Home Physical Sciences Effect of convection heat transfer on thermal energy storage unit
Article Open Access

Effect of convection heat transfer on thermal energy storage unit

  • EMAIL logo , , , and
Published/Copyright: December 31, 2018
Download Article (PDF)
Open Physics
From the journal Open Physics Volume 16 Issue 1

Abstract

Latent heat storage represents a promising technique to achieve net zero energy buildings. This work investigates the behaviour of phase change material (PCM) inside a rectangular enclosure, which represents the geometry of a latent heat storage system. The left side of the unit is exposed to a constant temperature (Th), while the other three walls are exposed to convection heat transfer boundary condition [h= 5, 10, and 15 W/(m2 K)] and different ambient temperatures (T∞ = 297◦ and 307◦K). The ambient temperatures were selected to be at/above the melting temperature of the studied PCM (coconut oil). To study the melting process of the PCM, the continuity, Navier-Stokes and energy equation were used. The Navier-Stokes equations were modified using the Carman-Kozeny relation. The finite element method was utilized to produce numerical results. The results are presented in terms of flow and thermal fields, Nusselt number (Nu), and the melt fraction (MF) of the PCM. The results show that, when T∞ = Tm, the melting rate of the PCM slows down with increasing the convection heat transfer coefficient. While the melting rate accelerates with increasing the convection heat transfer coefficient when T∞ > Tm.

Keywords: PCM; convection; Carman-Kozeny; melting process; Nusselt number
PACS: 44.25.+f; 47.10.ad; 47.11.Fg

Nomenclature

A( T) Parameter is defined in Eq. (6)

B( T) Parameter is defined in Eq. (7)

C Arbitrary constant in Eq. (6)

cp Specific heat at constant pressure (J/(kg K))

D Melting interface position from the left wall (m)

D( T) Gaussian function

g Gravitational acceleration (m/s2)

h Convection heat transfer (W/(m2 K))

hf Latent heat of fusion (J/kg)

k Thermal conductivity (W/(m K))

L Enclosure height (m)

MF Melt fraction

Nu Average Nusselt number

p Pressure (Pa)

q Arbitrary constant in Eq. (6)

T Temperature (K)

ΔT Range of melt temperature of the PCM (K)

t Time (s)

u Velocity component of the liquid PCM in the x-direction (m/s)

v Velocity component of the liquid PCM in the y-direction (m/s)

x Horizontal coordinate (m)

y Vertical coordinate (m)

Greek symbols

β Coefficient of volumetric thermal expansion (1/K)

μ Dynamical viscosity (Pa s)

ρ Density (kg/m3)

∞ Ambient

Subscripts

h Hot

l Liquid

m Melting

s Solid

1 Introduction

Conventional energy resources are currently diminishing because of a growing need for energy. For instance, substantial amounts of energy are required just for heating purposes. According to Natural Resources Canada, 63 % of the energy supply is used to heat spaces and 19% to heat water [1]. Additionally, it is these same CO2-emitting sources that are considered by many to be the main reason behind global warming. As a result, scientists are zealously working on ways to reduce dependency on these resources by searching for cleaner, renewable sources of energy.

Most alternate sources of energy will require some form of thermal energy storage. Latent heat storage represents a promising technique to achieve net zero energy buildings [2].

Joulin et al. [3] numerically studied the thermal behavior of PCM installed within solar passive walls. The study included 1-D and 2-D analyses for different aspect ratios. The vertical walls were differentially heated, and the horizontal walls were insulated. The authors reported that conduction dominates in the early stage of melting process then convection takes a growing role throughout the melting process.

Mbaye and Bilgen [4] performed a numerical study to investigate the impact of heat flux and the aspect ratio of the enclosure on the melting process of PCM. The vertical walls of the enclosure were subjected to constant heat flux and constant temperature. The authors found that the ratio of the heat flux entering and leaving the enclosure is not impacted by the aspect ratio at the beginning of the melting process; as melting continues the heat flux ratio increases with an increasing aspect ratio. Also, they found that the melting process accelerates as the aspect ratio decreases.

The melting of PCM inside a tall enclosure, that was subjected to constant heat flux at one vertical side, was numerically and experimentally studied by Pala and Joshi [5]. The authors found that at the beginning of the melting process, conduction dominates; after that, convection plays a significant role in the melting process. At the final stage of the melting process, sensible heating plays a role in the thermal storage and increases the temperature of the PCM.

Alawadhi [6] numerically investigated the impact of PCM on reducing heat gain. The PCM-filled cylinders were fixed inside a roof. The PCM type, quantity, and location inside a brick were studied. Convection boundary condition above and below the roof was applied. The authors found that using and locating the PCM in the center of the brick dramatically reduces the heat gain.

The main aim of this article is to study the effect of convection heat transfer conditions on the melting of PCM. The PCM fills a square enclosure. The left wall of the enclosure is isothermally heated, while the top, right, and bottom walls are exposed to convection boundary condition.

2 Physical and mathematical models

The energy storage system can be approximated by a 2-D enclosure. A schematic diagram of the enclosure system is shown in Figure 1. Initially, the solid form of the PCM occupies the enclosure. The initial temperature of the PCM is assumed to be equal to its melting temperature. The left vertical wall of the enclosure is maintained at constant temperature (Th) which is above the melting temperature (Tm) of the PCM. The remaining three walls are applied to convection boundary condition. The following assumptions are applied: the liquid phase of PCM is a Newtonian and incompressible fluid, all thermophysical properties of the PCM are assumed to be constant, the Boussinesq model is used in the buoyancy force term. In addition, in the energy equations, the internal heat generation and the viscous dissipation effect are neglected, and laminar fluid motion is assumed.

Figure 1 Schematic illustration of (a) the thermal storage system, (b) the physical model of the thermal storage unit
Figure 1

Schematic illustration of (a) the thermal storage system, (b) the physical model of the thermal storage unit

Conservation equations of mass (continuity), momentum, and energy (in the liquid and solid regions) are used to model the complete flow and thermal fields as shown below [3, 7].

(1)∂ρl∂t+∂ρlu∂x+∂ρlv∂y=0
(2)∂u∂t+u∂u∂x+v∂u∂y=1ρl−∂p∂x+μ∂2u∂x2+∂2u∂y2−A(T)u
(3)∂v∂t+u∂v∂x+v∂v∂y=1ρl−∂p∂y+μ∂2v∂x2+∂2v∂y2+gρβT−Tm−A(T)v
(4)∂T∂t+u∂T∂x+v∂T∂y=kl(ρcp)l∂2T∂x2+∂2T∂y2
(5)∂T∂t=ks(ρcp)s∂2T∂x2+∂2T∂y2.

In the above equations, ρl is the density of the liquid PCM, t is the time, u and v are the velocity components of the liquid PCM in the x and y-directions, p is the pressure, μ is the dynamic viscosity of PCM, g is the gravity, β is the coefficient of thermal expansion of the liquid PCM, T is the temperature, Tm is the melting temperature of the PCM, kl is the thermal conductivity of the liquid PCM, cpl is the specific heat at constant pressure of the liquid PCM, ks is the thermal conductivity of the solid PCM, ρs is the density of the solid PCM, and cps is the specific heat at constant pressure of the solid PCM.

In the momentum equations Eqs. (2) and (3), Kozeny-Carman relation is used to model the flow within the interface. The parameter A(T) in Eqs. (2) and (3) is defined to achieve a gradual reduction of the velocities of the liquid PCM from a finite value in the liquid zone to zero in the solid zone. To implement Kozeny-Carman relation, parameter A( T) is defined as [8]

(6)A(T)=C(1−B(T))2(B(T)3+q)

where C and q are arbitrary constants of value of 105 and 10−3, respectively. B( T) can be defined as [8]

(7)B(T)=0,T<(Tm−ΔT)T−Tm+ΔT2ΔT,(Tm−ΔT)<T<(Tm+ΔT)1,T>(Tm+ΔT)

where ΔT is the range of temperatures over which the melting process occurs. If the PCM is a pure material, ΔT is zero, and the mushy zone is thin. On the other hand, if the PCM is an impure material, ΔT is greater than zero, and the mushy zone is wider than that for pure material.

B(T) is zero when the temperature is lower than Tm, while it is one when the temperature is higher than Tm. Equations (6) and (7) can be used to calculate the thermophysical properties of the PCM, as follows [8]

(8)ρ(T)=ρs+(ρl−ρs)B(T)
(9)k(T)=ks+(kl−ks)B(T)
(10)cp(T)=cps+(cpl−cps)B(T)+hfD(T)
(11)μ(T)=μl(1+A(T)).

s and l stand for the solid and liquid phases of the PCM, respectively, and hf is the latent heat of fusion of the PCM. D(T), which is a Gaussian function, is used to determine the latent heat over a temperature range ΔT. D(T) can be calculated from [8]

(12)D(T)=e−T(T−Tm)2ΔT2πΔT2

The boundary and initial conditions of the thermal storage unit can be written as:

(13)lowerhorizontalwall:−k∂Tx,0,t∂y=hTx,0,t−T∞,u=v=0,upperhorizontalwall,−k∂Tx,L,t∂y=hTx,L,t−T∞,u=v=0,rightwall:−k∂Tx,L,t∂y=hTL,y,t−T∞,u=v=0,leftwall:T(0,y,t)=Th,u=v=0,interfacecondition:T(D,y,t)=Tm,ρhf∂D∂t=−k∂TD,y,t∂x−∂D∂y∂TD,y,t∂y,initialcondition:T(x,y,0)=Tm,u=v=0.

where h is the convection heat transfer coefficient, T∞ is the ambient temperature, L is the height of the unit, and D is the position of the melting interface starting from the left wall.

The averaged Nusselt number is calculated from [9]

(14)Nu=1ΔT∫0L−∂T∂xx=0dy

3 Numerical procedure

The governing equations, Eqs. (1-5), with the boundary and initial conditions are numerically solved using the finite element method. For this purpose, a numerical scheme was built using the commercial software COMSOL 4.3b. To avoid the results dependency on the mesh size, a careful examination is conducted. Four element sizes were tested, 2522 (fine), 6580 (finer), 16986 (extra fine), and 26544 (extremely fine) elements, as shown in Figure 2. The independency test is conducted for the case at h=10W/(m2 K) and T∞ = 297◦K. Insignificant differences are observed among the four cases. However, for the lower numbered elements, 2522, the solution witnesses a slight fluctuation. The two higher numbered cases, 16986 and 26544, consume more time to complete the solution. As a result, the finer meshing of 6580-elements is selected in the present work. The proposed discretization numerical scheme consists of 5964 triangular elements and 616 quadrilateral elements. The time step is 10 s. The simulation is aborted when the relative tolerance is smaller than 10−3 for the continuity, momentum, and energy equations.

Figure 2 The mesh independency test at h = 10 W/(m2K) and T∞ = 307◦K
Figure 2

The mesh independency test at h = 10 W/(m2K) and T∞ = 307◦K

4 Results and discussion

In the present work, a numerical study was conducted to investigate the impact of convection heat transfer condition on the melting process of PCM. Individually, the impact of the convection heat transfer coefficient and ambient temperature were investigated. Carman-Kozeny relation was used to simulate the liquid-solid interface. Coconut oil was chosen as the PCM because its melting temperature is close to the comfortable temperature range of occupied spaces, in air conditioning sector, of 293.5◦ - 296.5◦K (20.5◦ - 23.5◦C) [10]. The thermophysical properties of coconut oil are listed in Table 1.

Table 1

Thermophysical properties of coconut oil

Properties (units)Coconut oil
SolidLiquid
ρ (kg/m3)920918
μ (Pa s)-0.0268
cp (K/(kg K))3.7501.670
k (W/(m K))0.1660.166
β (1/K)0.7 × 10−3
hf (J/kg)103.000
Tm (K)297

A validation was performed to confirm the capability of the built model. The present model was validated by comparing the liquid-solid interface evolution gained from this model with the experimental results of Gau and Viskanta [11]. As shown in Figure 3, a good agreement was achieved. The insignificant discrepancies may have resulted from the theoretical assumptions.

Figure 3 Comparison of interface position during melting of gallium between Gau and Viskanta [11] and the present study
Figure 3

Comparison of interface position during melting of gallium between Gau and Viskanta [11] and the present study

Figure 4 shows the effect of the convection heat transfer coefficient on the flow and thermal fields at 1500 s, when T∞ = Tm = 297◦K, Figure 4(a), and T∞ > Tm = 307◦K, Figure 4(b). The studied convection heat transfer coefficients were [h= 5, 10, and 15 W/(m2 K)]. The arrows in Figure 4 represent the flow field, and the counters represent the thermal field where the blue region is the solid PCM and the colored region is the liquid PCM. As the PCM that is adjacent to the hot walls heats up and melts, it becomes lighter due to low density. The relatively low-density melted PCM rises along the hot wall assisted by the buoyancy force, then it heads right. The impermeability of the upper wall forces the melted PCM to move to the liquid-solid interface, convecting the thermal energy with it. As the warm melted PCM hits the cold liquid-solid interface, it transfers thermal energy to the solid PCM. As a result, the liquid PCM becomes less warm, and its density increases. The increment in the PCM density drives the melted PCM down along the liquid-solid interface. While flowing along the liquid-solid interface, the melted PCM keeps transferring the thermal energy to the liquid-solid interface [12]. However, the amount of transferred thermal energy to the liquid-solid interface from the melted PCM decreases along the liquid-solid interface. The nonuniform heat transferred results in a higher amount of melted PCM in the top of the unit. The value of convection heat transfer coefficient and the ambient temperature play a significant role on the rate of the melting process and the shape of the solid PCM throughout the melting process. For the same h, when T∞ = 297◦K (= Tm), Figure 4(a), the melting process solely starts from the left wall. However, when T∞ = 307◦K (> Tm), Figure 4(b), the melting process occurs starting from the left, top, and right walls. Although the unit was also exposed to ambient temperature from the bottom, insignificant effects result from exposing the bottom wall. The high temperatures of the side and top walls assist in initiating the melting of the PCM from the top and the right walls besides the melting due to heating from the left wall. With changing h, the rate of melting of the PCM differs according to T∞. When T∞ equals Tm, the rate of dissipating heat from the unit to the ambient increases with increasing h. As a result, the melting rate decreases with increasing h, as shown in Figure 4(a). The left wall of the unit is heated at a temperature that is higher than the melting temperature of the PCM. When the PCM melts, the melted PCM temperature becomes higher than the melting temperature. In the case when T∞ equals Tm, a heat loss occurs from the hot melted PCM to the ambient. The heat loss is increased by increasing h, which subsequently deaccelerates the melting rate. Figure 4(b) shows the acceleration in the melting process when increasing h under the condition of T∞ is higher than Tm. In this case, the rate of heating the PCM from the ambient increases due to increasing h. When the ambient temperature T∞ is higher than Tm, the ambient becomes a heat source to melt the solid PCM. As T∞ is higher, the melting becomes faster [13]. In addition, as h increases, the heat transfers faster from the ambient to the unit, and subsequently the PCM melts faster.

Figure 4 Flow (arrows) and thermal fields (contours) for different convection heat transfer coefficient (h = 5, 10, and 15 W/(m2K)) at 1500 s (a) T∞ = 297◦K, (b) T∞ (= 307◦K) > Tm
Figure 4

Flow (arrows) and thermal fields (contours) for different convection heat transfer coefficient (h = 5, 10, and 15 W/(m2K)) at 1500 s (a) T∞ = 297◦K, (b) T∞ (= 307◦K) > Tm

The effect of convection heat transfer coefficient h and the ambient temperature Tm on Nu, along the left wall of the unit, is shown in Figure 5. As soon as the melting starts, a thin layer of liquid PCM is formed, and conduction becomes the dominant heat transfer mode. Conduction is dominant in this stage of the melting process because the viscous force overcomes the buoyancy force due to the scant liquid PCM layer [14]. In this stage, the heat transfer decreases with time due to increasing the temperature of the vicinity liquid along the left wall. Then Nu arises when the liquid PCM layer becomes wider, and the viscous force decreases while the buoyancy force increases. As heating progresses, the temperature of the liquid PCM increases leading to a drop in the convection heat transfer rate, and as a result Nu reduces. Figure 5 shows that at the beginning of the melting process, both h and T∞ have an insignificant effect on Nu. After that, there is a positive effect of increasing h on improving Nu for both T∞ values. A similar improvement in Nu is obtained with increasing T∞ for all the studied values of h.

Figure 5 Effect of convection heat transfer coefficient and ambient temperature on Nu
Figure 5

Effect of convection heat transfer coefficient and ambient temperature on Nu

Figure 6 shows the effect of the convection heat transfer coefficient h and the ambient temperature T∞ on the melt fraction of the PCM, MF. The PCM melts faster at the early stages of the melting process because of the high temperature difference between the hot walls and the melted PCM adjacent to the hot walls [15]. However, the melting rate decreases with time as the liquid PCM temperature increases which reduces the heat transfer rate to the unit. When T∞ equals Tm, reducing h assists the PCM to melt faster. Where by reducing h, the heat losses from the unit decreases. While increasing h, in the case of T∞ is higher than Tm, the heat transfer rate to the unit increases. As a result, the melting rate increases.

Figure 6 Effect of convection heat transfer coefficient and ambient temperature on the melt fraction of the PCM
Figure 6

Effect of convection heat transfer coefficient and ambient temperature on the melt fraction of the PCM

5 Conclusions

A numerical study was performed to investigate the behavior of the melting process under convection heat transfer condition. Besides the governing equations of the fluid flow and energy, Carman-Kozeny relation was applied to simulate the liquid-solid interface. COMSOL 4.3b software, which is based on finite element method, was used to build the numerical model of the present study. It can be concluded that the melting rate increases with increasing h when T∞ is higher than Tm. In this case, the unit is exposed to a higher heating rate. When T∞ equals Tm, increasing h leads to increasing the heat loss from the unit. As a result, the melting rate of the PCM decreases.

References

[1] Heating equipment, 2013, http://www.nrcan.gc.ca/energy/products/categories/heating/13740Search in Google Scholar

[2] Lei J., Yang J., Yang E.-H., Energy performance of building envelopes integrated with phase change materials for cooling load reduction in tropical Singapore, Appl. Energy, 2016, 162, 207–217.10.1016/j.apenergy.2015.10.031Search in Google Scholar

[3] Joulin A., Younsi Z., Zalewski L., Rousse D.R., Lassue S., A numerical study of the melting of phase change material heated from a vertical wall of a rectangular enclosure, Int. J. Comput. Fluid Dyn., 2009, 23, 553–566.10.1080/10618560903203723Search in Google Scholar

[4] Mbaye M., Bilgen E., Phase change process by natural convection–diffusion in rectangular enclosures, Heat Mass Transf., 2001, 37, 35–42.10.1007/s002310000095Search in Google Scholar

[5] Pal D., Joshi Y.K., Melting in a side heated tall enclosure by a uniformly dissipating heat source, Int. J. Heat Mass Transf., 2001, 44, 375–387.10.1016/S0017-9310(00)00116-2Search in Google Scholar

[6] Alawadhi E.M., Thermal analysis of a building brick containing phase change material, Energy Build., 2008, 40, 351–357.10.1016/j.enbuild.2007.03.001Search in Google Scholar

[7] Sciacovelli A., Colella F., Verda V., Melting of PCM in a thermal energy storage unit: Numerical investigation and effect of nanoparticle enhancement, Int. J. Energy Res., 2013, 37, 1610–1623.10.1002/er.2974Search in Google Scholar

[8] Samara F., Groulx D., Biwole P.H., Natural convection driven melting of phase change material: comparison of two methods, in: Excerpt Proceeding COMSOL Conf., 2012. https://www.comsol.nl/paper/download/150959/groulx_paper.pdfSearch in Google Scholar

[9] Oztop H.F., Abu-Nada E., Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow, 2008, 29, 1326–1336.10.1016/j.ijheatfluidflow.2008.04.009Search in Google Scholar

[10] Wang S.K., Handbook of air conditioning and refrigeration, second, McGraw-Hill, 2001.Search in Google Scholar

[11] Gau C., Viskanta R., Melting and solidification of a pure metal on a vertical wall, J. Heat Transf., 1986, 108, 174–181.10.1115/1.3246884Search in Google Scholar

[12] Al-Jethelah M.S., Tasnim S.H., Mahmud S., Dutta A., Melting of nano-phase change material inside a porous enclosure, Int. J. Heat Mass Transf., 2016, 102, 773–787.10.1016/j.ijheatmasstransfer.2016.06.070Search in Google Scholar

[13] Mahdaoui M., Kousksou T., Blancher S., Msaad A.A., El Rhafiki T., Mouqallid M., A numerical analysis of solid–liquid phase change heat transfer around a horizontal cylinder, Appl. Math. Model., 2014, 38, 1101–1110.10.1016/j.apm.2013.08.002Search in Google Scholar

[14] Al-Jethelah M., Tasnim S.H., Mahmud S., Dutta A., Melting of nano-PCM in an enclosed space: Scale analysis and heatline tracking, Int. J. Heat Mass Transf., 2018, 119, 841–859.10.1016/j.ijheatmasstransfer.2017.11.106Search in Google Scholar

[15] Al-Jethelah M., Tasnim S.H., Mahmud S., Dutta A., Nano-PCM filled energy storage system for solar-thermal applications, Renew. Energy, 2018, 126, 137–155.10.1016/j.renene.2018.02.119Search in Google Scholar

Received: 2017-09-17
Accepted: 2018-10-27
Published Online: 2018-12-31

© 2018 M. S. M. Al-Jethelah et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

  1. Regular Articles
  2. A modified Fermi-Walker derivative for inextensible flows of binormal spherical image
  3. Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics
  4. Three-dimensional atom localization via probe absorption in a cascade four-level atomic system
  5. Determination of the energy transitions and half-lives of Rubidium nuclei
  6. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 1 - model development
  7. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 2 - model validation
  8. Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity
  9. Constructing analytic solutions on the Tricomi equation
  10. Feynman diagrams and rooted maps
  11. New type of chaos synchronization in discrete-time systems: the F-M synchronization
  12. Unsteady flow of fractional Oldroyd-B fluids through rotating annulus
  13. A note on the uniqueness of 2D elastostatic problems formulated by different types of potential functions
  14. On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics
  15. Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications
  16. Siewert solutions of transcendental equations, generalized Lambert functions and physical applications
  17. Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction
  18. A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis
  19. Dynamics of a dry-rebounding drop: observations, simulations, and modeling
  20. Modeling the initial mechanical response and yielding behavior of gelled crude oil
  21. Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation
  22. Solitary wave solutions of two KdV-type equations
  23. Applying industrial tomography to control and optimization flow systems
  24. Reconstructing time series into a complex network to assess the evolution dynamics of the correlations among energy prices
  25. An optimal solution for software testing case generation based on particle swarm optimization
  26. Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation
  27. Alternative methods for solving nonlinear two-point boundary value problems
  28. Global model simulation of OH production in pulsed-DC atmospheric pressure helium-air plasma jets
  29. Experimental investigation on optical vortex tweezers for microbubble trapping
  30. Joint measurements of optical parameters by irradiance scintillation and angle-of-arrival fluctuations
  31. M-polynomials and topological indices of hex-derived networks
  32. Generalized convergence analysis of the fractional order systems
  33. Porous flow characteristics of solution-gas drive in tight oil reservoirs
  34. Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method
  35. A Note on Koide’s Doubly Special Parametrization of Quark Masses
  36. On right-angled spherical Artin monoid of type Dn
  37. Gas flow regimes judgement in nanoporous media by digital core analysis
  38. 4 + n-dimensional water and waves on four and eleven-dimensional manifolds
  39. Stabilization and Analytic Approximate Solutions of an Optimal Control Problem
  40. On the equations of electrodynamics in a flat or curved spacetime and a possible interaction energy
  41. New prediction method for transient productivity of fractured five-spot patterns in low permeability reservoirs at high water cut stages
  42. The collinear equilibrium points in the restricted three body problem with triaxial primaries
  43. Detection of the damage threshold of fused silica components and morphologies of repaired damage sites based on the beam deflection method
  44. On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem
  45. Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
  46. Analysis of projectile motion in view of conformable derivative
  47. Computing multiple ABC index and multiple GA index of some grid graphs
  48. Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
  49. Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
  50. An activity window model for social interaction structure on Twitter
  51. Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
  52. On the zagreb polynomials of benzenoid systems
  53. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  54. The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
  55. Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
  56. Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
  57. Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
  58. Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
  59. An attempt to geometrize electromagnetism
  60. Structure of traveling wave solutions for some nonlinear models via modified mathematical method
  61. Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
  62. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
  65. Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  68. Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
  71. Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
  72. Temperature distribution around thin electroconductive layers created on composite textile substrates
  73. Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
  74. Analysis of spatial thermal field in a magnetic bearing
  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
  76. Assessment of short/long term electric field strength measurements for a pilot district
  77. Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
  78. Magnetic transmission gear finite element simulation with iron pole hysteresis
  79. Pulsed excitation terahertz tomography – multiparametric approach
  80. Low and high frequency model of three phase transformer by frequency response analysis measurement
  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
  82. Optimal design of a for middle-low-speed maglev trains
  83. Eddy current modeling in linear and nonlinear multifilamentary composite materials
  84. The visual attention saliency map for movie retrospection
  85. AC/DC current ratio in a current superimposition variable flux reluctance machine
  86. Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
  87. Cogging force reduction in linear tubular flux switching permanent-magnet machines
  88. Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
  89. Electro-magneto-hydrodynamic lubrication
  90. 3-D Electromagnetic field analysis of wireless power transfer system using K computer
  91. Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
  92. Design, fabrication and testing of electroadhesive interdigital electrodes
  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
https://doi.org/10.1515/phys-2018-0108
Keywords for this article
PCM; convection; Carman-Kozeny; melting process; Nusselt number
Creative Commons
BY-NC-ND 4.0

Articles in the same Issue

  1. Regular Articles
  2. A modified Fermi-Walker derivative for inextensible flows of binormal spherical image
  3. Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics
  4. Three-dimensional atom localization via probe absorption in a cascade four-level atomic system
  5. Determination of the energy transitions and half-lives of Rubidium nuclei
  6. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 1 - model development
  7. Three phase heat and mass transfer model for unsaturated soil freezing process: Part 2 - model validation
  8. Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity
  9. Constructing analytic solutions on the Tricomi equation
  10. Feynman diagrams and rooted maps
  11. New type of chaos synchronization in discrete-time systems: the F-M synchronization
  12. Unsteady flow of fractional Oldroyd-B fluids through rotating annulus
  13. A note on the uniqueness of 2D elastostatic problems formulated by different types of potential functions
  14. On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics
  15. Computational methods and traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation via two methods and its applications
  16. Siewert solutions of transcendental equations, generalized Lambert functions and physical applications
  17. Numerical solution of mixed convection flow of an MHD Jeffery fluid over an exponentially stretching sheet in the presence of thermal radiation and chemical reaction
  18. A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis
  19. Dynamics of a dry-rebounding drop: observations, simulations, and modeling
  20. Modeling the initial mechanical response and yielding behavior of gelled crude oil
  21. Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation
  22. Solitary wave solutions of two KdV-type equations
  23. Applying industrial tomography to control and optimization flow systems
  24. Reconstructing time series into a complex network to assess the evolution dynamics of the correlations among energy prices
  25. An optimal solution for software testing case generation based on particle swarm optimization
  26. Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation
  27. Alternative methods for solving nonlinear two-point boundary value problems
  28. Global model simulation of OH production in pulsed-DC atmospheric pressure helium-air plasma jets
  29. Experimental investigation on optical vortex tweezers for microbubble trapping
  30. Joint measurements of optical parameters by irradiance scintillation and angle-of-arrival fluctuations
  31. M-polynomials and topological indices of hex-derived networks
  32. Generalized convergence analysis of the fractional order systems
  33. Porous flow characteristics of solution-gas drive in tight oil reservoirs
  34. Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method
  35. A Note on Koide’s Doubly Special Parametrization of Quark Masses
  36. On right-angled spherical Artin monoid of type Dn
  37. Gas flow regimes judgement in nanoporous media by digital core analysis
  38. 4 + n-dimensional water and waves on four and eleven-dimensional manifolds
  39. Stabilization and Analytic Approximate Solutions of an Optimal Control Problem
  40. On the equations of electrodynamics in a flat or curved spacetime and a possible interaction energy
  41. New prediction method for transient productivity of fractured five-spot patterns in low permeability reservoirs at high water cut stages
  42. The collinear equilibrium points in the restricted three body problem with triaxial primaries
  43. Detection of the damage threshold of fused silica components and morphologies of repaired damage sites based on the beam deflection method
  44. On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem
  45. Ion acoustic quasi-soliton in an electron-positron-ion plasma with superthermal electrons and positrons
  46. Analysis of projectile motion in view of conformable derivative
  47. Computing multiple ABC index and multiple GA index of some grid graphs
  48. Terahertz pulse imaging: A novel denoising method by combing the ant colony algorithm with the compressive sensing
  49. Characteristics of microscopic pore-throat structure of tight oil reservoirs in Sichuan Basin measured by rate-controlled mercury injection
  50. An activity window model for social interaction structure on Twitter
  51. Transient thermal regime trough the constitutive matrix applied to asynchronous electrical machine using the cell method
  52. On the zagreb polynomials of benzenoid systems
  53. Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance
  54. The Greek parameters of a continuous arithmetic Asian option pricing model via Laplace Adomian decomposition method
  55. Quantifying the global solar radiation received in Pietermaritzburg, KwaZulu-Natal to motivate the consumption of solar technologies
  56. Sturm-Liouville difference equations having Bessel and hydrogen atom potential type
  57. Study on the response characteristics of oil wells after deep profile control in low permeability fractured reservoirs
  58. Depiction and analysis of a modified theta shaped double negative metamaterial for satellite application
  59. An attempt to geometrize electromagnetism
  60. Structure of traveling wave solutions for some nonlinear models via modified mathematical method
  61. Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation
  62. Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations
  63. Effect of magnetic field and heat source on Upper-convected-maxwell fluid in a porous channel
  64. Physical cues of biomaterials guide stem cell fate of differentiation: The effect of elasticity of cell culture biomaterials
  65. Shooting method analysis in wire coating withdrawing from a bath of Oldroyd 8-constant fluid with temperature dependent viscosity
  66. Rank correlation between centrality metrics in complex networks: an empirical study
  67. Special Issue: The 18th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  68. Modeling of electric and heat processes in spot resistance welding of cross-wire steel bars
  69. Dynamic characteristics of triaxial active control magnetic bearing with asymmetric structure
  70. Design optimization of an axial-field eddy-current magnetic coupling based on magneto-thermal analytical model
  71. Thermal constitutive matrix applied to asynchronous electrical machine using the cell method
  72. Temperature distribution around thin electroconductive layers created on composite textile substrates
  73. Model of the multipolar engine with decreased cogging torque by asymmetrical distribution of the magnets
  74. Analysis of spatial thermal field in a magnetic bearing
  75. Use of the mathematical model of the ignition system to analyze the spark discharge, including the destruction of spark plug electrodes
  76. Assessment of short/long term electric field strength measurements for a pilot district
  77. Simulation study and experimental results for detection and classification of the transient capacitor inrush current using discrete wavelet transform and artificial intelligence
  78. Magnetic transmission gear finite element simulation with iron pole hysteresis
  79. Pulsed excitation terahertz tomography – multiparametric approach
  80. Low and high frequency model of three phase transformer by frequency response analysis measurement
  81. Multivariable polynomial fitting of controlled single-phase nonlinear load of input current total harmonic distortion
  82. Optimal design of a for middle-low-speed maglev trains
  83. Eddy current modeling in linear and nonlinear multifilamentary composite materials
  84. The visual attention saliency map for movie retrospection
  85. AC/DC current ratio in a current superimposition variable flux reluctance machine
  86. Influence of material uncertainties on the RLC parameters of wound inductors modeled using the finite element method
  87. Cogging force reduction in linear tubular flux switching permanent-magnet machines
  88. Modeling hysteresis curves of La(FeCoSi)13 compound near the transition point with the GRUCAD model
  89. Electro-magneto-hydrodynamic lubrication
  90. 3-D Electromagnetic field analysis of wireless power transfer system using K computer
  91. Simplified simulation technique of rotating, induction heated, calender rolls for study of temperature field control
  92. Design, fabrication and testing of electroadhesive interdigital electrodes
  93. A method to reduce partial discharges in motor windings fed by PWM inverter
  94. Reluctance network lumped mechanical & thermal models for the modeling and predesign of concentrated flux synchronous machine
  95. Special Issue Applications of Nonlinear Dynamics
  96. Study on dynamic characteristics of silo-stock-foundation interaction system under seismic load
  97. Microblog topic evolution computing based on LDA algorithm
  98. Modeling the creep damage effect on the creep crack growth behavior of rotor steel
  99. Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs
  100. Chinese open information extraction based on DBMCSS in the field of national information resources
  101. 10.1515/phys-2018-0079
  102. CPW-fed circularly-polarized antenna array with high front-to-back ratio and low-profile
  103. Intelligent Monitoring Network Construction based on the utilization of the Internet of things (IoT) in the Metallurgical Coking Process
  104. Temperature detection technology of power equipment based on Fiber Bragg Grating
  105. Research on a rotational speed control strategy of the mandrel in a rotary steering system
  106. Dynamic load balancing algorithm for large data flow in distributed complex networks
  107. Super-structured photonic crystal fiber Bragg grating biosensor image model based on sparse matrix
  108. Fractal-based techniques for physiological time series: An updated approach
  109. Analysis of the Imaging Characteristics of the KB and KBA X-ray Microscopes at Non-coaxial Grazing Incidence
  110. Application of modified culture Kalman filter in bearing fault diagnosis
  111. Exact solutions and conservation laws for the modified equal width-Burgers equation
  112. On topological properties of block shift and hierarchical hypercube networks
  113. Elastic properties and plane acoustic velocity of cubic Sr2CaMoO6 and Sr2CaWO6 from first-principles calculations
  114. A note on the transmission feasibility problem in networks
  115. Ontology learning algorithm using weak functions
  116. Diagnosis of the power frequency vacuum arc shape based on 2D-PIV
  117. Parametric simulation analysis and reliability of escalator truss
  118. A new algorithm for real economy benefit evaluation based on big data analysis
  119. Synergy analysis of agricultural economic cycle fluctuation based on ant colony algorithm
  120. Multi-level encryption algorithm for user-related information across social networks
  121. Multi-target tracking algorithm in intelligent transportation based on wireless sensor network
  122. Fast recognition method of moving video images based on BP neural networks
  123. Compressed sensing image restoration algorithm based on improved SURF operator
  124. Design of load optimal control algorithm for smart grid based on demand response in different scenarios
  125. Face recognition method based on GA-BP neural network algorithm
  126. Optimal path selection algorithm for mobile beacons in sensor network under non-dense distribution
  127. Localization and recognition algorithm for fuzzy anomaly data in big data networks
  128. Urban road traffic flow control under incidental congestion as a function of accident duration
  129. Optimization design of reconfiguration algorithm for high voltage power distribution network based on ant colony algorithm
  130. Feasibility simulation of aseismic structure design for long-span bridges
  131. Construction of renewable energy supply chain model based on LCA
  132. The tribological properties study of carbon fabric/ epoxy composites reinforced by nano-TiO2 and MWNTs
  133. A text-Image feature mapping algorithm based on transfer learning
  134. Fast recognition algorithm for static traffic sign information
  135. Topical Issue: Clean Energy: Materials, Processes and Energy Generation
  136. An investigation of the melting process of RT-35 filled circular thermal energy storage system
  137. Numerical analysis on the dynamic response of a plate-and-frame membrane humidifier for PEMFC vehicles under various operating conditions
  138. Energy converting layers for thin-film flexible photovoltaic structures
  139. Effect of convection heat transfer on thermal energy storage unit
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Winner of the OpenAthens UX Award 2026
Winner of the OpenAthens UX Award 2026
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 1.4.2026 from https://www.degruyterbrill.com/document/doi/10.1515/phys-2018-0108/html
Scroll to top button