By Marcel Loncin
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Extra resources for Food engineering, principles and selected applications
If we move one plate with respect to the other establishing a constant velocity gradient with χ component dv /dz, the distance dx that the moving plate travels in a time dt is x dx = ^ Δζ dt dz The energy used is dQ = τA dx from which we derive dQ Vdt = τ- dv x dz or from Eq. t) This equation can be adapted to the case where the velocity gradients or the shear stresses are not constants, in particular to non-Newtonian fluids (Toor and Marchello, 1958). It should be noted that when the stresses are very important, changes of volume can have a notable influence on the temperature of the fluid.
4 21 Second Laws of Fick and Fourier the parallelopiped in time at are as follows: (1) for the vertical face, of area (dy dz) parallel t o yOz, and at abscissa x, -dydz ^p j^dt B (2) for the vertical face, parallel to yOz, and at abscissa χ 4- dx, — dy dz The difference between these two expressions gives the quantity of diffusing material accumulating in the parallelopiped because of diffusion in the χ direction: dx\ ° p dx) dVdt The volume dV of the parallelopiped has been substituted for the product (dx dy dz).
Equations Related to the Transfer of Mass, Heat, and Momentum During the derivation of Fick's first law, we considered that the concen trations were expressed as mass of solute Β per unit of total volume. The dimensions of the diffusivity @ were thus [l t~ It should be pointed out that in Fick's second equation the diffusivity is this same ^ a n d the dimen sions of the diffusivity are always [l t~ ] n o matter what units are used for the concentrations themselves, for example mass/volume (p ), mass fraction (x), mole fraction (X), mass of solute per mass of diluant (n), molarity, molality, partial pressure, etc.
Food engineering, principles and selected applications by Marcel Loncin