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Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model
Accepted Manuscript Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model R. Pakrouh, M.J. Hosseini, A.A. Ranjbar, R. Bahrampoury PII:
S0360-5442(17)31433-0
DOI:
10.1016/j.energy.2017.08.055
Reference:
EGY 11424
To appear in:
Energy
Please cite this article as: R.Pakrouh, M.J.Hosseini, A.A.Ranjbar, R.Bahrampoury, Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model, Energy (2017), doi: 10.1016/j.energy.2017.08.054 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model R. Pakrouha, M.J. Hosseinib,*, A.A. Ranjbara, R. Bahrampouryc a
School of Mechanical Engineering, Babol University of Technology, POB 484, Babol, Iran
c
Department of Mechanical Engineering, Golestan University, P. O. Box 155, Gorgan, Iran
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b
Department of Mechanical Engineering, K.N.Toosi University of Technology, Tehran, Iran
Abstract
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The present paper numerically investigates the performance of latent heat storage systems during solidification, which involves phase change material (PCM) capsules. Paraffin wax is
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considered as the PCM and water plays the role of heat transfer fluid (HTF). The simulation is conducted for two inlet temperatures, 30 °C and 40 °C, while the capsules’ diameter varies in the range of 10 mm to 60 mm. Among various existing models, the effective packed bed model which is not only able to provide temperature gradient data but also is capable of
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reporting entropy generation details. Results indicated that both reduction in capsules’ diameter and the HTF inlet temperature unfavorably increase the amount of the system’s irreversibility. However, it is demonstrated that the increase in the irreversibility does not
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essentially result in a reduction in performance of the latent heat storage system. In other words, the efficiency of the storage system is not a pure function of entropy generation
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number. In fact, the reduction in the diameter results in an improve in the second low efficiency while the inlet temperature reduction diminishes the efficiency. Results also implies that decisive parameters vary significantly only when the diameter reduces to 20 mm and further reduction doesn’t affect the system performance noticeably. Keywords: PCM, Entropy generation, Exergy analysis, Packed bed storage
*
Corresponding Author: (Seiyed Mohammad Javad Hosseini) Department of Mechanical Engineering, Golestan University, P.O. Box 155, Gorgan, Iran. Tel/Fax: +98 17 32440206, Email: [email protected]
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ACCEPTED MANUSCRIPT 1. Introduction The use of phase change materials (PCMs) in thermal energy storage (TES) systems is an attractive subject for researchers. Therefore, a large number of papers in this field has been published during the last few decades. Due to attractive characteristics of PCMs, such as
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large latent heat of fusion, constant phase change temperature and chemical stability, these materials find diverse applications like electronic cooling [1,2], air conditioning [3,4], smart textiles [5], biomedical [6] and solar energy storages [7,8]. PCM properties are subjected to
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small variation when is used repeatedly. The extent of the variation is dependent to the type of the PCM and the number of operation cycles. Dheep and Sreekumar [9] studied the effect
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of the number of the cycles on the properties of benzamide acid and sebacic acid. Results indicated that after 1000 cycles, the phase change temperatures change 0.02 and 0.85 percent. Among available geometries of latent heat thermal storage (LHTS) systems, packed bed storage systems with spherical capsules has been declared to be one of the most conventional
construction.
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and advantageous due to its large heat transfer area in a small volume and simple
Apparently the idea that the main objective of a thermal energy storage system is to store
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thermodynamic availability (exergy) and not to store energy is attributed to Bejan [10].
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Hence part of the studies is dedicated to thermodynamically investigate sensible as well as latent heat thermal storage systems. Jegadheeswaran et al. [11] presented a detailed review on different techniques used for exergy analysis of LHTS units. Li [12] performed a comprehensive review, which investigates the effect of different operating conditions and design parameters such as the HTF inlet temperature, HTF mass flow rate, reference temperature and PCM melting temperature on energy and exergy performance of the LHTS systems. MacPhee et al. [13] investigated the solidification process in an encapsulated ice TES unit numerically. They studied energy and exergy efficiencies for various inlet HTF
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ACCEPTED MANUSCRIPT temperatures, HTF flow rates and capsule geometries (spherical, cylindrical and slab). Their results showed that the energy analysis does not provide the sufficiently capable tool for assessment of the system’s performance, while the result, based on second law analysis, is found to be more reliable. In addition, they concluded that smaller amount of irreversibility
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and larger exergy efficiency is achievable as the HTF temperature approaches the solidification temperature or when the flow rate increases. Guelpa et al. [14] presented a detailed entropy generation analysis for solidification process in a radially finned shell and
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tube heat storage unit for the first time. They showed that second law analysis can be used as an effective tool for improving the design of LHTS systems. Recently Pizzolato et al. [15]
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employed the concept of the exergy destruction as a design tool for improving a thermocline TES system. They introduced the lifespan and the characteristic time of entropy generation for their case. They found that by inclusion of a porous layer and a solid baffle, the entropy generation can be reduced about 7% and 5% respectively. Mosaffa and Garousi [16]
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performed an exergoeconomic and environmental analysis on a latent heat thermal storage coupled with an air conditioning system. The effect of various operational parameters on charging and discharging processes of different PCMs is investigated and finally optimized
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from exergoeconomic and environmental viewpoints. Mosaffa et al. [17] implemented an
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advanced exergy analysis for a combination of LHTS unit and VCR (vapor compression refrigeration) as an air conditioning system. Their results indicated that the total rate of exergy destruction of an LHTS unit is an internal matter that is due to its irreversibilities. Rezaie et al. [18] employed the exergy analysis to investigate the function of the TES in a Friedrichshafen district energy (DE) system. They found that the overall energy and exergy efficiencies of the TES system are 60% and 19% less, respectively. They also showed that the energy efficiency of the TES system follows the same trend as the TES temperature. It is shown that the trends of exergy accumulation and exergy efficiency of the TES system are
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ACCEPTED MANUSCRIPT similar. Bindra et al. [19] developed a numerical heat transfer model, which takes the effects of energy storage density, heat losses and axial dispersion into account to investigate the exergetic performance of packed bed thermal storage systems. As shown in their work, as the heat losses to the surrounding or axial dispersion lowers the exergy efficiency rises. Bindra et
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al. [20] introduced the sliding flow method (SFM) to calculate the exergy losses that is due to the fluid pressure drop in a packed bed storage system. The results indicated that exergy destruction caused by the pressure drop cannot be passed up since for all possible packed bed
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storage systems the value of this parameter is in the range of 2-10%.
Since packed bed heat exchangers are known as efficient ones, implementation of a decision-
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making criterion based on second law of thermodynamics can be considered as usable investigation, due to frequent applications of heat storage systems in energy systems; simulation of such component through an efficient method, which provides entropy details, gives an appropriate tool for engineers to make the best decision. For investigation of the
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dynamic and thermal behavior of packed bed LHTS systems, some validated mathematical models can be found in the literature such as single-phase model, concentric dispersion model and continuous solid phase model. However, due to the complexity of PCM spheres
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arrangement, previous models are not capable to include some of the most important features
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such as size and arrangement of the spheres, shape of the flow passage, the flow field and thermal gradient inside the PCM spheres in detail. Therefore, previous works were not able to generate local distribution of entropy generation of latent energy storage systems. Effective packed bed model which has been introduced by Xia et al. [21] is able to overcome these limitations. In this paper, a CFD model has been developed based on the effective packed bed model that makes the calculation of local irreversibilities during solidification possible. In fact, the model has been used in the present study that is capable of providing entropy generation details in the packed bed system unlike previous studies.
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ACCEPTED MANUSCRIPT 2. Mathematical model 2.1. System description and governing equations Due to harmful consequences of fossil fuels emission to the atmosphere and since these sources of energy are going to deplete in next future decades, researchers are studying on
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utilization of renewable energy sources. Generally speaking, renewable energy sources are not available steadily during the day. For example at night, solar energy is not accessible, therefore heat storage systems can be employed to store the energy when it is present and
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release the energy to the system when the renewable energy is unavailable.
A schematic view of the PCM-based packed bed thermal storage system is shown in figure 1.
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The storage system is a cylinder that has a length of 850 mm and a diameter of 400 mm. The cylinder is packed with spheres of varying diameter, d, which provides a space for the HTF to pass. Water as the HTF and Paraffin Wax [22] as the PCM are taken into account. The thermophysical properties of the PCM are summarized in Table 1. Considering the reference,
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the data related to viscosity, conductivity, volumetric expansion coefficient are taken from the existing paraffin. However, the properties like thermal capacity and latent heat capacity are obtained using DSC analysis, the uncertainity of which is about ±3%.
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Since the investigated process is solidification, the whole storage system is initially kept at a
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homogenous temperature, T , above the phase change point. The current model can also be applied for non-homogenous initial conditions. The later condition happens when the solidification process initiates right after completion of the melting process. In this condition, although whole PCM is melt, there is no uniform temperature distribution and there yet exists temperature gradient. But in this study, since we concentrate on solidification, it is assumed that the melting process continues until the whole system approach the same temperature. The HTF at a temperature below the phase change point, T , enters the system at the bottom of the vertical cylinder, passes through the empty spaces left among the spheres and
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ACCEPTED MANUSCRIPT leaves the unit when it reaches the top end. This study focuses on the contribution of inlet temperature and spheres diameter variations on the performance of the system from thermodynamic point of view. Table 2 contains the working conditions and the range variation of the parameters investigated in this study.
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As mentioned above, the effective packed bed model [21] is implemented to simulate the phase change phenomenon. The model is able to display the temperature gradient in the PCM spheres as well as the HTF. This capability, makes it possible to calculate the amount of
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entropy generation in the packed bed LHTS. Figure 2 shows the computational domain selected for the current packed bed storage system, which is based on the effective packed
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bed model.
The basis of the effective packed bed model is on transformation of the three dimensional actual model to an effective two dimensional model. The transformation of these two should be conducted in a way that the void fraction remains the same. The amount of porosity in
ε
d S n ∙ π ∙ (2) =1− = S D∙H
4 d ∀ n ∙ 3 ∙ π ∙ (2) =1− = D ∀ π ∙ ( ) ∙ H 2
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ε
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both two and three dimensional scales are given below:
(1)
(2)
• •
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The assumptions considered in the simulation are as below: The model is solved for 2D and axisymmetric conditions.
Although the consequence of natural convection is subtle, the Boussinesq
approximation is included for a more accurate simulation. •
Void fraction (the ratio of empty spaces volume to the total container’s volume) is
assumed constant whose value is 0.36. •
The consequences of the capsule wall thermal resistance is neglected
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The flow of the HTF is assumed to be laminar and incompressible
•
Heat dissipation to the surrounding is neglected since the cylinder’s wall is assumed
to be a perfect insulation. •
The heat transfer among the capsules are disregarded because the heat transfer is not
•
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possible via point by point contacts.
The direct conversion of the three dimensional model to the two dimensional one
leads to connection of the capsules which blocks the flow of the HTF. Therefore, to
•
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overcome the restriction a small distance is considered among the capsules. The capsules are not floating and are fixed in their position.
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Considering the figure 2, the boundary condition equations are as below: Outflow boundary condition at the outlet: ∂V$ ∂T = 0 , =0 ∂y ∂y
(3)
Adiabatic condition at the external wall of the cylinder.
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∂T ∂T = = 0 , V$ = V& = 0 ∂y ∂x
(4)
For the HTF:
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Continuity:
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Considering the above assumptions, the governing equations on the HTF and the PCM are:
∇. )V* = 0
(5)
Momentum:
∂ )*. + ∇. ,ρV )*)V*. = −∇P + μ∇ )V* + ρg ,ρV )* ∂t
(6)
Energy: ∂T k )*. ∇T = ∇ ( +V T) ∂t ρC5
(7) 7
ACCEPTED MANUSCRIPT For the PCM: The conventional enthalpy-porosity method [23] is employed to solve the phase change problem. The governing equations for the PCM are:
Momentum: ∂ )*. + ∇. ,ρV )*)V*. = −∇P + μ∇ )V* + ρg ,ρV )*β(T − T7 ) + )*S ∂t
∂ )*ħ. = ∇. (k∇T) (ρħ) + ∇. ,ρV ∂t
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Energy:
(8)
(9)
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)* = 0 ∇. V
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Continuity:
(10)
The term )S* in the equation (9) is the source term, which is defined as below: )*S =
(1 − λ) )V* A λ + 0.001 ;<=>
(11)
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Where λ, is the liquid fraction, and the summation of the constant value at the denominator, 0.001, is considered to prevent zero value at the denominator. Amushy is the constant of the mushy zone and determines the extend of damping. Large values of the term results in higher
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slope of damping and very large values leads to some fluctuation in the solution. In the
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current study the value of this parameter is 105 that is determined via preliminary simulations. Considering the equation (10), total enthalpy of the material is the summation of sensible and latent enthalpies. A
ħ = h7 + @ C5 dT + λLD
(12)
AB
where λ is the liquid fraction whose value is in the range of 0 and 1. In this study, only one temperature is considered for the melting point.
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(13)
2.2. Numerical procedure and validation In this study, the simulation is conducted via a 2D in-house code [24]. The SIMPLE
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algorithm is employed to couple pressure and velocity in one equation. In order to solve the pressure correction equation, the presto scheme is taken into account. The QUICK differencing scheme is employed to discretize the energy and the momentum equations. The
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consequences of grid size are examined at initial stages of the simulation. The careful examination revealed that optimum value for the grid size is 0.002 m when the capsules’
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diameter is in the range of 15 mm-60 mm. However, it is found that when the diameter reduces to 10 mm, the optimum value becomes 0.0015 m. Similar examinations are also conducted to evaluate the optimum time step, which resulted in 0.01 s as the optimum step. Considering the convergence tolerance of 10IJ for the continuity and the momentum
every step.
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equations and 10IK for the energy equation, the convergence of the solution is examined for
The validation of effective packed bed model is extensively targeted by Xia et al [21] and
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Bellan et al [25]. The implemented code has also been validated in our previous publications
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[1, 24, 26]. Here, in order to assure about the accuracy of the implemented code for the effective pack bed model, a new validation based on another experimental results of Nallusamy [27] is presented. Figure 3 shows the PCM temperature at a specific height of the heat exchanger during the melting process. The diameter of studied tank is of 360 mm and its height is 460 mm. The number of the capsules of 55 mm diameter is 264. The capsules are uniformly arranged in 8 rows. The melting point of the PCM is 60±1 °C and the initial temperature is 32 °C. Water at 70 °C enter the tank at its top inlet. 9
ACCEPTED MANUSCRIPT 2.3. Formulations for energy and exergy analyses In order to evaluate the performance of a unit from energy point of view, different parameters have been defined. In this study, the heat release ratio, which is the ratio of the amount of released heat during the solidification process to the maximum heat gain capacity, is the
below [28]: η=
QN= Q;&
in which
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Q;& = M PC5, ,T − T5 . + LD + C5,= ,T5 − T .Q
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performance criterion. The corresponding correlations for calculating the ratio is brought
(14)
(15)
Another criterion is also included in this study, which considers latent heat. This decisive parameter is the latent heat ratio, which is the ratio of the amount of latent heat gain to the
ϕ=
Q QN=
in which
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Q = M (1 − λ)LD
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total heat release until the moment of consideration
(16)
(17)
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Contrary to the first law of thermodynamics, the second low assesses the quality of energy. Therefore, new concepts such as entropy and exergy are proposed. Based on non-equilibrium thermodynamics, the local entropy generation can be calculated using the following equation [29]: ρ
Ds = −∇. σ )* + SUVW Dt
(18)
Where σ is the entropy flux vector and SUVW is the rate of entropy generation per unit volume. The sources of the irreversibility in this study is mainly due to temperature gradient and
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ACCEPTED MANUSCRIPT viscosity. Therefore, the total rate of entropy generation can be estimated using the following equation: SUVW = SU>W + SUDW
(19)
Equation (19) can also be rewritten as following [30]: −J)))*Y . ∇T ∆: τ + T T
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SUVW =
(20)
where first and second terms on the right side of the equation are the rates of entropy
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generation per unit volume due to heat transfer and friction respectively which are calculated using the equations below. k ∂T ∂T ]^ _ + ^ _ ` T ∂x ∂y
SUDW =
2μ ∂u ∂v 1 ∂u ∂v ]^ _ + ^ _ + ^ + _ ` T ∂x ∂y 2 ∂y ∂x
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SU>W =
(21)
(22)
SUV = @ SUVW dV
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Rate of entropy generation is calculated by integrating of the SUVW over the volume. (23)
In order to estimate the magnitude of the terms included in equation (20), the following
SU>
SU> + SUD
(24)
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Be =
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equation is proposed [31]:
The dimensionless parameter Be is the Bejan number whose value is in the range of 0 to 1. In case of Be ≫ 0.5, the majority of the irreversibility is due to heat transfer and when the number is far below 0.5, friction constitutes a larger
portion of irreversibility. The
contribution of the two agents are almost equal when the Bejan number is around 0.5. Another concept, which is defined as the maximum work that a system can generate to reach the environment equilibrium state, is exergy. Contrary to energy, exergy is not conservative
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ACCEPTED MANUSCRIPT and can be destroyed by irreversibility. Therefore, the exergy balance for an adiabatic system may be written as below [32]: Exergy input − Exergy output = Exergy destroyed
(25)
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Second low efficiency or exergy efficiency is the criterion to assess the consequences of irreversibility on the system [32]. ExUi= ѱ(t) = 1 − ] ` ExU
(26)
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where ExUi= is the rate of exergy destruction that can be calculated using the Gouy-Stodola theorem as below [29]:
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ExUi= = T7 SUV
(27)
The quotient of the division of the equation (26), ExU , which is the rate of exergy input to the unit, is calculated using the following equation [14]: ExU = Qj . ^1 −
T7 _ T
(28)
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where Qj is the heat flux that passes through the PCM boundaries. The division enclosed by the brackets in equation (26) is called the entropy generation
ѱ(t) = 1 − N=
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number. Therefore the equation can be rewritten as below: (29)
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The above equation declares that by reducing the entropy generation number, the system’s efficiency improves. Therefore, entropy generation minimization has attracted scientists attention in optimization and improving energy systems. 3. Results As stated, irreversibility generally consists of two main sources of heat transfer and friction. Therefore, the contribution of each of these two on the entropy generation has to be determined. According to [14], the portion of friction is negligible in comparison with that of
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ACCEPTED MANUSCRIPT heat transfer. Figure 4 in which the significance and the trend of variation of each of these sources are shown approves the claim. According to the figure, the magnitude of the friction based entropy generation is in the order of 10Il that is negligible regarding the heat transfer based entropy. Therefore, the Bejan number is very close to 1 and the friction based entropy
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generation can be simply disregarded without adding any source of approximation. 3.1. Energy analysis
Figure 5a shows the total solidification time versus the capsules’ diameter for two different
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HTF inlet temperatures. According to the figure, although the amount of the PCM remains constant for all the cases, the total solidification time reduces significantly, as the spheres
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become smaller. The extent of this reduction is so significant that for the inlet temperature of 40 °C, the diameter reduction from 60 mm to 10 mm results in a solidification time decrease of 76.4 percent. This improvement is due to the consequent addition of 524 percent heat transfer surface. As the surface to volume ratio increases, the rate of heat transfer between the
shortens considerably.
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two mediums, the PCM and the HTF, promotes. As a result, the total solidification time
It is worth noting that the considerable reduction of solidification time as the diameter
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reduces may change the applications of the system; the application of the case that is able to
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release higher rate of heat in short time may differ from a heat storage unit that is designed to give less rate of heat but for longer period. The optimization of the operating time according to the application is another issue. In this study, the consequences of the parameters are considered only from thermodynamic point of view. Figure 5b shows the solid percentage versus time for different capsules’ diameters. The solid percentage refers to the average solidified PCM in the whole PCM. As the process progresses, the smaller capsules will lead to faster pace of solidification which is predictable. In fact, as the capsules’ diameter reduces, the heat transfer area increases which results in
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ACCEPTED MANUSCRIPT higher rate of solidification. The interesting point is that, for the diameter less than 20 mm, the solidification curves almost match on one another until 80 percent of solidification. The effect of sphere diameter variation on the heat flux and the heat release ratio is exhibited in figure 6(a) and 6(b) respectively. According to the figure, as the HTF enters the heat
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exchanger, due to the large temperature difference between the two mediums, the rate of heat transfer is large. The ascending trend in the heat flux is due to the inclusion of convective heat transfer to the initially dominant conduction heat transfer within the capsules. As the
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process progresses, it can be seen that the heat flux after reaching a maximum value reduces because the two medium temperature difference declines. As stated above, despite the
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amount of the PCM is constant, the heat transfer area increases as the diameter shrinks. Based on the Fourier equation, the increase in the area leads to higher rate of heat transfer. Considering figure 6(b), regardless of the ending stage of the solidification, it is apparent that the reduction in the diameter leads to higher rate of heat release of the PCM. It is noteworthy
ratio considerably. 3.2. Latent heat analysis
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that as the diameter undershoots 20 mm, extra reduction does not influence the heat release
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Figure 7 shows the latent heat ratio versus the solid percentage. According to the figure 7(a),
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regardless of the diameter, the ratio increases steeply at initial stages of the process when the first pieces of the solid are created. The increasing trend pursues until a maximum value after which a reduction trend appears that finishes to a final value. This final value is the ratio of the maximum latent heat to the heat released from the PCM up to the moment under consideration. The final values of the curves are close to each other (in range of 0.85 to 0.86). Since the ratio of the maximum latent heat to the thermal capacity of the system heat release is equal to 0.84, the obtained results show that, for the solidification process, all the units regardless of the diameter reach to the final point of heat release.
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ACCEPTED MANUSCRIPT According to the equation (16), the ratio of the latent heat to the total released heat is obtained. Therefore it is difficult to clearly separate these two by the available results and analyze the required parameter as the capsule’s size varies. Considering figure 7a, it can be
heat affects the total heat ratio more considerably.
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figured out that, until 0.2 of the solidification, as the diameter decreases, the released latent
Another interesting point is that the trend of the ratio is not identical for different diameters as the solid percentage varies. Considering the figure, as the diameter reduces, the latent heat
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ratio increases as the spheres become smaller, at the first 20 percent of the solidification process. After the point, the ratio reduces when the diameter of the capsules diminishes.
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According to the figure 7b, the same trend of variation for the dimeter variation is also observable for HTF inlet temperature variation. Until the solidification percentage of 0.2, the ratio of the latent heat for the inlet temperature of 30 °C is more. Therefore, during this stage, as the inlet temperature lowers, the latent heat affects the latent heat ratio more.
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As a specified amount of a PCM releases a determined maximum latent heat, it is expected that as the inlet temperature of the HTF reduces, the value of the latent heat ratio descends. According to figure 7b, as the HTF inlet temperature redcues from 40 °C to 30 °C, a
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reduction of 8 percent at the end of the process is observed. Therefore, the less the
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temperature difference between the inlet HTF and the initial temperature, the more acceptable latent heat performance is achievable. 3.3. Second law analysis
Figure 8 shows the outlet HTF temperature variation versus time and the solidification percentage. The ending of the curves is related to the completion of the solidification process. According to the figure, when all the PCM solidifies, the outlet temperature doesn’t approach the inlet temperature. In fact since there yet exists temperature gradient in the PCM, heat transfer continues even after solidification completion. According to the figure, at the initial
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ACCEPTED MANUSCRIPT stages of the process, the outlet temperature of the HTF reduces drastically. As the process progresses, the reduction persists with a less slope. In fact, the controlling factor of the outlet temperature is the interaction between the HTF and the PCM. This concept is better understood by figure 8b. According to the figure, for the 20 mm dimeter case and the cases of
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less diameters, the outlet temperature keeps constant for a long period and about the phase change temperature. As explained before, as the capsules’ diameter reduces, the heat transfer area increases and heat is absorbed by the HTF in a more uniform trend. In fact, when
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considering each of the cases during their solidification period, it becomes clear that the case of smaller capsule sizes result in more uniform and controllable outlet temperature.
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Figure 9a shows the PCM temperature variation during the whole solidification process for three point p1 (the lowest row) p2 (the middle row) and p3 (the highest row). According to the figure, the temperature variation for the both cases follow the same trend. Accordingly, when the heat transfer fluid reaches to the corresponding capsule, the temperature of it
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reduces and when it reaches to the solidification temperature, the temperature after the moment keeps constant for a considerable portion of time. After releasing all the latent heat,
the HTF.
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the temperature reduction continues until the approach of the point to the inlet temperature of
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Figure 9b shows the temperature variation of the central point of the capsules on the axis of the cylinder for different heights. According to the figure, during the first half of the process, the point of the 0.2 height of is only subjected to the inlet temperature variation and the point above the height are within melting range. As the process continues, the temperature gradient intensifies and after 350 minute, the temperature of the lower parts approaches the inlet HTF temperature and the temperature gradient decreases. Since the rate of entropy generation is dependent to the temperature value and its gradient (according to the equation 20), a comprehensive understanding of the entropy distribution is
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ACCEPTED MANUSCRIPT possible only when the temperature distribution is studied. The temperature distribution and the rate of entropy generation caused by heat transfer for the 60 mm sphere diameter are shown in figure 10. At initial stages, the maximum entropy generation is asigned to the thin layer of the solid PCM and the HTF at the tube boundaries. The superiority of these regions is
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related to their intense temperature gradient. Consequently, the general distribution of the entropy generation is found to be similar to the solidification front. On the other hand, the regions in which PCM melt is present have the least values of rate of entropy generation due
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to their higher temperature and less temperature gradient. As the process advances eg. at 10000th and 20000th seconds, the entropy distributes within the spheres while their intensity
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lowers. This trend results in a decrease in temperature gradient and a subsequent reduction in the rate of entropy generation. In fact, as time passes, the temperature of the spheres at the bottom of the container approaches to the dead state.
In order to explain how the entropy is distrubuted in the container, it is essential to pay
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attention to the corresponding equation (equation 21) in which it is stated that the rate of entropy genration is in a direct relation with square of the temperature gradient and has an inverse relationship with square of the temperature. However, since the range of temperatures
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distribution.
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is finite, the temperature gradient plays a more important role in entropy generation and its
The temperature contours and entropy generation distribution for 30 mm and 60 mm diameters are shown in figures 11 and 12. As mentioned in the description of figure 5, as the capsules’ diameter reduces, the heat transfer surface between the two mediums, the PCM and the HTF, increases. Therefore, as the surface to volume ratio increases, the rate of heat transfer and the temperature gradient rise. Thus, as the sphere diameter drops, the rate of entropy generation in the system increases which is also confirmed by figure 12. However, since the total solidification time of the 30 mm case is less, the PCM temperature approaches
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ACCEPTED MANUSCRIPT the fluid temperature earlier. Therefore, after 8000 seconds, the rate of entropy generation reduces in a trend that the 30 mm- case reaches to the dead state before the container filled with 60 mm spheres. The distribution of the rate of entropy generation for HTF inlet temperatures of 40 °C and 30
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°C is shown in figure 13. As mentioned above, the rate of the entropy generation is directly proportional to the square of the temperature gradient and is inversely proportional to the square of the temperature. As the inlet temperature reduces, not only the temperature gradient
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increases but also the maximum accessible temperature reduces. The consequence of these two supporting causes is a high rate of increase in the rate of entropy generation as the inlet
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temperature reduces. According to the figure, the rate of entropy generation of the case of 30 °C is considerably more than the other case until around 14000 s. However, since the inlet temperature reduction leads to shorter complete solidification time, the unit approaches to the corresponding dead state earlier. Therefore, it can be seen that, after about 14000 s the rate of
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entropy generation reduces for the case of 30 °C.
Figure 14 shows the global rate of entropy generation versus the solidification percentage. According to figure 14 (a), as the diameter of the spheres decreases, the rate of entropy
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generation climbs. Considering the figure, the rate of entropy generation generally increases
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at first over solidification percentage. However, the parameter reduces after a peak. To present some details, the trend of variation differs to some extend when the diameter is lower than 20 mm. In fact, for these diameters, the rate of entropy generation almost flattens out in a large range of solidification percentage. The consequence of inlet temperature on the global entropy generation is illustrated in figure 14 (b). As can be seen, as the temperature reduces, the rate of global entropy generation surges. The obtained results confirms the explanations presented for figure 13.
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ACCEPTED MANUSCRIPT The trend of variation of the entropy generation number is shown in figure 15 for different sphere diameters. As can be seen in the figure, for both of the inlet temperatures, the diameter reductions leads to a drop in entropy generation number, which according to the equation (29) results in an improvement in system’s efficiency. In fact, according to the equations (27)
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and (28), the variation of entropy generation is a function of three different parameters, the rate of entropy generation, heat flux and the PCM temperature. Since the rate of entropy generation is not the only factor that determines the efficiency, high rate of entropy
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generation is not essentially a defect of a system. As matter of fact, the promotion in the rate of exergy may be high enough to retrieve the consequences of the increase in entropy
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generation.
Similar deduction has also been presented by Guelpa et al. [14] for the efficiency of a PCM containing shell and tube heat exchanger. Considering the consequences of inlet temperature, it can be concluded that the temperature affects the heat exchanger in a different manner with
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respect to the diameter. In other words, as the HTF inlet temperature drops, the rate of entropy generation increases while the second law efficiency declines. This explanation can be confirmed when considering the figure 16. Figure 16 shows the time average of the rate of
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entropy generation and the second low efficiency. According to the figure, for both of the
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inlet temperatures, the diameter reduction leads to a gain in both the rate of entropy generation and the second low efficiency. In contrary to the diameter variation, for a fix diameter, a reduction in the inlet temperature leads to an increase in the average entropy generation and a reduction in second law efficiency. Moreover, figure 16 confirms that the diameter reduction to 20mm affects the efficiency considerably. However, further reduction does not influence the efficiency sensibly.
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ACCEPTED MANUSCRIPT In fact, it can be mentioned that within the range considered for the capsules’ diameter and the HTF inlet temperature, the system responses to variation of the temperature more pronouncedly from thermodynamic point of view. Another point that the figure reveals is that the effectiveness of the spheres diameter is more
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noticeable for lower inlet temperature. Based on the figure, considering the inlet temperature of 40 °C, the diameter reduction from 60 mm to 10 mm leads to 11 percent improvement in the efficiency. While similar reduction in the size of the spheres, leads to 21 percent
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improvement when the HTF inlet temperature is 30 °C. It can be concluded that the use of smaller capsules is more suitable for the lower inlet temperatures.
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It can also be concluded that, the 20 mm diameter capsule is the most appropriate size. As can be seen in the figure, by reducing the diameter to the values below 20 mm, although the second law efficiency almost keeps unvarying, the rate of entropy generation increases.” 4. Conclusion
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In this paper, the solidification process of a PCM is studied in a packed bed storage system from both first and second laws of thermodynamic viewpoints. In order to illustrate the rate of entropy generation in the system, the effective packed bed model is implemented. This
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model is able to demonstrate the occurring temperature gradients in the heat exchanger and
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therefore is an appropriate tool for evaluating and presenting the entropy distribution in the system. The capsules diameter and the HTF inlet temperature are the considered parameters in this study. Results are summarized as below: •
The diameter reduction leads to an increase in the surface to volume ratio which
results in an enhancement of the rate of heat transfer from the PCM to the HTF. •
A large value for the rate of entropy generation is not essentially equivalent to low
efficiency and the true criterion for the efficiency is the entropy generation number. On this bases a reduction in the spheres diameter inspite of increaseing the rate of entropy generation,
20
ACCEPTED MANUSCRIPT improve the system’s efficiency. Wheras, inlet temperature reduction increases the rate of entropy generation and reduces the system’s efficiency. •
Employing smaller capsules is more effective when the inlet temperature is lower in a
way that reducing the diameter from 60 mm to 10 mm for the inlet temperature of 40 °C
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results in 11 percent improvement in the efficiency while the same reduction in the diameter brings about 21 percent enhancement when the inlet temperature is 30 °C. •
Results revealed that the diameter reduction to 20 mm is considerably effective and
•
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more reduction does not leave a sensible consequence on the system’s performance.
The current study investigates the system from thermodynamic view point.
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Considering the aspect, the 20 mm case is the most advantageous one because as the capsules’ diameter reduces more, the second law efficiency doesn’t increase considerably while the rate of entropy generation still rises. In fact, without a more increment in entropy
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generation, an acceptable efficiency is available if this diameter is chosen.
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ACCEPTED MANUSCRIPT Nomenclatures
ħ k
LD )S*
T
T5 T7
Mass (kg) Total enthalpy (J/kg) Thermal conductivity (W/m. K) Latent heat of fusion (J/kg) Source term (N/m3) Temperature (K) Phase change temperature (K) Reference temperature (K)
T PCM average temperature (K)
Q
Qj
Velocity vector (m/s) Heat transfer (J)
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)V*
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M
Gravity (m/s2)
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g
Specific heat capacity (J/kg K)
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C5
Heat transfer rate (W)
Heat flux density (W/m2)
s
Specific entropy (J/kg.K)
SUVW
Rate of entropy generation per volume unit (W/m3. K)
N=
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SUV
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)))* JY
Rate of global entropy generation (W/K) Entropy generation number
H
Height of the cylinder (m)
D
Diameter of the cylinder (m)
d
PCM sphere diameter (m)
∀
Volume (m3)
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ACCEPTED MANUSCRIPT S
Area (m2)
Greek symbols
n
σ )*
Φ Ѱ τ
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m
Liquid fraction Dynamic viscosity (Pa.s) Density (kg/m3) Entropy flux vector (W/m2. K)
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λ
Expansion coefficient (1/K)
Latent heat ratio Second law efficiency Stress tensor (N/m2)
Reference
init
Initial state
in
Inlet state
s
solidus
l
liquidus
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ref
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Subscripts
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β
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ACCEPTED MANUSCRIPT References [1] Pakrouh R., Hosseini M.J., Ranjbar A.A., Bahrampoury R., A numerical method for PCM-based pin fin heat sinks optimization, Energy Conversion and Management, 103 (2015) 542-552.
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[2] Kalbasi R., Salimpour M. R., Constructal design of phase change material enclosures used for cooling electronic devices, Applied Thermal Engineering 84 (2015) 339-349.
[3] Kheradmand M., Azenha M., Aguiar J., Castro-Gomes J., Experimental and numerical
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studies of hybrid PCM embedded in plastering mortar for enhanced thermal behaviour of buildings, Energy 94 (2016) 250-261.
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[4] Zhang S., Wu W., Wang Sh., Integration highly concentrated photovoltaic module exhaust heat recovery system with adsorption air-conditioning module via phase change materials, Energy, In press, Available online 6 November 2016.
[5] Sánchez P, Sánchez-Fernandez MV, Romero A, Rodríguez JF, L S-S, Development of
16-21.
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thermo-regulating textiles using paraffin wax microcapsules, Thermochim Acta 498 (2010)
[6] Mondieig D, Rajabalee F, Laprie A, Oonk HAJ, Calvet T, Cuevas-Diarte MA, Protection
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of temperature sensitive biomedical products using molecular alloys as phase change
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material, Transfus Apher Sci 28 (2003) 143-148. [7] Li Q., Mostafavi Tehrani S., Taylor R., Techno-economic analysis of a concentrating solar collector with built-in shell and tube latent heat thermal energy storage, Energy 121 (2017) 220-237.
[8] Lu X., Ma R., Wang Ch., Yao W., Performance analysis of a lunar based solar thermal power system with regolith thermal storage, Energy 107 (2016) 227-233.
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ACCEPTED MANUSCRIPT [9] Raam Dheep G., Sreekumar A., Influence of accelerated thermal charging and discharging cycles on thermo-physical properties of organic phase change materials for solar thermal energy storage applications, Energy Conversion and Management, 105 (2015) 13-19. [10] Bejan, A., Two Thermodynamic Optima in the design of sensible heat units for thermal
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storage, J. Heat Transfer 100 (1978) 708-712
[11] Jegadheeswaran S., Pohekar S.D., Kousksou T., Exergy based performance evaluation of latent heat thermal storage system: A review, Renewable and Sustainable Energy Reviews 14
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(2010) 2580–2595.
[12] Li G., Energy and exergy performance assessments for latent heat thermal energy
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storage systems, Renewable and Sustainable Energy Reviews 51(2015) 926–954. [13] MacPhee D., Dincer I., Beyene A., Numerical simulation and exergetic performance assessment of charging process in encapsulated ice thermal energy storage system, Energy 41 (2012) 491-498.
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[14] Guelpa E., Sciacovelli A., Verda V., Entropy generation analysis for the design improvement of a latent heat storage system, Energy 53 (2013) 128-138. [15] Pizzolato., Sciacovelli A., Verda V., Transient local entropy generation analysis for the
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design improvement of a thermocline thermal energy storage, Applied Thermal Engineering
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76 (2015) 391-399.
[16] Mosaffa A.H., Garousi farshi L, Exergoeconomic and environmental analyses of an air conditioning system using thermal energy storage, Applied energy 162 (2016) 515-526. [17] Mosaffa A.H., Garousi Farshi L., Infante Ferreira C.A., Rosen M.A., Advanced exergy analysis of an air conditioning system incorporating thermal energy storage, Energy 77 (2014) 945-952. [18] Rezaie B., Reddy B.V., Rosen M.A., Exergy analysis of thermal energy storage in a district energy application, Renewable Energy 74 (2015) 848-854.
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ACCEPTED MANUSCRIPT [19] Bindra H., Bueno P., Morris J.F., Shinnar R., Thermal analysis and exergy evaluation of packed bed thermal storage systems, Applied Thermal Engineering 52 (2013) 255-263. [20] Bindra H., Bueno P., Morris J.F., Sliding flow method for exergetically efficient packed bed thermal storage, Applied Thermal Engineering 64 (2014) 201-208.
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[21] Xia L., Zhang P., Wang R.Z., Numerical heat transfer analysis of the packed bed latent heat storage system based on an effective packed bed model, Energy 35 (2010) 2022-2032. [22] Silva P.D., Goncalves L.C., Pires L., Transient behaviour of a latent-heat thermal energy
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store: numerical and experimental studies, Applied Energy 73 (2002) 83–98.
[23] Brent A.D., Voller V.R., Reid K.J., Enthalpy-porosity technique for modeling
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convection–diffusion phase change: application to the melting of a pure metal, Numerical Heat Transfer, Part B 13 (1988) 297–318.
[24] Hosseini M.J., Rahimi M., Bahrampoury R., Experimental and computational evolution of a shell and tube heat exchanger as a PCM thermal storage system, Int Communications in
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Heat and Mass Transfer 50 (2014) 128–136.
[25] Bellan S., Gonzalez-Aguilar J., Romero M., Rahman M., Goswami D., Stefanakos E.,
2015 ) 758 – 768
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Numerical investigation of PCM-based thermal energy storage system, Energy Procedia 69 (
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[26] Pakrouh R., Hosseini M.J., Bahrampoury R., A parametric investigation of a PCM-based pin fin heat sink, Mechanical Science 6 (2015) 65–73. [27] Nallusamy N., Sampatha S., Velraj R., Experimental investigation on a combined sensible and latent heat storage system integrated with constant/varying (solar) heat sources, Renewable Energy 32 (2007) 1206–1227. [28] Kaizawa A, Kamano H, Kawai A, Jozuka T, Senda T, Maruoka N. Thermal and flow behaviors in heat transportation container using phase change material. Energy Conversion and Management 49 (2008) 698-706.
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ACCEPTED MANUSCRIPT [29] Bejan A. Entropy generation minimization. Boca Raton: CRC; 1996. [30] De Groot SR, Mazur P. Non-equilibrium thermodynamics. New York: Dover Publications; 2011. [31] Benedetti P, Sciubba E., Numerical calculation of the local rate of entropy generation in
Energy Systems Division (Publication); 30 (1993) 81–91(AES).
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the flow around a heated finned-tube, American Society of Mechanical Engineers, Advanced
[32] Gunerhan H., Hepbasli A., Exergetic modeling and performance evaluation of solar
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water heating systems for building applications, Energy and Buildings 39 (2007) 509–516.
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Fig. 1. A schematic view of a packed bed storage unit
28
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Fig. 2. Schematic of the 2D axisymmetric computational domain
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ACCEPTED MANUSCRIPT 75 70
water flow rate = 2 lit /min
60 55 50 45 40
30
0
20
40
60
80
100
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35
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Experimental data [27] Present work
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PCM Temperature (°C)
65
120
140
Time (min)
Fig. 3. Comparison of PCM temperature between present work and Nallusamy et al. [27]
3.25E-07
3.20E-07
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0.3
3.15E-07
0.2
3.10E-07
. S heat . S friction
0.1
0
0
3.30E-07
3.05E-07
3.00E-07 1000
2000
3000
4000
5000
Time (s)
Fig. 4. Comparison of the magnitude of entropy generation terms
30
Entropy generation rate (W/K)
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Tinlet= 30 °C dp= 10 mm
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Entropy generation rate (W/K)
0.4
ACCEPTED MANUSCRIPT (a) 600
T inlet= 40 °C
Solidification time (min)
500
T inlet= 30 °C
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400
300
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200
0
10
20
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100
30
40
50
60
Sphere diameter (mm) (b) 100
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60
40
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Solid percentage (%)
80
dp= 60 mm dp= 45 mm dp= 30 mm
T inlet= 40 °C
dp= 20 mm dp= 15 mm dp= 10 mm
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20
0
0
100
200
300
400
500
600
Time (min)
Fig. 5. Variation of total solidification time versus PCM sphere diameter
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ACCEPTED MANUSCRIPT (a) 3000
2000
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Tinlet= 40 °C
1500
dp= 60 dp= 45 dp= 30 dp= 20 dp= 15
1000
dp= 10 mm
500
0
100
200
300
400
500
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0
mm mm mm mm mm
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Heat flux (W)
2500
600
700
Time (min)
(b) 1
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0.6
0.4
T inlet= 40 °C
dp= dp= dp= dp= dp= dp=
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Heat release ratio
,η
0.8
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0.2
0
0
100
200
300
400
500
60 mm 45 mm 30 mm 20 mm 15 mm 10 mm
600
Time (min)
Fig. 6. Effect of PCM sphere diameter on (a) heat flux and (b) heat release ratio
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ACCEPTED MANUSCRIPT (a) 1
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0.6
dp= 60 mm dp= 45 mm dp= 30 mm dp= 20 mm
0.4
Tinlet= 40 °C 0.2
0
20
40
60
80
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0
dp= 15 mm dp= 10 mm
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Latent heat ratio
,φ
0.8
100
Solid percentage (%)
(b)
1
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φFinal = 0.85
0.6
φFinal = 0.77
0.4
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Latent heat ratio
,φ
0.8
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0.2
0
0
0.2
T inlet= 40 °C T inlet= 30 °C
dp= 30 mm
0.4
0.6
0.8
1
Solid percentage (%)
Fig. 7. The latent heat ratio for various (a) PCM sphere diameter and (b) inlet HTF temperature
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ACCEPTED MANUSCRIPT (a) 58
dp= 60 dp= 45 dp= 30 dp= 20 dp= 15 dp= 10
Tinlet= 40 °C
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52
49
46
43
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Outlet Temperature (°C)
55
mm mm mm mm mm mm
0
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40 100
200
300
400
500
Time (min)
(b) 58
mm mm mm mm mm mm
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52
60 45 30 20 15 10
49
46
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Outlet Temperature (°C)
55
dp= dp= dp= dp= dp= dp=
T inlet= 40 °C
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43
40
0
0.2
0.4
0.6
0.8
1
Solid percentage (%)
Fig. 8. Variation of HTF outlet temperature for various sphere diameter
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ACCEPTED MANUSCRIPT (a) 60
dp= 60 mm
50
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PCM Temperature (°C)
55
45
Tinlet= 40 °C
40
35
30 100
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0
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Tinlet= 30 °C
600
Time (min)
(b) 55
dp= 60 mm
20 min
250 min
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300 min
49
46
350 min 400 min
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PCM Temperature (°C)
52
Tinlet= 40 °C
450 min
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43
500 min
40
0
0.2
0.4
0.6
0.8
1
y/H
Fig. 9. Temporal variation of the PCM at three different positions inside the tank
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ACCEPTED MANUSCRIPT . S heat (W/m3k)
40 41 42 43 44 45 46 47 48 49 50 51 52
0.010.05 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t=2000 s
t=10000 s
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Temperature (°C)
t=20000 s
t=30000 s
Fig. 10. Distribution of temperature(left hand) and entropy generation(right hand) for dp=60 mm and T = 40°C
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Temperature (°C)
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40 41 42 43 44 45 46 47 48 49 50 51 52
t=2000 s
t=5000 s
t=8000 s
t=12000 s
Fig. 11. Distribution of temperature for dp=60 mm (left hand) and dp=30 mm (right hand) for T = 40°C
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ACCEPTED MANUSCRIPT . S heat (W/m3k) 1
1.5
2
2.5
3
3.5
4
4.5
5
t=5000 s
t=8000 s
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t=2000 s
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0.010.05 0.1 0.5
t=12000 s
Fig. 12. Distribution of entropy generation for dp=60 mm (left hand) and dp=30 mm (right hand) for T = 40°C
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. S heat (W/m3k)
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0.010.05 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t=2000 s
t=8000 s
t=14000 s
t=20000 s
Fig. 13. Distribution of entropy generation for T = 40°C (left hand) and T = 30°C (right hand) for dp=60 mm
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ACCEPTED MANUSCRIPT (a)
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0.09
0.06 dp= 60 mm dp= 45 mm dp= 30 mm dp= 20 mm
0.03
dp= 15 mm dp= 10 mm 0
0
20
40
Tinlet= 40 °C 60
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Entropy generation rate (W/K)
0.12
80
100
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Solid percentage (%)
(b)
0.3
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dp= 30 mm
0.2
0.1
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Entropy generation rate (W/K)
0.4
0
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0
20
T inlet= 40 °C T inlet= 30 °C 40
60
Solid percentage (%)
80
100
Fig. 14. Global entropy generation rate SUV for various (a) PCM sphere diameters and (b) inlet HTF temperatures
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ACCEPTED MANUSCRIPT (a) 1
dp= 60 dp= 45 dp= 30 dp= 20 dp= 15 dp= 10
0.8
mm mm mm mm mm mm
Tinlet= 40 °C
Ns
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40
60
80
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0
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Solid percentage (%)
(b) 1
dp= dp= dp= dp=
0.8
60 45 30 20
mm mm mm mm
T inlet= 30 °C
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dp= 15 mm dp= 10 mm
Ns
0.6
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0.4
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0.2
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40
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Fig. 15. Entropy generation number Ns for (a) T = 40°C and (b) T = 30°C
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ACCEPTED MANUSCRIPT 100
80
(a)
70 60 50
{ =
40
{ =
30 20 10 0 15
20
30
rst uvwxyx z{|}x~xy
60
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0.225 0.2 0.175 0.15 0.125 0.1 0.075 0.05 0.025 0 10
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Global entropy generation rate
0.25
45
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10
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Second law efficiency
90
15
20
30
45
(b) { = { =
60
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rst uvwxyx z{|}x~xy
Fig. 16. Time averaged values of (a) second law efficiency and (b) entropy generation rate for different inlet
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HTF temperature
40
ACCEPTED MANUSCRIPT
Phase change temperature (˚C) Thermal conductivity (W/m.K) Dynamic viscosity (kg/m.s) Thermal expansion coefficient (1/K) Latent heat (J/kg)
value 10,15,20,30,45,60 30,40 57 15 6
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Table 2 Studied parameters and levels Parameters PCM sphere diameter (mm) HTF inlet temperature (˚C) Initial temperature (˚C) Reference temperature (˚C) HTF mass flow rate (lit/min)
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Specific heat (J/kg.K)
value 809.5 (solid) 771 (liquid) 2900 (solid) 2100 (liquid) 52.1 0.15 0.0055 0.00079 243500
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Table 1 Thermophysical properties of the PCM [22] Property Density (kg/m3)
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ACCEPTED MANUSCRIPT Highlights:
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SC M AN U TE D
•
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• •
Detailed entropy generation for PCM based packed bed storage was numerically investigated Simulations are performed with an effective packed bed model. Reduction in capsules diameter increases both the rate of entropy generation and the efficiency. Utilization of smaller capsules is more effective when the inlet HTF temperature is lower.
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•
S0360-5442(17)31433-0
DOI:
10.1016/j.energy.2017.08.055
Reference:
EGY 11424
To appear in:
Energy
Please cite this article as: R.Pakrouh, M.J.Hosseini, A.A.Ranjbar, R.Bahrampoury, Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model, Energy (2017), doi: 10.1016/j.energy.2017.08.054 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT
Thermodynamic analysis of a packed bed latent heat thermal storage system simulated by an effective packed bed model R. Pakrouha, M.J. Hosseinib,*, A.A. Ranjbara, R. Bahrampouryc a
School of Mechanical Engineering, Babol University of Technology, POB 484, Babol, Iran
c
Department of Mechanical Engineering, Golestan University, P. O. Box 155, Gorgan, Iran
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b
Department of Mechanical Engineering, K.N.Toosi University of Technology, Tehran, Iran
Abstract
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The present paper numerically investigates the performance of latent heat storage systems during solidification, which involves phase change material (PCM) capsules. Paraffin wax is
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considered as the PCM and water plays the role of heat transfer fluid (HTF). The simulation is conducted for two inlet temperatures, 30 °C and 40 °C, while the capsules’ diameter varies in the range of 10 mm to 60 mm. Among various existing models, the effective packed bed model which is not only able to provide temperature gradient data but also is capable of
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reporting entropy generation details. Results indicated that both reduction in capsules’ diameter and the HTF inlet temperature unfavorably increase the amount of the system’s irreversibility. However, it is demonstrated that the increase in the irreversibility does not
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essentially result in a reduction in performance of the latent heat storage system. In other words, the efficiency of the storage system is not a pure function of entropy generation
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number. In fact, the reduction in the diameter results in an improve in the second low efficiency while the inlet temperature reduction diminishes the efficiency. Results also implies that decisive parameters vary significantly only when the diameter reduces to 20 mm and further reduction doesn’t affect the system performance noticeably. Keywords: PCM, Entropy generation, Exergy analysis, Packed bed storage
*
Corresponding Author: (Seiyed Mohammad Javad Hosseini) Department of Mechanical Engineering, Golestan University, P.O. Box 155, Gorgan, Iran. Tel/Fax: +98 17 32440206, Email: [email protected]
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ACCEPTED MANUSCRIPT 1. Introduction The use of phase change materials (PCMs) in thermal energy storage (TES) systems is an attractive subject for researchers. Therefore, a large number of papers in this field has been published during the last few decades. Due to attractive characteristics of PCMs, such as
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large latent heat of fusion, constant phase change temperature and chemical stability, these materials find diverse applications like electronic cooling [1,2], air conditioning [3,4], smart textiles [5], biomedical [6] and solar energy storages [7,8]. PCM properties are subjected to
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small variation when is used repeatedly. The extent of the variation is dependent to the type of the PCM and the number of operation cycles. Dheep and Sreekumar [9] studied the effect
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of the number of the cycles on the properties of benzamide acid and sebacic acid. Results indicated that after 1000 cycles, the phase change temperatures change 0.02 and 0.85 percent. Among available geometries of latent heat thermal storage (LHTS) systems, packed bed storage systems with spherical capsules has been declared to be one of the most conventional
construction.
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and advantageous due to its large heat transfer area in a small volume and simple
Apparently the idea that the main objective of a thermal energy storage system is to store
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thermodynamic availability (exergy) and not to store energy is attributed to Bejan [10].
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Hence part of the studies is dedicated to thermodynamically investigate sensible as well as latent heat thermal storage systems. Jegadheeswaran et al. [11] presented a detailed review on different techniques used for exergy analysis of LHTS units. Li [12] performed a comprehensive review, which investigates the effect of different operating conditions and design parameters such as the HTF inlet temperature, HTF mass flow rate, reference temperature and PCM melting temperature on energy and exergy performance of the LHTS systems. MacPhee et al. [13] investigated the solidification process in an encapsulated ice TES unit numerically. They studied energy and exergy efficiencies for various inlet HTF
2
ACCEPTED MANUSCRIPT temperatures, HTF flow rates and capsule geometries (spherical, cylindrical and slab). Their results showed that the energy analysis does not provide the sufficiently capable tool for assessment of the system’s performance, while the result, based on second law analysis, is found to be more reliable. In addition, they concluded that smaller amount of irreversibility
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and larger exergy efficiency is achievable as the HTF temperature approaches the solidification temperature or when the flow rate increases. Guelpa et al. [14] presented a detailed entropy generation analysis for solidification process in a radially finned shell and
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tube heat storage unit for the first time. They showed that second law analysis can be used as an effective tool for improving the design of LHTS systems. Recently Pizzolato et al. [15]
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employed the concept of the exergy destruction as a design tool for improving a thermocline TES system. They introduced the lifespan and the characteristic time of entropy generation for their case. They found that by inclusion of a porous layer and a solid baffle, the entropy generation can be reduced about 7% and 5% respectively. Mosaffa and Garousi [16]
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performed an exergoeconomic and environmental analysis on a latent heat thermal storage coupled with an air conditioning system. The effect of various operational parameters on charging and discharging processes of different PCMs is investigated and finally optimized
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from exergoeconomic and environmental viewpoints. Mosaffa et al. [17] implemented an
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advanced exergy analysis for a combination of LHTS unit and VCR (vapor compression refrigeration) as an air conditioning system. Their results indicated that the total rate of exergy destruction of an LHTS unit is an internal matter that is due to its irreversibilities. Rezaie et al. [18] employed the exergy analysis to investigate the function of the TES in a Friedrichshafen district energy (DE) system. They found that the overall energy and exergy efficiencies of the TES system are 60% and 19% less, respectively. They also showed that the energy efficiency of the TES system follows the same trend as the TES temperature. It is shown that the trends of exergy accumulation and exergy efficiency of the TES system are
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ACCEPTED MANUSCRIPT similar. Bindra et al. [19] developed a numerical heat transfer model, which takes the effects of energy storage density, heat losses and axial dispersion into account to investigate the exergetic performance of packed bed thermal storage systems. As shown in their work, as the heat losses to the surrounding or axial dispersion lowers the exergy efficiency rises. Bindra et
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al. [20] introduced the sliding flow method (SFM) to calculate the exergy losses that is due to the fluid pressure drop in a packed bed storage system. The results indicated that exergy destruction caused by the pressure drop cannot be passed up since for all possible packed bed
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storage systems the value of this parameter is in the range of 2-10%.
Since packed bed heat exchangers are known as efficient ones, implementation of a decision-
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making criterion based on second law of thermodynamics can be considered as usable investigation, due to frequent applications of heat storage systems in energy systems; simulation of such component through an efficient method, which provides entropy details, gives an appropriate tool for engineers to make the best decision. For investigation of the
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dynamic and thermal behavior of packed bed LHTS systems, some validated mathematical models can be found in the literature such as single-phase model, concentric dispersion model and continuous solid phase model. However, due to the complexity of PCM spheres
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arrangement, previous models are not capable to include some of the most important features
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such as size and arrangement of the spheres, shape of the flow passage, the flow field and thermal gradient inside the PCM spheres in detail. Therefore, previous works were not able to generate local distribution of entropy generation of latent energy storage systems. Effective packed bed model which has been introduced by Xia et al. [21] is able to overcome these limitations. In this paper, a CFD model has been developed based on the effective packed bed model that makes the calculation of local irreversibilities during solidification possible. In fact, the model has been used in the present study that is capable of providing entropy generation details in the packed bed system unlike previous studies.
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ACCEPTED MANUSCRIPT 2. Mathematical model 2.1. System description and governing equations Due to harmful consequences of fossil fuels emission to the atmosphere and since these sources of energy are going to deplete in next future decades, researchers are studying on
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utilization of renewable energy sources. Generally speaking, renewable energy sources are not available steadily during the day. For example at night, solar energy is not accessible, therefore heat storage systems can be employed to store the energy when it is present and
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release the energy to the system when the renewable energy is unavailable.
A schematic view of the PCM-based packed bed thermal storage system is shown in figure 1.
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The storage system is a cylinder that has a length of 850 mm and a diameter of 400 mm. The cylinder is packed with spheres of varying diameter, d, which provides a space for the HTF to pass. Water as the HTF and Paraffin Wax [22] as the PCM are taken into account. The thermophysical properties of the PCM are summarized in Table 1. Considering the reference,
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the data related to viscosity, conductivity, volumetric expansion coefficient are taken from the existing paraffin. However, the properties like thermal capacity and latent heat capacity are obtained using DSC analysis, the uncertainity of which is about ±3%.
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Since the investigated process is solidification, the whole storage system is initially kept at a
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homogenous temperature, T , above the phase change point. The current model can also be applied for non-homogenous initial conditions. The later condition happens when the solidification process initiates right after completion of the melting process. In this condition, although whole PCM is melt, there is no uniform temperature distribution and there yet exists temperature gradient. But in this study, since we concentrate on solidification, it is assumed that the melting process continues until the whole system approach the same temperature. The HTF at a temperature below the phase change point, T , enters the system at the bottom of the vertical cylinder, passes through the empty spaces left among the spheres and
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ACCEPTED MANUSCRIPT leaves the unit when it reaches the top end. This study focuses on the contribution of inlet temperature and spheres diameter variations on the performance of the system from thermodynamic point of view. Table 2 contains the working conditions and the range variation of the parameters investigated in this study.
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As mentioned above, the effective packed bed model [21] is implemented to simulate the phase change phenomenon. The model is able to display the temperature gradient in the PCM spheres as well as the HTF. This capability, makes it possible to calculate the amount of
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entropy generation in the packed bed LHTS. Figure 2 shows the computational domain selected for the current packed bed storage system, which is based on the effective packed
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bed model.
The basis of the effective packed bed model is on transformation of the three dimensional actual model to an effective two dimensional model. The transformation of these two should be conducted in a way that the void fraction remains the same. The amount of porosity in
ε
d S n ∙ π ∙ (2) =1− = S D∙H
4 d ∀ n ∙ 3 ∙ π ∙ (2) =1− = D ∀ π ∙ ( ) ∙ H 2
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ε
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both two and three dimensional scales are given below:
(1)
(2)
• •
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The assumptions considered in the simulation are as below: The model is solved for 2D and axisymmetric conditions.
Although the consequence of natural convection is subtle, the Boussinesq
approximation is included for a more accurate simulation. •
Void fraction (the ratio of empty spaces volume to the total container’s volume) is
assumed constant whose value is 0.36. •
The consequences of the capsule wall thermal resistance is neglected
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ACCEPTED MANUSCRIPT •
The flow of the HTF is assumed to be laminar and incompressible
•
Heat dissipation to the surrounding is neglected since the cylinder’s wall is assumed
to be a perfect insulation. •
The heat transfer among the capsules are disregarded because the heat transfer is not
•
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possible via point by point contacts.
The direct conversion of the three dimensional model to the two dimensional one
leads to connection of the capsules which blocks the flow of the HTF. Therefore, to
•
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overcome the restriction a small distance is considered among the capsules. The capsules are not floating and are fixed in their position.
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Considering the figure 2, the boundary condition equations are as below: Outflow boundary condition at the outlet: ∂V$ ∂T = 0 , =0 ∂y ∂y
(3)
Adiabatic condition at the external wall of the cylinder.
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∂T ∂T = = 0 , V$ = V& = 0 ∂y ∂x
(4)
For the HTF:
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Continuity:
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Considering the above assumptions, the governing equations on the HTF and the PCM are:
∇. )V* = 0
(5)
Momentum:
∂ )*. + ∇. ,ρV )*)V*. = −∇P + μ∇ )V* + ρg ,ρV )* ∂t
(6)
Energy: ∂T k )*. ∇T = ∇ ( +V T) ∂t ρC5
(7) 7
ACCEPTED MANUSCRIPT For the PCM: The conventional enthalpy-porosity method [23] is employed to solve the phase change problem. The governing equations for the PCM are:
Momentum: ∂ )*. + ∇. ,ρV )*)V*. = −∇P + μ∇ )V* + ρg ,ρV )*β(T − T7 ) + )*S ∂t
∂ )*ħ. = ∇. (k∇T) (ρħ) + ∇. ,ρV ∂t
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Energy:
(8)
(9)
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)* = 0 ∇. V
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Continuity:
(10)
The term )S* in the equation (9) is the source term, which is defined as below: )*S =
(1 − λ) )V* A λ + 0.001 ;<=>
(11)
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Where λ, is the liquid fraction, and the summation of the constant value at the denominator, 0.001, is considered to prevent zero value at the denominator. Amushy is the constant of the mushy zone and determines the extend of damping. Large values of the term results in higher
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slope of damping and very large values leads to some fluctuation in the solution. In the
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current study the value of this parameter is 105 that is determined via preliminary simulations. Considering the equation (10), total enthalpy of the material is the summation of sensible and latent enthalpies. A
ħ = h7 + @ C5 dT + λLD
(12)
AB
where λ is the liquid fraction whose value is in the range of 0 and 1. In this study, only one temperature is considered for the melting point.
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ACCEPTED MANUSCRIPT λ = 0 if T < T5 λ = 1 if T > T5
(13)
2.2. Numerical procedure and validation In this study, the simulation is conducted via a 2D in-house code [24]. The SIMPLE
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algorithm is employed to couple pressure and velocity in one equation. In order to solve the pressure correction equation, the presto scheme is taken into account. The QUICK differencing scheme is employed to discretize the energy and the momentum equations. The
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consequences of grid size are examined at initial stages of the simulation. The careful examination revealed that optimum value for the grid size is 0.002 m when the capsules’
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diameter is in the range of 15 mm-60 mm. However, it is found that when the diameter reduces to 10 mm, the optimum value becomes 0.0015 m. Similar examinations are also conducted to evaluate the optimum time step, which resulted in 0.01 s as the optimum step. Considering the convergence tolerance of 10IJ for the continuity and the momentum
every step.
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equations and 10IK for the energy equation, the convergence of the solution is examined for
The validation of effective packed bed model is extensively targeted by Xia et al [21] and
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Bellan et al [25]. The implemented code has also been validated in our previous publications
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[1, 24, 26]. Here, in order to assure about the accuracy of the implemented code for the effective pack bed model, a new validation based on another experimental results of Nallusamy [27] is presented. Figure 3 shows the PCM temperature at a specific height of the heat exchanger during the melting process. The diameter of studied tank is of 360 mm and its height is 460 mm. The number of the capsules of 55 mm diameter is 264. The capsules are uniformly arranged in 8 rows. The melting point of the PCM is 60±1 °C and the initial temperature is 32 °C. Water at 70 °C enter the tank at its top inlet. 9
ACCEPTED MANUSCRIPT 2.3. Formulations for energy and exergy analyses In order to evaluate the performance of a unit from energy point of view, different parameters have been defined. In this study, the heat release ratio, which is the ratio of the amount of released heat during the solidification process to the maximum heat gain capacity, is the
below [28]: η=
QN= Q;&
in which
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Q;& = M PC5, ,T − T5 . + LD + C5,= ,T5 − T .Q
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performance criterion. The corresponding correlations for calculating the ratio is brought
(14)
(15)
Another criterion is also included in this study, which considers latent heat. This decisive parameter is the latent heat ratio, which is the ratio of the amount of latent heat gain to the
ϕ=
Q QN=
in which
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Q = M (1 − λ)LD
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total heat release until the moment of consideration
(16)
(17)
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Contrary to the first law of thermodynamics, the second low assesses the quality of energy. Therefore, new concepts such as entropy and exergy are proposed. Based on non-equilibrium thermodynamics, the local entropy generation can be calculated using the following equation [29]: ρ
Ds = −∇. σ )* + SUVW Dt
(18)
Where σ is the entropy flux vector and SUVW is the rate of entropy generation per unit volume. The sources of the irreversibility in this study is mainly due to temperature gradient and
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ACCEPTED MANUSCRIPT viscosity. Therefore, the total rate of entropy generation can be estimated using the following equation: SUVW = SU>W + SUDW
(19)
Equation (19) can also be rewritten as following [30]: −J)))*Y . ∇T ∆: τ + T T
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SUVW =
(20)
where first and second terms on the right side of the equation are the rates of entropy
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generation per unit volume due to heat transfer and friction respectively which are calculated using the equations below. k ∂T ∂T ]^ _ + ^ _ ` T ∂x ∂y
SUDW =
2μ ∂u ∂v 1 ∂u ∂v ]^ _ + ^ _ + ^ + _ ` T ∂x ∂y 2 ∂y ∂x
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SU>W =
(21)
(22)
SUV = @ SUVW dV
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Rate of entropy generation is calculated by integrating of the SUVW over the volume. (23)
In order to estimate the magnitude of the terms included in equation (20), the following
SU>
SU> + SUD
(24)
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Be =
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equation is proposed [31]:
The dimensionless parameter Be is the Bejan number whose value is in the range of 0 to 1. In case of Be ≫ 0.5, the majority of the irreversibility is due to heat transfer and when the number is far below 0.5, friction constitutes a larger
portion of irreversibility. The
contribution of the two agents are almost equal when the Bejan number is around 0.5. Another concept, which is defined as the maximum work that a system can generate to reach the environment equilibrium state, is exergy. Contrary to energy, exergy is not conservative
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ACCEPTED MANUSCRIPT and can be destroyed by irreversibility. Therefore, the exergy balance for an adiabatic system may be written as below [32]: Exergy input − Exergy output = Exergy destroyed
(25)
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Second low efficiency or exergy efficiency is the criterion to assess the consequences of irreversibility on the system [32]. ExUi= ѱ(t) = 1 − ] ` ExU
(26)
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where ExUi= is the rate of exergy destruction that can be calculated using the Gouy-Stodola theorem as below [29]:
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ExUi= = T7 SUV
(27)
The quotient of the division of the equation (26), ExU , which is the rate of exergy input to the unit, is calculated using the following equation [14]: ExU = Qj . ^1 −
T7 _ T
(28)
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where Qj is the heat flux that passes through the PCM boundaries. The division enclosed by the brackets in equation (26) is called the entropy generation
ѱ(t) = 1 − N=
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number. Therefore the equation can be rewritten as below: (29)
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The above equation declares that by reducing the entropy generation number, the system’s efficiency improves. Therefore, entropy generation minimization has attracted scientists attention in optimization and improving energy systems. 3. Results As stated, irreversibility generally consists of two main sources of heat transfer and friction. Therefore, the contribution of each of these two on the entropy generation has to be determined. According to [14], the portion of friction is negligible in comparison with that of
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ACCEPTED MANUSCRIPT heat transfer. Figure 4 in which the significance and the trend of variation of each of these sources are shown approves the claim. According to the figure, the magnitude of the friction based entropy generation is in the order of 10Il that is negligible regarding the heat transfer based entropy. Therefore, the Bejan number is very close to 1 and the friction based entropy
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generation can be simply disregarded without adding any source of approximation. 3.1. Energy analysis
Figure 5a shows the total solidification time versus the capsules’ diameter for two different
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HTF inlet temperatures. According to the figure, although the amount of the PCM remains constant for all the cases, the total solidification time reduces significantly, as the spheres
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become smaller. The extent of this reduction is so significant that for the inlet temperature of 40 °C, the diameter reduction from 60 mm to 10 mm results in a solidification time decrease of 76.4 percent. This improvement is due to the consequent addition of 524 percent heat transfer surface. As the surface to volume ratio increases, the rate of heat transfer between the
shortens considerably.
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two mediums, the PCM and the HTF, promotes. As a result, the total solidification time
It is worth noting that the considerable reduction of solidification time as the diameter
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reduces may change the applications of the system; the application of the case that is able to
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release higher rate of heat in short time may differ from a heat storage unit that is designed to give less rate of heat but for longer period. The optimization of the operating time according to the application is another issue. In this study, the consequences of the parameters are considered only from thermodynamic point of view. Figure 5b shows the solid percentage versus time for different capsules’ diameters. The solid percentage refers to the average solidified PCM in the whole PCM. As the process progresses, the smaller capsules will lead to faster pace of solidification which is predictable. In fact, as the capsules’ diameter reduces, the heat transfer area increases which results in
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ACCEPTED MANUSCRIPT higher rate of solidification. The interesting point is that, for the diameter less than 20 mm, the solidification curves almost match on one another until 80 percent of solidification. The effect of sphere diameter variation on the heat flux and the heat release ratio is exhibited in figure 6(a) and 6(b) respectively. According to the figure, as the HTF enters the heat
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exchanger, due to the large temperature difference between the two mediums, the rate of heat transfer is large. The ascending trend in the heat flux is due to the inclusion of convective heat transfer to the initially dominant conduction heat transfer within the capsules. As the
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process progresses, it can be seen that the heat flux after reaching a maximum value reduces because the two medium temperature difference declines. As stated above, despite the
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amount of the PCM is constant, the heat transfer area increases as the diameter shrinks. Based on the Fourier equation, the increase in the area leads to higher rate of heat transfer. Considering figure 6(b), regardless of the ending stage of the solidification, it is apparent that the reduction in the diameter leads to higher rate of heat release of the PCM. It is noteworthy
ratio considerably. 3.2. Latent heat analysis
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that as the diameter undershoots 20 mm, extra reduction does not influence the heat release
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Figure 7 shows the latent heat ratio versus the solid percentage. According to the figure 7(a),
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regardless of the diameter, the ratio increases steeply at initial stages of the process when the first pieces of the solid are created. The increasing trend pursues until a maximum value after which a reduction trend appears that finishes to a final value. This final value is the ratio of the maximum latent heat to the heat released from the PCM up to the moment under consideration. The final values of the curves are close to each other (in range of 0.85 to 0.86). Since the ratio of the maximum latent heat to the thermal capacity of the system heat release is equal to 0.84, the obtained results show that, for the solidification process, all the units regardless of the diameter reach to the final point of heat release.
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ACCEPTED MANUSCRIPT According to the equation (16), the ratio of the latent heat to the total released heat is obtained. Therefore it is difficult to clearly separate these two by the available results and analyze the required parameter as the capsule’s size varies. Considering figure 7a, it can be
heat affects the total heat ratio more considerably.
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figured out that, until 0.2 of the solidification, as the diameter decreases, the released latent
Another interesting point is that the trend of the ratio is not identical for different diameters as the solid percentage varies. Considering the figure, as the diameter reduces, the latent heat
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ratio increases as the spheres become smaller, at the first 20 percent of the solidification process. After the point, the ratio reduces when the diameter of the capsules diminishes.
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According to the figure 7b, the same trend of variation for the dimeter variation is also observable for HTF inlet temperature variation. Until the solidification percentage of 0.2, the ratio of the latent heat for the inlet temperature of 30 °C is more. Therefore, during this stage, as the inlet temperature lowers, the latent heat affects the latent heat ratio more.
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As a specified amount of a PCM releases a determined maximum latent heat, it is expected that as the inlet temperature of the HTF reduces, the value of the latent heat ratio descends. According to figure 7b, as the HTF inlet temperature redcues from 40 °C to 30 °C, a
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reduction of 8 percent at the end of the process is observed. Therefore, the less the
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temperature difference between the inlet HTF and the initial temperature, the more acceptable latent heat performance is achievable. 3.3. Second law analysis
Figure 8 shows the outlet HTF temperature variation versus time and the solidification percentage. The ending of the curves is related to the completion of the solidification process. According to the figure, when all the PCM solidifies, the outlet temperature doesn’t approach the inlet temperature. In fact since there yet exists temperature gradient in the PCM, heat transfer continues even after solidification completion. According to the figure, at the initial
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ACCEPTED MANUSCRIPT stages of the process, the outlet temperature of the HTF reduces drastically. As the process progresses, the reduction persists with a less slope. In fact, the controlling factor of the outlet temperature is the interaction between the HTF and the PCM. This concept is better understood by figure 8b. According to the figure, for the 20 mm dimeter case and the cases of
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less diameters, the outlet temperature keeps constant for a long period and about the phase change temperature. As explained before, as the capsules’ diameter reduces, the heat transfer area increases and heat is absorbed by the HTF in a more uniform trend. In fact, when
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considering each of the cases during their solidification period, it becomes clear that the case of smaller capsule sizes result in more uniform and controllable outlet temperature.
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Figure 9a shows the PCM temperature variation during the whole solidification process for three point p1 (the lowest row) p2 (the middle row) and p3 (the highest row). According to the figure, the temperature variation for the both cases follow the same trend. Accordingly, when the heat transfer fluid reaches to the corresponding capsule, the temperature of it
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reduces and when it reaches to the solidification temperature, the temperature after the moment keeps constant for a considerable portion of time. After releasing all the latent heat,
the HTF.
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the temperature reduction continues until the approach of the point to the inlet temperature of
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Figure 9b shows the temperature variation of the central point of the capsules on the axis of the cylinder for different heights. According to the figure, during the first half of the process, the point of the 0.2 height of is only subjected to the inlet temperature variation and the point above the height are within melting range. As the process continues, the temperature gradient intensifies and after 350 minute, the temperature of the lower parts approaches the inlet HTF temperature and the temperature gradient decreases. Since the rate of entropy generation is dependent to the temperature value and its gradient (according to the equation 20), a comprehensive understanding of the entropy distribution is
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ACCEPTED MANUSCRIPT possible only when the temperature distribution is studied. The temperature distribution and the rate of entropy generation caused by heat transfer for the 60 mm sphere diameter are shown in figure 10. At initial stages, the maximum entropy generation is asigned to the thin layer of the solid PCM and the HTF at the tube boundaries. The superiority of these regions is
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related to their intense temperature gradient. Consequently, the general distribution of the entropy generation is found to be similar to the solidification front. On the other hand, the regions in which PCM melt is present have the least values of rate of entropy generation due
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to their higher temperature and less temperature gradient. As the process advances eg. at 10000th and 20000th seconds, the entropy distributes within the spheres while their intensity
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lowers. This trend results in a decrease in temperature gradient and a subsequent reduction in the rate of entropy generation. In fact, as time passes, the temperature of the spheres at the bottom of the container approaches to the dead state.
In order to explain how the entropy is distrubuted in the container, it is essential to pay
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attention to the corresponding equation (equation 21) in which it is stated that the rate of entropy genration is in a direct relation with square of the temperature gradient and has an inverse relationship with square of the temperature. However, since the range of temperatures
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distribution.
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is finite, the temperature gradient plays a more important role in entropy generation and its
The temperature contours and entropy generation distribution for 30 mm and 60 mm diameters are shown in figures 11 and 12. As mentioned in the description of figure 5, as the capsules’ diameter reduces, the heat transfer surface between the two mediums, the PCM and the HTF, increases. Therefore, as the surface to volume ratio increases, the rate of heat transfer and the temperature gradient rise. Thus, as the sphere diameter drops, the rate of entropy generation in the system increases which is also confirmed by figure 12. However, since the total solidification time of the 30 mm case is less, the PCM temperature approaches
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ACCEPTED MANUSCRIPT the fluid temperature earlier. Therefore, after 8000 seconds, the rate of entropy generation reduces in a trend that the 30 mm- case reaches to the dead state before the container filled with 60 mm spheres. The distribution of the rate of entropy generation for HTF inlet temperatures of 40 °C and 30
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°C is shown in figure 13. As mentioned above, the rate of the entropy generation is directly proportional to the square of the temperature gradient and is inversely proportional to the square of the temperature. As the inlet temperature reduces, not only the temperature gradient
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increases but also the maximum accessible temperature reduces. The consequence of these two supporting causes is a high rate of increase in the rate of entropy generation as the inlet
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temperature reduces. According to the figure, the rate of entropy generation of the case of 30 °C is considerably more than the other case until around 14000 s. However, since the inlet temperature reduction leads to shorter complete solidification time, the unit approaches to the corresponding dead state earlier. Therefore, it can be seen that, after about 14000 s the rate of
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entropy generation reduces for the case of 30 °C.
Figure 14 shows the global rate of entropy generation versus the solidification percentage. According to figure 14 (a), as the diameter of the spheres decreases, the rate of entropy
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generation climbs. Considering the figure, the rate of entropy generation generally increases
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at first over solidification percentage. However, the parameter reduces after a peak. To present some details, the trend of variation differs to some extend when the diameter is lower than 20 mm. In fact, for these diameters, the rate of entropy generation almost flattens out in a large range of solidification percentage. The consequence of inlet temperature on the global entropy generation is illustrated in figure 14 (b). As can be seen, as the temperature reduces, the rate of global entropy generation surges. The obtained results confirms the explanations presented for figure 13.
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ACCEPTED MANUSCRIPT The trend of variation of the entropy generation number is shown in figure 15 for different sphere diameters. As can be seen in the figure, for both of the inlet temperatures, the diameter reductions leads to a drop in entropy generation number, which according to the equation (29) results in an improvement in system’s efficiency. In fact, according to the equations (27)
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and (28), the variation of entropy generation is a function of three different parameters, the rate of entropy generation, heat flux and the PCM temperature. Since the rate of entropy generation is not the only factor that determines the efficiency, high rate of entropy
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generation is not essentially a defect of a system. As matter of fact, the promotion in the rate of exergy may be high enough to retrieve the consequences of the increase in entropy
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generation.
Similar deduction has also been presented by Guelpa et al. [14] for the efficiency of a PCM containing shell and tube heat exchanger. Considering the consequences of inlet temperature, it can be concluded that the temperature affects the heat exchanger in a different manner with
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respect to the diameter. In other words, as the HTF inlet temperature drops, the rate of entropy generation increases while the second law efficiency declines. This explanation can be confirmed when considering the figure 16. Figure 16 shows the time average of the rate of
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entropy generation and the second low efficiency. According to the figure, for both of the
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inlet temperatures, the diameter reduction leads to a gain in both the rate of entropy generation and the second low efficiency. In contrary to the diameter variation, for a fix diameter, a reduction in the inlet temperature leads to an increase in the average entropy generation and a reduction in second law efficiency. Moreover, figure 16 confirms that the diameter reduction to 20mm affects the efficiency considerably. However, further reduction does not influence the efficiency sensibly.
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ACCEPTED MANUSCRIPT In fact, it can be mentioned that within the range considered for the capsules’ diameter and the HTF inlet temperature, the system responses to variation of the temperature more pronouncedly from thermodynamic point of view. Another point that the figure reveals is that the effectiveness of the spheres diameter is more
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noticeable for lower inlet temperature. Based on the figure, considering the inlet temperature of 40 °C, the diameter reduction from 60 mm to 10 mm leads to 11 percent improvement in the efficiency. While similar reduction in the size of the spheres, leads to 21 percent
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improvement when the HTF inlet temperature is 30 °C. It can be concluded that the use of smaller capsules is more suitable for the lower inlet temperatures.
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It can also be concluded that, the 20 mm diameter capsule is the most appropriate size. As can be seen in the figure, by reducing the diameter to the values below 20 mm, although the second law efficiency almost keeps unvarying, the rate of entropy generation increases.” 4. Conclusion
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In this paper, the solidification process of a PCM is studied in a packed bed storage system from both first and second laws of thermodynamic viewpoints. In order to illustrate the rate of entropy generation in the system, the effective packed bed model is implemented. This
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model is able to demonstrate the occurring temperature gradients in the heat exchanger and
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therefore is an appropriate tool for evaluating and presenting the entropy distribution in the system. The capsules diameter and the HTF inlet temperature are the considered parameters in this study. Results are summarized as below: •
The diameter reduction leads to an increase in the surface to volume ratio which
results in an enhancement of the rate of heat transfer from the PCM to the HTF. •
A large value for the rate of entropy generation is not essentially equivalent to low
efficiency and the true criterion for the efficiency is the entropy generation number. On this bases a reduction in the spheres diameter inspite of increaseing the rate of entropy generation,
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ACCEPTED MANUSCRIPT improve the system’s efficiency. Wheras, inlet temperature reduction increases the rate of entropy generation and reduces the system’s efficiency. •
Employing smaller capsules is more effective when the inlet temperature is lower in a
way that reducing the diameter from 60 mm to 10 mm for the inlet temperature of 40 °C
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results in 11 percent improvement in the efficiency while the same reduction in the diameter brings about 21 percent enhancement when the inlet temperature is 30 °C. •
Results revealed that the diameter reduction to 20 mm is considerably effective and
•
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more reduction does not leave a sensible consequence on the system’s performance.
The current study investigates the system from thermodynamic view point.
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Considering the aspect, the 20 mm case is the most advantageous one because as the capsules’ diameter reduces more, the second law efficiency doesn’t increase considerably while the rate of entropy generation still rises. In fact, without a more increment in entropy
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generation, an acceptable efficiency is available if this diameter is chosen.
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ACCEPTED MANUSCRIPT Nomenclatures
ħ k
LD )S*
T
T5 T7
Mass (kg) Total enthalpy (J/kg) Thermal conductivity (W/m. K) Latent heat of fusion (J/kg) Source term (N/m3) Temperature (K) Phase change temperature (K) Reference temperature (K)
T PCM average temperature (K)
Q
Qj
Velocity vector (m/s) Heat transfer (J)
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)V*
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M
Gravity (m/s2)
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g
Specific heat capacity (J/kg K)
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Heat transfer rate (W)
Heat flux density (W/m2)
s
Specific entropy (J/kg.K)
SUVW
Rate of entropy generation per volume unit (W/m3. K)
N=
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SUV
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)))* JY
Rate of global entropy generation (W/K) Entropy generation number
H
Height of the cylinder (m)
D
Diameter of the cylinder (m)
d
PCM sphere diameter (m)
∀
Volume (m3)
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ACCEPTED MANUSCRIPT S
Area (m2)
Greek symbols
n
σ )*
Φ Ѱ τ
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m
Liquid fraction Dynamic viscosity (Pa.s) Density (kg/m3) Entropy flux vector (W/m2. K)
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λ
Expansion coefficient (1/K)
Latent heat ratio Second law efficiency Stress tensor (N/m2)
Reference
init
Initial state
in
Inlet state
s
solidus
l
liquidus
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ref
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Subscripts
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β
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ACCEPTED MANUSCRIPT References [1] Pakrouh R., Hosseini M.J., Ranjbar A.A., Bahrampoury R., A numerical method for PCM-based pin fin heat sinks optimization, Energy Conversion and Management, 103 (2015) 542-552.
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[2] Kalbasi R., Salimpour M. R., Constructal design of phase change material enclosures used for cooling electronic devices, Applied Thermal Engineering 84 (2015) 339-349.
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studies of hybrid PCM embedded in plastering mortar for enhanced thermal behaviour of buildings, Energy 94 (2016) 250-261.
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[4] Zhang S., Wu W., Wang Sh., Integration highly concentrated photovoltaic module exhaust heat recovery system with adsorption air-conditioning module via phase change materials, Energy, In press, Available online 6 November 2016.
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thermo-regulating textiles using paraffin wax microcapsules, Thermochim Acta 498 (2010)
[6] Mondieig D, Rajabalee F, Laprie A, Oonk HAJ, Calvet T, Cuevas-Diarte MA, Protection
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of temperature sensitive biomedical products using molecular alloys as phase change
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material, Transfus Apher Sci 28 (2003) 143-148. [7] Li Q., Mostafavi Tehrani S., Taylor R., Techno-economic analysis of a concentrating solar collector with built-in shell and tube latent heat thermal energy storage, Energy 121 (2017) 220-237.
[8] Lu X., Ma R., Wang Ch., Yao W., Performance analysis of a lunar based solar thermal power system with regolith thermal storage, Energy 107 (2016) 227-233.
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ACCEPTED MANUSCRIPT [9] Raam Dheep G., Sreekumar A., Influence of accelerated thermal charging and discharging cycles on thermo-physical properties of organic phase change materials for solar thermal energy storage applications, Energy Conversion and Management, 105 (2015) 13-19. [10] Bejan, A., Two Thermodynamic Optima in the design of sensible heat units for thermal
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storage, J. Heat Transfer 100 (1978) 708-712
[11] Jegadheeswaran S., Pohekar S.D., Kousksou T., Exergy based performance evaluation of latent heat thermal storage system: A review, Renewable and Sustainable Energy Reviews 14
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(2010) 2580–2595.
[12] Li G., Energy and exergy performance assessments for latent heat thermal energy
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storage systems, Renewable and Sustainable Energy Reviews 51(2015) 926–954. [13] MacPhee D., Dincer I., Beyene A., Numerical simulation and exergetic performance assessment of charging process in encapsulated ice thermal energy storage system, Energy 41 (2012) 491-498.
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[14] Guelpa E., Sciacovelli A., Verda V., Entropy generation analysis for the design improvement of a latent heat storage system, Energy 53 (2013) 128-138. [15] Pizzolato., Sciacovelli A., Verda V., Transient local entropy generation analysis for the
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design improvement of a thermocline thermal energy storage, Applied Thermal Engineering
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76 (2015) 391-399.
[16] Mosaffa A.H., Garousi farshi L, Exergoeconomic and environmental analyses of an air conditioning system using thermal energy storage, Applied energy 162 (2016) 515-526. [17] Mosaffa A.H., Garousi Farshi L., Infante Ferreira C.A., Rosen M.A., Advanced exergy analysis of an air conditioning system incorporating thermal energy storage, Energy 77 (2014) 945-952. [18] Rezaie B., Reddy B.V., Rosen M.A., Exergy analysis of thermal energy storage in a district energy application, Renewable Energy 74 (2015) 848-854.
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ACCEPTED MANUSCRIPT [19] Bindra H., Bueno P., Morris J.F., Shinnar R., Thermal analysis and exergy evaluation of packed bed thermal storage systems, Applied Thermal Engineering 52 (2013) 255-263. [20] Bindra H., Bueno P., Morris J.F., Sliding flow method for exergetically efficient packed bed thermal storage, Applied Thermal Engineering 64 (2014) 201-208.
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[21] Xia L., Zhang P., Wang R.Z., Numerical heat transfer analysis of the packed bed latent heat storage system based on an effective packed bed model, Energy 35 (2010) 2022-2032. [22] Silva P.D., Goncalves L.C., Pires L., Transient behaviour of a latent-heat thermal energy
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store: numerical and experimental studies, Applied Energy 73 (2002) 83–98.
[23] Brent A.D., Voller V.R., Reid K.J., Enthalpy-porosity technique for modeling
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convection–diffusion phase change: application to the melting of a pure metal, Numerical Heat Transfer, Part B 13 (1988) 297–318.
[24] Hosseini M.J., Rahimi M., Bahrampoury R., Experimental and computational evolution of a shell and tube heat exchanger as a PCM thermal storage system, Int Communications in
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Heat and Mass Transfer 50 (2014) 128–136.
[25] Bellan S., Gonzalez-Aguilar J., Romero M., Rahman M., Goswami D., Stefanakos E.,
2015 ) 758 – 768
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Numerical investigation of PCM-based thermal energy storage system, Energy Procedia 69 (
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[26] Pakrouh R., Hosseini M.J., Bahrampoury R., A parametric investigation of a PCM-based pin fin heat sink, Mechanical Science 6 (2015) 65–73. [27] Nallusamy N., Sampatha S., Velraj R., Experimental investigation on a combined sensible and latent heat storage system integrated with constant/varying (solar) heat sources, Renewable Energy 32 (2007) 1206–1227. [28] Kaizawa A, Kamano H, Kawai A, Jozuka T, Senda T, Maruoka N. Thermal and flow behaviors in heat transportation container using phase change material. Energy Conversion and Management 49 (2008) 698-706.
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ACCEPTED MANUSCRIPT [29] Bejan A. Entropy generation minimization. Boca Raton: CRC; 1996. [30] De Groot SR, Mazur P. Non-equilibrium thermodynamics. New York: Dover Publications; 2011. [31] Benedetti P, Sciubba E., Numerical calculation of the local rate of entropy generation in
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the flow around a heated finned-tube, American Society of Mechanical Engineers, Advanced
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water heating systems for building applications, Energy and Buildings 39 (2007) 509–516.
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ACCEPTED MANUSCRIPT
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Fig. 1. A schematic view of a packed bed storage unit
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Fig. 2. Schematic of the 2D axisymmetric computational domain
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ACCEPTED MANUSCRIPT 75 70
water flow rate = 2 lit /min
60 55 50 45 40
30
0
20
40
60
80
100
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Experimental data [27] Present work
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PCM Temperature (°C)
65
120
140
Time (min)
Fig. 3. Comparison of PCM temperature between present work and Nallusamy et al. [27]
3.25E-07
3.20E-07
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0.3
3.15E-07
0.2
3.10E-07
. S heat . S friction
0.1
0
0
3.30E-07
3.05E-07
3.00E-07 1000
2000
3000
4000
5000
Time (s)
Fig. 4. Comparison of the magnitude of entropy generation terms
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Entropy generation rate (W/K)
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Tinlet= 30 °C dp= 10 mm
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Entropy generation rate (W/K)
0.4
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T inlet= 40 °C
Solidification time (min)
500
T inlet= 30 °C
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400
300
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200
0
10
20
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100
30
40
50
60
Sphere diameter (mm) (b) 100
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60
40
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Solid percentage (%)
80
dp= 60 mm dp= 45 mm dp= 30 mm
T inlet= 40 °C
dp= 20 mm dp= 15 mm dp= 10 mm
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20
0
0
100
200
300
400
500
600
Time (min)
Fig. 5. Variation of total solidification time versus PCM sphere diameter
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ACCEPTED MANUSCRIPT (a) 3000
2000
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Tinlet= 40 °C
1500
dp= 60 dp= 45 dp= 30 dp= 20 dp= 15
1000
dp= 10 mm
500
0
100
200
300
400
500
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mm mm mm mm mm
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Heat flux (W)
2500
600
700
Time (min)
(b) 1
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0.6
0.4
T inlet= 40 °C
dp= dp= dp= dp= dp= dp=
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Heat release ratio
,η
0.8
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0.2
0
0
100
200
300
400
500
60 mm 45 mm 30 mm 20 mm 15 mm 10 mm
600
Time (min)
Fig. 6. Effect of PCM sphere diameter on (a) heat flux and (b) heat release ratio
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ACCEPTED MANUSCRIPT (a) 1
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0.6
dp= 60 mm dp= 45 mm dp= 30 mm dp= 20 mm
0.4
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dp= 15 mm dp= 10 mm
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Latent heat ratio
,φ
0.8
100
Solid percentage (%)
(b)
1
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φFinal = 0.85
0.6
φFinal = 0.77
0.4
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Latent heat ratio
,φ
0.8
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0.2
0
0
0.2
T inlet= 40 °C T inlet= 30 °C
dp= 30 mm
0.4
0.6
0.8
1
Solid percentage (%)
Fig. 7. The latent heat ratio for various (a) PCM sphere diameter and (b) inlet HTF temperature
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ACCEPTED MANUSCRIPT (a) 58
dp= 60 dp= 45 dp= 30 dp= 20 dp= 15 dp= 10
Tinlet= 40 °C
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49
46
43
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Outlet Temperature (°C)
55
mm mm mm mm mm mm
0
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Time (min)
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mm mm mm mm mm mm
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60 45 30 20 15 10
49
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Outlet Temperature (°C)
55
dp= dp= dp= dp= dp= dp=
T inlet= 40 °C
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40
0
0.2
0.4
0.6
0.8
1
Solid percentage (%)
Fig. 8. Variation of HTF outlet temperature for various sphere diameter
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ACCEPTED MANUSCRIPT (a) 60
dp= 60 mm
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PCM Temperature (°C)
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45
Tinlet= 40 °C
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35
30 100
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Time (min)
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dp= 60 mm
20 min
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350 min 400 min
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PCM Temperature (°C)
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Tinlet= 40 °C
450 min
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500 min
40
0
0.2
0.4
0.6
0.8
1
y/H
Fig. 9. Temporal variation of the PCM at three different positions inside the tank
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ACCEPTED MANUSCRIPT . S heat (W/m3k)
40 41 42 43 44 45 46 47 48 49 50 51 52
0.010.05 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t=2000 s
t=10000 s
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Temperature (°C)
t=20000 s
t=30000 s
Fig. 10. Distribution of temperature(left hand) and entropy generation(right hand) for dp=60 mm and T = 40°C
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Temperature (°C)
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40 41 42 43 44 45 46 47 48 49 50 51 52
t=2000 s
t=5000 s
t=8000 s
t=12000 s
Fig. 11. Distribution of temperature for dp=60 mm (left hand) and dp=30 mm (right hand) for T = 40°C
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ACCEPTED MANUSCRIPT . S heat (W/m3k) 1
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4.5
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t=5000 s
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0.010.05 0.1 0.5
t=12000 s
Fig. 12. Distribution of entropy generation for dp=60 mm (left hand) and dp=30 mm (right hand) for T = 40°C
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. S heat (W/m3k)
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0.010.05 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t=2000 s
t=8000 s
t=14000 s
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Fig. 13. Distribution of entropy generation for T = 40°C (left hand) and T = 30°C (right hand) for dp=60 mm
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ACCEPTED MANUSCRIPT (a)
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0.09
0.06 dp= 60 mm dp= 45 mm dp= 30 mm dp= 20 mm
0.03
dp= 15 mm dp= 10 mm 0
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100
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Solid percentage (%)
(b)
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dp= 30 mm
0.2
0.1
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Entropy generation rate (W/K)
0.4
0
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20
T inlet= 40 °C T inlet= 30 °C 40
60
Solid percentage (%)
80
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Fig. 14. Global entropy generation rate SUV for various (a) PCM sphere diameters and (b) inlet HTF temperatures
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dp= 60 dp= 45 dp= 30 dp= 20 dp= 15 dp= 10
0.8
mm mm mm mm mm mm
Tinlet= 40 °C
Ns
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dp= dp= dp= dp=
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dp= 15 mm dp= 10 mm
Ns
0.6
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0.2
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Fig. 15. Entropy generation number Ns for (a) T = 40°C and (b) T = 30°C
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ACCEPTED MANUSCRIPT 100
80
(a)
70 60 50
{ =
40
{ =
30 20 10 0 15
20
30
rst uvwxyx z{|}x~xy
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0.225 0.2 0.175 0.15 0.125 0.1 0.075 0.05 0.025 0 10
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Global entropy generation rate
0.25
45
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10
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Second law efficiency
90
15
20
30
45
(b) { = { =
60
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rst uvwxyx z{|}x~xy
Fig. 16. Time averaged values of (a) second law efficiency and (b) entropy generation rate for different inlet
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HTF temperature
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Phase change temperature (˚C) Thermal conductivity (W/m.K) Dynamic viscosity (kg/m.s) Thermal expansion coefficient (1/K) Latent heat (J/kg)
value 10,15,20,30,45,60 30,40 57 15 6
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Table 2 Studied parameters and levels Parameters PCM sphere diameter (mm) HTF inlet temperature (˚C) Initial temperature (˚C) Reference temperature (˚C) HTF mass flow rate (lit/min)
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Specific heat (J/kg.K)
value 809.5 (solid) 771 (liquid) 2900 (solid) 2100 (liquid) 52.1 0.15 0.0055 0.00079 243500
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Table 1 Thermophysical properties of the PCM [22] Property Density (kg/m3)
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ACCEPTED MANUSCRIPT Highlights:
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Detailed entropy generation for PCM based packed bed storage was numerically investigated Simulations are performed with an effective packed bed model. Reduction in capsules diameter increases both the rate of entropy generation and the efficiency. Utilization of smaller capsules is more effective when the inlet HTF temperature is lower.
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•