Equation of heat transfer

The lack of hydrodynamic definition was recognized by Eucken (E7), who considered convective diffusion transverse to a parallel flow, and obtained an expression analogous to the Leveque equation of heat transfer (L5b, B4c, p. 404). Experiments with Couette flow between a rotating inner cylinder and a stationary outer cylinder did not confirm his predictions (see also Section VI,D). At very low rotation rates laminar flow is stable, and does not contribute to the diffusion process since there is no velocity component in the radial direction. At higher rotation rates, secondary flow patterns form (Taylor vortices), and finally the flow becomes turbulent. Neither of the two flow regimes satisfies the conditions of the Leveque equation. 

The equation of heat transfer does not contribute anything new even though the temperature field is not similar to the velocity field due to the presence of the buoyancy force. (This situation holds identically in turbulent and laminar flows.)... 

For the sake of simplification Xjeas. regarded as a sum of the consequtive processes of heating (Xheat.) hardening (Xhard.) of the aminoplastic. The time of heating is determined from the equation of heat transfer i.e. Eq. (2). 

The numerical solution of governing equations of heat transfer for the evaporator and condenser regions allows determining the temperature distribution along the MHP axial direction. 

To evaluate the rate of heat transfer at a boundary, a film, transmitting heat only by conduction is postulated. This fictitious film presents the same resistance to heat transfer as the complex turbulent and laminar regions near the wall. If on the hot side of the wall the fictitious layer had a thickness xi, the equation of heat transfer to the wall would be... 

The equation of heat transfer in a moving medium is similar to Eq. (3.1.1) of convective diffusion and has the form... 

The equation for the temperature distribution in the region of heat stabilization can be obtained from the equation of heat transfer in Supplement 6 (where the temperature depends only on the transverse coordinate Y and the convective terms are equal to zero, since Vx = V(Y) and Vy = Vz = 0). This equation has the form... 

The temperature field is solved using a backward implicit difference approximation. The nodal temperature and variable rate of temperature of element are given by the interpolation of shape function. In the solution of temperature field, the equation of heat transfer, initial conditions and boundary conditions have to be satisfied. Based on the variational principle, the problem can be converted into solving the extremum of a functional. The implicit difference equation is written as ... 

The equations of heat transfer in the rubber thickness and next to the surface are ... 

When a solid is heated or cooled, heat is transferred through the structure. The equations of heat transfer were initially formulated by Fourier. They predate and are of identical mathematical form to Pick s laws of diffusion (Sections 7.1 and 7.3). In the case of steady-state heat transfer, the one-dimensional heat transfer equation is ... 

Figure 7.6 presents the heat transfer factor versus Reynolds number for rotary drying processes and various materials. Figure 7.7 shows the ranges of variation of the heat transfer factor versus Reynolds number for the rotary drying process in comparison with other thermal processes, and Figure 7.8 presents the estimated equation of heat transfer factor for the rotary drying versus Reynolds number, in comparison with other thermal processes. 

Heat transfer in gas-fluidized bed can occur by conduction, convection, and radiation depending on the operating conditions. The contribution of the respective modes of heat transfer to the coefficient of heat transfer depends on particle classification, flow condition, fluidization regimes, type of distributor, operating temperature, and pressure. Heat transfer between a single particle and gas phase can be defined by the conventional equation of heat transfer ... 

The hg involves convection and gas radiation to or from a surface, and it is like two resistances in parallel, thus hg = he + hr. Similar to Ohm s Law, (/ = E/Rt), heat flux, q = Q/A = ATfR, or g — UAAT, which is the basic equation of heat transfer. Example 5.1 illustrates the method for calculating U, the overall coefficient of heat transfer. 

For simplicity s sake the ground (earth) is assumed to be a slab infinitely extended in the positive z-direction. The time-dependent equation of heat transfer then is... 

Localization. The reduction scheme of variables used in space is very general and is found in other energy varieties (see for instance the Fourier equation in the thermal domain—case study G2 Fourier s Equation of Heat Transfer in Chapter 10). The scheme relies on the scalar nature of state variables (at the global level) and uses a sequence of spatial operators that are the contra-gradient, the curl (also called rotational), and the divergence. 

Spatially reduced Ohm s law in electrodynamics, Fourier s equation of heat transfer, Fick s law for diffusion, and Newton s law in hydrodynamics are among the subjects treated and their comparison is enlightening. The various transfers tackled in this chapter are stationary diffusion,... 

With this lineic density, Newton s law as in Equation G4.1 is merely written as a dissipative relationship in a spatially reduced form, similar to the resistivity or conductivity relationships in many domains (see case studies G1 Reduced Ohm s Law and G2 Fourier s Equation of Heat Transfer for instance)... 

Heat transfer in porous media, like aerogels, is described by the equation of heat transfer. The involved heat transfer mechanisms are schematically illustrated in Figure 23.1. The principal discussion of the equation of heat transfer provides an insight into the nature of the physical material property thermal conductivity. Generally, the equation of heat transfer can be expressed as ... 

The specific heat not only describes the capacity to store heat, but also influences the dynamics of heat transfer within aerogels according to the equation of heat transfer (23.1) and (23.4). The higher the specific heat, the slower the heat propagation within a material, if the density and the effective thermal conductivity remain constant. In the late eighties and early nineties several authors investigated the specific heat of silica aerogels in detail [9,51, 52, 54, 64-67]. The focus of their research work was the study of the density of vibrational states g(a>) correlated to the specific heat and the solid thermal conductivity by phonons ... 

Basic Equations of Heat Transfer by Heat Conduction [ 1 -3]... 

As discussed in Section 6.4 for ammonia oxidation at a single Pt wire, that is, where the cylindrical wire is heated by an exothermic chemical reaction, the variation of temperature around a cylinder can nowadays be modeled by computer programs, for example, by the finite element method. The geometric structure is approximated by a meshing procedure that is used to define and break the model up into small elements. The differential equations of heat transfer and of the fluid dynamics (Navier-Stokes equations) are then numerically solved. The temperature gradients at the surface of the cylinder (Tcyi = const. = at... 

Temperature difference between particle and fluid can sometimes play an important role in the overall heat transfer. This may happen at high flow rates in the adiabatic operations. In such cases, the basic equations of heat transfer are given below. 

Ny varies generally with hot gas temperature, solid gas ratio, and gas velocity. Therefore, suitable drying tube volume must be calculated on the basis of the equation of heat transfer and heat balance. Ny gives the approximate value of the drying tube volume. The drying rate per unit volume is defined by... 

In this model is assumed that when u<0 there can be a region partially containing material in the solid phase, where the percentage of particles of firm substance depends on the temperature, therefore enthalpy at u<0 is a nonlinear function of temperature. For a description of the areas with partial content of solid substances more complex two-phase models exist, where the ratio between liquid and solid phases depends upon the time from the start of transition to a solid phase. Such models require calculation of two equations of heat transfer. These models are not considered as long-term current processes are assumed to take place. 

The formal solutions of the equations of heat transfer in solids and radiative transfer depend on the following properties of the near-surface material the complex dielectric constant (X), thermal conductivity k (ergs per centimeter per second per Kelvin), specific heat c (ergs per gram per Kelvin), and density p (grams per cubic centimeter). The analytic theory of heat transfer at planetary surfaces begins by assuming that the temperature at any point on the surface can be expanded in a Fourier series in time ... 

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