Role of heat and mass transfer in membrane distillation process

Desubnatron,81 (lI991) 91-106

Elsevler Science Publishers B.V., Amsterdam

S. Bandini,

C. Gostoli,

G.C.

Sarti

Dipartimento

di Ingegneria Chimica e di Process0 Universi tri di Bologna V. Risorgimento 2, I-40136 Bologna, Italy.

ABSTRACT Membrane Distillation is a separation process based on the evaporation through porous hydrophobic membranes. The membrane plays the role of physical support for the vapor-liquid interface. The process is characterized by simultaneous heat and mass transfer . In this work a simple criterion is presented to predict whether the overall permeation rate is mass or heat transfer controlled, simply based on the knowledge of the physical properties and of the transport coefficients of each intervening phase. The influence of the membrane properties and of the operating conditions on mass transfer rate and energy efficiency is discussed in some details. IW7?OWCTION Fresh water from sea water or salt solutions can be obtained by membrane distillation, a separation process based on the use of porous hydrophobic membranes in which a temperature difference is the driving force for mass transfer. In Various different process configurations can be considered. Direct Contact Membrane Distillation (DCMD) Cl-53 the external surfaces of the membrane are in direct contact with two liquid phases, kept at the feed and the permeate, different temperatures. A liquid-vapor interface is thus located at the pore entrances, where liquid vapor equilibria are estabilished; inside the pores only a gaseous phase is present - also called gas membrane - through which vapors are transported as long as a partial pressure difference is maintained. Different MD techniques have been developed: Gas Gap or Air Gap MD [6-83, Vacuum MD 19-101, Sweeping Gas MD [ill; in all cases the membrane plays the role of physical support for a vapor-liquid interface. 91

In this work only DCHD is considered, and thus the relevant resistances determining the transnembrane flux are physical associated to both heat and mass transfer through the membrane well as to heat transfer through the two liquid phases in direct contact with the membrane itself. Wide experimental results on DCMD have been pub1 ished, as different mathematical well as slightly models : qembrane conductivity, characteristics (thermal thickness, void and operating conditions as average temperature and fraction), in the two liquid phases greatly the fluidodynamics influence and satisfactory analysis evaporation rate. However, a detailed is still needed, based on which one could a priori know, for any what is the relative importance of operating conditions, heat and mass transfer resistances on the transmerbrane flux without the need of explicitly solving the model equations. From a practical point of view it is rather important to estabilish what actions should be undertaken in order to obtain significant improvements for the flux of a given DCMD system. In particular, one needs to decide first whether more effective results can be obtained by increasing the external heat transfer or rather by changing the membrane itself; case in the latter one has to recognize what membrane properties are required in void fraction, terms of thermal conductivity, tortuosity and thickness ; needless to say the hydrophobicity requirement is Therefore we are faced with the considered to be satisfied. problem of determining whether all resistances are equally important or else a prevailing resistance controls the overall review permeation rate. To that aim it is necessary to first the mathematical descriptions for the permeation rate in DCMD.

THEORY

OF DXRECT

CONTACT

MEMBRANE

DISTILLATION

Direct contact membrane distillation is actually a diffusion process through a gas membrane, which is the stagnant gas phase immobilized within the membrane pores. In the case of aqueous simply the diffusion of water vapor salt solutions through a stagnant film of air is taking place. Assuming that molecular diffusion is the prevailing mechanism of mass transport, the water mass flux is given by C81: #=ME

X

and

Pa

T”-’

Tb

-ii-

Qn

P - Ptrz P - PiI!1

The dependance of the diffusion coefficient temperature is explicitly included in

3 on pressure the theory as

93 P a/Tb=const., with b=2 111; generally speaking, an effective diffusivity should be used in Eq. (1). in order to account for both molecular and Knudsen mechanisms for diffusion through the The analysis here presented remains unaltered membrane pores. coefficient or an whether 0 represents the molecular diffusion effective diffusivity. linearized For dilute solutions and small DT Eq.(l) can be law and of the Clausius-Clapeyron by making use of Raoult’s equation in linear form; the resulting linear expression for the mass flux is:

(2)

N=K(I;TT-lTT”j in which D-f0 =-

R-P MAR

Dx

(3)

l-J&

is the boiling point elevation mass transfer coefficient given

and K is by:

a sort

of thermoosmotic

M Am PT P- PL(l

- xal

(41

The above linearized model neglects only terrs which are third order in (DT/T.) and leads to vanishing errors for the temperature differences across the membrane up to 15-C 183; the linear model can be used with confidence for DCMD, since, in view of the small membrane thickness, the temperature difference DT is rather small. A heat flux parallels the mass flux through the membrane given by: NA

+ha1;IT=UL(mb

-tn)

(5)

in which hr=kr/B and lk=(l/hr + l/hi)-~ are the heat transfer coefficient of the membrane and of the 1 iquid layers respectively; as a consequence of heat transfer the actual temperature difference DT across the membrane is reduced with respect to the bulk temperature difference DTs; the ratio t = DT/M’b has been called temperature polarization coefficient c33. Finally, the explicit dependance of mass flux on the bulk temperature difference can be introduced. By making use of Eqs.(P) and (5) the water flux can be expressed as:

94

N= A positive

K UL lh+h+KA

flux

(mb

is obtained

-

hm

+ UL

UL DPI

(61

for

Apparently, owing to the heat transfer resistance through the 1 iquid phases, in order to have a water flux out of a salt the bulk temperature difference to be overcome is solution, larger than the boiling point elevation.

THE

ROLE

OF #EAT

AND hi&S

TRANSER

The evaporation rate appears to be influenced by heat and mass transfer properties of the membrane, embodied by the coefficients K and h., as well as by the heat transfer process through the liquid phases as described by the overall 1iquid heat transfer coefficient UL. our main aim is to derive a simple In the present work, criterion to deterrine what is the rate controlling resistance dimensionless of the DCMDprocess, simply based on relevant the need of either solving the transport nuzabers, without equations which describe the process or else without the need of some prior experimental tests. some preliminary observations are in order. To that extent, First of all no real conclusion can be drawn simply based on the temperature polarization coefficient, indeed a value of T below unity does not necessarily indicate a heat transfer control. As a clearcut example, consider the case of a partition with high thermal conductivity but a negligibly small void fraction: based on the value of r, which may be much below unity, the could be drawn that the process wrong conclusion rate is exclusively controlled by heat transfer while instead it is self evident that the permeation rate is very sensitive to the mass transfer resistance of the membrane. Finally, we notice that in the present case the very usual problem of the relative importance of the determining intervening resistances has some remarkably new features, insofar as there is a mass transfer resistance through the membrane which is in series with the heat transfer resistance and in parallel with the heat through the liquid phases, transfer resistance through the membrane. The relevant resistances are encountered in three different phases in series and in addition they are not dimensionally homogeneous; thus one

apply the well known procedure used for series of cannot On the other hand the simultaneous homogeneous resistances. heat and mass transfer resistances which are considered in the wet bulb theory both refer to a single fluid phase, which is not the case here. the problem from a more Therefore, we need to approach The relative importance of the fundamental point of view. can be albeight very different in nature, various resistances, properly determined by considering their respective sensitivity factors on the distillate flux, namely what percentage change in the overall mass flux is obtained as a consequence of a unit Thus, in order to percentage change of any single resistance. hm and K we are led to the compare the flux dependance upon UL, following definitions for the sensitivity factors:

eu =

M/N dUL/UL

9h =

M/N clhr/hr

on =

WN WK

I= I“‘,i I= K,km

UL,hn

d&N

aaz uL d&N

akhh, dezN a&K

(91

(101

All the quantities ii are the proper dimensionless numbers needed in order to determine the relative weight of all the single transport stages and to recognize what is, if there is any, the controlling resistance. As an example, in the case of very fast heat transfer through the liquid phases (UL-m), the effective temperature difference DT approaches the bulk temperature difference DTb and, according the flux linearly increases with K. Correspondingly to Kq.(2), $14 1, or in other words the process rate is completely controlled by the mass transfer resistance through the merbrane. On the other hand, for low UL values DT is greatly reduced with respect to DTb, a substantial increase of mass flux can be achieved by increasing the ratio DT/DTb, which depends on both UL and h.. The influence of UL and hr is described by %u and Ph respectively. It is worth noting that clu ~0 and )h
96 Suitable expressions for Bi can be easily obtained from (6) in the following form: 1 4iJ = 1

Rf

4

1

R2

4

R2

4

+

RI/Rz

OJTbm”

1

4

&

RI

On =

1 4

-

1) -I

-

1)

(111

1

- R2 Oh =

Eq.

R2

4

RI/Rz

-

(mb/lrr’

-I

(12)

R2

4

RI

(13)

R2

4

two of the above #i's Apparently, only independent froa each other insofar as relationship holds: ou

Ok

4

4

4w

=

1

parameters are the following

(14)

although single terms may be significantly out of the range 09. Three dimensionless numbers enter the analysis: the ratio IYTa/JYl'O between the bulk temperature difference and the boiling point elevation, and Ri and R2 defined as follows: Rl

=-

UL

KA

R2 = $-

i15)

In a qualitative way we can say that RI and R2 properly compare the membrane mass transer rate with the heat transfer rate in the external phases and in the membrane respectively. While RI compares transport properties of the membrane with properties of the external liquid phases, R2 involves only properties of the membrane and is independent of membrane thickness.

The DCMD of pure water is a special case considered as a reference situation, although it may be directly of interest for As it will be the concentration of aqueous suspensions. clearly shown in the next session, the general case has not only quantitative but also qualitative differences with respect to Here the analysis semplifies to some the pure water case. extent since the boiling point elevation is zero and the last On the term in the r.h.s. of Eqs. (11) and (12) vanishes. contrary, the sensitive parameter #n remains given by Eq.(13).

97 i.e. the percentage in particular that here *u
‘I=

NA

NA +hdr=

1 1 + R2

The membrane properties determining the thermal efficiency for DCMDof pure water are the void fraction E, the tortuosity thermal conductivity of the factor X and the membrane mater ial. Interestingly, the thermal efficiency is independent of the membrane thickness 6; indeed, as 8 increases both h. and K decrease in the same way. In Fig.1 R2 is reported as a function of void fraction l with the average temperature as a parameter. For commercially available membranes and realistic operating the conditions, order of magnitude of R2 is unity, corresponding to 50% thermal efficiency. The calculated ii values for the DCMDof pure water are reported in Figs. 2 and 3 with R2 as a parameter. For low Ri values @u approaches unity, i.e. the process rate is highly sensitive to changes in the external heat transfer coefficient. The same result can be obtained by recognizing that the temperature polarization coefficient goes to zero. Indeed, only in the case of the DCMDof pure water one can show that:

However a high iv value does not necessarily imply that the process is enterely controlled by the external heat transfer resistance. Indeed, also changes in K and in hr may give rise to significant changes on the mass flux, depending upon R2. As an example, consider the reference case of Rz=l, when R1=0.1 in addition to Ov*l, we also have that Tn=0.5; namely an increase in K of, say, 50% reflects into an increase of 25% in the flux.

98

.-

0.

1

.5 &

Fig-l. D.C.M.D. of pure water: ratio between heat losses and minimum energy demand versus void fraction, at different average bulk temperatures. km = 0.22 W/mK, X = 2.

When the changes in K are obtained through changes in the membrane properties which affect also hr, one should account for the effects of both quantities on the flux. For instance, an increase in g gives rise to an increase in K and to a decrease in hr and both effects add up to a further positive increase of the flux; on the other hand a decrease in the membrane thickness 8 gives rise to an increase of both K and h. and the two effects on the flux cancel with each other, i.e. the flux is independent of the membrane thickness. For high Rt values, on the contrary, S,n-4 while both ?h and 1cuapproach zero: now, the process rate is entirely controlled by the mass transfer resistance through the membrane. In terms of the temperature polarization coefficient we have r=l. The Rr range where mass transfer controls depends upon the values of Rz; e.g. from Fig.2 we can say that the requirement is met for In these ranges the RHO when Rz=l, for Rt>lOO when Rz=lO. permeate flux is proportional to K and thus changes linearly with l/8. We notice in closure that in all cases when different the influence of K always membrane properties are considered, Thus there is no overcomes the influence of hm (#w > (?hl).

Fig.2.

D.C.M.D. of the liquid

pure water: the role played by heat phase (UL) on the evaporation rate.

transfer

&AL

Fig.3. D.C.M.D. transfer through

’ KA of pure water: the role played the membrane on the evaporation

by heat rate.

and mass

100 way of increasing the permeate flux by increasing the membrane That conclusion holds for the pure water case only; thickness. it will be shown in the following that in DCMD of salt there is an optimum value for on the contrary, solutions, membrane thickness.

of the DCHD of salt solutions the In the more general case representing the relative effect on the sensitivity factors ii, permeate flux due to changes in the single resistances, are given by Eqs. (ll), (12) and (13) and are expressed in terms of all the three parameters RI, R2 and DTb/DTO. First of all we notice that ?n is given by the same expression both for pure water and for salt solution feeds, in terms of the dimensionless numbers Rr and R2. The physical meaning of the latter parameters has been already discussed above; we simply any point out that in the present case R2 does not represent more the exact ratio between the energy loss, h.DT , and the minimum energy demand for the evaporation, N X, insofar as, in view of Eq.(2), the driving force for the permeate flux is given by DT-DTO and not simply by DT. The major differences between the DCMDof salt solutions and of pure water are encountered in the range of small Rr values, i.e. when the external heat transfer is very significant. Noteworthy, the unity value is no longer the upper bound for all ii’s; in particular Ou can be even much larger than unity, due to the effect of the last term in the right hand side of Eq.(ll). Values larger than unity for @u imply that a given percentage change in UL can result in a much larger percentage what is not obtainable * for change in the transmembrane flux, Physically this stems from the fact that the pure water feeds. actual driving force for the mass flux is DT-DTO, while changes in UL affect directly the temperature difference across the membrane, DT. Thus, an appreciable change in DT may well in the result in a much larger change, on a percentage basis, real driving force DT-DTO; this occurs in particular, when DT is Apparently that is the case when the of the order of DTO. external heat transfer controls the overall heat transfer rate namely for low values of UL/h.=Rr/Rs. Since only the positive fluxes out of the salt solutions are the condition given by Eq.(7) must hold, and of some interest, temperature thus for any given feed, membrane and bulk difference the external heat transfer coefficient UL must exceed In terms of the relevant a minimum threshold value.

dimensionless

quantities

we have: R2

RI

>

RImin

=

mb/r;rrO

-

1

The influence of the external heat transfer resistance becomes higher and higher as Rr approaches its mAmum value; when small permeate flux is obtained and, a negligibly RI-- RImin correspondingly, an extremly high sensitivity with respect to UL formally represented by a diverging value of Ou: obtained, t”,m. The behaviour of Tu in terms of Rr and DTb/DTO IS represented in Fig.4, for the typical value of R2=1. Other remarkable considerations are in order on the DCHD Parallel to the large behaviour in the range of low Rl values. sensitivity on the external heat transfer coefficient, for salt solutions a corresponding high sensitivity is observed on the membrane heat transfer coefficient hr. Indeed, the second term in the r.h.s. of Eq.(ll) which is responsible for the mentioned exactly the same as the effects on Pu, is, apart from the sign, Therefore, last term entering the expression for #h, Eq.(12). in this range there is the possibility of strongly affecting the permeate flux also by changing the heat transfer resistance of in view of the negative sign of the membrane. In particular, increase a decrease of h. results in a much larger percent *b, in the distillate flux. Finally, we remind that &r may not be really negligible in the same Rr range, depending upon the value of R2, > or ultimately, on the value of both average temperature and void fraction. We can thus conclude that, analogously to the pure water case, for low Ri values there is not a single controlling resistance, but rather there is a possible significant effect of the mass transfer through the membrane and a large sensitivity to both external and internal heat transfer resistances; the latter effects are here highly amplified with respect to the pure water case. We notice, however, while the heat flux that when Ri+Rr.in, remains finite the mass flux vanishes and the process is characterized by a rather poor erlergy efficiency; thus in a good engineering design those conditions should be avoided. The analysis becomes much simpler in the range of Ri>>Rr.ln; in such cases both QIJ and Ih become negligibly small and 9n approaches unity, thus indicating a mass transfer control for the DCMD from salt solutions. The actual Rr value at which this occurs depends upon the value of R2; however, based on Figs. 4 and 3, which still holds true for &I, we can say that in all cases when Rr>lO significant changes in the transmembrane flux could only be obtained by changes in the mass transfer

102 resistance, while only minor effects are associated to changes in the heat transfer resistances. From a practical view point it is rather interesting to inspect what are the ultimate effects obtained by changing the membrane Clearly a decrease in the membrane thickness thickness alone. plays two opposite roles, one positive associated to an increase negative in the mass transfer coefficient and the other associated to an increase in the membrane heat transfer coefficient. While in the pure water case the two effects compensate with each other, in the low Ri range, the same is not true for salt solutions due to the larger sensitivity with respect to h.. thus the intriguing presence of an optimum membrane thickness is obtained. The effect of changes in 6 is analyzed by introducing the sensitivity factor @a of the distillate flux with respect to the thickness, i.e.:

From Eqs. has:

(4).

(9), (IO), in view of the definition of h., one

In Fig. 5 the sensitivity factor (-$a) is plotted as a function of Rl at various R2, for the reference case of DTb/DT"=lOO. Remarkably, for low Rr values #a is positive, namely in that range an increase in the membrane thickness results in an increase even higly appreciable in the permeate flux. On the other hand, for high Rr values Pa turns negative, meaning that an increase in the membrane thickness leads to a decrease in the distillation rate. This latter behaviour parallels what is also observed in the pure water case, and is well understood by reminding that in the high Ri range the process is mass transfer controlled and an increase in the thickness gives rise to an increase in the mass transfer resistance. The optimum value of the membrane thickness which maximizes the permeate flux corresponds to the value of Rr at which O&=0. On the other hand there is also a minimum value of 8 corresponding to the minimum value of RI, given from Eq.(l8). As an example, for DCMD at TI = 5OOC and DTb=2OOC through a PTFE membrane (ks=0.22 W/mK) with l=O.6, X=2 and U~=1000 W/m2K the minimum and optimum values for the membrane thickness are 2.2~ and 24 nm respectively for a 35000 ppm NaCl solution; the values increase with the solute concentration, e.g. at 100,000

Pig.4. D.C.M.D. of aqueous salt solutions: the role played by heat transfer in the liquid phase (UL) on the evaporation rate, at different DTb/DTO.

1

Fig.5. The role played by the membrane thickness on the evaporation rate in DCMD of aqueous salt solutions (-) and pure water (----).

104

The thermal ppm they are 7.2 and 45 w respectively. for instance, efficiency is lower than in the pure water case; the thermal efficiency is 49% in pure in the conditions above, while for 35000 ppm and 100,000 pp~ salt solutions water case, in both cases using the it reduces to 43% and 38% respectively,

optimum membranethickness. For salt solutions the independent water case.

thermal efficiency

of the membrane thickness

as it

occurs

is no longer for the pure

The DCMD has been analyzed with the particular aim to recognize, based on suitable dimensionless parameters, what actions should be undertaken in order to obtain signif icant improvements for the distillate flux of a given DCMDsystem. For any given operating conditions the process rate results from the combined effects of the mass and heat transfer resistances within the membrane as well as of the heat transfer resistances within the feed and permeate streams. It has been shown that a satisfactory analysis of the effects of the various resistances cannot be performed by using simply the temperature polarization coefficient. Therefore the proper sensitivity factors of the distillate flux with respect to i)the convective heat transfer rate, ii)the membrane heat transfer coefficient and iii)the membrane mass transfer coefficient have been introduced and evaluated. In addition to the ratio DTb/DTO, two other significant dimensionless quantities Ri and Rz enter the analysis, defined in Eqs.(l5) and (16); roughly they compare the external and the membrane heat transfer rate with the membrane permeability. For high Ri values, say for Ri>lO, the transmembrane flux is controlled by the mass transfer resistance so that higher fluxes can be achieved by more permeable or by thinner membranes. In this range the salt concentration has a minor effect, indeed nearly the same behaviours are observed for the DCMD of pure water and of salt solutions, in terms of sensitivity factors. On the contrary for low Ri values, say for Ri
105 permeation rate; one can for instance increase the external heat for which the sensitivity factor is even transfer coefficient, Alternatively or simultaneously we have two larger than 1. to decrease the membrane heat one is additional options: by increasing the membrane thickness, transfer coefficient e.g. the other is to increase the meabrane heat transfer coefficient K e.g. by increasing the temperature or the void fraction of the These conclusions cannot be obtained simply based on membrane. the temperature polarization coefficient. It is remarkable to notice that for salt solutions there is a of the membrane which needs to be exceeded in minimum value order to obtain a positive distillate flux, as well as an optimum membrane thicknesss which maximizes the permeation rate. This properly holds olny for salt solutions and is no longer true for the case of pure water. The effect of the membrane thickness is very peculiar; indeed DCMD seems to be the unique membrane process in which the membrane should be sufficiently thick in order to be effective. is simply The energy efficiency for the DCMDof pure water , related to R2 thruogh Rq.(16), and is independent of the for salt solutions the thermal thickness 6; on the contrary, efficiency varies with the membrane thickness. ACKNOWLEZWEMENTS This work has been partially supported grant N. 89/00881.72/115.12703.

by MURST 40% and by CNR

NOTATION

3 K n N P P T UL Lao

h. k : E X \

diffusion coefficient mass transfer coefficient, Rq.(4) molecular weight mass flux pressure vapor pressure of pure water temperature overall liquid heat transfer coefficient boiling point elevation = k./6, membrane heat transfer coefficient thermal conductivity ionic mole fraction membrane thickness void fraction tortuosity factor latent heat of vaporization per unit mass

106 SubscriPts 1 2 % m S

w

warm side cold side air bulk average value solid water

REFERENCES

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