Convective heat transfer in a porous enclosure saturated by nanofluid with different heat sources

1 Introduction

The problem of convective heat transfer in porous media has been major topic for research studies due to its fundamental nature and the wide spectrum of engineering applications such as geothermal energy systems, oil recovery, subsoil water filtration, nuclear waste storage, and chemical separation processes. On focusing the applications, Cheng [1] intensively analyzed the theoretical and experimental investigations of convective heat transfer in porous media with regard to applications in geothermal systems. Since the studies of Darcy model revealed that the non-Darcy effects are significant, Vafai and Tien [2] have performed an extensive discussion in porous media with these non-Darcian effects. Free convection in fluid saturated heat generating porous medium in a square enclosure has been studied by Das and Sahoo [3]. It is concluded that the presence of heat generation in porous medium affects the temperature gradients and fluid velocity markedly. Saeid and Pop [4] observed that in order to get maximum heat transfer, the heater is placed in the middle of the vertical wall for long heater and low Ra whereas it is positioned in the upper half of the vertical wall for short heater and high Ra. Partially cooled and inclined porous rectangular enclosures were studied by Oztop [5]. He has found that the inclination angle and aspect ratio are the dominant parameters on heat transfer and fluid flow in the porous enclosures. Oztop and Varol [6] numerically studied the effects of sinusoidal temperature distribution on side wall and inclination angle in porous square lid-driven cavity. It is found that an enhanced heat transfer is observed with an increase in the inclination angle. Basak et al. [7] studied on mixed convection flows in a cavity filled with porous medium with various wall thermal boundary conditions.

Nanofluids are new class of heat transfer fluids and are engineered by suspending nanometer-sized particles in convectional heat transfer fluids. The concept of nanofluids is an innovative idea and presents new challenges and opportunities for thermal scientists and engineers. Since nanofluids have attracted a great deal of attention and are used in many fields such as automotive, electronics, and biomedicine due to their better thermal performance, it was first introduced by Choi [8]. Eastman et al. [9] observed the increased heat transfer in nanofluids and have shown that when the water and ethylene glycol are used along with Cu, CuO, Al₂O₃, carbon nanotubes, and TiO₂ nanoparticles has tendency to increase the heat transfer up to 40%. Choi et al. [10] verified that the additional of nanoparticles in base fluids is to increase the thermal conductivity of the fluid up to approximately two times. Khanafer and Vafai [11] presented a numerical study on heat transfer enhancement in a 2-D enclosure utilizing nanofluids. Maiga et al. [12] numerically studied the heat transfer enhancement in tube and disk by using nanofluids. They have proven that the inclusion of nanoparticles is not only increasing the heat transfer rate but also the wall shear stress as much as 7 times that of the base fluid. A single phase approach on mixed convective heat transfer of nanofluid in horizontal tube with uniform heat flux has been used by Akbari and Behzadmehr [13]. Oztop and Abu-Nada [14] numerically studied natural convection of nanofluid in partially heated enclosure. Aminossadati and Ghasemi [15] provided a numerical simulation of natural convective heat transfer in a nanofluid-filled enclosure with localized heat source along the bottom wall. Kumar and Prasad [16] studied the analysis of flow and thermal field in nanofluid using single phase thermal dispersion model. They found that considerable improvement in thermal conductivity is observed as a result of increase in the shape factor. Muthtamilselvan et al. [17] conducted a numerical analysis on the heat transfer enhancement of copper-water nanofluids in a lid-driven cavity. They have found that both the aspect ratio and solid volume fraction affect the fluid flow and heat transfer rate in the enclosure. Yadav et al. [18] studied the thermal instability of rotating nanofluid layer. It is observed that the joint behaviour of Brownian motion and thermal thermophoresis of nanoparticles creates destabilizing effect, which can reduce the critical Rayleigh number by as much as one to two orders of magnitude as compared to that of regular fluids without nanoparticles. The combined effect of a vertical magnetic field and the boundaries on the onset of convection in an electrically conducting nanofluid layer heated from below has been examined using linear stability theory by Yadav et al. [19]. Linear and non-linear analyses of transient behaviour of Soret-driven buoyancy convection in a vertically orientated Hele-Shaw cell with nanoparticles suspension was investigated theoretically as well as numerically by Yadav and Kim [20]. Influence of magnetic field on the onset of nanofluid convection has been analyzed by Yadav et al. [21, 22]. The influence of rotation on the onset of natural convection made by purely internal heating in a horizontal layer of nanofluid has been investigated numerically by Yadav [23].

Only few literatures have been conducted to analyze the convection heat transfer of nanofluids in porous media. Ghazvini and Shokouhmand [24] focused on nanofluid-cooled microchannel heat sink using fin model and porous media approach. They showed that an increase in porosity is to enhance the dimensionless temperature in both fin and porous media approaches. Rosca et al. [25] made a numerical study on Non-Darcy mixed convective heat transfer over a horizontal plate embedded in a porous medium saturated by a nanofluid. Rohni et al. [26] analyzed combined convection boundary layer flow along a vertical cylinder embedded in a porous medium saturated by a nanofluid. Natural convection of nanofluids in porous media has been numerically investigated by Bourantas et al. [27]. In the presence of a porous medium and for different Ra, the overall heat transfer rate is increased as the solid volume fraction increases. Zhang et al. [28] presented a numerical study on MHD flow and radiation heat transfer of nanofluids in porous media with the effects of variable surface heat flux and chemical reaction. It is found that velocity and temperature fields are strongly affected in the presence of magnetic field and radiation effects. Nguyen et al. [29] made a numerical analysis on laminar natural convective flow in non-Darcy porous enclosure filled with nanofluid. Their result has revealed that increasing the solid volume fraction decreases the overall heat transfer rate in Darcy flow regime at a high Rayleigh number and low Darcy number. Convective Flow and Heat Transfer from Wavy Surfaces: Viscous Fluids, Porous Media, and Nanofluid have been addressed by Shenoy et al. [30]. Yadav et al. [31–33] studied distinctly the convection of nanofluid in a porous medium. Muthtamilselvan and Sureshkumar [34, 35 numerically investigated on mixed convection in lid-driven porous cavity filled with nanofluid.

A review of the existing literature has indicated that convective heat transfer in a porous enclosure [1–7] and convective heat transfer in a nanofluid-filled enclosure [8–23] are carried out separately for past few decades. A very little work [24–35] has been done on convective heat transfer in a porous medium filled with nanofluid, which indicates that there lacks fundamental information about the convective heat transfer in a porous medium filled with nanofluid. In some of engineering applications, especially for cooling of electronic equipments or electronic cards, natural convection in a cavity can be produced due to the presence of multiple heat sources. Chu et al. [36] conducted an experimental and numerical study to analyze the effects of heater size, location, aspect ratio and boundary conditions on natural convection in a rectangular air filled enclosure. They indicated that heater size and location are important parameters on flow and temperature field and heat transfer. The problem of temperature and flow field in a partially heated enclosure for different conditions has been studied extensively in literature [4–15]. To the best of the author’s knowledge, there is no such study in the literatures indicating the mixed convection flow of nanofluid in porous cavity with various lengths and different locations of the heat source. To fulfill the necessity, the aim of the present study is to investigate the effects of different length and different locations of the heat source along the left wall and to examine the influence of the Richardson number, porosity, Darcy number, and solid volume fraction on mixed convection flow in a square porous enclosure saturated by nanofluid.

2 Mathematical Analysis

Fig. 1 presents a schematic diagram of physical configuration of the present study. The system is considered to be unsteady, laminar, incompressible mixed convective flow and heat transfer in 2-D square porous cavity of size L filled with nanofluid. The top and bottom walls are considered to be thermally insulated. In the sidewalls, the left wall is heated fully or partially with higher temperature θ_h whereas the right wall is kept with a lower temperature θ_c, such that θ_h>θ_c. The top wall of the cavity is moving from left to right with constant speed U₀. The porous cavity is filled with nanofluid, which is made of a base fluid (water) and spherical nanoparticles(Cu). The solid particles in nanofluid are ultra fine (< 100nm) and its size and shape are assumed to be uniform [11–37]. The fluid in the cavity is Newtonian and incompressible. The porous medium is assumed to be isotropic, homogeneous, and in thermodynamic equilibrium with the fluid. In our study, the single phase model is considered. The basis of single phase model is the fact that solid particles in nanofluid are ultra fine and are easily fluidized. These particles can thus be approximately considered to behave like a fluid. Under the assumptions that there exist no motion slip between the discontinuous phase of the dispersed ultra fine particles and the continuous liquid and the local thermal equilibrium exists between the nanoparticles and the base fluid, the nanofluid can be treated as a pure fluid [12, 16]. The fluid physical properties are assumed to be constant except the density variation in the buoyancy term. The Boussinesq approximation is valid. The thermophysical properties of the base fluid and nanoparticles are shown in Table 1. The governing equations consisting of mass, momentum and energy equations for the mixed convection of the nanofluid in the 2-D porous cavity can be written in the dimensional form as follows:

Fig. 1

A schematic view of the considered cavity in the present study.

where u is the Darcy velocity vector, p is the fluid pressure, θ is the temperature, and ϵ is the porosity of the medium. The permeability K and Forchheimer’s coefficient F can be written as

where a = 150 and b = 1.75 are Ergun’s constants [38] to parameterize the microscopic geometry of the porous materials. σ = [ϵ(ρC_P)_f + (1 − ϵ)(ρC_P)_s]/(ρC_P)_f$ is the specific heat ratio. Since the fluid and porous medium are in thermodynamic equilibrium, we have (ρC_P)_f = (ρC_P)_s and σ = 1.

The appropriate dimensional forms of the boundary conditions are given as follows:

For t = 0; u = v = 0, θ = θ_c, 0 ≤ (x, y) ≤ L

For t > 0; u = v = 0, ∂θ∂y=0,y=0

Case 1. Increasing heat souce length from bottom:

Case 2. Increasing heat source length from top:

Case 3. Heat source placement:

Where y₁ = L/6, 3L/6, and 5L/6 indicates the center of the heating portion from the bottom of the enclosure. In case 1, the heat source is extended from the left-bottom of the cavity and the normalized length, L_H, of the heat source is varied as L/3, 2L/3, and L of the cavity height. In case 2, the heat source is extended from the left-top of the cavity and the normalized length, L_H, of the heat source is varied similar to case 1, i.e. L/3, 2L/3, and L of the cavity height. In case 3, the length of the heat source, L_H, is maintained constant at L/3 and its position is changed as bottom, middle, and top of the left wall so that y₁ is L/6, 3L/6, and 5L/6 from the lower left corner. However, the heat source length L_H = 1/3 at the bottom in case 1 and the position of the heat source when y₁ = L/6 at the bottom in case 3 are the same. Similarly, the heat source length L_H = 1/3 at the top in case 2 and the position of the heat source when y₁ = 5L/6 at the top in case 3 are the identical. The full length of the heat source extended from left-bottom in case 1 and the whole length of the heat source extended from left-top in case 2 are the same. Therefore, there are six different configurations among three cases.

The above dimensional form of mass, momentum and energy equations can be converted to non-dimensional form by using the following dimensionless parameters:

After non-dimensionalization, the above dimensional form of governing equations can now be written as follows:

The non-dimensional boundary conditions of the considered problem are given as follows:

For τ = 0; U = V = 0, T = 0, 0≤(X,Y)≤ 1

For τ > 0; U = V = 0, ∂T∂Y=0,Y=0

Case 1:   T = 1, X = 0, 0 ≤ Y ≤ y₂ + 1/6

Case 2: T = 1, X = 0, 2 - 1/6 ≤ Y ≤ 1

Case 3: T = 1, X = 0, y₂ - 1/6 ≤ Y ≤ y₂+1/6

where y₂ = 1/6, 3/6, and 5/6.

The properties of the nanofluids are given in Table 2.

The local and average heat transfer rates of the cavity can be presented by means of the local and average Nusselt numbers. The local Nusselt number for the consider problem is calculated along the left heated wall and is defined as Nu = −(∂T/∂X)_X=0. The average Nusselt number (Nu_avg) for overall heat transfer rate is obtained by integrating the local Nusselt number and is expressed as follows:

where L_H is the length of the heat source.

Many researchers have cited the classical Maxwell [39] model in which only the particle volume concentration and the thermal conductivities of the particle are considered but the Brownian motion of the nanoparticles and the effect of solid like nano-layers formed around nanoparticles are not taken into the account. To execute the above need the modified Maxwell [40] model considers a nano-layer with a solid like structure formed by the liquid molecules close to solid surface. As far as the modified Maxwell model is concerned, k_eff is the effective thermal conductivity of the nanofluid for spherical nanoparticles, which is given as follows:

where σ is the ratio of the thickness of nano-layer to the original radius of nanoparticles (h_nl/r_s) and k_eq is the equivalent thermal conductivity of nanoparticles and their layers:

where η is the ratio of thermal conductivity of nano-layer upon the thermal conductivity of the nanoparticles (η = k_nl/k_s). In this study, it is assumed that h_nl = 2nm, r_s = 3nm and k_nl = 100k_f. Yu and Choi [40] obtained that for these conditions, the result of the modified Maxwell model are in good agreement with experimental results.

3 Numerical Technique and Validation

3.1 Numerical Method

The governing equations (4)–(7) subject to the boundary conditions are discretized by the finite volume method (FVM) on a uniform staggered grid arrangement using SIMPLE algorithm of Patankar [41]. The third order accurate deferred QUICK scheme of Hayase et al. [42] and central difference scheme are applied for the convection and diffusion terms in the both momentum and energy equations. The uniform grid is selected in both X and Y directions. The solution domain consists of a number of control volumes at which discretization equations are applied. The governing equations are transferred into a system of algebraic equations through integration over each control volume. The solution of the discretized equations for each variable is obtained by TDMA line-by-line method. The global convergence criterion for overall termination of the solution procedure is taken as 10⁻⁶. Here it is set as the sum of temperature error for all the nodes except the boundary reduced to a convergence criteria 10⁻⁶

∑i,j=1imax,jmax(Ti,jn+1−Ti,jn)<10−6

The grid independence tests are carried out using the grid points from 21×21 to 101×101 for Ri = 100, Pr = 6.2, ϵ = 0.4, Da = 10⁻², L_H = 1,y₂ = 3/6 and χ = 0.06. The grid independence test which is shown in Fig. 2 demonstrated that an 81×81 grid system is enough to obtain the desired accuracy of results.

Fig. 2

Average Nusselt number for different mesh sizes at Ri = 100, Da = 10⁻², ϵ = 0.4, χ = 0.06, and L_H = 1,y₂ = 3/6.

3.2 Code Validation

In order to check the accuracy of the present results, the present computational code has been validated by two published work in the literature. The direct validation of the present numerical model against the numerical results of Khanafer and Chamkha [44] was performed. The results of mid-plane velocity profiles are presented in Fig. 3 and an excellent agreement is observed between the two numerical results. The results for combined convection in lid driven inclined square enclosure filled with nanofluid were also compared with Abu-Nada and Chamkha [45]. Table 3 shows that the comparison of the average Nusselt number between two numerical results. In this case, the agreement is found to be excellent. Hence, the results compared with the previous literature provide confidence to the accuracies of the present numerical solutions.

Fig. 3

Comparison of the mid-plane velocity profiles of Khanafer and Chamkha [44].

Table 3

Comparison of the average Nusselt number between the present numerical results and Abu-Nada and Chamkha [45].

Ri	χ	Present Nuavg	Abu-Nada and Chamkha [45]
0.2	0.0	2.5315	2.6443
0.2	0.02	2.6282	2.7321
0.2	0.05	2.7756	2.8665
0.5	0.0	2.1220	2.1831
0.5	0.02	2.2038	2.2548
0.5	0.05	2.3277	2.3651
2	0.0	1.5911	1.6070
2	0.02	1.6522	1.6594
2	0.05	1.7448	1.7414
5	0.0	1.3213	1.3258
5	0.02	1.3767	1.3751
5	0.05	1.4623	1.4534

4 Results and Discussion

The numerical results show the effects of different lengths (L_H = 1, 1/3, and 2/3) and various positions (y₂ = 1/6, 3/6, and 5/6) of the heat source along the left side wall, as well as the effects of the Richardson number, the Darcy number, the porosity, and the solid volume fraction in a nanofluid-saturated porous enclosure. The copper-water nanofluid is chosen as working fluid with the Prandtl number Pr = 6.2. The dimensionless parameters controlling the fluid flow and heat transfer are varied in the range of 0.01 ≤ Ri ≤ 100, 10⁻⁴ ≤ Da ≤ 10⁻¹, 0.2 ≤ ϵ ≤ 0.8, and 0 ≤ χ ≤ 0.06. The computational results are discussed in the form of isotherms, streamlines, mid-plane velocity profiles, and local as well as average Nusselt numbers for various physical parameters.

Fig. 4 displays typical temperature and fluid flow contours for different lengths and locations of the heat source at Ri = 100, Da = 10⁻², ϵ = 0.4, and χ = 0.06. For L_H = 1 and y₂ = 3/6, the hot fluid along the left wall rises towards the top wall and turns to fall on the right cold wall forming a clockwise rotating cell. As a result, the isotherms are more packed along the hot and cold walls and are almost parallel to the insulated walls. When the length of the heater is changed to L_H = 2/3 with y₂ = 2/6, the structure of the recirculating cell appears as in the previous configuration. This is because the amount of fluid circulating in the cavity is almost the same for both cases. Consequently, a similar effect in isotherms is observed except near the hot wall in Fig. 4(b). The flow pattern appears diagonally from left bottom to right top as the heat source length decreases from L_H = 1 to L_H = 1/3. This can be clearly seen in Fig. 4(c). On changing the positions of the heating portion with fixed length (L_H = 1/3), the clustered isotherms move the core region of the recirculating eddy according to the heating positions, namely, y₂ = 1/6, 3/6, and 5/6. Moreover, thin boundary layers are formed along the left sidewall of the cavity only near the respective positions of the heat source (see Figs. 4(c)–(e)). When the heat source length L_H = 1/3 with y₂ = 5/6 is varied to L_H = 2/3 with y₂ = 4/6, the isotherms are spread almost in the whole cavity and reveals the enhancement of heat transfer. The same fluid flow is observed as occurred in Fig. 4(b). These contour plots confirm that the fluid flow is dependent of the heat source length and location when the natural convection dominates.

Fig. 4

Isotherms (left) and Streamlines (right) for different size and locations of the heat sourec with Ri = 100, Da = 10⁻², ϵ = 0.4, and χ = 0.06.

Fig. 5 shows the isotherms and streamlines in order to discuss the effects of solid volume fraction (χ) for L_H = 1 and y₂ = 3/6 while the remaining parameters are set as constant. The heat distribution in the porous enclosure is due to convection whether the nanoparticles are present (χ = 0.06) or absent (χ = 0). For χ = 0, a clockwise rotating cell appears inside the enclosure. In the presence of the nanoparticles, i.e., when χ = 0.06, the flow strength of the rotating cell decreases. This is a consequence of enhancement in the fluid viscosity due to the increase of nanoparticles. This can be verified in Table 4. From this table, it can be understood that when χ = 0.06 (fluid with nanoparticles), the increase in viscosity is about 16% as to compare with χ = 0 (fluid without nanoparticles). The isotherms show that a slight thicker thermal boundary layers are developed for χ = 0.06. This reveals the fact that an increase in the solid volume fraction results in an improvement in thermal conductivity (see Table 4). Furthermore, more fluid is heated with the increase of solid volume fraction of nanoparticles.

Fig. 5

Isotherms (left) and Streamlines (right) for different solid volume fraction with Ri = 1, L_H = 1, y₂ = 3/6, Da = 10⁻², and ϵ = 0.4.

In order to discuss the effect of Darcy number, the streamlines and isotherms for L_H = 1 and y₂ = 3/6 with Ri = 1, ϵ = 0.4, and χ = 0.06 are presented in Fig. 6. Since the resistance from the boundary friction is less at Da = 10⁻¹(high permeability), the fluid circulation is prominent within the enclosure. Further, it leads to convective mechanism. The flow behaviour inside the porous cavity is described by a primary vortex and a small secondary vortex appeared near right bottom corner. The corresponding isotherms indicate that the thermal boundary layers are formed along the heated wall with a better temperature distributions. This can be seen in Fig. 6(a). However, as the Darcy number is decreased from 10⁻¹ to 10⁻⁴, the boundary frictional resistance is gradually increased, and hence the fluid flow inside the enclosure is inhibited remarkably except the region near the top surface. Since the flow retarding effect of the porous medium is increased at 10⁻⁴(low permeability), the convective heat transfer mechanism is suppressed. As a consequence, the isotherms are almost parallel to the vertical walls except the small region close to the top surface where the induced convective activities are significant. These are revealed in Fig. 6(b).

Fig. 6

Isotherms (left) and Streamlines (right) for different Darcy number with Ri = 1, L_H = 1,y₂ = 3/6, ϵ = 0.4, and χ = 0.06.

Fig. 7 illustrates the velocity profiles for different Ri, lengths and locations of the heat source with the fixed value of Da = 10⁻², ϵ = 0.4, and χ = 0.06. When Ri = 100, the velocity profiles demonstrate a single vortex form for different lengths and locations of the heat source. It can be understood that changing the length of the heat source slightly affects the flow field inside the enclosure, as shown in Fig. 7(a). On the other hand, a major influence on the flow field is observed for various locations of the heating positions. Moreover, the magnitude of the velocity for the bottom location of the heat source is greater than those for the middle and top locations of the heat source. These can be seen in Fig. 7(b). On decreasing Ri, i.e., when Ri ≤ 1, the influence of length and location of the heat source on the flow field is not remarkable. It is due to the fact that the buoyancy force induced by the heat source is overwhelmed by the mechanical effect of the moving lid. It can be verified clearly in Figs. 7(c)–(d). These velocity profiles confirm that for Ri = 1 and 0.01, the fluid flow in porous enclosure is independent of the heat source lengths and its locations. But, there is a considerable change in the flow behaviour when Ri = 100, i.e., the natural convection dominates for different lengths and locations of the heat source.

Fig. 7

V-velocity profiles at mid-plane of the cavity for different L_H and y₂ at (a-b) Ri = 100, (c) Ri = 1, (d) Ri=0.01.

The effects of Ri, Da, ϵ, and χ on the velocity profiles for the fixed length of the heat source, L_H = 1, with the location y₂ = 3/6 are depicted in Fig. 8. From Fig. 8(a), it is observed that the velocity profiles turn out to be steeper with the decrease of Ri, which indicates that the magnitude of U-velocity decreases on decreasing the Richardson number in the mid-section of the cavity. In Fig. 8(b), for Da = 10⁻⁴, as the low permeability of porous medium remarkably reduces the motion of nanofluid, the magnitude of the horizontal velocity becomes steeper, which indicates the conduction dominated flow regime inside the enclosure. As Da increases from 10⁻⁴ to 10⁻¹, the magnitude of the velocity increases and shows the transition from conduction mode to convection mode. Physically, as Da increases, the degree of porosity of the porous media also increases, which provides free passage to the fluid within the enclosure. The effects of porosity on the horizontal velocity are exhibited in Fig. 8(c). It should be noted that the values of the horizontal velocity for ϵ = 0.2 are greater than that of in the ϵ = 0.8. Meanwhile, a similar effect in the fluid flow is observed between the porosity ϵ = 0.6 and 0.8. Figs. 8(b) and 8(c) reveal the fact that the permeability of porous medium plays a major role in the fluid motion and demonstrates more domination in the flow velocity than the porosity. Fig. 8(d) shows the velocity profiles for different volume fraction while the remaining parameters are set to be fixed. It is found that a slight effect on velocity profiles is observed as the solid volume fraction increases, which indicates that an increase in the solid volume fraction significantly affects the fluid flow inside the porous cavity.

Fig. 8

U-velocity profiles at mid-plane of the cavity for different (a) Ri, (b) Da, (c) ϵ, and (d) χ.

Fig. 9 presents the effects of Ri, Da, ϵ, and χ on the local Nusselt number for different lengths and locations of the heat source. Fig. 9(a) and (b) exhibit the local Nusselt number for different Ri and various lengths and locations of the heat source. It is obvious that for the low values of Ri = 0.01 and heat source length L_H = 1/3, the heat transfer rate is enhanced significantly. For the reduced lengths of the heat source, i.e., L_H = 2/3 and 1/3, a high heat transfer rate is observed near the unheated space (the insulated portion) of the left vertical wall whereas such behaviour is not found in the fully heated wall. In addition, for Ri = 0.01, the curve for the variations of the local Nusselt number is almost a parabolic type in the fully heated wall and it gets flattered with the increase in Ri, as seen in Fig. 9(a). On changing the location of the heat source from bottom to middle and then to top with fixed length L_H = 1/3, the heat transfer rate increases for all Ri (Fig. 9(b)). The effects of Da for different lengths and locations of the heat source are depicted in Figs. 9(c) and (d). In Fig. 9(c), extending the length of the heat source from the bottom of the cavity reduces the heat transfer rate for different Darcy numbers. Meanwhile, a high heat transfer rate is obtained for high permeability when compared with low permeability for different heat source lengths, as seen in Fig. 9(c). The reason of this behaviour is due to the fact that the resistance of porous medium is almost neglected in the higher permeability. The heat transfer rate increases with the increase of Darcy number for all the three locations of the heat source (see Fig. 9(d)). Fig. 9(e)–(f) shows the local Nusselt number of porosity for various lengths and locations of the heat source. In the local Nusselt number plots, the minima and maxima are obtained in the extreme edges of the heat source when the lengths of the heat source are L_H = 2/3 and 1/3. It is due to the fact that the buoyancy force is boosted up near the unheated space above the heating portion. But, such behaviour is not observed in the fully heated wall, as shown in Fig. 9(e). On increasing the porosity from 0.2 to 0.8, the heat transfer rate decreases slightly for the middle location whereas it increases for the bottom and top locations. These can be verified in Fig. 9(e)–(f). It can be understood from Fig. 9(c)–(f) that the permeability of the porous medium creates a strong effect in the heat transfer rate than the porosity. The effect of solid volume fraction for different Darcy numbers is shown in Fig. 9(g). In general, the local Nusselt numbers start with low value at the bottom end of the left wall and increase to a high value near the top end of the left wall for all the considered values of χ. This local Nusselt number plot reveals the fact that the temperature distribution is more prominent near the left top corner as the top lid moves from left to right. Moreover, it can be understood that an increase in the solid volume fraction increases the local Nusselt number for all Da considered. The local Nusselt numbers are high for Da = 10⁻² whereas it is very low when Da = 10⁻⁴.

Fig. 9

Local Nusselt numbers: (a-b) the effects of Ri, (c-d) the effects of Da, (e-f) the effects of ϵ, with different L_H and y₂, and (g) the effects of χ with different Da.

The variations of average Nusselt number against time are exhibited in Fig. 10 for various governing parameters. For all the parameters, the average Nusselt number abruptly decreases and attains the steady state condition after some time. As the length of the heat source is decreased from 1 to 1/3 gradually, the overall heat transfer rate increases constantly (Fig. 10(a)). The steady state is achieved almost equally for all the different lengths of the heat source. Next, the influence of location of the heat source on the average Nusselt number against time is presented in Fig. 10(b). For the top location of the heat source, a high heat transfer rate is obtained and the time to attain the steady state is little bit high compared with the remaining locations of the heat source, as seen in Fig. 10(b). The time history of the average Nusselt number for different Ri is illustrated in Fig. 10(c). The values of average Nusselt number for all Ri show a quick decline and attain the steady state after sometime. Moreover, for Ri = 0.01, the time to attain the steady state is increased. The average Nusselt number versus time for different porosity is displayed in Fig. 10(d). An increase in porosity is to decrease the overall heat transfer rate slightly and leads to the steady state more quickly. Fig. 10(e) presents the average Nusselt number versus time for different solid volume fraction. It is seen that when the fluid is pure, i.e., χ = 0, the values of average Nusselt number have a slight oscillation after a quick decline. But, such behaviour is vanished gradually with an increase in the solid volume fraction. The time to reach the steady state is increased with the increase of solid volume fraction. In Fig. 10(f), It is observed that an increase in the value of Da results in an increase in the average Nusselt number. The time for attaining the steady state is earlier for Da = 10⁻³ than that of in the remaining Darcy number. These can be seen in Fig. 10(f).

Fig. 10

Time history of average Nusselt numbers for different (a-b) L_H and y₂, (c) Ri, (d) ϵ, (e) χ, and (f) Da.

The variations of average Nusselt number against Ri, Da, ϵ, and χ for different lengths and locations of the heat source are shown in Fig. 11. It is seen that for any fixed length and location of the heat source, the total heat transfer rate is increased with a decrease in Ri. At Ri = 100, a better heat transfer rate is obtained for L_H = 1/3 with y₂ = 3/6. But, when Ri = 0.01, an enhanced heat transfer rate is observed for L_H = 1/3 with y₂ = 5/6. These are depicted in Fig. 11(a). There are variations in the average Nusselt number with Darcy numbers when the lengths and locations of the heat source are changed; see Fig. 11(b). The overall heat transfer rate decreases with a decrease in Darcy number for different lengths and locations. This suggests that the high permeability becomes more dominant in the overall heat transfer rate than the low permeability. The average Nusselt number for the top location of the heat source shows a better enhancement in the overall heat transfer rate as to compare with the other cases for all the values of Da. Fig. 11(c) displays the influence of porosity on the overall heat transfer rate with the effects of various lengths and locations of the heat source. For all ϵ, the overall heat transfer rate is augmented with a decrease in the length of the heat source and when the locations of the heat source vary from bottom to top. It is important to note that an increase in the porosity decreases the overall heat transfer rate remarkably except the bottom and top locations of the heat source with fixed length L_H = 1/3. The possible reason behind the decrease of the average Nusselt number can be verified by having a look at the local Nusselt number (see Fig. 9(e)–(f)). In Fig. 11(d), it is observed that the overall heat transfer rate generally increases when the nanoparticles are added into the base fluid. This enhancement of heat transfer is evident for Da = 10⁻⁴, where heat conduction dominates. A further increase in Da enhances the flow velocity due to the higher permeability of the porous medium, and it also changes the flow structure. This leads to a high heat transfer rate in the enclosure. In the higher permeability (Da = 10⁻¹), the addition of nanoparticles in base fluid enhances the overall heat transfer rate appreciably when compared with the low permeability (Da = 10⁻⁴). It is due to the fact that the higher permeability of the porous medium allows the nanofluid to flow easily in the enclosure whereas the intensity of fluid flow is retarded in the low permeability. A fluid with nanoparticles (nanofluid) gives a better heat transfer enhancement than the pure fluid (fluid without nanoparticles) for all Da considered. The variation of average Nusselt number with Da and χ is presented in Table 5. An increase in the solid volume fraction from 0 to 0.06 increases the average Nusselt number for all Darcy number varied. However, a high heat transfer rate (57.26%) is obtained when Da = 10⁻¹ and χ = 0.06.

Fig. 11

Average Nusselt numbers: (a) the effects of Ri, (b) the effects of Da, (c) the effects of ϵ, with different L_H and y₂, and (d) the effects of χ with different Da.

5 Conclusions

The effects of different heat source lengths and locations on mixed convection of nanofluid in a square lid-driven enclosure fully saturated with porous medium have been numerically investigated. The effects of heat source length and location on the fluid flow are significant only in the natural convection mode whereas it does not make an impact on the fluid flow in the forced and mixed convection modes. Among the different lengths of the heat source, a better heat transfer rate is obtained only for the length L_H = 1/3 of the heat source. In the case of location of the heat source, the overall heat transfer rate is enhanced when the location of the heat source is at the top of the hot wall. The time to attain the steady state is almost the same for all different types of lengths and locations of the heat source. The steady state is achieved more quickly in the low permeability than that of in the high permeability. The permeability of the porous medium plays a significant role in convection of nanofluid than porosity. On increasing the Darcy number, the overall heat transfer rate is enhanced regardless of the length and location of the heat source. For the nanofluid, the time to reach the steady state is high whereas it is low for the base fluid (without nanoparticles). By adding the nanoparticles to the base fluid in the porous medium, the heat transfer rate is increased. The variation of the average Nusselt number is linear with the increase of solid volume fraction for all Da considered. The inclusion of nanoparticles in the base fluid increases the overall heat transfer rate remarkably for Da = 10⁻¹ when compared with Da = 10⁻⁴. A better heat transfer rate (57.26%) is obtained for Da = 10⁻¹ and χ = 0.06