Equation convective heat conduction

In a system with homogeneous reactions (e.g. reactive absorption), mass and heat transfer is described by the following convective diffusion and convective heat conduction equations (Kenig, 2000) ... 

The thermogravitational motion is described in the Boussinesq approximation in which the variable density in the equations of motion (5.9.1)—(5.9.3) and in the convective heat conduction equation (5.9.4) is taken into account only in the Archimedes term (the last term in (5.9.2)). This term is proportional to the temperature deviation T from the mean value. The thermocapillary motion... 

In order to account for the heat loss through the metallic body of the cone, a heat conduction equation, obtained by the elimination of the convection and source terms in Equation (5.25), should also be incorporated in the governing equations. 

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. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The heat conduction equation is first order in time, and thus the initial condition cannot involve any derivatives (it is limited to a specified temperature). However, the heal conduction equation is second order in space coordinates, and thus a boundary condition may involve first derivalives at the boundaries as well as specified values of temperature. Boundary conditions most commonly encountered in practice are the specified temperature, specified heat flux, convection, and radiation boundary conditions. 

Air at Tg acts on top surface of the rectangular solid shown in Fig. P5-123 with a convection heat transfer coefficient of h. The correct steady-state finite-difference heat conduction equation for node 3 of this solid is... 

Equation (9 16) is known as the steady-state heat conduction equation and is completely analogous to the creeping-motion equation of Chaps. 7 and 8. It can be seen that convection plays no role in the heat transfer process described by (9 16) and (9 17). Thus the form of the velocity field is not relevant, and in spite of the initial assumption (9 15), there is no dependence of 0o on the Reynolds number of the flow. The solution of (9 16) and (9-17) depends on only the geometry of the body surface, represented in (9 17) by S. 

Conjugated eonduetion-convection problems are among the elassieal formulations in heat transfer that still demand exact analytical treatment. Since the pioneering works of Perelman (1961) [14] and Luikov et al. (1971) [15], such class of problems continuously deserved the attention of various researchers towards the development of approximate formulations and/or solutions, either in external or internal flow situations. For instance, the present integral transform approach itself has been applied to obtain hybrid solutions for conjugated conduction-convection problems [16-21], in both steady and transient formulations, by employing a transversally lumped or improved lumped heat conduction equation for the wall temperature. 

The problem can be solved effectively by converting the convection-diffusion equation into the well studied heat conduction equation by introducing the stream fimction P as a new variable. In terms of the stream function the velocity components in spherical coordinates z and 0 are,... 

With R] = 0 Eq. (3.3.18) is termed the convective diffusion equation. When, in addition, u = 0, the equation reduces to the ordinary diffusion equation, which is also referred to as Pick s second law of diffusion. It is applicable to diffusion in solids or stationary liquids and has the same form as the heat conduction equation in stationary media with constant thermal conductivity. 

These equations repeat those previously set down. Flete, u is the kinematic viscosity, and a is the thermal diffusivity. The subscripts have been dropped in the convective diffusion equation, and D can be the binary diffusion coefficient, the effective electrolytic diffusion coefficient, or the diffusion coefficient of the fth species. The molar concentration is to be interpreted in the same context. In the energy equation, sometimes referred to as the heat conduction equation in the form written, heat flux due to interdiffusion and due to viscous dissipation have been neglected as small. Heat sources are also absent. 

To obtain the above equations in a closed form, a series of continuum equations still need to be established. At the interfaces between the heater, sensor, and the surrounding solid material, there is a continuum equation which characterizes the heat conduction at the two sides. A convective heat transfer equation similar to Eq. 10 should be used for the interfaces between a solid surface and the surrounding flowing fluid. 

At all channel wall surfaces, the no-slip boundary condition is applied to the velocity field (the Navier-Stokes equation), the fixed zeta-potential boundary condition is imposed on the EDL potential field (the Poisson-Boltzmann equation), and the insulation boundary condition is assigned to the applied electric field (the Laplace equation), and the no-mass penetration condition is specified for the solute mass concentration field (the mass transport equation). In addition, the third-kind boundary condition (i. e., the natural convection heat transfer with the surrounding air) is applied to the temperature field at all the outside surfaces of the fabricated channels to simultaneously solve the energy equation for the buffer solution together with the conjugated heat conduction equation for the channel wall. 

Heat may be transferred from the fuel elements by conduction, convection, and radiation. For all but high-temperature gas-cooled reactors, the last of these is of little significance under normal operating conditions. Convection is the process by which the heat from the surface of the fuel element is carried into the coolant and subsequently removed from the reactor. Before this happens, however, the heat generated in the fuel must travel by conduction from the interior of the element to its surface. In the present section, we confer the application of the standard heat conduction equations to determine the temperature distribution within fuel elements of various geometries. 

The objective is to develop a model that can predict the heat transfer coefficient h or the Nusselt number. Equation (4.20) says nothing about the heat transfer coefficient h, and we need to conduct experiments to develop a model of heat flux that includes convection, heat conduction, viscosity, heat capacity, etc., so that we can predict the... 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

When a gas reacts with a solid, heat will be transfened from the solid to the gas when the reaction is exothermic, and from gas to solid during an endothermic reaction. The energy which is generated will be distributed between the gas and solid phases according to the temperature difference between the two phases, and their respective thermal conductivities. If the surface temperature of the solid is T2 at any given instant, and that of the bulk of the gas phase is Ti, the rate of convective heat transfer from the solid to the gas may be represented by the equation... 

Thus, inserting the above expressions for the heat conducted from the cell and the heat convected from the cell, together with the heat evolved from the cell, from equation (50) in equation (44),... 

For the analysis, a steady-state fire was assumed. A series of equations was thus used to calculate various temperatures and/or heat release rates per unit surface, based on assigned input values. This series of equations involves four convective heat transfer and two conductive heat transfer processes. These are ... 

Suppose the bottom temperature of the liquid is maintained at 25 °C for a thin pool. Let us consider this case where the bottom of the pool is maintained at 25 °C. For the pool case, the temperature is higher in the liquid methanol as depth increases. This is likely to create a recirculating flow due to buoyancy. This flow was ignored in developing Equation (6.33) only pure conduction was considered. For a finite thickness pool with its back face maintained at a higher temperature than the surface, recirculation is likely. Let us treat this as an effective heat transfer coefficient, between the pool bottom and surface temperatures. For purely convective heating, conservation of energy at the liquid surface is... 

Another problem related to the validity of equation 9.9 is that equation 9.6 applies only to heat conduction. If T — 12 is large, some significant fraction of heat will be transferred by convection and radiation and thus will not be monitored by the thermopile. Consequently, the use of partial compensating Peltier or Joule effects was essential in the experiments involving Calvet s calorimeter, whose thermopiles had a fairly low thermal conductivity. 

Radiation is the rate of heat transfer by electromagnetic waves emitted by matter. Unlike conduction and convection, radiation does not require an intervening medium to propagate. The basic rate of radiation heat-transfer equation between a high temperature (Th) black body and a low temperature Tf) black body is Stefan-Boltzmann s law ... 

Combination of Equations 1 and 2 allows calculation of the rate of heat transfer from the growing crystal surface to the bulk solution. Under heat balance conditions, this rate of heat generation must be balanced by the amount of heat removed from the crystallizer by convection and conduction. This will be determined by the overall heat transfer coefficient, U, between the bulk solution and the refrigerant including convective resistances between the fluid and both sides of the crystallizer wall (refrigerant side and product side) as well as the conductive resistance across the crystallizer wall. 

---