Heat conduction equation unsteady state

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. 

In Sec. 4-4, this solution will be presented in graphical form for calculation purposes. For now, our purpose has been to show how the unsteady-heat-conduction equation can be solved, for at least one case, with the separation-of-variables method. Further information on analytical methods in unsteady-state problems is given in the references... 

Mathematical modeling of mass or heat transfer in solids involves Pick s law of mass transfer or Fourier s law of heat conduction. Engineers are interested in the distribution of heat or concentration across the slab, or the material in which the experiment is performed. This process is represented by parabolic partial differential equations (unsteady state) or elliptic partial differential equations. When the length of the domain is large, it is reasonable to consider the domain as semi-infinite which simplifies the problem and helps in obtaining analytical solutions. These partial differential equations are governed by the initial condition and the boundary condition at x = 0. The dependent variable has to be finite at distances far (x = ) from the origin. Both parabolic and elliptic partial... 

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. 

For heat conduction in solids, where the velocity terms are zero, Eq. (6.6) simplifies considerably. When combined with Eq, (6.7), it gives the well-known three-dimensional unsteady-state heat conduction equation... 

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

The initial and boundary conditions associated with the partial differential equations must be specified in order to obtain unique numerical solutions to these equations. In general, boundary conditions for partial differential equations are divided into three categories. These are demonstrated below, using the one-dimensional unsteady-state heat conduction equation... 

Classic examples of parabolic differential equations are the unsteady-state heat conduction equation... 

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

There is also another key parameter linked to the choice of the material for the reactor. First, the choice is obviously determined by the reactive medium in terms of corrosion resistance. However, it also has an influence on the heat transfer abilities. In fact, the heat transport depends on the effusivity relative to the material, deflned by b = (XpCp) the effusivity b appears in the unsteady-state conduction equation. 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

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. 

When heat is conducted through a solid under unsteady-state conditions, the following general equation applies ... 

Application of the law of conservation of energy to a three-dimensional solid, with the heat flux given by (5-1) and volumetric source term S (W/m3), results in the following equation for unsteady-state conduction in rectangular coordinates. 

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. 

General solutions of unsteady-state conduction equations are available for certain simple shapes such as the infinite slab, the infinitely long cylinder, and the sphere. For example, the integration of Eq. (10.16) for the heating or cooling of an infinite slab of known thickness from both sides by a medium at constant... 

Hence, the local mass transfer coefficient scales as the two-thirds power of a, mix for boundary layer theory adjacent to a solid-liquid interface, and the one-half power of A, mix for boundary layer theory adjacent to a gas-liquid interface, as well as unsteady state penetration theory without convective transport. By analogy, the local heat transfer coefficient follows the same scaling laws if one replaces a, mix in the previous equation by the thermal conductivity. 

After regarded U-vertical pipe as an equivalent pipe, soil temperature field which around it is a cylinder temperature field, in the circular direction there is no temperature gradient, the temperature distribution of the concrete around can be regarded as an axisymmetric problems. The styles of differential equations of heat conductivity are axisymmetric and unsteady state ... 

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... 

A number of problems such as the diying of porous solids of various shapes fell into this category. Solutions to such similar problems may be found in Crank, Mathematics of Dijfiision, and also in Catslaw and Jaeger, Heal Conduction in Solids, since the unsteady-state diffusirm equation often takes on the same form as that for heat conduction. 

To derive the equation for unsteady-state condition in one direction in a solid, we refer to Fig. 5.1-1. Heat is being conducted in the x direction in the cube Ax, Ay, Az in size. For conduction in the x direction, we write... 

Unsteady-state conduction in a cylinder. In deriving the numerical equations for unsteady-state conduction in a flat slab, the cross-sectional area was constant throughout. In a cylinder it changes radially. To derive the equation for a cylinder. Fig. 5.4-3 is used where the cylinder is divided into concentric hollow cylinders whose walls are Ax m thick. Assuming a cylinder 1 m long and making a heat balance on the slab at point n, the rate of heat in — rate of heat out = rate of heat accumulation. 

In Section 5.1 an unsteady-state equation for heat conduction was derived,... 

Since this equation is identical mathematically to the unsteady-state heat-conduction Eq. (5.10-10),... 

Cooling of the plastic part in the mold is essentially a problem in unsteady state heat transfer. The equation describing this (if density and thermal conductivity are constant) is... 

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... 

On the basis of their initial and boundary conditions, partial differential equations may be further classified into initial-value or boundary-value problems. In the first case, at least one of the independent variables has an open region. In the unsteady-state heat conduction problem, the time variable has the range 0 r >, where no condition has been specified at r = eo therefore, this is an initial-value problem. When the region is closed for all independent variables and conditions are specified at all boundaries, then the problem is of the boundary-value type. An example of this is the three-dimensional steady-state heat conduction problem described by the equation... 

When using finite differences to solve the unsteady heat conduction problem, another approach involves writing finite difference equations at each grid point (node) only for the spatial variables while leaving the time derivative intact. This leads, generally, to a large number of simultaneous ODEs, which can be solved by, for example, a Runge-Kutta method. However, one must be careful since this set of ODEs can be stiff. Consider the same one-dimensional, unsteady state heat conduction problem as solved in Examples 8.1 and 8.2. This problem is solved by the method of lines in the next example. 

---