Proceedings of CHT-15 ICHMT International Symposium on Advances in Computational Heat Transfer May 25-29, 2015, Rutgers University, Piscataway, USA CHT-15-077
THE EFFECT OF THE MUSHY-ZONE CONSTANT ON SIMULATED PHASE CHANGE HEAT TRANSFER Ali C. Kheirabadi and Dominic Groulx§ Dept. of Mechanical Engineering, Dalhousie University, Halifax NS, Canada §Correspondence author. Fax: +1 (902) 423-6711 Email:
[email protected]
ABSTRACT This paper presents a numerical study aimed at understanding the impact of the mushy zone constant, Amush, on simulated phase change heat transfer. This parameter is found in the CarmanKoseny equation which is used in the enthalpy-porosity formulation for modeling phase change; this approach models fluid flow within the mushy region as flow through a porous medium. The melting of lauric acid inside a rectangular thermal storage unit was simulated in COMSOL 4.4 and FLUENT 15.0; with Amush and the melting temperature range, ∆T, being varied per study. The simulated melt front positions were directly compared to experimental results presented by Shokouhmand and Kamkari [2013]. Results showed that Amush is an important parameter for accurately modelling phase change heat transfer; in particular, high Amush values corresponded to slower melting rates and the smallest Amush values resulted in unphysical predictions of the melt front development. Additionally, it was concluded that Amush and ∆T are not independent of one another in their roles of accurately modeling the melting rate; different values of ∆T would require different values of Amush to achieve the same melt front development. Further efforts are required to identify ideal values for these parameters, as well as to determine the extent to which these parameters hold for different materials and physical setups. It is anticipated that this paper will lead to further discussion on the significance of the mushy zone as a numerical technique for accurately modelling phase change heat transfer. NOMENCLATURE Amush A B C Cp(T) D D(T) L S(T) r
Mushy –zone constant [kg/s] µ(T) constant [-] µ(T) constant [K] µ(T) constant [1/K] Modified heat capacity [J/kg·K] µ(T) constant [1/K2] Gaussian function [-] Latent heat of fusion [J/kg] Source term [kg/s] Sigmoid function constant [1/K]
Greek symbols ∆Hf Specific latent enthalpy [J/kg] ∆T Mushy-zone temperature range [K] ε Carman-Koseny equation constant [-] µ(T) Modified viscosity [kg/m·s] φ(T) Melt fraction Subscripts 0 Initial condition, or reference value B Buoyancy PCM Phase change material pg plexiglass
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INTRODUCTION The use of phase change materials (PCMs) in thermal storage systems has received a considerable amount of attention within the heat transfer and renewable energy communities in recent years. This is due to the large heat storage capacity and isothermal behavior associated with PCMs undergoing solidliquid phase transitions [Pielichowska and Pielichowski 2014]. Major examples of the applications of PCMs include thermal storage of solar energy [Mazman et al. 2009, Joseph et al. 2014], electronic component temperature control [Ling et al. 2014], temperature regulation in residential and commercial buildings [Rastogi et al. 2015, Sharif et al. 2015] and performance enhancement of solar collection technologies [Nkwetta and Haghighat 2014]. Nonlinear behaviors such as natural convection and latent heat absorption dominate the heat transfer processes within a melting PCM body; therefore playing a significant role in the transient heat storage capability of PCMs. Numerical methods capable of accurately predicting these nonlinear behaviors are crucial for the design optimization of intermittent thermal storage processes involving PCMs [Dutil et al. 2011]. Solid-liquid phase change heat transfer is of the type of problem known as a “moving boundary” problem. With very few solutions known analytically, the vast majority of these problems are solved numerically; either through the use of a deformable mesh/grid [Lacroix and Voller 1990], or more commonly, through the use of the enthalpy or modified specific heat methods [Ogoh and Groulx 2012]. These methods account for the additional latent heat of fusion required for energy conservation during phase change. It has also been shown through numerous experimental studies that natural convection plays a dominating role within the liquid phase of a melting body [Liu and Groulx 2014]. Consequently, the addition of buoyancy forces are required to model natural convection during general phase change processes of various PCMs. One prominent numerical method that has been developed for solid-liquid phase change problems involving natural convection is the enthalpy-porosity formulation [Voller and Prakash 1987]. The popularity of this technique has grown due to its applicability to a broad range of phase change problems, as well as its compatibility with existing numerical solvers [Voller et al. 1987]; the enthalpyporosity formulation currently serves as the solidification and melting model within ANSYS FLUENT. Similar to the prevalent enthalpy formulation [Al-abidi et al. 2013], this technique assumes phase change to occur over a finite temperature range; thus generating an artificial mushy region through which the melt fraction of a fluid element varies from zero (solid phase) to 1 (liquid phase). Naturally, the velocity of the fluid element within the mushy region should also vary from zero (solid phase) to the natural convection velocity (liquid phase). The enthalpy-porosity formulation deals with this velocity transition by modelling flow within the mushy region as flow through a porous medium. A sink term, in the form of the Carman-Koseny equation, is added to the Navier-Stokes equations to mimic the effect of damping within the mushy region. A similar suite of equations can also be programmed within COMSOL Multiphysics to solve for transient phase change heat transfer with natural convection [Groulx and Biwole 2014]. In this case, the enthalpy-porosity formulation serves as a basis on which small modifications are applied to capture other aspects of the physics involved; the addition of the sink term also facilitates convergence using the finite element method [Samara et al. 2012]. The treatment of fluid flow from natural convection within the mushy zone has been the subject of numerous discussions at recent heat transfer conferences. Although the Carman-Koseny equation is used in the majority of available models, a constant within this formulation, called the “mushy-zone” constant (Amush), is still the subject of frequent inquiry. Various commercial software guidelines and researchers have designated values to Amush ranging from 103 to nearly 1010; a variation of 7 orders of magnitude.
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This paper presents a study on the impact of varying the mushy-zone constant, Amush, on the overall simulation results of a melting PCM body. A simple rectangular geometry is employed for which experimental results are published [Shokouhmand and Kamkari 2013]; the PCM used is dodecanoic acid, more commonly known as lauric acid, for which thermo-physical properties are well known [Desgrosseilliers et al. 2013]. The impact of the value of Amush on the physicality of the results is therefore assessed. Simulations are performed using two different software: ANSYS FLUENT 15.0 which has a predefined phase change physics where the constant Amush is an input, and COMSOL Multiphysics 4.4 in which the entire set of physical equations needed to account for natural convection and phase change interface movement must be added by the user, again with constant Amush playing a definite role. The numerical results are compared in terms of melting interface position over time, and from the comparison to experimental results, light is shed on the physical impact of the “mushy-zone” constant. PHYSICAL MODEL As mentioned in a recent review paper on numerical modeling of PCMs, there is less and less research carried out where numerical results are compared to an appropriate experimental counterpart for validation [Dutil et al. 2011]. The research work presented in this paper is based on the experimental study presented by Shokouhmand and Kamkari [2013]; where the melting of lauric acid in a rectangular cavity was quantified by visually tracking the shape of the melting interface over time. The three dimensional cavity was surrounded by plexiglass on five sides and an aluminum plate on the remaining side; water was circulated through the aluminum plate to maintain constant temperatures of 55, 60 or 70°C. The results obtained from this setup enable simplification of the system to a quasi-two dimensional numerical model. Geometry The experimental apparatus was modeled in two dimensions in COMSOL Multiphysics 4.4 and ANSYS FLUENT 15.0 as shown in Fig. 1. The geometry consists of a 0.05 × 0.12 m cavity filled with lauric acid; enclosed within 0.025 m thick plexiglass on the bottom, left and top sides. The outer surfaces of the plexiglass are insulated. In the experimental setup, the entire right side of the unit was sealed with a 35 mm thick aluminum plate which was maintained at a constant temperature. Therefore, for modeling purposes, the entire right side of the system is simply maintained at a constant temperature Tw = 343.15 K (70°C). The entire system is initially at a constant temperature T0 = 299.15 K (26°C). Material Properties The PCM used in this study is lauric acid, who’s thermophysical properties are presented in Table 1. The values of specific heats*, thermal conductivities, densities, and latent heat of fusion from Shokouhmand and Kamkari [2013] agree with vigorous experimental data presented in other papers [Desgrosseilliers et al. 2013]; as a result these values are used. The temperature for the onset of melting presented (43.55°C) agrees with accepted values and experimentally observed values [Murray and Groulx 2014] measured in the author’s lab; the actual temperature for onset of melting is taken as Tm = 316.65 K (43.5°C). The remaining properties are taken from Yaws’ Handbook of Thermodynamics and Physical Properties [2003]. The thermophysical properties used for plexiglass are listed in Table 2. NUMERICAL MODEL The enthalpy formulation requires a single domain in which the same set of governing equations are used to model both solid and liquid phases of a PCM [Dutil et al. 2011]. The transition from solid to liquid, and vice versa, occurs over a finite temperature range (∆T) generating an artificial mushy region at the solid-liquid interface. The fluid velocity within the mushy region varies from zero (at the solid *
Values of specific heat of the solid are measured starting at 1,900 J/kg·K at 20°C.
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boundary) to the natural convection velocity (at the liquid boundary) as the melt fraction varies from 0 to 1.
Figure 1. Model geometry and boundary conditions.
Table 1 Thermophysical Properties of Lauric Acid Tm
cp,s cp,l ks kl
316.65 K 940 kg/m3 885 kg/m3 2,180 J/kg∙K* 2,390 J/kg∙K 0.16 W/m∙K 0.14 W/m∙K 0.008 kg/m∙s 0.004 kg/m∙s 0.0008 K-1 187,200 J/kg
℃ ℃
L
Table 2 Thermophysical Properties of Plexiglass 1180 kg/m3 1464 J/kg∙K 0.19 W/m∙K
cp,pg kpg
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The following subsections present the governing equations and user-defined functions/material properties used in the ANSYS FLUENT and COMSOL Multiphysics for modeling such a method. In both cases, phase change is quantified through Eq. (1) for the melt fraction: ( )=
0
,
,
!( " !∆ / ) ∆
1
−
∆
−
∆
+
∆
