Heat Conduction and Molecular Diffusion

Heat can be transferred by conduction, convection, or radiation and/or combinations thereof. Heat transfer within a homogeneous solid or a perfectly stagnant fluid in the absence of convection and radiation takes place solely by conduction. According to Fourier s law, the rate of heat conduction along the y-axis per unit area perpendicular to the y-axis (i.e., the heat flux q, expressed as W in - or kcal m 2 h ) will vary in proportion to the temperature gradient in the y direction, dt/dy (°C m or K m ), and also to an intensive material property called heat or thermal conductivity k (W m K or kcal h m °C ). Thus, 

The negative sign indicates that heat flows in the direction of negative temperature gradient, namely, from warmer to colder points. Some examples of the approximate values of thermal conductivity (kcalh m °C ) at 20 °C are 330 for copper, 0.513 for liquid water, and 0.022 for oxygen gas at atmospheric pressure. Values of thermal conductivity generally increase with increasing temperature. 

According to Fick s law, the flux of the transport of component A in a mixture of A and B along the y axis by pure molecular diffusion, that is, in the absence of convection, J(kgh m ) is proportional to the concentration gradient of the diffusing component in the y direction, dC /dy (kg m ) and a system property called diffusivity or the diffusion coefficient of A in a mixture of A and B, 

It should be noted that D g is a property of the mixture of A and B, and is defined with reference to the mixture and not to the fixed coordinates. Except in the case of equimolar counter-diffusion of A and B, the diffusion of A would result in the movement of the mixture as a whole. However, in the usual case where the concentration of A is small, the value of ) g is practically equal to the value defined with reference to the fixed coordinates. 

Values of diffusivity in gas mixtures at normal temperature and atmospheric pressure are in the approximate range of 0.03-0.3 m- h and usually increase with temperature and decrease with increasing pressure. Values of the liquid phase diffusivity in dilute solutions are in the approximate range of 0.2-1.2 X 10 5 m h , and increase with temperature. Both gas-phase and liquid-phase diffusivities can be estimated by various empirical correlations available in reference books. 

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For a deflagration the energy from the reaction is transferred to the unreacted mixture by heat conduction and molecular diffusion. These processes are relatively slow, causing the reaction front to propagate at a speed less than the sonic velocity. 

In the following, flows that are caused by molecular processes or by radiation shall be considered. The flows driven by molecular processes are heat conduction and molecular diffusion. 

Note that use is made of the solution from aprevious problem that was solved in dimensionless form. Also, the analogy between heat conduction and molecular diffusion can best be exploited when dimensionless variables are utilized. 

The mechanism of flame propagation into a stagnant fuel-air mixture is determined largely by conduction and molecular diffusion of heat and species. Figure 3.1 shows the change in temperature across a laminar flame, whose thickness is on the order of one millimeter. 

Heat is produced by chemical reaction in a reaction zone. The heat is transported, mainly by conduction and molecular diffusion, ahead of the reaction zone into a preheating zone in which the mixture is heated, that is, preconditioned for reaction. Since molecular diffusion is a relatively slow process, laminar flame propagation is slow. Table 3.1 gives an overview of laminar burning velocities of some of the most common hydrocarbons and hydrogen. 

From Fig. 16.8 we see that in ordinary pipe flow for regions away from the wall the eddy viscosity is typically about 100 times the molecular viscosity (i.e., the Reynolds stresses are about 100 times the stresses due to molecular viscosity), that the eddy viscosity is a strong function of position and Reynolds number, and that it is difficult to calculate values of the eddy viscosity near the center of the pipe. From Eq. 16.15 we see that the sum of the eddy and molecular viscosities is equal to Tl dVJdy) at the center of the pipe both quantities are zero. To obtain the correct limit in this ratio as both numerator and denomnator approach zero requires more precise experimental measurements of and y than are currently available. We may infer from Fig. 16.8 that in this type of pipe flow the heat transfer and mixing will be of the order of 100 times the heat transfer and mixing due to molecular thermal conductivity and molecular diffusion. 

Later on, ZeFdovich and Frank-Kamenetskii [533] showed that the heat conductivity and the diffusion equations are identical provided the mean diffusion coefficient is equal to the thermal diffusion coefficient valid for close masses of diffusing molecules responsible for molecular heat transfer. It follows from the identity of these equations that the temperature field, i.e. the temperature treated as a function of the x coordinate (T = T(x)), and the concentration field n = n(x) are also similar... 

There is apparently an inherent anomaly in the heat and mass transfer results in that, at low Reynolds numbers, the Nusselt and Sherwood numbers (Figures. 6.30 and 6.27) are very low, and substantially below the theoretical minimum value of 2 for transfer by thermal conduction or molecular diffusion to a spherical particle when the driving force is spread over an infinite distance (Volume 1, Chapter 9). The most probable explanation is that at low Reynolds numbers there is appreciable back-mixing of gas associated with the circulation of the solids. If this is represented as a diffusional type of process with a longitudinal diffusivity of DL, the basic equation for the heat transfer process is ... 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

Since the width of this zone is significantly greater than the length of the molecular mean free path, we should speak not of energy transfer by direct impact, but of heat conduction and other dissipative processes in the gas—diffusion and viscosity—related to the gradients of the temperature, concentration and the velocity along the normal to the wave front. 

The balance between conduction and diffusion still operates for a much larger isolated wet object, provided radiation is excluded. This is the basis of the wet bulb thermometer method for measuring humidity. The actual rate of evaporation now is not as simply determined and is influenced by wind. The wet bulb temperature is almost independent of wind condition, owing to a convenient accident. Heat conduction is a diffusion process, and the diffusion coefficient for water vapor in air (0.24 sq. cm./sec.) is numerically close to the diffusion coefficient of temperature in air (thermal conductivity/specific heat = 0.20 sq. cm./sec.). Hence, the exact way in which each molecular diffusion process merges into the more rapid eddy diffusion process is not important because no matter how complex the transition is, it must be quantitatively similar for the two processes. 

In the past, combustion modeling was directed towards ffuid mechanics that included global heat release by chemical reaction. The latter was often described simply with the help of thermodynamics, assuming that the chemical reactions are much faster than the other processes like diffusion, heat conduction, and flow. However, in most cases chemistry occurs on time scales which are comparable with those of flow and molecular transport. As a consequence, detailed information about the individual elementary reactions is required if transient processes like ignition and flame quenching or pollutant formation shall be successfully modeled. The fundamental concept of using elementary reactions to describe a macroscopic... 

This definition of has the disadvantage that e is not a simple property of the fluid, such as the molecular viscosity, but is also a function of the flow rate and position in the flow. It has the advantage that it lets us easily formulate the ratio of the Reynolds stresses to viscous stresses. In addition, in calculations of heat and mass transfer, we may introduce a similar eddy thermal conductivity and eddy diffusivity. Under some circumstances these three eddy properties are identical, and under all circumstances they are at least of the same order of nagnitude. So this approach helps to apply fluid flow data to the solution of... 

Viscous stress is an extremely important variable, and this quantity is identified by the Greek letter r. Viscous stress represents molecular transport of momentum, analogous to heat conduction and diffusion. All molecular transport mechanisms correspond to irreversible processes that generate entropy under realistic conditions. When fluids obey Newton s law of viscosity, there is a linear relation between viscous stress and velocity gradients. All fluids do not obey Newton s law of viscosity, but almost aU gases and low-molecular-weight liquids are Newtonian. 

As shown in Chapter 2, transfer of heat by conduction is due to random molecular motions, and there is an obvious analogy between the two processes of heat conduction and of mass transport by diffusion. Pick in 1855 recognized this fact, and put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Pourier (1822). 

Heat delivery. Convection and conduction from hot gas sweeping by is the leading mode of heat transfer to a drying coating to supply the latent heat of vaporization of solvent. Except when solvent evaporation is so very rapid as to produce an appreciable convective velocity away from the surface, in turbulent gas flow the mechanisms of heat transfer to and solvent transfer away from the evaporating surface are virtually identical combinations of convective action with thermal conduction on the one hand and molecular diffusion on the other. This is reflected in useful correlations, like Colburn s, of the mass transfer coefficient with the more easily measured heat transfer coefficient in turbulent flow. It is also the reason that the now fairly extensive literature on the performance and design of driers focuses on heat transfer coefficients and heat delivery rates. 

Transfer of heat by conduction is due to random molecular motions, and thus there is an analogy between the heat conduction and the matter diffusion processes. Historically, Fourier first established the mathematical theory of heat conduction in 1822, putting it on a quantitative basis and a few decades later (1855), Fick recognised the analogy and adopted a similar mathematical theory for the diffusion of a substance. 

Consider a cylindrical tower filled with a solid-supported catalyst. The feed is liquid and flows upward through the reactor. The reactor operates adiabatically. The geometric variables are reactor diameter D [L] reactor length L [L], which is the height of the catalyst mass and solid-supported catalyst pellet or extrudate diameter Jp [L]. The material variables are fluid viscosity /x [L MT ], fluid density p [L M], fluid—solid heat capacity Cp [L MT 0 ], fluid—solid heat conductivity k [LMT 0 ], and molecular diffusivity Z>Difr [L T ]. The process variables are... 

The Lewis number is a dimensionless group composed of four parameters fluid density (p), fluid heat capacity (Cp), molecular diffusivity (D), and fluid thermal conductivity (k). 

An elementary molecular theory for nonequilibrium processes in a hard-sphere gas can be based on the gas kinetic theory that was presented in Chapter 9. We will apply this theory to self-diffusion and give the results of its application to heat conduction and viscous flow. 

Physical property data basic information required over the ranges of temperature, pressure and composition to be encountered in the process includes the density, viscosity, thermal conductivity, specific heat and molecular diffusivities of fluids together with the specific heat of the adsorbent and the bulk voidage and bulk density of the adsorbent bed. Many more properties of the system may need to be obtained if rigorous approaches to the design problem are adopted. 

Pick s law States that the molecular diffusion of water vapor in a gas without appreciable displacement of the gas is analogous to the conduction of heat, and is governed by a similar type of law. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

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