Conduction-Convection Problems

Conduction usually occurs in conjunction with convection, and if the temperatures are high, they also occur with radiation. In some practical situations where radiation cannot be readily estimated, convection heat transfer coefficients can be enhanced to include the effect of radiation. Combined conduction and convection led to the concept of thermal resistances, analogous to electrical resistances, which can be solved similarly. 

We will now consider the problem of calculating the heat transfer from a hot fluid to a composite plane of refractory wall and through an outer steel shell. 

Convection from the freeboard to the inside refractory surface will follow Newton s law of cooling as 

This is followed by one-dimensional steady state conduction through the refractory lining 

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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. 

Note that similar methods can be applied also to the solution of systems arising from the time-dependent heat conduction - convection problems. There are also other space decomposition methods. Let us mention the displacement decomposition technique for solving the elasticity problems and composite grid technique, for solving problems, which need a local resolution. More details can be found in Blaheta (2002) and Blaheta et al. (2002b). 

In many of the applications of heat transfer in process plants, one or more of the mechanisms of heat transfer may be involved. In the majority of heat exchangers heat passes through a series of different intervening layers before reaching the second fluid (Figure 9.1). These layers may be of different thicknesses and of different thermal conductivities. The problem of transferring heat to crude oil in the primary furnace before it enters the first distillation column may be considered as an example. The heat from the flames passes by radiation and convection to the pipes in the furnace, by conduction through the... 

The electric field is strongest at the center of the conducting spot and vanishes on the symmetry cell boundary at the membrane. A circulation is expected to result with a typical vortex size4 Rb- The appropriate electro-convective problem will be formulated in 6.5. 

To demonstrate Pawlowski s matrix transformation technique, an example will be used in which a forced convection problem, where a fluid with a viscosity p, a density p, a specific heat Cp and a thermal conductivity k, is forced past a surface with a characteristic size D at an average speed u. The temperature difference between the fluid and the surface is described by AT = Tf — Ts and the resulting heat transfer coefficient is defined by h. 

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. 

The heat transport is treated similar to the transport of mass. There are three ways for heat transport thermal conduction, convection and radiation. If the heat transport is limited, hot spots can occur in the catalyst bed, causing deactivation of the catalyst. Moreover, if a hot-spot occurs, the temperature at which is the reaction occurs is unknown. However, in typical laboratory equipment (small reactor, small particle size, diluted catalyst, and limited conversion), this is usually not a problem. 

We may summarize our introductory remarks very simply. Heat transfer may take place by one or more of three modes conduction, convection, and radiation. It has been noted that the physical mechanism of convection is related to the heat conduction through the thin layer of fluid adjacent to the heat-transfer surface. In both conduction and convection Fourier s law is applicable, although fluid mechanics must be brought into play in the convection problem in order to establish the temperature gradient. 

We shall defer part of our analysis of conduction-convection systems to Chap. 10 on heat exchangers. For the present we wish to examine some simple extended-surface problems. Consider the one-dimensional fin exposed to a surrounding fluid at a temperature T as shown in Fig. 2-9. The temperature of the base of the fin is T0. We approach the problem by making an energy balance on an element of the fin of thickness dx as shown in the figure. Thus... 

Reconsider Prob. 6-45. Using tlie re.sulis of this problem, obtain a relation for the volumetric heat generation rate in W/iiP. Then express the convection problem as an equivalent conduction problem in the oil layer. Verify your model by solving the conduction problem and obtaining a relation for the maximum temperature, which should be identical 10 the one obtained in the convcclion analysis. 

Gases are nearly transparent to radiatioo, and thus heat transfer through a gas layer is by simultaneous convection (or conduction, if the gas is quiescent) and radiation. Natural convection heat transfer coefficients are typically very low compared to those for forced convection. Therefore, radiation is usually disregarded in forced convection problems, but it must be considered in natural convection problems that involve a gas. This is especially the case for surfaces with high emissivities. For example, about half of the heat transfer through the air. space of a double-pane window is by radiation, The total rate of heat transfer is determined by adding the convection and radiation components,... 

This chapter illustrated the use of FEMLAB for problems of heat conduction, heat conduction and convection, and mass diffusion and convection. Problems included heat conduction in a 2D plane, several microfluidic devices (T-sensor and serpentine mixer), and heat effects in orifice flow. Specific methods demonstrated in FEMLAB include ... 

Thermal problems may be similarly divided into two classes. Some of these can be solved by employing only the laws of thermodynamics they are called thermodynamically determined problems. Some others, however, require knowledge beyond these laws these are called thermodynamically undetermined problems. Gas dynamics and heat transfer are two major thermodynamically undetermined disciplines. In addition to the general laws of thermodynamics and fluid mechanics, gas dynamics depends on equation of state while heat transfer requires knowledge on conduction, convection, and radiation phenomena, which we shall now introduce. Each of these phenomena relates heat to temperature, the same way that stress must be related to strain in mechanics. 

Thus, unlike k (which is a thermal property), A is merely a definition and depends on flow (conditions). That is, unlike thermal conductivity, the heat transfer coefficient cannot be tabulated and needs to be determined for each flow condition. Accordingly, Chapters 5 and 6 are devoted to elaboration of Eqs. (1.60) and (1.61) and the solution of convection problems in teams of a heat transfer coefficient. Here, for some appreciation, an order-of-magnitude range of each heat transfer coefficient corresponding to natural or forced convection in different fluids is given in Table 1.2. The order-of-magnitude difference between the A values for natural convection and forced convection resulting from flow of the same fluid should be noted. 

The formulation of convection problems, to be outlined by including the effect of fluid motion into conduction, presents no real difficulties. Let us proceed to the formulation of these problems by the help of some conduction problems formulated in the preceding chapters. Recall the original conduction problem,... 

Clearly, qc may also be expressed by means of conduction in the solid, which leads to the definition of the Biot number [recall Eq. (3.1)]. Note the fundamental difference in the use of Eqs, (3.1) and (5.8). In conduction problems, k and Too are given, and Eq. (3.1) is employed as a boundary condition. Because of their complexity, however, convection problems are usually solved in terms of simpler boundary conditions unrelated to h (such as specified temperature or heat flux), and Eq. (5.8) is utilized for the evaluation of h. 

In Chapters 2 and 3 we have already introduced the concept of penetration depth for an approximate solution of conduction problems (recall Section 2.4.1, and Exs. 2.11 and 3.9). This concept, which we utilized to determine the steady or unsteady penetration depth of heat (or thermal boundary layer) in solids and stagnant fluids, actually applies to all diffusion processes, such as diffusion of momentum, mass, electricity, and neutrons, as well as diffusion (or conduction) of heat It is a convenient tool for an approximate solution of conduction problems and is indispensable for convection problems, which are considerably more complicated than conduction problems. 

Problems involving gas radiation are significantly more complicated than those involving conduction, convection, and/or enclosure radiation. Consequently, the following discussion is restricted to one-dimensional problems. 

All BU produce heat as a result of metabolic processes, and so the major problem faced by many BU is the removal of excess heat. BU that are not provided with the ability to rid themselves of waste heat will eventually die. We have already seen in Section 2.7 that BU may lose heat by conduction, convection, radiation, and evaporation. Of these, conduction is not usually significant because BU... 

In the pure convection problem, heat transfer through the wall is characterized by an appropriate thermal boundary condition directly or indirectly specified at the wall-fluid interface. In a pure convection problem, the solution of the temperature problem for the solid wall is not needed the velocity and temperature are determined only in the fluid region. However, the heat transfer through the sohd walls of the microchannel by conduction may have significant normal and/or peripheral as well as axial components, or the wall may be of ncmuniform thickness. In these cases the temperature problem for the solid wall needs to be analyzed simultaneously with that for the fluid in order to calculate the real wall-fluid interface heat flux distribution. In this case the wall-fluid heat transfer is referred to as conjugate heat tranter. 

One practical strategy for working toward this end is to require students to routinely classify problem sets in addition to solving them. Students would have to examine textbooks in a new way, with the goal of understanding how chapters and sections differ from one another, yet are related. Consider the implications of asking students in a heat transfer course not only to solve conduction and convection problems but to be able to explain what makes these different from one another, what sorts of assumptions each makes, and what sorts of considerations get left out when one uses them in practical applications. 

The heat supplied by the barrel heaters has to be conducted through the entire thickness of the barrel and through the entire thickness of the melt film before it can reach the solid bed. Problems with this energy transport are considerable heat losses by conduction, convection, and radiation. Another, probably more severe, problem is the low thermal conductivity of the polymer. The heat has to be transferred across the entire melt film thickness. Therefore, the conductive heat flux will be small, particularly when the melt film thickness is large. Increasing the barrel temperature can accelerate the heating process however, this temperature is limited by the possibility of degradation of the polymer. 

A simplified heat model applicable to many convection problems is derived from conservation of energy based on the following assumptions (i) constant thermal conductivity, k (ii) negligible viscous dissipation, O (iii) negligible compressibility effect and (iv) negligible radiation heat transfer rate. TTie energy equation for such a model is derived as... 

To analyze and predict these problems, FE techniques have been integrated into mold design procedures. The results are compared with the data derived from the experimental hybrid mold, to aid the optimization of conduction, convection and radiation boundary conditions of the hybrid mold [3]. 

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