted internl het exchnge between the solid nd fluid phses W m Q o het exchnge between the solid nd fluid phses through therml conduction t the wll W m Pr Prndtl number Re p prticle Reynoldumber =n+0.5/ t time s T temperture K T 0 initil temperture K u fluid velocity m s U function of nd, defined by Eqs. 6 nd 7 V function of, defined by Eqs. 6 nd 7 W Lplce trnsformtion of U x longitudinl coordinte m y trnsverse coordinte m Greek Symbols interfcil re per unit volume of the porous medium m porosity = f c f / s c s, prmeter defined by Eq. 5 nondimensionl trnsverse coordinte, defined by Eq. 5 =H/k s,eff /h i, nondimensionl hlf height of the chnnel, defined by Eq. 5 =xh i / f c f u, nondimensionl xil length scle, defined by Eq. 5 =T T w /T 0 T w, nondimensionl temperture, defined by Eq. 5 dynmic viscosity kg m s density kg m 3 het exchnge rtio, defined by Eq. 49 w dimensionless totl het flux t the wll, defined by Eq. 46 =h i t/ s c s, nondimensionl time, defined by Eq. 5 f nondimensionl chrcteristic time for fluid phse Journl of Het Trnsfer MAY 0, Vol. 33 / 0560-
s nondimensionl chrcteristic time for solid phse Subscripts d fully developed f fluid phse in inlet NC without convection term in the fluid phse s solid phse, stedy stte t totl w wll o initil Superscripts trnsverse verge References Lee, D. Y., nd Vfi, K., 999, Anlyticl Chrcteriztion nd Conceptul Assessment of Solid nd Fluid Temperture Differentils in Porous Medi, Int. J. Het Mss Trnsfer, 4, pp. 43 435. Alzmi, B., nd Vfi, K., 00, Constnt Wll Het Flux Boundry Conditions in Porous Medi under Locl Therml Non-Equilibrium Conditions, Int. J. Het Mss Trnsfer, 45, pp. 307 3087. 3 Amiri, A., nd Vfi, K., 998, Trnsient Anlysis of Incompressible Flow Through Pcked Bed, Int. J. Het Mss Trnsfer, 4, pp. 459 479. 4 Schumnn, T. E. W., 99, Het Trnsfer: Liquid Flowing Through Porous Prism, J. Frnklin Inst., 08, pp. 405 46. 5 Riz, M., 977, Anlyticl Solution for Single- nd Two-Phse Models of Pcked-Bed Therml Storge Systems, ASME J. Het Trnsfer, 99, pp. 489 49. 6 Spig, G., nd Spig, M., 98, A Rigorous Solution to A Het Trnsfer Two Phse Model in Porous Medi nd Pcked Beds, Int. J. Het Mss Trnsfer, 4, pp. 355 364. 7 Kuznetsov, A. V., 994, An Investigtion of Wve of Temperture Difference Between Solid nd Fluid Phses in Porous Pcked Bed, Int. J. Het Mss Trnsfer, 37, pp. 3030 3033. 8 Kuznetsov, A. V., 997, A Perturbtion Solution for Heting Rectngulr Sensible Het Storge Pcked Bed With Constnt Temperture t the Wlls, Int. J. Het Mss Trnsfer, 40, pp. 00 006. 9 Hendl, R., Quesnel, W., nd Sghir, Z., 008, Anlyticl Solution of the Therml Behvior of Circulting Porous Bet Exchnger, Fluid Dyn. Mter. Process., 4, pp. 37 43. 0 Besley, D. E., nd Clrk, J. A., 984, Trnsient Response of Pcked Bed for Therml Energy Storge, Int. J. Het Mss Trnsfer, 7, pp. 659 669. Yng, K., nd Vfi, K., 00, Anlysis of Temperture Grdient Bifurction in Porous Medi An Exct Solution, Int. J. Het Mss Trnsfer, 53, pp. 436 435. Vfi, K., nd Tien, H. C., 989, A Numericl Investigtion of Phse Chnge Effects in Porous Mterils, Int. J. Het Mss Trnsfer, 3, pp. 6 77. 3 Vfi, K., nd Sozen, M., 990, Anlysis of Energy nd Momentum Trnsport for Fluid Flow Through Porous Bed, ASME J. Het Trnsfer,, pp. 690 699. 0560- / Vol. 33, MAY 0 Trnsctions of the ASME