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SOLUTION - Fundamentals Of Heat And Mass Transf...


In physics and engineering contexts, especially in the context of diffusion through a medium, it is more common to fix a Cartesian coordinate system and then to consider the specific case of a function u(x, y, z, t) of three spatial variables (x, y, z) and time variable t. One then says that u is a solution of the heat equation if




SOLUTION - Fundamentals of Heat and Mass Transf...



Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of α and solutions of the heat equation with α = 1. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case α = 1.


By the second law of thermodynamics, heat will flow from hotter bodies to adjacent colder bodies, in proportion to the difference of temperature and of the thermal conductivity of the material between them. When heat flows into (respectively, out of) a material, its temperature increases (respectively, decreases), in proportion to the amount of heat divided by the amount (mass) of material, with a proportionality factor called the specific heat capacity of the material.


where c \displaystyle c is the specific heat capacity (at constant pressure, in case of a gas) and ρ \displaystyle \rho is the density (mass per unit volume) of the material. This derivation assumes that the material has constant mass density and heat capacity through space as well as time.


If the medium is not the whole space, in order to solve the heat equation uniquely we also need to specify boundary conditions for u. To determine uniqueness of solutions in the whole space it is necessary to assume additional conditions, for example an exponential bound on the growth of solutions[2] or a sign condition (nonnegative solutions are unique by a result of David Widder).[3]


Solutions of the heat equation are characterized by a gradual smoothing of the initial temperature distribution by the flow of heat from warmer to colder areas of an object. Generally, many different states and starting conditions will tend toward the same stable equilibrium. As a consequence, to reverse the solution and conclude something about earlier times or initial conditions from the present heat distribution is very inaccurate except over the shortest of time periods.


The following solution technique for the heat equation was proposed by Joseph Fourier in his treatise Théorie analytique de la chaleur, published in 1822. Consider the heat equation for one space variable. This could be used to model heat conduction in a rod. The equation is


A fundamental solution, also called a heat kernel, is a solution of the heat equation corresponding to the initial condition of an initial point source of heat at a known position. These can be used to find a general solution of the heat equation over certain domains; see, for instance, (Evans 2010) for an introductory treatment.


Since the heat equation is linear, solutions of other combinations of boundary conditions, inhomogeneous term, and initial conditions can be found by taking an appropriate linear combination of the above Green's function solutions.


The mathematical model for multi-dimensional, steady-state heat-conduction is a second-order, elliptic partial-differential equation (a Laplace or Poisson Equation). Typical heat transfer textbooks describe several methods for solving this equation for two-dimensional regions with various boundary conditions. Analytical solutions usually involve an infinite series of transcendental functions. This series must be truncated and evaluated at an array of locations to give an approximate estimate of the temperatures found over the 2-D region. Some texts also include detailed graphical methods using various paper and pen tools for estimating temperature and heat-flow lines for 2-D problems, but these latter methods have become largely obsolete due to the widespread use of computers and associated numerical algorithms (although the principles on which graphical methods are based are often useful in checking the validity of numerical solutions).


Analysis We take the air-conditioning duct as the system. This is a control volume since mass crosses the system boundaryduring the process. We observe that this is a steady-flow process since there is no change with time at any point and thus ,there is only one inlet and one exit and thus , and heat is lost from the system. The energy balance for this steady-flow systemcan be expressed in the rate form as


1-52 Prob. 1-51 is reconsidered. The amount of heat loss through the glass as a function of the window glass thicknessis to be plotted.Analysis The problem is solved using EES, and the solution is given below.


The present chapter discusses a boundary element formulation for transient heat conduction using time-dependent fundamental solutions. This formulation can be viewed as a direct extension of potential theory since the proper fundamental solution of the diffusion equation is used to obtain an equivalent boundary integral equation. Numerical techniques are then employed to solve the integral equation in discrete form through a time-marching procedure. 041b061a72


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