DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION

1. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random molecular motion moves any quantity downthe concentration gradient, i.e. from high concentration to low concentration. Diffusion does not require flow, but it operates in the presence of flow. Consider the illustrated container of water. A dilute concentration of dye (molecules) is placed in the lower half of the container. In time, molecular action cause the dye-free fluid to mix with the dye-laden fluid, so that the concentration eventually becomes uniform.

2. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random molecular motion moves any quantity downthe concentration gradient, i.e. from high concentration to low concentration. Diffusion does not require flow, but it operates in the presence of flow. Consider the illustrated container of water. A dilute concentration of dye (molecules) is placed in the lower half of the container. In time, molecular action cause the dye-free fluid to mix with the dye-laden fluid, so that the concentration eventually becomes uniform.

3. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random molecular motion moves any quantity downthe concentration gradient, i.e. from high concentration to low concentration. Diffusion does not require flow, but it operates in the presence of flow. Consider the illustrated container of water. A dilute concentration of dye (molecules) is placed in the lower half of the container. In time, molecular action cause the dye-free fluid to mix with the dye-laden fluid, so that the concentration eventually becomes uniform.

4. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random molecular motion moves any quantity downthe concentration gradient, i.e. from high concentration to low concentration. Diffusion does not require flow, but it operates in the presence of flow. Consider the illustrated container of water. A dilute concentration of dye (molecules) is placed in the lower half of the container. In time, molecular action cause the dye-free fluid to mix with the dye-laden fluid, so that the concentration eventually becomes uniform.

5. FD,con,3 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION In the case below the dye is diffusing in the x3 direction. Let c denote the concentration of dye. Note that c is a decreasing function of x3, so that The diffusive flux of dye in the vertical direction isfrom high concentration to low concentration, which happens to be upward in this case. c The simplest assumption we can make for diffusion is the linear Fickian form: where FD,con,3 denotes the diffusive flux of contaminant (in this case dye) in the x3 direction, x3 where Dc denotes the kinematic molecular diffusivity of the contaminant.

6. FD,con,3 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION The units of c are quantity/volume. For example, in the case of dissolved salt this would be kg/m3, and in the case of heat it would be joules/m3. The units of FD,con,3 should be quantity (crossing)/face area/time. In the case of dissolved salt, this would be kg/m2/s, and in the case of heat it would be joules/m2/s. The units of Dc are thus c These units happen to be the same as those of the kinematic viscosity of the fluid, i.e. . x3 In the case of heat, Dc is denoted as Dh and FD,con,3 is denoted as FD,heat, 3.

7. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION The 3D generalization of the Fickian forms for diffusivity are where c is the concentration of the contaminant (quantity/volume). The concentration of heat per unit volume (Joules/m3) is given as cp. Thus where k = cpDh denotes the thermal conductivity. The dimensionless Prandtl number Pr and Schmidt number Sc are defined as This comparison is particularly useful because we will later identify the kinematic viscosity with the kinematic diffusivity of momentum.

8. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Some numbers for heat Heat in air Heat in water In the above relations  denotes the dynamic viscosity of water.

9. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Some values of Dc and Dh are given as follows.

10. ni dA DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Consider a control volume that is fixed in space, through which fluid can freely flow in and out. In words, the equation of conservation of contaminant is: /t(quantity of contaminant in control volume) = net inflow rate of contaminant in control volume + Net rate of production of contaminant in control volume Contaminant concentration is denoted as c (quantity/volume). Contaminant can be produced internally by e.g. a chemical reaction (that produces heat or some some species of molecule). Let S denote the rate of production of contaminant per unit volume per unit time (quantity/m3/s). Where S is negative it represents a sink (loss rate) rather then source (gain rate) of contaminant. The net inflow rate includes both convective and diffusive flux terms. Translating words into an equation,

11. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION But by the divergence theorem Thus the conservation equation becomes or since the volume is arbitrary,

12. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Now So the conservation equation reduces to a convection-diffusion equation with a source term: If the fluid is incompressible, i.e. ui/xi = 0, the relation reduces to

13. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Special case of heat, for which c  cp and Dc  Dh, S  Sh or thus