Heat conduction unsteady-state

The heat transfer problem which must be solved in order to calculate the temperature profiles has been posed by Lee and Macosko(lO) as a coupled unsteady state heat conduction problem in the adjoining domains of the reaction mixture and of the nonadiabatic, nonisothermal mold wall. Figure 5 shows the geometry of interest. The following assumptions were made 1) no flow in the reaction mixture (typical molds fill in <2 sec.) ... 

The simulation example DRY is based directly on the above treatment, whereas ENZDYN models the case of unsteady-state diffusion, when combined with chemical reaction. Unsteady-state heat conduction can be treated in an exactly analogous manner, though for cases of complex geometry, with multiple heat sources and sinks, the reader is referred to specialist texts, such as Carslaw and Jaeger (1959). 

Unsteady-State Heat Conduction and Diffusion in Spherical and Cylindrical Coordinates... 

We first recall the physical situation to facilitate this, we draw a sketch (see Fig. 1). At high oven temperatures, the heat is transferred from the heating elements to the meat surface by both radiation and heat convection. From there, it is transferred solely by the unsteady-state heat conduction that surely represents the rate-limiting step of the whole heating process (Fig. 1). 

The higher the thermal conductivity 2 of the body, the faster the heat spreads out. The higher its volume-related heat capacity pCp, the slower the heat transfer. Therefore, unsteady-state heat conduction is characterized by only one material property, the thermal diffusivity, a = A/pCp of the body. 

The elegant solution of this first example should not tempt the reader to believe that dimensional analysis can be used to solve every problem. To treat this example by dimensional analysis, the physics of unsteady-state heat conduction had to be understood. Bridgman s (2) comment on this situation is particularly appropriate ... 

Sect. 5.4), the heat transfer process can be modeled using classical unsteady state heat conduction theory [142-144]. From the mathematical solutions to heat conduction problems, a thermal diffusivity can be extracted from measurements of temperatures vs. time at a position inside a gel sample of well-defined geometry. 

Fig. 5.5 Temperature profiles for unsteady-state heat conduction in finite flat plates T(x, 0) = To, T( b, t) = 7j. [Reprinted by permission from H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Oxford University Press, New York, 1973.]...

We will discuss the solution of steady-state and unsteady-state heat conduction problems in this chapter, using the finite-difference method.. The finite-difference method comprises the replacement of the governing equations and corresponding boundary conditions by a set of algebraic equations. The discussion here is not meant to be exhaustive in its mathematical rigor. The basics are presented, and the solution of the finite-difference equations by numerical methods are discussed. The solution of convection problems using the finite-difference method is discussed in a later chapter. 

A full treatment of unsteady-state heat conduction is not in the field of this text. A derivation of the partial differential equation for one-dimensional heat flow and the results of the integration of the equations for some simple shapes are the only subjects covered in this section. It is assumed throughout that k is independent of temperature. 

This equation is often known as Pick s second law. As expected, a very similar equation can be derived for unsteady-state heat conduction in an infinite slab flncropera et a1.. 2011T... 

In many cases, unsteady-state heat conduction is occurring but the rate of heat generation is zero. Then Eqs. (5.1-8) and (5.1-9) become... 

UNSTEADY-STATE HEAT CONDUCTION IN VARIOUS GEOMETRIES... 

Sec. 5.3 Unsteady-State Heat Conduction in Various Geometries... 

Figure 5.3-3, calculated using Eq. (5.3-7), is a convenient plot used for unsteady-state heat conduction into a semiinfinite solid with surface convection. If conduction into the solid is slow enough or h is very large, the top line with h /ailk = oo is used. 

Figure 5.3-5. Unsteady-state heat conduction in a large flat plate. [From H. P. Gurney and J. Lurie, Ind. Eng. Chem., 15, 1170 (/92i).]... 

Figure 5.3-6. Chan for delermin mg temperature at the center of a large flat plate for unsteady-state heat conduction. [From H. P. Heisler, Trans. A.S.M.E., 69, 227 ( 947). With permission.]...

Figure 5.3-9. Unsteady-state heat conduction in a sphere. [From H. P. Gurney and J. 

Numerical calculation methods for unsteady-state heat conduction are similar to numerical methods for steady state discussed in Section 4.15. The solid is subdivided into sections or slabs of equal length and a fictitious node is placed at the center of each section. Then a heat balance is made for each node. This method differs from the steady-state method in that we have heat accumulation in a node for unsteady-state conduction. 

These rates of chilling or cooling are governed by the laws of unsteady-state heat conduction discussed in Sections 5.1 to 5.4. The heat is removed by convection at the... 

Other coordinate systems. Fourier s second law of unsteady-state heat conduction can be written as follows. 

Relation between mass- and heat-transfer parameters. In order to use the unsteady-state heat-conduction charts in Chapter 5 for solving unsteady-state diffusion problems, the dimensionless variables or parameters for heat transfer must be related to those for mass transfer. In Table 7.1-1 the relations between these variables are tabulated. For K 1.0, whenever appears, it is given as Kk, and whenever c, appears, it is given as cJK. 

Unsteady-state diffusion in more than one direction. In Section 5.3F a method was given for unsteady-state heat conduction to combine the one-dimensional solutions to yield solutions for several-dimensional systems. The same method can be used for unsteady-state diffusion in more than one direction. Rewriting Eq. (5.3-11) for diffusion in a rectangular block in the x, y, and z directions. 

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