Mass Transfer Enhancement in Double‐Pass,Parallel‐Plate Mass Exchangers under Asymmetric Wall Fluxes

A device involving mass exchangers with a permeable membrane inserted to divide the open duct into two subchannels for conducting double‐pass laminar countercurrent operations under asymmetric wall mass fluxes, which results in a substantial improvement of the mass transfer, has been developed and investigated theoretically. The resultant partial differential equations for such a double‐pass forced‐convection mass transfer problem are referred to as conjugated Graetz problems and solved analytically in this work by using an eigenfunction expansion in terms of power series for the homogeneous part and an asymptotic solution for the inhomogeneous part. The theoretical predictions of mass transfer efficiency improvement in double‐pass, parallel‐plate mass exchangers are obtained by suitably adjusting the permeable membrane location. They are represented graphically and compared with the results from an open duct of a single‐pass operation (without a permeable membrane inserted). The increment of power consumption is also discussed.     

Device Performance Improvement of Double‐Pass Concentric Circular Mass Exchangers under Uniform Wall Fluxes

A double‐pass concentric circular mass exchanger under uniform wall fluxes is produced by inserting a permeable barrier into a circular tube to improve the device performance. The mathematical formulation was developed theoretically for such double‐pass, forced‐convection, mass‐transfer problems which are referred to as conjugated Graetz problems. The analytical solutions are obtained by the linear superposition of an asymptotic solution and a homogeneous solution which are linear in the axial direction and solved with the use of an eigenfunction expansion in a power series. The analytical results show that the mass‐transfer rate of a double‐pass mass exchanger can be improved compared to that of a single‐pass mass exchanger by suitably adjusting the permeable barrier position. Moreover, the ratio of mass‐transfer efficiency improvement and power consumption increment is also shown to make good economic sense.     

Heat Transfer Phenomena of Power Law Fluids in Double‐Pass Heat Exchangers with Isoflux Conditions

A new design of conjugated heat transfer in double‐pass parallel‐plate laminar countercurrent operations of power law fluids under wall isoflux was investigated experimentally and theoretically. The analytical solutions were obtained with a superposition model by introducing an eigenfunction expansion in terms of a power series for the homogeneous part and an asymptotic solution for the inhomogeneous part. The influence of the power law index on the average Nusselt numbers with the various design and operating parameters is also delineated. The theoretical predictions of the experimental results are represented graphically. The heat transfer performance was considerably improved when compared with a single‐pass parallel‐plate heat exchanger (without inserting a solid separator sheet). Suitable adjustments of the solid separator sheet position can effectively enhance the heat transfer efficiencies for such a recycling double‐pass device, as compared with the efficiencies of single‐ and double‐pass devices.     

A Theoretical Study of the Recycle Effect on Multi‐Pass Mass Exchangers with Three Idealized Membranes Inserted

A new device of multi‐pass mass exchanger was developed by inserting three idealized membranes to divide an open conduit into four subchannels. The mathematical formulations of the device with external recycle, as referred to a well‐known conjugated Graetz problem, were conducted to improve the mass transfer efficiency. The eigenfunction expansion technique with the eigenfunction expanding in terms of an extended power series was used to solve such a conjugated Graetz problem analytically. The theoretical predictions were presented graphically and compared with those in the single‐pass operations (without any idealized membrane inserted and external recycle) of the same working dimensions.     

Recycle effect on membrane extraction through cross-flow rectangular modules

The effects of recycle on membrane extraction through a cross-flow rectangular module have been studied both theoretically and experimentally. Theoretical analysis of mass transfer in cross-flow device with and without recycle is analogous to heat transfer in cross-flow heat exchangers. Experiments were carried out with the use of a membrane sheet made of microporous polypropylene as a permeable barrier to extract acetic acid from aqueous solution by methyl isobutyl ketone. Theoretical predictions are in agreement with the experimental results. In contrast to a device without recycle, improvement in mass transfer is achievable if cross-flow membrane extraction is operated in a device of same size with recycle which provides the increase of fluid velocity, resulting in reduction of mass-transfer resistance. It is shown that recycle can enhance mass transfer, especially for large feed-concentration solution with high distribution coefficient operated under high reflux ratio.     

MASS-TRANSFER FLOW IN A PARALLEL-PLATE CHANNEL WITH PERMEABLE BARRIERS INSERTED FOR COUNTERCURRENT MULTI-PASS OPERATIONS

The mass-transfer flow of a multi-pass counterflow laminar mass exchanger by inserting three permeable barriers with external refluxes was investigated theoretically, and the mathematical formulation was developed as well. With the use of an orthogonal expansion technique and the eigenfunction expanding in power series, the analytical solutions were obtained for finding the outlet concentration and mass transfer efficiency. The influences of the design parameters, the channel thickness ratio (β) and barrier characteristics (γ), and the operating parameters, the mass-transfer Graetz number (Gz _m ) and recycle ratio (R), are examined. Theoretical predictions of the new multi-pass operations are compared with those in the single- (without the permeable barrier inserted) and double-pass operations under the same working dimensions and represented graphically. The results show that the influences of the recycle ratio and channel thickness ratio in the present device with external refluxes are dominant in low and high mass-transfer Graetz number regions, respectively. Therefore, an optimal economic feasibility of multi-pass mass exchangers was found when the recycle ratio and channel thickness ratio are suitably selected under the operating conditions of interest.     

MASS-TRANSFER FLOW IN A PARALLEL-PLATE CHANNEL WITH PERMEABLE BARRIERS INSERTED FOR COUNTERCURRENT MULTI-PASS OPERATIONS

The mass-transfer flow of a multi-pass counterflow laminar mass exchanger by inserting three permeable barriers with external refluxes was investigated theoretically, and the mathematical formulation was developed as well. With the use of an orthogonal expansion technique and the eigenfunction expanding in power series, the analytical solutions were obtained for finding the outlet concentration and mass transfer efficiency. The influences of the design parameters, the channel thickness ratio (β) and barrier characteristics (γ), and the operating parameters, the mass-transfer Graetz number (Gz_m) and recycle ratio (R), are examined. Theoretical predictions of the new multi-pass operations are compared with those in the single- (without the permeable barrier inserted) and double-pass operations under the same working dimensions and represented graphically. The results show that the influences of the recycle ratio and channel thickness ratio in the present device with external refluxes are dominant in low and high mass-transfer Graetz number regions, respectively. Therefore, an optimal economic feasibility of multi-pass mass exchangers was found when the recycle ratio and channel thickness ratio are suitably selected under the operating conditions of interest.     

MASS TRANSFER ENHANCEMENT OF CONJUGATED GRAETZ PROBLEMS IN MULTI-PASS PARALLEL-PLATE MASS EXCHANGERS WITH EXTERNAL RECYCLE

A new external recycle design at the ends of multi-pass parallel plate mass exchangers with uniform wall concentration is theoretically investigated in this study. The analytical solutions were obtained using the orthogonal expansion technique and the eigenfunction expanding in power series. The results are graphically represented. The influences of the recycle ratio, mass-transfer Graetz number, channel thickness ratio, and characteristic parameter of the permeable barrier on the mass transfer rate are presented in this study. Compared to the single-pass device, considerable mass transfer improvement is obtainable by employing the new external recycle design at both ends.     

ANALYSIS OF DIALYSIS IN CROSS-FLOW PARALLEL-PLATE MEMBRANE MODULES WITH FEED-STREAM RECYCLE FOR IMPROVED PERFORMANCE

The effect of external recycle on the performance of dialysis in cross-flow rectangular membrane modules with feed-stream recycle was investigated theoretically. The analysis of mass transfer in parallel-plate devices with and without recycle is analogous to heat transfer in parallel-plate heat exchangers. In contrast to a device without recycle, improvement in mass transfer is achievable if cross-flow parallel-plate dialysis is operated in a device of the same size with recycle, which provides increased fluid velocity, resulting in reduction of the mass-transfer resistance.     

THE EFFECTS OF RECYCLE ON LAMINAR COUNTERFLOW DOUBLE-PASS MASS EXCHANGERS

A device of mass exchangers with inserting an impermeable sheet or a permeable barrier to divide an open duct into two channels with uniform wall concentration and external refluxes at the ends, resulting in substantially improving the mass transfer, has been designed and investigated theoretically. Considerable improvement in mass transfer is obtainable by employing such double-pass devices with inserting an impermeable sheet and a permeable barrier, instead of using an open conduit with single-pass operations. The effects of reflux ratio and permeable-barrier (or impermeable-sheet) location on the enhancement of transfer e ciency as well as on the increment of power consumption, has been also discussed. The results are represented graphically and compared with those in an open duct (without a permeable barrier or an impermeable sheet inserted and thus with single-flow operations).     

The effects of recycle on laminar counterflow double-pass mass exchangers

A device of mass exchangers with inserting an impermeable sheet or a permeable barrier to divide an open duct into two channels with uniform wall concentration and external refluxes at the ends, resulting in substantially improving the mass transfer, has been designed and investigated theoretically. Considerable improvement in mass transfer is obtainable by employing such double-pass devices with inserting an impermeable sheet and a permeable barrier, instead of using an open conduit with single-pass operations. The effects of reflux ratio and permeable-barrier (or impermeable-sheet) location on the enhancement of transfer e ciency as well as on the increment of power consumption, has been also discussed. The results are represented graphically and compared with those in an open duct (without a permeable barrier or an impermeable sheet inserted and thus with single-flow operations).     

A study of mass transfer efficiency in a parallel-plate channel with external refluxes

The phenomenon of mass transfer through a parallel-plate channel with uniform wall concentration and external refluxes has been investigated by use of an orthogonal calculation technique. Considerable improvement is achieved when the external refluxes and barrier position are suitably adjusted. Analytical results show that recycle can enhance the mass transfer efficiency for high inlet flow rate compared with that in a single-pass device (without a permeable barrier inserted).     

Effect of External Recycle on Mass Transfer in Countercurrent Double-Pass Cross-Flow Rectangular Dialyzers

A study on mass transfer in countercurrent cross-flow rectangular dialyzers with external recycle was investigated. Theoretical analysis of mass transfer in cross-flow dialyzers is analogous to heat transfer in cross-flow heat exchangers. In contrast to a single-pass cross-flow device without recycle, improvement in mass transfer of about 30% is achievable if a double-pass device of the same size with external recycle is employed, which provides the increase of fluid velocity, resulting in increase of mass-transfer coefficient. Higher effectiveness may be expected when dialysis is operated for the system in which the mass-transfer resistances in the liquid films are comparable with that in the membrane.     

Application of internal reflux in the raffinate phase for membrane extraction in cross-flow rectangular modules

The effect of internal reflux in the raffinate phase for membrane extraction through a cross-flow rectangular module was investigated. Theoretical analysis of mass transfer in cross-flow devices with or without recycle was analogous to heat transfer in cross-flow heat exchangers. In contrast to a device without reflux, considerable improvement in mass transfer is achievable if cross-flow membrane extractors are operated with reflux which provides the increase of fluid velocity, resulting in reduction of mass-transfer resistance. It is concluded that reflux can enhance mass transfer, especially for large flow rate and feed-concentration solution operated under high reflux ratio.     

Modified correction‐factor analysis for mass transfer in double‐pass rectangular membrane modules

Mass transfer in mass exchangers, such as membrane extractors and dialysers, can be analogous to heat transfer in heat exchangers with correction‐factor analysis. In present study, the expressions of mass‐transfer rate and correction‐factor charts for mass transfer in parallel‐flow double‐pass rectangular membrane modules were modified by the modified correction‐factor analysis, instead of the conventional correction‐factor analysis. They are explicit, instead of implicit in the conventional correction‐factor analysis. The results can be readily calculated without using trial‐and‐error method. Experimental results confirm the predicted values for membrane extraction of acetic acid from aqueous solution by methyl isobutyl ketone, as well as for dialysing from aqueous solution, in the double‐pass devices. © 2012 Canadian Society for Chemical Engineering     

Simplified MATLAB Solution Schemes of Heat Transfer Equations

A simplified generalized finite difference solution using MATLAB has been developed for steady‐state heat transfer in a bar, slab, cylinder, and sphere. Solutions are given for all types of boundary conditions: temperature and flux boundary conditions. A generalized solution for 2D heat transfer in a slab is also developed. Solutions have been compared with analytical solutions and there is very good agreement between computed and analytical solutions. The solution can easily be extended to similar types of situations in fluid flow and mass transport operations.     

On the examination of recycle on heat (and mass) transfer in concentric tubes

The effect of recycle on heat (and mass) transfer in concentric circular tubes has been investigated by finite difference methods. Theoretical results show that recycle can enhance the heat transfer rate for large Graetz numbers compared with that in an open tube (without an inner tube inserted and without recycle). Competition between a preheating effect and a residence-time effect can be used to explain the heat transfer behavior. The heat transfer rate can be further augmented, with nearly no increase in total pressure drop, by employing flow pattern B instead of flow pattern A.     

EXTERNAL RECYCLE ON UPWARD-TYPE DOUBLE-PASS SOLAR AIR HEATERS

The external recycle effect on collector efficiency in an upward-type double-pass flat-plate solar air heater is investigated theoretically. Application of recycle to a solar air heater has two conflicting effects: the desirable effect of increasing the heat transfer coefficient and the undesirable effect of decreasing the heat transfer driving force (temperature difference). The theoretical predictions show that more than 150% improvement in collector efficiency is obtainable if external recycling is carried out in a way that the advantage effect compensates for the disadvantage effect. The collector performance improvement increases with increasing recycling ratio, especially when operating at a lower airflow rate with higher inlet air temperature. The collector efficiency obtained with a double-pass device with recycling improves the efficiency obtained with a single-pass device of the same size with recycling.     

An analytical study of heat-transfer efficiency in laminar counterflow concentric circular tubes with external refluxes

The mathematical formulation of a new device of counterflow heaters or coolers in concentric circular tubes with uniform wall temperature and external refluxes is developed and the analytical solution has been investigated with the use of an orthogonal expansion technique by expanding eigenfunctions in terms of power series.Comparisons of enhancement in heat-transfer efficiency are made with single-pass operations of the same size (without an inner tube inserted). Considerable improvement in the heat-transfer efficiency for large Graetz numbers is obtainable by introducing the recycle-effect concept in designing such double-pass operations. Analytical results show that suitable adjustment of impermeable-sheet location could have a substantially improving the heat-transfer efficiency. The increment of power consumption has been also discussed.     

Membrane extraction in rectangular modules with external recycle

The analysis of solvent extraction in the membrane modules of cocurrent, countercurrent and cross-flows with external recycle, has been carried out by mass-transfer theory, analogous to heat transfer in recycled heat exchangers. Considerable improvement is achievable by recycle operation if the increase in mass-transfer coefficient by applying the recycle effect can compensate for the decrease in the driving force of mass transfer due to the dilution of inlet feed solution. Under comparable conditions most solute is extracted in countercurrent-flow arrangement, the least in cocurrent-flow. However, the separation efficiencies of the three flow-type devices approach in unity for lower feed concentration and/or larger flow rate and/or larger reflux ratio.