Conduction equation transient

One-Dimensional Conduction The one-dimensional transient conduction equations are (for constant physical properties)... 

The first type of model considers the heat transfer surface to be contacted alternately by gas bubbles and packets of closely packed particles. This leads to a surface renewal process whereby heat transfer occurs primarily by transient conduction between the heat transfer surface and the particle packets during their time of residence at the surface. Mickley and Fairbanks (1955) provided the first analysis of this renewal mechanism. Treating the particle packet as a pseudo-homogeneous medium with solid volume fraction, e, and thermal conductivity (kpa), they solved the transient conduction equation to obtain the following expression for the average heat transfer coefficient due to particle packets,... 

Using the slug-flow model, show that the boundary-layer energy equation reduces to the same form as the transient-conduction equation for the semi-infinite solid of Sec. 4-3. Solve this equation and compare the solution with the integral analysis of Sec. 6-5. 

One-Dimensional Conduction Lumped and Distributed Analysis The one-dimensional transient conduction equations in rectangular (b = 1), cylindrical (b = 2), and spherical (b = 3) coordinates, with constant k, initial uniform temperature 7), S = 0, and convection at the surface with heat-transfer coefficient h and fluid temperature 77, are... 

When temperatures of materials are a function of both time and space variables, more complicated equations result. Equation (5-2) is the three-dimensional unsteady-state conduction equation. It involves the rate of change of temperature with respect to time 3t/30. Solutions to most practical problems must be obtained through the use of digital computers. Numerous articles have been published on a wide variety of transient conduction problems involving various geometrical shapes and boundaiy conditions. 

Two-Dimensional Conduction The governing differential equation for two-dimensional transient conduction is... 

McAdams (Heat Transmission, 3d ed., McGraw-HiU, New York, 1954) gives various forms of transient difference equations and methods of solving transient conduction problems. The availabihty of computers and a wide variety of computer programs permits virtually routine solution of complicated conduction problems. 

The heating of the liquid can be approximated by the transient conduction of heat to a slab of finite thickness 8, with the conventional differential equation,... 

Following the procedure used with the one-dimensional FEM model and using the constant strain triangle element developed in the previous section, we can now formulate the finite element equations for a transient conduction problem with internal heat generation rate per unit volume of Q. The governing equation is given by... 

The simplest model of this kind can be represented by one in which an isolated particle surrounded by gas is in contact with or in the vicinity of the heating surface for a certain time, during which the heat transfer between the particle and the heating surface takes place by transient conduction, as shown in Fig. 12.4. In terms of the model, the Fourier equation of thermal conduction can be expressed as... 

In the emulsion phase/packet model, it is perceived that the resistance to heat transfer lies in a relatively thick emulsion layer adjacent to the heating surface. This approach employs an analogy between a fluidized bed and a liquid medium, which considers the emulsion phase/packets to be the continuous phase. Differences in the various emulsion phase models primarily depend on the way the packet is defined. The presence of the maxima in the h-U curve is attributed to the simultaneous effect of an increase in the frequency of packet replacement and an increase in the fraction of time for which the heat transfer surface is covered by bubbles/voids. This unsteady-state model reaches its limit when the particle thermal time constant is smaller than the particle contact time determined by the replacement rate for small particles. In this case, the heat transfer process can be approximated by a steady-state process. Mickley and Fairbanks (1955) treated the packet as a continuum phase and first recognized the significant role of particle heat transfer since the volumetric heat capacity of the particle is 1,000-fold that of the gas at atmospheric conditions. The transient heat conduction equations are solved for a packet of emulsion swept up to the wall by bubble-induced circulation. The model of Mickley and Fairbanks (1955) is introduced in the following discussion. 

The equation of thermal energy (Eq. 2.9-16) for transient conduction in solids without internal heat sources reduces to... 

Of a more complete approach are the zone models [3], which consider two (or more) distinct horizontal layers filling the compartment, each of which is assumed to be spatially uniform in temperature, pressure, and species concentrations, as determined by simplified transient conservation equations for mass, species, and energy. The hot gases tend to form an upper layer and the ambient air stays in the lower layers. A fire in the enclosure is treated as a pump of mass and energy from the lower layer to the upper layer. As energy and mass are pumped into the upper layer, its volume increases, causing the interface between the layers to move toward the floor. Mass transfer between the compartments can also occur by means of vents such as doorways and windows. Heat transfer in the model occurs due to conduction to the various surfaces in the room. In addition, heat transfer can be included by radiative exchange between the upper and lower layers, and between the layers and the surfaces of the room. 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

To analyze a transient heat-transfer problem, we could proceed by solving the general heat-conduction equation by the separation-of-variables method, similar to the analytical treatment used for the two-dimensional steady-state problem discussed in Sec. 3-2. We give one illustration of this method of solution for a case of simple geometry and then refer the reader to the references for analysis of more complicated cases. Consider the infinite plate of thickness 2L shown in Fig. 4-1. Initially the plate is at a uniform temperature T, and at time zero the surfaces are suddenly lowered to T = T,. The differential equation is... 

We continue our discussion of transient heat conduction by analyzing systems which may be considered uniform in temperature. This type of analysis is called the lumped-heat-capacity method. Such systems are obviously idealized because a temperature gradient must exist in a material if heat is to be conducted into or out of the material. In general, the smaller the physical size of the body, the more realistic the assumption of a uniform temperature throughout in the limit a differential volume could be employed as in the derivation of the general heat-conduction equation. 

Consider an insulated composite rod, which is formed of two parts of equal length. The thermal conductivities of parts a and b are a and kh, for 0 < x < 1/2. The nondimensional transient, one-dimensional heat conduction equations over the length x of the rod are expressed by the following equations... 

The coverage of Chapter 4, Transient Heat Conduction, is now expanded to include (1) the derivation of the dimensionless Biot and Fourier numbers by nondimensionalizing the heat conduction equation and the boundary and initial... 

Noting that the area A is constant for a plane wall, the one-dimensional transient heal conduction equation in a plane wall becomes... 

Now consider a sphere, with density p, specific heat c, and outer radius R. The area of the sphere normal to the direclion of heat transfer at any location is A — 4vrr where r is the value of the radius at that location. Note that the heat transfer area A depends on r in this case also, and thus it varies with location. By considering a thin spherical shell element of thickness Ar and repeating tile approach described above for the cylinder by using A = 4 rrr instead of A = InrrL, the one-dimensional transient heat conduction equation for a sphere is determined to be (Fig. 2-17)... 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

C Write down the one-dimensiotial transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, and indicate what each variable represents. 

Consider a semi-inlinite solid with constant thermophysical properties, no internal heat generation, uniform theimal cnnditinn.s on its exposed surface, and initially a uniform temperature of Tj throughout. Heat tfansfec in this case occurs only in the direction uormal to the surface (the x direction), and thus it is one-dimensional. Differential equations are independent of the boundary or initial conditions, and thus Eq. 4—lOa for one-dimensional transient conduction in Cartesian coordinates applies. The depth of the solid is large (x expressed mathematically as a boundary condition as T x —> , 0 = T,. 

The separation of variables technique docs not work in this case since the medium is infinite. But another clever approach that converts the partial differential equation into an ordinary diSerendal equation by combining the Bvo independent variables x and t into a single variable rj, called the similarity variable, works well. For transient conduction in a semi-infinite medium, it is defined as... 

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