mass transfer in multiphase systems - Greenleaf University

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MASS TRANSFER IN MULTIPHASE SYSTEMS: VOLATILE ORGANIC COMPOUND REMOVAL IN THREE-PHASE SYSTEMS This appendix shows how the transport equations (conservation of mass used for illustration) are the same regardless of the observer. The basic development (Bird 1960) is that there are three types of concentration derivatives: As a fixed observer of flow quantifying the concentration of some quantity of mass in a stream. For this, it is simply C/t, the partial of C with respect to t holding x, y, and z constant. As a random moving observer in the stream, the derivatives must include the motion: dC C C dx C dy C dz = + + + dt t x dt y dt z dt (83) As an observer flowing with the stream, the substantial derivative is as follows: DC C C C C = + v + v + v x y z Dt t x y z (84) The substantial derivative for a moving body with the flow is explained in reference to the relations for a fixed position in the following. Extensive development and analysis is used from the masterful work by Anderson in computational fluid dynamics (CFD). Similar analysis below and many other mathematical tools are available in (Anderson 1995). Conservation of mass For a fluid particle moving between 2 points, a Taylor series provides 2 1 ( x2 x1) ............ x t (85) Dividing by (t 2 -t 1) D t t v ...... t t x t Dt lim 2 1 2 1 2 1 (86) The substantial derivative is shown below in operator form: D Dt v t (87) f f ( x, y, z, t) (88) Any function f can be shown using calculus of several variables, e.g., df f f dx f dy f dz dt t x dt y dt z dt (89) 51
MASS TRANSFER IN MULTIPHASE SYSTEMS: VOLATILE ORGANIC COMPOUND REMOVAL IN THREE-PHASE SYSTEMS Divergence V vtndS vtdS dV vtdS (90) (91) DV Dt vdV (92) Shrinking the control volume down to δV: D V Dt V vdV (93) Assume δV is small enough that so that the divergence doesn’t change (i.e., it becomes a constant if δV is small enough and therefore comes outside the integral): D V Dt v V (94) The divergence is the volume rate of change per unit volume of a moving fluid element, i.e.: 1 D V V Dt v (95) Case I, Control Volume Fixed The net amount leaving the volume element = the rate of mass decrease 52
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MASS TRANSFER IN MULTIPHASE SYSTEMS
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Samuel Clay Ashworth ALL RIGHTS RES
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CURRICULUM VITAE Samuel C. Ashworth
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model results using Polymath, Excel
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Lead process engineer and assistant
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Determination of Viable Processes f
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ACKNOWLEDGMENT The work within was
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5.0 INTERPRETATIONS, CONCLUSIONS, A
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NOMENCLATURE a a,b, etc. Parameter
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Sh v Sherwood number Velocity, L/t
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mass transfer in multiphase systems - Greenleaf University

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