Influence of Residence-Time Distribution on a Surface-Renewal Model of Constant-Pressure Cross-Flow Microfiltration

Abstract

-This work examines the influence of the residence-time distribution (RTD) of surface elements on a model of cross-flow microfiltration that has been proposed recently (Hasan et al., 2013). Along with the RTD from the previous work (Case 1), two other RTD functions (Cases 2 and 3) are used to develop theoretical expressions for the permeate-flux decline and cake buildup in the filter as a function of process time. The three different RTDs correspond to three different startup conditions of the filtration process. The analytical expressions for the permeate flux, each of which contains three basic parameters (membrane resistance, specific cake resistance and rate of surface renewal), are fitted to experimental permeate flow rate data in the microfiltration of fermentation broths in laboratory-and pilot-scale units. All three expressions for the permeate flux fit the experimental data fairly well with average root-mean-square errors of 4.6% for Cases 1 and 2, and 4.2% for Case 3, respectively, which points towards the constructive nature of the model -a common feature of theoretical models used in science and engineering.

A frequency quantum interpretation of the surface renewal model of mass transfer

“…We note that, in the general equation for the unsteady-state age distribution provided by Chung et al [ 8 ] of which equation (2.15) is a special case, the form of the steady-state age distribution has to be known or assumed a priori in order to derive the specific form of the unsteady-state age distribution, unlike in this work. In a recent paper, Zhang & Chatterjee [ 41 ] used this age distribution in the derivation of analytical expressions for the permeate flux decline and cake build-up in constant pressure, crossflow microfiltration. It is once again observed that, as t p → ∞, equation (2.15) reduces to the steady-state Danckwerts age distribution.…”

“…This distribution can be thought of as being a combination of ideas contained in the distributions discussed earlier, and has also been used by Zhang & Chatterjee [ 41 ] to model permeate flux decline and cake build-up in constant pressure, crossflow microfiltration. For the benefit of the reader, we recapitulate their derivation.…”

A surface‐renewal model for constant flux cross‐flow microfiltration

“…As demonstrated by Zhang and Chatterjee, using different speculative hypotheses about the behavior of liquid elements on the membrane surface, which correspond to different startup conditions, different RTD functions [i.e., f ( t , t p )] can be derived. These can then be used in eqs.…”

“…and to develop expressions for the age‐averaged pressure drop across the cake and cake‐mass buildup. In this study, the Danckwerts distribution function will be used to represent the ages of surface elements, i.e., where S (assumed to be constant) is the rate of renewal of liquid elements at the membrane surface and is a hydrodynamic parameter. It increases with velocity of the main flow and can also be looked upon as a “scouring” term that represents the removal of deposited material from the membrane wall, which will depend upon the level of flow instability.…”

“…The surface-renewal concept has been used to theoretically model constant pressure, cross-flow microfiltration and ultrafiltration by a number of workers. [1][2][3][4][5][6][7][8][9] Compared to the film and boundary-layer models of cross-flow membrane filtration, the surface-renewal model has the potential to more realistically describe the transfer of dissolved/suspended solids due to random hydrodynamic impulses generated at the membrane-liquid interface, e.g., due to membrane roughness or by the use of spacers or turbulence promoters.…”