1 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION Molecular diffusion is a process by which random molecular motion moves any quantity down the 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 Molecular diffusion is a process by which random molecular motion moves any quantity down the 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. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
3 Molecular diffusion is a process by which random molecular motion moves any quantity down the 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. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
4 Molecular diffusion is a process by which random molecular motion moves any quantity down the 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. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
5 In the case below the dye is diffusing in the x 3 direction. Let c denote the concentration of dye. Note that c is a decreasing function of x 3, so that The diffusive flux of dye in the vertical direction is from high concentration to low concentration, which happens to be upward in this case. The simplest assumption we can make for diffusion is the linear Fickian form: where F D,con,3 denotes the diffusive flux of contaminant (in this case dye) in the x 3 direction, c x3x3 where D c denotes the kinematic molecular diffusivity of the contaminant. F D,con,3 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
6 The units of c are quantity/volume. For example, in the case of dissolved salt this would be kg/m 3, and in the case of heat it would be joules/m 3. The units of D c are thus These units happen to be the same as those of the kinematic viscosity of the fluid, i.e.. c x3x3 In the case of heat, D c is denoted as D h and F D,con,3 is denoted as F D,heat, 3. F D,con,3 The units of F D,con,3 should be quantity (crossing)/face area/time. In the case of dissolved salt, this would be kg/m 2 /s, and in the case of heat it would be joules/m 2 /s. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
7 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/m 3 ) is given as c p . Thus where k = c p D h 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. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
8 Some numbers for heat Heat in air kg/m 3 J/kg/ K N s/m 2 m 2 /s J/s/m/ K m 2 /s CC KK cpcp kDhDh Pr E E E E E E E E E E E E E E E E E E E E E E E E-01 Heat in water kg/m 3 J/kg/ K N s/m 2 m 2 /s J/s/m/ K m 2 /s CC KK cpcp kDhDh Pr E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 In the above relations denotes the dynamic viscosity of water. DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
9 Some values of D c and D h are given as follows. gases and vapors in air at 25 C = 1.54e-5 m 2 /s m 2 /s substanceDcDc Sc H2H2 7.12E CO E Ethyl alcohol1.19E Benzene8.80E dissolved solutes in water at 20 C = 1.004e-6 m 2 /s m 2 /s substanceDcDc Sc H2H2 5.13E E+02 O2O2 1.80E E+02 CO E E+02 N2N2 1.64E E+02 NaCl1.35E E+02 Glycerol7.20E E+03 Sucrose4.50E E+03 DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
10 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/m 3 /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, dA nini DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
11 But by the divergence theorem Thus the conservation equation becomes or since the volume is arbitrary, DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
12 Now So the conservation equation reduces to a convection-diffusion equation with a source term: If the fluid is incompressible, i.e. u i / x i = 0, the relation reduces to DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION
13 Special case of heat, for which c c p and D c D h, S S h or thus DIFFUSIVE FLUX, HEAT & CONTAMINANT CONSERVATION