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time (min)

Fig. 27. Experimental results of a batch DCMD process in which an aqueous solution of NaCl is used as the feed.

4.3. Mathematical model of VMD

Recently, vacuum membrane distillation (VMD) has received much attention in concentrating the aqueous solution or removing trace amount of volatile organic species from water. The difference between VMD and DCMD lies in two aspects: how the driving force for mass transfer through the membrane is exerted and how the permeated vapor is condensed or removed. In DCMD, the driving force for mass transfer through the membrane is the vapor pressure difference of volatile component across the membrane, which is caused by the temperature difference between the feed and the permeate liquid. And the permeated vapor is condensed in the permeate liquid stream within the membrane module. In VMD, however, the process is driven by both the temperature difference and the total pressure difference between two sides of the membrane. The total pressure difference arises from the vacuum pulled by a vacuum pump in the cold chamber of membrane module. The permeated vapor is sucked out of the cold chamber by the vacuum pump and condensed in an external heat exchanger.

If hot water is used as the feed and flows in the direction tangent over the membrane, the temperature difference at two sides of the membrane brings about a water vapor pressure gradient within the membrane pores. This driving force produces a mass flux through the membrane, and in this case both Knudsen and molecule diffusion are involved in this process. However, since VMD generally operates at the total pressure of the order of 10 to 30 kPa which is usually below the vapor pressure of water, only trace amount of air exists in the membrane pores. Therefore, the mass transfer resistance caused by molecule-molecule collision can be neglected and the diffusion process within the pores is dominated by Knudsen diffusion. According to Eq. (13), the mass flux related to this mechanism is:

* 3 t V InRT A ^ y where VP a is the water vapor pressure gradient within the membrane pores. In addition to Knudsen diffusion, the total pressure difference arising from the vacuum causes a convective mass transport through the pores, known as Poiseuille flow. Becuase almost no air exists within membrane pores, the water vapor in the pores is pure gas and thus the total pressure difference acting as the driving forece for Poiseuille flow also is VP a- According to Eq. (11), the mass flux related to this mechanism is:

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