MASS TRANSPORT Macroscopic Balances for Multicomponent Systems

Transcription

1 TRANSPORT PHENOMENA MASS TRANSPORT Macroscopic Balances for Multicomponent Systems

2 Macroscopic Balances for Multicomponent Systems 1. The Macroscopic Mass Balance 2. The Macroscopic Momentum and Angular Momentum Balances 3. The Macroscopic Energy Balance 4. The Macroscopic Mechanical Energy Balance

3 Macroscopic Mass Balance

4 Macroscopic Mass Balance

5 Macroscopic Momentum and Angular Momentum Balances The macroscopic statements of the laws of conservation of momentum and angular momentum for a fluid mixture, with gravity as the only external force, are For most mass transfer processes these terms are so small that they can be safely negleted.

6 Macroscopic Energy Balance For a fluid mixture, the macroscopic statement of the law of conservation of energy is The Q term accounts for addition of energy to the system as a result of mass transfer. It may be of considerable importance, particularly if material is entering through the bounding surface at a much higher or lower temperature than that of the fluid inside the flow system, or if it reacts chemically in the system.

7 Macroscopic Mechanical Energy Balance The macroscopic statement of the law of conservation of mechanical energy is The additional term B 0 accounts for the mechanical energy transport across the mass transfer boundary.

8 Unsteady Operation of a Packed Column EXAMPLE There are many industrially important processes in which mass transfer takes place between a fluid and a granular porous solid. In this operation, a solution containing a single solute A at mole fraction x A1 in a solvent B is passed at a constant volumetric flow rate w/ρ through a packed tower. The tower packing consist of a granular solid capable of adsorbing A from the solution. At the start of the percolation, the interstices of the bed are filled with pure liquid B, and the solid is free of A. he percolating fluid displaces this solvent evenly so that the solution concentration of A is always uniform over any cross section. Problem: Develop an expression for the concentration of A in the column as a function of time and of distance down the column. A fixed-bed absorber (a) pictorial representation of equipment; (b) a typical effluent curve.

9 Unsteady Operation of a Packed Column EXAMPLE Solution. We think of the two phases as being continuous and existing side by side. We define the contact area per unit packed volume of column as a. Now, however, one of the phases is stationary, and unsteady-state conditions prevail. Because of this locally unsteady behavior, the macroscopic mass balances are applied locally over a small column increment of height Δz. We may use macroscopic mass balance in molar units and the assumption of dilute solutions to state that the molar rate of flow of solvent, W B, is essentially constant over the length of the column and the time of operation.

10 Unsteady Operation of a Packed Column We now proceed to use macroscopic mass balance to write the mass conservation relations for species A in each phase for a column increment of height Δz: The symbols have the following meaning: ε volume fraction of column occupied by the liquid S cross-section area of column c As phase x A mole of adsorbed A per unit volume of the solid bulk mole fraction of A in the liquid phase x A0 interfacial mole fraction of A in the fluid phase, assumed to be in equilibrium with c As fluid-phase mass transfer coefficient k 0 x

11 Unsteady Operation of a Packed Column For the fluid phase, in the column increment under consideration, becomes Here с is the total molar concentration of the liquid. Equation above may be rewritten by the introduction of a modified time variable, defined by It may be seen that, for any position in the column, t' is the time measured from the instant that the percolating solvent "front" has reached the position in question. After substitution we get:

12 Unsteady Operation of a Packed Column The two equations are to be solved simultaneously along with the interphase equilibrium distribution, x A0 = mc As, in which m is a constant. The boundary conditions are: Before solving these equations, it is convenient to rewrite them in terms of the following dimensionless variables:

13 Unsteady Operation of a Packed Column In terms of these variables, the differential equations and boundary conditions take the form: with the boundary conditions Y(ζ, 0) = 0 and X(0, τ) = 1. The solution to above equations for these boundary conditions is: