D. Ramakrishna

Bio: D. Ramakrishna is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Natural convection & Nusselt number. The author has an hindex of 10, co-authored 10 publications receiving 240 citations.

Visualization of Heat Transport during Natural Convection Within Porous Triangular Cavities via Heatline Approach

Abstract: In this article, natural convection in a porous triangular cavity has been analyzed. Bejan's heatlines concept has been used for visualization of heat transfer. Penalty finite-element method with biquadratic elements is used to solve the nondimensional governing equations for the triangular cavity involving hot inclined walls and cold top wall. The numerical solutions are studied in terms of isotherms, streamlines, heatlines, and local and average Nusselt numbers for a wide range of parameters Da (10−5–10−3), Pr (0.015–1000), and Ra (Ra = 103–5 × 105). For low Darcy number (Da = 10−5), the heat transfer occurs due to conduction as the heatlines are smooth and orthogonal to the isotherms. As the Rayleigh number increases, conduction dominant mode changes into convection dominant mode for Da = 10−3, and the critical Rayleigh number corresponding to the on-set of convection is obtained. Distribution of heatlines illustrate that most of the heat transport for a low Darcy number (Da = 10−5) occurs from the top...

A comprehensive heatline based approach for natural convection flows in trapezoidal enclosures: Effect of various walls heating

Abstract: The present numerical study deals with natural convection flow in closed trapezoidal enclosures. The detailed analysis is carried out in two cases: (1) linearly heated side walls; (2) linearly heated left wall and cold right wall. In both the cases bottom wall is uniformly heated and top wall is well insulated. A penalty finite element method with bi-quadratic elements is used to obtain the results in the form of isotherms, streamlines and heatlines and local and average Nusselt numbers. Numerical results are obtained for various values of Rayleigh number Ra ( 10 3 ≤ R a ≤ 10 5 ), Prandtl number Pr ( 0.015 ≤ Pr ≤ 1000 ) and inclination angles (φ = 45°, 60° and 90°). Results signify that, at low Ra ( R a = 10 3 ) heat transfer is conduction dominant. At R a = 10 5 , multiple circulations of streamlines and heatlines results in enhanced convection. For linearly heated side walls (case 1), symmetric pattern in fluid flow and heat flow is observed. Enhanced thermal transport is observed from bottom wall to top portion of side walls via dense heatlines along the vertical center line. It is found that, less intense circulations occurs in square cavity (φ = 90°) compared to other cavities φ = 45°, 60°. In case 2, the cold right wall receives larger amount of heat from bottom wall compared to that of linearly heated left wall. The formation of boundary layer on the walls is explained based on heatlines. The local and average Nusselt numbers are also illustrated using heatlines. It is found that, Nub distribution exhibits sinusoidal variation at P r = 1000 in case 1. It is also found that, Nul and Nur display wavy pattern at higher Ra for all Pr in case 2. Finally, it is concluded that, overall heat transfer rates are larger for square cavity (φ = 90°) compared to other angles (φ = 45°, φ = 60°) irrespective of heating patterns for side walls.

Analysis of heatlines during natural convection within porous square enclosures: Effects of thermal aspect ratio and thermal boundary conditions

Abstract: Numerical investigation of natural convection within porous square enclosures has been performed for various thermal boundary conditions based on thermal aspect ratio on bottom and side walls. Penalty finite element analysis with bi-quadratic elements is used to solve the governing 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 1 ), Pr (0.015–1000) and Ra ( Ra = 10 3 –10 5 ). At low Darcy number ( Da = 10 −5 ), heatlines are perpendicular to the isotherms indicating conduction dominant heat transfer. As Da increases to 10 −3 , convection is initiated and the thermal mixing has been observed at the central regime for all A s. At low Prandtl number ( Pr = 0.015) with high Darcy number ( Da = 10 −2 and Da = 10 1 ), multiple circulations are observed in streamlines and heatlines and they suppressed for higher Prandtl number ( Pr = 1000). Isotherms are highly compressed along bottom wall at higher Prandtl numbers ( Pr = 0.7 and 1000) at A = 0.1 and 0.5. Temperature gradient is found to be high at the center of the bottom wall for A = 0.1 due to dense heatlines at that zone and that decreases as A increases from 0.1 to 0.9, irrespective of Pr , Da . Also, the temperature gradient is smaller at the top portion of side walls for A = 0.1 due to sparse heatlines along those zones and that is high for A = 0.9 due to dense heatlines. Distribution of heatlines illustrate that significant heat transport occurs from hot bottom wall to the top portion side walls at higher Darcy number ( Da = 10 1 ). It is found that Nu b attains maximum at X = 0.5 and minimum at corners for Da = 10 −5 , whereas that exhibits sinusoidal variation for Da = 10 −3 and Da = 10 1 irrespective of Pr and A . It is also found that Nu l follows wavy pattern at low Prandtl number ( Pr = 0.015) with higher Darcy number ( Da = 10 1 ) irrespective of A due to larger gradients of heatfunctions at several locations of left wall. The average Nusselt number show that the overall heat transfer rate is high at A = 0.1 compared to that of A = 0.5 and A = 0.9 irrespective of Da and Pr due to larger gradients of heatfunctions at A = 0.1.

Analysis of thermal efficiency via analysis of heat flow and entropy generation during natural convection within porous trapezoidal cavities

Abstract: Thermal management via distributions of heatlines and entropy generation for natural convection within trapezoidal cavities in presence of hot left wall, cold right wall and adiabatic horizontal walls has been studied in this article. Heat flow visualization has been carried out via heatline concept. Galerkin finite element method has been used to analyze streamlines, isotherms, heatlines, entropy generation due to fluid friction and heat transfer over wide range of parameters ( 10 - 5 ⩽ Da ⩽ 10 - 3 , 0.015 ⩽ Pr ⩽ 1000 at Ra = 10 6 ). At low Darcy number ( Da = 10 - 5 ), conduction dominant heat transfer is found based on low magnitudes of streamlines and heatlines. Heatlines indicate that heat transfer occurs from hot left wall to cold right wall and thermal mixing is found inside the cavity. The thermal mixing is enhanced as Da increases from 10 - 5 to 10 - 3 . The thermal gradients are high near the lower portion of left wall and near upper portion of right wall for Da ⩾ 10 - 4 irrespective of φ and Pr and thus, thermal boundary layer thickness is small along those zones. The maximum entropy generation due to fluid friction ( S ψ , max ) occurs along the left wall for φ = 30 ° and 90 ° irrespective of Pr whereas that occurs along the right wall for φ = 60 ° at Da = 10 - 3 . The maximum entropy generation due to heat transfer ( S θ , max ) occurs at the left edge of bottom wall irrespective of Pr and Da for φ = 30 ° and 60 ° whereas that occurs at the left edge of bottom wall and right edge of top wall for φ = 90 ° with Da = 10 - 5 and 10 - 4 . At φ = 90 ° with Da = 10 - 3 , S θ , max occurs along both side walls for Pr = 0.015 whereas that occurs along left wall for Pr = 1000 . It is found that total entropy generation is high for Pr = 1000 compared to that of Pr = 0.015 at higher Da. It is also found that the trapezoidal cavities with φ = 60 ° and 90 ° correspond to less entropy generation with significant heat transfer rates at Da = 10 - 3 for Pr = 0.015 and Pr = 1000 and thus the trapezoidal cavities with φ ⩾ 60 ° may be the optimal design for thermal processing of Pr = 0.015 and Pr = 1000 fluids.

Visualization of heat transport due to natural convection for hot materials confined within two entrapped porous triangular cavities via heatline concept

Abstract: This paper analyzes the detailed heat transfer within two entrapped porous triangular cavities involving cold inclined walls and hot horizontal walls. A penalty finite element analysis with bi-quadratic elements is performed to investigate the results in terms of isotherms, streamlines and heatlines and local and average Nusselt numbers. The parameters for this study are Darcy number, Da ( 10 - 5 – 10 - 3 ) , Prandtl number, Pr ( 0.015 – 1000 ) and Rayleigh number ( 10 3 – 5 × 10 5 ) . It has been found that at small Darcy number ( Da = 10 - 5 ) , heat transfer is primarily conduction dominant and heatlines are found to be orthogonal to the isotherms. The presence of multiple circulations in streamlines and heatlines are observed within the lower triangle at small Prandtl number Pr = 0.015 with high Darcy number ( Da = 10 - 3 ) whereas only single pair of circulations are observed for higher Prandtl numbers. The convective cells in heatlines gradually become enhanced as Pr increases from 0.015 to 1000. In contrast, variation of Prandtl number gives negligible change in heating pattern within the upper triangle and intensity of streamlines and heatlines are less irrespective of Prandtl number. Heat transfer rates are estimated in terms of local ( Nu l , Nu h ) and average Nusselt numbers ( Nu l ¯ , Nu h ¯ ) . Heat transfer rates are also explained based on heatlines. Local Nusselt numbers with spatial distribution exhibit monotonic trend irrespective of Da and Pr for the upper triangle whereas wavy distribution of local Nusselt number occur for the lower triangle. For Da = 10 - 3 , average Nusselt numbers ( Nu h ¯ and Nu l ¯ ) increase exponentially with Ra at higher Rayleigh numbers. But, overall lower heat transfer rates are observed for the upper triangle. Finally, it is concluded that lower triangle has always has higher heat recovery capacity compared to upper triangle. To achieve efficient heat transfer, fluids with high Prandtl numbers are recommended for the lower triangle whereas any fluid with any Prandtl number may be acceptable for the upper triangle.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Introduction to the Finite Element Method

Abstract: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Convection Heat Transfer

Abstract: Geometry: shape, size, aspect ratio and orientation Flow Type: forced, natural, laminar, turbulent, internal, external Boundary: isothermal (Tw = constant) or isoflux (q̇w = constant) Fluid Type: viscous oil, water, gases or liquid metals Properties: all properties determined at film temperature Tf = (Tw + T∞)/2 Note: ρ and ν ∝ 1/Patm ⇒ see Q12-40 density: ρ ((kg/m) specific heat: Cp (J/kg ·K) dynamic viscosity: μ, (N · s/m) kinematic viscosity: ν ≡ μ/ρ (m/s) thermal conductivity: k, (W/m ·K) thermal diffusivity: α, ≡ k/(ρ · Cp) (m/s) Prandtl number: Pr, ≡ ν/α (−−) volumetric compressibility: β, (1/K)

Magnetohydrodynamic free convection of Al2O3–water nanofluid considering Thermophoresis and Brownian motion effects

Abstract: In this study MHD effect on natural convection heat transfer in an enclosure filled with nanofluid is investigated. The transport equations used in the analysis took into account the effect of Brownian motion and thermophoresis parameters. The Navier Stokes equations in their vorticity-stream function form are used to simulate the flow pattern, isotherms and concentration. The governing equations are solved via Control Volume based Finite Element Method. The inner and outer circular walls are maintained at constant temperatures while two other walls are thermally insulated. The heat transfer between cold and hot regions of the enclosure cannot be well understood by using isotherm patterns so heatline visualization technique is used to find the direction and intensity of heat transfer in a domain. Effect of Hartmann number (Ha = 0, 30, 60 and 100), buoyancy ratio number (Nr = 0.1–4) and Lewis number (Le = 2, 4, 6 and 8) on streamline, isotherm, isoconcentration and heatline are examined. Also a correlation for Nusselt number corresponding to active parameters is presented. The results indicate that Nusselt number is an increasing function of buoyancy ratio number but it is a decreasing function of Lewis number and Hartmann number. Also it can be concluded that as buoyancy ratio number increases the effects of other active parameters are more pronounced.

Entropy generation during Al2O3/water nanofluid flow in a solar collector: Effects of tube roughness, nanoparticle size, and different thermophysical models

Abstract: In this paper, an analytical study is performed on the entropy generation and heat transfer due to nanofluid flow in a flat plate solar collector. The working fluid considered in this work is Al 2 O 3 /water nanofluid with four different particle sizes, including 25, 50, 75, and 100 nm and volume concentrations up to 4%. Effects of tube roughness, nanoparticle size, and different thermophysical models are investigated on the Nusselt number, heat transfer coefficient, outlet temperature of the collector, entropy generation, and Bejan number. In addition, the effects of solar radiation and ambient temperature on entropy generation are examined. The results are presented for constant mass flow rates ranging from 0.1 to 0.8 kg/s. It is found that when the mass flow rate is considered to be constant for all working fluids, the Nusselt number and heat transfer coefficient have different trends. It is observed that uncertainties in thermophysical models and tube roughness have considerable effects on the values of heat transfer coefficient and Nusselt number. The findings show that with an increase in the volume fraction of nanofluid, the outlet temperature increases while with increasing the nanoparticle size a very insignificant decrease is observed in the outlet temperature. It is seen that the trend of changes in the outlet temperature is exactly in opposite to the Nusselt number trend. The analysis of entropy generation concludes that the entropy generation decreases with increasing the nanofluid concentration. It is found that the tube roughness increases the entropy generation and its effect is more visible at high mass flow rates while the effects of uncertainties in thermophysical models on entropy generation are not significant in any mass flow rate and volume fraction. Finally, a critical mass flow rate is determined under two different intensities of solar radiation and ambient temperature so that for the values higher than the critical mass flow rate the effects of roughness on entropy generation become important and should be considered.