Non-equilibrium thermodynamic analysis of coupled heat and moisture transfer across a membrane

Zhijie Shen,Jingchun Min

Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,Department of Engineering Mechanics,Tsinghua University,Beijing 100084,China

Keywords:Membrane Non-equilibrium thermodynamics Heat transfer Mass transfer Coupling effect

ABSTRACT Non-equilibrium thermodynamics theory is used to analyze the transmembrane heat and moisture transfer process,which can be observed in a membrane-type total heat exchanger(THX).A theoretical model is developed to simulate the coupled heat and mass transfer across a membrane,total coupling equations and the expressions for the four characteristic parameters including the heat transfer coefficient,molardriven heat transfer coefficient,thermal-driven mass transfer coefficient,and mass transfer coefficient are derived and provided,with the Onsager’s reciprocal relation being confirmed to verify the rationality of the model.Calculations are conducted to investigate the effects of the membrane property and air state on the coupling transport process.The results show that the four characteristic parameters directly affect the transmembrane heat and mass fluxes:the heat and mass transfer coefficients are both positive,meaning that the temperature difference has a positive contribution to the heat transfer and the humidity ratio difference has a positive contribution to the mass transfer.The molar-driven heat transfer and thermal-driven mass transfer coefficients are both negative,implying that the humidity ratio difference acts to reduce the heat transfer and the temperature difference works to diminish the mass transfer.The mass transfer affects the heat transfer by 1%–2% while the heat transfer influences the mass transfer by 7%–14%.The entropy generation caused by the temperature difference-induced heat transfer is much larger than that by the humidity difference-induced mass transfer.

1.Introduction

Membrane-based dehumidification has caused increasing attention because of its low cost and less energy consumption compared to the conventional cooling-based dehumidification.In the membrane dehumidification process,the membrane serves as a barrier between a highly humid air and a less humid air,water vapor transports from the highly to less humidity air across membrane,causing dehumidification of the highly humid air [1–3].A typical application of the membrane-based humidify control technology is the membrane-type total heat exchanger(THX),which is used to recover both heat and moisture from the exhaust indoor air to treat the supply outdoor air for an air-conditioned space in summer.Such exchanger contains a core made of water vapor permeable membrane that allows both heat and moisture to transfer across it,accompanied by the adsorption and desorption of water vapor at the membrane surfaces.

Many researches have been done on THX [4–13].Zhang and Jiang [4] developed a theoretical model to evaluate the performance of a crossflow THX,Niu and Zhang [5] numerically investigated the effects of different membrane materials under common conditions and reported that the membrane material with a linear adsorption curve yielded the best performance of THX,and Zhang and Niu [6] further presented the effectiveness correlations for estimating the THX performance based on the number of transfer units (NTU) approach.Min and Su [7–9] conducted a series of works on the performance analysis of THX,they investigated the effects of membrane spacing and thickness,membrane properties and outdoor air state.Min and Duan[10,11]compared four different methods to evaluate the performance of a THX,and pointed out that the numerical method with consideration of the adsorption heat was the best method to calculate such performance,especially when the heat and moisture transfer in different directions.Al-Wakedet al.[12] studied the performance enhancement of THX for different numbers of flow channels,flow configurations,weather conditions,and airflow rates.Leeet al.[13]experimentally investigated the moisture transfer characteristics with a THX core made of different paper membranes.

The non-equilibrium thermodynamics (NET) is widely used to comprehend and quantitatively describe coupling phenomena,which are irreversible in nature [14].The NET theory evaluates the rate of entropy generation resulting from the irreversibility taking place in a coupling transport process,which can be characterized by thermodynamic forces and thermodynamic fluxes.Linear phenomenological equations (LPEs) reveal the relations between them,with the thermodynamic forces being related to the thermodynamic fluxes in a linear form with phenomenological coefficients.Such coefficients must obey the Onsager’s reciprocal relations,in other words,the coefficient matrix must be symmetrical.

The NET theory is often used to analyze various membrane transport phenomena such as the forward osmosis [15],osmotic distillation [16],gas adsorption [17],pervaporation [18–20],etc.Fanget al.[15] studied the transport phenomena of solvent and solute through a membrane in hydraulic and osmotic pressure driven membrane processes on the basis of NET with three membrane parameters including the water permeability,reflection coefficient,and solute permeability.Wang and Min [16,17]modeled and analyzed the simultaneous heat and mass transfer during membrane osmotic distillation as well as the gas adsorption processes at a non-equilibrium steady state based on the NET theory.Toikkaet al.[18,19] attempted to describe the pervaporation data using a NET approach and pointed out that the thermodynamic description,analysis and approximation of pervaporation data were a way to develop the theory of pervaporation.Kuhnet al.[20]studied the mass and heat transports for water pervaporation through a zeolite membrane,they derived the transport equations for coupled heat and mass transfer from the framework of NET,and stated that a NET model was more capable of revealing the physical nature than the commonly used Maxwell–Stefan model.In addition,Demirel and Sandler [21] expressed the phenomenological equations with the resistance coefficients that are capable of reflecting the extent of interaction between the heat and mass transfer using dissipation-phenomenological equations.Keulenet al.[22] proposed a NET model to describe the transport phenomena in membrane distillation against a pressure difference at liquid-membrane interfaces through a hydrophobic membrane,and their results showed that 85% of the energy was dissipated and a lower interface heat transfer resistance might benefit the system performance.Castel and Favre [23] performed critical analysis and evaluation of the energy efficiency concept for membrane separation using the NET approach.

Simultaneous heat and mass transfer can be analyzed without adopting the NET theory,as in Wuet al.[24] and Tang and Min[25].The advantage of using the NET approach is that it can help to clarify the coupling effects of the combined heat and mass transfer,making it possible to quantitatively evaluate the contribution of each thermodynamic force to each thermodynamic flux,with the magnitudes and signs of the four coupling characteristic parameters representing the intensity and direction of such contributions.Simultaneous heat and mass transfer occur in the membrane-based humidity control and energy recovery applications such as the membrane-type total heat exchanger(THX),however,no report has been seen in literature to analyze the coupling relations of such transports from the viewpoint of the nonequilibrium thermodynamics,which is especially suitable for analyzing the coupling phenomena.

The present research deals with the coupled heat and moisture transfer across a dense membrane,with the THX as an application background.A theoretical model is developed to describe the transmembrane heat and moisture transfer and their coupling relations based on the NET theory,and equations for calculating the four coupling characteristic parameters (phenomenological coefficients) and entropy generation rate are derived to evaluate the coupling effect and process irreversibility.The model considers not only the conductive transfer in membrane but also the convective transfer between the air fluid and membrane surfaces,it further considers the effects of adsorption and desorption heats at the membrane surfaces.The uniqueness of the present research is to take the THX as an application background and to include both the conductive and convective transfer in the model,unlike the previous works that usually neglected the convective transfer to facilitate the NET model establishment.Calculations are conducted to investigate the influences of membrane property and air state on the transmembrane heat and moisture transfer characteristics.

2.Theoretical Model

2.1.Physical model and overall heat and mass transfer model

Consider the transmembrane heat and moisture transfer process as illustrated in Fig.1.A membrane separates the humid air fluids,Air 1 and Air 2,which have different temperatures and humidities,with Air 1 possessing higher temperature and humidity ratio than Air 2,i.e.T10>T20andW10>W20.Convective heat and mass transfer takes place between the air fluids and membrane surfaces while conductive heat and mass transfer occurs in membrane.Heat and moisture transfer from Air 1 to Air 2 due to the temperature and water vapor concentration gradients.Along with the mass transfer,adsorption and desorption of water vapor take place at the membrane surfaces,the former generates the heat of adsorption while the latter consumes heat,making the heat transferred across the membrane to exceed that by the convection on each side.

Based on the transport and solution-diffusion theories,the overall mass and heat transfer across membrane can be represented by

whereJis the mass flux across membrane,kandhare the convective mass and heat transfer coefficients,θ is the moisture content of membrane,Dwmis the moisture diffusivity in membrane,λmis the thermal conductivity in membrane,δ is the thickness of membrane,Lis the heat of adsorption or desorption,qSandqLare the convective and adsorptive heat fluxes,andqtotis the total heat flux that is the sum ofqSandqL.Subscripts 0,1,2 and m express the bulk air,side of Air 1,side of Air 2,and membrane.

Fig.1.Schematic diagram of heat and moisture transfer across a membrane.

According to the heat and mass transfer Chilton-Colburn analogy theory[26],the convective heat and mass transfer coefficients can be related by

whereLeis the Lewis number,whose value is about 0.85 for a temperature range of 0-40 °C [27].

2.2.Basic theory of non-equilibrium thermodynamics

Based on the non-equilibrium thermodynamic theory [21],the entropy generation rate of coupled heat and mass transfer across membrane can be represented by

where σtotis the total entropy generation rate per unit area,σqand σJare the entropy generation rates per unit area induced by the heat and mass transfer,XqandXJare the thermodynamic forces,given by

andqandJare the heat and mass fluxes corresponding toXqandXJ.In Eq.(7),μ is the chemical potential.

When the temperature difference between the two surfaces of membrane is small,the thermodynamic forces corresponding to the transmembrane conductive heat and mass transfer can be expressed as

From Eqs.(6) and (8),the entropy generation rate of coupled heat and mass transfer across membrane can be written as

where ΔTmis the temperature difference between the two surfaces of membrane,given by ΔTm=T1–T2,Tmis the average membrane temperature,calculated byTm=(T1+T2)/2,and (Δμm)Tis the chemical potential difference between the two surfaces of membrane at a fixed temperature,given by (Δμm)T=(μ1–μ2)T.Based on the nonequilibrium thermodynamic theory,the transmembrane heat and mass fluxes can be further represented by

whereLis the phenomenological coefficient,which reflects the phenomenological relation between the thermodynamic flux and force.

At atmospheric pressure ofP0,the chemical potential is a function of temperature and mole fraction,and the chemical potential of water vapor can be expressed as [28]

where μΘis the chemical potential of pure water vapor,which is temperature dependent,Rgis the gas constant of water vapor,andxis the mole fraction of water vapor.

The following equations apply for humid air at the atmospheric pressure [29]

where φ is the relative humidity,Wis the humidity ratio,Pvis the water vapor partial pressure,andPsis the saturation water vapor pressure,Tis the temperature in K,withC1=-5800,C2=1.3915,C3=-0.0486,C4=4.1764×10-5,C5=-1.4452×10-8andC6=6.5460.Eq.(14) is valid for 273.15–473.15 K [29].Since the water vapor partial pressure is generally much smaller than the atmospheric pressure,Eq.(13) can be simplified by

So the humidity ratio is approximately proportional to the mole fraction of water vapor.Taking logarithm on both sides of Eq.(15)yields

From Eqs.(11) and (16),the chemical potential difference at isothermal condition can be expressed as

where ΔWmis the difference of humidity ratios at the two surfaces of membrane,given by ΔWm=W1–W2.When ΔWmis small,Eq.(17) can be further represented by

Substituting Eq.(18) into Eq.(10) gives

2.3.Mass transfer

As mentioned above,the mass transfer through a dense membrane obeys the solution-diffusion theory,i.e.,the water vapor absorbs at the membrane surface on one side,then diffuses through the membrane,and eventually desorbs from the membrane surface on the other side.The driving force of mass transfer through a membrane is the moisture content of membrane,which can be expressed as a function of humidity ratio and temperature in form of

Using the Taylor expansion for mass flux equation with ignorance of the second order infinitesimal yields

Substitution of Eq.(21) into Eq.(1) gives

By comparing Eq.(19) and Eq.(22),we get the phenomenological coefficients

2.4.Heat transfer

Because of the existences of moisture adsorption and desorption at membrane surfaces,the heat transferred through the membrane includes not only the convection heat but also the adsorption heat due to the mass transfer,which can be assumed to be equal to the latent heat of vaporization of water.From of Eqs.(1)–(4) and (21),we obtain

Comparison of Eqs.(19) and (25) yields the other two phenomenological coefficients

Eqs.(23),(24),(26) and (27) are the expressions for the four phenomenological coefficients.

2.5.Verification of Onsager’s reciprocal relation

The moisture adsorption and desorption at membrane surfaces can be regarded as liquefaction and vaporization,the Clausius–Clapeyron equation [28] can thus apply,i.e.

Combining Eqs.(12) and (28) and deriving water vapor partial pressure with respect to temperature for constant relative humidity yields

From Eq.(15),we have

From the partial differential relations,we can further obtain

Meanwhile,the adsorption characteristics of water vapor at membrane surface can be represented by [30]

wherewmaxis the maximum moisture uptake,andCis the adsorption constant,which reflects the characteristics of adsorption and determines the shape of adsorption curve.From Eq.(32),the following expression can be written

From the partial differential relations,the following relationship holds

Substituting Eq.(35) into Eq.(23) yields

Comparison of Eqs.(36) and (27) suggestsLWT=LTW,so each individual phenomenological parameter included in the model can meet the requirement of the Onsager reciprocal relation,supporting the reasonability of the model.

2.6.Overall coupling equations

Similar to Eq.(19),the overall coupling heat and mass transfer equations can also be expressed in a form of

where ΔT=T10–T20,ΔW=W10–W20,and α,α′,β′and β are the coupling characteristic parameters,named as the heat transfer coefficient,molar-driven heat transfer coefficient,thermal-driven mass transfer coefficient,and mass transfer coefficient,respectively,they reflect the coupling relations between the heat and mass transfer across membrane.

Meanwhile,the heat and mass transfer resistances through membrane can be express as [31]

whereRT,mandRW,mare the heat and mass transfer resistances through membrane.Considering the convective heat and mass transfer between membrane and air on both sides,the overall heat and mass transfer resistances can be derived in form of

Substituting Eq.(21) into Eq.(1),we obtain

Transforming Eq.(42) by extracting the membrane moisture resistance yields

Since the ratio of membrane to overall resistance is equal to that of membrane to overall driving force,using Eqs.(38) and(40),Eq.(43) can be transformed as

Eq.(44) shows that the humidity ratio and temperature differences can both affect the mass transfer,although the contribution of the latter may be much smaller than that of former.It also reflects the coupling relationship between the mass and heat transfer from the view of the mass transfer.By comparing Eqs.(44) and (37),we can get the mass transfer coefficient and thermal-driven mass transfer coefficient as below

Huet al.[31,32] analyzed the effect of adsorption heat on heat transfer for membrane transport,which can be represented by

Substituting Eq.(44) into Eq.(47) yields

Eq.(48) indicates that the temperature and humidity ratio differences can both affect the heat transfer,although the contribution of the latter may be much smaller than that of former.It also reflects the coupling relationship between the heat and mass transfer from the view of the heat transfer.By comparing Eqs.(48) and (37),the heat transfer coefficient and molar-driven heat transfer coefficient can be achieved as follows

3.Parameter Selection and Model Calculation

Table 1 presents the values taken for the membrane and air parameters,these values are specified based on our previous studies on the membrane-type total heat exchanger [7–11].The air fluid with high temperature (T10) corresponds to the outdoor air in summer,while that with lower temperature (T20) corresponds to the indoor air in an air-conditioned space,which has more stable temperature and humidity as compared to the outdoor air.The convective heat transfer coefficient is taken ash1=h2=50.3 W?m-2?K-1,and the convective mass transfer coefficient can then be calculated to bek1=k2=0.0556 kg?m-2?s-1according to Eq.(5).Calculations are conducted for various conditions as presented in Table 1,with theT10=308.15 K air temperature,φ10=70% relative humidity,C=2.5 membrane adsorption constant,wmax=0.25 kg?kg-1maximum water uptake,λm=0.1-W?m-1?K-1thermal conductivity,andDwm=2.5×10-7kg?m-1?s-1in-membrane moisture diffusivity as the representative condition.On investigation of the effect of each parameter,only the parameter concerned is altered,with all the other parameters being fixed at the representative condition.The calculations useP0=101325 Pa,cpa=1.005 kJ?kg-1?K-1andL=2500 kJ?kg-1.

Table 1 Values taken for membrane and air parameters

The overall heat and moisture transfer across membrane can also be calculated from Eqs.(1)–(4),and the results generated by the NET model agree with those by Eqs.(1)–(4) within 5% for all cases listed above.The reason for this discrepancy is mainly because of the formulae simplification and linearization made during the NET model establishment,e.g.,as seen in Eq.(21).As stated early,the advantage of using the NET approach is that it can help to clarify the coupling effect of the combined heat and mass transfer,making it possible to quantitatively evaluate the contribution of each thermodynamic force to each thermodynamic flux,with the magnitudes and signs of the four coupling characteristic parameters representing the intensity and direction of such contributions.So,the results given by the NET model are of significance and importance.

4.Calculation Results and Discussion

Fig.2 shows the variations of four characteristic parameters with membrane adsorption property including the adsorption constant (C) and maximum moisture uptake (wmax),and Fig.2(a)-(d)are for the heat transfer coefficient (α),molar-driven heat transfer coefficient (α′),thermal-driven mass transfer coefficient (β′),and mass transfer coefficient (β),respectively.In the calculations,only the adsorption constant and maximum moisture uptake are varied,with all the other parameters being fixed at the representative condition,which is defined in Section 3.It can be seen from Eq.(37)that the four characteristic parameters directly affect the transmembrane heat and mass fluxes:a larger characteristic parameter means a greater contribution of the driving force to the flux,and a positive parameter means a positive contribution and a negative parameter implies a negative contribution.From Fig.2(a) and (d),the heat and mass transfer coefficients are both positive,this is natural and logical because the temperature difference causes the heat transfer while the humidity ratio difference induces the mass transfer.From Fig.2(b) and (c),the molar-driven heat transfer and thermal-driven mass transfer coefficients are both negative,implying that the humidity ratio difference acts to reduce the heat transfer and the temperature difference works to diminish the mass transfer.

It is also seen from Fig.2 that,the heat transfer coefficient in Fig.2a changes very slightly with the adsorption constant (C) and maximum moisture uptake (wmax),implying that the membrane adsorption property has only minimal influence on the heat transfer.The primary reason is that the membrane adsorption property is directly related to the mass transfer rather than the heat transfer.For the heat transfer coefficient,the driving force is the temperature difference and the flux is the heat flux,neither of them has a direct relationship with the membrane adsorption property.The heat transfer coefficient has values closer to but less than half the convective heat transfer coefficient ofh1=h2=50.3 W?m-2-?K-1,this is appropriate because the conductive heat transfer resistance in membrane is much smaller than the convective heat transfer resistances on the two sides of membrane,as addressedby Min and Su[9].The above result provides an additional support for the reliability of the present calculations.In contrast,the other three coefficients in Fig.2(b),(c),and (d) are all affected significantly by the membrane adsorption property.With increasing maximum moisture uptake (wmax),the absolute values of the four coefficients in Fig.2(a)-(d) all tend to increase.

With increasing adsorption constant(C),extreme values appear in Fig.2.This may be attributed to the variations of the adsorption curve withC,which affects the four coefficients shown by Fig.2.By taking the partial derivative of the mass transfer coefficient β toCto be zero,i.e.

we obtain

whereCoptis the optimal adsorption constant,which depends on the mean relative humidity of the air fluids on the two sides of membrane.The optimal adsorption constant has a value of aboutC=2.5 for all cases.According to Eq.(52),the higher the mean relative humidity,the larger the optimal adsorption constant,implying that the membrane with largeCshould be used in humid climate while that with a smallerCshould be employed in a dryer climate.Similar discussion can be found in Min and Wang [33].

As indicated by Eq.(37),the heat transfer may be affected by both the heat transfer coefficient in Fig.2(a) and the molardriven heat transfer coefficient in Fig.2(b),while the mass transfer may be affected by both the thermal-driven mass transfer coefficient in Fig.2(c) and the mass transfer coefficient in Fig.2(d).Fig.3 depicts various heat and mass fluxes for different membrane adsorption properties,with Fig.3(a) comparing the heat fluxes caused by the temperature difference(qT)and those by the humidity ratio difference (qW),and Fig.3(b) comparing the mass fluxes induced by the temperature difference(JT)and those by the humidity ratio difference (JW).In Fig.3(a),qSexpresses the convective heat flux (sensible heat flux),which is equal to the sum ofqTandqW.qThas a positive and predominant contribution toqSwhileqWhas a negative and insignificant contribution toqS,withqWaccounting for 1%–2% ofqS.As a result,qSis mainly controlled byqT,which shows little variation with the membrane adsorption constant and maximum moisture uptake,the reason is that the heat transfer coefficient changes very slightly with them,as observed in Fig.2(a).In Fig.3(b),Jexpresses the convective mass flux,which is equal to the sum ofJTandJW.JThas a negative and minor contribution toJwhileJWhas a positive and major contribution toJ,withJTaccounting for 7%–14% ofJ.As a result,Jis primarily determined byJW,which shows a maximum at a certain value ofC.

Fig.2.Variations of characteristic parameters with membrane adsorption property.

Fig.3.Comparison of heat and mass fluxes for different membrane adsorption properties.

Fig.4.Variations of entropy generation rates caused by heat and mass transfer across membrane with membrane adsorption property.

Fig.4 illustrates the variations of the entropy generation rates caused by the heat and mass transfer across membrane with the membrane adsorption property,of which Fig.4(a) is for the entropy generation rate caused by the temperature difference induced heat transfer,σq,Fig.4(b) is for the entropy generation rate caused by the humidity ratio difference induced mass transfer,σJ,Fig.4(c) is for the total entropy generation rate,σtot,which is equal to the sum of σq,and σJ,and Fig.4(d)shows a direct comparison of σq,σJand σtotforwmax=0.25 kg?kg-1maximum moisture uptake.From Fig.4(a),the entropy generation rate caused by the heat transfer(σq)is affected very gently by the membrane adsorption property:as the adsorption constant (C) increases,σqfirst decreases and then increases;as the maximum moisture uptake(wmax) increases,σqtends to decrease.From Fig.4(b),the entropy generation rate caused by the mass transfer (σJ) is affected significantly by the membrane adsorption property:asCincreases,σqfirst increases and then decreases;aswmaxincreases,σqtends to increase.The variations of σqand σJin Fig.4(a)and 4(b)are similar to those ofqTandJin Fig.3(a) and 3(b),because larger heat and mass fluxes may lead to larger entropy generation rates.From Fig.4(c),the total entropy generation rate is more insensitive to the membrane adsorption property because the effect of the membrane adsorption property on σqis opposite to that on σJ,making the resultant of σqand σJ,i.e.σtot,to be more insensitive to the membrane adsorption property.Fig.4(d) directly compares σq,σJ,and σtot,it is seen that σqis much larger than σJ,so σtotis mainly controlled by σq.

Fig.5.Variations of characteristic parameters with membrane transport property.

As we all know,the entropy generation is a measure of irreversibility,it reflects the loss of the ability to do work.The present research deals with the coupled heat and mass transfer across membrane,it does not involve the issue of work and its loss,so more attention should be paid to the four characteristic parameters,which directly affect the transmembrane heat and mass transfer.Fig.5 illustrates the variations of the four characteristic parameters with the membrane transport property including the membrane thermal conductivity (λm) and in-membrane moisture diffusivity(Dwm),and Fig.5(a),5(b),5(c)and 5(d)are for the heat transfer coefficient (α),molar-driven heat transfer coefficient (α′),thermaldriven mass transfer coefficient (β′),and mass transfer coefficient(β),respectively.In the calculations,only λmandDwmare changed,with all the other parameters being fixed at the representative condition,which is defined in Section 3.From Fig.5(a),the heat transfer coefficient varies very slightly with λmandDwm,the reason is that the membrane thermal resistance accounts for only a small amount(10%or less)of the total thermal resistance,which is the sum of the membrane resistance and the convective resistances on both sides of membrane.From Fig.5(b)and 5(c),the molar-driven heat transfer coefficient and thermal-driven mass transfer coefficient are both negative,meaning that the humidity ratio difference has a negative contribution to the heat transfer and the temperature difference has a negative contribution to mass transfer,and the larger their absolute values,the more obvious their effects.The absolute values of the molar-driven heat transfer coefficient and thermal-driven mass transfer coefficient are affected by λmandDwmin a similar manner,both of them increase with reducing λmor increasingDwm.From Fig.5(d),the mass transfer coefficient shows little change with λmbut increases withDwm,the main reason for the little change with λm,again,is due to the small membrane thermal resistance relative to the total thermal resistance.

Fig.6 corresponds to Figs.2 and 5,it is for the temperature(T10)and relative humidity(φ10)of the air fluid with higher temperature and humidity,which mimics the outdoor air,as seen in Fig.1.From Fig.6(a),bothT10and φ10has minimal influence on the heat transfer coefficient.AlthoughT10may affect the heat transfer through ΔT=T10–T20,the heat transfer coefficient itself shows little variation.From Fig.6(b) and 6(c),asT10goes up,the absolute value of the molar-driven heat transfer coefficient decreases for low φ10but turns to increase for higher φ10,and that of the molar-driven mass transfer coefficient,on the whole,tends to increase.As φ10increases,the absolute values of the molar-driven heat transfer coefficient and thermal-driven mass transfer coefficient both increase.From Fig.6(d),asT10rises,the mass transfer coefficient decreases for low φ10but turns to increase for higher φ10;as φ10increases,the mass transfer coefficient increases.As stated earlier,the magnitude and sign of each characteristic parameter actually reflect the extent and orientation that each driving force exerts on each flux,and the larger the absolute value of the parameter,the more significantly the driving force affects the flux.

Fig.6.Variations of characteristic parameters with air temperature and relative humidity.

5.Conclusions

A theoretical model describing the coupled heat and mass transfer across a membrane has been developed based on the non-equilibrium thermodynamic theory,expressions for the four coupling characteristic parameters including the heat transfer coefficient,molar-driven heat transfer coefficient,thermal-driven mass transfer coefficient,and mass transfer coefficient are derived,and the Onsager’s reciprocal relation is confirmed to verify the reliability of the model.The conclusions are as follows:

(1) The four characteristic parameters directly affect the transmembrane heat and mass fluxes:the heat and mass transfer coefficients are both positive,meaning that the temperature difference has a positive contribution to the heat transfer and the humidity ratio difference has a positive contribution to the mass transfer.The molar-driven heat transfer and thermal-driven mass transfer coefficients are both negative,implying that the humidity ratio difference acts to reduce the heat transfer and the temperature difference works to diminish the mass transfer,although their effects are very weak.

(2) The heat transfer coefficient varies very slightly with the membrane property and air state,while the molar-driven heat transfer coefficient,thermal-driven mass transfer coefficient,and mass transfer coefficient are all affected moderately by the membrane property and air state.

(3) The convective heat flux consists of the heat flux caused by the temperature difference and that by the humidity ratio difference,and the former is substantially larger than the latter.The convective mass flux is comprised of the mass flux induced by the temperature difference and that by the humidity ratio difference,and the former is obviously smaller than the latter.

(4) The total entropy generation consists of the entropy generation caused by the temperature difference induced heat transfer and that by the humidity difference induced mass transfer,and the former is much larger than the latter.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research is funded by Beijing Natural Science Foundation(3182015).

Nomenclature

Cabsorption constant of membrane

cpspecific heat at constant pressure,kJ?kg-1?K-1

Dwmmoisture diffusivity in membrane,kg?m-1?s-1

hconvective heat transfer coefficient,W?m-2?K-1

Jmass flux,kg?m-2?s-1

JTmass flux driven by temperature gradient,kg?m-2?s-1

JWmass flux driven by humidity ratio gradient,kg?m-2?s-1

kconvective mass transfer coefficient,kg?m-2?s-1

Lphenomenological coefficient or adsorption heat,J?kg-1

Ppressure,Pa

P0atmospheric pressure,Pa

Pssaturation pressure of water vapor,Pa

Pvpartial pressure of water vapor,Pa

qheat flux,W?m-2

qLadsorption heat flux,W?m-2

qSconvective heat flux,W?m-2

qTheat flux driven by temperature gradient,W?m-2

qWheat flux driven by humidity ratio gradient.W?m-2

RT,mheat transfer resistance through membrane,m2?K?W-1

RT,OVoverall heat transfer resistance,m2?K?W-1

RW,mmass transfer resistance through membrane,m2?s?kg-1

RW,OVoverall mass transfer resistance,m2?s?kg-1

Rggas constant of water vapor,J?kg-1?K-1

Tabsolute temperature,K

Whumidity ratio,kg?kg-1

wmaxmaximum moisture uptake,kg?kg-1

Xthermodynamic force

α heat transfer coefficient,W?m-2?K-1

Α′molar-driven heat transfer coefficient,W?m-2

β mass transfer coefficient,kg?m-2?s-1?K-1

Β′thermal-driven mass transfer coefficient,kg?m-2?s-1

δ thickness of membrane,m

θ moisture content of membrane surface,kg?kg-1

λ heat conductivity,W?m-1?K-1

μ chemical potential,J?kg-1

σ entropy generation rate,W?m-2?K-1

φ relative humidity

Subscripts

m membrane or mean

OV overall

opt optimal

tot total

1 membrane surface on side of high temperature and humidity

10 air with high temperature and humidity

2 membrane surface on side of low temperature and humidity

20 air with low temperature and humidity