On the heat transfer correlation for membrane distillation

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On the heat transfer correlation for membrane distillation

Chi-Chuan Wang

⇑

Department of Mechanical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history: Received 17 June 2009

Received in revised form 5 March 2010 Accepted 21 November 2010 Available online 8 January 2011

Keywords:

Membrane distillation Heat transfer coefﬁcient Wilson plot

a b s t r a c t

The present study examines the heat transfer coefficients applicable for membrane distillation. In the available literatures, researchers often adopt some existing correlations and claim the suitability of these correlations to their test data or models. Unfortunately this approach is quite limited and questionable. This is subject to the influences of boundary conditions, geometrical configurations, entry flow condi-tions, as well as some influences from spacer or support. The simple way is to obtain the heat transfer coefficients from experimentation. However there is no direct experimental data for heat transfer coef-ficients being reported directly from the measurements. The main reasons are from the uncertainty of permeate side and of the comparatively large magnitude of membrane resistance. Additional minor influ-ence is the effect of mass transfer on the heat transfer performance. In practice, the mass transfer effect is negligible provided the feed side temperature is low. To increase the accuracy of the measured feed side heat transfer coefficient, it is proposed in this study to exploit a modified Wilson plot technique. Through this approach, one can eliminate the uncertainty from permeate side and reduce the uncertainty in mem-brane to obtain a more reliable heat transfer coefficients at feed side from the experimentation.

1. Introduction

Membrane distillation (MD) is a hybrid of thermal distillation and membrane processes. It features a thermally driven process via a microporous membrane separating the warm and cold solu-tion. The vapor pressure difference is generated through the non isothermal process amid feed side and permeate side, thereby va-por molecules will transva-port through from the warm feed side into the hydrophobic membrane pores provided the vapor pressure is

being established.Fig. 1 is a schematic showing the typical MD

membrane and its heat/mass transport mechanisms. Evaporation occurs at the warm feed side, yet the vapor molecules migrate across the non wetted pores to condense at the permeate side. The overall heat transfer process is described subsequently.

From this schematic, heat is ﬁrst transferred from the warm solution across the thermal boundary layer:

qf ¼ hfðTf Tm1Þ ð1Þ

Here qfis the convective heat transfer flux and hfis the heat transfer coefficient at the feed side. The heat transfer within the membrane (qm) is divided in parallel into conduction across the membrane material (qc) as well as the heat subject to vapor flowing through the membrane (qv), yet the relevant heat transfer equation inside the membrane is given by[1]:

qm¼ Jk þ km

d ðTm1 Tm2Þ ð2Þ

where J is the evaporative mass ﬂux across the membrane and k is the heat of vaporation. The effective thermal conductivity of the membrane can be determined as the following

km¼ ð1

e

Þksþ

e

kg ð3Þ

where

e

denotes porosity of the membrane material, and ksand kg are the thermal conductivity for solid (polymer) and gas in the pore. Analogously, the heat transfer from the membrane to the per-meate side is given by:

qp¼ hpðTm2 TpÞ ð4Þ

At steady state conditions, the heat ﬂux is expressed by:

q ¼ qf ¼ qm¼ qp ð5Þ

Rearranging Eqs.(1)–(4)yield

q ¼ Tf Tp 1 hfþ 1 km dþ Jk DTm þ1 hp ð6Þ

As clearly seen from Eq.(6), three resistances prevails for the heat transport from warm feed side to cold permeate side, namely the convective resistance of feed side (1/hf) and permeate side (1/hp)

and the membrane resistance 1

km dþ

Jk

DTm

. Normally the feed side

⇑Address: EE474, 1001 University Road, Hsinchu 300, Taiwan. Tel.: +886 3 5712121x55105; fax: +886 3 5820634.

E-mail address:ccwang@mail.nctu.edu.tw

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resistance is larger than that of permeate side, yet it plays a crucial role in overall heat transport process of MD.

Of the literatures concerning with the heat transfer coefﬁcients applicable to the feed side or permeate side, none of the existing

literatures were really dealt with the data reduction of the col-lected data. Even for those researchers who perform substantial experiments in MD, there were no direct thermal resistances re-ported from their measurements. In fact, all the research works, Nomenclature

A surface area (m2₎

cp speciﬁc heat (kJ/kg K)

D diameter (m)

h heat transfer coefﬁcient (W/m2_K)

J evaporative ﬂux (kg/m2_s)

k thermal conductivity (W/m K)

_

m mass ﬂow rate (kg/s)

Nu Nusselt number

Pr Prandtl number

q heat ﬂux (W/m2₎

_

Q heat transfer rate (W)

R resistance (m2K/W)

T temperature (°C)

U overall heat transfer coefﬁcient (W/m2K)

v velocity (m/s)

Greek symbols

e

porosity

d membrane thickness (m)

DTLM log mean temperature difference (K)

DTm temperature difference across membrane (K)

k heat of evaporation (kJ/kg)

l

dynamic viscosity (kg/m s)

q

density (kg/m3₎

Subscripts

ave average value

f feed side

g gas phase

in inlet

m membrane

m1 membrane in contact with the feed side

m2 membrane in contact with the permeate side

out outlet p permeate side s solid phase t total

T

f

T

m1

T

_m2

T

p

1/h

f

1/h

p

Membrane

Feed side

Permeate

_side

£_

/k

m

(T

m1

-T

m2

)/J£f

Vapor flow

T

f

T

p

T

m2

T

m1

Membrane

Permeate

side

Feed side

(a) Temperature variation for MD Process

(b) Thermal Resistance Network

Fig. 1. Schematic showing the temperature variation for membrane distillation across and the associated thermal resistance network. (a) Temperature variation for MD process. (b) Thermal resistance network.

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including those theoretical modeling or experimental work, adopts an approach by selecting certain correlations from the literatures. Some typical correlations being selected is tabulated inTable 1 which were summarized by Phattaranawik et al.[2]. The research-ers then test the correlations against some experimental data either from their own measurement or from other literatures and claim the most suitable one of the selected correlations. This ap-proach seems feasible but may actually mislead the readers about the ‘‘most suitable correlation’’ itself. For the correlations tabulated inTable 1, Eq. (L1) is regarded as the most used in the literatures (e.g.,[3–5]). Martínez-Díez and Vázquez-González[3]found that various correlations may lead to signiﬁcant departure between measurements and predictions, and Eq. (L1) is in satisfactory

agreement with the experimental results. Banat et al. [4] found

Eq. (L1) is applicable for laminar flow condition, and for turbulent flow condition the Dittus & Boleter type Eq. (T10) applies. Based on the heat and Mass analogy, Martínez-Díez and Vázquez-González [5]extends the applicability of Eq. (L1) to predict mass transfer subject to concentration polarization, and satisfactory agreement was reported. In addition, the heat transfer performance is related to the geometry and operation condition. For instance, natural con-vection (or mixed concon-vection) may cast certain influences on the overall transport process. Kadir et al.[6]examined the heat trans-fer and friction correlations of a rectangular channel subject to finned surface. They found the heat transfer performance is depending on hydraulic diameter, the distance between fins in

the ﬂow direction and ﬁn arrangement. Dogan and Sivrioglu [7]

investigated mixed convection heat transfer from longitudinal fins inside a horizontal channel in the natural convection dominated region for a wide range of Rayleigh numbers and different fin heights and spacings. Their experimental results implicated study show that the dimensionless optimum fin spacing occurs for an as-pect ratio between 0.08 and 0.12.

For laminar flow condition, as noted, geometry and boundary conditions can significantly affect the heat transfer correlation. For instance, for circular duct under fully thermally developed condition, the Nusselt number for constant temperature condition is 3.66 while it is 4.36 for constant heat flux condition. In typical MD operation, the boundary conditions are neither constant temperature nor constant heat flux. Meanwhile, influence of developing may prevail hydraulically, thermally, and simulta-neously. These will certainly reinforce the uncertainty of selecting

correlations. For further comparison about the tabulated correla-tions, one can see the results inFig. 2showing the variation of heat transfer performance with the rise of the Reynolds number under laminar flow condition. As shown in the figure, significant differ-ences of the predicted Nusselt number are encountered. For a Rey-nolds number around 2000, the calculation by Eq. (L4) is about 3.3 times higher than that of Eq. (L1) at a feed temperature of 70 °C, yet the difference increases to about 4 times at an even lower temper-ature of 20 °C as appeared inFig. 2b. Note that the exponents of the Reynolds number for Eq. (L4) and Eq. (L6) are 0.64 and 0.73, respectively, which are unexpected high pertaining to laminar flow condition. Excluding the unexpected over-prediction by Eqs. (L4) and L6, the other correlation like Eq. (L3) is still about two times higher than that of Eq. (L1). The significant departure of the corre-lations may be associated with the real experimentation. For in-stance, the membrane is usually accompanied with some or with some supporting means which inevitably lead to a higher heat transfer performance. Contrast to that of laminar flow correlation, the correlations valid for turbulent flow shows only mild

differ-ence. As shown in Fig. 3, the maximum deviation is around

30–40%. In summary of the comparison, one can conclude a diver-sity amid the existing correlations particularly for laminar ﬂow

Table 1

Common correlations used to estimate the heat transfer coefﬁcient in membrane distillation. (From Phattaranawik et al.[2]).

Equation Laminar or turbulent Equation no.

Nu ¼ 1:86 RePr L=D 1=3 _L _L1 Nu ¼ 4:36 þ_{1þ0:0011ðRePrðD=LÞÞ}0:036RePrðD=LÞ 0:8 L L2 Nucooling¼ 11:5ðRePrÞ0:23ðD=LÞ0:5 L L3 Nuheating¼ 15ðRePrÞ0:23ðD=LÞ0:5 Nu ¼ 0:13Re0:64_Pr0:38 _L _L4 Nu ¼ 1:95 RePr L=D 1=3 _L _L5 Nu ¼ 0:097Re0:73_Pr0:13 _L _L6 Nu ¼ 3:66 þ 0:104RePrðD=LÞ 1þ0:0106ðRePrðD=LÞÞ0:8 L L7 Nu ¼ 1 þ6D L ðf =8ÞRePr 1:07þ12:7ðf =8Þ1=2_ðPr2=3_1Þ _T _T8 Nu ¼ 1 þ6D L ðf =8ÞðRe1000ÞPr 1þ12:7ðf =8Þ1=2_ðPr2=3_1Þ T T9 Nu ¼ 0:023 1 þ6D L Re0:8_Pr1=3 _T _T10 Nu ¼ 0:036Re0:8_Pr1=3 D L 0:055 _T _T11 Nu ¼ 1 þ6D L ðf =8ÞRePr 1:2þ13:2ðf =8Þ1=2_ðPr2=3_1Þ T T12 Nu ¼ 0:027 1 þ6D L Re0:8_Pr1=3 lbulk lsur 0:14 _T _T13

Re

0 500 1000 1500 2000 2500

Nu

0 10 20 30 40 L1 L2 L3 L4 L5 L6 L7 L5 L2 L1 L6 L4 L7 L3

(a) Evaluations are performed at a feed

temperature of 20 °C

Re

0 500 1000 1500 2000 2500

Nu

0 5 10 15 20 25 30 35 L1 L2 L3 L4 L5 L6 L7 L6 L1 L5 L3 L7 L2 L4

(b) Evaluations are performed at a feed

temperature of 70 °C

Fig. 2. Variation of Nusselt number vs. Reynolds number subject to laminar ﬂow condition. The Eqs. (L1. . .L7) are tabulated from Table 1. (a) Evaluations are performed at a feed temperature of 20 °C. (b) Evaluations are performed at a feed temperature of 70 °C.

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condition. And it should be stressed that most applications may be involved in laminar ﬂow condition which make the correctness of the laminar ﬂow correlation especially crucial.

Things get even more complicated as the ﬂow passages are not in circular shape such as rectangular or plate channels. The im-posed heating condition may just appear in the membrane side while other sides are either insulated or in contact with other channels. This eventually provides some uncertainties when apply-ing existapply-ing correlations. However, the existapply-ing literatures still pro-vides limited information about the selection or data reduction of this resistance. Therefore the objective of the present effort is to summarize and to recommend the correct way to obtain the feed side resistance in the subsequent discussion.

2. Data reduction of the thermal resistance at feed side and permeate side

Despite considerable experimental work being made during the past, as aforementioned, there is no heat transfer coefﬁcient re-ported directly from the data reduction of measurements. Prior to discussing this unusual consequence, let’s review the typical data reduction method in heat transfer experiments. Consider a

typical counter ﬂow arrangement without any mass transfer being taken place across the membrane as seen inFig. 1a, the total heat transfer rate used in the calculation for the membrane module (or heat exchanger) is from the mathematical average of the feed side and permeate side, namely,

_ Qf ¼ _mfcp;fðTf ;in Tf ;outÞ ð7Þ _ Qp¼ _mpcp;pðTp;out Tp;inÞ ð8Þ _ Qave¼ ð _Qf þ _QpÞ=2 ð9Þ

In a well-controlled experimental condition, the heat un-balance amid both sides should be less than 2% or 3%. From the rating equation for the overall heat transfer rate, one can obtain the overall resistance as:

1 UA¼

DT

LM _ Qave ð10Þ

whereDTLMis the log mean temperature difference, designated as

DT

LM¼

ðTf ;in Tp;outÞ ðTf ;out Tp;inÞ ln Tf ;inTp;out

Tf ;outTp;in

ð11Þ

The overall resistance (1/UA) product is related to the individual resistance: 1 UA¼ 1 hfAfþ d kmAmþ 1 hpAp ð12Þ

Consider a symmetrical ﬂat membrane having the same surface area in both side, the foregoing equation can be simpliﬁed as:

1 U¼ 1 hf þ d km þ1 hp ð13Þ

In this regard, if one of the thermal resistance is known (e.g. 1/hp), the associated thermal resistance in the feed side can be obtained by subtracting the wall resistance and the permeate resistance from then overall resistance, i.e.,

1 hf ¼1 U d km 1 hp ð14Þ

Through this simple data reduction, the heat transfer coefficient can be easily estimated from Eq.(14). Regardless the process is quite straightforward and is easily implemented for typical experimenta-tion; the success of this approach relies heavily on two important matters. The first one is the correctness of the correlation being used to back out in the permeate side. To resolve this issue, one usu-ally intentionusu-ally places the permeate side in a more well-known and controlled conditions. For instance, the permeate side is placed in the tube side while the feed side is located at the shell side. With this arrangement, the permeate side heat transfer coefficient, hp, is evaluated from the Dittus–Boelter type correlation or Gnielinski semi-empirical correlation with considerable accuracy. However, one should bear in mind that the Dittus–Boelter correlation is appli-cable in turbulent flow whereas the Gnielinski equation is valid pro-vided Re > 2300. The second issue that affects this approach is even more vital, that are the magnitude of the thermal resistances of per-meate side and wall surface. Apparently the overall thermal resis-tance 1/U is larger than individual resisresis-tances. To improve the accuracy of the calculated feed side resistance from Eq.(14), it is important to minimize the corresponding wall and permeate resis-tance. Smaller resistances will improve the accuracy of feed side resistance during subtraction. This is associated with the error propagation during subtraction. A larger thermal resistance results in larger uncertainty during subtraction, yet the uncertainty is further amplified when the individual resistance is above 50%. In

Re

0 3000 6000 9000 12000 15000 18000

Nu

30 60 90 120 150 T8 T9 T10 T11 T12 T13 T8 T9 T10 T11 T12 T13

(a) Evaluations are performed at a feed

temperature of 20 °C

Re

0 3000 6000 9000 12000 15000 18000

Nu

20 40 60 80 100 T8 T9 T10 T11 T12 T13 T8 T10 T9 T12 T11,T13

(b) Evaluations are performed at a feed

temperature of 70 °C

Fig. 3. Variation of Nusselt number vs. Reynolds number subject to transition and turbulent ﬂow condition. The Eqs (T8. . .T13) are tabulated from Table 1. (a) Evaluations are performed at a feed temperature of 20 °C. (b) Evaluations are performed at a feed temperature of 70 °C.

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practice, it would be better to control the individual resistance to be less than 25%. Typical experimental techniques to minimize the thermal resistance of the permeate side are raising the ﬂuid ﬂow velocity or put augmentation such as inserts, roughness, or added surfaces at the permeate side.

3. Additional uncertainty caused by the membrane process The data reduction method described in previous section is actu-ally more appropriate for conventional heat exchanger. There are two additional uncertainties associated with the membrane pro-cess. The ﬁrst one is the effect of mass transfer across the membrane that affects the heat transfer at feed side as well as per-meate side. The effect of mass transfer on heat transfer is termed Ackemann correction factor, Qtaishat et al.[8]and Phattaranawik and Jiraratananon[9]had examined its inﬂuence on the overall heat transfer performance. The presence of mass transfer on the overall heat transfer rate is rather small. The analysis by Qtaishat et al.[8] showed that the effect of mass transfer increased with the rise of feed temperature, yet the effect in the feed side is larger than that of permeate side. The maximum increased values from mass trans-fer on the overall heat transtrans-fer is 4.3% and 1.43%, respectively at an average feed side temperature of 55.93 °C and a permeate side tem-perature of 18.28 °C. Analogously results were also reported by

Phattaranawik and Jiraratananon [9]. A maximum inﬂuence of

about 13% is encountered at a feed temperature of 368 K. The appreciable effect of 13% is related to the high feed temperature where the inﬂuence mass transfer is enormous as it approaches the boiling point. Normally this is unlikely to happen for a typical MD process. As a consequence, it can be concluded that the effect of mass transfer on heat transfer is small provided tests are con-ducted away from the saturation temperature.

The second uncertainty with the membrane is from the mem-brane itself. As seen from Eq.(14)where the feed side resistance is obtained by subtracting the membrane and permeated side resistance from the overall resistance. Again, the uncertainty rises considerably with the membrane resistance. Note that typical heat exchanger usually employ metals whose thermal conductivity can easily surpass tens or hundreds of W/m K, yielding a rather small wall resistance. However, typical thermal conductivity applicable to MD spans amid 0.04–0.06 W/m K which yields a quite percent-age of the total resistance even for a very thin membrane. In fact, for a typical membrane thickness of 150

l

m, a simple estimation reveals that the membrane resistance could easily surpass 50% or 60% at a lower feed temperature of 20 °C of the total resistance while the resistance is reduced to 20–30% when the feed tempera-ture is raised over 70 °C. The results suggest a dramatic inaccuracy associated with direct subtraction approach for membranes, espe-cially the data in low temperature. To increase the accuracy of the direct subtraction, the easiest way is simply to perform tests at a more elevated temperature but again it has the drawback of losing important data at lower temperature. Another way is to adopt the hybrid membrane. A more thin functionalized membrane layer is placed at the feed side and a support layer with large porosity is adhered to the thin layer to reduce the conduction loss.

4. Proposed data reduction procedure to obtain the heat transfer coefﬁcient at feed side

Based on the forgoing discussion, it seems quite difficult to ob-tain the correct heat transfer coefficient from the experiments under membrane distillation process. The direct subtraction method has some inherited flaws, yet existing correlations may be inapplicable. To this regard, the present author proposes the Wilson plot method to obtain the heat transfer coefficient from

the experimental data with a much higher accuracy. The method has been brieﬂy cited for calculation of mass transfer coefﬁcient

(permeability, e.g. [10,11]) but is never fully addressed to

achieve the heat transfer coefficient in membrane distillation. The subsequent outline is a modification to conventional Wilson plot method applicable to obtain the heat transfer coefficient. Through this approach, one can eliminate the uncertainty from permeate side and reduce the uncertainty in membrane to obtain a more reliable heat transfer coefficients at feed side from the experimentation.

1. Conduct experiments for membrane distillation, obtain the four terminal temperatures at the feed side and permeate side. Hence one can calculate the associate heat transfer rate and the logarithmic mean temperature difference, respectively:

Q ¼ UADTLMTD¼ _mcp;fðTf ;o Tf ;iÞ ð15Þ

DT ¼

ðTf ;o Tp;iÞ ðTf ;i Tp;oÞ

ln Tf ;oTp;i Tf ;iTp;o

ð16Þ

Notice that it is suggested that the temperature difference in feed side or permeate side should be at least 2 °C as suggested for common heat exchanger test to minimize the measurement uncertainty (Wang et al.[12]).

2. The overall resistance can be easily estimated as:

Rt¼ 1 U¼ 1 hf þ_k 1 m d þ Jk DTm þ1 hp ¼ Rfþ Rmþ Rp ð17Þ

3. Measure the ﬂux, J, from the experiment.

4. To obtain the feed side resistance (heat transfer coefficient) via Wilson plot, it is a must to maintain a fixed permeate resistance throughout the test. This can be made possible by fixing an average permeate temperature and a fixed flow rate.

5. Unlike the convectional Wilson plot where the wall resistance (membrane resistance) stays unchanged, the membrane resis-tance is actually varying from experiment to experiment pertaining to ﬂow conditions. Therefore, the membrane resis-tance must be subtracted from the overall resisresis-tance.

6. The heat transfer performance in feed side is generally assumed to be the following form:

Nu ¼ C0ReaPrb ð18Þ

Note that an iteration process must be fulfilled before finally ar-rived at the correct exponent values of a and b. Initially, one can presume some values of a and b. As a consequence, the heat feed side transfer coefficient is obtained:

hf ¼ C0

q

f

v

fD

l

f !a

l

_fcp;f kf b kf D¼ C0k 1b f

q

a fc b p;f

l

baf D a1

_v

a f ¼ C1

v

af ð19Þ

7. Estimate the effective thermal conductivity,

km¼ ð1

e

Þksþ

e

kg ð20Þ

This term is rather insensitive to change of temperature and is independent of ﬂow condition.

8. From the experimental measurement,

_ Q ¼ _mcp;fðTf ;o Tf ;iÞ ¼ _Qm¼ km d

DT

mþ Jk A ð21Þ

The temperature across membraneDTmis then obtained. 9. Calculate the membrane resistance

Rm¼_k 1 m d þ Jk DTm ð22Þ

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10. Rearranging the overall resistance equation in the following, Rt¼1 U¼ 1 C1

v

af þ_k_m 1 d þ Jk DTm þ C2 ð23Þ

The foregoing equation can thus be arranged to have the fol-lowing form: 1 U 1 km d þ Jk DTm ¼ 1 C1

v

af þ C2 ð24Þ

The above equation takes the form as

y ¼ mx þ c ð25Þ with y ¼1 U 1 km d þ Jk DTm m ¼ 1 C1 x ¼

v

a f c ¼ C2

11. By changing the velocity in the feed side, one can have a plot of y verse x on a linear scale as shown inFig. 4. However, if the pot is not linear, one should adjust the corresponding values of a and b to achieve the linearity. The slope m and the inter-cept c are then determined from the plot. With m, the heat transfer coefﬁcient in the feed side can be obtained.

5. Conclusion

The present study discusses the heat transfer coefﬁcients applicable for membrane distillation. In the available literatures, most researchers adopt some existing correlations and test the applicability of these correlations to their test data or models. Unfortunately this approach is quite limited and extrapolation of this approach to other conditions is quite questionable. This is

subject to the influences of boundary conditions, geometrical configurations, entry flow conditions, as well as some influences from spacer or support.

The simple way is to obtain the heat transfer coefficients from experimentation. However there is no direct experimental data for heat transfer coefficients being reported directly from the mea-surements. The reasons for this unusual situation are attributed to some causes mainly because it is very difficult to obtain heat trans-fer coefficients from direct subtraction of the individual resistances from the overall resistances. The main reasons are from the uncer-tainty of permeate side and the comparatively large magnitude of membrane resistance. Additional minor influence is the effect of mass transfer on the heat transfer performance. However, the mass transfer effect is negligible provided the feed side temperature is low. To increase the accuracy of the measured feed side heat trans-fer coefficient, it is proposed in this study to exploit a modified Wilson plot technique. Through this approach, one can eliminate the uncertainty from permeate side and reduce the uncertainty in membrane to obtain a more reliable heat transfer coefficient at feed side from the experimentation.

Acknowledgements

This work was supported by the Energy Bureau and Department of Industrial Technology, both from Ministry of Economic Affairs, Taiwan.

References

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-1 slope m = Intercept =

y

x

C0 C₁

Fig. 4. Schematic of the Wilson plot technique to obtain the feed side heat transfer coefﬁcients.