Heat transfer concentrated diffusion flux equations

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

Olir discussion on diffusion will be restricted primarily to binary systems containing only species A and B. We now wish to determine how the molar diffusive flux of a species (i.e., Ja) is related to its concentration gradient. As an aid in the discussion of the transport law that is ordinarily used to describe diffusion, lesll similar laws ftom other trans K)it processes. For example, in conductive heat transfer the constitutive equation relating the heat flux q and the temperature gradient is Fourier s law ... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

INTERPRETATION OF DIFFUSION EQUATIONS. Equation (21.16) is the basic equation for mass transfer in a nonturbulent fluid phase. It accounts for the amount of component A carried by the convective bulk flow of the fluid and the amount of A being transferred by molecular diffusion. The vector nature of the fluxes and concentration gradients must be understood, since these quantities are characterized by directions and magnitudes. As derived, the positive sense of the vectors is in the direction of increasing b, which may be either toward or away from the interface. As shown in Eq. (21.6), the sign of the gradient is opposite to the direction of the diffusion flux, since diffusion is in the direction of lower concentrations, or downhill, like the flow of heat down a temperature gradient. 

The eddy diffusivity depends on the fluid properties but also on the velocity and position in the flowing stream. Therefore Eq, (21.30) cannot be directly integrated to determine the flux for a given concentration difference. This equation is used with theoretical or empirical relationships for e f in fundamental studies of mass transfer, and similar equations are used for heat or momentum transfer in developing analogies between the transfer processes. Such studies are beyond the scope of this text, but Eq. (21.30) is useful in helping to understand the form of some empirical correlations for mass transfer. 

This chapter addresses the three fundamental transport properties characteristic of Chemical Engineering heat transfer, momentum transfer, and mass transfer. The underlying physical properties that represent each of these phenomena are thermal conductivity, viscosity, and diffusivity and the equations describing them have a similar form. Heat flux through conduction is expressed as a temperature gradient with units of W m . Note that heat flux, mass flux, etc. are physical measures expressed with respect to a surface (m ). Momentum flux in laminar flow conditions is known as shear stress and has units of Pa (or N m ) it equals the product of viscosity and a velocity gradient. Finally, molar flux (or mass flux) equals the product of diffusivity and a concentration gradient with units of mol m s These phenomena are expressed mathematically as shown in Table 7.1. 

Compared with heat transfer, the process of moisture transport is slower by a factor of approximately 10. For example, moisture equilibration of a 12 mm thick composite, at 350 K, can take 13 years whereas thermal equilibration only takes 15 s. Fick adapted the heat conduction equation of Fourier, and his (Pick s) second law is generally considered to be applicable to the moisture diffusion problem. The one-dimensional Fickian diffusion law, which describes transport through the thickness, and assumes that the moisture flux is proportional to the concentration gradient, is ... 

These are, of course, precisely the type of simple expressions one wishes to have on hand because it allows us to calculate flux from the embedded object to its surrotmdings. Equation 2.27 requires knowledge only of the bounding concentrations and the diffusivity. The geometry of the system is accounted for through the so-called shape factor S, which has the dimensions of length and is extensively tabulated in standard handbooks of heat transfer. A short version is given in Table 2.4. 

The flux Ja is typically measured by pan evaporation, lake evaporation, cooling pond heat transfer or an eddy correlation technique that relates turbulent diffusion of water vapor to mass flux. The concentration of water vapor at elevation z, Ca(z), is determined from simultaneous measurements of temperature and relative humidity and an application of Equations 9.56 and 9.57. At the water surface (z = 0), it is assumed that the air temperature is equal to water temperature and that relative humidity is 100%, so water temperature measurements and Equation 9.56 are sufficient to determine Ca(z = 0). If the analog of heat transfer is used, then... 

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