Solids transient heat conduction

Example 5.2 Semi-infinite Solid with Constant Thermophysical Properties and a Step Change in Surface Temperature Exact Solution The semi-infinite solid in Fig. E5.2 is initially at constant temperature Tq. At time t — 0 the surface temperature is raised to T. This is a one-dimensional transient heat-conduction problem. The governing parabolic differential equation... 

Transient Heat Conduction in Semi-Infinite Solids 240 Contact of Two Semi-Infinite Solids 245... 

TRANSIENT HEAT CONDUCTION IN SEMI-INFINITE SOLIDS... 

Transient Heat Conduction in Semi-Infinite Solids... 

Transient mass difliision in a stationary medium is analogous to transient heat transfer provided that the solution is dilute and thus the density of the medium p is constant. In Chapter 4 we presented analytical and graphical solutions for one-dimensional transient heat conduction problems in solids with constant properties, no heat generation, and uniform initial temperature. The analogous one-dimensional transient mass diffusion problems satisfy these requirements ... 

Solidifying processes are important in cryogenics, food and process industries and also in metallurgy. A main point of interest is the speed at which the boundary between the solid and the liquid phase moves. Prom this the time required for solidifying layers of a given thickness can be calculated. The modelling of these processes belongs to the held of transient heat conduction, as the enthalpy of... 

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. 

The particular solution of Eq. (21.41) is the same as that for transient heat conduction to a semi-infinite solid, Eq. (10.26). 

The temperature distribution in the solid polymer sample can well be apvproximated by the Fourier equation for transient heat conduction within a medium of constant thermal diffusivity, i.e. 

The melt compounding process comprises an energy balance on the particle surface. In the first phase, particles are in powder form, and the polymer melt mixed with the additives is seen as a continuum. The heat flow distribution by the melt, through transient heat conduction, leads to an increase in the temperatnre of the solid particles for a specific period of time. The heat flow can therefore occur on the melt side if the radial temperatnre around a polymer is known. The polymer to be melted is embedded in the melt in the completely filled melting zone. The additives and fillers are directly wetted by the melt and are incorporated. 

We begin our treatment of transient heat conduction by analyzing a simplified case. In this situation we consider a solid which has a very high thermal conductivity or very low internal conductive resistance compared to the external surface resistance, where convection occurs from the external fluid to the surface of the solid. Since the internal resistance is very small, the temperature within the solid is essentially uniform at any given time. 

It is assumed that the wall temperature varies with time only along its axial direction and is uniform throughout its thickness at any axial location. This condition is very closely obtained in any solid undergoing transient heat conduction when the Biot number, h (volume of solid/area wetted) /Jb, is less than 0.100. For the cases examined in this paper the maximum Biot number of the wall is 0.012. 

Assuming this interaction to be similar to that of transient heat conduction into a semiinfinite solid, estimates have been made of the temperature distributions in the liquid and the amount of pressurizing gas condensed. 

The complete governing equations in such cases must allow for transient heat conduction inside the solid particle, and hence will be a rather involved... 

Considep two-dimensional transient heat transfer in an L-shaped solid body that is initially at a uniform temperalure of 90°C and whose cross section is given in Fig. 5-51. The thermal conductivity and diffusivity of the body are k = 15 W/m C and a - 3.2 x 10 rriVs, respectively, and heat is generated in Ihe body at a rate of e = 2 x 10 W/m. The left sutface of the body is insulated, and the bottom surface is maintained at a uniform temperalure of 90°C at all times. A1 time f = 0, the entire top surface is subjected to convection to ambient air at = 25°C with a convection coefficient of h = 80 W/m C, and the right surface is subjected to heat flux at a uniform rate of r/p -5000 W/m. The nodal network of the problem consists of 15 equally spaced nodes vrith Ax = Ay = 1.2 cm, as shown in the figure, Five of the nodes are at the bottom surface, and thus their temperatures are known. Using the explicit method, determine the temperature at the top corner (node 3) of the body after 1,3, 5, 10, and 60 min. 

Consider two-diuieiisional transient heat transfer in an L-shaped solid bar that is initially at a uniform temperature of IdO C and whose cross section is given in the figure. The thermal conductivity and diftiisivity of the body are 1 = 15 V/m - C and a - 3.2 X 10 ni%, respectively, and heat is... 

The diffusion coefficients in solids are typically very low (on the order of 10 to 10" mVs), and thus the diffusion process usually affects a thin layer at the surface. A solid can conveniently be treated as a semi-infinite medium during transient mass diffusion regardless of its size and shape when the penetration depth is small relative to the thickness of the solid. When this is not the case, solutions for one dimensional transient mass diffusion through a plane wall, cylinder, and sphere can be obtained from the solution.s of analogous heat conduction problems using the Heisler charts or one term solutions pieseiited in Chapter 4. 

In this chapter we will deal with steady-state and transient (or non steady-state) heat conduction in quiescent media, which occurs mostly in solid bodies. In the first section the basic differential equations for the temperature field will be derived, by combining the law of energy conservation with Fourier s law. The subsequent sections deal with steady-state and transient temperature fields with many practical applications as well as the numerical methods for solving heat conduction problems, which through the use of computers have been made easier to apply and more widespread. 

In section 2.5.3 it was shown that the differential equation for transient mass diffusion is of the same type as the heat conduction equation, a result of which is that many mass diffusion problems can be traced back to the corresponding heat conduction problem. We wish to discuss this in detail for transient diffusion in a semi-infinite solid and in the simple bodies like plates, cylinders and spheres. 

Vertical Surfaces. Analyses and measurements related to transient heat transfer on vertical surfaces have been reviewed by Ede [79], For negligible solid heat capacitance, results of analyses are discussed for step changes in wall temperature and in surface heat flux, and for periodic specified temperature or flux, as well as for other prescribed variations. Most recent analyses have confirmed the estimate of Siegel [252] that the local time for departure from the conduction regime at distance position x from the leading of a vertical flat plate after a step change in wall temperature is... 

We introduced the idea of change of variables in Section 10.1. The coupling of variables transformation and suitable initial conditions often lead to useful particular solutions. Consider the case of an unbounded solid material with initially constant temperature Tg in the whole domain 0 < x < oo. At the face of the solid, the temperature is suddenly raised to 7 (a constant). This so-called step change at the position jc = 0 causes heat to diffuse into the solid in a wavelike fashion. For an element of this solid of cross-sectional area A, density p, heat capacity C, and conductivity k, the transient heat balance for an element Ax thick is... 

The quasisteady-state approach can be applied in transient simulations, in a similar fashion to the ID models discussed in Section 3.1. The only retained transient term is that of solid heat conduction in Eq. (3.29). Quasisteady-state and fully transient 2D channel simulations have been recendy compared for fuel-lean hydrogen hetero-Zhomogeneous combustion over Pt in Brambiha et al. (2014). Requirements for the appUcability of quasisteady-state approach have been elaborated in Schneider et al. (2008) and Karagiannidis and Mantzaras (2010). 

The ID transient energy balance of the solid phase is described by the heat conduction equation ... 

If the nonuniformities in the temperature and species concentrations among the catalyst channels need to be considered, a 2D or 3D modeling approach is required. In this case, the transient energy balance equation of the monolith should be extended to two or three dimensions. The heat conduction equation of the solid phase is formulated in polar coordinates for 2D simulations (13.18) and in Cartesian coordinates for 3D simulations (13.19). 

Heat conduction, convection, boiling heat transfer, radiation, transient heat transfer, forced flow in pipes and packed beds, mass transfer by diffusion, and diffusion in porous solids. 

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