Entropy Generation in a Partly Porous Heat Exchanger

Transcription

1 Entropy Gnration in a Partly Poros Hat Exanr N. Alloa and S. Cik Falté d Géni Méaniq t Géni ds proédés,usthb B. P. 3, El Alia, Bab Ezzoar 16111, Alria Abstrat T ntropy nration d to at transfr and to flid frition as bn analyzd nmrially for an annlar at xanr partly or flly filld wit a poros mdim. T oal of tis work is to find t optimal paramtrs in ordr to minimiz prssr drop, to nan at transfr and to rd t ntropy nration. T fft of t poros layr tiknss, t prmability and t fftiv trmal ondtivity ar invstiatd. T rslts sow tat for rtain onditions, t minimization of total ntropy nration dpnds on t tiknss of t poros sbstrat and its trmopysial proprtis. Kywords : ntropy nration, poros mdia, dobl pip at xanr 1. Introdtion Nowadays, sond law analysis of trmodynami systms as bom a prominnt topi in trmal ninrin. Bjan (1994) as writtn a novl trmodynami book ddiatd to t ntropy nration tro at and flid flow. Baytas (1999) as stdid t ntropy nration in tiltd satratd poros avity for laminar natral onvtion at transfr. T rsar of ford onvtion in a partially poros annlar dt as bn ondtd in rnt yars (Cik t al., 1995; Boadf t al., 1999). Yt, only fw nmbr of stdis onsidrd ntropy nration in ford onvtiv at transfr wit poros mdia. Rntly, a pakd dt wit niform atin (Dmirl, 1995), and an annlar pakd bd (Dmirl and Karaman, 000) av bn analyzd trmodynamially. T sop of tis papr is on minimization of ntropy nration in a dobl pip at xanr of innr and otr diamtrs d i and d rsptivly. T ot flid flows in t innr ylindr and t old on in t annlar ap. A poros sbstrat is attad to t xtrnal srfa of t innr ylindr and t otr ylindr is wll inslatd (Fir 1). T fft of t poros layr tiknss, t prmability and t fftiv trmal ondtivity ar invstiatd.

2 r old flid ot flid Poros sbstrat X adiabati wall adiabati wall Fir 1 : Smati of pysial domain. Matmatial Formlation T flow is assmd stady, laminar and inomprssibl. In addition, t proprtis ar assmd onstant and t poros mdim is onsidrd omonos, isotropi and is satratd wit a flid in loal trmal qilibrim wit t solid matrix. T flow is ovrnd by t Navir-Stoks qations in t flid rion and is modld by t Dary-Brinkman-Forimr qation in t poros rion. T dimnsionlss ovrnin qations ar : Continity qation : U 1 ( rv) + = 0 X r r (1) Momntm qation : innr ylindr : U U + V U = R dp U ρ r () dx R R v r annlar ap : poros rion : U U + V U = dp r U U CF U dx R R r r r R Da R Da v (3) flid rion : U U + V U = dp r U (4) dx R r Enry qation : innr ylindr : U θ + V θ = 1 1 r θ (5) R Pr R α r

3 annlar ap : poros rion R U θ + V θ = 1 r θ (6) R Pr r flid rion : U θ + V θ = 1 1 r θ (7) R Pr r Loal ntropy nration qation : S & = S& + β S& p T (8) wr : &S + P = U U, X r S & θ θ + T = X r (9) and β k i ( T in T in ) [ θ ( T T ) + T ] = (10) i i in in in T sbsript i stands for in t ot sid and for in t old sid. Wil in t poros rion, i indiats t fftiv trmal ondtivity in k i. T sbsript in dnots t inlt vals. T total ntropy nration is allatd by intration : ri 0 ot sid : S = S& rdr (11) r ri old sid : S = S& rdr (1) wr t dimnsionlss qantitis ar dfind as : P= ρ p in CF= ρ εf T T θ= T T in in K in in ρ R= ρ Rρ= ρ α Rα= α in D Da= K D R R v= = k k p Pr = k Rv= U= in k and ar t trmal ondtivity and t dynami visosity. T trms and T dnots t vloity and tmpratr of t flid. K is t prmability of t poros sbstrat, D =d -d i and in is t inlt vloity of t old flid. T sbsripts and

4 indiat rsptivly t old flid and t ot flid, wil t sbsript orrsponds to fftiv vals of t poros mdim. T assoiatd bondary onditions ar: no slip at walls prsribd inlt vloitis and tmpratrs trmally inslatd xtrnal ylindr ontinity of vloitis, tmpratrs and niqnss of t flxs at t flid-poros intrfa 3. Nmrial prodr T st of qations (1) to (7) wit t bondary onditions is solvd by a ontrol volm mtod as sstd by Patankar (1980). T obtaind systm of albrai qations is tn solvd wit t lin by lin mtod. A niform rid wit diffrnt stp sizs in a rion and a total nmbr of nods qal to (48x350) is tilizd. T onvrn ritria of t itrativ pross is mt wn t absolt val of rlativ rror on t flow rat at a nod is lss tan 10-3 and wn t absolt rror on t at flx transfrrd btwn t two flids ovr t ntir xanr is lss tan T vloity and t tmpratr filds fond wit t implmntd od ar in ood armnt wit tos fond by Boadf t al. (1999). 4. Rslts and disssion W av onsidrd in tis stdy t diamtr ratio is qal to, t flow rats of idntial flids ar t sam in bot dts; t fftiv visosity in t poros mdim is qal to t flid visosity (Brinkman assmption) ; t Prandtl nmbr is qal to 4, t inrtia offiint in t poros mdim CF is qal to 0.35, t Rynolds nmbr R=500 and t dimnsionlss flow lnt is qal to 50. Fir sows tat t dimnsionlss ntropy nration d to prssr drop inras wit t poros layr tiknss (). Tis inras is vry important wn t prmability of t poros matrix is vry wak (Dary nmbr Da=10-6 ). Tis variation is d to t prssr radint tat volvs in t sam way tat t tiknss of t poros layr and in an invrs trnd of t Dary nmbr. On t otr and, t dimnsionlss ntropy nration d to tmpratr radint (fir 3) inrass ntil a maximm val orrspondin to a ritial tiknss of t poros layr tat dpnds on t prmability. Byond tis ritial tiknss, t ntropy nration starts drasin and boms vn lowr tan t flid as. Unxptdly, t at xanr narly flly poros (abov 95%) prsnts t bst prformans. It is xplaind by t onvtiv flx tat inrass from t ritial tiknss and tn boms vry important wn t annlar spa is narly flly filld wit t poros mdim.

5 Sps 1.E+04 1.E+03 Da=1E-1 Da=1E- Da=1E-3 Da=1E-4 Da=1E-6 Sts 8.0E E-01 1.E+0 1.E E-01 Da=1E-1 Da=1E- Da=1E-3 Da=1E-4 Da=1E-6 5.0E Fir : ntropy nration d to Fir 3 : ntropy nration d to prssr drop as fntion of at t at transfr as fntion of at t xit of t at xanr xit of t at xanr old sid, R =1 old sid, R =1 T dimnsionlss ntropy nration d to flid flow frition and to at transfr dpnds on t β paramtr tat is fntion of t inlt onditions and t flid proprtis. W av onsidrd as an xampl tr ass of diffrnt flids: watr, toln and bnzn. W av fond tat t β paramtr varis btwn 10 4 and For β=10 5, fir 4 sows t fft of t poros layr tiknss on t total ntropy nration for diffrnt Dary nmbrs. Tis ntropy nration volvs lik t ntropy nration d to at transfr. Tis mans tat most of t irrvrsibility is d to at transfr for tis val of β. 8.0E E+04 Da=1E-6 Da=1E-4 Da=1E-3 Da=1E- Da=1E-1 7.0E+04 Ss 6.5E E E E Fir 4 : total ntropy nration as fntion of at t xit of t at xanr, old sid, R=1, β=10 5

6 1.E+05 1.E+05 1.E+04 Ss Ss 1.E+03 1.E+0 R=1 R=5 R=10 R=100 1.E+04 R=1 R=5 R=10 R=100 1.E E Fir 5 : S in t xit of t at Fir 6 : S in t xit of t at xanr, old flid, β=10 5 Da=10 - xanr, old flid, β=10 5 Da=10-6 W obsrv tro t firs 5 and 6 tat t total ntropy nration drass wit t inras of t trmal ondtivity of t poros matrix. Tis rdtion boms sbstantial from a rtain tiknss tat dpnds on t Dary nmbr. Tis fft is d t ondtiv flx tat boms important ompard to t onvtiv flx. Wn t poros mdim is mor ondtin tan t flid (R >5), t rdtion of t total ntropy nration is not vry important if t poros tiknss is lss tan 60%. 5. Conlsion For wll osn onditions, rslts of t analysis of ntropy nration d to flid flow frition and to at transfr in paralll at xanr wit a poros mdim in t annlar ap sowd tat t insrtion of poros sbstrat improvs t at transfr as rportd by Boadf t al. (1999) and rds t total ntropy nration if t tiknss is abov a ritial val tat dpnds on t prmability of t poros matrix and wn t trmal ondtivity ratio is ratr tan 1 Rfrns Bjan A, 1994, Entropy Gnration Tro Hat and Flid Flow, Jon Wily and Sons, In, Nw York. Baytas A.C., 000, Entropy nration for natral onvtion in an inlind poros avity, Int. J. of Hat and Mass Transfr, vol. 43, N 1, pp Boadf K., S. Cik, A. Bomdin, G. Lariat, 1999, Efft of poros sbstrat addition on at xanr ffiiny, ImE Transations 565/01/99, pp Cik S., A. Bomdin, K. Boadf, G. Lariat, 1995, Analytial soltion of non- Darian ford onvtion in an annlar dt partially filld wit a poros mdim, Int. J. of Hat and Mass Transfr, vol. 38, N 9, pp Dmirl Y, 1995, Trmodynami optimisation of onvtiv at transfr in a pakd dt, Enry, vol. 0, N 10, pp Dmirl Y., R. Karaman, 000, Trmodynami analysis of onvtiv at transfr in an annlar pakd bd, Int. J. of Hat and Flid Flow, vol. 1, pp Patankar S.V, 1980, Nmrial at transfr and flid flow, M GrawHill.