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Hajar Lagziri Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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Hanae El Fakiri Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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Abdelmajid El Bouardi Department of Physics, Faculty of Science, University of Abdelmalek Essaadi, Tetouan, Morocco

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Abstract

The thermo convective instability of the Darcy-Benard problem (DB) using Robin (third-kind) thermal conditions is investigated here. We consider a viscous Newtonian fluid saturating a porous layer in which the layer is sandwiched between two impermeable boundaries. The upper and the lower walls are modelled in the form of the Neumann (second-kind) and the Robin (third-kind) thermal conditions, respectively. The difference in the temperature distribution between both phases allows the lack of a local thermal equilibrium model to be present. As a consequence, the third kind of thermal condition brings about one extra dimensionless parameter of the Biot number to the usual one of the inter-heat transfer coefficient and the thermal conductivity ratio. The normal modes method adopted in a linear stability analysis gives rise to perturbed governing equations. The eigenvalue problem is handled numerically as a result of the perturbed governing equations leading to the marginal stability condition.

Abstract

The thermo convective instability of the Darcy-Benard problem (DB) using Robin (third-kind) thermal conditions is investigated here. We consider a viscous Newtonian fluid saturating a porous layer in which the layer is sandwiched between two impermeable boundaries. The upper and the lower walls are modelled in the form of the Neumann (second-kind) and the Robin (third-kind) thermal conditions, respectively. The difference in the temperature distribution between both phases allows the lack of a local thermal equilibrium model to be present. As a consequence, the third kind of thermal condition brings about one extra dimensionless parameter of the Biot number to the usual one of the inter-heat transfer coefficient and the thermal conductivity ratio. The normal modes method adopted in a linear stability analysis gives rise to perturbed governing equations. The eigenvalue problem is handled numerically as a result of the perturbed governing equations leading to the marginal stability condition.

1 Introduction

The Darcy-Benard problem (DB) [1], which embodies the thermo convective instability in porous media, becomes the cornerstone of many industrial and environmental applications. For instance, the onset of the convection cell accelerates the dissolution and the mixing of CO2 in the aqueous phase, which, in turn, increases its storage lifespan in deep geological layers [2]. The emergence of this phenomenon does not only favor the process of CO2 sequestration but also the extraction of the thermal energy from the geothermal reservoirs where the underground heating condition occurs. The main process behind the thermal instability in these fields is the prevailing effect of buoyant forces in comparison with resistive mechanisms that can be generated by viscous dissipation, thermal conductivity or other physical properties [3]. Whatever the medium in which instability arises, the motion of the fluid particles caused by buoyancy effects is always driven either by the concentration of solute particles or by the vertical temperature gradient [4, 5]. However, the materials manufactured by porous media are more effective to hasten the heat transfer process than the case of Newtonian clear fluid. The Rayleigh number Rc appears in the former configuration at the critical value of Rc39 while in the latter one at Rc1707 [6, 7].

The prevailing effects of the local thermal non-equilibrium (LTNE) render the thermal boundary conditions to be mismatched with the standard forms defined in the equilibrium one. As the temperature profile for the solid structure diverges from the fluid phase, several thermal boundaries are required to investigate the Darcy-Benard problem [8, 9]. On the other hand, whatever the type of these boundaries in a LTNE the marginal stability curves may tend to behave the same way as the standard one if the two dimensionless numbers defined by the internal heat transfer coefficient H and the thermal conductivity ratio γ resulted in the discrepancy between thermal conductivities of phases and the loss or gain of heat from one phase to another have specific values or limits. For instance, there are two possible conditions at which the transition from LTNE to LTE is acceptable: an infinite limit of H with a nonzero value of γ or finite and non-vanishing values of H with large γ. The former case assures a rapid and higher amount of heat transferred between phases while the latter corresponds to a case in which the saturated fluid holds higher thermal conductance than the solid skeleton. A small H or γ can reverse the approach of the LTE towards a non-equilibrium one [10–13].

The majority of the analysis carried out in the Darcy-Benard problem is focused on the case where the local thermal equilibrium condition (LTE) takes place. In other words, the convection cell in a horizontal saturated porous layer is handled only when a Newtonian saturating fluid has a temperature field equal to a solid matrix. All these instability analyses are studied either with impermeable or permeable bounding planes beside different positions of the porous layer [14–16]. However, the feature of the LTE is not always true as we can encounter in some cases a disparity in the thermal conductivity or/and a weakness in the heat transfer coefficient, which can yield a relaxation of the local thermal equilibrium regime. From a practical standpoint, the local thermal non-equilibrium (LTNE) model has a vital effect on the cooling process of countless engineering devices [10, 17]. On the other hand, the approach of the LTNE can highlight difficulties in the modelling of thermal boundaries, especially in the case of isoflux [18–20]. The objective behind this work is to analyze the convective instability in the case where two impermeable walls are supposed to be modelled in the form of Newton's cooling law equation for Robin (third-kind) thermal conditions and uniform heat flux (Model A) for Neumann (second-kind) thermal conditions.

2 Mathematical model

The setup assumed here is the archetype of the Darcy-Benard configuration (DB) with two rigid walls [1]. In addition, two different temperature fields are allowed, which enables violation of local thermal equilibrium. The upper wall is heated by a heater to yield uniform heat flux (Model A) while the lower one is determined as Newton's cooling law equation. For definiteness, our investigation has been performed with porosity φ = 1/2. A sketch of the system geometry is drawn in Fig. 1. Besides, we presume that Darcy's law and the Oberbeck-Boussinesq approximation are applicable. The governing equations in the dimensionless forms are written as [19, 12]:
×v=Ra×(Tfez),
χTst=2Ts+Hγ(TfTs),
Tft+vTf=2TfH(TfTs).
Fig. 1.
Fig. 1.

The sketch of the Darcy-Benard configuration (DB)

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

While the dimensionless boundary conditions used in Eqs (3) are summed up in Fig. 1 as:
z=0:Tfz=Bf(Tf1),Tsz=γBf(Ts1),w=0,
z=1:γTfz+Tsz+1+γ=0,Ts=Tf,w=0.

The notation of the vector v indicates the velocity field, T indicates the temperature [K] and the subscripts “s” and “f” indicate a saturating fluid phase and a solid matrix, respectively. In contrast, the dimensionless inter-heat transfer coefficient, thermal conductivity ratio, Biot number, and modified Rayleigh number in a porous medium are briefly the parameters noted as H, γ, Bf and Ra.

Eqs (1) and (2) are a virtue of scaling quantities alongside the curl operator applied to Darcy's law. The proposed scaling quantities are [19, 12]:
x*xl,t*tl2αf,v*(u,v,w)φαfl,Ts,f*Tw+ΔTTs,f,λf=αf(ρC)f,λs=αs(ρC)s,
T=qwlλm,λm=(1φ)λs+φλf,am=λm(ρC)f,γ=φλf(1φ)λs,χ=αfαs,
H=hl2φλf,Rα=(1+γ)βgTKlγαmvf,Bf=hw,flλf

The superscript of the star notation refers to dimensional variables. α is the thermal diffusivity [m2 s−1], Tw is the temperature of the lower wall [K], ρ is the density [kg m−3], h is the inter-phase volumetric heat transfer coefficient [W (m3 K−1)], ez unit vector in the z-direction, C is the heat capacity per unit of mass [J (kg K−1)], qw is the heat flux at the upper wall [W m−2], νf is the kinematic viscosity [m2 s−1], g is the modulus of the gravitational acceleration [m s−2], β is thermal expansion coefficient [K−1], t is the time [s] and 1 is the thickness of the layer [m]. The subscript “w” means the wall while the symbol “m” denotes the effective. λ is the thermal conductivity [W (m K−1)], hw,f is the superficial heat transfer coefficient of the fluid phase [W (m2 K−1)], K is the permeability of the medium [m2].

3 Basic state

The basic profile assumed for Eqs (1) and (2) is a steady state with zero velocity that fluid exists in a solid skeleton. Consequently, the basic profile is:
vb=0
Ts,b=(2Bf(1z)γΛ+(1+γ)32Λ1+γ)coshΛBf(1+γ(γ+Bf(z1)(1+γ))sinh(z1)ΛBf(2γΛcoshΛ+Bfγ(1+γ)sinhΛ
Tf,b=(2Bf(1z)γΛ+(1+γ)32Λ1+γ)coshΛ+Bf(1+γ(γ+Bf(z1)(1+γ))sinh(z1)ΛBf(2γΛcoshΛ+Bfγ(1+γ)sinhΛ

Where: Λ=H(1+γ)

We have used the subscript “b” as a symbol of the basic flow.

The first derivative of Tf,b that will be adopted in the governing equations is
Tf,b=1+(1+γ)Λcosh[(1+y)Λ]2γΛcosh[Λ]+Bfγ(1+γ)sinh[Λ]

3.1 Basic state for local thermal equilibrium cases

The cases where the temperature of solid structure matches the fluid phase can emerge through two different limits:
  1. a)The limiting case of H with γO(1) that refers to the basic temperature profile of both phases as
Tsb=Tfb=1+γ+2γBf(1z)2γBf.
  1. b)The case of γ with HO(1) whose basic temperature profile is,
Tsb=Tfb=1z1Bf.

3.2 Basic state for local thermal non-equilibrium cases

There are two limits in which we can have two autonomous temperature profiles: H0 with γO(1) or γ0 with HO(1).The basic temperature profile for both phases in the former limit is
Tfb=1+γ+Bf2(1+z)γ(1+γ)+Bf(z+(1+z)γ+γ2)Bfγ(2+Bf+Bfγ),
Tsb=1+Bf+γ+Bf(1+Bf)(1+z)γ+Bf(Bf(1+z)+z)γ2Bfγ(2+Bf+Bfγ).
While in the latter one it is
Tfb=eHz(e2He2Hz+eHz(1+Bf(1+z))+eH(2+z)(1+BfBfz))(1+coth[H])2Bf,
Tsb=11Bfz.

The basic temperature profile for all cases is expressed following the condition of φ = 1/2. Therefore, the Biot number of the solid phase can be replaced by the fluid one through the relation of Bs=γBf. For instance, some authors [20] have adopted this relation as a condition in the numerical method to look for the critical value of BsBf without mentioning anything related to the basic temperature profile for φ = 1/2. This in turn exhibits a discrepancy between our basic temperature profiles and the one cited in [20], knowing that they have employed the same thermal boundary conditions as we did.

4 Stability analysis

The basic solution is investigated by using the perturbation process which defines the main flow of each variable as a sum between the basic state and the distributed flow. More precisely,
Tf,s=Tfs,b+εTfs,v=vb+εv
According to linearization method, all the second orders of ε are cancelled in Eqs (1) and (2). Thus, we can obtain:
×v=Ra×(Tfez),
χTst=2Ts+Hγ(TfTs),
Tft+wTf,b=2Tf+H(TsTf),
z=0:Tfz=Bf(Tf1),Tsz=γBf(Ts1),w=0
z=1:γTfz+Tsz+1+γ=0,Ts=Tf,w=0.
The governing equations are invariable with respect to rotation around the z-axis. We consider velocity perturbations in the two-dimensional plane (x, z), in the form of perturbed stream function ψ(x,z):
u=ψz,w=ψx
The normal modes method allows us to tackle the perturbed equations by expressing the small secondary flow as
{ψ(x,z,t),Tfs(x,z,t)}={iΨ(z),θfs(z)}ei(kxωt)
Hence the symbols of Ψ(z) and θ(z) are used to describe the perturbation amplitude functions. The wave number is defined by the symbol k while the growth rate and the angular frequency are noted with ωi=Im{ω} and ωr=Re{ω} respectively. The complex parameter ω is defined as the sum of the imaginary and real parts of ω.
The linear stability analysis is focused only on those modes whose growth rate is neither growing nor decaying with time. This feature implies the condition of ωi=0. Therefore, substituting Eqs (14) and (15) into Eqs (13) with ωi=0 yields that,
Ψk2Ψ+kRaθf=0
θsk2θs+γH(θfθs)+iωrχθs=0
θfk2θf+H(θsθf)+kTfΨ+iωrθf=0
z=0:θf=Bfθf,θs=γBfθs,Ψ=0
z=1:γθf+θs=0,θs=θf,Ψ=0

The two primes in Eqs (16) mean a second derivative with respect to z.

5 The principle of exchange of instabilities

As the principal of exchange of instabilities cannot be proven analytically owing to the variable coefficients that arise in the expression of Tf,b(z) in Eq. (7), we will be obligated to verify its validity through the numerical solution presented in section (6) by using the eigenvalue problem written in Eqs (16). To satisfy this principle we must obtain numerically the value of ωr=Re{ω}=0. In other words, the fluid layer has to be without oscillatory instability behaviour.

According to the outcomes of Table 1 the condition of ωr=Re{ω}=0 is fulfilled as all the values are nearer to zero, thus, the principle exchange of instabilities is held. Consequently, we can neglect ωr from the eigenvalue problem Eqs (16) and write that
Ψk2Ψ+kRaθf=0
θsk2θs+γH(θfθs)=0
θfk2θf+H(θsθf)+kTfΨ=0
z=0:θf=Bfθf,θs=γBfθs,Ψ=0
z=1:γθf+θs=0,θs=θf,Ψ=0
Table 1.

The real values of ωr calculated by the numerical method of the Runge-Kutta solver and shooting method

H=102 and Bf=10H=102 and Bf=10
γ=101γ=10γ=101γ=10
kωrkωrkωrkωr
12.260839×102111.049347×101811.657385×101819.989764×106
29.074241×102224.309620×101828.040071×101629.948204×106
33.932491×102139.037235×101832.885237×101539.582209×106
47.540579×102141.569522×101743.826143×101449.666727×106
51.298220×102052.674431×101757.646567×101559.269420×106
62.027311×102064.297010×101764.574078×101569.209937×106

6 Numerical solutions

The numerical solutions proposed for the eigenvalue problem Eqs (17) are based on the two combined procedures carried out between the Runge-Kutta solver and the finding-root algorithm of the shooting method. To apply both of them, we need to switch Eqs (17) into an initial value problem by providing other boundary conditions to Eq. (17d),
Ψ(0)=1,θf=s1,θs=s2

We take into consideration the normalisation condition of Ψ(0)=1. The two unknown's parameters s1 and s2 are defined as values for θs(0) and θf(0), in the meantime they are assigned as one of those input parameters fixed in the shooting method. This latter is also implemented via the function Find Root of the Mathematica10 software. In general, the process employed here allows us to develop accurate values for s1, s2 and Ra. The resulting function of Ra(k) is employed to draw the marginal stability curves for every input parameter (H, γ, Bf).

Table 2 shows a comparison of the critical values noted for work [20] in the case where Robin and Neumann thermal boundary conditions are used with the presence of the free surface, and the critical values calculated for our problem with the same thermal conditions but with rigid impermeable walls. If we look at the modified Rayleigh number defined in this paper Eq. (3) is the same as the one employed in [12] and not the one used by [20]. The incongruence in the definition will make the comparison of the results a little bit unclear or hardly seen. To overcome this discrepancy, we calculate the critical values for both Rc and Rac by considering the same conditions γ, Bf and H, mentioned in [20]. From Table 1 we can remark that the values of Rc and Rac in our problem differ from one another. This is somehow contrasted with what arises in the Brinkman problem whose Rc and Rac are nearly equal (see [11]). As a consequence, the parameter γ has a more vital impact on Darcy's configuration than Brinkman one. On the other hand, the results in [20] point out that as much as γ decreases, the Rc goes closer to zero, in other words, the basic flow becomes alone unstable when γ0. Overall, this limiting case makes the effect of other parameters on the instability behaviour unseen, and this why we have taken by preference the definition of Rac rather than Rc. The difference between the critical values in both studied cases does not depend only on the definition of Rc but also on the hydrodynamic boundary conditions employed by each problem. As is well known the instability in a porous layer emerges at low critical values in the case of an open boundary than an impermeable wall. This in general upholds the finding results in Table 2.

Table 2.

Comparison of Celli et al. [20] results with the study problem

Bf = 1 and H = 0.1
The work of Celli et al. [20]The studied problem
γkcRcγkcRcRac=1+γγRc
11.405615.0164412.0617012.2272424.45449
0.51.517453.681890.52.234598.8284926.48548
0.251.623502.380600.252.386315.6260228.13013

7 Results and discussion

The frames in Figs 25 present the trends of Rac and kc versus Bf with various values of γ while the marginal curves drawn for γ and H with fixed Bf are displayed in Fig. 6. The dashed line describes the critical value of Rac and kc obtained for cases close to the LTE model. We recall that stability always exists in the region situated below the concave shape of the neutral curves, while instability emerges in the area above the concave curves. In Fig. 6 the right frame reveals that decreasing the values of γ tend the marginal stability curves to move continuously upward, therefore prescribing a more stable state. In addition, the results that emerge from Figs 25 ensure a non-monotonic increase of the critical values with respect to γ. Furthermore, the effect of γ→0 does not seem to neglect the influence of H as Rac = ∞ with H→∞. This feature enables the basic state to behave as adiabatic conditions. Regardless of the effect of H, the limit of γ→∞ leads the values of Rac and kc to be nearly as Rac = 27.10, kc = 2.33 apart from the ones in Fig. 3 and in Fig. 5 whose critical values are different, this refers to the definition of Rac. Therefore, a higher value of γ can be regarded as one of the destabilizing factors in this case. Otherwise, the growth of H can bridge the gap between LTNE and LTE when γ has a finite value and Bf →∞. In other words, the parameters of H, Bf and γ can have stabilising or destabilising effect as they are based on the thermal conductivity of the solid and fluid phase besides the inter-phase volumetric heat transfer coefficient h.

Fig. 2.
Fig. 2.

The plots of Rac and kc versus Bf for H = 1/10

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 3.
Fig. 3.

The trends of Rac and kc versus Bf for H = 10

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 4.
Fig. 4.

The trends of Rac and kc versus Bf for H = 1/100

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 5.
Fig. 5.

The plots of Rac and kc versus Bf for H = 100

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

Fig. 6.
Fig. 6.

Marginal stability curves relative to H→0 (right frame) and γ→0 (left frame)

Citation: International Review of Applied Sciences and Engineering 14, 3; 10.1556/1848.2022.00577

8 Conclusion

The studied configuration of the modified Darcy-Benard problem is an approach to the design of metal foams saturated with Newtonian fluid. The solid matrix of the metal foams has thermal conductivity infinitely higher than its counterpart fluid phase, which means the presence of the LTNE model. This design is used as a heat exchanger to enhance the transfer of heat from the cooling Newtonian fluid to the solid body, as a result it can be a good alternative to the finned surfaces. According to the numerical results, the stability effects are much more dominant in γ→0 than in γ→∞ when no heat is transferred between the two phases. Besides, the curves Ra relative to H→0 exhibit more sensitivity via the value of γ when Bf →∞. Broadly speaking, the behaviour of γ, H and Bf can hasten the performance of heat exchange in metal foams between the porous layer and a saturating fluid.

Acknowledgement

This research was not funded by any grant.

References

  • [1]

    C. W. Horton and F. T. Rogers, “Convection currents in a porous medium,” J. Appl. Phys., vol. 16, p. 367, 1945. https://doi.org/10.1063/1.1707601.

    • Search Google Scholar
    • Export Citation
  • [2]

    B. Wen, A. Daria, L. Zhang, and A. Hesse Marc, “Carbon dioxide dissolution in a closed porous medium at low pressure,” J. Fluid Mech., vol. 854, no. 10, pp. 5687, 2018. https://doi.org/10.1017/jfm.2018.622.

    • Search Google Scholar
    • Export Citation
  • [3]

    D. A. Nield and T. Simmons Craig, “A brief introduction to convection in porous media,” Transport Porous Media, vol. 130, no. 10, pp. 237250, 2018. https://doi.org/10.1007/s11242-018-1163-6.

    • Search Google Scholar
    • Export Citation
  • [4]

    R. Dubey and P. V. S .N. Murthy, “The onset of double-diffusive convection in a Brinkman porous layer with convective thermal boundary conditions,” AIP Adv., vol. 9, 2019, Paper no. 045322. https://doi.org/10.1063/1.5087037.

    • Search Google Scholar
    • Export Citation
  • [5]

    A. Bouachir, M. Mamon, R. Rebhi, and S. Benissaadi, “Linear and nonlinear stability analyses of double-diffusive convection in a vertical brinkman porous enclosure under soret and dufour effects,” Fluids, vol. 6, no. 8, pp. 292302, 2021. https://doi.org/10.3390/fluids6080292.

    • Search Google Scholar
    • Export Citation
  • [6]

    A. D. Nield and A. Bejan, Chapter 6 – Convection in Porous Media: Internal Natural Convection: Heating from below, 4th ed. New York, Springer-Verlag, 2013, pp. 221329, https://doi.org/10.1007/978-1-4614-5541-7_6.

    • Search Google Scholar
    • Export Citation
  • [7]

    C. W. Horton and F. T. Rogers, “Convection currents in a porous medium,” J. Appl. Phys., vol. 16, no. 1, pp. 367377, 1945. https://doi.org/10.1063/1.1707601.

    • Search Google Scholar
    • Export Citation
  • [8]

    M. Parhizi, M. Torabi, and A. Jain, “Local thermal non-equilibrium (LTE) model for developed flow in porous media with spatially-varying biot number,” Int. J. Heat Mass Tran., vol. 164, pp. 19, 2021. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120538.

    • Search Google Scholar
    • Export Citation
  • [9]

    V. Kambiz and Y. Kun, “A note on local thermal non-equilibrium in porous media and heat flux bifurcation phenomenon in porous media,” Transport Porous Media, vol. 96: 169172, 2013. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120538.

    • Search Google Scholar
    • Export Citation
  • [10]

    F.-K. Arman, N. Meysam, and M. Yasser, “Pulsating Flow in a Channel Filled with a Porous Medium under local thermal non-equilibrium condition: an exact solution,” J. Therm. Anal. Calorim., vol. 145, no. 7, pp. 27532775, 2020. https://doi.org/10.1007/s10973-020-09843-0.

    • Search Google Scholar
    • Export Citation
  • [11]

    A Postelnicu and D. A. S. Rees, “The onset of Darcy-Brinkman convection in a porous layer using a thermal nonequlibrium model - Part I: stress-free boundaries,” Int. J. Energy Res., vol. 27, no. 10, pp. 961973, 2003. https://doi.org/10.1002/er.928.

    • Search Google Scholar
    • Export Citation
  • [12]

    N. Banu and D. A. S. Rees, “Onset of Darcy-Benard convection using a thermal Nonequilibrium model,” Int. J. Heat Mass Tran., vol. 45 , no. 11: 22212228, 2002. https://doi.org/10.1016/S0017-9310(01)00331-3.

    • Search Google Scholar
    • Export Citation
  • [13]

    A Barletta and D. A. S Rees, “Local thermal non-equilibrium effects in the Darcy-Benard instability with isoflux boundary conditions,” Int. J. Heat Mass Tran., vol. 55, pp. 384394, 2012. https://doi.org/10.1016/j.ijheatmasstransfer.2011.09.031.

    • Search Google Scholar
    • Export Citation
  • [14]

    A. Barletta, M. Celli, and D. A. S Rees, “Buoyant flow and instability in a vertical cylindrical porous slab with permeable Boundaries,” Int. J. Heat Mass Tran., vol. 157, pp. 210320, 2020. https://doi.org/10.1016/j.ijheatmasstransfer.2020.119956.

    • Search Google Scholar
    • Export Citation
  • [15]

    A. Barletta and D. A. S Rees, “On the onset of convection in a highly permeable vertical porous layer with open boundaries,” AIP Phys. Fluids, vol. 31, 2019, Paper no. 074106. https://doi.org/10.1063/1.5110484.

    • Search Google Scholar
    • Export Citation
  • [16]

    A. Barletta and M. Celli, “The Horton-Rogers-Lapwood problem for an inclined porous layer with permeable boundaries,” Roy. Soc. Publish. Proc. A, vol. 474, 2018. https://doi.org/10.1098/rspa.2018.0021.

    • Search Google Scholar
    • Export Citation
  • [17]

    F. Al-Sumaily Gazy, A. Al Ezzi, A. Dhahad Hayder, M. C. Thompson, and T. Yusaf, “Legitimacy of the local thermal equilibrium hypothesis in porous media: a comprehensive Review,” J. Energy, vol. 14, no. 23, pp. 247, 2021. https://doi.org/10.3390/en14238114.

    • Search Google Scholar
    • Export Citation
  • [18]

    H. Lagziri and M. Bezzazi, “Robin boundary effects in the Darcy-Rayleigh problem with local thermal non equilibrium model,” Transport Porous Media, vol. 129, no. 1, pp. 701720, 2019. https://doi.org/10.1007/s11242-019-01301-2.

    • Search Google Scholar
    • Export Citation
  • [19]

    A. Barletta, M. Celli, and H. Lagziri, “Instability of a horizontal porous layer with local thermal non-equilibrium: effects of free surface and convective boundary conditions,” Int. J. Heat Mass Transfer, vol. 89, no. 1, pp. 7589, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.026.

    • Search Google Scholar
    • Export Citation
  • [20]

    M. Celli, H. Lagziri, and M. Bezzazi, “Local thermal non-equilibrium effects in the Horton-Rogers-Lapwood problem with a free surface,” Int. J. Therm. Sci., vol. 116, no. 1, 254264, 2017. https://doi.org/10.1016/j.ijthermalsci.2017.03.001.

    • Search Google Scholar
    • Export Citation
  • [1]

    C. W. Horton and F. T. Rogers, “Convection currents in a porous medium,” J. Appl. Phys., vol. 16, p. 367, 1945. https://doi.org/10.1063/1.1707601.

    • Search Google Scholar
    • Export Citation
  • [2]

    B. Wen, A. Daria, L. Zhang, and A. Hesse Marc, “Carbon dioxide dissolution in a closed porous medium at low pressure,” J. Fluid Mech., vol. 854, no. 10, pp. 5687, 2018. https://doi.org/10.1017/jfm.2018.622.

    • Search Google Scholar
    • Export Citation
  • [3]

    D. A. Nield and T. Simmons Craig, “A brief introduction to convection in porous media,” Transport Porous Media, vol. 130, no. 10, pp. 237250, 2018. https://doi.org/10.1007/s11242-018-1163-6.

    • Search Google Scholar
    • Export Citation
  • [4]

    R. Dubey and P. V. S .N. Murthy, “The onset of double-diffusive convection in a Brinkman porous layer with convective thermal boundary conditions,” AIP Adv., vol. 9, 2019, Paper no. 045322. https://doi.org/10.1063/1.5087037.

    • Search Google Scholar
    • Export Citation
  • [5]

    A. Bouachir, M. Mamon, R. Rebhi, and S. Benissaadi, “Linear and nonlinear stability analyses of double-diffusive convection in a vertical brinkman porous enclosure under soret and dufour effects,” Fluids, vol. 6, no. 8, pp. 292302, 2021. https://doi.org/10.3390/fluids6080292.

    • Search Google Scholar
    • Export Citation
  • [6]

    A. D. Nield and A. Bejan, Chapter 6 – Convection in Porous Media: Internal Natural Convection: Heating from below, 4th ed. New York, Springer-Verlag, 2013, pp. 221329, https://doi.org/10.1007/978-1-4614-5541-7_6.

    • Search Google Scholar
    • Export Citation
  • [7]

    C. W. Horton and F. T. Rogers, “Convection currents in a porous medium,” J. Appl. Phys., vol. 16, no. 1, pp. 367377, 1945. https://doi.org/10.1063/1.1707601.

    • Search Google Scholar
    • Export Citation
  • [8]

    M. Parhizi, M. Torabi, and A. Jain, “Local thermal non-equilibrium (LTE) model for developed flow in porous media with spatially-varying biot number,” Int. J. Heat Mass Tran., vol. 164, pp. 19, 2021. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120538.

    • Search Google Scholar
    • Export Citation
  • [9]

    V. Kambiz and Y. Kun, “A note on local thermal non-equilibrium in porous media and heat flux bifurcation phenomenon in porous media,” Transport Porous Media, vol. 96: 169172, 2013. https://doi.org/10.1016/j.ijheatmasstransfer.2020.120538.

    • Search Google Scholar
    • Export Citation
  • [10]

    F.-K. Arman, N. Meysam, and M. Yasser, “Pulsating Flow in a Channel Filled with a Porous Medium under local thermal non-equilibrium condition: an exact solution,” J. Therm. Anal. Calorim., vol. 145, no. 7, pp. 27532775, 2020. https://doi.org/10.1007/s10973-020-09843-0.

    • Search Google Scholar
    • Export Citation
  • [11]

    A Postelnicu and D. A. S. Rees, “The onset of Darcy-Brinkman convection in a porous layer using a thermal nonequlibrium model - Part I: stress-free boundaries,” Int. J. Energy Res., vol. 27, no. 10, pp. 961973, 2003. https://doi.org/10.1002/er.928.

    • Search Google Scholar
    • Export Citation
  • [12]

    N. Banu and D. A. S. Rees, “Onset of Darcy-Benard convection using a thermal Nonequilibrium model,” Int. J. Heat Mass Tran., vol. 45 , no. 11: 22212228, 2002. https://doi.org/10.1016/S0017-9310(01)00331-3.

    • Search Google Scholar
    • Export Citation
  • [13]

    A Barletta and D. A. S Rees, “Local thermal non-equilibrium effects in the Darcy-Benard instability with isoflux boundary conditions,” Int. J. Heat Mass Tran., vol. 55, pp. 384394, 2012. https://doi.org/10.1016/j.ijheatmasstransfer.2011.09.031.

    • Search Google Scholar
    • Export Citation
  • [14]

    A. Barletta, M. Celli, and D. A. S Rees, “Buoyant flow and instability in a vertical cylindrical porous slab with permeable Boundaries,” Int. J. Heat Mass Tran., vol. 157, pp. 210320, 2020. https://doi.org/10.1016/j.ijheatmasstransfer.2020.119956.

    • Search Google Scholar
    • Export Citation
  • [15]

    A. Barletta and D. A. S Rees, “On the onset of convection in a highly permeable vertical porous layer with open boundaries,” AIP Phys. Fluids, vol. 31, 2019, Paper no. 074106. https://doi.org/10.1063/1.5110484.

    • Search Google Scholar
    • Export Citation
  • [16]

    A. Barletta and M. Celli, “The Horton-Rogers-Lapwood problem for an inclined porous layer with permeable boundaries,” Roy. Soc. Publish. Proc. A, vol. 474, 2018. https://doi.org/10.1098/rspa.2018.0021.

    • Search Google Scholar
    • Export Citation
  • [17]

    F. Al-Sumaily Gazy, A. Al Ezzi, A. Dhahad Hayder, M. C. Thompson, and T. Yusaf, “Legitimacy of the local thermal equilibrium hypothesis in porous media: a comprehensive Review,” J. Energy, vol. 14, no. 23, pp. 247, 2021. https://doi.org/10.3390/en14238114.

    • Search Google Scholar
    • Export Citation
  • [18]

    H. Lagziri and M. Bezzazi, “Robin boundary effects in the Darcy-Rayleigh problem with local thermal non equilibrium model,” Transport Porous Media, vol. 129, no. 1, pp. 701720, 2019. https://doi.org/10.1007/s11242-019-01301-2.

    • Search Google Scholar
    • Export Citation
  • [19]

    A. Barletta, M. Celli, and H. Lagziri, “Instability of a horizontal porous layer with local thermal non-equilibrium: effects of free surface and convective boundary conditions,” Int. J. Heat Mass Transfer, vol. 89, no. 1, pp. 7589, 2015. https://doi.org/10.1016/j.ijheatmasstransfer.2015.05.026.

    • Search Google Scholar
    • Export Citation
  • [20]

    M. Celli, H. Lagziri, and M. Bezzazi, “Local thermal non-equilibrium effects in the Horton-Rogers-Lapwood problem with a free surface,” Int. J. Therm. Sci., vol. 116, no. 1, 254264, 2017. https://doi.org/10.1016/j.ijthermalsci.2017.03.001.

    • Search Google Scholar
    • Export Citation
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Senior editors

Editor-in-Chief: Ákos, Lakatos University of Debrecen (Hungary)

Founder, former Editor-in-Chief (2011-2020): Ferenc Kalmár University of Debrecen (Hungary)

Founding Editor: György Csomós University of Debrecen (Hungary)

Associate Editor: Derek Clements Croome University of Reading (UK)

Associate Editor: Dezső Beke University of Debrecen (Hungary)

Editorial Board

  • Mohammad Nazir AHMAD Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

    Murat BAKIROV Center for Materials and Lifetime Management Ltd. Moscow Russia

    Nicolae BALC Technical University of Cluj-Napoca Cluj-Napoca Romania

    Umberto BERARDI Ryerson University Toronto Canada

    Ildikó BODNÁR University of Debrecen Debrecen Hungary

    Sándor BODZÁS University of Debrecen Debrecen Hungary

    Fatih Mehmet BOTSALI Selçuk University Konya Turkey

    Samuel BRUNNER Empa Swiss Federal Laboratories for Materials Science and Technology

    István BUDAI University of Debrecen Debrecen Hungary

    Constantin BUNGAU University of Oradea Oradea Romania

    Shanshan CAI Huazhong University of Science and Technology Wuhan China

    Michele De CARLI University of Padua Padua Italy

    Robert CERNY Czech Technical University in Prague Czech Republic

    György CSOMÓS University of Debrecen Debrecen Hungary

    Tamás CSOKNYAI Budapest University of Technology and Economics Budapest Hungary

    Eugen Ioan GERGELY University of Oradea Oradea Romania

    József FINTA University of Pécs Pécs Hungary

    Anna FORMICA IASI National Research Council Rome

    Alexandru GACSADI University of Oradea Oradea Romania

    Eric A. GRULKE University of Kentucky Lexington United States

    Janez GRUM University of Ljubljana Ljubljana Slovenia

    Géza HUSI University of Debrecen Debrecen Hungary

    Ghaleb A. HUSSEINI American University of Sharjah Sharjah United Arab Emirates

    Nikolay IVANOV Peter the Great St.Petersburg Polytechnic University St. Petersburg Russia

    Antal JÁRAI Eötvös Loránd University Budapest Hungary

    Gudni JÓHANNESSON The National Energy Authority of Iceland Reykjavik Iceland

    László KAJTÁR Budapest University of Technology and Economics Budapest Hungary

    Ferenc KALMÁR University of Debrecen Debrecen Hungary

    Tünde KALMÁR University of Debrecen Debrecen Hungary

    Milos KALOUSEK Brno University of Technology Brno Czech Republik

    Jan KOCI Czech Technical University in Prague Prague Czech Republic

    Vaclav KOCI Czech Technical University in Prague Prague Czech Republic

    Imra KOCSIS University of Debrecen Debrecen Hungary

    Imre KOVÁCS University of Debrecen Debrecen Hungary

    Angela Daniela LA ROSA Norwegian University of Science and Technology

    Éva LOVRA Univeqrsity of Debrecen Debrecen Hungary

    Elena LUCCHI Eurac Research, Institute for Renewable Energy Bolzano Italy

    Tamás MANKOVITS University of Debrecen Debrecen Hungary

    Igor MEDVED Slovak Technical University in Bratislava Bratislava Slovakia

    Ligia MOGA Technical University of Cluj-Napoca Cluj-Napoca Romania

    Marco MOLINARI Royal Institute of Technology Stockholm Sweden

    Henrieta MORAVCIKOVA Slovak Academy of Sciences Bratislava Slovakia

    Phalguni MUKHOPHADYAYA University of Victoria Victoria Canada

    Balázs NAGY Budapest University of Technology and Economics Budapest Hungary

    Husam S. NAJM Rutgers University New Brunswick United States

    Jozsef NYERS Subotica Tech College of Applied Sciences Subotica Serbia

    Bjarne W. OLESEN Technical University of Denmark Lyngby Denmark

    Stefan ONIGA North University of Baia Mare Baia Mare Romania

    Joaquim Norberto PIRES Universidade de Coimbra Coimbra Portugal

    László POKORÁDI Óbuda University Budapest Hungary

    Antal PUHL (1950–2023) University of Debrecen Debrecen Hungary

    Roman RABENSEIFER Slovak University of Technology in Bratislava Bratislava Slovak Republik

    Mohammad H. A. SALAH Hashemite University Zarqua Jordan

    Dietrich SCHMIDT Fraunhofer Institute for Wind Energy and Energy System Technology IWES Kassel Germany

    Lorand SZABÓ Technical University of Cluj-Napoca Cluj-Napoca Romania

    Csaba SZÁSZ Technical University of Cluj-Napoca Cluj-Napoca Romania

    Ioan SZÁVA Transylvania University of Brasov Brasov Romania

    Péter SZEMES University of Debrecen Debrecen Hungary

    Edit SZŰCS University of Debrecen Debrecen Hungary

    Radu TARCA University of Oradea Oradea Romania

    Zsolt TIBA University of Debrecen Debrecen Hungary

    László TÓTH University of Debrecen Debrecen Hungary

    László TÖRÖK University of Debrecen Debrecen Hungary

    Anton TRNIK Constantine the Philosopher University in Nitra Nitra Slovakia

    Ibrahim UZMAY Erciyes University Kayseri Turkey

    Tibor VESSELÉNYI University of Oradea Oradea Romania

    Nalinaksh S. VYAS Indian Institute of Technology Kanpur India

    Deborah WHITE The University of Adelaide Adelaide Australia

International Review of Applied Sciences and Engineering
Address of the institute: Faculty of Engineering, University of Debrecen
H-4028 Debrecen, Ótemető u. 2-4. Hungary
Email: irase@eng.unideb.hu

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2022  
Scimago  
Scimago
H-index
9
Scimago
Journal Rank
0.235
Scimago Quartile Score Architecture (Q2)
Engineering (miscellaneous) (Q3)
Environmental Engineering (Q3)
Information Systems (Q4)
Management Science and Operations Research (Q4)
Materials Science (miscellaneous) Q3)
Scopus  
Scopus
Cite Score
1.6
Scopus
CIte Score Rank
Architecture 46/170 (73rd PCTL)
General Engineering 174/302 (42nd PCTL)
Materials Science (miscellaneous) 93/150 (38th PCTL)
Environmental Engineering 123/184 (33rd PCTL)
Management Science and Operations Research 142/198 (28th PCTL)
Information Systems 281/379 (25th PCTL)
 
Scopus
SNIP
0.686

2021  
Scimago  
Scimago
H-index
7
Scimago
Journal Rank
0,199
Scimago Quartile Score Engineering (miscellaneous) (Q3)
Environmental Engineering (Q4)
Information Systems (Q4)
Management Science and Operations Research (Q4)
Materials Science (miscellaneous) (Q4)
Scopus  
Scopus
Cite Score
1,2
Scopus
CIte Score Rank
Architecture 48/149 (Q2)
General Engineering 186/300 (Q3)
Materials Science (miscellaneous) 79/124 (Q3)
Environmental Engineering 118/173 (Q3)
Management Science and Operations Research 141/184 (Q4)
Information Systems 274/353 (Q4)
Scopus
SNIP
0,457

2020  
Scimago
H-index
5
Scimago
Journal Rank
0,165
Scimago
Quartile Score
Engineering (miscellaneous) Q3
Environmental Engineering Q4
Information Systems Q4
Management Science and Operations Research Q4
Materials Science (miscellaneous) Q4
Scopus
Cite Score
102/116=0,9
Scopus
Cite Score Rank
General Engineering 205/297 (Q3)
Environmental Engineering 107/146 (Q3)
Information Systems 269/329 (Q4)
Management Science and Operations Research 139/166 (Q4)
Materials Science (miscellaneous) 64/98 (Q3)
Scopus
SNIP
0,26
Scopus
Cites
57
Scopus
Documents
36
Days from submission to acceptance 84
Days from acceptance to publication 348
Acceptance
Rate

23%

 

2019  
Scimago
H-index
4
Scimago
Journal Rank
0,229
Scimago
Quartile Score
Engineering (miscellaneous) Q2
Environmental Engineering Q3
Information Systems Q3
Management Science and Operations Research Q4
Materials Science (miscellaneous) Q3
Scopus
Cite Score
46/81=0,6
Scopus
Cite Score Rank
General Engineering 227/299 (Q4)
Environmental Engineering 107/132 (Q4)
Information Systems 259/300 (Q4)
Management Science and Operations Research 136/161 (Q4)
Materials Science (miscellaneous) 60/86 (Q3)
Scopus
SNIP
0,866
Scopus
Cites
35
Scopus
Documents
47
Acceptance
Rate
21%

 

International Review of Applied Sciences and Engineering
Publication Model Gold Open Access
Submission Fee none
Article Processing Charge 1100 EUR/article
Regional discounts on country of the funding agency World Bank Lower-middle-income economies: 50%
World Bank Low-income economies: 100%
Further Discounts Limited number of full waiver available. Editorial Board / Advisory Board members: 50%
Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription Information Gold Open Access

International Review of Applied Sciences and Engineering
Language English
Size A4
Year of
Foundation
2010
Volumes
per Year
1
Issues
per Year
3
Founder Debreceni Egyetem
Founder's
Address
H-4032 Debrecen, Hungary Egyetem tér 1
Publisher Akadémiai Kiadó
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 2062-0810 (Print)
ISSN 2063-4269 (Online)

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