Heat flux by conduction

If the reactive gas produced at the burning surface of an energehc material reacts slowly in the gas phase and generates a luminous flame, the distance Lg between the burning surface and the luminous flame front is termed the flame stand-off distance. In the gas phase shown in Fig. 3.9, the temperature gradient appears to be small and the temperature increases relatively slowly. In this case, heat flux by conduction, the first term in Eq. (3.41), is neglected. Similarly, the rate of mass diffusion, the first term in Eq. (3.42), is assumed to be small compared with the rate of mass convection, the second term in Eq. (3.42). Thus, one gets... 

The subscript 0 in this equation signifies zero mass transfer at the surface (/(0) = 0). The subscript 2 indicates that the enthalpy potential contributing to the heat flux by conduction alone is dependent only on the specific heat of the air and not that of the coolant. 

In the above equation, represents the heat flux by conduction through the porous medium, and represents the heat flux by convection of the fluid mixture. The y direction is assumed to be of unit length. 

In this expression the flux Jg = (q - 2/ pj Jd/VT where q is the heat flux by conduction andJ/jj is the diffusion flux Jdj = P being the rate of diffusion,... 

Fig. 5.25 Temperature gradient, conductive heat flux, convective heat flux, and heat flux by chemical reaction as a function of distance from the burning surface at 3 MPa (initial temperatures 243 K and 343 K) for BAMO/ THF = 60 40 copolymer.

Rate of Heat Transfer. Fourier s Law may be integrated and solved for a number of geometries to relate the rate of heat transfer by conduction to the temperature driving force. Equations are given below that allow the calculation of steady-state heat flux and temperature profiles for a number of geometries. 

Comparing Equation (4) with Equation (5), i = qy, AV = AT and Rn = (bn/kn). Thus, we can think of the (b/k) values as thermal resistances, the resistance of the material layer to heat transfer by conduction. From Equation (4), it can be seen that the heat flux is equal to the overall temperature difference AT, divided by the sum of the thermal resistance of each layer L(b/k). The concept of thermal resistance of a layer is very useful and we will return to it later. 

Equation (20) may be compared to Equation (4) for the heat flux through a composite plate. As we have seen, (6w/fcw) is the thermal resistance of the wall to heat transfer by conduction. Similarly, (1//+) and (1 /Ac) are the resistances to heat transfer offered by the hot and cold films, respectively. 

In gas-filled windows there are three heat transfer mechanisms conduction and convection through the gas layer and radiation between the surroundings and the glass surfaces. The heat flow by conduction is minimized by using a fairly thick gas layer with a low conductivity. With even thicker layers, the effect of convection becomes important. Conduction and radiation cause similar heat fluxes, with heat transfer coefficients of a few watts per square metre per kelvin. 

Many practical mass transfer problems involve the diffusion of a species through a plane-parallel medium that does not involve any homogeneous chemical reactions under one-dimensional steady conditions. Such mass transfer problems are analogous to the steady one-dimensioiial heat conduction problems in a plane wall with no heal generation and can be analyzed similarly. In fact, many of the relations developed in Chapter 3 can be used for mass transfer by replacing temperature by mass (or molar) fraction, thermal conductivity by pD g (or CD ), and heat flux by mass (or molar) flux (Table 14-8). 

Recall our earlier analysis of the mass transfer process in Section 8.1 when we found that the were constant through the film.) The energy flux E is related to the conductive heat flux by Eq. 11.1.4 which, when combined with Eq. 11.2.8, simplifies to... 

In summary, the heat transport by conduction is generally important in reaction engineering applications. The thermal radiation flux is important in particular cases. The multi-component mixture specific contributions to the total energy flux are usually negligible. 

The physical meaning of the terms (or group of terms) in the entropy equation is not always obvious. However, the term on the LHS denotes the rate of accumulation of entropy within the control volume per unit volume. On the RHS the entropy flow terms included in show that for open systems the entropy flow consists of two parts one is the reduced heat flow the other is connected with the diffusion flows of matter jc, Secondly, the entropy production terms included in totai demonstrates that the entropy production contains four different contributions. (The third term on the RHS vanishes by use of the continuity equation, but retained for the purpose of indicating possible contributions from the interfacial mass transfer in multiphase flows, discussed later). The first term in totai arises from heat fluxes as conduction and radiation, the third from diffusion, the fourth is connected to the gradients of the velocity field, giving rise to viscous flow, and the fifth is due to chemical reactions. 

The situation is analogous to momentum flux, where the relative Importance of turbulent shear to viscous shear follows the same general pattern. Under certain ideal conditions, the correspondence between heat flow and momentum flow is exact, and at any specific value of rjr the ratio of heat transfer by conduction to that by turbulence equals the ratio of momentum flux by viscous forces to that by Reynolds stresses. In the general case, however, the correspondence is only approximate and may be greatly in error. The study of the relationship between heat and momentum flux for the entire spectrum of fluids leads to the so-called analogy theory, and the equations so derived are called analogy equations. A detailed treatment of the theory is beyond the scope of this book, but some of the more elementary relationships are considered. 

Introduction. When a fluid is flowing in laminar flow and mass transfer by molecular diffusion is occurring, the equations are very similar to those for heat transfer by conduction in laminar flow. The phenomena of heat and mass transfer are not always completely analogous since in mass transfer several components may be diffusing. Also, the flux of mass perpendicular to the direction of the flow must be small so as not to distort the laminar velocity profile. 

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. 

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