MOLECULAR TRANSPORT OF HEAT

The transfer of heat in a fluid may be brought about by conduction, convection, diffusion, and radiation. In this section we shall consider the transfer of heat in fluids by conduction alone. The transfer of heat by convection does not give rise to any new transport property. It is discussed in Section 3.2 in connection with the equations of change and, in particular, in connection with the energy transport in a system resulting from work and heat added to the fluid system. Heat transfer can also take place because of the interdiffusion of various species. As with convection this phenomenon does not introduce any new transport property. It is present only in mixtures of fluids and is therefore properly discussed in connection with mass diffusion in multicomponent mixtures. The transport of heat by radiation may be ascribed to a photon gas, and a close analogy exists between such radiative transfer processes and molecular transport of heat, particularly in optically dense media. However, our primary concern is with liquid flows, so we do not consider radiative transfer because of its limited role in such systems. 

Thermal conduction is the molecular transport of heat energy caused by a temperature gradient. The heat is transported from regions of high temperature towards those at lower temperatures, i.e. in the opposite direction to the temperature gradient. The corresponding law is FOURIER S law. The same phenomenon of transport of energy... 

A pure gas whose temperature T is a function of the spatial coordinates is considered. The temperature gradient causes a molecular transport of heat energy which obeys the following relationship ... 

Heat transport and Fourier s law. Fourier s law for molecular transport of heat or heat conduction in a fluid or solid can be written as follows for constant density p and heat capacity Cp. 

There are several significant problems in chemical engineering that require a fundamental understanding of differential equations in order to fully appreciate the underlying transport phenomena. In this book, differential equation means an equation containing derivatives of an unknown function to be determined [1]. For example, Fourier s law [2—4] for the molecular transport of heat in a fluid or a solid can be written as a first-order differential equation... 

As the fluid s velocity must be zero at the solid surface, the velocity fluctuations must be zero there. In the region very close to the solid boundary, ie the viscous sublayer, the velocity fluctuations are very small and the shear stress is almost entirely the viscous stress. Similarly, transport of heat and mass is due to molecular processes, the turbulent contribution being negligible. In contrast, in the outer part of the turbulent boundary layer turbulent fluctuations are dominant, as they are in the free stream outside the boundary layer. In the buffer or generation zone, turbulent and molecular processes are of comparable importance. 

If the temperature of a water mass is altered after having been isolated from the sea surfece, deviations from saturation can result. Such alterations are a consequence of heat transport via either conduction (molecular diffusion of heat) or turbulent mixing of adjacent water masses. In the case of the former, hydrothermal systems are important subsurface heat sources. If two water masses of different temperature imdergo turbulent mixing, the temperature of the admixture will differ from that of the source waters. 

Conduction (1) The transport of heat via molecular processes. (2) The transport of electrons causing an electrical current to flow. 

Waves of chemical reaction may travel through a reaction medium, but the ideas of important stationary spatial patterns are due to Turing (1952). They were at first invoked to explain the slowly developing stripes that can be exhibited by reactions like the Belousov-Zhabotinskii reaction. This (rather mathematical) chapter sets out an analysis of the physically simplest circumstances but for a system (P - A - B + heat) with thermal feedback in which the internal transport of heat and matter are wholly controlled by molecular collision processes of thermal conductivity and diffusion. After a careful study the reader should be able to ... 

The above description is of a thermally propagating steady-state wave. It must be emphasized, however, that the basic feature of a thermal mechanism is not altered by the superposition of molecular diffusion onto the diffusional transport of heat. This applies not only to interdiffusion of reactants and products but also to the diffusion of chain carriers participating in the chemical reaction, provided that the chains are unbranched. The reason for this is that in a wave driven by a diffusion process, the source strength of an entering mass element must continue to grow despite the drain by the adjacent sink region. This growth can occur only if the reaction rate is increased by a product of the reaction, which may be temperature as well as a material product. 

Electrodes and solid catalysts applied in the synthesis of chemicals or in emission control are, generally, hierarchical systems comprising dimensions ranging from millimeter to nanometer scale, allowing for mass and heat transport within a reactor, molecular transport of reactants and products through a pore system, and chemical reactions on nanostructured, frequently multifunctional surface sites as illustrated in Figure 4.2.2. Catalyst preparation always yields a catalyst precursor, whereas the active phase is only formed in contact with the feed of the substrate molecules in... 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

It will be recalled that a is the molecular diffusivity of heat. In turbulent flow one might assume that the heat transport could be represented by... 

Instabilities exist in individual flame fronts and lead to the formation of cells and of various unstable modes depending on molecular transport of chemical species and heat [379 266]. 

B. Havskjold, T. Ikeshoji and S. Kjelstrup Ratkje, On the Molecular Mechanism of Thermal Diffusion in Liquids, Mol. Phys. 80 (1993) 1389 B. Havskjold and S. Kjelstrup Ratkje, Criteria for Local Equilibrium in a System with Transport of Heat and Mass, J. Stat. Phys. 78 (1995) 463. 

The conventional parameterizations used describing molecular transport of mass, energy and momentum are the Fick s law (mass diffusion), Fourier s law (heat diffusion or conduction) and Newton s law (viscous stresses). The mass diffusivity, Dc, the kinematic viscosity, i/, and the thermal diffusivity, a, all have the same units (m /s). The way in which these three quantities are analogous can be seen from the following equations for the fluxes of mass, momentum, and energy in one-dimensional systems [13, 135] ... 

One consequence of the continuum approximation is the necessity to hypothesize two independent mechanisms for heat or momentum transfer one associated with the transport of heat or momentum by means of the continuum or macroscopic velocity field u, and the other described as a molecular mechanism for heat or momentum transfer that will appear as a surface contribution to the macroscopic momentum and energy conservation equations. This split into two independent transport mechanisms is a direct consequence of the coarse resolution that is inherent in the continuum description of the fluid system. If we revert to a microscopic or molecular point of view for a moment, it is clear that there is only a single class of mechanisms available for transport of any quantity, namely, those mechanisms associated with the motions and forces of interaction between the molecules (and particles in the case of suspensions). When we adopt the continuum or macroscopic point of view, however, we effectively spht the molecular motion of the material into two parts a molecular average velocity u = (w) and local fluctuations relative to this average. Because we define u as an instantaneous spatial average, it is evident that the local net volume flux of fluid across any surface in the fluid will be u n, where n is the unit normal to the surface. In particular, the local fluctuations in molecular velocity relative to the average value (w) yield no net flux of mass across any macroscopic surface in the fluid. However, these local random motions will generally lead to a net flux of heat or momentum across the same surface. 

A transport of mass or diffusion of mass will take place in a fluid mixture of two or more species whenever there is a spatial gradient in the proportions of the mixture, that is, a concentration gradient. Mass diffusion is a consequence of molecular motion and is closely analogous to the transport of heat and momentum in a fluid. 

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. 

Numerical solutions to simple thermal energy transport problems in the absence of radiative mechanisms require that the viscosity fi, density p, specific heat Cp, and thermal conductivity k are known. Fourier s law of heat conduction states that the thermal conductivity is constant and independent of position for simple isotropic fluids. Hence, thermal conductivity is the molecular transport property that appears in the linear law that expresses molecular transport of thermal energy in terms of temperature gradients. The thermal diffusivity a is constructed from the ratio of k and pCp. Hence, a = kjpCp characterizes diffusion of thermal energy and has units of length /time. 

Two extreme views of vibrational energy and heat flow may be useful in describing these properties in protein molecules. Proteins are large on the molecular scale and so it is tempting to ascribe macroscopic properties to them, for instance a coefficient of thermal conductivity to describe the flow of heat. However, proteins have of course a discrete set of vibrations, and a rather detailed description of energy flow among the vibrational states may be needed to characterize the transport of heat. In this respect, we might expect linear response predictions of heat flow to break... 

The laws which govern the phenomena of molecular transport of quantities of matter, of heat and of momentum, seen in the paragraphs above, can be presented in the following general form ... 

The diffusivities D, a, v characterize the ease of the molecular transports of the three quantities matter, heat and momentum, respectively. These diffusivities can be expressed in the same units, i.e. m s in basic SI units. 

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

Introduction. In molecular transport of momentum, heat, or mass there are many similarities, which were pointed out in Chapters 2 to 6. The molecular diffusion equations of Newton for momentum, Fourier for heat, and Fick for mass are very similar and we can say that we have analogies among these three molecular transport processes. There are also similarities in turbulent transport, as discussed in Sections 5.7C and 6.1A, where the flux equations were written using the turbulent eddy momentum diffusivity e, the turbulent eddy thermal diffusivity a, and the turbulent eddy mass diffusivity. However, these similarities are not as well defined mathematically or physically and are more difficult to relate to each other. 

As noted in the introduction, every chemical reaction requires heat and mass transfer, whereas apart from molecular transport mechanisms, heat or mass can be transported by a fluid (movement involving momentum transfer). The same transport phenomena govern the units in which no chemical reaction is taking place. Hence, although for the design of a chemical reactor, one needs chemical knowledge (in particular chemical kinetics and chemical thermodynamics), what is... 

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