Thermal energy, molecular flux

Equation 1.70 shows that the molar diffusional flux of component A in the y-direction is proportional to the concentration gradient of that component. The constant of proportionality is the molecular diffusivity 2. Similarly, equation 1.69 shows that the heat flux is proportional to the gradient of the quantity pCpT, which represents the. concentration of thermal energy. The constant of proportionality klpCp, which is often denoted by a, is the thermal diffusivity and this, like 2, has the units m2/s. 

Reactions (1) and (2) essentially convert solar radiant energy into thermal energy. The parameters which determine the rate of ozone formation (UV photon flux, atomic and molecular oxygen number density and the total gas number density) are not constant with altitude and so the ozone concentration and hence Tg varies with altitude. The net result is that Tg increases thoughout the stratosphere until a maximum is reached at the stratopause whence Tg begins to decrease again. 

Use Fourier s law to calculate the molecular flux of thermal energy in the radial direction, specifically air = R. 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

Step 6. Rearrange the expression for the molecular flux of thermal energy to obtain the equation of change for specific entropy. 

IDENTIFICATION OF THE MOLECULAR FLUX OF THERMAL ENERGY IN THE EQUATION OF CHANGE FOR TOTAL ENERGY... 

COUPLING BETWEEN DIFFUSIONAL MASS FLUX AND MOLECULAR FLUX OF THERMAL ENERGY IN BINARY MIXTURES THE ONSAGER RECIPROCAL RELATIONS... 

IDENTIFICATION OF FOURIER S LAW IN THE MOLECULAR FLUX OF THERMAL ENERGY AND THE REQUIREMENT THAT THERMAL CONDUCTIVITIES ARE POSITIVE... 

The linear laws given by equations (25-60) are rearranged to solve for the molecular flux of thermal energy ... 

The molecular flux of thermal energy, given by equation (26-5), is rearranged as follows ... 

The molecular flux of thermal energy in equation (26-10) is written in terms of the diffusional mass flux of component A and the temperature gradient as... 

By definition of the mass-average velocity v of the mixture, all diffusional mass fluxes with respect to v must sum to zero. Hence, Ja = —jn for binary mixtures. The final expression for the molecular flux of thermal energy in binary mixtures, neglecting the diffusion-thermo (i.e., Dufour) effect, is... 

The multicomponent equation of change for specific internal energy, given by (27-4), which is consistent with the first law of thermodynamics and the definition of the molecular flux of thermal energy via the entropy balance, reduces to ... 

These conditions on the diffusional mass flux and the molecular flux of thermal energy, the latter of which includes Fourier s law and the interdiffusional contribution, allow one to relate temperature and reactant molar density within the pellet. If n is the local coordinate measured in the direction of n, then equations (27-20) and (27-21) can be combined as follows ... 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

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