Applications of Mathematical Heat Transfer and Fluid Flow Models in Engineering and Medicine by Dorfman Abram S.;

Author:Dorfman, Abram S.; [Dorfman, Abram S.]
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2016-11-23T00:00:00+00:00

The first and second equations (4.89) present the balance of heat and mass (moisture) transfer, respectively, for porous body. The left-hand side of these equations defines the quantity of heat or mass of the body element that changes during the unit of time. This change occurs due to conduction in x and y direction of heat introduced by two first terms in the right-hand side of the first equation and due to mass diffusion represented by the terms containing the derivatives and in the right-hand part of the second equation. The additional terms determine: the heat associated with evaporation caused by changing the moisture content (the last term in the first equation) and the mass change induced by thermal diffusion (two terms with derivatives and in the second equation). Estimation of the magnitude of terms in equations (4.89) shows that the terms determining the heat and mass transfer in the direction are relatively small and these may be neglected. Then, the previous equations and the usual initial and symmetry conditions take the form

4.90

The conjugate conditions consist of four equations. Three of these are equalities of temperatures, vapor densities and mass fluxes defined on the interface from coolant (+) and body (−) sides. The forth condition is the balance on the interface: the difference between heats incoming from coolant and absorbing by material (two terms on the left-hand side) is used for evaporation (the right-hand side part)

4.91

The temperatures and are determined from the boundary layer energy equation (1.11) and system (4.90), respectively. The vapor density at the material surface from coolant side is defined from equation for vapor concentration at the surface and two relations and gained by considering the air and vapor as ideal gases. Since these three equations consist of three unknown , , and , simple algebra yields a relation for desired density

4.92

To determine the vapor density at the surface from body side, the equation of desorption isotherm (Com. 4.26) should be used, where is the saturated vapor density. Substituting from the first equation (4.92) and from the equation of desorption isotherm gives the conjugate condition in the form of the second expression (4.92) that relates the temperature and moisture content at the material surface to relative density of vapor at the surface .

The heat flux at the surface from coolant side is defined as a difference between the incoming coolant heat (the first term) and the heat being carrying away by transverse vapor flux that is found as a sum of diffusion and convective fluxes

4.93

where is the vapor enthalpy at the surface. The last result (4.93) is obtained after substitution of expression (4.88) for the transverse velocity and taken into account that . Similarly defined is the heat flux at the surface from a body side that is also found as a sum of the conductive heat and heat taken by flux of vapor. The last one that comes across a body consists of the diffusion and thermal diffusion fluxes

4.94

Substituting vapor fluxes and and heats and into two last equations (4.

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