Surface energy flux vector

Here, S = 5 gj is the surface stress, f r the external body force per unit mass of material surface, Sa- the specific internal surface energy, q . = q a the surface heat flux vector, rjg. the surface entropy density, So- = i a the surface entropy flux vector, and h /d the surface entropy production. 

Combination of the material balance Equation 3.30 and the selected equations, which relate the flux vectors for all species to their concentrations, gives a set of equations to be solved subject to the appropriate boundary conditions at the particle surface. If particles cannot be considered as isothermal, the energy balance and energy flux equations are also required. 

D-23). Then F becomes the energy per second transported across a surface of unit area that is, F is the heat-flux vector q [see equations (D-28) and (D-29) with neglected]. Hence equation (31) becomes... 

In terms of a continuum theory the heat flux vector represents the direction and magnitude of the energy flow at a position indicated by the vector x. ft can also be dependent on time t. The heat flux q is defined in such a way that the heat flow dQ through a surface element d.4 is... 

Electromagnetic theory shows [6-12] that the flux of the electromagnetic field energy normal to a unit surface containing the vectors of the electric and magnetic field is equal to the vector product... 

Energy is transferred from the hotter to the cooler body, and so heat is said to flow towards the cooler one. Although energy is a scalar quantity, the flux of energy (energy flow through a surface) is a vector — direction matters. The above equation is thus more precisely written in calculus notation as ... 

The energy flux of a plane, monochromatic wave is represented by the Poynting vector as discussed in Section 1.2. In this section we relate the Poynting vector to other quantities used in the description of the radiative transfer of energy in planetary atmospheres and from surfaces. 

Table 1.1 Conjugate pairs of variables in work terms for the fundamental equation for the internal energy U. Here/is force of elongation, Z is length in the direction of the force, <7 is surface tension, As is surface area, , is the electric potential of the phase containing species i, qi is the contribution of species i to the electric charge of a phase, E is electric field strength, p is the electric dipole moment of the system, B is magnetic field strength (magnetic flux density), and m is the magnetic moment of the system. The dots indicate scalar products of vectors.

Matrix characterization of the radiative energy balance at each surface zone is facilitated via definition of three M X 1 vectors the radiative surface fluxes Q = [ i], with units of watts and the vectors H = [if,] and W = [Wi] both having units of W/m2. The arrays H and W define the incident and leaving flux densities, respectively, at each surface zone. The variable W is also referred to in the literature as the radiosity or exitance. Since W = el-E + pIH, the radiative flux at each surface zone is also defined in terms of E, II, and W by three equivalent matrix relations, namely,... 

The flux J of a quantity y> such as the momentum or the kinetic energy is calculated as follows If the flux is to be across the arbitrary surface element dS, the center of the first molecule must lie in the parallelopiped surface element, and the vector k must intersect the surface element. The flux in the direction of positive n is then given by... 

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