Unsteady-state diffusion conduction

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... 

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. 

Unsteady state diffusion processes are of considerable importance in chemical engineering problems such as the rate of drying of a solid (H14), the rate of absorption or desorption from a liquid, and the rate of diffusion into or out of a catalyst pellet. Most of these problems are attacked by means of Fick s second law [Eq. (52)] even though the latter may not be strictly applicable as mentioned previously, these problems may generally be solved simply by looking up the solution to the analogous heat-conduction problem in Carslaw and Jaeger (C2). Hence not much space is devoted to these problems here. 

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. 

Danckwerts PV (1950) Unsteady-state diffusion or heat conduction with moving boundary. Trans Faraday Soc 46(9) 701-712... 

Except for this section and Section 18.7. the solutions of the unsteady diffusion equation in one to three dimensions are beyond the scope of this book. Solutions to Eqs. (15-12c. d, e), the corresponding two-and three-dimension equations, and the equivalent heat conduction equations have been extensively studied for a variety of boundary conditions (e.g., Crank. 197S Cussler. 2009 Incropera et al 2011). Readers interested in unsteady-state diffusion problems should refer to these or other sources on diffusion. 

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. 

Charts for diffusion in various geometries. The various heat-transfer charts for unsteady-state conduction can be used for unsteady-state diffusion and are as follows. 

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. 

Eq. (6.10) is the three-dimensional unsteady-state diffusion equation, which has the same form as the respective heat conduction equation (6.8). 

Parabolic The heat equation 3T/3t = 3 T/3t -i- 3 T/3y represents noneqmlibrium or unsteady states of heat conduction and diffusion. 

Steady-state periodic heating and unsteady-state methods can be applied to measure the thermal conductivity and diffusivity of coal. Methods such as the compound bar method and calorimetry have been replaced by transient hot-wire/line heat source, and transient hot plate methods that allow very rapid and independent measurements of a and X. In fact, such methods offer the additional advantage of measuring these properties not only for monolithic samples but also for coal aggregates and powders under conditions similar to those encountered in coal utilization systems. 

Convection of heat via blood depends primarily on the local blood flow in the tissue and the vascular morphology of the tissue. Thermal diffusion is determined by thermal conductivity in the steady state, and thermal diffusivity in the unsteady state. In addition to these transport parameters, we need to know the volumes and geometry of normal tissues and tumor. In general, tumor volume changes as a function of time more rapidly than normal tissue volume. In special applications, such as hyperthermia induced by electromagnetic waves or radiofrequency currents, we need electromagnetic properties of tissues—the electrical conductivity and the relative dielectric constant. In the case of ultrasonic heating, we need to specify the acoustic properties of the tissue—velocity of sound and attenuation (or absorption) coefficient. 

Unsteady-state or transient heat conduction commonly occurs during heating or cooling of grains. It involves the accumulation or depletion of heat within a body, which results in temperature changes in the kernel with time. The rate at which heat is diffused out of or into a kernel or layer of kernels is dependent on the thermal diffusivity coefficient, a, of the grain ... 

Equation (7.10) is the representation of a transient (unsteady-state), one-dimensional heat conduction that must be satisfied at all points within the material. The combination of properties, a = k/Cpp, which has units of m /s, is known as the thermal diffusivity, and is an important parameter in transient conduction problems, a is a measure of the efficiency of energy transfer relative to thermal inertia. For a given time under similar heating conditions, thermal effects will... 

In the common case of cylindrical vessels with radial symmetry, the coordinates are the radius of the vessel and the axial position. Major pertinent physical properties are thermal conductivity and mass diffusivity or dispersivity. Certain approximations for simplifying the PDEs may be justifiable. When the steady state is of primary interest, time is ruled out. In the axial direction, transfer by conduction and diffusion may be negligible in comparison with that by bulk flow. In tubes of only a few centimeters in diameter, radial variations may be small. Such a reactor may consist of an assembly of tubes surrounded by a heat transfer fluid in a shell. Conditions then will change only axially (and with time if unsteady). The dispersion model of Section P5.8 is of this type. 

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