Chapter 1 Introduction Nodal methods were first introduced and developed to solve neutron diffusion equations in 1970’s [2]. The success of Nodal methods in the field of neutronics was stimulated into the heat flow and fluid flow problems in 1980 [3]. The Nodal Integral Method scheme is developed by approximately satisfying the governing differential equations on finite size brick-like elements. These differential equations are obtained by discretizing the space of independent variables [4]. In the early development of nodal schemes these brick-like elements were referred to as nodes and hence the scheme was called nodal. Nodes in Nodal Integral Method is however similar to the elements of the finite element approach, but the nodes have finite volumes and not only the points in the space of independent variables [5]. An error analysis on NIM methodology is performed to establish the convergence of the solution of two-dimensional heat diffusion equation to the exact solution [1]. Motivation Some of the fluid flow, heat flow and particle flow problems are the most complex problems. It is difficult to find the analytic solution of these problems due to their extra-ordinary complexity. Scientists and engineers are facing this problem from decades and then they turned to the numerical methods. By developing increasingly better numerical schemes, researchers have been able to solve increasingly more complex flow problems [11]. The heat diffusion equations are used to describe the flow of heat. In chapter 2 we use Nodal Integral Method to solve the heat diffusi... ... middle of paper ... ... (2.11) T ̿_(i,j)=(〖T ̅^y〗_(i,j)+〖T ̅^y〗_(i-1,j))/2-a^2/3 〖S ̅^y〗_(0 i,j) (2.12) Substituting the value of 〖S ̅^x〗_(0 i,j) and 〖S ̅^y〗_(0 i,j) from the equations (2.11) and (2.12) in the equations (2.6), (2.7) and (2.8) , finally we will get the following three equations. 〖T ̅^x〗_(i,j-1)+4 〖T ̅^x〗_(i,j)+〖T ̅^x〗_(i,j+1)=3 (T ̿_(i,j)+T ̿_(i,j+1) ) (2.13) 〖T ̅^y〗_(i-1,j)+4 〖T ̅^y〗_(i,j)+〖T ̅^y〗_(i+1,j)=3 (T ̿_(i,j)+T ̿_(i,j)) (2.14) 3((〖T ̅^y〗_(i-1,j)+ 〖T ̅^y〗_(i,j))/(2 a^2 ))+3 ((〖T ̅^x〗_(i,j-1)+ 〖T ̅^x〗_(i,j))/(2 b^2 )) - (3/(a^2+b^2 )) T ̿_(i,j)=-S ̿_(i,j)/k (2.15) Equations (2.13), (2.14) and (2.15) represent the averaged temperature variable, averaged in x-direction, y-direction and cell averaged respectively.
(Eq. 7) (Eq. 8) are both used to calculate the heat of the solution and the heat of the calorimeter.
The thermometer’s original temperature before coming in contact with an outside object is represented by T. ∆T/∆t is the average temperature of the digital thermometer. represents the temperature of the heat flowing object. In this lab, the temperature of the air is represented by Tair=T. To= Thand is the temperature of the hand.
i.e. K ̇(t)=sY(t)-δK(t), L ̇(t)=nL(t) and A ̇(t)=gA(t) it is important to consider the new assumptions that concern the newly added inputs.
...ion, Scriver CR, Beaudet AL, Sly WS, Valle D (eds), McGraw-Hill, New York, pp. 4353-4392
The weakest feature of the paper is that although the formulas, presented by authors, are in general correct, but they do not support the conclusions the author extract from them, and mistake is hidden in the interpretation.
leads to an increase in unit cell volume. This can be seen from the data in the article. When temperature rises
In this section, the steady state isothermal flow model developed by Atti (2006) and Garfield (2009) is presented. The model is for one-dimensional flow based on the continuity and momentum equations presented in section 3.2.1 and 3.2.2 respectively.
Heat transfer from high temperature heated surfaces finds considerable application in engineering. Because of its large number of applications in industries, considerable efforts have been made by researchers to investigate various aspects of the heat transfer and its fundamental principles involved. Fluid flow problems involving heat transfer viz. in presence of convention and radiation represents an idealization of many meaningful problems in engineering practice. Due to the presence of higher level of temperature required in many system like boiler, nuclear reactor; the effect of radiation heat transfer increases. So, there becomes a need of including radiative effect of the participating medium and also their boundary conditions. Keeping this in mind, an attempt was made to investigate the heat transfer in the Indian Pressurized Heavy Water Reactor (IPHWR) during Loss of Coolant Accident (LOCA) with low steam flow. This study will help in estimating the safe working limits for the heat dissipation in the reactor.
In conclusion, it is the engineers’ preference to choose one technique for calculation over the other, although there are some minor differences in the final output of the functions. The examples throughout this document highlight the benefits of Mesh Analysis against Thevenin Analysis.
...he principle numbers of Froude, Reynolds and Weber. Mathematical model predicts the heat and mass transfer in numerical framework for both transports phenomena of relevance to the industry continuous casting tundish system. Additionally, it has an excellent agreement outlet temperature respond the step input temperatures in the inlet stream of water in the tundish model. The simulations of 8x8 grid and 16x16 grid are applied to obtain significant difference between the TAV maps in which both grids are computed by software represent the specific flow of the fluid in the model and the steel caster as the actual size system. Therefore, the physical and mathematical modeling is used as a guidance to build a model before the prototype is constructed in terms of calculation, measurement and determination of specific fluid flow, heat and mass transfer in the water model.
Let us consider the heat flow in a bar or rod along the x-axis. Consider a rod of homogeneous material of density is ρ (gm⁄〖cm〗^3 ) and having a constant cross-sectional area A (〖cm〗^2 ). Let c be the specific heat and k be thermal conductivity of material. We suppose that the sides of the bar are insulated and the loss of heat from the sides by conduction or radiations is negligible. Take an end of the bar as the origin and the direction of heat flow as the positive x-axis.
Because to solve a problem analytically can be very hard and spend a lot of time, global, polynomial and numerical methods can be very useful. However, in last decades, numerical methods have been used by many scientists. These numerical methods can be listed like The Taylor-series expansion method, the hybrid function method, Adomian decomposition method, The Legendre wavelets method, The Tau method, The finite difference method, The Haar function method, The...
My aim in this piece of work is to see the effect of temperature on the rate of a reaction in a solution of hydrochloric acid containing sodium thiosulphate.
...ke the vanishing determinant, a fixed value of λ = λk, is chosen accordingly. Therefore, at λ = λk, the coefficients of the unknown amplitude Aj in equation (2.12) will become fixed and then it will be possible in obtaining the solution Ajk (the additional subscript k will be used to indicate the correspondence with the particular values of λk). Such a system of equations does not determine the Ajk uniquely but gives their ratios. A convenient mathematical solution designated by the quantities mjk are defined in terms of an arbitrary solution A_jk^' by the formula
Conduction is a mode of heat transfer where heat energy is transported from more energetic particles to less energetic particles. The basic equation that describes heat transfer through conduction is Fourier’s law, as shown below.