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Multi-scale modelling of membrane distillation: some theoretical considerations Robert William Field, Ho Yan Wu, and Jun Jie Wu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie302363e • Publication Date (Web): 06 Feb 2013 Downloaded from http://pubs.acs.org on February 19, 2013
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Industrial & Engineering Chemistry Research
Multi-scale modelling of membrane distillation: some theoretical considerations Robert W Field1,* Ho Yan Wu1 and Jun Jie Wu2
1 Department of Engineering Science, University of Oxford, Oxford, OX1 3 PJ, UK 2 School of Engineering and Computing Sciences, Durham University, DH1 3LE, UK
Prepared for Special Issue to honor Prof. Giulio Sarti - I&EC Research Abstract Firstly it is shown that the effective thickness of the membrane is the sum of the actual thickness, k0/UL and ⁄ where k0 is the thermal conductivity of the membrane matrix, λ is the latent of vaporization of water, C is a parameter (defined as flux per unit thickness of membrane per unit of temperature driving force) and UL is a coefficient combining the feed side and permeate side film heat transfer coefficients. For typical conditions the sum of the additional terms exceeds 100µm which clearly shows that the flux is not inversely proportional to membrane thickness. Also to a first approximation the thermal efficiency is independent of membrane thickness. This work and the development of an overall mass transfer coefficient for direct contact membrane distillation build upon the pioneering work of Giulio Sarti. Secondly a re-assessment of the traditional method for combining the Knudsen diffusion coefficient and the molecular diffusion coefficient suggests that the traditional sum of resistances approach engages in some double counting and thereby overestimates the resistance and consequently underestimates the flux.
Introduction Writing in the mid 1990’s, Lawson and Lloyd1 noted that many researchers were devoting their efforts towards determining new applications for membrane distillation (MD) in areas where the benefits of MD, especially the high rejection and low operating temperatures, made the process attractive. They drew a contrast with desalination applications. Now whilst MD is currently not a viable alternative to RO it is certainly viewed as a potential alternative especially in conjunction with solar energy and/or for decentralized treatment of brackish waters. This renewed interest is reflected in the developing literature including two recent reviews2,3. With this renewal of interest, it is timely to address the following questions: (i) What is the appropriate expression for the overall mass transfer coefficient, Keff from bulk feed to bulk permeate? (ii) Is the value of Keff useful for module design? (iii) At the pore scale, what is the correct expression for diffusion in the transition regime where both Knudsen diffusion and molecular diffusion are important? So firstly there will be a re-examination of the linear model for MD of Gostoli, Sarti & Matulli4 and consideration given to thermal efficiency, i.e. whether a compromise has to be made between a thinner membrane favouring the desired high mass transport of vapour (but also a greater heat flux), and a thicker membrane limiting heat loss (but also reducing mass transport). The interesting question, “What is the
* Corresponding Author
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limiting flux as the membrane thickness tends to a small value of the order of 10um?” is also addressed towards the end of the paper. Prior to that, the second question is discussed. Thirdly the current method for combining the Knudsen diffusion coefficient and the molecular diffusion coefficient will be reassessed starting from basic kinetic theory. There are four main MD configurations namely, direct contact (DCMD), air gap, vacuum and sweeping gas MD. The first part of this paper is concerned with DCMD because of the desire to link to some of Sarti’s work4, because DCMD has the simplest design and because the method introduced can be extended. In DCMD, both sides of the membrane are in contact with liquid; the aqueous feed on one side and the liquid permeate on the other. The permeate flux across the membrane in any MD system depends not only on the properties of the membrane (including pore size, pore size distribution, porosity and thickness of the membrane)1,3 but also on the heat transfer coefficients and the thermal conductivity of the membrane. In the case of DCMD, the film heat transfer coefficients for both the feed side and the permeate side are important, as illustrated in Figure 1. An interesting question is: “What is the limiting flux as the membrane thickness tends to a value say one order of magnitude larger than the maximum pore diameter?” The answer will be provided later.
Development of an overall MD mass transfer coefficient, Keff Symbols are similar to those used previously4. Consider a small section of a module. There are three expressions for the heat flux. Respectively for DCMD: heat transfer across/through the membrane; to the membrane from the bulk feed; and from the membrane to the bulk permeate. The flux through the membrane is N. Q Q λ (1) Q h T T (2) Q h T T (3) Symbols have their usual meaning and are listed in the Nomenclature. One can allow for the salinity of the feed in terms of a correction to the temperature driving force and for sufficiently small temperature differences across the membrane, the driving force of the partial pressure difference in water vapour pressure can be linearly related to the difference in temperature, ∆ and ∆ . The effective driving force is4: ∆ ∆ , and the flux is given by:
∆ ∆ (4) where K is the flux of water vapour through the membrane per unit of temperature difference across the membrane. As it is desirable to relate the flux at a particular point in a module to the local difference in bulk temperatures, an overall MD mass transfer coefficient, Keff is defined.
∆! ∆! (5) where Keff is the flux of water vapour through the membrane per unit of temperature difference between the bulk liquids and ∆! is the corresponding local bulk temperature difference. For ease of exposition, salinity will be ignored (i.e. ∆! " 0 and two heat transfer coefficients will be defined:
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Industrial & Engineering Chemistry Research
% 1⁄' 1⁄' (
)* +
, -.
(6)
( * 0 /
(7)
The overall thermal conductivity ko depends upon the thermal conductivity of the polymer, the thermal conductivity of the vapour/gases in the pores of the membrane and the porosity of the membrane. Khayet3 mentions alternative expressions for calculating ko. Now from (2) and (3):
11⁄' 1⁄' ∆! ∆ i.e. 1 % ∆! ∆ (8)
In other works one might find written as ! ! . Also
-2
1
∆
,
(9)
Combine (8) and (9) to obtain an equation in ∆! and ∆: -
) ,. 6 0 ∆ 6 ∆7
From (4) and (5) with ∆T89: ∆T 9: 0,
∆
Combining (10) and (11) and eliminating ∆T8,
∆!
-
(11)
∆>
∆ ;>Λ
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(b) d≈5Λ
(c) d>Λ, (b) d≈5Λ, (c) d
