I am also thankful to all the staff of Department of Mathematics, BUET for such help. In the thesis, we numerically investigated and described the results of the influence of different values of the buoyancy parameter, the Rayleigh number ( ), the Prandtl number ( ) and the volumetric ratio of nanoparticles ( ). The results were discussed and visually presented using streamlines and isothermal lines for the velocity and temperature contours and the average Nusselt number for the heat transfer rate.
The Rayleigh number, Prandtl number, and nanosolid volume ratio all have a significant influence on the convective flow regime, and the value of the average Nusselt number changes as these parameters change. Ra106, the average Nusselt number increases to 18.6% for sinusoidal temperature and 21.6% for constant temperature distribution at the hot bottom wall of the trapezoidal enclosure using water-Fe3O4 nanofluid. The water-Fe3O4 nanofluid achieves a greater heat transfer rate (5.83% for sinusoidal temperature and 7.38% for constant temperature distribution) than the base fluid (water).
The solid volume fraction increases by 0% to 3%, the average Nusselt number increases by 12.67% for sinusoidal temperature and 15.57% for constant temperature distribution using water-Cu nanofluid at the hot bottom wall of the trapezoidal enclosure. Compared to water-Fe3O4 nanofluid, the Cu-water nanofluid achieves a higher rate of heat transfer 6.77% for sinusoidal temperature and 7.63% for constant temperature distribution at bottom wall using solid volume fraction.
Introduction
- Introduction
- Keywords definition and description
- Heat transfer
- Nanoparticles
- Nanofluids
- Convection heat transfer
- Natural convection heat transfer
- Viscosity
- Thermal conductivity
- Thermal diffusivity
- Nanoparticles solid volume fraction
- Dimensionless Parameter
- Rayleigh number
- Prandtl number
- Nusselt number
- Literature Review
- Objectives
- Possible Outcomes
- Scope of the Thesis
The fundamental modes of heat transfer are conduction or diffusion, convection (free and forced) and radiation. Nanofluid thermophysical properties are required for the calculation of the heat transfer coefficient and the Nusselt number. Transfer of heat from one place to another due to the molecular movement of fluids (Air or liquids) is known as convection heat transfer.
Heat transfer occurs faster in materials with high thermal conductivity than in materials with low thermal conductivity. The obtained results showed a high dependence of the heat transfer on the slope angle, flow fields, Rayleigh number and aspect ratio. 22] investigated the buoyancy-induced heat transfer within a nanofluid-filled trapezoidal cavity with variable thermal conductivity and viscosity.
They reported that the effect of the viscosity was more dominant than the thermal conductivity in heat transfer enhancement and heat transfer enhancement is not so noticeable for the use of nanofluid. They found that the magnetic field shrinks the convective heat transfer and flow strength of nanofluid. They showed that the heat transfer coefficient is increased as the Reynolds number and volume fraction of solid nanoparticles are increased.
The heat transfer performance of nanofluids depends on several essential parameters such as nanofluids size, shape, constructive materials of base fluids, concentration, etc. Roy [42] studied the heat transfer characteristic between a square enclosure and a circular, elliptical, or rectangular cylinder. To find the heat transfer in a trapezoidal cavity for ferrosoferric oxide (Fe3O4)-water and Cu-water nanofluids.
To compare the heat transfer performance of the mentioned individual nanofluids using different volume fractions of solid matter. Changes in various flow parameters and heat regulation are expected to have a noticeable effect on the heat transfer and fluid flow structure in the confined space. The model can be used to discuss the effect of nanofluid effective viscosity on natural convection heat transfer in a trapezoidal enclosure.
This chapter also contains a literature review of previous studies of heat transfer using different types of fluids and nanofluids that are relevant to the present work. The results have been shown in isothermal lines, streamlined to better understand the heat transfer mechanism through trapezoidal enclosures.
Numerical Analysis
- Introduction
- Finite Element Method
- Problem Formulation
- Physical model
- Mathematical modeling
- Thermo-physical properties
- Numerical Analysis
- Grid sensitivity test
- Meshing
- Validation of the numerical scheme
The degree of the polynomial depends on the number of nodes assigned to the element. A finite element matrix equation must be constructed, relating the node values of the unknown function to other parameters. Various approaches can be used to transform the physical formulation of the problem into its finite element discrete analogue.
If the physical formulation of the problem is known as a differential equation, then the most popular finite element formulation method is the Galerkin method. If the physical problem can be formulated as the minimization of a function, then the variational formulation of the finite element equations is usually used. The lower surface is set to a temperature Th while the upper surface of the enclosure is cooled to a constant temperature Tc.
Therefore, thermophysical properties of base fluids and nanoparticles can be simply defined as the properties of the fluid system as well as the material properties. To solve the governing dimensionless equation and the boundary conditions, the Galerkin weighted residual system of the finite element method was used. Six nodes triangular elements are used in this work for the development of the finite element equations.
The Newton-Raphson iteration technique was adapted to solve a set of global nonlinear algebraic equations in matrix form. One of the most important aspects of the simulation is its independence from the number of meshes used in the geometry. This means that the findings are independent of the number of elements in the domain.
Comparisons with previously published results are necessary to validate the correctness of the numerical results and the validity of the mathematical model developed during the current investigation. The results of the present numerical code are compared with those published by Basak et al. As a result, validation increases confidence in the numerical code to continue with the aforementioned objective of the current investigation.
Results and Discussions
- Introduction
- Effect of Rayleigh Number
- Effect of Prandtl Number
- Effect of Solid Volume Fraction
- Nusselt Number
- Comparison
- Comparison with Basak et al. [51]
- Comparison with Weheibi et al. [51]
The flow inside the cavity is strongly generated by nanoparticles along with the increasing values of the buoyancy-driven parameter, as seen in the picture. The effects of the Prandtl numbers ( , , , ) are shown in figure 3.3 with respect to streamlines with constant temperature and sinusoidal temperature in terms of the constant parameters ϕ = 1% of water-Fe3O4 nanofluid and Ra = 104. The effect of streamline for constant temperature lower wall (column-a) and sinusoidal temperature lower wall (column-b) is shown in figure 3.5 and isothermal contour for constant temperature lower wall (column-a) and sinusoidal temperature lower wall (column-b) ) is shown in the figure 3.6 shown for the different volumetric ratios ( , , , ) of ferrospheric oxide nanoparticle with the fixed value of the parameters.
For constant temperature, the direction of the streamlines of the right large vortex is counterclockwise, while the other vortex is clockwise. In the case of nanofluid, the velocity layer is more active in the vicinity of the wall. Nanoparticles mixing with the base fluid play an important role in heat transfer, this phenomenon is shown in the last column of figure 3.6.
Without nanoparticles in solution, the isotherms are loosely connected with dashed lines indicating the convective nature of the heat transfer. For increasing the nanoparticle volume ratio from 0% to 3%, the average heat transfer rate increased by 5.83% for sinusoidal temperature and 7.38% for constant temperature distribution on the bottom surface of the trapezoidal cavity. In conclusion, although the fluid flow rate decreases by increasing the volume ratio of nanoparticles, but the average heat transfer rate increased.
At the beginning, the value of the average Nusselt number is 5.9 for sinusoidal temperature and 6 for constant temperature. For the Rayleigh number to , the value of the average Nusselt number increases by 18.6% for sinusoidal temperature distribution of the heated bottom wall and 21.6% for constant temperature. In the initial stage, the value of the average Nusselt number is 5.5 for sinusoidal temperature and 5.6 for constant temperature.
For the Prandtl number 1 to, the value of the average Nusselt number increases about 12.73% for sinusoidal temperature distribution and 14.28% for constant temperature of the heated bottom surface. The value of the average Nusselt number is increased to increase the volume fractions of nanoparticles. For the nanoparticle volume fractions 0 to 3%, the value of the average Nusselt number increases by approx. 5.83% for sinusoidal temperature and 7.38% for constant temperature using water-Fe3O4 nanofluid, and the value of the average Nusselt number increases by approx. 12.63% for sinusoidal temperature. and 15.57% for constant temperature using water-Cu nanofluid.
Also included in this table is the percent inaccuracy of the values between these two studies. This table also shows the percentage of inaccuracy of the values between these two studies.
Conclusions and Future Research
Conclusions
Future Research
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