Conduction heat transfer Laplace equation

This is the simplest model of an electrocatalyst system where the single energy dissipation is caused by the ohmic drop of the electrolyte, with no influence of the charge transfer in the electrochemical reaction. Thus, fast electrochemical reactions occur at current densities that are far from the limiting current density. The partial differential equation governing the potential distribution in the solution can be derived from the Laplace Equation 13.5. This equation also governs the conduction of heat in solids, steady-state diffusion, and electrostatic fields. The electric potential immediately adjacent to the electrocatalyst is modeled as a constant potential surface, and the current density is proportional to its gradient ... 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

At all channel wall surfaces, the no-slip boundary condition is applied to the velocity field (the Navier-Stokes equation), the fixed zeta-potential boundary condition is imposed on the EDL potential field (the Poisson-Boltzmann equation), and the insulation boundary condition is assigned to the applied electric field (the Laplace equation), and the no-mass penetration condition is specified for the solute mass concentration field (the mass transport equation). In addition, the third-kind boundary condition (i. e., the natural convection heat transfer with the surrounding air) is applied to the temperature field at all the outside surfaces of the fabricated channels to simultaneously solve the energy equation for the buffer solution together with the conjugated heat conduction equation for the channel wall. 

The biharmonic and Laplace s equations are the governing differential equations for many two dimensional problems in linear elasticity and heat transfer. Among these problems are those which arise in the theory of thin plates, two dimensional thermal stresses, torsion and bending of prismatic bars, and two dimensional heat conduction. 

Models for phenomena such as heat conduction, fluid flow, and diffusional mass transfer are also based on Laplace s equation. Consequently, many solutions to the potential distribution problems or the analogous problems in other fields are available. The current distribution can be obtained from the potential distribution through Ohm s law [Eq. (22)]. 

In the second chapter we consider steady-state and transient heat conduction and mass diffusion in quiescent media. The fundamental differential equations for the calculation of temperature fields are derived here. We show how analytical and numerical methods are used in the solution of practical cases. Alongside the Laplace transformation and the classical method of separating the variables, we have also presented an extensive discussion of finite difference methods which are very important in practice. Many of the results found for heat conduction can be transferred to the analogous process of mass diffusion. The mathematical solution formulations are the same for both fields. 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

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