Non-uniform Heating Of Bottom Wall Research Articles

Effects of uniform or non-uniform heating at bottom wall on MHD mixed convection in a porous cavity saturated by nanofluid

Abstract A numerical investigation on the effects of uniform and non-uniform heating of bottom wall on mixed convective heat transfer in a square porous chamber filled with nanofluid in the appearance of magnetic field is carried out. Uniform or sinusoidal heat source is fixed at the bottom wall. The top wall moves in either positive or negative direction with a constant cold temperature. The vertical sidewalls are thermally insulated. The finite volume approach based on SIMPLE algorithm is followed for solving the governing equations. The different parameters connected with this study are Richardson number (0.01 ≤ Ri ≤ 100), Darcy number (10−4 ≤ Da ≤ 10−1), Hartmann number (0 ≤ Ha ≤ 70), and the solid volume fraction (0.00 ≤ χ ≤ 0.06). The results are presented graphically in the form of isotherms, streamlines, mid-plane velocities, and Nusselt numbers for the various combinations of the considered parameters. It is observed that the overall heat transfer rate is low at Ri = 100 in the positive direction of lid movement, whereas it is low at Ri = 1 in the negative direction. The average Nusselt number is lowered on growing Hartmann number for all considered moving directions of top wall with non-uniform heating. The low permeability, Da = 10−4 keeps the flow pattern same dominating the magnetic field, whereas magnetic field strongly affects the flow pattern dominating the high Darcy number Da = 10−1. The heat transfer rate increases on enhancing the solid volume fraction regardless of the magnetic field.

Numerical Simulation of MHD Natural Convection Flow Within Porous Trapezoidal Cavity With Heated Triangular Obstacle

A steady laminar two-dimensional finite element based Galerkin weighted residual numerical approach (GWRNA) has been adopted to solve magnetohydrodynamic (MHD) natural convective flow filled with porous medium in a trapezoidal cavity with various aspect ratio of heated triangular obstacle $$ (1/3 \le AR \le 1) $$ . The boundary location of the cavity consists of non-uniform heating of bottom wall whilst the two vertical walls are maintained at a cold temperature and the top wall is thermal insulated. For the simulation purpose, wide range of parameters, such as, Hartmann number (Ha = 50), Prandtl number (0.026 ≤ Pr ≤ 1000), Rayleigh number (103 ≤ Ra ≤ 106) and Darcy number (10−5 ≤ Da ≤ 10−1) are used in the cavity with various AR. The study demonstrated that the aforementioned parameters have strong influence on the stream functions and isotherms for different AR. A stronger flow circulation is observed for higher Ra and Da. Besides these, a symmetry for fluid flow simulations and isotherm contours are viewed. The result too unveils that the heat transfer rates as well as local and average Nusselt numbers strongly depend on various AR and abovementioned parameters.

Finite Element Analysis of Magnetohydrodynamic Mixed Convection in a Lid-Driven Trapezoidal Enclosure Having Heated Triangular Block

A numerical research on magnetohydrodynamic mixed convection flow in a lid-driven trapezoidal enclosure at non-uniform heating of bottom wall has been studied numerically. The enclosure consists of insulated top wall and cold side walls, too. It also contains a heated triangular block (Rot = 0° - 90°) located somewhere inside the enclosure. The boundary top wall of the enclosure is moving through uniform speed U0. The geometry of the model has been represented mathematically by coupled governing equations in accordance with proper boundary conditions and then a two-dimensional Galerkin finite element based numerical approach has been adopted to solve this paper. The numerical computations have been carried out for the wide range of parameters Prandtl number (0.5 ≤ Pr ≤ 2), Reynolds number (60 ≤ Re ≤ 120), Rayleigh number (Ra = 103) and Hartmann number (Ha = 20) taking with different rotations of heated triangular block. The results have been shown in the form of streamlines, temperature patterns or isotherms, average Nusselt number and average bulk temperature of the fluid in the enclosure at non-uniform heating of bottom wall. It is also indicated that both the streamlines, isotherm patterns strongly depend on the aforesaid governing parameters and location of the triangular block but the thermal conductivity of the triangular block has a noteworthy role on the isotherm pattern lines. Moreover, the variation of Nuav of hot bottom wall and θav in the enclosure is demonstrated here to show the characteristics of heat transfer in the enclosure.

Mixed convection of heat generating nanofluid in a lid-driven cavity with uniform and non-uniform heating of bottom wall

Steady state two-dimensional mixed convection in a lid-driven square cavity filled with Cu–water nanofluid is investigated numerically in the presence of internal heat generation. In the present investigation, bottom wall is uniformly and non-uniformly heated while two vertical walls are fixed and they are thermally insulated. The top wall is moving from left to right at a constant speed. The governing equations are normalized and solved numerically with boundary conditions by finite volume approach using third order accurate upwinding scheme (deferred QUICK). Effects of the pertinent physical parameters are investigated in terms of the flow and temperature fields, as well as Nusselt number distributions. The presented results show that the solid volume fraction plays a significant role on the flow and thermal fields and the Nusselt number distributions for different flow configurations. It is found that Richardson number strongly affect the fluid flow and heat transfer in the cavity. For Ri<1, the forced convection becomes dominant in the entire cavity, the natural convection relatively weak.

MHD free convection within trapezoidal cavity with non-uniformly heated bottom wall

The paper is connected to investigate numerical simulation of two-dimensional laminar steady-state on MHD free convection within trapezoidal cavity with non-uniformly heated bottom wall. The cavity consists of non-uniformly heated bottom wall, insulated top wall and isothermal side walls with inclination angles (ϕ). Heat flow patterns in the presence of free convection within trapezoidal enclosures have been analyzed with heatlines concept. The fluid is concerned for the wide range of Rayleigh number (Ra) from 103 to 107 and Prandtl number (Pr) from 0.026, 0.7, 1000 with various tilt angles Φ=45°, 30° and 0° (square). The physical problems are represented by non-dimensional governing equations along with the corresponding boundary conditions and discretized by using Galerkin weighted residual method of finite element formulation. Results are presented in terms of streamlines, isotherms, local Nusselt number along distance and average Nusselt number along Rayleigh number, Ra for non-uniform heating of the bottom wall, governing parameters namely, Prandtl number Pr, and at the three values of Rayleigh number Ra, varying from 103 to 107, covering free convection dominated regimes. It is shown that the average and local Nusselt number at the non-uniform heating of bottom wall of the cavity is depending on the dimensionless parameters and also tilts angles.

Heatline analysis on natural convection for nanofluids confined within square cavities with various thermal boundary conditions

Natural convection of nanofluids in presence of hot and cold side walls (case 1) or uniform or non-uniform heating of bottom wall with cold side walls (case 2) have been investigated based on visualization of heat flow via heatfunctions or heatlines. Galerkin finite element method has been employed to solve momentum and energy balance as well as post processing streamfunctions and heatfunctions. Various nanofluids are considered as Copper–Water, TiO2–Water and Alumina–Water. Enhancement of heat transfer with respect to base fluid (water) has been observed for all ranges of Rayleigh number (Ra). Dominance of viscous force or buoyancy force are found to play significant roles to characterize the heat transfer rates and temperature patterns which are also established based on heatlines. In general, convective closed loop heatlines are present even at low Rayleigh number (Ra=103) within base fluid, but all nanofluids exhibit dominant conductive heat transport as the flow is also found to be weak due to dominance of viscous force for case 1. On the other hand, convective heat transport at the core of a circulation cell, typically represented by closed loop heatlines, is more intense for nanofluids compared to base fluid (water) for case 2 at Ra=105. It is also found that heatlines with larger heatfunctions values for nanofluids coincide with heatlines with smaller heatfunction values for water at walls. Consequently, Nusselt number which is also correlated with heatfunctions show larger values of nanofluids for all ranges of Ra. Average Nusselt numbers show that larger enhancement of heat transfer rates for all nanofluids at Ra=105 and Alumina–Water and Copper–Water exhibit larger enhancement of heat transfer rates.

A Peclet number based analysis of mixed convection for lid-driven porous square cavities with various heating of bottom wall

Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniform and non-uniform heating of bottom wall. The relevant parameters in the present study are Darcy number (Da=10−5−10−5), Grashof number (Gr=103−105), Prandtl number (Pr=0.026−10) and Reynolds number (Re=1−102). The influence of convection is analyzed with Peclet number (Pe=Re.Pr). It is observed that the temperature profiles are symmetric for low values of Pe or Pr even in the presence of asymmetric flow fields irrespective of Da. The flow distribution affects significantly temperature distributions at high Pe irrespective of Da. Effect of Peclet numbers have been further investigated for both natural convection and forced convection dominant regimes at high Da. Strong coupling between flow fields and temperature are observed at high Pe. It is interesting to observe that large isothermal mixing zone at Pr=10 reduces the overall flow strength compared to Pr=0.026 case. Local Nusselt numbers show almost uniform and low values for low Peclet numbers and localized enhanced heat transfer rates are observed for high Peclet numbers at Da=10−3.

Analysis of heatlines for natural convection within porous trapezoidal enclosures: Effect of uniform and non-uniform heating of bottom wall

In this paper, the numerical investigation of natural convection in a porous trapezoidal enclosures has been performed for uniformly or non-uniformly heated bottom wall. Penalty finite element analysis with bi-quadratic elements is used for solving the Navier–Stokes and energy balance equations. The numerical solutions are studied in terms of streamlines, isotherms, heatlines, local and average Nusselt numbers for a wide range of parameters Da(10 −5–10 −3), Pr(0.015–1000) and Ra( Ra = 10 3–10 6). At low Darcy number ( Da = 10 −5), heat transfer is primarily due to conduction for all φ’s as seen from the heatlines which are normal to the isotherms. As Da increases to 10 −4, convection is initiated and the thermal mixing has been observed at the central regime for all φ’s. Distribution of heatlines illustrate that most of the heat transport for high Darcy number ( Da = 10 −3) occurs from hot bottom wall to the top portion of cold side walls. It has been found that secondary circulations appear at the top corners of the cavity for φ = 45°, 60° and bottom corners of the cavity for φ = 90° with Pr = 0.015, Da = 10 −3 and Ra = 10 6. The physical interpretation of local and average Nusselt numbers are illustrated using heatlines. For uniformly heated bottom wall with cold side walls, Nu b values are maximum near the corners of bottom wall for all φ’s irrespective to Da and Pr. In contrast, for non-uniformly heated bottom wall, the local Nusselt number ( Nu b ) is found to be minimum near the corners of bottom wall and that is also found to be a sinusoidal variation with distance at high Da for all angles ( φ). The Nu s distribution is similar in both cases along the side wall except near the junction of hot and cold wall for all φ’s. Overall, heat transfer analysis for bottom and side walls is presented in terms of average Nusselt numbers ( Nu b ¯ , Nu s ¯ ) . The critical Ra numbers corresponding to the onset of convection are obtained at Da = 10 −3 for Pr(0.015–1000). For Da = 10 −3, average Nusselt numbers ( Nu b ¯ and Nu s ¯ ) increase exponentially beyond the critical Ra. Overall, the heat transfer rate is large for square cavity ( φ = 90°) compared to other angles ( φ) irrespective of heating patterns.

Natural convection flows in porous trapezoidal enclosures with various inclination angles

Simulations were carried out using penalty finite element analysis with bi-quadratic elements to investigate the influence of uniform and non-uniform heating of bottom wall within a trapezoidal enclosure of various inclination angles (φ). Parametric study has been carried out for a wide range of Rayleigh number (Ra)(103⩽Ra⩽106), Prandtl number (Pr)(0.026⩽Pr⩽988.24) and Darcy number (Da)(10-3⩽Da⩽10-5). Numerical results are presented in terms of stream functions, isotherm contours and Nusselt numbers. The heat transfer is primarily due to conduction at lower values of Darcy number (Da) and convection dominant heat transfer is observed at higher Da values. The intensity of circulation increases with increase in Darcy number. Increase in the intensity of circulations and larger temperature gradient are also observed with increase in φ from 0° to 45° especially at larger Pr and Ra. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating at all Rayleigh and Darcy numbers, but average Nusselt number is lower for non-uniform heating. Local heat transfer rates are found to be relatively greater for φ=0°. It is observed that the local heat transfer rate at the central portion of bottom wall is larger for non-uniform heating case. Average Nusselt number plots show higher heat transfer rates at the bottom wall for φ=0° as compared to φ=45° and φ=30°. It is observed that the average heat transfer rate at the bottom wall is found to be invariant with respect to φ at higher Ra for non-uniform heating. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and the power law correlations between average Nusselt number and Rayleigh numbers are presented for convection dominated regimes.

Heat flow analysis for natural convection within trapezoidal enclosures based on heatline concept

Heat flow patterns in the presence of natural convection within trapezoidal enclosures have been analyzed with heatlines concept. In the present study, natural convection within a trapezoidal enclosure for uniformly and non-uniformly heated bottom wall, insulated top wall and isothermal side walls with inclination angle φ have been investigated. Momentum and energy transfer are characterized by streamfunctions and heatfunctions, respectively, such that streamfunctions and heatfunctions satisfy the dimensionless forms of momentum and energy balance equations, respectively. Finite element method has been used to solve the velocity and thermal fields and the method has also been found robust to obtain the streamfunction and heatfunction accurately. The unique solution of heatfunctions for situations in differential heating is a strong function of Dirichlet boundary condition which has been obtained from average Nusselt numbers for hot or cold regimes. Parametric study for the wide range of Rayleigh number ( Ra ) , 10 3 ⩽ Ra ⩽ 10 5 and Prandtl number ( Pr ) , 0.026 ⩽ Pr ⩽ 1000 with various tilt angles φ = 45 ° , 30 ° and 0 ° (square) have been carried out. Heatlines are found to be continuous lines connecting the cold and hot walls and the lines are perpendicular to the isothermal wall for the conduction dominant heat transfer. The enhanced thermal mixing near the core for larger Ra is explained with dense heatlines and convective loop of heatlines. The formation of boundary layer on the walls has a direct consequence based on heatlines. The local Nusselt numbers have also been shown for side and bottom walls and variation of local Nusselt numbers with distance have also been explained based on heatlines. It is found that average heat transfer rate does not vary significantly with φ for non-uniform heating of bottom wall.

Analysis of mixed convection flows within a square cavity with uniform and non-uniform heating of bottom wall

A penalty finite element analysis with bi-quadratic elements is performed to investigate the influence of uniform and non-uniform heating of bottom wall on mixed convection lid driven flows in a square cavity. In the present investigation, bottom wall is uniformly and non-uniformly heated while the two vertical walls are maintained at constant cold temperature and the top wall is well insulated and moving with uniform velocity. A complete study on the effect of Gr shows that the strength of circulation increases with the increase in the value of Gr irrespective of Re and Pr. As the value of Gr increases, there occurs a transition from conduction to convection dominated flow at Gr = 5 × 10 3 and Re = 1 for Pr = 0.7 . A detailed analysis of flow pattern shows that the natural or forced convection is based on both the parameters Ri ( Gr Re 2 ) and Pr. As the value of Re increases from 1 to 10 2, there occurs a transition from natural convection to forced convection depending on the value of Gr irrespective of Pr. Particularly for higher value of Grashof number ( Gr = 10 5 ), the effect of natural convection is dominant upto Re = 10 and thereafter the forced convection is dominant with further increase in Re. As Pr increases from 0.015 to 10 for a fixed Re and Gr ( Gr = 10 3 ), the inertial force gradually becomes stronger and the intensity of secondary circulation gradually weakens. The local Nusselt number ( Nu b ) plot shows that the heat transfer rate is very high at the edges of the bottom wall and then decreases at the center of the bottom wall for the uniform heating and that contrasts lower heat transfer rate at the edges for the non-uniform heating of the bottom wall. It is also observed that Nu l shows non-monotonic behavior with both uniform and non-uniform heating cases for Re = 10 at higher value of Pr. The average Nusselt number plot for the left or right wall shows a kink or inflexion at Gr = 10 4 for highest value of Pr. Thus the overall power law correlation for average Nusselt number may not be obtained for mixed convection effects at higher Pr.

Finite element simulation of natural convection flow in a trapezoidal enclosure filled with porous medium due to uniform and non-uniform heating

The phenomena of natural convection in a trapezoidal enclosure filled with porous matrix has been studied numerically. A penalty finite element analysis with bi-quadratic elements is performed to investigate the influence of uniform and non-uniform heating of bottom wall while two vertical walls are maintained at constant cold temperature and the top wall is well insulated. Parametric study for the wide range of Rayleigh number Ra ( 10 3 ⩽ Ra ⩽ 10 5 ) , Prandtl number Pr ( 0.015 ⩽ Pr ⩽ 1000 ) and Darcy number ( 10 - 3 ⩽ Da ⩽ 10 - 5 ) shows consistent performance of the present numerical approach to obtain the solutions in terms of stream function and isotherm contours. For parameters studied in the above range, a symmetry is observed for temperature and flow simulations. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh and Darcy numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It is observed that the conduction is dominant irrespective of Ra for Da = 10 - 5 . As Rayleigh number increases, there is a change from conduction dominant region to convection dominant region for Da = 10 - 3 . The correlations between average Nusselt number and three parameters (Rayleigh number ( Ra), Prandtl number ( Pr) and Darcy number ( Da)) are also obtained.

Natural convection and flow simulation in differentially heated isosceles triangular enclosures filled with porous medium

A finite element analysis is performed to investigate the effects of uniform and non-uniform heating of bottom wall on natural convection flows within isosceles triangular enclosures filled with porous medium. The detailed analysis is carried out in two cases depending on various thermal boundary conditions: (I) two inclined walls are maintained at constant cold temperature while the bottom wall is uniformly heated; (II) two inclined walls are maintained at constant cold temperature while the bottom wall is non-uniformly heated. The present numerical procedure adopted in this investigation yields consistent performance over a wide range of parameters of Darcy number, Da ( 10 - 5 ⩽ Da ⩽ 10 - 3 ), Rayleigh number, Ra ( 10 3 ⩽ Ra ⩽ 10 6 ) and Prandtl number, Pr ( 0.026 ⩽ Pr ⩽ 1000 ) in all the cases mentioned above. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. It is observed that at small Darcy numbers, the heat transfer is primarily due to conduction irrespective of Pr. As the Darcy number increases, there is a change from conduction dominant regime to convection dominant regime. Flow circulations are also found to be strong functions of Pr at large Da ( Da = 10 - 3 ) and multiple circulation cells occur at small Pr with Ra = 10 6 . Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case, but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. As average Nusselt number is same on both the inclined walls, the average Nusselt number for bottom wall is 2 times that of the inclined wall which is well matched in two cases considered for verifying the thermal equilibrium of the system. The correlations are proposed for average Nusselt number as functions of Ra for various Darcy and Prandtl numbers.

Finite Element Simulation on Natural Convection Flow in a Triangular Enclosure Due to Uniform and Nonuniform Bottom Heating

A penalty finite element analysis with biquadratic elements has been carried out to investigate natural convection flows within an isosceles triangular enclosure with an aspect ratio of 0.5. Two cases of thermal boundary conditions are considered with uniform and nonuniform heating of bottom wall. The numerical solution of the problem is illustrated for Rayleigh numbers (Ra), 103⩽Ra⩽105 and Prandtl numbers (Pr), 0.026⩽Pr⩽1000. In general, the intensity of circulation is found to be larger for nonuniform heating at a specific Pr and Ra. Multiple circulation cells are found to occur at the central and corner regimes of the bottom wall for a small Prandtl number regime (Pr=0.026−0.07). As a result, the oscillatory distribution of the local Nusselt number or heat transfer rate is seen. In contrast, the intensity of primary circulation is found to be stronger, and secondary circulation is completely absent for a high Prandtl number regime (Pr=0.7–1000). Based on overall heat transfer rates, it is found that the average Nusselt number for the bottom wall is 2 times that of the inclined wall, which is well, matched in two cases, verifying the thermal equilibrium of the system. The correlations are proposed for the average Nusselt number in terms of the Rayleigh number for a convection dominant region with higher Prandtl numbers (Pr=0.7 and 10).

Role of ‘Bejan’s heatlines’ in heat flow visualization and optimal thermal mixing for differentially heated square enclosures

Heat flow patterns in the presence of natural convection have been analyzed with Bejan’s heatlines concept. Momentum and energy transfer are characterized by streamfunctions and heatfunctions, respectively such that streamfunctions and heatfunctions satisfy the dimensionless forms of momentum and energy balance equations, respectively. Finite element method has been used to solve the velocity and thermal fields and the method has also been found robust to obtain the streamfunction and heatfunction accurately. The unique solution of heatfunctions for situations in differential heating is a strong function of Dirichlet boundary condition which has been obtained from average Nusselt numbers for hot or cold regimes. The physical significance of heatlines have been demonstrated for a comprehensive understanding of energy distribution and optimal thermal management via analyzing three cases. Case 1 involves the uniform and non-uniform heating of bottom wall with cooled side walls. The studies illustrate that the heat flow primarily occurs from the central regime of the bottom wall to a very small regime of the top portion of side walls. A large portion of central regime of cold side walls do not receive significant amount of heat. In order to maximize the thermal energy distribution, the distributed heating at the middle portions of the bottom and side walls have been considered in case 2 and heatlines clearly depict the distributions of heat from the hot walls to the large regimes of the cold wall. Further case 3 illustrates the enhanced heat flows in presence of heated bottom and left side walls. Heatline is found as an effective numerical tool to visualize energy distribution in order to establish a suitable heating strategy.

Natural convection flows in a trapezoidal enclosure with uniform and non-uniform heating of bottom wall

A penalty finite element analysis with bi-quadratic elements is performed to investigate the influence of uniform and non-uniform heating of bottom wall on natural convection flows in a trapezoidal cavity. In the present investigation, bottom wall is uniformly and non-uniformly heated while two vertical walls are maintained at constant cold temperature and the top wall is well insulated. Parametric study for the wide range of Rayleigh number ( Ra), 10 3 ⩽ Ra ⩽ 10 5 and Prandtl number ( Pr), 0.07 ⩽ Pr ⩽ 100 shows consistent performance of the present numerical approach to obtain the solutions in terms of stream functions and the temperature profiles. For certain values of the parameters studied in the above range, a symmetry is observed while representing the flow patterns in terms of stream functions. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. The power law correlations between average Nusselt number and Rayleigh numbers are presented. The effect of Prandtl number in the variation of local and average Nusselt numbers is more significant for Prandtl numbers in the range 0.07–0.7 than 10–100.