^{2}K is found to be for inverted hybrid square-semicircular notched fin array. This value of heat transfer coefficient is about 8% higher than compared to that with plain rectangular plate fin array (7.25 W/m^{2}K) subjected to same operating conditions. Average heat transfer coefficients for inverted plate fin array combinations are high as compared with the non-inverted combinations. Also, hybrid plate fin arrays, whether inverted or not, show higher heat transfer coefficient values as compared with uniform notched fin arrays.

Cooling of electronic devices has drawn more attention of the researchers due to their increased use in almost all technological advancements. Researchers across the globe are working on different configurations of fins for heat transfer enhancement especially in natural convection process. It is observed that fins with notches give the best results reducing the cost of material and increasing the convective heat transfer rate. Heat transfer enhancement techniques have been extensively developed to improve the thermal performance of heat exchanger systems with a view to reducing the size and cost of the systems.

Basim Freegah et al.^{ }^{ }studied transfer enhancement in plate-fin heat sinks with fillet profile using six different geometries of heat sink. The authors found that plate-fin heat sinks with corrugated half-round pins in vertical arrangement subjected to parallel flow show better thermal performance over other configurations. Charles and Wang ^{2}reported experimental results for natural convection heat transfer in three fin patterns viz. rectangular, trapezoidal and inverted trapezoidal of aluminium alloy. Authors found that heat transfer coefficient for inverted trapezoidal fin array is 10% and 25% higher than that of rectangular and normal trapezoidal fin design respectively. Kharche and Farkade ^{3}^{ }reported experimental results for vertical rectangular finned array with and without notches. It was found that heat transfer rate of un-notched fins was less than the notched fins by about 18%. Mohan et al.^{ }^{ }experimentally investigated heat transfer on plate fin heat exchangers with wavy fins. The authors derived general correlations for friction factor and colburn factor using multiple linear regression analysis. Higher heat transfer coefficient with reasonable pressure drop was observed by using wavy fins. Dahiyaet al.^{ }^{ }conducted numerical studies on heat transfer through micro channel heat sinks with three different inlet and outlet flow arrangements. The authors found highest heat transfer coefficient in divergent-convergent combination of heat sinks as compared to other combinations. Aggarwal et al.^{ }^{ }^{ }^{ }^{10} reported CFD studies of natural convection heat transfer in three types of tapered fins with taper angles of 1°, 2° and 3°, subjected to different heating power. Fins with 2° taper angle showed better heat transfer performance at all heat inputs than the remaining two types of fins.

From the detailed review of literature, it is found that the rate of heat transfer can be increased passively by increasing the surface area, by using the inserts, extended surfaces or fins. In the present work, an attempt is made to study and analyze heat transfer enhancement passively using un-notched and notched plate fins.

In the present work, computational studies are conducted to analyze performance of seven types of vertical plate fin arrays subjected to natural convection heat transfer. Three types of vertical plate fins viz. plain rectangular fin, rectangular fin with square notch and rectangular fin with semi-circular notch are used with their seven combinations. These seven combinations of plate fin arrays under consideration are: plain rectangular, rectangular square notched, rectangular semi-circular notched, rectangular inverted square notched, rectangular inverted semi-circular notched, hybrid rectangular square-semi-circular notched and hybrid rectangular inverted square-semi-circular notched.

3D CFD simulations of all seven combinations of plate fin arrays are performed using SolidWorks Flow Simulation 2019^{ }

Size of each rectangular plain fin is: 80 mm X 43 mm X 2 mm. Lengths and thicknesses of all types of fins are 80 mm and 2 mm respectively. The height of each vertical fin other than plain rectangular one is taken in such a way that surface areas of all seven types of fin geometries are same. By keeping the surface area constant for all fin arrays, its effect on convective heat transfer enhancement is nullified. Dimensions of the three mainly used fin geometries are presented in

3D CAD models of all seven types of fin geometries are prepared using SolidWorks 2019 software.

Mesh generation in SolidWorks Flow Simulation 2019 is accomplished by using Cartesian-based meshes that are independent of the model geometry. This technique generates cubic cells that are adjacent to each other as well as the boundary of the computational domain. The cells are also oriented along the Cartesian coordinates of the CAD geometry. Since Cartesian-based meshes cannot generally form fit to the surfaces of the geometry, partial cells consisting of fluid and solid volume will be generated at the fluid/solid boundary. The mesh can then be refined by splitting the cubic cells into a maximum of 8 similarly shaped cuboids to properly represent surface boundaries of the geometry. Since the CAD model is native to the program, it can easily identify solid edges or boundaries of the solid model and can then refine the mesh more accurately ^{12}.

After creating suitable dense mesh each individual cell or a control volume is ready for calculation. First, the area and normal vector direction for each face of each cell are calculated using the CAD geometry. Then the appropriate equations can be applied to each cell depending on their type fluid flow equations for fluidic cells, and heat conduction and direct electrical current for solid cells. Navier-Stokes’s equations in the Cartesian coordinate system are used to formulate the fundamental conservation laws of mass, momentum, and energy^{ }

Conservation of mass:

At steady state, the first term in equation (1) will be zero hence, this equation reduces to

Conservation of momentum:

Where, u is the fluid velocity, ρ is the fluid density, and ρg_{i} is buoyancy force per unit mass.

At steady state, in the absence of turbulence, equation (3) reduces to

Conservation of energy:

Where, E is energy, h is the thermal enthalpy,

For the present case of steady state energy transfer without turbulence, equation (5) reduces to

For simulation of heat transfer through solid region, following energy conservation equation is used:

Where, Cp is specific heat, _{i}is the Eigen values of the thermal conductivity tensor.

At steady state, equation (8) change to,

While selecting the boundary conditions, heat source of 38.4 W is given at heater position in the base plate. The same wattage is provided while performing validation experiments with plain rectangular plate fin array. The other initial conditions as presented in^{3} for all fin arrays and the base plate, asbestos cement sheet (k = 0.58 W/mK, Cp = 840 J/kgK) as insulation at four side faces and one bottom face of the base plate, radiation heat transfer: not considered, time dependent nature: not considered, gravity: 0 m/s^{2} along X and Y direction & -9.81 m/s^{2} along Z direction. Default fluid: air, flow type: laminar.

SolidWorks Flow Simulation software provides multiple solver options such as Direct Sparse, Intel Direct Sparse, Large Problem Direct Sparse, and FFEPlus iterative solver. For the present simulation, FFEPlus iterative solver is selected which is a default solver option under automatic solver selection. This solver solves the set of algebraic equations using approximate technique viz. advanced matrix reordering. The solution is assumed and the associated errors are evaluated during each iteration^{ }^{-6} for accurate results.

During preprocessing, the 3D model of plain rectangular plate fin array is defined, meshing is performed and boundary conditions are set. During processing, solver is defined, a convergence criterion is set and the simulations are performed with the given initial and boundary conditions. The post processing includes deriving the results in the form of contours, vector plots and surface plots.

^{2}K is observed at the heater surface and also at the base plate. This is due to the negligible temperature difference between the hot surfaces and the local ambient temperature close these surfaces. The highest heat transfer coefficient of the order of 49 W/m^{2}K is observed on the side faces of the vertical plate fins due to sufficient temperature difference the surface and the ambient. Also, natural convection currents in the form of plumes are prominently taking the heat away for these surfaces. The average heat transfer coefficient for the plate fin array is found to be 7.251 W/m^{2}K.

The CFD results for the heat transfer coefficient at plain rectangular plate fin array are validated by comparing them with the available experimental results performed by the authors. Experiments are performed on the plain aluminium plate fin array subjected to constant heat flux condition by applying voltage of 60 V and corresponding current of 0.64 A. The resulting input power available at resistance heaters is 38.4 W. Temperatures are measured at the base of the fins, at the vertical surface as well as at the tip of the fins. Overall heat transfer coefficient is calculated using the concepts of fin efficiency and by using modified Rayleigh’s equation^{ }^{2}K. Difference between the experimental and CFD result for heat transfer coefficient is about 3% which is very small. Experimental result for heat transfer coefficient is smaller as compared to the corresponding computational result. The main reason behind this trend is neglect of radiative heat loss in computational studies, but in actual practice, there is always some radiative heat loss. Hence, the present CFD model used for simulation of natural convection heat transfer through plate fin array is found to be valid.

Using the same process that is used for simulation of plain rectangular fin array, 3D CFD simulations are performed on the remaining six combinations of plate fin arrays. These fin arrays include: rectangular square notched, rectangular semi-circular notched, rectangular inverted square notched, rectangular inverted semi-circular notched, hybrid rectangular square-semi-circular notched and hybrid rectangular inverted square-semi-circular notched. Once the simulations are complete, post processing is conducted to study variations in surface temperature, velocity and heat transfer coefficient for different plate fin arrays.

^{2}K is found to be for inverted hybrid square-semicircular notched fin array (^{2}K for inverted square notched fin array (^{2}K for hybrid square-semicircular notched fin array (^{2}K for inverted semi-circular notched fin array (^{2}K for semicircular notched fin array (^{2}K for square notched fin array (

Average heat transfer coefficients for inverted plate fin array combinations are high as compared with the non-inverted combinations. Also, hybrid plate fin arrays, whether inverted or not, show higher heat transfer coefficient values as compared with uniform notched fin arrays. These results highlight importance of inverted hybrid fin arrays for effectively transferring the heat from the heat sinks by natural convection (

In the present work, computational studies are conducted on seven types of vertical plate fin array combinations. SolidWorks Flow Simulation software is used for conducting this research. Computational studies can be effectively used to reduce expenses on additional experimentation.

3D CFD simulation is first conducted on plain vertical plate fin array. Computational value of average heat transfer coefficient is found to be 7.251 W/m^{2}K, which is about 3% higher than the corresponding experimental value i.e. 7.03 W/m^{2}K.The main reason behind lower experimental value is neglect of radiative heat loss in computational studies, but in actual practice, these is always some radiative heat loss.

Among all fin combinations, the highest average heat transfer coefficient of 7.82 W/m2K is found to be for inverted hybrid square-semicircular notched fin array. Average heat transfer coefficients for inverted plate fin array combinations are high as compared with the non-inverted combinations.

Hybrid plate fin arrays, whether inverted or not, show higher heat transfer coefficient values in most of the cases as compared with uniform notched fin arrays.

Rectangular plain plate fin array shows higher heat transfer coefficient as compared to the rectangular plate fin array with square notches for the same operating conditions. That is, providing notches on plain geometry might not be sometimes feasible for heat transfer enhancement.

Limitations and future scope: The CFD results obtained in the present work are validated only for plain plate fin array. Detailed experiments can be performed on remaining six combinations of the plate fin arrays and the experimental results can be compared with the CFD results obtained in the present work. Also, all experiments can be performed at different input power values so as to investigate their effect on natural convection heat transfer in all combinations of plate fin arrays.