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Numerical and experimental study on the solidification of PCM around a vertical axially finned isothermal cylinder
Applied Thermal Engineering 21 (2001) 53±77
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Numerical and experimental study on the solidi®cation of PCM around a vertical axially ®nned isothermal cylinder K.A.R. Ismail*, C.L.F. Alves, M.S. Modesto Depto de Engenharia TeÂrmica e de Fluidos, FEM-UNICAMP CP 6122, CEP 13083-970, Campinas (SP), Brazil Received 18 October 1999; accepted 24 December 1999
Abstract This paper presents the results of a numerical and experimental investigation realized on ®nned tubes with the objective of using them in thermal storage systems. The model is based upon the pure conduction mechanism of heat transfer, the enthalpy formulation approach and the control volume method. The ®nite dierence approximation and the alternating direction scheme are used to discretize the basic equations and the associated boundary and initial conditions. The model was validated by comparison with available results and additional experimental measurements realized by the authors. The number of ®ns, ®n length, ®n thickness, the degree of super heat and the aspect ratio of the annular spacing are found to in¯uence the time for complete solidi®cation, solidi®ed mass fraction and the total stored energy. The results con®rm the importance of the ®ns in delaying the undesirable eects of natural convection during the phase change processes. Also, this study indicates the strong in¯uence of the annular space size, the radial length of the ®n and the number of ®ns on the solidi®ed mass fraction and the time for complete phase change. Based upon experimental observations and the tendencies of the numerical results, a metallic tube ®tted with four±®ve ®ns of constant thickness equal to the tube wall thickness and of radial length around twice the tube diameter should be a compromise solution between eciency, increase in the heat ¯ow rate and the loss of available storage capacity. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Solidi®cation; Fusion; PCM; Axially ®nned tubes; Energy storage
* Corresponding author. Tel.: +55-019-788-3376; fax: +55-019-239-3722. E-mail address: [email protected] (K.A.R. Ismail). 1359-4311/01/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 0 ) 0 0 0 0 2 - 8
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Nomenclature Biot number = hri =ks ) speci®c heat (J kgÿ1Kÿ1) heat capacity per unit volume = rc (J mÿ3 K) heat capacity per unit volume including the phase change (J mÿ3 K) dimensionless heat capacity per unit volume including the phase change parameters whose value varies between 0 and 1 for the implicit/explicit, alternating direction implicit and Cranck±Nicolson formulations H enthalpy per unit volume (J mÿ3) K thermal conductivity (W mÿ1 Kÿ1) thermal conductivity including the phase change (W mÿ1 Kÿ1) k T ~ dimensionless thermal conductivity including the phase change k T L latent heat (J kgÿ1) dimensionless ®n length = rf ÿ rw = re ÿ rw lf m parameter whose value varies between 0 and 1 number of ®ns nf Q heat ¯ux (W) R radial coordinate (m) R dimensionless radial coordinate = r=ri rf ®n extreme radius (m) external radius of the tube (m) rw internal radius of the tube (m) ri radius of external cylinder or radius of the symmetry circle (m) re radial position of the solid±liquid interface (m) rs phase change temperature (K) Tm tube wall temperature (K) Tw time for complete solidi®cation tc d Dirac delta function D means variation DT half the phase change temperature range (K) f tangential coordinate (8) half ®n angle (8) ff half angle between two successive ®ns (8) fm Z u unity function l latent heat per unit volume (J mÿ3) y dimensionless temperature = Tm ÿ T = Tm ÿ Tw r density (kg mÿ3) t dimensionless time = ks t=rw cs x dimensionless phase change temperature range = DT = Tm ÿ Tw c 0 dimensionless superheating parameter = Ti 0 ÿ Tm = Ti 0 ÿ Tw Bi C C C T ~ C T f, g
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Subscripts f ®n l liquid s solid w wall Abbreviations PCM Phase Change Material ADI Alternating Direction Implicit AR Aspect ratio = re ÿ rw =rw 1. Introduction Latent heat thermal storage systems are usually compact in comparison with sensible heat systems and have a constant temperature of operation. For these reasons and many others, the latent heat storage systems are becoming more popular and many units are already in application. One important feature of the latent heat storage is that the heat exchanger is integrated within the storage unit and hence it is necessary to optimize the heat transfer area and reduce the duration of the charging and discharging cycles. One of the most common methods used to improve thermal performance of these systems is the use of ®nned tubes as the heat transfer elements. These ®ns can be axial or radial and are usually attached to the tubes. Sparrow et al. [1] investigated experimentally a tube with four ®ns. In their work they observed the presence of natural convection in the liquid phase which led to delay or even complete interruption of the solidi®cation process. They also noticed that the use of ®ns can delay the commencement of natural convection and help delay the domination of natural convection on the heat conduction during the solidi®cation process. By how much can the number of ®ns and their geometry aect the process? This aspect of the problem was not investigated. In their experiments, the temperature of the tube was kept constant at a value higher than the phase change temperature, a condition which always permits the presence of natural convection. Later, Sparrow et al. [2] studied the process of transition which occurs during a solidi®cation process controlled by natural convection to a process dominated by conduction and observed that the transition is a function of the temperature dierence between the tube wall temperature and the initial temperature of the superheated liquid PCM phase. They noticed that natural convection dominates the process during the ®rst instances and consequently delays the solidi®cation process. The use of ®nned tubes can accelerate the transition of natural convection to pure conduction. Formulation of phase change problems by the enthalpy method can be found in Ockendon and Hodkins [3], Elliot and Ockendon [4], Meyer [5], Baxter [6], Voller and Cross [7], Bonacina et al. [8], and Ismail [29,30]. The work due to Sparrow and Hsu [9] may be considered as being one of the early studies to analyze the two dimensional solidi®cation around a cylinder. They used in their model the immobilization technique, neglected the thermal resistance of the tube wall, and considered the PCM as semi-in®nite in the radial direction and initially at the phase-change temperature. In
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their work the possible eects of the type discretization used for treating the pseudo-convective terms were not shown explicitly and in the numerical treatment they used the ®nite volume approach and the central-dierences. Later Shamsundar [10] solved the same problem as speci®ed by Sparrow and Hsu [9]. Sinha and Gupta [11] reported the results of an experimental study realized on the solidi®cation of water around a horizontal copper cylinder cooled by a ¯ow of refrigerant at isothermal conditions. The material of the external tube was glass to allow optical measurements of the interface position around the tube for dierent wall temperatures. They developed a model based upon the front immobilization technique and the ®nite volume approach and found good agreement between the model predictions and their experimental results. Later, Lacroix and Voller [12] realized a comparative study between the enthalpy method together with the ®nite volumes approach and the method of Body Fitted Coordinates when applied to a rectangular cavity ®lled with PCM and concluded that the enthalpy method needs a ®ne grid when treating materials with unique phase change temperature, such as pure materials while the Body Fitted Coordinates is restrictive when the geometry is complex. In another study of a comparative nature realized by Viswanath and Jaluria [13], they applied the enthalpy method and the front immobilization technique using, together with both methods, the ®nite volume approach to solve the fusion of PCM in a rectangular cavity. Recently, Blackwell and Hogan [14] used the Landau transform method associated with the ®nite volumes method to study the thermal ablation problem in cylindrical geometry. Sastri et al. [15] used the interface immobilization method together with the ®nite volumes approach to study numerically the problem of solidi®cation in an annular space with the phase change material at an initial temperature higher than the solidi®cation temperature. The heat conduction equations in the liquid and solid phases as well as the interface conditions were solved using a fully implicit scheme. They obtained a correlation of the time for complete solidi®cation in terms of the Biot number and the geometry of the annular space. Cao and Faghri [16,17] solved numerically the fusion problem of a storage unit of annular geometry for space applications. In their work, they considered the thermal resistance of the tube wall and solved simultaneously the equations of motion and energy for the working ¯uid, the tube wall and the PCM. The model used is based upon the enthalpy method while the SIMPLE scheme was used to solve the equation of motion. In their second study they modeled the same problem but used the k±e model to represent the turbulence ¯ow. Charach et al. [18] solved the solidi®cation of PCM, initially at the solidi®cation temperature in a semi-in®nite domain around a cooled cylinder. By using series expansion the results of Shamsundar [10] were extended to situations of quasi-steady regime. Bellecci and Conti [19] realized numerical simulation of a storage system for solar energy application having annular geometry and with forced convection inside the tube. Lacroix [20] solved numerically the problem of fusion of paran in annular vertical geometry using water as the working ¯uid. He used the concept of eective thermal conductivity for the PCM liquid phase, determined experimentally with objective of including the eects of free convection in the pure conduction model. He used the enthalpy method for the PCM while the working ¯uid was included through an energy balance in which he used constant temperature correlation for the Nusselt number and obtained reasonable agreement between the model predictions and the experimental results. Later, Zhang and Faghri [21] used the integral energy method to solve the fusion problem in
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an annular space with water as the working ¯uid in a laminar ¯ow regime while the phase change material used is paran initially at the fusion temperature. The integral method was used in the PCM while the temperature, the interface position and Nusselt number were calculated at each time increment by an iterative procedure. In the studies realized by Choi and Kim [22] they analyzed experimentally the performance of a latent heat storage with ®nned and ®nless tubes. Zhang and Faghri [23] studied the annular space ®lled with PCM and with the inner cylinder ®tted with internal ®ns. Gonc° alves [24] studied the solidi®cation in the annular space ®lled with PCM and ®tted with internal ®ns on the outer cylinder and external ®ns on the inner cylinder. Ismail [29,30] presented a comprehensive review of literature on the subject as well as the results of many experimental and numerical studies on phase change heat transfer into and around simple and complex geometries realized during the last 25 years at the laboratory of Thermal Energy Storage and Heat Pipes of The Mechanical Engineering Faculty, The State University of Campinas, Brazil. In the present study, a model for solidi®cation around a vertical axially ®nned tube submerged in a PCM has been developed. The model is based upon the enthalpy method and the control volumes approach. In this study the number of ®ns, ®ns' height, thickness, the degree of superheat and the aspect ratio of the annular space were investigated. In order to validate the numerical predictions and the proposed model, speci®c experiments were realized and the results were compared with the numerical predictions and good agreement was found. The results of this study are of great interest in the design, analysis of latent heat storage systems and cold storage applications. The ®ns are found to help in enhancing the energy transfer process to and from the PCM mass. The results presented are for the case of constant wall temperature, a condition which can be attained by a heat pipe or a metallic tube carrying a heat transfer ¯uid at a high Biot number. This condition, that is constant wall temperature, is not a restrictive one. It is possible to include a temperature gradient along the tube length and incorporate it in this basic model. More details can be found in Ismail [29].
2. Formulation of problem The objective of the present study is to investigate the solidi®cation process around an isothermal ®nned tube and how it is aected by the ®ns, geometry and number as also by the operational conditions. In this case, we consider a ®nned tube submersed into PCM enclosed by an external cylindrical surface thermally insulated or by a symmetry surface in the case of multi-tubes placed inside a PCM tank. Fig. 1 shows the ®nned tube and the details of the geometry of the symmetrical region, where the three sides are thermally insulated while the tube wall is maintained at constant temperature. Considering that the heat transfer process is controlled only by pure conduction, one can write the conduction equation for the solid and liquid phases, respectively, as: @Ts 1@ @Ts 1 @ ks @Ts rks 1 rs cs r @r r @f r @f @t @r
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@Tl 1@ @Tl 1 @ kl @Tl rkl rl cl r @r r @f r @f @t @r
The energy balance equation at the solid±liquid interface can be written as:
Fig. 1. Details of the ®nned tube and the symmetry region.
2
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@Ts @Tl 1 @ rs ÿ kl ks 1 2 rs @f @r @r
2!
rs l
@rs @t
59
3
The speci®c heat and the latent heat per unit volume for the solid±liquid and the latent heat per unit volume, respectively, as: Cs rs cs ;
Cl rl cl ; l rs L
while the enthalpy per unit volume can be written for the phase change material as: H T C T dT lZ T ÿ Tm T
Z u
1 2
when ur0 when u < 0
4
5
6
taking into account the sudden increase of the enthalpy by l when phase change occurs, the speci®c heat of the phase change material becomes: C T
dH T C T ld T ÿ Tm dT
C T
Cs Cl
when T < Tm when T > Tm
7
8
where d T ÿ Tm is the Dirac function. Using Eqs. (1), (2) and (8) become: ! T @T k @T 1 @ @T 1 @ C T rkÅ T r @f @t r @r @r r @f
9
where:
8 > TRTm ÿ DT ks T ; > > < k T ; TrTm DT l k T > kl ÿ ks > > T ÿ Tm ÿ DT; Tm ÿ DT < T < Tm DT : kl T 2DT
assuming a linear variation of k 8 > Cs T ; > > < C T ; l C T > l Cs > > Cl ; : 2DT 2
10
with T in the phase change temperature range and TRTm ÿ DT TrTm DT Tm ÿ DT < T < Tm DT
in the phase change range is the result of the integral: where the value of C T
11
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Tm DT Tm ÿDT
C~ T dT l
Tm Tm ÿDT
Cs T dT
Tm DT Tm
Cl T dT
assuming that Cs T and Cl T do not depend upon T in the interval Tm ÿ DT < T < Tm DT: Bonacina et al. [8] and Shamsundar and Srimivesan [25] demonstrated that Eq. (9) converges to Eq. (3) when the temperature approaches the phase change temperature Tm. Hence, one can conclude that Eq. (9) alone is the controlling equation for this phase change problem. In order to solve the two dimensional heat conduction in the ®n, one must use Eq. (9) with the corresponding ®n material properties such as density, speci®c heat and thermal conductivity. In this case one can write: C T ra ca Ca k T ka
12
To facilitate the numerical treatment and enable better investigation of the parameters of the problem, the following dimensionless variables are adopted: y
Tm ÿ T Tm ÿ Tw
k~ y
k~ y and
R
r ks t DT ; t ; x 2 ri Tm ÿ Tw Cs ri
8 1; > > > < kl =ks ;
> kl ÿ ks y > > 1ÿ ; :1 2ks x
yRx yr ÿ x ÿx < y < x
ka for the fin ks
8 > 1; yRx > > < C =C ; yr ÿx 1 s C~ y > l Cl Cs > > ; ÿx < y < x : 2Cs DT 2Cs Ca C~ y Cs
9 > > > > > > > > = > > > > > > > > ;
9 > > > > > > > > = > > > > > > > > ;
When the new dimensionless variables are substituted in Eq. (9), it becomes: ! ~ @y 1 @ @y 1 @ k y @y Rk y C~ y @t R @R @R R @f R @f
13
14
15
16
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and the corresponding boundary conditions in terms of the new variables are: y 1, for R 1
17
@y rm 0, for R @f ri
18
@y 0, for f 0 @f
19
@y 0, for f fm @f
20
The discretization of Eq. (16) is realized by using the control volume technique. In this case the domain of the problem is divided into a convenient number of control volumes and Eq. (16) is integrated over R, f and t to obtain, after some mathematical manipulations, the required algebraic equations. The same is done with respect to the boundary conditions of the problem. Details of the mathematical manipulations are omitted here for the sake of brevity .With the algebraic equations in hand, it is possible to use for their representation explicit formulation, implicit formulation, Crank Nicholson formulation, ADI formulation or conjugate ADI with Crank Nicholson. Details of these methods can be found in Patankar [26] and Smith [27]. In the present study, we adopted the ADI formulation because it is unconditionally stable. More details about the numerical solution can be found in Gonc° alves [24]. The numerical code was optimized by numerical experimentation. The criterion of convergence was tested for precision of 10%, and in the range of 1±0.1%. The values of 1 and 0.2% produced coincident curves of solidi®ed mass fraction against time. For 0.1% precision, six times the computational time used for the case of 10% precision was required. The maximum number of iterations of 10, 20 and 30 were tested. The three values produced coincident curves of solidi®ed mass fraction against time and we adopted for the present study the value of 30, using 20% more computational time than the case of 20 iterations. The minimum number of iterations was also investigated using the values of 3, 4 and 6. For the same reasons, we adopted the case of six iterations as minimum. The values of maximum and minimum number of iterations are based upon Nogotov [28] where he suggests that the initial time increment to be multiplied by 1.3 if the convergence is reached before the minimum number of iterations is achieved and then used for the next time increment. Also the time increment must be divided by 2.0 if the precision is not achieved after the maximum number of iterations is attained. The number of grid points tested in the vertical direction was 21, 33 and 41. The results obtained seem to be coincident and we adopted the value of 33 which consumed 100% more computational time than the case of 21 grid points. The horizontal grid includes the ®n region, after ®n region and the ®nal part of the domain and consequently the number of points is dierent for each region. We tested the sets (10, 10, 30), (15, 15, 45) and (20, 20, 60); and the corresponding curves of solidi®ed mass fraction against time were coincident for the three sets. We adopted the set of points (15, 15, 45) which consumed 50% more than the ®rst set and 100% less than the last set.
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3. Experimental setup In order to validate the model and the numerical predictions an experimental setup was designed, constructed and instrumented to obtain necessary data suitable for comparisons with the proposed model. The basic components of the experimental rig shown in Fig. 2 are: 1. 2. 3. 4.
Container for phase change material; Thermally insulated box for keeping the PCM container; Tube heat exchanger and the test tube around which the phase change occurs; Thermally controlled water bath to heat-up the PCM and ensure the initial temperature for the tests;
Fig. 2. Layout and details of the experimental rig.
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Fig. 3. Solid±liquid interface for dierent time intervals, case of four ®ns.
5. Water pump for water circulation; 6. A rig for measuring the PCM thickness distribution. The phase change material used in the experiments is paran, whose national code is 130/135 Type 1, whose phase change temperature range is 55±578C, has 5% of oil and has the following thermophysical properties:
Fig. 4. Solid±liquid interface for dierent time intervals, case of three ®ns.
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Fig. 5. Solid±liquid interface for dierent time intervals, case of ®ve ®ns.
Tm DT rl rs cl kl L
55.88C 1.398C 771.2 (kg/m3) 771.2 (kg/m3) cs 2176 (J/kg) ks 0.089 (J/s m K) 232.4 (MJ/kg)
To realize an experimental test it is necessary to ensure that the water bath is at the desired uniform temperature, place the PCM container in the water bath, and circulate water in the
Fig. 6. Eect of the initial super heat of the liquid phase on the solidi®cation front.
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Fig. 7. Eect of the initial liquid phase super heat on the solidi®cation front.
heat exchanger to keep the ®nned tube at uniform temperature below the PCM melting temperature. When all temperatures are stable and at nearly the desired values, 1. Remove the PCM container from the water bath and place it inside the adiabatic chamber. 2. When the temperature of the PCM is at the desired value remove the top cover of the PCM container, lower down the heat exchanger with the ®nned tube and place back the top cover. 3. Start the time measurement.
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Fig. 8. Comparison between numerical predictions and experimental results.
Fig. 9. Eect of the ®n length on the solidi®cation time.
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Fig. 10. Variation of the solidi®cation time with the ®n length.
When the experiment's time is achieved, stop the clock, remove the tube from the PCM tank and cool it immediately. Place the tube in the thickness measuring apparatus and measure the angular distribution of the PCM thickness.
Fig. 11. Interface position for dierent ®n length.
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The important temperatures in this study are the initial liquid phase temperature, Ti 0, the phase change temperature, Tm ,and the refrigerant tube wall temperature, Tw : We de®ne a dimensionless parameter denoted here as the initial superheat factor and is given m ÿTw and the values used in the experiment are: 0.0; 0.1 and 0.2. byC 0 Ti T0i ÿT 0 w rrwe ÿ 1, and the The aspect ratio of the annular space AR is de®ned as:AR re ÿr rw w experimental value used here is 2.481. The ®n length is de®ned as: lf rref ÿr ÿrw , and the experimental values used are lf 0:087; 0.196 and 0.348. The angular half-width of the ®n is be given by: ff sin ÿ1 2rtf f and the values tested are 0.550, 0.450 and 0.360. The number of ®ns nf used is 2, 3, 4 and 5. While the dimensionless time for conversion is t 4:653117844 10 ÿ4 t s: 4. Results and discussion An extensive experimental testing program was realized to determine the eects of the geometry of the ®n and the operational conditions on the thermal performance of the tube,
Fig. 12. Eect of the ®n thickness on the solidi®cation time.
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solidi®ed PCM and the solidi®cation rate. Some of these results are presented below to show these eects and to be used later on for validating the numerical model. Figs. 3±5 show the solid±liquid interfaces for dierent time intervals and tubes of dierent number of ®ns. Fig. 3 shows the case of the tube with four ®ns, ®n length of 8 mm and initial temperature of PCM of 658C. It is easy to verify the evolution of the solidi®ed mass patterns and the thickness distribution around the tube and ®ns. Similar results, but for dierent number of ®ns, are presented in Figs. 4 and 5. Fig. 6 shows the eect of the initial superheat of the PCM liquid phase on the solidi®cation front for the case of tube with four ®ns, ®n length of 8 mm and for a solidi®cation time of 30 min, when the initial liquid PCM temperature is of 55.8, 59.8 and 64.88C, respectively. As can be seen when the initial liquid PCM temperature is equal or near the solidi®cation temperature, the process of solidi®cation is obviously quicker. Similar results for the case of tubes with three and ®ve ®ns are shown in Fig. 7. Fig. 8 shows comparisons between the numerical predictions and the experimental results. The comparative results shown in Fig. 8(a) and (b) show good agreement between the numerical predictions and the experiments. The dierences indicated in case (c) are attributed to the large experimental gradient of the interface near the ®ns. In the numerical solution, we
Fig. 13. Eect of the ®n thickness on the solidi®cation time.
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Fig. 14. Interface position for dierent ®n thicknesses.
Fig. 15. Eect of the ®n number on the solidi®cation time.
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Fig. 16. Eect of the number of ®ns on the solidi®cation time.
adopted linear gradients and in future studies this can be corrected by imposing better representation of these gradients in the numerical model. The dierences in cases Fig. 8(d) and (e) can be attributed to the appearance of dentrits over the solid±liquid interface for long solidi®cation times which led to increasing the surface area and hence increase the heat transfer rate. This situation does not exist in the numerical model, as the liquid±solid interface is always considered a smooth surface. Even with these apparent dierences the variation in the solidi®ed mass fraction is less than 4% in all the cases examined. Fig. 9 shows the eect of the ®n length on the solidi®cation time. One can observe, that an increase in the ®n length leads to reducing the solidi®cation time. The time for the complete solidi®cation can be observed in Fig. 10 indicating a strong reduction in its value with the increase in ®n length. The instantaneous interface positions for two values of ®n length are shown in Fig. 11. One can observe the progressive increase in the solidi®ed mass fraction with time and also the increase in the solidi®ed mass when the ®n length is increased. The in¯uence of the thickness of the ®n is shown in Fig. 12. As can be seen the increase in ®n thickness leads to the increase of the solidi®ed mass fraction and reduction of the solidi®cation time. One can also notice that the in¯uence of the ®n thickness is small.
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Fig. 17. Interface positions for dierent number of ®ns.
Fig. 18. In¯uence of the aspect ratio of annular space on the solidi®cation time.
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Fig. 19. In¯uence of the aspect ratio of annular space on the solidi®cation time.
Fig. 20. Interface positions for dierent aspect ratios of annular space.
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Fig. 21. Eect of super heating on the solidi®cation time.
The time for complete solidi®cation is also shown in Fig. 13. The positions of the interface fronts at dierent time intervals during the solidi®cation process are shown in Fig. 14. The eect of varying the number of ®ns on the solidi®ed mass fraction is shown in Fig. 15. As can be veri®ed, the increase of the number of ®ns leads to increasing the solidi®ed mass fraction and reducing the time for complete solidi®cation, as is shown in Fig. 16. The con®guration of the solidi®cation interfaces for dierent time intervals is shown in Fig. 17 for dierent number of ®ns. Again, one can observe that increasing the number of ®ns leads to increasing the solidi®ed mass and reduces the time for the complete solidi®cation. The volume of the annular space between the inner tube carrying the refrigerant ¯uid and the external symmetry surface has a strong in¯uence on the solidi®ed mass fraction and the time for complete solidi®cation. One can observe from Fig. 18 that if this ratio, denominated here as the aspect ratio of the annular space, AR, is increased from 2 to 5, the dimensionless time for the complete solidi®cation is increased from about 0.5 to 4. Fig. 19 shows clearly this strong eect. The solid±liquid interface for dierent values of the annular space aspect ratio, AR, are shown in Fig. 20. The in¯uence of the temperature dierence between the phase change temperature and tube wall temperature can be seen in Fig. 21, where the temperature dierence is varied from 10 to 258C. As can be seen, the temperature dierence has a strong eect on the solidi®ed mass
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Fig. 22. Eect of super heating on the solidi®cation time.
fraction and also on the time for complete solidi®cation as shown in Fig. 22. The liquid±solid interfaces for two values of temperature dierence and at dierent time intervals are shown in Fig. 23. 5. Conclusions The proposed thermal model and the numerical method of solution adopted produced numerical predictions of the solid-liquid interface that agree generally well with the experimental results, except when the ®n is very long or after long intervals of solidi®cation periods. The ®rst situation leads to a big inclination of the experimentally determined frontiers near the ®n. This eect is not adequately represented in the numerical model. The second situation can be explained by the increase of the heat transfer area due to the dentrits increase. In all the experiments dentrits were visible indicating that the heat transfer is controlled by conduction. The parameters analyzed numerically include the ®n length, ®n thickness, number of ®ns, the aspect ratio of the annular space and the temperature dierence between the phase change temperature and the wall temperature of the tube. The results indicate that these parameters
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Fig. 23. The interface position for dierent degrees of super heat.
have signi®cant eect on the time for complete solidi®cation. From the results presented one can observe that the ®n thickness has a relatively small in¯uence on the solidi®cation time and that the ®n length as well as the number of ®ns aects strongly the time for complete solidi®cation and the solidi®cation rate. The aspect ratio of the annular space has a strong eect on the time for solidi®cation and also the time for complete solidi®cation. The temperature dierence has an opposite eect on the solidi®cation of PCM and the time for complete solidi®cation seems to decrease with the increase of this temperature dierence. Additional numerical and experimental work is being realized in The Laboratory of Thermal Storage and Heat Pipes on plastic ®nned tubes ®tted with axial or radial ®ns of constant thickness. The results will be used in designing domestic cold storage for air conditioning applications. Acknowledgements The authors wish to thank Fapesp and CNPq for the ®nancial support to the research project of the ®rst author and Capes for the scholarships for the co-authors. References [1] E.M. Sparrow, E.D. Larson, J.W. Ramsey, Int. J. Heat Transfer 24 (1981) 273. [2] E.M. Sparrow, J.W. Ramsey, J.S. Harris, J. Heat Transfer 103 (1981) 7.
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[3] J.R. Ockendon, W.R. Hodkins, Moving Boundary Problems in Heat Flow and Diusion, Clarendon Press, Oxford, 1975. [4] C.M. Elliot, J.R. Ockendon, Weak and Variational Methods for Moving Boundary Problems, Pitman Advanced Publishing Program, Boston, 1982. [5] G.H. Meyer, SIAM J. Numerical Analyses 10 (3) (1973) 522. [6] D.C. Baxter, J. Heat Transfer 84 (1962) 545. [7] V. Voller, M. Cross, Int. J. Heat Mass Transfer 24 (1981) 545. [8] C. Bonacina, G. Comini, A. Fasano, M. Primicerio, Int. J. Heat Mass Transfer 16 (1973) 1825. [9] E.M. Sparrow, C.F. Hsu, Int. J. of Heat and Mass Transfer 24 (8) (1981) 1345. [10] N. Shamsundar, Int. J. of Heat and Mass Transfer 25 (10) (1982) 1614. [11] T.K. Sinha, J.P. Gupta, Int. J. of Heat and Mass Transfer 25 (11) (1982) 1771. [12] M. Lacroix, V.R. Voller, Numerical Heat Transfer 17 (B) (1990) 25. [13] R. Viswanath, Y. Jaluria, Numerical Heat Transfer 24 (B) (1993) 77. [14] B.F. Blackwell, R.E. Hogan, Journal of Thermophysics and Heat Transfer, 8(2) (1993) 282. [15] M.K. Sastri, S.S. Sablani, S.P. Venkateshan, Journal of Thermophysics and Heat Transfer 4 (3) (1989) 398. [16] Y. Cao, A. Faghri, Int. J. Heat Mass Transfer 34 (1) (1990) 93. [17] Y. Cao, A. Faghri, J. Heat Transfer 114 (1992) 1019. [18] C. Charach, Y. Keizman, M. Sokolov, Int. Communications in Heat and Mass Transfer 18 (1991) 639. [19] C. Bellecci, M. Conti, Int. J. Heat Mass Transfer 36 (1992) 2157. [20] M. Lacroix, Numerical Heat Transfer 15 (B) (1993) 191. [21] Y. Zhang, A. Faghri, Int. J. Heat and Mass Transfer 39 (4) (1996) 717. [22] J.C. Choi, S.D. Kim, Energy: the International Journal 17 (1992) 1153. [23] Y. Zhang, A. Faghri, Int. J. Heat and Mass Transfer 39 (15) (1996) 3165. [24] M.M. Gonc° alves, Armazenadores de Calor Latente de Geometria Anular com Aletas Alternadas, Doctorate thesis, The State University of Campinas, Campinas, S.P., Brasil (1996). [25] N. Shamsundar, R. Srimivesan, Journal Solar Engineering 192 (1980) 263. [26] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. [27] G.D. Smith, Numerical Solution of Partial Dierential Equations, Oxford University Press, London, 1969. [28] E.F. Nogotov, Applications of Numerical Heat Transfer, McGraw-Hill, New York, 1978. [29] K.A.R. Ismail, Modeling of Thermal Processes; Fusion and Solidi®cation, ISBN 85-900609-3-4, The State University of Campinas, Brasil, 1999 (in Portuguese). [30] K.A.R. Ismail, Cold Storage Systems; Fundamentals and Modeling, ISBN 85-900609-2-6, The State University of Campinas, Brasil, 1998 (in Portuguese).
www.elsevier.com/locate/apthermeng
Numerical and experimental study on the solidi®cation of PCM around a vertical axially ®nned isothermal cylinder K.A.R. Ismail*, C.L.F. Alves, M.S. Modesto Depto de Engenharia TeÂrmica e de Fluidos, FEM-UNICAMP CP 6122, CEP 13083-970, Campinas (SP), Brazil Received 18 October 1999; accepted 24 December 1999
Abstract This paper presents the results of a numerical and experimental investigation realized on ®nned tubes with the objective of using them in thermal storage systems. The model is based upon the pure conduction mechanism of heat transfer, the enthalpy formulation approach and the control volume method. The ®nite dierence approximation and the alternating direction scheme are used to discretize the basic equations and the associated boundary and initial conditions. The model was validated by comparison with available results and additional experimental measurements realized by the authors. The number of ®ns, ®n length, ®n thickness, the degree of super heat and the aspect ratio of the annular spacing are found to in¯uence the time for complete solidi®cation, solidi®ed mass fraction and the total stored energy. The results con®rm the importance of the ®ns in delaying the undesirable eects of natural convection during the phase change processes. Also, this study indicates the strong in¯uence of the annular space size, the radial length of the ®n and the number of ®ns on the solidi®ed mass fraction and the time for complete phase change. Based upon experimental observations and the tendencies of the numerical results, a metallic tube ®tted with four±®ve ®ns of constant thickness equal to the tube wall thickness and of radial length around twice the tube diameter should be a compromise solution between eciency, increase in the heat ¯ow rate and the loss of available storage capacity. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Solidi®cation; Fusion; PCM; Axially ®nned tubes; Energy storage
* Corresponding author. Tel.: +55-019-788-3376; fax: +55-019-239-3722. E-mail address: [email protected] (K.A.R. Ismail). 1359-4311/01/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 0 ) 0 0 0 0 2 - 8
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Nomenclature Biot number = hri =ks ) speci®c heat (J kgÿ1Kÿ1) heat capacity per unit volume = rc (J mÿ3 K) heat capacity per unit volume including the phase change (J mÿ3 K) dimensionless heat capacity per unit volume including the phase change parameters whose value varies between 0 and 1 for the implicit/explicit, alternating direction implicit and Cranck±Nicolson formulations H enthalpy per unit volume (J mÿ3) K thermal conductivity (W mÿ1 Kÿ1) thermal conductivity including the phase change (W mÿ1 Kÿ1) k T ~ dimensionless thermal conductivity including the phase change k T L latent heat (J kgÿ1) dimensionless ®n length = rf ÿ rw = re ÿ rw lf m parameter whose value varies between 0 and 1 number of ®ns nf Q heat ¯ux (W) R radial coordinate (m) R dimensionless radial coordinate = r=ri rf ®n extreme radius (m) external radius of the tube (m) rw internal radius of the tube (m) ri radius of external cylinder or radius of the symmetry circle (m) re radial position of the solid±liquid interface (m) rs phase change temperature (K) Tm tube wall temperature (K) Tw time for complete solidi®cation tc d Dirac delta function D means variation DT half the phase change temperature range (K) f tangential coordinate (8) half ®n angle (8) ff half angle between two successive ®ns (8) fm Z u unity function l latent heat per unit volume (J mÿ3) y dimensionless temperature = Tm ÿ T = Tm ÿ Tw r density (kg mÿ3) t dimensionless time = ks t=rw cs x dimensionless phase change temperature range = DT = Tm ÿ Tw c 0 dimensionless superheating parameter = Ti 0 ÿ Tm = Ti 0 ÿ Tw Bi C C C T ~ C T f, g
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Subscripts f ®n l liquid s solid w wall Abbreviations PCM Phase Change Material ADI Alternating Direction Implicit AR Aspect ratio = re ÿ rw =rw 1. Introduction Latent heat thermal storage systems are usually compact in comparison with sensible heat systems and have a constant temperature of operation. For these reasons and many others, the latent heat storage systems are becoming more popular and many units are already in application. One important feature of the latent heat storage is that the heat exchanger is integrated within the storage unit and hence it is necessary to optimize the heat transfer area and reduce the duration of the charging and discharging cycles. One of the most common methods used to improve thermal performance of these systems is the use of ®nned tubes as the heat transfer elements. These ®ns can be axial or radial and are usually attached to the tubes. Sparrow et al. [1] investigated experimentally a tube with four ®ns. In their work they observed the presence of natural convection in the liquid phase which led to delay or even complete interruption of the solidi®cation process. They also noticed that the use of ®ns can delay the commencement of natural convection and help delay the domination of natural convection on the heat conduction during the solidi®cation process. By how much can the number of ®ns and their geometry aect the process? This aspect of the problem was not investigated. In their experiments, the temperature of the tube was kept constant at a value higher than the phase change temperature, a condition which always permits the presence of natural convection. Later, Sparrow et al. [2] studied the process of transition which occurs during a solidi®cation process controlled by natural convection to a process dominated by conduction and observed that the transition is a function of the temperature dierence between the tube wall temperature and the initial temperature of the superheated liquid PCM phase. They noticed that natural convection dominates the process during the ®rst instances and consequently delays the solidi®cation process. The use of ®nned tubes can accelerate the transition of natural convection to pure conduction. Formulation of phase change problems by the enthalpy method can be found in Ockendon and Hodkins [3], Elliot and Ockendon [4], Meyer [5], Baxter [6], Voller and Cross [7], Bonacina et al. [8], and Ismail [29,30]. The work due to Sparrow and Hsu [9] may be considered as being one of the early studies to analyze the two dimensional solidi®cation around a cylinder. They used in their model the immobilization technique, neglected the thermal resistance of the tube wall, and considered the PCM as semi-in®nite in the radial direction and initially at the phase-change temperature. In
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their work the possible eects of the type discretization used for treating the pseudo-convective terms were not shown explicitly and in the numerical treatment they used the ®nite volume approach and the central-dierences. Later Shamsundar [10] solved the same problem as speci®ed by Sparrow and Hsu [9]. Sinha and Gupta [11] reported the results of an experimental study realized on the solidi®cation of water around a horizontal copper cylinder cooled by a ¯ow of refrigerant at isothermal conditions. The material of the external tube was glass to allow optical measurements of the interface position around the tube for dierent wall temperatures. They developed a model based upon the front immobilization technique and the ®nite volume approach and found good agreement between the model predictions and their experimental results. Later, Lacroix and Voller [12] realized a comparative study between the enthalpy method together with the ®nite volumes approach and the method of Body Fitted Coordinates when applied to a rectangular cavity ®lled with PCM and concluded that the enthalpy method needs a ®ne grid when treating materials with unique phase change temperature, such as pure materials while the Body Fitted Coordinates is restrictive when the geometry is complex. In another study of a comparative nature realized by Viswanath and Jaluria [13], they applied the enthalpy method and the front immobilization technique using, together with both methods, the ®nite volume approach to solve the fusion of PCM in a rectangular cavity. Recently, Blackwell and Hogan [14] used the Landau transform method associated with the ®nite volumes method to study the thermal ablation problem in cylindrical geometry. Sastri et al. [15] used the interface immobilization method together with the ®nite volumes approach to study numerically the problem of solidi®cation in an annular space with the phase change material at an initial temperature higher than the solidi®cation temperature. The heat conduction equations in the liquid and solid phases as well as the interface conditions were solved using a fully implicit scheme. They obtained a correlation of the time for complete solidi®cation in terms of the Biot number and the geometry of the annular space. Cao and Faghri [16,17] solved numerically the fusion problem of a storage unit of annular geometry for space applications. In their work, they considered the thermal resistance of the tube wall and solved simultaneously the equations of motion and energy for the working ¯uid, the tube wall and the PCM. The model used is based upon the enthalpy method while the SIMPLE scheme was used to solve the equation of motion. In their second study they modeled the same problem but used the k±e model to represent the turbulence ¯ow. Charach et al. [18] solved the solidi®cation of PCM, initially at the solidi®cation temperature in a semi-in®nite domain around a cooled cylinder. By using series expansion the results of Shamsundar [10] were extended to situations of quasi-steady regime. Bellecci and Conti [19] realized numerical simulation of a storage system for solar energy application having annular geometry and with forced convection inside the tube. Lacroix [20] solved numerically the problem of fusion of paran in annular vertical geometry using water as the working ¯uid. He used the concept of eective thermal conductivity for the PCM liquid phase, determined experimentally with objective of including the eects of free convection in the pure conduction model. He used the enthalpy method for the PCM while the working ¯uid was included through an energy balance in which he used constant temperature correlation for the Nusselt number and obtained reasonable agreement between the model predictions and the experimental results. Later, Zhang and Faghri [21] used the integral energy method to solve the fusion problem in
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an annular space with water as the working ¯uid in a laminar ¯ow regime while the phase change material used is paran initially at the fusion temperature. The integral method was used in the PCM while the temperature, the interface position and Nusselt number were calculated at each time increment by an iterative procedure. In the studies realized by Choi and Kim [22] they analyzed experimentally the performance of a latent heat storage with ®nned and ®nless tubes. Zhang and Faghri [23] studied the annular space ®lled with PCM and with the inner cylinder ®tted with internal ®ns. Gonc° alves [24] studied the solidi®cation in the annular space ®lled with PCM and ®tted with internal ®ns on the outer cylinder and external ®ns on the inner cylinder. Ismail [29,30] presented a comprehensive review of literature on the subject as well as the results of many experimental and numerical studies on phase change heat transfer into and around simple and complex geometries realized during the last 25 years at the laboratory of Thermal Energy Storage and Heat Pipes of The Mechanical Engineering Faculty, The State University of Campinas, Brazil. In the present study, a model for solidi®cation around a vertical axially ®nned tube submerged in a PCM has been developed. The model is based upon the enthalpy method and the control volumes approach. In this study the number of ®ns, ®ns' height, thickness, the degree of superheat and the aspect ratio of the annular space were investigated. In order to validate the numerical predictions and the proposed model, speci®c experiments were realized and the results were compared with the numerical predictions and good agreement was found. The results of this study are of great interest in the design, analysis of latent heat storage systems and cold storage applications. The ®ns are found to help in enhancing the energy transfer process to and from the PCM mass. The results presented are for the case of constant wall temperature, a condition which can be attained by a heat pipe or a metallic tube carrying a heat transfer ¯uid at a high Biot number. This condition, that is constant wall temperature, is not a restrictive one. It is possible to include a temperature gradient along the tube length and incorporate it in this basic model. More details can be found in Ismail [29].
2. Formulation of problem The objective of the present study is to investigate the solidi®cation process around an isothermal ®nned tube and how it is aected by the ®ns, geometry and number as also by the operational conditions. In this case, we consider a ®nned tube submersed into PCM enclosed by an external cylindrical surface thermally insulated or by a symmetry surface in the case of multi-tubes placed inside a PCM tank. Fig. 1 shows the ®nned tube and the details of the geometry of the symmetrical region, where the three sides are thermally insulated while the tube wall is maintained at constant temperature. Considering that the heat transfer process is controlled only by pure conduction, one can write the conduction equation for the solid and liquid phases, respectively, as: @Ts 1@ @Ts 1 @ ks @Ts rks 1 rs cs r @r r @f r @f @t @r
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@Tl 1@ @Tl 1 @ kl @Tl rkl rl cl r @r r @f r @f @t @r
The energy balance equation at the solid±liquid interface can be written as:
Fig. 1. Details of the ®nned tube and the symmetry region.
2
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@Ts @Tl 1 @ rs ÿ kl ks 1 2 rs @f @r @r
2!
rs l
@rs @t
59
3
The speci®c heat and the latent heat per unit volume for the solid±liquid and the latent heat per unit volume, respectively, as: Cs rs cs ;
Cl rl cl ; l rs L
while the enthalpy per unit volume can be written for the phase change material as: H T C T dT lZ T ÿ Tm T
Z u
1 2
when ur0 when u < 0
4
5
6
taking into account the sudden increase of the enthalpy by l when phase change occurs, the speci®c heat of the phase change material becomes: C T
dH T C T ld T ÿ Tm dT
C T
Cs Cl
when T < Tm when T > Tm
7
8
where d T ÿ Tm is the Dirac function. Using Eqs. (1), (2) and (8) become: ! T @T k @T 1 @ @T 1 @ C T rkÅ T r @f @t r @r @r r @f
9
where:
8 > TRTm ÿ DT ks T ; > > < k T ; TrTm DT l k T > kl ÿ ks > > T ÿ Tm ÿ DT; Tm ÿ DT < T < Tm DT : kl T 2DT
assuming a linear variation of k 8 > Cs T ; > > < C T ; l C T > l Cs > > Cl ; : 2DT 2
10
with T in the phase change temperature range and TRTm ÿ DT TrTm DT Tm ÿ DT < T < Tm DT
in the phase change range is the result of the integral: where the value of C T
11
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Tm DT Tm ÿDT
C~ T dT l
Tm Tm ÿDT
Cs T dT
Tm DT Tm
Cl T dT
assuming that Cs T and Cl T do not depend upon T in the interval Tm ÿ DT < T < Tm DT: Bonacina et al. [8] and Shamsundar and Srimivesan [25] demonstrated that Eq. (9) converges to Eq. (3) when the temperature approaches the phase change temperature Tm. Hence, one can conclude that Eq. (9) alone is the controlling equation for this phase change problem. In order to solve the two dimensional heat conduction in the ®n, one must use Eq. (9) with the corresponding ®n material properties such as density, speci®c heat and thermal conductivity. In this case one can write: C T ra ca Ca k T ka
12
To facilitate the numerical treatment and enable better investigation of the parameters of the problem, the following dimensionless variables are adopted: y
Tm ÿ T Tm ÿ Tw
k~ y
k~ y and
R
r ks t DT ; t ; x 2 ri Tm ÿ Tw Cs ri
8 1; > > > < kl =ks ;
> kl ÿ ks y > > 1ÿ ; :1 2ks x
yRx yr ÿ x ÿx < y < x
ka for the fin ks
8 > 1; yRx > > < C =C ; yr ÿx 1 s C~ y > l Cl Cs > > ; ÿx < y < x : 2Cs DT 2Cs Ca C~ y Cs
9 > > > > > > > > = > > > > > > > > ;
9 > > > > > > > > = > > > > > > > > ;
When the new dimensionless variables are substituted in Eq. (9), it becomes: ! ~ @y 1 @ @y 1 @ k y @y Rk y C~ y @t R @R @R R @f R @f
13
14
15
16
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61
and the corresponding boundary conditions in terms of the new variables are: y 1, for R 1
17
@y rm 0, for R @f ri
18
@y 0, for f 0 @f
19
@y 0, for f fm @f
20
The discretization of Eq. (16) is realized by using the control volume technique. In this case the domain of the problem is divided into a convenient number of control volumes and Eq. (16) is integrated over R, f and t to obtain, after some mathematical manipulations, the required algebraic equations. The same is done with respect to the boundary conditions of the problem. Details of the mathematical manipulations are omitted here for the sake of brevity .With the algebraic equations in hand, it is possible to use for their representation explicit formulation, implicit formulation, Crank Nicholson formulation, ADI formulation or conjugate ADI with Crank Nicholson. Details of these methods can be found in Patankar [26] and Smith [27]. In the present study, we adopted the ADI formulation because it is unconditionally stable. More details about the numerical solution can be found in Gonc° alves [24]. The numerical code was optimized by numerical experimentation. The criterion of convergence was tested for precision of 10%, and in the range of 1±0.1%. The values of 1 and 0.2% produced coincident curves of solidi®ed mass fraction against time. For 0.1% precision, six times the computational time used for the case of 10% precision was required. The maximum number of iterations of 10, 20 and 30 were tested. The three values produced coincident curves of solidi®ed mass fraction against time and we adopted for the present study the value of 30, using 20% more computational time than the case of 20 iterations. The minimum number of iterations was also investigated using the values of 3, 4 and 6. For the same reasons, we adopted the case of six iterations as minimum. The values of maximum and minimum number of iterations are based upon Nogotov [28] where he suggests that the initial time increment to be multiplied by 1.3 if the convergence is reached before the minimum number of iterations is achieved and then used for the next time increment. Also the time increment must be divided by 2.0 if the precision is not achieved after the maximum number of iterations is attained. The number of grid points tested in the vertical direction was 21, 33 and 41. The results obtained seem to be coincident and we adopted the value of 33 which consumed 100% more computational time than the case of 21 grid points. The horizontal grid includes the ®n region, after ®n region and the ®nal part of the domain and consequently the number of points is dierent for each region. We tested the sets (10, 10, 30), (15, 15, 45) and (20, 20, 60); and the corresponding curves of solidi®ed mass fraction against time were coincident for the three sets. We adopted the set of points (15, 15, 45) which consumed 50% more than the ®rst set and 100% less than the last set.
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3. Experimental setup In order to validate the model and the numerical predictions an experimental setup was designed, constructed and instrumented to obtain necessary data suitable for comparisons with the proposed model. The basic components of the experimental rig shown in Fig. 2 are: 1. 2. 3. 4.
Container for phase change material; Thermally insulated box for keeping the PCM container; Tube heat exchanger and the test tube around which the phase change occurs; Thermally controlled water bath to heat-up the PCM and ensure the initial temperature for the tests;
Fig. 2. Layout and details of the experimental rig.
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Fig. 3. Solid±liquid interface for dierent time intervals, case of four ®ns.
5. Water pump for water circulation; 6. A rig for measuring the PCM thickness distribution. The phase change material used in the experiments is paran, whose national code is 130/135 Type 1, whose phase change temperature range is 55±578C, has 5% of oil and has the following thermophysical properties:
Fig. 4. Solid±liquid interface for dierent time intervals, case of three ®ns.
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Fig. 5. Solid±liquid interface for dierent time intervals, case of ®ve ®ns.
Tm DT rl rs cl kl L
55.88C 1.398C 771.2 (kg/m3) 771.2 (kg/m3) cs 2176 (J/kg) ks 0.089 (J/s m K) 232.4 (MJ/kg)
To realize an experimental test it is necessary to ensure that the water bath is at the desired uniform temperature, place the PCM container in the water bath, and circulate water in the
Fig. 6. Eect of the initial super heat of the liquid phase on the solidi®cation front.
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Fig. 7. Eect of the initial liquid phase super heat on the solidi®cation front.
heat exchanger to keep the ®nned tube at uniform temperature below the PCM melting temperature. When all temperatures are stable and at nearly the desired values, 1. Remove the PCM container from the water bath and place it inside the adiabatic chamber. 2. When the temperature of the PCM is at the desired value remove the top cover of the PCM container, lower down the heat exchanger with the ®nned tube and place back the top cover. 3. Start the time measurement.
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Fig. 8. Comparison between numerical predictions and experimental results.
Fig. 9. Eect of the ®n length on the solidi®cation time.
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67
Fig. 10. Variation of the solidi®cation time with the ®n length.
When the experiment's time is achieved, stop the clock, remove the tube from the PCM tank and cool it immediately. Place the tube in the thickness measuring apparatus and measure the angular distribution of the PCM thickness.
Fig. 11. Interface position for dierent ®n length.
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The important temperatures in this study are the initial liquid phase temperature, Ti 0, the phase change temperature, Tm ,and the refrigerant tube wall temperature, Tw : We de®ne a dimensionless parameter denoted here as the initial superheat factor and is given m ÿTw and the values used in the experiment are: 0.0; 0.1 and 0.2. byC 0 Ti T0i ÿT 0 w rrwe ÿ 1, and the The aspect ratio of the annular space AR is de®ned as:AR re ÿr rw w experimental value used here is 2.481. The ®n length is de®ned as: lf rref ÿr ÿrw , and the experimental values used are lf 0:087; 0.196 and 0.348. The angular half-width of the ®n is be given by: ff sin ÿ1 2rtf f and the values tested are 0.550, 0.450 and 0.360. The number of ®ns nf used is 2, 3, 4 and 5. While the dimensionless time for conversion is t 4:653117844 10 ÿ4 t s: 4. Results and discussion An extensive experimental testing program was realized to determine the eects of the geometry of the ®n and the operational conditions on the thermal performance of the tube,
Fig. 12. Eect of the ®n thickness on the solidi®cation time.
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69
solidi®ed PCM and the solidi®cation rate. Some of these results are presented below to show these eects and to be used later on for validating the numerical model. Figs. 3±5 show the solid±liquid interfaces for dierent time intervals and tubes of dierent number of ®ns. Fig. 3 shows the case of the tube with four ®ns, ®n length of 8 mm and initial temperature of PCM of 658C. It is easy to verify the evolution of the solidi®ed mass patterns and the thickness distribution around the tube and ®ns. Similar results, but for dierent number of ®ns, are presented in Figs. 4 and 5. Fig. 6 shows the eect of the initial superheat of the PCM liquid phase on the solidi®cation front for the case of tube with four ®ns, ®n length of 8 mm and for a solidi®cation time of 30 min, when the initial liquid PCM temperature is of 55.8, 59.8 and 64.88C, respectively. As can be seen when the initial liquid PCM temperature is equal or near the solidi®cation temperature, the process of solidi®cation is obviously quicker. Similar results for the case of tubes with three and ®ve ®ns are shown in Fig. 7. Fig. 8 shows comparisons between the numerical predictions and the experimental results. The comparative results shown in Fig. 8(a) and (b) show good agreement between the numerical predictions and the experiments. The dierences indicated in case (c) are attributed to the large experimental gradient of the interface near the ®ns. In the numerical solution, we
Fig. 13. Eect of the ®n thickness on the solidi®cation time.
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Fig. 14. Interface position for dierent ®n thicknesses.
Fig. 15. Eect of the ®n number on the solidi®cation time.
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Fig. 16. Eect of the number of ®ns on the solidi®cation time.
adopted linear gradients and in future studies this can be corrected by imposing better representation of these gradients in the numerical model. The dierences in cases Fig. 8(d) and (e) can be attributed to the appearance of dentrits over the solid±liquid interface for long solidi®cation times which led to increasing the surface area and hence increase the heat transfer rate. This situation does not exist in the numerical model, as the liquid±solid interface is always considered a smooth surface. Even with these apparent dierences the variation in the solidi®ed mass fraction is less than 4% in all the cases examined. Fig. 9 shows the eect of the ®n length on the solidi®cation time. One can observe, that an increase in the ®n length leads to reducing the solidi®cation time. The time for the complete solidi®cation can be observed in Fig. 10 indicating a strong reduction in its value with the increase in ®n length. The instantaneous interface positions for two values of ®n length are shown in Fig. 11. One can observe the progressive increase in the solidi®ed mass fraction with time and also the increase in the solidi®ed mass when the ®n length is increased. The in¯uence of the thickness of the ®n is shown in Fig. 12. As can be seen the increase in ®n thickness leads to the increase of the solidi®ed mass fraction and reduction of the solidi®cation time. One can also notice that the in¯uence of the ®n thickness is small.
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Fig. 17. Interface positions for dierent number of ®ns.
Fig. 18. In¯uence of the aspect ratio of annular space on the solidi®cation time.
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Fig. 19. In¯uence of the aspect ratio of annular space on the solidi®cation time.
Fig. 20. Interface positions for dierent aspect ratios of annular space.
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Fig. 21. Eect of super heating on the solidi®cation time.
The time for complete solidi®cation is also shown in Fig. 13. The positions of the interface fronts at dierent time intervals during the solidi®cation process are shown in Fig. 14. The eect of varying the number of ®ns on the solidi®ed mass fraction is shown in Fig. 15. As can be veri®ed, the increase of the number of ®ns leads to increasing the solidi®ed mass fraction and reducing the time for complete solidi®cation, as is shown in Fig. 16. The con®guration of the solidi®cation interfaces for dierent time intervals is shown in Fig. 17 for dierent number of ®ns. Again, one can observe that increasing the number of ®ns leads to increasing the solidi®ed mass and reduces the time for the complete solidi®cation. The volume of the annular space between the inner tube carrying the refrigerant ¯uid and the external symmetry surface has a strong in¯uence on the solidi®ed mass fraction and the time for complete solidi®cation. One can observe from Fig. 18 that if this ratio, denominated here as the aspect ratio of the annular space, AR, is increased from 2 to 5, the dimensionless time for the complete solidi®cation is increased from about 0.5 to 4. Fig. 19 shows clearly this strong eect. The solid±liquid interface for dierent values of the annular space aspect ratio, AR, are shown in Fig. 20. The in¯uence of the temperature dierence between the phase change temperature and tube wall temperature can be seen in Fig. 21, where the temperature dierence is varied from 10 to 258C. As can be seen, the temperature dierence has a strong eect on the solidi®ed mass
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Fig. 22. Eect of super heating on the solidi®cation time.
fraction and also on the time for complete solidi®cation as shown in Fig. 22. The liquid±solid interfaces for two values of temperature dierence and at dierent time intervals are shown in Fig. 23. 5. Conclusions The proposed thermal model and the numerical method of solution adopted produced numerical predictions of the solid-liquid interface that agree generally well with the experimental results, except when the ®n is very long or after long intervals of solidi®cation periods. The ®rst situation leads to a big inclination of the experimentally determined frontiers near the ®n. This eect is not adequately represented in the numerical model. The second situation can be explained by the increase of the heat transfer area due to the dentrits increase. In all the experiments dentrits were visible indicating that the heat transfer is controlled by conduction. The parameters analyzed numerically include the ®n length, ®n thickness, number of ®ns, the aspect ratio of the annular space and the temperature dierence between the phase change temperature and the wall temperature of the tube. The results indicate that these parameters
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Fig. 23. The interface position for dierent degrees of super heat.
have signi®cant eect on the time for complete solidi®cation. From the results presented one can observe that the ®n thickness has a relatively small in¯uence on the solidi®cation time and that the ®n length as well as the number of ®ns aects strongly the time for complete solidi®cation and the solidi®cation rate. The aspect ratio of the annular space has a strong eect on the time for solidi®cation and also the time for complete solidi®cation. The temperature dierence has an opposite eect on the solidi®cation of PCM and the time for complete solidi®cation seems to decrease with the increase of this temperature dierence. Additional numerical and experimental work is being realized in The Laboratory of Thermal Storage and Heat Pipes on plastic ®nned tubes ®tted with axial or radial ®ns of constant thickness. The results will be used in designing domestic cold storage for air conditioning applications. Acknowledgements The authors wish to thank Fapesp and CNPq for the ®nancial support to the research project of the ®rst author and Capes for the scholarships for the co-authors. References [1] E.M. Sparrow, E.D. Larson, J.W. Ramsey, Int. J. Heat Transfer 24 (1981) 273. [2] E.M. Sparrow, J.W. Ramsey, J.S. Harris, J. Heat Transfer 103 (1981) 7.
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