Parametric investigation of the enhancing effects of finned tubes on the solidification of PCM

International Journal of Heat and Mass Transfer 152 (2020) 119485

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/hmt

Parametric investigation of the enhancing effects of finned tubes on the solidification of PCM Felipe S dos Santos a, Kamal A.R. Ismail a,∗, Fatima A.M. Lino a, Ahmad Arabkoohsar b, Taynara G.S. Lago a a

State University of Campinas, Faculty of Mechanical Engineering, Mendeleiev street, 200, Cidade Universitária “Zeferino Vaz" Barão Geraldo 13083-860, Campinas, SP, Brazil Aalborg University, Fredrik BajersVej 5, 9100 Aalborg, Denmark

b

a r t i c l e

i n f o

Article history: Received 30 July 2019 Revised 17 January 2020 Accepted 5 February 2020

Keywords: PCM Finned tube Solidification Interface position Interface velocity Time for complete phase change Correlations

a b s t r a c t This paper presents the results of a study on the enhancement of solidification around finned tubes and the development of correlations to predict their thermal performance. The effects of the geometrical and operational parameters on the solidification process and thermal performance are investigated. A numerical code to predict the solidification around radial finned tubes based on pure conduction and the enthalpy method is developed and validated against experimental results showing good agreement. Results of additional experiments were also used to develop correlations for the interface position, interface velocity and the time for complete solidification. The fin diameter, and low tube wall temperature enhance the interface position and velocity, and reduce the time for complete solidification. Experiments showed that there is an optimum fin diameter for which the solidified phase change material (PCM) and stored energy are the highest. The proposed correlations for the interface position, interface velocity and the time for complete phase change seem to agree well with experimental results within maximum deviation of 4%, 7% and 1.03%, respectively. Hence, the correlations can be used for overall and quick estimates of solidification of PCM around radial finned tubes. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction The continuous increase of the energy demands to satisfy the needs of the growing worldwide populations and the enhanced industrial activities increase the environment impacts and depletion of natural resources. The severe impacts on the environment alerted the world population and the governmental and nongovernmental organizations about the necessity to use better the available energy resources and search for renewable sources of dominated technology and small cost. Solar energy appears at the top of the list of the renewables of dominated technology. Since solar energy is of intermittent nature, energy storage is therefore indispensable for its better utilization. Thermal energy can be stored in the form of sensible heat, latent heat or in combination. Latent heat energy storage systems are effective and more attractive due to their high storage capacity and nearly isothermal behavior during the charging and discharging processes. One of its main dis-

∗

Corresponding author. E-mail addresses: [email protected] (K.A.R. Ismail), [email protected] (F.A.M. Lino), [email protected] (A. Arabkoohsar), [email protected] (T.G.S. Lago). https://doi.org/10.1016/j.ijheatmasstransfer.2020.119485 0017-9310/© 2020 Elsevier Ltd. All rights reserved.

advantages is the low thermal conductivity which usually leads to low charging and discharging heat rates and hence decreasing the overall performance of the system. This negative impact can be attenuated by employing enhancement techniques such as dispersion of high conductivity particles in the PCM, micro-encapsulation of the PCM, use of finned surfaces among others [1,2]. There are many reviews handling PCM in the different ranges of temperature, their general characteristics, techniques for encapsulation, modeling and simulation of their thermal behaviors in walls, tanks, spheres [3–5]. Since the poor thermal conductivity of PCM is usually the strong barrier for their wide use, many research work and technical reviews were destined to present new concepts, application and simulation studies, numerical and experimental investigations, thermal conductivity inserts and additives including fins, heat pipes, porous materials and nano-particles [6]. Broadly PCM can be divided into metallic PCM such as tin and non metallic PCMs such as paraffins and glycols which have low thermal conductivity. Fins with high thermal conductivity and various geometric configurations are widely introduced into latent heat thermal energy storage systems to improve their performance. Dhaidan and Khodadadi [7] presented a review on the improvements in performance of latent heat energy storage systems due

2

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Nomenclature Specific heat (Jkg−1 K−1 ) Heat capacity per unit volume (Jm−3 K) Heat capacity per unit volume including phase change (Jm−3 K) H(T) Enthalpy per unit volume (Jm−3 ) k Thermal conductivity (Wm−1 K−1 ) k˜ (T ) Thermal conductivity including phase change (Wm−1 K−1 ) L Latent heat (Jkg−1 ) M Mass (kg) r Radial coordinate (m) R Dimensionless radial coordinate = r/rw rw External radius of the tube (m) re Radius of the symmetry circle (m) rf Radius of fin (m) t Time (s) T Temperature (K) Tinitial Initial temperature (K) Tfinal Final temperature (K) Tm Phase change temperature (K) T Half phase change temperature range (K) δ (T − Tm ) Dirac delta function  Elemental distance along the tube η (T − Tm ) Unit step function λ Latent heat per unit volume (Jm−3 ) ϕ Dimensionless temperature = (Tm − T )/(Tm − Tw ) ρ Density (kgm−3 ) τ Dimensionless time = ks t/r2 w Cs ξ Dimensionless half phase change temperature range = (T )/(Tm − Tw ) z Axial coordinate (m) zf Half fin thickness (m) zt Distance from half fin thickness to the position in the middle between two successive fins (m) c C(T) C˜(T )

Subscripts f fin i Interface position l Liquid phase s Solid phase w wall W Water z Along axial coordinate to utilizing high thermal conductivity fins. They reviewed analytical, computational, and experimental studies focussed on improving the performance of latent heat energy storage systems that utilize fins to promote heat transfer. A variety of phase change materials including pure/commercial paraffins, carbonate mixtures, polyethylene glycol etc. was presented and commented. Many studies are available on fins fixed to flat surfaces and tubes to increase the heat transfer area and consequently the heat transfer rate between the PCM and the surface. Thermal models, numerical solutions, effects of the spacing between fins, flow parameters such as Reynolds number and inlet temperature of the working fluid were also investigated and reported by Erek et al.[8], Rahimi et al. [9], Hosseini et al. [10], Joybari et al. [11]. CFD simulations were used in treating finned tubes submersed in PCM as in Tayet al.[12] who also validated experimentally their CFD model of a vertical finned tube heat exchanger. Parry et al.[13] reported on the development of a computationally efficient numerical simulation model for a shell-and-tube thermal energy storage system. Comparison of the heat transferred during charging and

discharging phases predicted by both the one-dimensional and the refined two-dimensional model agreed to within 8.5%, indicating the usefulness of the one-dimensional model. Internal and external extended surfaces were investigated both experimentally and numerically to identify the parameters most important to be taken into consideration when dimensioning a storage tank for a specific use. These parameters include the extended area, geometry of the surface facing the PCM, working temperature, number of fins and the spacing between them. These parameters affect strongly the interface velocity, PCM solidified mass, stored energy and the time for complete phase change [14–21]. Simultaneous charging and discharging processes received a lot of attention in modeling, simulation and experimental verification to ensure adequate and efficient operation of the storage systems [22–25]. Modeling of finned tubes, inclusion of convection together with conduction in the modeling process, different numerical treatments and experimental validation were treated in some investigations. Detailed discussion of the effects of fins on the interface position, on the interface velocity and on the time for complete phase change were addressed and the effects of natural convection on these parameters were also investigated [22–30] In the present study, a thermal model for the solidification of PCM around a horizontal radial finned tube is developed and solved numerically. The formulation of the model is based on pure heat conduction and the enthalpy method while the finite difference approach is used to discretize the system of equations and the associated boundary, initial and final conditions. A home-built numerical code is developed, and validated against experimental results. An extensive number of experimental tests were conducted to investigate the effects of the diameter of fin, the tube wall temperature and mass flow rate of the heat transfer fluid on the interface position, the interface velocity, the solidified mass fraction and the time for complete phase change. The experimental results were also used to develop and validate the proposed correlations for predicting the interface position, the interface velocity and the time for complete solidification. One contribution from the present study is the development of a home-built numerical code based on the two dimensional model for the solidification of PCM around finned tube submersed in liquid and its validation against experimental results. Another contribution is the detailed experimental investigation of the geometrical and working parameters that influence the process of phase change around finned tubes. The experimental treatment allowed quantifying the influence of the fin area, and the tube wall temperature on the time for complete phase change, interface velocity and the solidified mass and stored energy. These experimental results are useful for validation of numerical models and other comarative aspects. Based on these experimental data correlations were developed and validated against independent experimental results. The developed correlations are useful for predicting the thermal performance and a handy design tool for finned tube latent heat storage systems.

2. Formulation of the problem The problem under consideration presented in Fig. 1, is composed of a horizontal tube fitted with external radial fins and submersed in liquid PCM at its phase change temperature. A circulating cold fluid at temperature less than the phase temperature of the PCM flows along the tube forming a solid layer of PCM over the tube and fins surfaces. To develop the thermal model for this problem some simplifying assumptions are considered including that the PCM is pure and of well-defined phase change temperature, thermo physical properties vary with temperature, constant tube

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

3

Fig. 1. Details of the finned tube problem.

wall temperature and heat transfer process dominated by conduction. The problem of phase change (solidification) around finned tube submersed in liquid PCM can be formulated as a two dimensional problem dominated by conduction and described by the equation of heat conduction in the solid PCM, heat conduction in the PCM liquid phase, heat balance at the interface and heat conduction in the fin. Among available methods for the numerical solution two possible methods can be considered. The first method is to solve the four equations simultaneously, each subject to its boundary and initial conditions. This route will probably need some additional simplifications, possible energy balance between the inter regions and approximations especially in relation to the fin which will be in contact with the liquid and solid phases of the PCM during the solidification process. This will make the problem much more difficult in addition to the continuous tracking of the interface. Solutions available for this case are scarce and are of limited applications, subject to simplifications such as zero fin thickness, admitting a temperature profile along the fin, mean temperature along the fin, etc. The second possible route is to formulate the problem by the enthalpy method which offers significant advantages and make the numerical solution easier and relatively manageable. The two

equations for the solid and liquid phases collapse to one equation valid for the two regions of the solid and liquid PCM since in the method the enthalpy is considered instead of the temperature [34]. When the temperatures of the liquid and solid approach the phase change temperature, the energy balance equation, becomes the same as the resultant heat conduction equation [35]. This equivalence was proved in the available literature and is well accepted. The fin is represented by a two dimensional heat conduction equation similar to the above for the PCM. If the properties of fin are molded in the same form as in the enthalpy method, then the same equation can be used to handle the total domain containing the fin and PCM. In applying the heat conduction equation the corresponding thermo physical properties will be utilized according to the treated region. Hence, the enthalpy method will be used to treat the thermal model of the problem of phase change around a horizontal finned tube. In the formulation this issue will be taken into consideration to facilitate the mathematical and the numerical treatments. The energy equation in cylindrical coordinates for the PCM solid phase is:

ρs c s

    ∂ Ts 1 ∂ ∂ Ts ∂ ∂ Ts = r ks + ks ∂t r ∂r ∂r ∂z ∂z

(1)

4

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

The energy equation for the PCM liquid phase is:



ρl c l

∂ Tl 1 ∂ ∂T = r kl l ∂t r ∂r ∂r





+

∂ ∂ Tl k ∂z l ∂z



(2)

The boundary conditions at the interface [35] can be written as:



ks ∂∂Trs − kl ∂∂Trl



 2 

1 + ∂∂ zs

= ρs L ∂∂ ts

Tl = Ts = Tm

, at r = s(t )

(3)

Consider the region representing the domain of the problem extending from half of the fin width (z = zi = 0) to half the distance between two successive fins (z = zt ), where the lower boundary coincides with the external diameter of the tube, while the upper boundary coincides with the radius of the symmetry circle. This domain is shown in Fig. 1b. The boundary conditions associated with this domain and the respective explanations are presented below.

On the tube wall: r = rw , T = TW On the symmetry circle: r = re ,

∂T =0 ∂r

Start of the symmetry region (seeFig.1b ):z = 0,

∂T =0 ∂z



C (T )dT + λη (T − Tm )

(6)

T

where λ is the latent heat per unit volume and ηis a unit step function. The thermal capacity per unit volume is defined as:

C˜(T ) =

dH (T ) = C (T ) + λδ (T − Tm ) dT

(7)

where δ (T − Tm ) is the Dirac delta function and the behavior of the equivalent thermal capacity in the phase change material is given by:

(4b)

C (T ) = Cs when T < Tm − T Thermal capacity of solid C (T ) = Cl when T > Tm + T Thermal capacity of liquid

(4c)

Substituting Eq. (7) into Eq. (6) and integrating over the phase change temperature range one can obtain:

Eq. (4a) refers to the condition of constant wall temperature, the second condition Eq. (4b) refers to the symmetry circle through which there no heat transfer (adiabatic) while the third Eq. (4c) and fourth Eq. (4d) conditions refer to the start and end of the symmetry region considered in the present analysis. The initial and final conditions can be written as:



T(r, z, t = 0 )≥ Tm + T PCM in the liquid state T r, z, t = t f ≤ Tm − T PCM in the solid state

H (T ) =

(4a)

∂T End of the symmetry region(see Fig.1b ) : z = zt , = 0 (4d) ∂z



enthalpy method and in this way the energy equation which describes the two phases of PCM and the interface can also describe the fin. In other words, the same equation describes the complete domain including PCM and fin. Consequently the boundary conditions are specified for the boundaries of the complete domain. Considering the above discussion we adopted the enthalpy method as a route for the solution of the finned tube submersed in liquid PCM as formulated above. Following Bonacina et al. [34] we define the enthalpy per unit volume of PCM as:

(5)

where T is half of the phase change temperatura range and tf is the time at the end of the phase change process. To solve this problem numerically as pointed out before, we have to add one more heat conduction equation for the fin, establish the boundary and initial conditions for each equation and solve the system simultaneously. This procedure is very complicated and time consuming and rarely used in the literature (to the authors knowledge). To make the numerical solution more manageable and less complicated the Landau-type immobilization technique can be used, but the analytical treatment to put the equations in a numerically executable form is usually very complicated as done by Padmanabhan and Murthy [31]. Approximate analytical solutions were also obtained by different authors using different simplifying assumptions such as fin of zero thickness, preestablished or approximate temperature distribution along the fin as in [32,33]. It is noteworthy that one of the main difficulties in the numerical solution for such problems is the need to track the location of the phase-change interface continuously during the solution process, so that the interfacial conditions can be applied there. A technique which alleviates this difficulty is the enthalpy method, in which a single two dimensional heat conduction equation by using the material enthalpy instead of the temperature can represent the entire domain, including both phases and the interface [34,35]. This fact is very important since it reduces significantly the computing time. Additionally, since the two dimensional conduction equation for the fin has the same form as the equation for PCM, one can elaborate the fin properties in the same form as in the

H (T ) =

Tm +T

C˜(T )dT =λ +

Tm −T

Tm

Tm −T

Cs (T )dT +

(8)

Tm +T

Cl (T )dT

(9)

Tm

The above equation describes the variation of the enthalpy during the phase change process. In the same manner the thermal conductivity is given as



k˜ (T ) =

ks ( T ) kl ( T )

T ≤ Tm − T T ≤ Tm + T



(10)

According to Bonacina et al. [34] and Ozisik [35], Eqs. (1) and (2) can be represented by:

C˜(T )

    ∂T 1 ∂ ˜ ∂T ∂ ˜ ∂T = r k (T ) + k (T ) ∂t r ∂r ∂r ∂z ∂z

(11)

This means that both the solid and liquid PCM which are of different properties are represented by the same heat conduction Eq. (11). Also, Ozisik [35] proved that Eq. (11) represents the energy balance equation at the interface solid/liquid when the temperature approximates the phase change temperature. The heat conduction in the fin is represented by an equation similar to Eq. (11). Hence, by defining the values of the heat capacity and the thermal conductivity of the fin C˜(T ) and k˜ (T ) in the same form as in the enthalpy method, Eq. (11) can represent the fin in the total domain. The values of C˜(T ) and k˜ (T ) for the fin are given by,

C˜(T ) = ρ f c f = C f k˜ (T ) = k f

(12)

In this manner Eq. (11) represents the liquid PCM, the solid PCM, the heat balance at the interface and the fin. The complete domain of the solution of the energy equation is described earlier and shown by the dashed lines in Fig. 1b. The energy equation is applied in its generic form to the complete domain as indicated by the dashed lines and for each region of the domain (fin, solid phase or liquid phase) and the corresponding properties are used. The solution of the energy equation uses as a strategy identifying each part of the domain (solid, liquid PCM or fin) and solves the energy equation with the respective local properties. In this manner, the temperature field is obtained for the entire domain.

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

To facilitate the numerical solution, the following dimensionless parameters are adopted [36]:

ϕ=

Tm −T ; Tm −Tw

k¯ =

r ; rw

Z=

z ; rw

C¯ =

k ; ks ks t = C r2 ; s w kls = kksl ; C C f s = Csf ;

C ; Cs Cl Cls = Cs ;

τ

Bi =

R=

St eT = Cρs s LT =

(13)

Cs T

λ ;

T ; Tm −Tw kf k f s = ks

ξ=

hrw ; k

Introducing the new dimensionless variables, the dimensionless thermal conductivity is expressed as:

k¯ (ϕ ) =

⎧ 1; ⎪ ⎪ ⎨kls

  1 + 12 (kls − 1 ) 1 − ϕξ ; ⎪ ⎪ ⎩ kf ks

⎫ ϕ ≤ 1 − 2ξ ⎪ ⎪ ⎬ ϕ≥1 1 − 2ξ < ϕ < 1⎪ ⎪ ⎭ F in

(14)

In a similar manner, the dimensionless thermal capacity is written as:

⎧ 1; ⎪ ⎨C ; ls C¯ (ϕ ) = 1  1 + 1 + C ; ls 2 St eT ⎪ ⎩ Cf Cs

⎫ ϕ ≤ 1 − 2ξ ⎪ ⎬ ϕ≥1 1 − 2ξ < ϕ < 1⎪ ⎭

(15)

F in

Introducing the new variables in Eqs. (11), (4) and (5) one can obtain the final forms presented below:

    ∂C 1 ∂ ∂ϕ ∂ ˜ ∂ϕ = Rk˜ (ϕ ) + k (ϕ ) ∂τ R ∂ R ∂R ∂Z ∂Z       ∂ϕ ∂ϕ ∂ϕ   ϕW = 0|R=1 , = 0 , = 0 , = 0 ∂R ∂ Z ∂ Z R=R Z=Z =0 Z=Z

C˜(ϕ )

e

i

(16)

(17)

t

The initial and final conditions are respectively,

ϕ = 1, ϕ = 1 − 2ξ

(18)

where the initial condition corresponds to the thermal equilibrium at the initial state when the dimensionless temperature is ϕ = 1. The final condition corresponds to final state when all PCM is solidified and at the dimensionless temperature given by ϕ = 1 − 2ξ . The transformation of Eq. (16) into a system of algebraic equations is done by using the finite difference technique and the alternating direction implicit method because of its unconditional stability and easiness of programming. The details are omitted here for brevity. The set of equations of the model and the boundary and initial conditions are implemented in a thoroughly tested home-built computational code. The grid points along the radial and axial directions were varied having fine grid distribution near the fin and the tube surface because of the strong temperature gradients. The input parameters used in the simulations are finned and finless copper tubes of 500 mm length and 15 mm tube diameter, thickness of 3 mm and spacing between fins of 60 mm, the PCM used is water and the range of phase change is 0.1 °C. The convergence criterion used is that the residue of the temperature variable is less than or equal to 10−4 . The sequence of the main steps of the developed numerical code is shown in Fig. 2. 2.1. Mesh optimization The axial grid points are distributed in the fin region, in the region nearer to the fin and over the region until the symmetry line. This is done to account for the strong temperature gradients near the cooling surfaces, that is, fin and tube regions. The grid distribution is identified as (N-M-MM) where N is the number of grid

5

points in the fin region, M is the number of points in the near fin region and MM is the number of points up to the symmetry line of the domain. For each region the number of grid points is varied while keeping the values for the other two regions unchanged. The numerical results were compared to choose the values that produced marginal variation in the interface position and satisfied the convergence criterion adopted for this case as 10−4 mm. The values of N tested are: N = 5, 10, 15, 20. The value of 15 was found to produce variation which satisfies the above convergence criterion. The values of M tested are: M = 10, 20, 30, 40, 50, 60, and the value of 50 was found satifactory. The values of MM tested are: MM = 10, 20, 30, 40, and the value of 30 satisfied the above convergence criterion. The combination of 15 points in the fin zone, 50 points in the zone after the fin and 30 points in the final zone, denoted as 15–50–30 configuration in comparison with other configuration showed good precision and relatively smaller time. To determine the adequate radial grid which satisfies the convergence criterion the grid was varied as 20, 70, 100 and 120. The grid of 70 points satisfied the convergence criterion. The time grid was varied as 10−3 , 10−4 , 10−5 , 10−7 , 10−9 , 10−12 , and the adequate time grid size was found to be 10−12 . Fig. 3a shows one test result for the axial grid distribution of mesh: 15–50–30 showing the solid liquid interface lines for different intervals of time during the solidification process. It is interesting to see the local effect of the fin in enhancing the solidification of PCM. Other test results are omitted for brevity. Fig. 3b shows the test results for the radial grid. The grid of 70 points was chosen for time and precision considerations. 2.2. Validation of the model and code To validate the numerical model and code the predictions were compared with experimental results from [37] under the same conditions for the case of bare tube at wall temperature of −15.9 °C, (Fig. 4a). The maximum deviation between the experiments and predictions is of the order of 3.55%, indicating that the model and code can predict adequately the solidification of PCM around finless tube. Fig. 4b shows another comparison of the solidification around bare tube for wall temperature of −10.5 °C, [37]. The maximum deviation between the experiments and predictions is of the order of 9.31%, confirming the adequacy of the model and code to describe solidification around cooled tubes submersed in PCM. Fig. 4c shows the variation of the radial interface position with time for the case of finned tube. As can be observed at the beginning, the rate of variation of the interface position is high but decreases with time due to the increase of the thermal resistance caused by the continuously increasing PCM solidified mass. The initial difference between the experimental results and predictions is due to the fact the initial liquid PCM temperature is not exactly zero but about approximately 0.5 °C which makes the initial gradient smaller than the numerical value and provokes some differences. Towards the end of the process, the experimental results seem to reach steady state while the numerical predictions shows a slightly increasing tendency which can be attributed to small thermal losses from the equipment. These losses do not exist in the numerical model. As a whole the agreement is good showing a maximum deviation of about 6.98%. Fig. 4d shows a comparison between the predicted interface velocity and the experimental measurements. As can be seen the comparison shows a deviation of about 7.97% indicating that the code can very well predict the interface velocity for the case of finned tubes submersed in PCM. One can observe that initially the interface velocity is high due to the small thermal resistance offered by the small solid PCM layer. With the increase of time the

6

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 2. Flow chart of the computational code.

solidified layer increases and consequently the thermal resistance resulting in decreasing the interface velocity. This continues until the thermal resistance is so high that the interface velocity is negligibly small to cause any further effect on the phase change process. 3. Experimental rig and test procedure The general scheme of the experimental system shown in Fig. 5 is composed of a conventional compression refrigeration circuit, a secondary fluid circuit for cooling the working fluid (Ethanol) and a coiled tube heat exchanger submersed in the secondary fluid tank. Five fins each of thickness 3 mm are installed

on each tube occupying about 300 mm of the central section of the tube. Initial tests showed that there is no variation of temperature along the finned tube and it behaved as an isothermal tube. All temperature measurements and photographs were focused on the third fin for symmetry as can be seen in Fig. 6. The test tube described above is connected to the secondary fluid circuit where Ethanol is cooled by the refrigerant flowing through the coiled tube heat exchanger. The temperature and mass flow rate of the secondary fluid are controlled as required. The PCM test tank is of rectangular form built from 15 mm thickness acrylic sheet with the test tube extending across the tank. The tank is filled with PCM (water) whose initial temperature can be varied as desired. High resolution digital camera is installed

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

7

Fig. 3. Mesh independence tests for the axial and radial grids.(a) Mesh: 15-50-30 (b) Optimization of the radial grid points.

to photograph the finned tube and the reference scale. The reference scale helps converting the image dimensions into real values. Calibrated thermocouples of the type T are fixed at entry and exit of the finned tube, at different positions in the PCM test tank, along the finned tube and in the secondary fluid tank. The thermocouples and orifice plate are calibrated and error analysis and propagation in the results are made. The final results indicate uncertainty in the thermocouples of ± 0.5 °C, image conversion precision of ± 0.1 mm while that of the mass flow rate is ± 10−4 kg/s. Measurements were taken when the desired testing conditions are achieved including the temperature of the working fluid in the finned tube, temperature of the Ethanol tank, temperature of the PCM, and the mass flow rate of the secondary fluid. Under these conditions all initial conditions are registered and the chronometer is switched on. During the first hour, all the readings of the measurement points are registered and the finned tube is photographed every 2 minutes. During the second and third hours measurements are registered every 15 min. Starting the fourth hour the time interval is increased to 30 min until the end of the experiment. The experiment is considered terminated when no change in temperature or interface position is registered at three successive time intervals. The registered photographs of the solidliquid interface with time are processed by using the Tracker software and the dimensions in the image are converted to physical dimensions with the help of the reference scale, Fig. 6.

Fig. 4. Validation of the numerical predictions against the experimental results.(a) Variation of the interface position with time for bare tube. (b) Variation of the interface position with time for bare tube.(c) Variation of the interface position with time for finned tube.(d) Variation of the interface velocity with time for finned tube.

8

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 5. Test bench scheme. 1: Electric motor; 2: Compressor; 3: Set point; 4: Heat exchanger; 5: Flow meter; 6: Pump; 7: Ethanol tank; 8: Oil filter; 9: Solenoid valve; 10: PCM tank; 11: Signal acquisition board; 12: Computer; 13: Finned tube; 14: Digital camera; 15: Valve; 16: Condensing unit; 17: T-type thermocouples; 18: Lamp.

Fig. 6. Tracker software with the finned tube positioned for the digitalization of the interface position.

The overall experimental error for the radial position is ± 0.6 mm, for the interface velocity is ± 0.0127 mm/min, for the solidified mass is ± 0.0232 kg and for the stored energy ± 0.024 kJ. 4. Results and discussion 4.1. Experimental measurements on finless tube Fig. 7 shows the variation of the radial interface position with time for the case of bare tube and for different tube wall temperatures. As can be seen, initially the rate of change of radial interface position with time is big but decreases with time. This is due to the fact that initially the thickness of the solidified PCM is small and hence the thermal resistance between the liquid PCM and the tube surface is small leading to high heat flow rate and consequently more solidification of PCM. With the increase of time, the solidified layer thickness increases and consequently the heat transfer rate and the solidification rate decrease. This continues until a stage is reached where the solidified PCM layer is too big, the thermal resistance is so high that the rate of heat removal

from the liquid PCM is extremely small tending to zero. At this stage the solidification process nearly stops and the corresponding time is called the time for complete phase change. One can observe that during the last three successive time intervals there is no noticeable increase of the radial interface position. One can also observe the increase of the interface position with the decrease of the tube wall temperature. This can be attributed to the increase of the temperature gradient which increases the heat transfer rate and consequently the interface position. Fig. 8 shows the variation of the interface velocity with time for different tube wall temperatures. As can be seen, the initial interface velocity is high reaching about 1.9 mm/min and after about 270 min drops to about 0.2 mm/min. This behavior is attributed to the fact that initially the solidified PCM layer is small and consequently the thermal resistance between the liquid PCM and the tube surface is small. This leads to high heat transfer rate and high solidification velocity. With the increase of time the solidified thickness and the thermal resistance increase and the interface velocity decreases until it drops to almost zero towards the end of the phase change process as can be observed at about 850 min.

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 7. Variation of the interface position with time for different tube wall temperatures for the case of finless tube.

Fig. 8. Variation of the interface velocity with time for different tube wall temperatures.

Figs. 9 and 10 show the variation of the solidified mass and the stored energy with the mass flow rate of the secondary fluid for three tube wall temperatures. As can be seen, the increase of the mass flow rate of the secondary fluid increases the PCM solidified mass. This increase is mainly due to the increase of Reynolds and Nusselt numbers, which increases the heat transfer coefficient and consequently the rate of heat removal from the liquid PCM and the PCM solidified layer. Since the stored energy is proportional to the solidified PCM mass a similar behavior is expected as in Fig. 10. The reduction of the tube wall temperature increases the PCM solidified layer, however the effect is more pronounced than the increase of the mass flow rate as can be seen. Considering the two temperatures of −11.8 °C and −2.9 °C, the increase of the PCM solidified mass due to this change of wall temperature is about 61%, almost three times more than the effect due the variation of the mass flow rate. This effect is mainly due to the increase of the temperature gradient between the tube wall temperature and the liquid PCM, which increases the heat transfer rate and enhances the PCM solidified mass. This effect is also found in the case of stored energy, Fig. 10. Fig. 11 shows the variation of the time for complete phase change with the mass flow rate for the case of finless tube. As

9

Fig. 9. Variation of the solidified PCM mass with the mass flow rate of the secondary fluid for different tube wall temperature for the case of finless tube.

Fig. 10. Variation of the stored energy with mass flow rate of the secondary fluid for different tube wall temperature for the case of finless tube.

Fig. 11. Variation of the time for complete phase change with the mass flow rate of the secondary fluid.

10

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 12. Variation of the time for complete phase change with the tube wall temperature.

Fig. 14. Variation of the interface position with time for different mass flow rates of the secondary fluid for the case of fin of 60 mm diameter.

Fig. 15. Variation of the interface position with time for different tube wall temperatures for the case of fin of 100 mm diameter.

Fig. 13. Variation of Reynolds number with the mass flow rate of the circulating secondary fluid.

can be seen, the increase of the mass flow rate reduces the time for complete phase change. The change of the mass flow rate from the minimum value to the maximum value reduces the time for complete phase change by 11.7, 10.3 and 7.7% for tube wall temperatures of −11.8, −7.8 and −2.9 °C, respectively. This effect is mainly due to the increase of Reynolds and Nusselt numbers which increases the heat transfer coefficient and consequently the heat transfer rate, enhances the interface velocity and reduces the time for complete solidification. As can be seen in Fig. 12, reducing the tube wall temperature decreases the time for complete solidification. This effect is mainly due to the increase of the temperature gradient, which enhances the interface velocity and, consequently decreases the time for complete solidification. The reduction in this case is about 29%. This confirms that the reduction in temperature is more significant than the increase of Reynolds number, due to the increase of the mass flow rate as in Fig. 13.

4.2. Experimental measurements on finned tubes Solidification tests were done on finned tubes to show the effect of the fins on the phase change process. Fig. 14 shows the variation of the radial interface position with time for different mass flow rates of the secondary fluid and for a tube with 60 cm fin diameter. As can be seen initially the rate of change of the radial position with time is big but decreases with time. This is due to the fact that the initial solidified layer is small and hence the thermal resistance is small resulting in high heat flow rate and consequently more solidified PCM. With the increase of time, the solidified mass thickness and consequently the thermal resistance increase. This process continues until a stage is reached where the solidified PCM layer is too big, the thermal resistance is so high that the rate of heat removal from the liquid PCM is extremely small tending to zero. The effect of the increase of the mass flow rate of the secondary fluid on the interface position can be attributed to the increase of the Reynolds and Nusselt numbers which increases the heat transfer coefficient and the PCM solidification rate. Fig. 15 shows the variation of the interface position with time for fin diameter of 100 mm. One can observe that initially the rate of change of the interface position is big due to the small PCM

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 16. Variation of the interface velocity with time for different mass flow rates of the secondary fluid for the case of fin of 60 mm diameter.

solidified layer and consequently the small thermal resistance. As the time increases, the solidified layer increases and the heat transfer rate decreases due to the increase of the thermal resistance. Towards the end of the phase change process the solidified mass is thick and this reduces significantly the rate of heat transfer causing practically no more increase of the solidified mass. It is possible to observe the strong influence of the tube wall temperature on the interface position. The reduction of the tube wall temperature increases the temperature gradient between the tube surface and the liquid PCM enhancing the heat transfer rate and increasing the interface position. The wall temperature is an important project parameter for the control and adjustment of the time for complete phase change according to the process. Fig. 16 shows the variation of the interface velocity with time. As can be observed the initial interface velocity is high reaching about 1.7 mm/min and after about 50 min drops down to about 0.1 mm/min. This behavior is caused by the PCM solidified layer which is initially thin and consequently the thermal resistance is small causing both big heat transfer rate and high solidification velocity. With the increase of time the thickness of the solidified layer increases, the thermal resistance increases and the interface velocity decreases until it drops to almost zero towards the end of the phase change process. Fig. 17 shows the variation of the interface velocity for the case of fin diameter of 100 mm. One can observe the influence of the change of the tube wall temperature on the interface velocity. As can be seen reducing the tube wall temperature increases the temperature gradient between the tube wall and the liquid PCM and enhances the heat transfer rate and increases the interface velocity. Fig. 18 shows the variation of the solidified PCM mass with the mass flow rate of the secondary fluid. The increase of the mass flow rate increases the Reynolds and Nusselt numbers causing the increase of the heat transfer coefficient and the solidified mass. One can observe that the influence of the mass flow rate is much less than the effect of the tube wall temperature, confirming the previous comments. The total stored energy is the sum of the energy stored in the form of sensible heat of water (in the liquid and solid states), kJ, and the energy stored in the form of latent heat, kJ, as described by the equations below:

Qtotal = QS(W ) + QL(W )

(19)

11

Fig. 17. Variation of the interface velocity with time for different tube wall temperatures for the case of fin of 100 mm diameter.

Fig. 18. Variation of the solidified PCM mass with the mass flow rate of the secondary fluid for different tube wall temperatures.

Qtotal == M(W ) c(W ) (Tm − Tinitial ) + ML + M(W ) c(W ) (T f inal − Tm ) (20) where the subscript w refers to water. For the calculation of the solidified mass one can refer to Fig. 6 where the image of the finned tube and the solidified mass are shown. The calculation procedure is outlined below. Divide the length of the finned tube in a convenient number of small segments. Consider one of these elementary segments along the tube axis of length z and localize its intersection with the interface profile along the radial direction, whereri is the average radius of the segment. Then, the solidified mass of this segment (excluding the tube) can be calculated as:

     M z = π r i 2 − r w 2  z ρs

(21a)

Repeat this procedure for all segments along the whole length of the tube and sum up to give:

Ms =

 

    π r i 2 − r w 2  z ρs

(21b)

12

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

Fig. 19. Variation of the stored energy with the mass flow rate of the secondary fluid for different tube wall temperatures.

Fig. 21. Variation of the time for complete phase change with the tube wall temperature for the case of fin of 90 mm diameter.

Fig. 22. Variation of the measured interface position with the tube wall temperatures for different fin diameters. Fig. 20. Variation of the time for complete phase change with the mass flow rate for different tube wall temperatures for the case of fin of 60 mm diameter.

The total volume of the nf fins each of thickness 2zf is

Vf = n f

  2   π r f − rw 2 2z f

(21c)

And the total mass is:

Mf = nf

  2    π r f − r w 2 z ρ f

(21d)

Hence, the net solidified mass is given by:

Ml = Ms − M f

(22)

Fig. 19 shows the influence of the variation of the mass flow rate on the stored energy in the PCM. The results have the same tendency as the solidified PCM mass. The time for complete phase change for the case of fin diameter 60 mm is shown in Fig. 20. It is found that the change of the mass flow rate from the minimum to the maximum value reduces the time for complete phase change by 9.1, 8.0 and 7.4% for tube wall temperatures of −11.3, −9.3 and −3.9 °C, respectively. Fig. 21 shows the variation of the time for complete solidification with the tube wall temperature for the case of fin of 90

mm diameter. As can be seen, reducing the tube wall temperature strongly affects the time for complete phase change. The results can be explained as before. Fig. 22 shows the variation of the interface position with the tube wall temperature for three fin diameters in comparison with the finless tube after eight hours of continuous solidification. The interface position decreases with the increase of the tube wall temperature and this behavior can be explained as before. Also, the increase of the fin diameter increases the heat transfer area and, consequently increases the interface position. It is found that the tube with 100 mm fin diameter enhances the interface position by about 47% with reference to the tube without fins. Fig. 23 shows the variation of the interface velocity with the tube wall temperature for four fin diameters in comparison with the finless tube. The interface velocity decreases with the increase of the tube wall temperature as shown before, and that the increase of the fin diameter increases the heat transfer area and the interface velocity. The tube with fin of 100 mm showed an increase of about 148% in the interface velocity in comparison with the finless tube.

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

13

Fig. 23. Variation of the measured interface velocity with the tube wall temperatures for different fin diameters. Fig. 25. Comparison of the numerical predictions with the measured interface position for fin diameter of 60 mm.

Fig. 24. Variation of the measured PCM solidified mass with the tube wall temperatures for different fin diameters.

Fig. 24 shows the variation of the PCM solidified mass with the tube wall temperature for five fin diameters in comparison with the finless tube after eight hours of continuous solidification. The PCM solidified mass is found to decrease with the increase of the tube wall temperature as before and that the increase of the fin diameter increases the interface position and the PCM solidified mass. In comparison with the finless tube, the tube with fin of 100 mm produces an increase of about 126% in the PCM solidified mass. 4.3. Comparison of the numerical predictions with experimental results In this section, the numerical predictions are compared with experimental results to demonstrate the capacity and precision of the numerical code and its adequacy for numerical simulations. Fig. 25 shows a comparison of the predicted interface position with the experimental results. The predicted numerical results have the same behavior as the experiments. The numerical model seems to overestimate the interface position in the first 300 min of the simulations showing maximum deviation of about 25%. This can be attributed to thermal losses, which exist in the real experiment but not in the numerical simulations causing a reduction of

Fig. 26. Comparison of the numerical predictions with the measured interface velocity for fin diameter of 60 mm.

the interface position. Towards the end of the process, more than 500 min, the maximum deviation between the experiments and prediction is of the order of 9.0%. Fig. 26 shows a comparison between the predicted interface velocity and the experimental measurements. The agreement is good confirming that the numerical model can predict satisfactorily the interface velocity. The maximum difference between the experiments and predictions is of the order of 2.5%. Fig. 27 shows a comparison of the predicted solidified mass and the solidified mass determined experimentally. The results show that the numerical predictions have similar tendencies as the experiments but overestimates the solidified mass. This is due to thermal losses occurring in the real experiment resulting in a maximum difference of 7.5%. Fig. 28 shows a comparison of the predicted stored energy and the experimental measurements. The numerical model shows similar tendencies as the experiments but overestimates somewhat the stored energy due to thermal losses occurring in the experiment. The maximum difference between the experiments and prediction is of the order of 7.5%.

14

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

variables. The interface position, interface velocity and the time for complete solidification are considered as dependent variables. Some of the present experimental data were used to develop the correlations by formulating a number of equations equal to the number of unknown constants in the equations, then the linear system of equations is solved simultaneously. The curves that most closely approximated the experimental data were the exponential and potential models. The potential model showed a significant regression at 95% confidence level, and high correlation coefficient, R2 . The interface position (Pinter f ace ) is correlated with the fin diameter, mass flow rate, tube wall temperature and the corresponding solidification time as:

Pinter f ace = 4.37D0.74 M0.13 |Tw |0.73t −0.27

Fig. 27. Comparison of the numerical predictions with the measured PCM solidified mass for fin diameter of 60 mm.

(23)

where D: diameter of fin (mm),M: mass flow rate (kg/s), Tw : tube wall temperature (°C), t: time (min). The interface velocity (Vinter f ace ) is correlated to the fin diameter, mass flow rate, tube wall temperature and the solidification time interval by:

Vinter f ace = 6.92 × 10−3 D0.48 M0.21 |Tw |1.11t −0.31

(24)

The correlation of the complete solidification time (τ ) based on experimental data is determined in a similar way as:

τ = 12.5D0.12 M−0.197 |Tw |1.29

Fig. 28. Comparison of the numerical predictions with the measured stored energy for fin diameter of 60 mm.

Fig. 29 shows a comparison between the numerical predictions and the experimental results for the case of finless tube submersed in liquid PCM. Fig. 29a shows the variation of the interface position with time. As can be seen the comparison shows a deviation of about 11.45%. The predicted interface velocity is compared with the experimental results in Fig. 29b, where a deviation of about 14.4% is found. 4.4. Development of the correlations In order to facilitate the use of the experimental results for predictions and quick estimates of latent heat storage performance systems, a set of correlations was developed for some of the important phase change parameters such as the interface position which permits calculating the solidified mass and consequently latent heat stored, the interface velocity and total solidification time which allow calculating the heat transfer rate and the total time required for fully charging the latent storage tank. The development of the correlations is based on the tube wall temperature Tw (°C), the mass flow rate of the cooling fluid M(kg/s), the time t(s) and D (mm) is the diameter of fin as the independent

(25)

To validate the developed correlations, different sets of experiments were used for this purpose. The parameters of the experimental tests were inserted in the correlations and the predicted results were compared with the real experiments as shown in Figs. 30–32. Fig. 30 shows a comparison of the experimental interface position with the numerical prediction from the correlation. As can be seen, the agreement is good showing a maximum deviation of about 4%, indicating that the correlation is adequate for predicting the PCM interface position. To validate the correlation for the interface velocity a different set of experimental parameters is used to generate Fig. 31. The results indicate that the agreement is good with a maximum deviation of 7%. The noticeable diviation at high tube wall temperatures can be attributed to possible measurement errors not accounted for in the simulations. The correlation for the complete solidification time is validated by comparison with independent experimental results as in Fig. 32. As can be seen the agreement is good showing a maximum deviation of about 1.03%. 5. Conclusions The thermal model based on the enthalpy method for the solidification of PCM around a radial finned tube is used to construct a home-built numerical code, which is optimized, validated and used to investigate the effects of the tube wall temperature, the diameter, number, and material of the fin on the interface position, the interface velocity, the solidified mass and the time for complete phase change. The detailed experimental investigation of the geometrical and working parameters on the process of phase change around finned tube allowed quantifying the influence of the fin area, the most effective fin diameter and the tube wall temperature. These experimental data can be useful for the validation of numerical results and codes and can be helpful for storage designers. The results show that the fin diameter increases the interface position, velocity of the interface, solidified mass and reduces the time for complete solidification. Also the results show that there is

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

15

Fig. 29. Comparison of the numerical predictions with the measurements for bare tube.(a) Variation of the interface position for a bare tube.(b) Variation of the interface velocity for a bare tube.

an optimum fin diameter beyond which there is no improvement of the thermal performance of finned tubes. The reduction of the tube wall temperature has significant effect on the phase change parameters due to the increased temperature gradient, which enhances the phase change process increasing the interface velocity, and reducing the time for complete phase change.

The high mass flow rate enhances the interface position and velocity and reduces the time for complete phase change but the effects are not as much as those, due to the reduction of the wall temperature. The proposed correlations for the interface position, interface velocity and the complete phase change time seem to agree well with experimental results within maximum error of 4%, 7% and

16

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485

1.03%, respectively. Hence, these correlations can be used for quick estimates of solidification of PCM around radial finned tubes and also for design purposes. Declaration of Competing Interest The authors declare that there is no conflict of interest. CRediT authorship contribution statement Felipe S dos Santos: Formal analysis, Data curation. Kamal A.R. Ismail: Methodology, Project administration, Writing - original draft. Fatima A.M. Lino: Validation, Writing - review & editing. Ahmad Arabkoohsar: Writing - review & editing. Taynara G.S. Lago: Data curation. Acknowledgments

Fig. 30. Comparison of the correlation predictions with the experimental results: variation of the interface position with the tube wall temperature.

The second author wishes to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the PQ Research Grant 304372/2016-1. The last author wishes to thank CNPq for the doctorate scholarship. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer. 2020.119485. References

Fig. 31. Comparison of the correlation predictions with the experimental results: variation of the interface velocity with the tube wall temperature.

Fig. 32. Comparison of the correlation predictions with the experimental results: variation of the time for complete solidification with the mass flow rate.

[1] L. Liu, D. Su, Y. Tang, G. Fang, Thermal conductivity enhancement of phase change materials for thermal energy storage: a review, Renew. Sustai. Energy Rev. 62 (2016) 305–317. [2] N.I. Ibrahim, F.A. Al-Sulaiman, S. Rahman, B.S. Yilbas, A.Z. Sahin, Heat transfer enhancement of phase change materials for thermal energy storage applications: a critical review, Renew. Sustai. Energy Rev. 74 (2017) 26–50. [3] L. Fan, J.M. Khodadadi, Thermal conductivity enhancement of phase change materials for thermal energy storage: a review, Renew. Sustain. Energy Rev. 15 (2011) 24–46. [4] M.M. Joybari, F. Haghighat, J. Moffat, P. Sr, Heat and cold storage using phase change materials in domestic refrigeration systems: the state-of-the-art review, Energy Build. 106 (2015) 111–124. [5] A.M. Hayder, A.B. Sapit, Q.A. Abed, Review of solar thermal storage techniques, J. Eng. Appl. Sci. 12 (21) (2017) 6103–6119. [6] J.P. da Cunha, P. Eames, Thermal energy storage for low and medium temperature applications using phase change materials – A review, Appl. Energy 177 (2016) 227–238. [7] N.S. Dhaidan, J.M. Khodadadi, Improved performance of latent heat energy storage systems utilizing high thermal conductivity fins: a review, J. Renew. Sustain. Energy 9 (2017) 034103-1 –034103-43. [8] A. Erek, Z. Ilken, M.A. Acar, Experimental and numerical investigation of thermal energy storage with a finned tube, Int. J. Energy Res. 29 (2005) 283–301. [9] M. Rahimi, A.A. Ranjbar, D.D. Ganji, K. Sedighi, M.J. Hosseini, Experimental Investigation of Phase Change inside a Finned-Tube Heat Exchanger, Hindawi Publishing Corporation, J. Eng. 11 (2014) 641954 Article ID. [10] M.J. Hosseini, M. Rahimi, R. Bahrampoury, Thermal analysis of PCM containing heat exchanger enhanced with normal annular fins, Mech. Sci. 6 (2015) 221–234. [11] F.H M.M.Joybari, A.A.Al-A S.Seddegh, Heat transfer enhancement of phase change materials by fins under simultaneous charging and discharging, Energy Convers. Manage. 152 (2017) 136–156. [12] N.H.S. Tay, F. Bruno, M. Belusko, A. Castell, L.F. Cabeza, Experimental validation of a CFD model on a vertical finned tube heat exchanger phase change thermal energy storage system, proceedings of the 12th international conference on energy storage, Innostock (2012). [13] A.J. Parry, P.C. Eames, F.B. Agyenim, Modeling of thermal energy storage shell-and-tube heat exchanger, Heat Transfer Eng. 35 (1) (2014) 1–14. [14] A.H. Mosaffa, F. Talati, M.A. Rosen, H.B. Tabrizi, Approximate analytical model for PCM solidification in a rectangular finned container with convective cooling boundaries, Int. Commun. Heat Mass Transfer 39 (2012) 318–324. [15] M.D. Muhammad, O. Badr, H. Yeung, Validation of a CFD melting and solidification model for phase change in vertical cylinders, Heat Transfer, Part A 68 (5) (2014) 501–511. [16] A.H.B F.Tahrour, C.B M.Aksas, 3-D numerical study and comparison of eccentric and concentric annular-finned tube heat exchangers, J. Eng. Sci. Technol. 10 (11) (2015) 1508–1524.

F.S. dos Santos, K.A.R. Ismail and F.A.M. Lino et al. / International Journal of Heat and Mass Transfer 152 (2020) 119485 [17] I. Jmal, M. Baccar, Numerical study of PCM solidification in a finned tube thermal storage including natural convection, Appl. Therm. Eng. 84 (2015) 320–330. [18] A.M. Abdulateef, J. Abdulateef, S. Mat, K. Sopian, B. Elhub, M.A. Mussa, Experimental and numerical study of solidifying phase-change material in a triplex– tube heat exchanger with longitudinal/triangular fins, Int. Commun. Heat Mass Transfer 90 (2018) 73–84. [19] H. Niyas, S. Prasad, P. Muthukumar, Performance investigation of a lab–scale latent heat storage prototype – Numerical results, Energy Convers. Manage. 135 (2017) 188–199 a). [20] M.D. Muhammad, O. Badr, Performance of a finned, latent-heat storage system for high temperature applications, Appl. Therm. Eng. 116 (2017) 799–810. [21] H. Niyas, C.R.C. Rao, P. Muthukumar, Performance investigation of a lab-scale latent heat storage prototype – Experimental results, Sol. Energy 155 (2017) 971–984 b). [22] H. Jannesari, N. Abdollahi, Experimental and numerical study of thin ring and annular fin effects on improving the ice formation in ice-on-coil thermal storage systems, Appl. Energy 189 (2017) 369–384. [23] M.J. Hosseini, A.A. Ranjbar, M. Rahimi, R. Bahrampoury, Experimental and numerical evaluation of longitudinally finned latentheat thermal storage systems, Energy Build. 99 (2015) 263–272. [24] M.K. Rathod, J. Banerjee, Thermal performance enhancement of shell and tube Latent Heat Storage Unit using longitudinal fins, Appl. Therm. Eng. 75 (2015) 1084–1092. [25] N. Kayansayan, M.A. Acar, Ice formation around a finned-tube heat exchanger for cold thermal energy storage, Int. J. Therm. Sci. 45 (2006) 405–418. [26] D. Liu, C. Yang, H. Yang, Experimental and numerical study of a tube-fin cool storage heat exchanger, Appl. Therm. Eng. 149 (2019) 712–722. [27] M. Yaghoubi, M. Mahdavi, An investigation of natural convection heat transfer from a horizontal cooled finned tube, Exp. Heat Transfer 26 (2013) 343–359.

17

[28] H.M.S. Al-Sarrach, M.A.S G.Y.Kahwaji, Modeling coupled conduction–convection ice formation on horizontal axially finned and unfinned tubes, J. Fluids Eng. 139 (2017) 1–13. [29] A. Abdi, V. Martin, J.N.W. Chiu, Numerical investigation of melting in a cavity with vertically oriented fins, Appl. Energy 235 (1) (2019) 1027–1040. [30] M. Lacroix, M. Benmadda, Numerical simulation of natural convection-dominated melting and solidification from a finned vertical wall, Numer. Heat Transfer, Part A Appl. 31 (1) (1997) 71–86. [31] P.V. Padmanabhan, M.V.K. Murthy, Outward phase change in a cylindrical annulus with axial fins on the inner tube, Int. J. Heat Mass Transfer 29 (12) (1986) 1855–1868. [32] A.H. Mosaffa, F. Talati, H. Basirat Tabrizi, M.A. Rosen, Analytical modeling of PCM solidification in a shell and tube finned thermal torage for air conditioning systems, Energy Build. 49 (2012) 356–361. [33] T. Bauer, Approximate analytical solutions for the solidification of PCMs in fin geometries using effective thermophysical properties, Int. J. Heat Mass Transfer 54 (23–24) (2011) 4923–4930. [34] C. Bonacina, G. Comini, A. Fasano, M. Primicerio, Numerical solution of phase change problems, Int. J. Heat Mass Transfer 16 (10) (1973) 1825–1832. [35] M.N. Ozisik, Heat Conduction, 2nd edition, Wiley Interscience, 1993. [36] F.S. Santos, Numerical and experimental investigation of heat transfer with phase change in radially finned tubes, MSc thesis, State University of Campinas, Brazil, 2018 (in Portuguese). [37] C.R.E.S. Nóbrega, Numerical and experimental study of phase change around axial finned tubes for application in latent heat storage systems, PhD in Mechanical Engineering, State University of Campinas, Brazil, 2019 (in Portuguese).