- Email: [email protected]
Combined radiation-convection in an air filled enclosure with in-line heaters
International Communications in Heat and Mass Transfer 110 (2020) 104399
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Combined radiation-convection in an air filled enclosure with in-line heaters ⁎
T
S. Saravanan , N. Raja Centre for Differential Equations and Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Natural convection Thermal radiation Finite volume method Discrete heater
The effect of two fundamental arrangements of discrete heaters on the combined thermal radiation and natural convection in a square enclosure is investigated numerically. The heaters are placed either side by side or one above the other within the air filled enclosure. The nonlinear conservation equations for the resulting flow were solved by a finite volume based computational method. The steady state results which provide qualitative information on the nature of fluid flow are discussed. The present results are of use in some geometrical considerations for the optimal design of air cooled electronics.
1. Introduction Thermal management of electric modules such as ICs, PCBs, resistors and other micro-electronic components in closed cabins continues to remain a great challenge in the past few decades. The unavoidable heat from these modules causes an adverse effect on performance and reliability. In most cases the electronic components are often assembled inside sealed rectangular cabins as found in CPUs, super computers, TV monitors, low powered mobiles, etc. Hence the continuous effective cooling of such rectangular cabins becomes important and various techniques are being used in industries to achieve this (Murshed and Castrio [1]). Among them, the mechanism of natural convection air cooling is a desirable one due to its reduced cost, reliability and maintenance free nature. The literature review shows several investigations conducted on natural convection in air-filled rectangular enclosures containing solid obstacles (Dagtekin and Oztop [2], Kandaswamy et al. [3], Deng et al. [4], Rahimi et al. [5], Pordanjani et al. [6]). In all these studies the embedded solid obstacles act as heated elements, representing the micro-electronic components. Normally in several electronic cooling applications, air is used as the working medium, simply because of its noiseless way of thermal control. In such applications the temperature of inner heat sources can be significantly reduced by enhancing convection and therefore natural convection air cooling is preferable than using any other expensive conductive material. As far as air cooled technologies for electronic components are concerned, the radiative heat exchange from the surfaces involved also plays a key role and significantly affects the convective flow patterns. Some studies in the literature (see for example Sun et al. [7], Martyushev and Sheremet [8], Saravanan and Sivaraj [9], Miroshnichenko
⁎
Corresponding author. E-mail address: [email protected] (S. Saravanan).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104399
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
and Sheremet [10] and Saravanan and Raja [11]) report that the omission of surface radiation in such problems leads to erroneous predictions and also show that the surface radiation considerably improves the cooling process in a great manner. In recent years, most of the studies dealing with the problem of combined surface radiationconvection consider single heated element inside the enclosure. However in practical situations the presence of multiple micro-electronic components and their relative position and size play an important role in thermal distribution and one should pay keen attention on these for better cooling options. To the authors' knowledge the combined effect of radiation-convection in an enclosure with two or more heated elements has not been investigated. In the present study we choose a fundamental and essential model that could focus on this issue. Accordingly we placed two discrete heaters inside the closed enclosure through two different in-line arrangements as in Rahimi et al. [5]. We hope the results obtained in this study will be helpful in providing qualitative suggestions to improve the thermal design of electronic packages. 2. Problem formulation Fig. 1 displays the physical configuration considered and is typical of those encountered in electronics industry. It contains a square enclosure of length L with two isothermally heated square blocks of length L/5 kept at a higher temperature Th. The heaters are placed at the enclosure center in-line either horizontally (i.e., side by side - Case (i)) or vertically (i.e., one above the other - Case (ii)). The vertical walls of the enclosure are cold, maintained at Tc and the horizontal walls are perfectly insulated. Moderate temperatures alone are considered so that the medium (air) remains non-participating. The walls of the enclosure/
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
In order to obtain the fluid flow in terms of streamlines, the dimensionless stream function Ψ is calculated through
y,v Insulated
L
∂Ψ , ∂Y
U=
V=−
∂Ψ ∂X
(9)
3. Solution procedure
Tc
L/5
Th
L/5
Th
The net radiation method described in Siegel and Howell [12] is employed to determine the net radiative fluxes at the surfaces. For this purpose the walls of the enclosure and the heaters are replaced by N non-overlapping radiative surface elements. On introducing the scales Qrd, k = qrd, k/σTh4, Rk = qo, k/σTh4 and Θk = Tk/Th, we obtain the dimensionless form of net radiative flux, outgoing and incoming radiative components respectively of the kth element in the form
Tc
L/5
L/5
Qrd, k = Rk − Ik , 0
L
Insulated
k = 1, 2. .…,N
(10)
N
x,u
Rk = εk Θk4 + (1 − εk )
∑ Fkj Rj ,
k = 1, 2, ..…,N (11)
j=1
Fig. 1. Stencil of the physical configuration.
and N
heater are assumed to be gray, opaque and diffuse reflectors and emitters of thermal radiation. The gravity g is acting parallel to the yaxis and the Boussinesq approximation is invoked. We introduce the scales (X, Y) = (x, y)/L, (U, V) = (u, v)L/α, P = L2p/ρα2, τ = αt/L2, θ = (T − Tc)/(Th − Tc) and Qrd = qrd/σTh4. Then the equations governing the unsteady laminar flow in dimensionless form are
∂U ∂V =0 + ∂X ∂Y
Ik =
∂V ∂V ∂V ∂P ∂ 2V ∂ 2V ⎞ +U +V =− + Pr ⎛ 2 + + RaPrθ ∂τ ∂X ∂Y ∂Y ∂Y 2 ⎠ ⎝ ∂X
(3)
∂θ ∂θ ∂θ ∂ 2θ ∂ 2θ +U +V = + ∂τ ∂X ∂Y ∂X 2 ∂Y 2
(4)
⎜
1 2Ls
2
∑ (Lkj,i − Lkj,2+i) i=1
⎟
Here Lkj, 1, Lkj, 2 and Lkj, 3, Lkj, 4 are the shortest lengths of the crossed and uncrossed strings respectively, joining the outer edges of the surfaces k and j. Ls similarly represents the length of the kth surface. The obstruction and shadow effects of the heaters are taken into account while calculating the configuration factors. The finite volume method is used to solve the governing conservation Eqs. (1)–(6). We employed a uniform and regular mesh of finite volumes spread over the enclosure. The SIMPLE algorithm of Patankar [13] was used to take care of the coupling between pressure and velocity fields. The power law scheme was adopted to simplify the diffusive and convective terms. Then the solution of the resulting set of algebraic equations is obtained through the line-by-line procedure of the Thomas algorithm. As we are interested in the steady state results alone it was considered when the convergence criteria
⎟
subjected to the following conditions
τ > 0:
(12)
The constant Fkj in the above equations stands for the configuration factor from the kth to jth surface element. It represents the fraction of the radiation energy leaving the kth surface and striking the jth surface. We follow Hottel's crossed string method (Hottel and Saroffim [14]) to calculate the configuration factor as follows:
Fkj = (2)
⎜
U = V = 0, θ = 0 at 0 ≤ X , Y ≤ 1 U = V = 0, θ = 0 at X = 0, 1 and 0 ≤ Y ≤ 1 ∂θ U = V = 0, = NRC Qrd at Y = 0 ∂Y ∂θ U = V = 0, = −NRC Qrd at Y = 1 ∂Y U = V = 0, θ = 1 on the heaters
(5)
m−1 ∣ ∑ ∣ϕim , j − ϕi, j
(6)
i, j
where Pr = ν/α is the Prandtl number, Ra = gβΔTL3/(να), the Rayleigh number, NRC = σTh4L/(kΔT), the radiation-conduction number and Qrd = qrd/(σTh4), the net radiative flux. One should notice that for the present problem with radiatively non-participating medium, the coupling between radiation and convection is seen only on the thermal boundary conditions at the insulated walls. The local convective and radiative Nusselt numbers calculated at the isothermal cold walls are introduced to measure the total heat transfer rate:
Nucv =
qcv qcd
∂θ = , ∂X
Nurd =
qrd qcd
= −NRC Qrd
≤ 10−6
∑ ∣ϕim ,j ∣ i, j
is achieved. Here ϕ represents the variables U, V or θ, the superscript m refers the iteration number and (i, j) refers the space coordinates. For validation purpose, the present numerical model is compared with the experimental results of Ramesh and Venkateshan [15] who dealt with a similar situation in an obstacle free enclosure (see Table 1) and fairly Table 1 Comparison of Nu with the experimental study of Ramesh and Venkateshan [15] for different values of ε.
(7)
Their averages are then evaluated by integration. The rate of heat transfer across the enclosure is found by summing the average convective and radiative Nusselt numbers, i.e.,
Nu = Nucv + Nurd
k = 1, 2, ..…,N .
j=1
(1)
∂U ∂U ∂U ∂P ∂ 2U ∂ 2U ⎞ +U +V =− + Pr ⎛ 2 + ∂τ ∂X ∂Y ∂X ∂ X ∂Y 2 ⎠ ⎝
τ = 0:
∑ Fkj Rj ,
Gr = 106
(8) 2
ε
Nu (Experimental study)
Nu (Present study)
Difference in %
0.05 0.85
8.283 16.754
8.4817 17.7725
2.39 6.07
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
the augmented convective flux impinging on the top wall causes energy loss from it through outward radiation. In other words, the radiative heating and convective cooling at the bottom and top walls respectively are well noticed in both Cases (i,ii). However the two different heater arrangements produce their own patterns of Qrd at the top wall, which are well distinct from those corresponding to low Ra. In Case (i) the outgoing radiation gradually decreases as one move towards the center indicating a reduction in convective influx. In Case (ii), two maxima are seen exactly corresponding to the edges of the heater. The relevant isotherms and streamlines are given in Figs.3 and 4 for Ra = 103 and 107, respectively. From the streamline pattern corresponding to Ra = 103, a weaker convection is seen inside the enclosure and therefore the effect of radiation is not significant on the flow field. For an increased Ra(=107) thermal boundary layers are formed along top portion of the cold walls. Moderate convection cells around the heaters with primary vortices in regions between the heaters and the side walls can be seen. Moreover, one may notice the emergence of two secondary cells with low intensity which transfer the heat energy to the top wall. When ε = 1, the increased fluid flux from the bottom makes these secondary cells visibly stronger. One may notice that the convective influx from the secondary cells decreases as one move towards the center on the top wall and hence the radiative outflux also decreases (see Fig. 2(b)). One may look at our earlier paper (Saravanan and Raja [11]) to have a more clear understanding of the role of secondary cells on the surface radiation in a similar study. In Case (ii), the heater arrangement retards free flow of the fluid inside the enclosure due to buoyancy effect. However when ε = 1, the additional fluid flux moves well up to the ceiling dominating the bulk resistance offered by the obstacles and this can be well observed via |Ψ|max. Also in general, a sort of homogenization of the temperature field can be seen inside the enclosure when ε = 1. We end up this section by recording the average Nusselt numbers for different values of Ra and ε in Table 3.
Table 2 Grid independent results of Nu for Ra = 107 and ε = 0.5. Grid size
82 × 82
102 × 102
122 × 122
142 × 142
Case (i) Case (ii)
24.4424 23.7534
24.6145 23.9721
24.7196 24.1755
24.7646 24.2115
good agreement is observed. Moreover the grid independent study of the solution was also made and the deviation of Nu between 122×122 and 142×142 grid systems is found to be less than 0.2%(Table 2). Hence the grid system is fixed at 122×122. 4. Results and discussion The present investigation is made to examine the effect of two different in-line heater arrangements on the resulting coupled radiationconvection in a closed square enclosure. The difference in temperature was maintained at 20K and the average temperature of the working medium was set at 303.15K. The effect of ε = 0 and 1 representing the pure convection and black body radiation respectively are discussed. The steady state result in the form of isotherms and streamlines are presented in a single plot exploiting the symmetry observed. In order to understand the radiative heat exchange, Qrd along the insulated walls are plotted in Fig. 2. Almost identical patterns are seen at the bottom and top insulated walls in both Cases when Ra = 103. This is expected due to weak convection. However radiative fluxes change depending on the heater arrangement. In Case (i), Qrd is remains negative almost throughout the insulated walls implying the dominance of inward radiation over the outward one. On the other hand in Case (ii), the inward radiation is dominant only near the side walls. One may anticipate this trend based on the underlying geometry. At this point one may compare the results with those of single discrete heater reported in Saravanan and Sivaraj [9]. When Ra is increased to 107, the resulting convection notably alters the net radiative fluxes on the insulated walls in both Cases and are shown in Fig. 2 (b). In such a situation, the extra heat received via the inward radiative flux is clear throughout the bottom wall. This excess energy intensifies the otherwise existing natural convective mechanism within the enclosure. Thus
5. Conclusion The effect of arrangements of in-line heaters, side by side or one above the other, on the combined radiation-convection in an air filled enclosure has been studied. The two different arrangements produce
Fig. 2. Dimensionless radiative fluxes along the insulated walls for (a) Ra = 103 and (b) Ra = 107. Solid line (——-) for Case (i) and dotted line (⋯⋯) for Case (ii). 3
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
Fig. 3. Isotherms and streamlines for Ra = 103 and ε = 1.
Fig. 4. Isotherms and streamlines for Ra = 107 and for different ε.
4
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
References
Table 3 Values of Nucv and Nurd with different Ra and ε for both Case (i) and Case (ii). Case
Ra
ε
Nucv
Nurd
Case (i)
103 105
1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.0 0.0 0.5 1.0 0.0 0.5 1.0
3.7403 5.2190 5.2771 5.3592 14.7487 14.6832 14.6542 2.3468 4.5723 4.5290 4.5195 14.3264 13.9288 13.8129
1.0407 0.0 2.1208 4.9538 0.0 10.0364 23.4640 1.1239 0.0 2.2380 5.1008 0.0 10.2467 23.6767
107
Case (ii)
103 105
107
[1] S.M.S. Murshed, C.A.N. Castro, A critical review of traditional and emerging techniques and fluids for electronics cooling, Renew. Sust. Energ. Rev. 78 (2017) 821–833. [2] I. Dagtekin, H.F. Oztop, Natural convection heat transfer by heated partitions within enclosure, Int. Comm. Heat Mass Transfer 28 (2001) 823–834. [3] P. Kandaswamy, J. Lee, A.K. Abdul Hakeem, S. Saravanan, Effect of baffle-cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles, Int. J. Heat Mass Transf. 51 (2008) 1830–1837. [4] Q.-H. Deng, G.-F. Tang, Y. Li, M.Y. Ha, Interaction between discrete heat sources in horizontal natural convection enclosures, Int. J. Heat Mass Transf. 45 (2002) 5117–5132. [5] A. Rahimi, A. Kasaeipoor, E.H. Malekshah, L. Kolsi, Natural convection analysis by entropy generation and heatline visualization using lattice Boltzmann method in nanofluid filled cavity included with internal heaters-Empirical thermo-physical properties, Int. J. Mech. Sci. 133 (2017) 199–216. [6] A.H. Pordanjani, A. Jahanbakhshi, A.A. Nadooshan, M. Afrand, Effect of two isothermal obstacles on the natural convection of nanofluid in the presence of magnetic field inside an enclosure with sinusoidal wall temperature distribution, Int. J. Heat Mass Transf. 121 (2018) 565–578. [7] H. Sun, E. Chenier, G. Lauriat, Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body, Appl. Therm. Eng. 31 (2011) 1252–1262. [8] S.M. Martyushev, M.A. Sheremet, Conjugate natural convection combined with surface thermal radiation in an air filled cavity with internal heat source, Int. J. Therm. Sci. 76 (2014) 51–67. [9] S. Saravanan, C. Sivaraj, Surface radiation effect on convection in a closed enclosure driven by a discrete heater, Int. Comm. Heat Mass Transfer 53 (2014) 34–38. [10] I.V. Miroshnichenko, M.A. Sheremet, Radiation effect on conjugate turbulent natural convection in a cavity with a discrete heater, Appl. Math. Comput. 321 (2018) 358–371. [11] S. Saravanan, N. Raja, Effect of variable sidewall temperatures on the combined surface radiation-convection in a discretely heated enclosure, Trans. of the ASME, J. Heat Transf. 140 (2018) 0945031–0945035. [12] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, fourth ed., Taylor and Francis Group, New York, 2002. [13] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis group, New York, 1980. [14] H.C. Hottel, A.F. Saroffim, Radiative Heat Transfer, McGraw Hill, New York, 1980. [15] N. Ramesh, S.P. Venkatesan, Effect of surface radiation and natural convection in a square enclosure, J. Thermophys. Heat Transf. 13 (1999) 299–301.
two distinct patterns of radiative fluxes at the insulated walls even for low Rayleigh numbers. When convection becomes strong for high Rayleigh numbers, the thermal radiation produces two opposite effects at the bottom and top insulated walls: additional radiative heating at the bottom and additional convective cooling at the top. When the heaters remain side by side, surface radiation plays a prominent role in altering the flow pattern, by making the secondary cells just below the top wall dominant. On the other hand the influence of surface radiation is minimal when the heaters are placed one above the other. However the resulting heat transfer rate and the flow pattern are found to be advantageous in this Case. Acknowledgment The authors thank University Grants Commission, India, for its support through the DRS SAP in Differential equations and Fluid dynamics.
5
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Combined radiation-convection in an air filled enclosure with in-line heaters ⁎
T
S. Saravanan , N. Raja Centre for Differential Equations and Fluid Dynamics, Department of Mathematics, Bharathiar University, Coimbatore 641 046, India
A R T I C LE I N FO
A B S T R A C T
Keywords: Natural convection Thermal radiation Finite volume method Discrete heater
The effect of two fundamental arrangements of discrete heaters on the combined thermal radiation and natural convection in a square enclosure is investigated numerically. The heaters are placed either side by side or one above the other within the air filled enclosure. The nonlinear conservation equations for the resulting flow were solved by a finite volume based computational method. The steady state results which provide qualitative information on the nature of fluid flow are discussed. The present results are of use in some geometrical considerations for the optimal design of air cooled electronics.
1. Introduction Thermal management of electric modules such as ICs, PCBs, resistors and other micro-electronic components in closed cabins continues to remain a great challenge in the past few decades. The unavoidable heat from these modules causes an adverse effect on performance and reliability. In most cases the electronic components are often assembled inside sealed rectangular cabins as found in CPUs, super computers, TV monitors, low powered mobiles, etc. Hence the continuous effective cooling of such rectangular cabins becomes important and various techniques are being used in industries to achieve this (Murshed and Castrio [1]). Among them, the mechanism of natural convection air cooling is a desirable one due to its reduced cost, reliability and maintenance free nature. The literature review shows several investigations conducted on natural convection in air-filled rectangular enclosures containing solid obstacles (Dagtekin and Oztop [2], Kandaswamy et al. [3], Deng et al. [4], Rahimi et al. [5], Pordanjani et al. [6]). In all these studies the embedded solid obstacles act as heated elements, representing the micro-electronic components. Normally in several electronic cooling applications, air is used as the working medium, simply because of its noiseless way of thermal control. In such applications the temperature of inner heat sources can be significantly reduced by enhancing convection and therefore natural convection air cooling is preferable than using any other expensive conductive material. As far as air cooled technologies for electronic components are concerned, the radiative heat exchange from the surfaces involved also plays a key role and significantly affects the convective flow patterns. Some studies in the literature (see for example Sun et al. [7], Martyushev and Sheremet [8], Saravanan and Sivaraj [9], Miroshnichenko
⁎
Corresponding author. E-mail address: [email protected] (S. Saravanan).
https://doi.org/10.1016/j.icheatmasstransfer.2019.104399
0735-1933/ © 2019 Elsevier Ltd. All rights reserved.
and Sheremet [10] and Saravanan and Raja [11]) report that the omission of surface radiation in such problems leads to erroneous predictions and also show that the surface radiation considerably improves the cooling process in a great manner. In recent years, most of the studies dealing with the problem of combined surface radiationconvection consider single heated element inside the enclosure. However in practical situations the presence of multiple micro-electronic components and their relative position and size play an important role in thermal distribution and one should pay keen attention on these for better cooling options. To the authors' knowledge the combined effect of radiation-convection in an enclosure with two or more heated elements has not been investigated. In the present study we choose a fundamental and essential model that could focus on this issue. Accordingly we placed two discrete heaters inside the closed enclosure through two different in-line arrangements as in Rahimi et al. [5]. We hope the results obtained in this study will be helpful in providing qualitative suggestions to improve the thermal design of electronic packages. 2. Problem formulation Fig. 1 displays the physical configuration considered and is typical of those encountered in electronics industry. It contains a square enclosure of length L with two isothermally heated square blocks of length L/5 kept at a higher temperature Th. The heaters are placed at the enclosure center in-line either horizontally (i.e., side by side - Case (i)) or vertically (i.e., one above the other - Case (ii)). The vertical walls of the enclosure are cold, maintained at Tc and the horizontal walls are perfectly insulated. Moderate temperatures alone are considered so that the medium (air) remains non-participating. The walls of the enclosure/
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
In order to obtain the fluid flow in terms of streamlines, the dimensionless stream function Ψ is calculated through
y,v Insulated
L
∂Ψ , ∂Y
U=
V=−
∂Ψ ∂X
(9)
3. Solution procedure
Tc
L/5
Th
L/5
Th
The net radiation method described in Siegel and Howell [12] is employed to determine the net radiative fluxes at the surfaces. For this purpose the walls of the enclosure and the heaters are replaced by N non-overlapping radiative surface elements. On introducing the scales Qrd, k = qrd, k/σTh4, Rk = qo, k/σTh4 and Θk = Tk/Th, we obtain the dimensionless form of net radiative flux, outgoing and incoming radiative components respectively of the kth element in the form
Tc
L/5
L/5
Qrd, k = Rk − Ik , 0
L
Insulated
k = 1, 2. .…,N
(10)
N
x,u
Rk = εk Θk4 + (1 − εk )
∑ Fkj Rj ,
k = 1, 2, ..…,N (11)
j=1
Fig. 1. Stencil of the physical configuration.
and N
heater are assumed to be gray, opaque and diffuse reflectors and emitters of thermal radiation. The gravity g is acting parallel to the yaxis and the Boussinesq approximation is invoked. We introduce the scales (X, Y) = (x, y)/L, (U, V) = (u, v)L/α, P = L2p/ρα2, τ = αt/L2, θ = (T − Tc)/(Th − Tc) and Qrd = qrd/σTh4. Then the equations governing the unsteady laminar flow in dimensionless form are
∂U ∂V =0 + ∂X ∂Y
Ik =
∂V ∂V ∂V ∂P ∂ 2V ∂ 2V ⎞ +U +V =− + Pr ⎛ 2 + + RaPrθ ∂τ ∂X ∂Y ∂Y ∂Y 2 ⎠ ⎝ ∂X
(3)
∂θ ∂θ ∂θ ∂ 2θ ∂ 2θ +U +V = + ∂τ ∂X ∂Y ∂X 2 ∂Y 2
(4)
⎜
1 2Ls
2
∑ (Lkj,i − Lkj,2+i) i=1
⎟
Here Lkj, 1, Lkj, 2 and Lkj, 3, Lkj, 4 are the shortest lengths of the crossed and uncrossed strings respectively, joining the outer edges of the surfaces k and j. Ls similarly represents the length of the kth surface. The obstruction and shadow effects of the heaters are taken into account while calculating the configuration factors. The finite volume method is used to solve the governing conservation Eqs. (1)–(6). We employed a uniform and regular mesh of finite volumes spread over the enclosure. The SIMPLE algorithm of Patankar [13] was used to take care of the coupling between pressure and velocity fields. The power law scheme was adopted to simplify the diffusive and convective terms. Then the solution of the resulting set of algebraic equations is obtained through the line-by-line procedure of the Thomas algorithm. As we are interested in the steady state results alone it was considered when the convergence criteria
⎟
subjected to the following conditions
τ > 0:
(12)
The constant Fkj in the above equations stands for the configuration factor from the kth to jth surface element. It represents the fraction of the radiation energy leaving the kth surface and striking the jth surface. We follow Hottel's crossed string method (Hottel and Saroffim [14]) to calculate the configuration factor as follows:
Fkj = (2)
⎜
U = V = 0, θ = 0 at 0 ≤ X , Y ≤ 1 U = V = 0, θ = 0 at X = 0, 1 and 0 ≤ Y ≤ 1 ∂θ U = V = 0, = NRC Qrd at Y = 0 ∂Y ∂θ U = V = 0, = −NRC Qrd at Y = 1 ∂Y U = V = 0, θ = 1 on the heaters
(5)
m−1 ∣ ∑ ∣ϕim , j − ϕi, j
(6)
i, j
where Pr = ν/α is the Prandtl number, Ra = gβΔTL3/(να), the Rayleigh number, NRC = σTh4L/(kΔT), the radiation-conduction number and Qrd = qrd/(σTh4), the net radiative flux. One should notice that for the present problem with radiatively non-participating medium, the coupling between radiation and convection is seen only on the thermal boundary conditions at the insulated walls. The local convective and radiative Nusselt numbers calculated at the isothermal cold walls are introduced to measure the total heat transfer rate:
Nucv =
qcv qcd
∂θ = , ∂X
Nurd =
qrd qcd
= −NRC Qrd
≤ 10−6
∑ ∣ϕim ,j ∣ i, j
is achieved. Here ϕ represents the variables U, V or θ, the superscript m refers the iteration number and (i, j) refers the space coordinates. For validation purpose, the present numerical model is compared with the experimental results of Ramesh and Venkateshan [15] who dealt with a similar situation in an obstacle free enclosure (see Table 1) and fairly Table 1 Comparison of Nu with the experimental study of Ramesh and Venkateshan [15] for different values of ε.
(7)
Their averages are then evaluated by integration. The rate of heat transfer across the enclosure is found by summing the average convective and radiative Nusselt numbers, i.e.,
Nu = Nucv + Nurd
k = 1, 2, ..…,N .
j=1
(1)
∂U ∂U ∂U ∂P ∂ 2U ∂ 2U ⎞ +U +V =− + Pr ⎛ 2 + ∂τ ∂X ∂Y ∂X ∂ X ∂Y 2 ⎠ ⎝
τ = 0:
∑ Fkj Rj ,
Gr = 106
(8) 2
ε
Nu (Experimental study)
Nu (Present study)
Difference in %
0.05 0.85
8.283 16.754
8.4817 17.7725
2.39 6.07
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
the augmented convective flux impinging on the top wall causes energy loss from it through outward radiation. In other words, the radiative heating and convective cooling at the bottom and top walls respectively are well noticed in both Cases (i,ii). However the two different heater arrangements produce their own patterns of Qrd at the top wall, which are well distinct from those corresponding to low Ra. In Case (i) the outgoing radiation gradually decreases as one move towards the center indicating a reduction in convective influx. In Case (ii), two maxima are seen exactly corresponding to the edges of the heater. The relevant isotherms and streamlines are given in Figs.3 and 4 for Ra = 103 and 107, respectively. From the streamline pattern corresponding to Ra = 103, a weaker convection is seen inside the enclosure and therefore the effect of radiation is not significant on the flow field. For an increased Ra(=107) thermal boundary layers are formed along top portion of the cold walls. Moderate convection cells around the heaters with primary vortices in regions between the heaters and the side walls can be seen. Moreover, one may notice the emergence of two secondary cells with low intensity which transfer the heat energy to the top wall. When ε = 1, the increased fluid flux from the bottom makes these secondary cells visibly stronger. One may notice that the convective influx from the secondary cells decreases as one move towards the center on the top wall and hence the radiative outflux also decreases (see Fig. 2(b)). One may look at our earlier paper (Saravanan and Raja [11]) to have a more clear understanding of the role of secondary cells on the surface radiation in a similar study. In Case (ii), the heater arrangement retards free flow of the fluid inside the enclosure due to buoyancy effect. However when ε = 1, the additional fluid flux moves well up to the ceiling dominating the bulk resistance offered by the obstacles and this can be well observed via |Ψ|max. Also in general, a sort of homogenization of the temperature field can be seen inside the enclosure when ε = 1. We end up this section by recording the average Nusselt numbers for different values of Ra and ε in Table 3.
Table 2 Grid independent results of Nu for Ra = 107 and ε = 0.5. Grid size
82 × 82
102 × 102
122 × 122
142 × 142
Case (i) Case (ii)
24.4424 23.7534
24.6145 23.9721
24.7196 24.1755
24.7646 24.2115
good agreement is observed. Moreover the grid independent study of the solution was also made and the deviation of Nu between 122×122 and 142×142 grid systems is found to be less than 0.2%(Table 2). Hence the grid system is fixed at 122×122. 4. Results and discussion The present investigation is made to examine the effect of two different in-line heater arrangements on the resulting coupled radiationconvection in a closed square enclosure. The difference in temperature was maintained at 20K and the average temperature of the working medium was set at 303.15K. The effect of ε = 0 and 1 representing the pure convection and black body radiation respectively are discussed. The steady state result in the form of isotherms and streamlines are presented in a single plot exploiting the symmetry observed. In order to understand the radiative heat exchange, Qrd along the insulated walls are plotted in Fig. 2. Almost identical patterns are seen at the bottom and top insulated walls in both Cases when Ra = 103. This is expected due to weak convection. However radiative fluxes change depending on the heater arrangement. In Case (i), Qrd is remains negative almost throughout the insulated walls implying the dominance of inward radiation over the outward one. On the other hand in Case (ii), the inward radiation is dominant only near the side walls. One may anticipate this trend based on the underlying geometry. At this point one may compare the results with those of single discrete heater reported in Saravanan and Sivaraj [9]. When Ra is increased to 107, the resulting convection notably alters the net radiative fluxes on the insulated walls in both Cases and are shown in Fig. 2 (b). In such a situation, the extra heat received via the inward radiative flux is clear throughout the bottom wall. This excess energy intensifies the otherwise existing natural convective mechanism within the enclosure. Thus
5. Conclusion The effect of arrangements of in-line heaters, side by side or one above the other, on the combined radiation-convection in an air filled enclosure has been studied. The two different arrangements produce
Fig. 2. Dimensionless radiative fluxes along the insulated walls for (a) Ra = 103 and (b) Ra = 107. Solid line (——-) for Case (i) and dotted line (⋯⋯) for Case (ii). 3
International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
Fig. 3. Isotherms and streamlines for Ra = 103 and ε = 1.
Fig. 4. Isotherms and streamlines for Ra = 107 and for different ε.
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International Communications in Heat and Mass Transfer 110 (2020) 104399
S. Saravanan and N. Raja
References
Table 3 Values of Nucv and Nurd with different Ra and ε for both Case (i) and Case (ii). Case
Ra
ε
Nucv
Nurd
Case (i)
103 105
1.0 0.0 0.5 1.0 0.0 0.5 1.0 1.0 0.0 0.5 1.0 0.0 0.5 1.0
3.7403 5.2190 5.2771 5.3592 14.7487 14.6832 14.6542 2.3468 4.5723 4.5290 4.5195 14.3264 13.9288 13.8129
1.0407 0.0 2.1208 4.9538 0.0 10.0364 23.4640 1.1239 0.0 2.2380 5.1008 0.0 10.2467 23.6767
107
Case (ii)
103 105
107
[1] S.M.S. Murshed, C.A.N. Castro, A critical review of traditional and emerging techniques and fluids for electronics cooling, Renew. Sust. Energ. Rev. 78 (2017) 821–833. [2] I. Dagtekin, H.F. Oztop, Natural convection heat transfer by heated partitions within enclosure, Int. Comm. Heat Mass Transfer 28 (2001) 823–834. [3] P. Kandaswamy, J. Lee, A.K. Abdul Hakeem, S. Saravanan, Effect of baffle-cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles, Int. J. Heat Mass Transf. 51 (2008) 1830–1837. [4] Q.-H. Deng, G.-F. Tang, Y. Li, M.Y. Ha, Interaction between discrete heat sources in horizontal natural convection enclosures, Int. J. Heat Mass Transf. 45 (2002) 5117–5132. [5] A. Rahimi, A. Kasaeipoor, E.H. Malekshah, L. Kolsi, Natural convection analysis by entropy generation and heatline visualization using lattice Boltzmann method in nanofluid filled cavity included with internal heaters-Empirical thermo-physical properties, Int. J. Mech. Sci. 133 (2017) 199–216. [6] A.H. Pordanjani, A. Jahanbakhshi, A.A. Nadooshan, M. Afrand, Effect of two isothermal obstacles on the natural convection of nanofluid in the presence of magnetic field inside an enclosure with sinusoidal wall temperature distribution, Int. J. Heat Mass Transf. 121 (2018) 565–578. [7] H. Sun, E. Chenier, G. Lauriat, Effect of surface radiation on the breakdown of steady natural convection flows in a square, air-filled cavity containing a centered inner body, Appl. Therm. Eng. 31 (2011) 1252–1262. [8] S.M. Martyushev, M.A. Sheremet, Conjugate natural convection combined with surface thermal radiation in an air filled cavity with internal heat source, Int. J. Therm. Sci. 76 (2014) 51–67. [9] S. Saravanan, C. Sivaraj, Surface radiation effect on convection in a closed enclosure driven by a discrete heater, Int. Comm. Heat Mass Transfer 53 (2014) 34–38. [10] I.V. Miroshnichenko, M.A. Sheremet, Radiation effect on conjugate turbulent natural convection in a cavity with a discrete heater, Appl. Math. Comput. 321 (2018) 358–371. [11] S. Saravanan, N. Raja, Effect of variable sidewall temperatures on the combined surface radiation-convection in a discretely heated enclosure, Trans. of the ASME, J. Heat Transf. 140 (2018) 0945031–0945035. [12] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, fourth ed., Taylor and Francis Group, New York, 2002. [13] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Taylor and Francis group, New York, 1980. [14] H.C. Hottel, A.F. Saroffim, Radiative Heat Transfer, McGraw Hill, New York, 1980. [15] N. Ramesh, S.P. Venkatesan, Effect of surface radiation and natural convection in a square enclosure, J. Thermophys. Heat Transf. 13 (1999) 299–301.
two distinct patterns of radiative fluxes at the insulated walls even for low Rayleigh numbers. When convection becomes strong for high Rayleigh numbers, the thermal radiation produces two opposite effects at the bottom and top insulated walls: additional radiative heating at the bottom and additional convective cooling at the top. When the heaters remain side by side, surface radiation plays a prominent role in altering the flow pattern, by making the secondary cells just below the top wall dominant. On the other hand the influence of surface radiation is minimal when the heaters are placed one above the other. However the resulting heat transfer rate and the flow pattern are found to be advantageous in this Case. Acknowledgment The authors thank University Grants Commission, India, for its support through the DRS SAP in Differential equations and Fluid dynamics.
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