CHAPTER 3 NUMERICAL AND EXPERIMENTAL INVESTIGATIONS OF SOLIDIFICATION IN A CYLINDRICAL PCM STORAGE UNIT

Transcription

1 46 CHAPER 3 NUMERICAL AND EXPERIMENAL INVESIGAIONS OF SOLIDIFICAION IN A CYLINDRICAL PCM SORAGE UNI he design of a PCM based stoage system along with the flow of heat tansfe fluids (HF) involves solidification and melting henomena, fo which the analytical solutions ae not available excet in vey few cases with simle geomety and bounday conditions. Hence, a lot of numeical methods have been develoed by eseaches to study the efomance of PCM solidification/melting along with HF flow in the stoage unit. In the esent wok, an enthaly based tansient numeical model is develoed to investigate and edict the efomance of a PCM contained in the annula otion of a cylindical containe. he two dimensional temeatue vaiations along the and z diections ae consideed fo the analysis. Since the PCM selected fo the study is aaffin, which has a shot ange of hase change temeatues, the govening equation and bounday conditions ae discetized using the imlicite finite diffeence method and ae solved using MALAB softwae. An exeiment is also conducted fo the said configuation to validate the numeical model. 3.1 COMPUAIONAL FORMULAION A sectional view of the model consideed fo the numeical analysis is shown in Figue 3.1. he double walled ie shown, has an inne adius i

2 47 though which the ai is ciculated, and the annula otion has a thickness R = o - i in which the PCM is filled. he oute wall of the ie is comletely insulated. he heat tansfe between the ai and the PCM takes lace though the inne suface of the imbedded ie and also fom the to and bottom sufaces of the imbedded ie. he following assumtions ae made in the ensuing analysis: i) he themo-hysical oeties of the liquid and the solid hases of the PCM ae the same, excet fo the density. All the oeties emain constant with esect to temeatue. ii) iii) iv) he buoyancy foce fom the volume change due to the hase change is neglected. Since the mass of the PCM is small enough, conduction is the dominating mode of heat tansfe. he tube wall esistance is neglected and the ai is assumed to be in diect contact with the PCM. he ai flow though the inne adius has a heating / cooling effect u to the oute adius o whee the symmety bounday conditions ae being used. v) he H - elationshi duing hase change is assumed to be linea. vi) vii) he convection at the to and bottom sufaces is equal. he mean velocity of the heat tansfe fluid is assumed to be constant. viii) Initially the PCM is in a sue heated liquid state.

3 GOVERNING EQUAION In accodance with the above assumtions, the govening equations fo the PCM egion duing solidification ae H 1 k k, ( i < < o and 0 < z < L ) (3.1) t z z z = L S L m HF f HF PCM z i f (t) o Figue 3.1 Sectional views of the hysical model showing the PCM egion of analysis = i and = o esectively, coesond to the adius of the tube and the oute adius of the PCM egion. z = 0 and z = L esectively, coesond to the bottom and to sufaces of the PCM egion.

4 49 he H- elations fo the thee egions ae H = c ; ( m ) (Solid) H = c + 2 ( - m + ) ; ( m ) < < ( m + ) (Inteface) H = c + ; ( m ) (Liquid) (3.2) he flow of heat tansfe fluid though the ie is consideed fictionless (lug flow).he govening enegy equation fo the HF duing solidification is t ( u) z HF 2 z2 (3.2.a) Howeve in the esent wok, while solving the temeatue vaiation of the HF simultaneously with the PCM equation the HF is linked with the bounday though the known convective heat tansfe coefficient evaluated fom the Nusselts coelation and also the diffusion along the flow diection is neglected. 3.3 INIIAL AND BOUNDARY CONDIIONS Initially (at t = 0), the PCM is at the temeatue, = ini (3.3) he bounday condition at = o is 0, ( 0 < z < L ) (3.4) he bounday condition at = i is

5 50 i HF a a a L) z,o ( A h A k (0 < z < L ) (3.5) he bounday condition at z = 0 is o i HF b b b 0) z, ( A h A k ( i < < o ) (3.6) he bounday condition at z = L is o i HF t t t L) z, ( A h A k ( i < < o ) (3.7) Howeve, this being a tansient analysis, the instantaneous nodal enegy balances ae stated subsequently and ae emloyed in the numeical scheme. 3.4 NON-DIMENSIONALISAION he govening equation of the heat flow and the bounday conditions ae non-dimensionalised using the following non-dimensional aametes. s * 2 i s m m t c c H )) 0, ( ( L z c Ste i HF m a )) 0, ( ( z c Ste o i HF m b

6 51 Ste t c ( (, z L)) m HF, i o h L b Bib, k h a i Bia, k h L t Bit, k R, i z Z, L L K i Govening Equation he Stefan numbe defined in the above equation is a vaiable along the axial diection. Howeve, Ste b and Ste t ae constants with a high value at the bottom and a low value at the to of the model consideed, due to diffeent HF temeatue values. Similaly, the Biot numbe Bi a coesonds to the convection in the axial diection, wheeas Bi t and Bi b coesonds to the convection at the to and bottom of the model. he govening equation is nomalised using the above non-dimensional aametes. he nomalised fom of the equation (3.1) is 2 * 1 1 R (1< R< 1.4 and 0 < Z < 1) (3.8) 2 2 R R R K Z R = 1 and R = 1.4 esectively coesond to the adius of the inne tube and the oute adius of the PCM egion. Z = 0 and Z = 1 esectively coesond to the bottom and to of the PCM egion. thee egions ae he non-dimensional Enthaly-emeatue elations fo the ; 0 (Solid) 2c 2c ; 0 < < 1+ 2 c (Inteface)

7 52 1 ; 1+ 2 c (Liquid) (3.9) he discetised enegy equation fo the HF is obtained fom the enegy balance fo the contol volumes discetised along the flow diection. Consideing the convective heat flow fom PCM wall. Hence the discetised enegy equation fo the HF is given diectly in the next section Initial and bounday conditions (3.8) is witten as he non-dimensional fom of the initial condition fo Equation When 0 : ini (3.10) he tansfomed steady state bounday conditions - equations (3.4), (3.7) ae witten as At R = 1.4, 0 R ; ( R = 1.4, 0.0 < Z < 1.0) R At R = 1, Bi a ste a At Z = 0, Z Bi b ste b t ste * Z At Z = 1, Bi t ; ( R = 1.0, 0.0 < Z < 1.0) ; ( Z = 0.0, 1.0 < R < 1.4) ; ( Z = 1.0, 1.0 < R < 1.4) (3.11) It is seen fom the fomulation that all the enegy equations ae couled, and ae to be solved with the attendant bounday conditions simultaneously. he Enthaly-emeatue elations ae now genealized, as given in Velaj et al (1999):

8 53 ' 0.5 (1 a) whee ' 1 a 1 a,with 2c a (3.12) It can be seen that ' 0 coesonds to the solid hase, ' 1 to the liquid hase and duing hase change '. Since anges fom 0 1 a to (1+ a) in the inteface egion, 1 a 1 ' 1 eesents the solid faction. 1 a eesents the liquid faction and 3.5 DISCREISAION he discetization ocedue fo a cone node, inteio and suface nodes is esented in this discussion. he finite diffeence imlicit scheme is emloyed fo the time maching oblem and the esultant fomulations ae ogammed using MALAB. he two-dimensional gid adoted fo the analysis of the PCM inteio node is shown in Figue 3.2. Figue 3.2 Gid adoted fo the PCM inteio nodes

9 54 he non-dimensionalised govening equation (3.8) fo the aaffin egion is discetized using the finite diffeence imlicit scheme and the esultant equation is given as * ABCD C m1 i,j A m1 'm1 m1 'm1 B i,j1 i,j1 i, j1 i,j1 * m1 'm1 m1 ' m1 'm 1 m D i j 1, (ABCD i 1,j i1,j i1,j ) i,j i,j (3.13) whee A 1 R R R R 2R R B 1 1 C 1 1 Z 1 Z K D 1 K he dimensional and non-dimensional foms of the discetized equation alicable to the bounday sufaces and the cone nodes as shown in Figue A1.1 ae given in ables A1.1 and A1.2 esectively, in the Aendix. he enegy balance equation fo the HF flowing though the tube is witten fo the one dimensional contol volumes of same the width (z) divided along the flow diection adjacent to the PCM nodes, consideing the enegy flow to the PCM and neglecting the wall esistance of the containe. he discetised dimensional and non dimensional equations ae given in able A1.3. he aametes, non dimensional themal diffusivity and fozen volume faction, ae also evaluated using the following ocedue. he non dimensional themal diffusivity () and fozen volume faction ae also evaluated using the following ocedue. he non dimensional themal diffusivity () is the atio of the themal diffusivity of the PCM at any state to

10 55 the themal diffusivity of the PCM in the solid state. In ode to eesent conveniently in the equation, * is intoduced which is the eciocal of. P PS ( 1 ' ) Pl' P PS (1 ' ) Pl PS ' Pl 1 ' 1 PS 1 ' D 1 ; D = Pl PS (3.14) Fozen Volume o Solidified Volume is given by calculating the aamete (1+ ' ) and multilying by volume fo each node. Ratio of solidified volume to total volume gives the fozen volume faction o solid faction fo each node. Fozen volume e unit length of the tube is given by summation of all solidified volume fo all the nodes to total volume of the PCM e unit tube length. Fozen Volume Faction e unit tube length, VF = otal solidified volume otal volume 3.6 COMPUAIONAL PROCEDURE he govening equation and bounday conditions ae solved numeically using the Gauss-Siedel iteative method, and the comutational ocedue in this two-dimensional oblem ovides the equied accuacy. he solution methodology is exlained below. he initial non dimensional temeatue distibution () is used to calculate an initial non dimensional

11 56 enthaly () value of the PCM at each node on the comutational gid. he ocedue at each time ste is as follows: i) Using the enegy equation fo the HF (Aendix. A4.1), the temeatue of ai flowing though the tube is found by using the initial values of the PCM temeatue. m1 m 1 ii) An initial guess fo the new, Ste values of the PCM is made by simly assuming the coesonding m, Ste m values of the evious time ste and the fist iteation is efomed. iii) he iteative loo is now stated. Fo each iteation, the calculations fo the aaffin egion nodes ae efomed. While solving fo a aticula node, the latest comuted m1 m1 values of,ste ae used fo the suounding nodes in the equation. Using equations ' and 0.5 ' (1 a) 1 a 1 a ae udated fo the PCM nodes. 2c & a the values of, ' he values obtained ae comaed with the fist guess (o the evious iteation esults) to see if they satisfy the convegence citeia. If the citeion is not satisfied, iteation is efomed again. (ste 3 is eeated).

12 57 iv) he solidified volume faction is calculated at this time ste. v) A check is made to see if all the PCM nodes ae solidified. If not, the solution is advanced to the next time ste and the entie ocedue is eeated. he themo hysical oeties of the PCM consideed in the analysis is shown in able 2.5. he heat tansfe coefficients of the ai flowing though the tube and fo the ai at the bottom and to of the module ae calculated using the Dittus-Boelte equation (fo the heating of fluids). Fo the ai flowing though the tube at a velocity of 2 m/sec the heat tansfe coefficient value obtained fom the Dittus-Boelte equation is W/m 2 K and the heat tansfe coefficient value obtained at the to and the bottom of the module fo a velocity of 0.3 m/sec is 2.34 W/m 2 K. he samle calculation fo the evaluation of h a is given below. Velocity of ai u = 2 m/s Kinematic Viscosity, = 1.5 x 10-6 m 2 /s Diamete of the hole Reynold s numbe, D = m ud Re = Pandtl numbe, P = Nusselt numbe, Nu = (Re) 0.8 (P) 0.4 Nu = h 1 D/k = 31.96

13 58 Inut the oeties of ai Inut the oeties of the PCM Inut the geomety of the containing chambe Inut the time ste value and zeo definition fo convegence t = 0 ansfe the values of t[0] i,j, i,j t[0] t[0], i,j Ste i, fom the evious time ste to the esent time ste fist iteation Fo t > 0 x = 0 ansfe the values of t[0] i,j, i,j t[0], t[0] i,j,* t[0] i,j, Ste i, fom the evious iteation to the esent iteation x = x+1 * Calculate m 1 m 1 i, j, Ste i, j fom discetised non dimensional equation fo all the nodes. Calculate fo all nodes ' m i, j ' 1 a 1 a (1 a ) m 1 m 1 ' m 1 i, j * m 1 i, j 1 m 1 i, j i, j i, j m 1 ' m 1 Pl i, j 1 i, j PS 1 t = t +1 NO Check fo i,j [x] - i,j [x-1] e1 Calculate VF YES YES NO Is VF<100% OUPU: VF and emeatue of the PCM at secified time stes Sto Figue 3.3 Comutational Pocedue-flow chat

14 59 Figue 3.4 Gid Indeendence test Figue 3.4a PCM comutational domain and nodes selected fo analysis

15 60 heefoe, h a = W/m 2 K Similaly, fo a velocity of 0.3 m/s, h t = h b = 2.34 W/m 2 K he convegence citeia used fo a PCM node is x x1 e 1 (3.15) whee x indicates the cuent iteation, x-1 indicates the evious iteation and e 1 is the small toleance value. he above said comutational ocedue is efomed by witing a MALAB ogam. he comutational ocedue is also deicted in a flow chat shown in Figue 3.3. Seveal tial uns ae efomed on the model by vaying the gid size with the solidification time as the citeion fo the gid indeendence test and the esults ae shown in Figue 3.4. Consistent numeical esults ae obtained fo gid sizes geate than Hence, the comutations ae efomed assuming the node matix as shown in Figue 3.4a with t 1 second fo the geometical dimension consideed in the analysis. Howeve the esults fo vaious aametes ae esented fo the 4 nodal locations (2,2), (5,5),(9,9) and (9,18) shown in the figue. he value of e 1 is selected as E -03, so that convegence is established within a few iteations fo each time ste. 3.7 EXPERIMENAL INVESIGAION he exeimental set u consists of a hollow cylindical shell filled with the PCM (aaffin) in the annulus, and the inne tube is made as the assage fo the flow of heat tansfe fluid. he PCM shell has an oute diamete of 10.5 cm and a deth of 150 mm (mass of PCM 0.55 kg). It is insulated with one inch thick olystyene foam. he inne tube is made of coe and has a diamete of 7.5 cm and thickness of 1 mm. Cold ai at a equied temeatue geneated fom the climate simulato is allowed to ass

16 61 though an insulated duct which is connected to a shell and tube aangement. wo RD sensos (eo ± 0.3 ºC) ae laced in two locations in the PCM egion with one nea the oute wall and the othe nea the inne wall of the shell. Anothe RD is located nea the HF enty oint to the shell to measue the inlet ai temeatue. he velocity of the ai enteing the climatic simulato is measued using a vane tye anemomete (eo ± 0.01 m/s), which is used to evaluate the velocity of ai enteing the tube using the continuity equation. Figue 3.5 Schematic of the exeimental set u Duing the exeiment, initially, the temeatue of the PCM in the annula otion of the shell is maintained at a constant temeatue above the melting temeatue of the PCM (32º C). he cold ai at a temeatue of 22º C geneated fom the climatic simulato, is ciculated though the inne tube. he velocity of the ai is adjusted in such a way that the ai assing though the inne tube of the exeimental unit has a velocity of 2 m/s using

17 62 by the dame. he temeatue of the PCM at the 2 RD locations is continuously monitoed and ecoded in the data acquisition system. he exeiment is continued until the temeatue of the PCM at the RD location nea the oute wall eaches a temeatue below its solidification temeatue. Seveal exeiments ae conducted fo the eeatability of the eadings. 3.8 RESULS AND DISCUSSIONS Initially, the numeical model is validated by comaing the esults of the temeatue vaiation at two selected RD locations in the PCM egion obtained fom the exeiment, with the esult of the numeical model. he two RDs in the exeimental set-u coesond to the nodal locations (2, 2) 8.3 mm 0.83 mm and (9, 18) 37.5 mm 15 mm in the comutational domain consideed in the analysis. he temeatue vaiation with esect to time obtained both exeimentally and numeically fo the two selected locations is shown in Figues 3.6.a and 3.6.b. It is seen fom the temeatue vaiation that the numeical esults ae in good ageement with the exeimental esults. he solidification at a aticula location is comleted when the temeatue of the PCM is educed below 25º C. Node (2, 2) comletes its solidification at 6500 seconds wheeas node (9, 18) comletes its solidification at seconds. Since the esults ae in good ageement with the exeimental esults the numeical model is used fo futhe analysis and vaious contou gahs and othe esults obtained fom the numeical esults ae discussed in the following section. Figues 3.7.a and 3.7.b show the temeatue contous of the PCM duing the sensible and latent cooling of the PCM esectively, at 4 diffeent time intevals. In the comutational domain consideed in the analysis, the heat tansfe coefficient is evaluated fo the given geometic and flow conditions and the values at the left face, bottom face and to face ae 10.16, 2.34 and 2.34 W/m 2 K esectively, while the ight side face is consideed as adiabatic.

18 63 (a) node (2,2) (bottom left cone) (b) fo Node (9,18) (Insulated bounday) Figue 3.6 heoetical and exeimental temeatue vaiation with time duing solidification a) fo Node (2,2) b) fo Node (9,18)

19 64 he above said bounday conditions ae used in the analysis along with the heat tansfe fluid inlet temeatue of 22 C and the initial temeatue of 32 C fo the PCM domain. It is seen fom the contous that the PCM in the cones of the cylinde ae cooled faste than the PCM in the othe egion. he PCMs in these egions ae exosed to moe heat tansfe aea as heat can be tansfeed fom both vetical and hoizontal sufaces which makes it a two dimensional heat tansfe. he selected PCM in the esent study has a hase change temeatue in the ange of 25º C to 27º C (i.e., 26 ± 1º C ). he inteio egion of the PCM attains the hase change temeatue of 27º C only afte 2000 seconds. Usually in the hase change heat tansfe oblems, if the PCM neae to the heat tansfe suface is solidified comletely, sub cooling may occu, that will educe the temeatue otential diffeence between the heat tansfe fluid and the inteio PCM. In the esent oblem the temeatue diffeence between the suounding heat tansfe fluid and the hase change temeatue is only 4 to 5º C. Hence at this otential diffeence fo the selected PCM and the maximum dimension of the cylinde used in the esent study, does not ossess the state of sub cooling. Figue 3.7.a shows the temeatue contous of the PCM duing the hase change ocess at 4 time eiods with an inteval of 200 seconds. he solidification begins at the lowe and ue left cone of the cylinde whee the PCM is exosed to a moe convective heat tansfe suface aea, and the solidification font moves fom these two cones towads the cente of the ight side face (i.e. node (9,18). Futhe, the solidification font movement is faste fom the lowe cone (i.e. the lowe at of the cylinde), as this egion is suounded by lowe temeatue ai than the ue egion. he ai which entes the bottom egion is heated, as it flows though the tube by eceiving

20 65 the heat fom the PCM. Hence, the temeatue diffeence between the ai and the PCM in the ue egion is less, comaed to that in the lowe egion. Comlete solidification occus in the fathest node afte seconds fom the stat of the solidification ocess. Figue 3.7c shows the vaiation of the non dimensional temeatue contou at vaious time intevals duing the hase change ocess. he non dimensional temeatue eesents the atio of the sensible heat value above the solidus temeatue to the latent heat value of the PCM. he contous shown in the figue show the temeatue vaiation afte a non dimensional time () of his coesonds to a dimensional time of 3000 seconds. It is aleady seen fom the dimensional temeatue [( m ) (m )] C contou that the entie egion of analysis attains the hase change temeatue afte 3000 seconds. In the esent analysis, since the ange of the hase change temeatue consideed in the analysis is 2 C (i.e. between the liquidus temeatue (m + ) and solidus temeatue (m - )), the maximum faction of the sensible heat available in the PCM duing this hase change is Hence, in the esent contous at the time intevals vaying fom 3000 to seconds, value anges between to 0.0. he non dimensional temeatue of 0.0 indicates that the aticula egion has attained the solidus temeatue. Figue 3.7d shows the non dimensional enthaly contous at vaious time intevals duing the hase change ocess. At the stat of the exeiment,, which is the atio of the initial enthaly value of the PCM with esect to the solidus temeatue, to the latent heat value of the PCM, is he contou shown in the figue coesonds to the dimensional time afte 3000 seconds. he value of = which coesonds to the liquidus

21 66 temeatue, and the value of 0 coesonds to the solidus temeatue. Since at these time intevals, the entie egion is at the hase change temeatue, value vaies fom to 0.0. he negative value of in the contou eesents the sub cooled egion. It is seen fom the figue that a small otion of the bottom left cone is sub cooled when = (9000 seconds). Figue 3.7.e shows the vaiation of the non dimensional themal diffusivity, *, at vaious time intevals. * eesents the atio ( s ) of the themal diffusivity of the solid PCM to the themal diffusivity of the PCM at any instant. he aaffin consideed in the esent analysis has a solid themal diffusivity of m 2 /s and liquid themal diffusivity of m 2 /s. Hence initially, when the PCM is in the liquid egion, * value is When the PCM is changing its hase, * value is evaluated based on the faction of the PCM solidified in the hase change egion. When the PCM attains the solidus temeatue at a aticula egion, * eaches the value of 1, and it emains 1 even when the temeatue falls below the solidus temeatue. Hence, * contou with a value of 1 distinguishes the solidified and un solidified PCM egion. It is seen fom the contou that comlete solidification is seen at the left bottom at = (6000 seconds). When = (12000 seconds) the majo otion (80%) is solidified and only a small otion at the to ight is still in the hase change egion. In ode to visualize the continuous temeatue do with esect to time in the PCM egion, the temeatue vaiation with esect to time duing sensible and latent cooling is shown in Figue 3.8 (fo 4 selected nodes given in the figue inset). It is obseved fom the figue that the temeatue do duing the sensible cooling of the PCM is faste fo all the nodes. he node (2,2) which is vey close to the bottom and side sufaces attained its hase change temeatue aoximately afte 600 seconds, wheeas the nodes (5,5), (9,9) and (9,18) attained thei hase change ocess at aound 1400, 1500 and

22 seconds esectively. he mateial comleted its hase change ocess at aound 6000 seconds, 9600 seconds,12200 seconds and seconds esectively fo the nodal locations (2,2), (5,5), (9,9) and (9,18). he nodes (2,2) and (5,5) eceive the heat tansfe effect fom both the bottom sufaces wheeas the nodes (9,9) and (9,18) have no effect fom the bottom sufaces, and the sole effect is fom the tube wall suface. It is seen fom the figue that thee is no aeciable change in the temeatue vaiations fo the nodes (9,9) and (9,18), though (9,18) is located only at half the distance fom the tube wall suface, comaed to the nodal location of (9,18). his shows that the esistance in the PCM is much less comaed to the tube wall suface convective esistance. Hence, thee is a ossibility of futhe inceasing the heat tansfe by inceasing the suface convective heat tansfe coefficient fo the esent geometical configuation. Figue 3.9 shows the vaiation of the non-dimensional enthaly with esect to time fo the 4 selected nodes. It is aleady exlained that the value vaies fom to duing the solidification ocess. Initially, when the PCM loses its sensible heat above the solidification temeatue, the decease in is at a faste ate. he node (2,2) undegoes the hase change ocess afte 600 seconds ( = ). Duing this eiod the decease in the enthaly values of the inteio nodes is vey low as the temeatue diving otential fo heat tansfe deceases within the PCM egion. Futhe, it is seen fom the gah that all the nodes exeience two diffeent enthaly gadients with esect to time within the hase change egion. his is again due to the change in temeatue diving otential exeienced by the nodes in the PCM egion duing the hase change ocess. When the nodes nea the wall egion undego a hase change, all the inteio nodes exeience vey low temeatue diving otential. Howeve, afte the solidification of a aticula node, the adjacent node away fom the

23 68 heat tansfe suface eceives highe as the solidified node stats sub cooling oviding highe, fo heat tansfe which esults in a highe sloe in the non dimensional enthaly with esect to time. his ocess oceeds as the solidification font moves away fom the heat tansfe suface till the comlete solidification of the exteio nodes. Figue 3.10 shows the non-dimensional themal diffusivity * with esect to time. In the esent analysis the oety vaiation of the liquid and solid PCM ae consideed, without consideing the oety vaiation with esect to temeatue in the same hase. Hence * which is defined as the atio of the solid themal diffusivity to the themal diffusivity of the PCM at any instant, vaies between 1.0 and 1.52, deending on the solid faction of the PCM in the node consideed. It is seen fom the figue, that initially, the * value is 1.52 which is the atio of the solid themal diffusivity to the liquid themal diffusivity. Duing the hase change ocess at a aticula node, the vaiation of * exactly matches with the vaiation in the. hough the vaiation of * is indeendent of the temeatue within the hase change egion the solid faction/liquid faction in a aticula node is a function of the temeatue. Hence, the *, which is the function of the solid/liquid faction follows the same atten as that of the duing the hase change ocess. Howeve, the vaiation of the non dimensional themal diffusivity * could not be seen beyond the solidus and liquidus temeatue as seen in the non dimensional enthaly vaiation. Hence, this aamete is useful in detemining the faction solidified fo a aticula node duing the hase change ocess, and distinguishing the solidified and unsolidified node.

24 Figue 3.7 emeatue Contou lots (a) the temeatue of the PCM at vaious time intevals duing sensible cooling (b) emeatue of the PCM at vaious time intevals duing solidification (latent cooling) (c) Non dimensional temeatue (d) Non dimensional enthaly (e) Non dimensional diffusivity *. (Fo h a = W/m 2 K h b = h t = 2.34 W/m 2 K) 69

25 70 i = 22C, initial = 32C, V tube = 2 m/s, Re = h a = W/m 2 K h b = h t = 2.34 W/m 2 K Bi a = Figue 3.8 Vaiation of the dimensional and non dimensional emeatue of the PCM with esect to time Figue 3.9 Vaiation of the Non dimensional enthaly fo vaious times

26 71 Figue 3.10 Vaiation of the Non dimensional diffusivity * with time 1 Solid Faction i = 22C initial = 32C V tube = 2 m/s i = 22C, initial = 32C, V tube = 2 m/s, Re = h a = W/m 2 K h b = h t = 2.34 W/m 2 K VF ime in seconds Figue 3.11 Vaiation of the solid faction with time duing solidification

27 72 Figue 3.11 shows the vaiation of the solid faction with esect to time, which is the atio of the total solidified volume to the total volume of the liquid. he solid faction is zeo u to 1100 seconds, which imlies that the solidification has not stated and the PCM is in the sensible cooling ocess till that time eiod. he value of the solid faction eaches one at seconds, which means that the entie volume is solidified at this time. he simulation is teminated at seconds when the solid faction eaches a value of 1. he numeical esults obtained fom the esent simulation ae used in the eliminay design of a modula heat exchange containing seveal cylindical holes aanged in a unifom atten, which ae the assages fo the heat tansfe fluid suounded by the PCM on the shell side of the module. A detailed CFD analysis has been caied out on one such module and the esults of the analyses ae esented in the next chate.