Two-dimensional modeling of a cross flow plate and frame membrane humidifier for fuel cell applications

Journal of Membrane Science 409–410 (2012) 285–301

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Two-dimensional modeling of a cross flow plate and frame membrane humidifier for fuel cell applications Mayank Sabharwal a , Christian Duelk b , Divesh Bhatia a,∗ a b

Fuel Cell System Modeling, Electrical and Electronics Dept, Mercedes-Benz Research and Development India, Pine Valley, Embassy Golf Links, Bangalore 560071, India Fuel Cell System Development, Group Research and Advanced Engineering, GR/AFS Neue Strasse 95, 73230 Kirchheim/Teck-Nabern, Germany

a r t i c l e

i n f o

Article history: Received 3 August 2011 Received in revised form 28 March 2012 Accepted 28 March 2012 Available online 6 April 2012 Keywords: Membrane Humidifier model Cross flow Plate and frame Fuel cell

a b s t r a c t We present a steady-state two-dimensional model of a cross flow plate-and-frame membrane humidifier for a fuel cell system. Sensitivity analysis of the model is performed for various operating conditions and geometric parameters. The analysis shows that the water transfer rate increases with an increase in the velocities at the dry and wet sides, wet side inlet pressure, wet side inlet relative humidity, dry side inlet temperature and the number of plates. The relative humidity at the dry side outlet is found to increase with an increase in the wet side velocity, pressure at the dry and wet sides, wet side inlet relative humidity and the number of plates. Simulations are performed to study the performance of a system consisting of a humidifier and a fuel cell stack. It is observed that low current densities combined with a low stoichiometric ratio result in a high water transfer rate, water recovery ratio and relative humidity at the stack inlet. Maintaining a high operating pressure or using a high number of plates in the humidifier improves the performance of the humidifier, but they result in a respective increase in the parasitic losses, and the cost and size of the humidifier. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The fuel cell technology has gained significant importance as a substitute for conventional energy sources over the recent decades. The wide range of applicability ranging from cell phones to automobiles to utility scale power plants combined with high efficiency, excellent part-load performance, low emissions of regulated pollutants and a wide size support range make the fuel cell a promising technology. This has drawn the attention of researchers and manufacturers who are working towards implementation of this new important technology [1]. The polymer electrolyte membrane fuel cell (PEMFC) is the most commonly used fuel cell technology in the automobile industry. The underlying principle is the extraction of chemical energy by the reaction of hydrogen and oxygen to produce water and convert the same to electrical energy. The fuel cell system comprises of the fuel cell stack, fuel processor, thermal management system, air management system, power management system and water management system. Each of these plays an essential role for the optimal working of the fuel cell.

∗ Corresponding author. Tel.: +91 80 6768 8211; fax: +91 80 6768 7111. E-mail addresses: [email protected], [email protected] (D. Bhatia). 0376-7388/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2012.03.066

Water management system is an important component of the fuel cell system because it is responsible for supplying appropriate amount of water to the air entering the cathode. External gas-togas humidification from the cathode exhaust to the cathode inlet is achieved by means of a humidifier. The humidifier optimally uses the water produced in the cell reaction to maintain the desired humidity at the cathode inlet. Efficient operation of a fuel cell stack occurs within a narrow range of humidity. A low water content of the gases entering the cathode results in a low ionic conductivity of the membrane, thereby increasing the ohmic losses [2]. In contrast, high humidity could result in water condensation and channel flooding. If the water produced in the catalyst is not removed effectively, the water residing in the pores of the gas diffusion layer blocks the pathway of reactants, thus resulting in voltage losses [3]. Hence, maintaining the relative humidity of the gases entering the cathode is a crucial aspect of fuel cell system development. Weber and Darling [4] identified water transport plates as a means of managing the optimal water requirements of the stack. The performance of the plates was analyzed and the plates were found to be effective for operation of a PEMFC below 100 ◦ C. Huizing et al. [5] proposed a design methodology for the plateand-frame humidifier based on the ratio of the residence time and the characteristic diffusion time. The algorithm was validated by rapid prototyping of different flow channel geometries. Chen and Peng [6] used thermodynamic modeling to study the dynamics of a membrane humidifier. They studied the dynamic behaviour

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of the pressure, flow rate, temperature and relative humidity and the effect of channel flow structures, dimensions and cross-section geometry. They developed a control strategy comprising of a proportional controller to reduce the transient effects on the system performance. Similar dynamic models were also developed and validated by Chen et al. [7], Park et al. [8] and Zaremba and Jennings [9]. Zhang and Huang [10] developed a model for a counter-current hollow fiber membrane module and studied the convective and diffusive resistances for mass and heat transfer. Zhang [11] performed simulations for a cross flow membrane based enthalpy exchanger to study the variation of Sherwood and Nusselt numbers within the humidifier. Recently, detailed CFD studies of the humidifier for cocurrent, counter and cross flows were performed by Zhang et al. [12]. They found the boundary conditions and the Nusselt and Sherwood numbers for various flow arrangements. Zhang and Niu [13] used the effectiveness method to obtain correlations for the moisture content and temperature profiles in a cross-flow plate-and-frame humidifier. However, the model did not consider the effect of heat conduction and heat losses. These correlations were further improved by Kadylak et al. [14] by relaxing certain assumptions. Rather than using the Clausius–Clapeyron equation for the saturation vapour pressure, they used the Goff–Gratch and Hyland–Wexler equations, which are considered to be the most accurate. In addition, they used a non-linear correlation between absolute and relative humidity in contrast to the linear relation used by Zhang and Niu [13]. However, the effect of water transfer from wet side to the dry side on the heat transfer rate was not considered in the correlations developed by Zhang and Niu [13] and Kadylak et al. [14]. Kadylak and Merida [15] further proposed water recovery ratio as a good performance measurement parameter based on their experimental studies on a counter flow membrane humidifier test bench. They observed a low water uptake below a relative humidity of 80%, beyond which a steep increase in the uptake was observed. Temperature effects were found to be almost three times more pronounced than the pressure effects. Cave and Merida [16] studied the effect of flow rates on the temperature and concentration profiles in a single channel membrane humidifier. Their experimental results showed that heat losses affected the overall performance significantly. However, most of the models in the literature neglect the effect of heat losses on the humidifier performance. Zhang et al. [17] reported that cross flow configuration of a humidifier minimizes the pressure drops in the streams. This becomes important if we consider the entire fuel cell system where the pressure drop across each component cumulatively affects the overall efficiency of the fuel cell system. The present work outlines the modeling of a cross flow plate-and-frame humidifier for a fuel cell system. To our knowledge, the two dimensional variation of concentration and temperature in a cross flow humidifier has not been studied in the literature. In the present work, a two dimensional model is proposed which considers the effect of concentration, velocity, pressure and temperature on each other as well as on the performance of the humidifier. Mass transport by convection, mass and heat transfer between the dry and wet side streams, heat conduction within the streams and the dependence of transport properties on the stream conditions are incorporated into the model. The effect of water transfer from the wet side to the dry side on the heat transfer rate is incorporated into the model, which is neglected in some of the previous modeling studies. Since heat losses affect the humidifier performance quite significantly [16], they are taken into consideration as well. The integration of the humidifier and a fuel cell stack is studied by performing a sensitivity analysis on the pressure, current density and stoichiometric ratio. Sensitivity analysis of the various geometric parameters and operating conditions provides key insight into the operation of the humidifier and various design considerations.

Table 1 Geometric dimensions of the cross-flow plate-and-frame humidifier. Parameter

Numerical value

Li Wi d wc wl

0.5 m 0.5 m 0.5 mm 5 mm 1.04 mm

2. Model development 2.1. Geometry details Fig. 1(a) illustrates the typical geometry and the flow directions for a cross flow plate-and-frame humidifier. There are several channels through which the flow takes place. A magnified view of the channels and the space between them is provided in the inset. The geometric dimensions are given in Kadylak et al. [14] and are summarized in Table 1. The width of the channels is 5 mm, whereas the height of the channel is 0.5 mm [17]. The ratio of the channel area and the total cross-sectional area (including the land area) was found to be 0.83. For our modeling study, we approximate the total flow through channels by one single flow region and the schematic is shown in Fig. 1(b). The flow entering the dry side of the humidifier is usually compressed air devoid of any moisture. The wet side inlet comprises a mixture of air and water vapour, which comes from the fuel cell stack. This stream has a high relative humidity because of the presence of water, which is produced by the electrochemical reaction of H2 and O2 in the fuel cell stack. The humidifier optimizes the water requirement of the fuel cell system by enabling the exchange of water between the cathode outlet and the dry side inlet of the humidifier. In a humidifier, a number of plates of the form shown in Fig. 1(a) are stacked on top of each other to achieve the load capacity and performance metric required by the fuel cell system. 2.2. Assumptions The model is based on the following assumptions: (i) steady-state conditions, (ii) all gases obey the ideal gas law, (iii) diffusion of water within the flow channels is ignored both in the flow direction and the direction transverse to the flow. Water is assumed to diffuse only from the wet side to the dry side through the membrane, (iv) diffusion of air across the membrane is ignored, (v) the water is assumed to diffuse at the same temperature as that of the wet side, (vi) flow in different channels has been approximated by flow in a bulk space of equivalent width, (vii) the specific heats are assumed constant, (viii) the thermal conductivity of air and diffusivity of water in air are assumed constant, (ix) the pressure gradient is assumed to be a linear function of the velocity, and (x) the condensation of water vapour is not considered. 2.3. Model equations The mole balance and energy balance equations were written in two dimensions for the dry and wet sides. The pressure gradient was assumed to be a linear function of the velocity, and is discussed later. Since the concentration, pressure, and temperature are interdependent on each other, the equations for these variables were

M. Sabharwal et al. / Journal of Membrane Science 409–410 (2012) 285–301

a

287

Li

i

wl

W

Dry side flow direction

d wc

Channels for dry side flow

Membrane Wet side flow direction

Channels for wet side flow

b

L Dry Side Inlet

W

Wet Side Outlet

Dry Side Outlet (to cathode inlet)

Membrane Wet side inlet (from cathode outlet)

y

Water Transfer

x Fig. 1. (a) Schematic of a cross flow plate-and-frame humidifier. (b) Schematic of the humidifier used in the modeling study.

combined together using the ideal gas law to obtain the final set of equations. The water mole balance equations for the dry and wet sides are given as follows: ∂cd c ∂pd (cw − cd )(pd − cd RTd ) c ∂T = d + Kc − d d pd ∂x ud Hd pd Td ∂x ∂x

(1)

∂cw (cw − cd )(pw − cw RTw ) cw ∂pw cw ∂Tw = − Kc − pw ∂y uw Hw pw Tw ∂y ∂y

(2)

Here, cd and cw are the concentrations of water at the dry and wet sides, respectively, pd and pw are the pressures of the gas at the dry and wet sides, respectively, ud and uw are the velocities of the gaseous flow at the dry and wet sides, respectively, Td and Tw are the temperatures of the gases at the dry and wet sides, respectively, Hd and Hw are the heights of the flow areas for the dry and wet sides, respectively, R is the universal gas constant, and Kc is the overall mass transfer coefficient between the wet and dry sides. The first term on the right hand side of Eqs. (1) and (2) represents the effect of pressure drop on the concentration gradient. The second term represents the water transfer from the wet side to the dry side, whereas, the third term represents the effect of the temperature gradient on the concentration gradient. The relations for different differentials like ∂pd /∂x, ∂pw /∂y, ∂Td /∂x and ∂Tw /∂y obtained in Eqs. (1) and (2) are given later. The variation of velocities at the dry and wet sides in the axial and transverse direction is given by Eqs. (3) and (4), respectively, as follows: ∂ud u ∂p u ∂T Kc RTd (cw − cd ) + d d =− d d + pd ∂x pd Hd Td ∂x ∂x

(3)

∂uw uw ∂pw uw ∂Tw Kc RTw (cw − cd ) + =− − pw ∂y pw Hw Tw ∂y ∂y

(4)

The velocity variation was obtained using an equation for the dependence of the total molar flow rate on the various operating conditions. It can be observed from Eqs. (3) and (4) that the variation in velocities due to the pressure drop, molar flux of water and temperature is considered in the model. However, we expect that the effect of water flux on the velocity would be negligible as compared to the effects of pressure drop and temperature change. This is because the molar flow rate of water transferred between the wet and dry sides is negligible as compared to that of air. The dependence of pressure gradient in the axial and transverse directions is given by Eqs. (5) and (6), respectively, where Kpd and Kpw are constants. ∂pd = −Kpd ud ∂x

(5)

∂pw = −Kpw uw ∂y

(6)

The energy balance equations for the dry and wet sides are given by (Tw − Td )[Hf + Kc Cp,H2 O (cw − cd )] + kHd (∂2 Td /∂y2 ) ∂Td = Hd (ud cd Cp,H2 O + udi cair,di Cp,air ) ∂x

(7)

−Hf (Tw − Td ) + kHw (∂2 Tw /∂x2 ) ∂Tw = Hw (uw cw Cp,H2 O + uwi cair,wi Cp,air ) ∂y

(8)

Here, Hf is the overall heat transfer coefficient, cair is the concentration of air, Cp,H2 O is the specific heat capacity of water, Cp,air is the specific heat capacity of air, k is the thermal conductivity of the gas and the subscript ‘i’ denotes the inlet conditions. The terms ∂2 Td /∂y2 and ∂2 Tw /∂x2 arise from considering the thermal conduction along the transverse direction within the dry and wet side streams, respectively. The first term within the square brackets

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in the numerator of Eq. (7) represents the heat transfer between the wet and dry sides. As described earlier, Hf represents the overall heat transfer coefficient and Kc Cp,H2 O (cw − cd ) represents a coefficient for the heat transferred due to water transfer from the wet side to the dry side. The heat transferred due to the mass transfer of water has been neglected in the literature correlations for a cross-flow plate-and-frame humidifier [13,14]. In Eq. (8), the mass transfer term is not present because it is assumed that the water is transferred at the temperature of the wet side. In the current work, the mass transfer coefficients for the dry and wet sides are obtained using the asymptotic Sherwood number calculated by Zhang [11] for a cross-flow membrane-based enthalpy exchanger and is given as Sh = 7.81

(9)

Here, Sh represents the Sherwood number and was obtained for a humidifier with an aspect ratio of 50 [11]. The same value was used for the wet and dry sides as the two flow regions are essentially identical. By solving the momentum, energy and concentration equations simultaneously, Zhang [11] calculated the variation of Sherwood number with the position in a cross-flow humidifier. We used these values and estimated the length over which the Sherwood number varies with the axial position. We found that for the operating conditions used in the current study, the region of variable Sherwood number was less than 3% of the length of the humidifier. Hence, we use the asymptotic value of the Sherwood number given by Eq. (9). The mass transfer coefficient is calculated as follows [11]: kc = Sh

D  g

d˝

(10)

Here, kc is the convective mass transfer coefficient, d is the hydraulic diameter and Dg is the diffusivity of water in air, for which a value of 2.12 × 10−5 m2 s−1 is used in this study. The overall resistance for mass transfer is calculated by adding the convective resistance of the dry and wet sides and the diffusive resistance of the membrane. The overall mass transfer coefficient is given by [14] 1 Kc = ((1/kc,d ) + (1/kc,w ) + (ım /Dm ))

(11)

where kc,d is the convective mass transfer coefficient on the dry side, kc,w is the convective mass transfer coefficient on the wet side, ım is the membrane thickness and Dm is the diffusivity of water in the membrane. The hydraulic diameter for the flow geometry is defined as d˝d =

4A˝d P˝d

(12a)

d˝w =

4A˝w P˝w

(12b)

where Ad and Aw are the flow cross-section areas for the dry side and wet side, respectively, whereas, Pd and Pw are the wetted perimeters on the dry side and wet side, respectively. The effect of a supporting diffusion media is not considered in the current work. It was observed that for the operating conditions and parameter values used in the current study, the convective resistance of the dry and wet sides was approximately three times higher than the diffusive resistance of the membrane. Thus, the convective transfer from the gas phase to the membrane was the rate-limiting step in the overall water transfer process. This implies that if the addition of diffusion media is considered in the model, its effect on the overall mass transfer coefficient would not be significant and the trends obtained in the current work would not be affected. However, if the diffusion media is thick or the diffusivity of water through the diffusion media is low, the resistance offered by the diffusion media

to the mass transfer would be significant and cannot be neglected. Under such circumstances, the model would have to be modified to account for the resistance offered by the diffusion media. The diffusivity of water in the membrane (Dm ) was evaluated using the following expression [6–8,18]:



Dm = D exp 2416

 1

303

−



1 Tmem

(13)

where Tmem is the membrane temperature and the coefficient D␭ is an empirical constant, defined as follows [6–8,19]:

D =

⎧ −1 10 ⎪ ⎪ ⎪ ⎪ −6 ⎨ 10 (1 + 2(m − 2))

m < 2 2 ≤ m ≤ 3

⎪ 10−6 (3 − 1.67(m − 3)) 3 < m < 4.5 ⎪ ⎪ ⎪ ⎩ −6 1.25 × 10

(14)

m ≥ 4.5

Here, m is the membrane water content and can be defined as m = 0.043 + 17.81am − 39.85a2m + 36a3m

(15)

where am is the membrane relative humidity and can be determined as am =

d + w 1 × 2 100

(16)

Here, d is the relative humidity at the dry side and w is the relative humidity at the wet side. Similar to the mass transfer coefficient, the heat transfer coefficient is found using the asymptotic Nusselt number (Nu) found by Zhang [11] and is given as Nu = 7.78

(17)

The convective heat transfer coefficient for the dry and wet side (hf ) is given by [11] hf = Nu

 k  d˝

(18)

where k is the thermal conductivity of the gas. Similar to the overall mass transfer coefficient, the overall heat transfer coefficient is calculated from the convective heat transfer resistance of the dry side (1/hf,d ), convective heat transfer resistance of the wet side (1/hf,w ) and the resistance of the membrane (ım /km ). The overall heat transfer coefficient is given as [20] Hf =

1 ((1/hf,d ) + (1/hf,w ) + (ım /km ))

(19)

where km is the thermal conductivity of the membrane, and subscripts d and w denote the dry and wet sides, respectively. The inlet conditions for the various variables are specified and are given by the following equations: At x = 0, cd = cdi , Td = Tdi , pd = pdi , ud = udi

(20)

At y = 0, cw = cwi , Tw = Twi , pw = pwi , uw = uwi

(21)

The heat losses from the edges of the plates are represented by boundary conditions which are given as follows: At x = 0, k

∂Tw = Hl (Tw − Tamb ) ∂x

At x = L, −k At y = 0, k

∂Tw = Hl (Tw − Tamb ) ∂x

∂Td = Hl (Td − Tamb ) ∂y

At y = W, −k

∂Td = Hl (Td − Tamb ) ∂y

(22) (23) (24) (25)

M. Sabharwal et al. / Journal of Membrane Science 409–410 (2012) 285–301

W

Dry inlet

Wet outlet

L

(0,W)

289

Dry outlet

Membrane

Heat loss from wet side (0,0)

y

(L,0)

Wet inlet

Heat loss from dry side x Fig. 2. Schematic of the humidifier showing the heat losses from the dry and wet sides (

In Eqs. (22)–(25), W is the effective width of the plate, L is the effective length of the plate, Tamb is the ambient temperature and Hl is the coefficient of heat loss to the surroundings. For ease of interpretation, Fig. 2 gives a schematic of the humidifier, showing the heat losses from the dry and wet sides. 3. Numerical solution Finite differencing was used to discretize Eqs. (1)–(8). The first order derivatives were discretized using backward differencing and the second order derivatives were discretized using central differencing. Upon discretization, the differential equations were reduced to a set of non-linear equations with temperature, water concentration, velocity and pressure on the dry and wet sides as the unknown variables. Both the x and y axes were discretized into 20 equal segments, which resulted in a system of 7056 non-linear equations. These set of equations were solved using the solver ‘fsolve’ in MATLAB® . A major challenge was to reduce the computation time in order to easily implement the model in system or vehicle models. In order to achieve this, suitable initial guesses were provided to the solver. For this purpose, a simple model was developed neglecting transverse conduction, heat losses and the axial and transverse variations in the pressure and velocity. The model equations were then reduced to linear equations upon discretization. The obtained set of linear equations was solved using matrix algebra to get the initial guesses for the detailed model consisting of non-linear equations. Secondly, the location of the non-zero elements in the Jacobian matrix was provided to the solver as an input, thereby reducing the computer memory requirements and computation time. This is because the number of non-zero elements of the Jacobian matrix was very less as compared to the total number of elements in the matrix. The discretized equation set was solved for various geometric parameters and operating conditions. Table 2 outlines the list of parameters used for the simulations. Here, an effective length and

Table 2 List of parameters used for the simulations. Parameter

Numerical value

L W Hd Hw Np ım km

0.415 m 0.415 m 0.5 mm 0.5 mm 30 0.02 mm 0.18 Wm−1 K−1

represents heat flow,

represents mass flow).

width were calculated based on the total flow area considering the channel width and number of channels reported by Kadylak et al. [14]. Np in Table 2 represents the number of plates. Solution of the equations provided a two-dimensional variation of the unknown variables. Numerical integration of the two-dimensional profiles at the outlet of the humidifier was performed to calculate measurable quantities such as molar flow rates. These were also used to verify the mass balance for water and air in the simulation studies. 4. Results and discussion Sensitivity analysis was performed to study the effect of operating conditions and geometric parameters. A set of base operating conditions was chosen such that they were indicative of a fuel cell system. The operating conditions are given in Table 3. To study the effect of a particular parameter, its value was changed keeping the other parameters constant. In Table 3, N˙ air,di and N˙ air,wi are the molar flow rates of air (without water) at the dry and wet side inlets, respectively, whereas, di and wi are the relative humidities at the dry and wet side inlets, respectively. The relative humidity at the dry side inlet was kept at 0% in conjunction with a fuel cell system where the air entering the dry side is usually devoid of any moisture. The wet side pressure was chosen to be lower than the dry side pressure to account for the pressure drop across the humidifier dry side and the fuel cell stack before the gas enters the wet side of the humidifier. Unless otherwise explicitly mentioned, the calculations for the sensitivity analysis are performed at the conditions given in Table 3. For the set of conditions given in Table 3, the velocities at the dry and wet sides are 4.13 and 6.26 m s−1 , respectively. In the range of sensitivity analysis, the Reynolds number for both the dry and wet sides was less than 1000 indicating that the flow is laminar. The Peclet number for the dry side was of the order of 104 , whereas, that on the wet side was of the order of 105 . This is consistent with the assumption of neglecting the diffusion terms in the flow direction.

Table 3 Base operating conditions at the inlets used for the sensitivity analysis. Parameter

Numerical value

N˙ air,di N˙ air,wi

0.8 mol s−1 0.8 mol s−1 1.8 bar 1.5 bar 75 ◦ C 75 ◦ C 0% 80%

pdi pwi Tdi Twi di wi

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90

Temperature at wet side / oC

Concentration of H2O on dry side / mol m-3

12 10 8 6 4

85

80

2 0

0.4 0.4

75

0.3 0.3

Wet side flow direction / m

0.2

0.4

0.2 0.1

0.1 0

0

0.2

Dry side flow direction / m

Wet side flow direction / m

0

0

0.1

0.2

0.3

0.4

Dry side flow direction / m

Fig. 3. Two-dimensional variation of water concentration at the dry side. Fig. 4. Two-dimensional variation of temperature at the wet side.

The transverse Peclet number for the dry and wet sides are defined in Eqs. (27) and (28), respectively, and are given as: Petr,d = Petr,w =

W 2 /Dg L/ud L2 /Dg W/uw

(27) (28)

It was observed that the transverse Peclet number on the dry side was of the order of 104 , whereas, that on the wet side was of the order of 105 . The high value of transverse Peclet number indicates that the transverse diffusion time for mass transfer in the flow region is much higher than the axial convection time. Hence, in the current work, the diffusion of water in the direction transverse to the flow is neglected. Fig. 3 shows the steady-state water concentration profile at the dry side of the humidifier. The operating conditions are the same as those given in Table 3. From Fig. 3, it is observed that the dry side water concentration increases along the dry side flow direction, which is due to the diffusion of water from the wet side. However, the dry side water concentration decreases along the wet side flow direction. This is because the water transfer leads to a decrease in its concentration at the wet side along the direction of the wet side flow. This results in a lower water concentration gradient between the wet and dry sides along the wet side flow direction, which in turn results in a decrease in the dry side water concentration along the wet side flow direction. Fig. 4 shows the two-dimensional steady-state temperature profile at the wet side of the humidifier. For these simulations, the dry side inlet was maintained at 90 ◦ C, the wet side inlet temperature was maintained at 75 ◦ C, and the other conditions were the same as those given in Table 3. The variation in the temperature profile is more pronounced as compared to the other variables. The temperature at the wet side increases along the wet side flow direction because of the heat transferred from the dry side. However, the temperature at the wet side decreases along the dry side flow direction. This is due to a decrease in the dry side temperature along the dry side flow direction, due to the heat transferred upstream. This results in lower temperature gradients between the dry and wet sides. Along the edges, the temperature decreases because of heat losses to the surroundings, which have been taken into account in the model. For sensitivity analysis, the model was simulated for a varied set of inlet conditions to analyze the effect of various geometric and operating parameters on the performance of the humidifier. The various operating parameters like the flow rates at the dry and wet

sides, pressures at the dry and wet sides, temperatures at the dry and wet sides and the relative humidity at the wet side were varied and their effect on the performance was analyzed. The effect of the geometric parameters like the number of plates was also analyzed. The performance parameters analyzed in this study are the concentration of water at the dry side outlet, the relative humidity at the dry side outlet, water transfer rate and the water recovery ratio. The relative humidity and the concentration of water at the dry side outlet are the most important parameters in the fuel-cell industry, as a narrow range of the relative humidity results in an optimum performance of the fuel cell stack. The water recovery ratio (WRR) is defined as the ratio of the molar rate of water transferred from the wet to the dry side and the molar flow rate of water entering the wet side. WRR =

N˙ H2 O,wi − N˙ H2 O,wo N˙ H2 O,wi

(29)

Here, N˙ H2 O,wi is the molar flow rate of water at the wet side inlet, and N˙ H O,wo is the molar flow rate of water at the wet side outlet. 2

The water recovery ratio becomes a critical parameter when the wet side water content becomes a limiting factor in the process of humidification. In the current work, the dew point temperature of the gases at the dry side outlet has also been analyzed in some cases. However, it should be mentioned that a single performance metric might not be sufficient to analyze the humidifier. For example, it will be shown later that an increase in the relative humidity at the dry side outlet might not imply an increase in the water transfer rate. 4.1. Effect of dry side molar flow rate To study the effect of dry side velocity on the performance of the humidifier, the total molar flow rate at the dry side inlet was varied between 0.2 and 1.2 mol s−1 . Fig. 5 shows the variation of the water concentration and relative humidity at the dry side outlet with the total molar flow rate at the dry side inlet. It is observed that the concentration of water and the relative humidity at the dry side outlet decrease with an increase in the total molar flow rate at the dry side inlet. This is because of a decrease in the residence time of the dry side gas with an increase in the dry side molar flow rate. Another factor which affects the water concentration and relative humidity at the dry side outlet is the pressure drop across the dry side. As expected, the pressure drop increases with an increase in the velocity at the dry side. The calculated pressure drop ranges

291

12

100 90

10

80 70

8

60 6

50 40

Concentration Relative humidity

4

30 20

2

Relative humidity at dry outlet / %

Water concentration at dry outlet / mol m

-3

M. Sabharwal et al. / Journal of Membrane Science 409–410 (2012) 285–301

10 0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Total molar flow rate at dry side inlet / mol s-1 Fig. 5. Variation of water concentration and relative humidity at the dry side outlet with total molar flow rate at the dry side inlet.

between 50 and 250 mbar, which is typical of a humidifier used in a fuel cell system. Increase in the pressure drop results in a lower partial pressure of water vapour at the dry side outlet and consequently, a lower relative humidity. Park and Oh [21] performed experiments on a water-to-gas humidifier with a Nafion® membrane and they too showed that an increase in gas velocity results in a decrease in the relative humidity. Fig. 6 shows the variation of the water transfer rate and water recovery ratio with the total molar flow rate at the dry side inlet. It is observed that the water transfer rate and the water recovery ratio increase with an increase in the total molar flow rate at the dry side inlet. As the dry side total molar flow rate increases, the water concentration at the dry side decreases, which was also observed in Fig. 5. A lower water concentration at the dry side results in an increase in the concentration gradient between the dry side and wet side. This leads to an increase in the water transfer rate. Since the flow rate of water entering the wet side is maintained constant, the water recovery ratio also increases with an increase in the dry side molar flow rate. Park et al. [8] showed by a modeling study that an increase in the flow rate at the dry side results in an increase in the vapour transfer rate. Cave and Merida [16] performed experiments

on a single channel humidifier and they observed an increase in the water recovery ratio with an increase in the dry side molar flow rates. 4.2. Effect of wet side molar flow rate The wet side velocity was varied by changing the total molar flow rate at the wet side between 0.25 and 1.51 mol s−1 , keeping the relative humidity constant at 80%. To maintain a constant relative humidity, the ratio of the molar rates of water vapour and air was maintained constant. Fig. 7 shows the variation of water concentrations at the dry and wet side outlets and the water transfer rate with the total molar flow rate at the wet side inlet. It is observed that the concentration of water at the dry side outlet, the concentration of water at the wet side outlet and the water transfer rate increase with an increase in the wet side molar flow rate. For a fixed relative humidity, an increase in the wet side total molar flow rate results in an increase in the water molar flow rate. This results in maintaining a high wet side water concentration, despite the water transfer to the dry side. The increased water concentration on the wet side leads to a higher concentration gradient between the wet

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Fig. 6. Variation of water transfer rate and water recovery ratio with total molar flow rate at the dry side inlet.

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Fig. 7. Variation of the water concentrations at the dry and wet side outlets and water transfer rate with the total molar flow rate at the wet side inlet.

and dry sides. This results in an increase in the water transfer rate and water concentration at the dry side outlet. It must be noted here that at high total molar flow rates, the water transfer rate and the water concentration at the dry side outlet approach an asymptotic value. At high total molar flow rates, the wet side outlet concentration tends to approach the wet side inlet concentration. Thus, the concentration gradient between the wet and dry side saturates at high molar flow rates, resulting in an asymptotic approach of the water transfer rate and the dry side outlet water concentration. Fig. 8 shows the variation of relative humidity at the dry side outlet and water recovery ratio with the total molar flow rate at the wet side inlet. It is observed that the relative humidity at the dry side outlet increases and the water recovery ratio decreases with an increase in the total molar flow rate at the wet side inlet. As explained earlier, the water concentration at the dry side outlet increases with an increase in the total molar flow rate at the wet side inlet. This results in an increase in the relative humidity at the dry side outlet. The decrease in the water recovery ratio with an increase in the total molar flow rate at the wet side inlet indicates that the water transfer rate does not increase at the same

rate as the increase in the molar flow rate of water at the wet side inlet. An interesting observation from the results in Fig. 8 is that at the lowest molar flow rate, the water recovery ratio is almost unity. This is in good agreement with the limiting case of the wet side velocity tending to zero. Low wet side velocities correspond to high residence times, thus resulting in transfer of most of the water entering the wet side. 4.3. Effect of dry side pressure The dry side inlet pressure was varied between 1.5 and 2 bar keeping the other operating conditions the same as those given in Table 3. In a fuel cell system, the wet side inlet pressure is dependent on the dry side inlet pressure. This is because of the pressure drop across the dry side and the stack before the gas enters the wet side of the humidifier. Therefore, the wet side inlet pressure is less than the dry side inlet pressure. However, for sensitivity analysis, the dry side pressure is varied keeping the wet side pressure constant. The combined effect of the dry and wet side pressure is discussed later in Section 4.8.1. 1

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0 0

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Total molar flow rate at wet side inlet / mol s

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Fig. 8. Variation of relative humidity at the dry side outlet and water recovery ratio with the total molar flow rate at the wet side inlet.

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Pressure drop

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Relative humidity

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Relative humidity at dry outlet / %

Pressure drop across dry side / mbar

250

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0 1.4

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2.1

Pressure at dry side inlet / bara Fig. 9. Variation of pressure drop across the dry side and the relative humidity at the dry side outlet with the dry side inlet pressure.

between the dry and wet sides, which reduces the water transfer rate. Also, the dry side velocity decreases with an increase in the dry side pressure. As explained earlier in Section 4.1, the water transfer rate decreases with a decrease in the dry side velocity. Thus, an increase in the water concentration at the dry side and a decrease in the dry side velocity result in a decrease in the water transfer rate. It is also observed from Fig. 10 that the water recovery ratio decreases with an increase in the pressure at the dry side inlet. This can be related to the decrease in the water transfer rate with an increase in the pressure at the dry side for fixed wet side inlet conditions. 4.4. Effect of wet side pressure The effect of wet side pressure is studied by varying the wet side inlet pressure between 1.2 and 1.7 bar, keeping the total molar flow rate constant and the water vapour mole fraction at the wet side inlet as 0.21. Fig. 11 shows the variation of the inlet and outlet velocities at the wet side and the water transfer rate with the wet

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Pressure at dry side inlet / bara Fig. 10. Variation of the water transfer rate and water recovery ratio with the dry side inlet pressure.

Water recovery ratio

Water transfer rate / mol s-1

Fig. 9 shows the variation of pressure drop across the dry side and the relative humidity at the dry side outlet with the dry side inlet pressure. It is observed that the calculated dry side pressure drop lies between 140 and 210 mbar. The pressure drop across the dry side decreases with an increase in the pressure at the dry side inlet because of the decrease in the dry side inlet velocity. The decrease in the pressure drop results in an increase in the total pressure at the dry side outlet, thus increasing the partial pressure of water. This results in an increase in the relative humidity at the dry side outlet with an increase in the dry side pressure, as is observed in Fig. 9. Fig. 10 shows the variation of the water transfer rate and water recovery ratio with the dry side inlet pressure. It is observed that the water transfer rate decreases with an increase in the pressure at the dry side inlet. The effect of the dry side inlet pressure on the water transfer rate can be studied by analyzing the water concentration and the velocity at the dry side. An increase in the dry side pressure results in an increase in the water concentration at the dry side. This results in a decrease in the concentration gradient

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Inlet velocity

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Pressure at wet side inlet / bara Fig. 11. Variation of the inlet and outlet velocities at the wet side and the water transfer rate with the wet side inlet pressure.

side inlet pressure. With an increase in the pressure at the wet side inlet for a fixed temperature and total molar flow rate, the velocity at the wet side inlet decreases. A decrease in the inlet velocity results in a decrease in the outlet velocity as well. It is observed from Fig. 11 that for low wet side inlet pressures, the outlet velocity is higher than the inlet velocity. This is because of lower pressure at the outlet as compared to the pressure at the inlet. However, with an increase in the wet side inlet pressure, the decrease in the wet side velocity results in a decrease in the pressure drop across the wet side. Hence, the difference in the inlet and outlet velocities decreases with an increase in the pressure at the wet side inlet. Moreover, as observed in Fig. 11, the water transfer rate increases with an increase in the wet side inlet pressure. This is because of an increase in the water concentration at the wet side. This results in a decrease in the molar flow rate at the wet side outlet, which further results in a lower wet side outlet velocity as compared to the wet side inlet velocity. Fig. 12 shows the variation of the relative humidity at the dry side outlet and the water recovery ratio with the wet side inlet pressure. Both the water recovery ratio and the relative humidity at the dry side outlet increase with an increase in the wet side inlet pressure. The increase in the relative humidity is because of an increased water transfer rate, thus resulting in a high partial pressure of water at the dry side outlet. The increase in the water recovery ratio is because of an increase in the water transfer rate, while keeping the molar flow rate of water vapour at the wet side inlet constant. 4.5. Effect of wet side relative humidity The relative humidity at the wet side is an important factor in the operation of the humidifier. The relative humidity at the wet side inlet was varied between 50% and 100% to analyze its effect on the performance of the humidifier. Although not shown here, it was found that the water transfer rate increases with an increase in the relative humidity at the wet side inlet. This is because for a high relative humidity at the wet side inlet, more amount of water enters the wet side, creating higher concentration gradients, and thereby increasing the water transfer rate. With an increase in the water transfer rate, the water concentration at the dry side outlet increases, thus resulting in an increase in the relative humidity at the dry side outlet. However, the water recovery ratio decreases slightly over the range of analysis because the rate of increase of

water transfer is less than the rate of increase of water flow rate at the wet side inlet. 4.6. Effect of temperature Temperature is an important parameter in the humidifier operation as it affects almost all the other operating variables like the velocities, concentrations, and relative humidities. The dry side temperature was varied between 60 and 85 ◦ C, while the wet side temperature was maintained constant at 75 ◦ C, and the other operating conditions were the same as those given in Table 3. Fig. 13 shows the variation of the water transfer rate and the water recovery ratio with the dry side inlet temperature. With an increase in the dry side inlet temperature, the velocity at the dry side increases which decreases the residence time. On the other hand, the concentration of water decreases with an increase in the temperature, thus resulting in a high concentration gradient between the dry and wet sides. The combined effect of these opposing factors results in a negligible effect of temperature on the water transfer rate, as is observed from Fig. 13. It is also observed from Fig. 13 that the water recovery ratio is almost constant. This is expected because the water transfer rate remains almost constant for fixed wet side inlet conditions. Fig. 14 shows the variation of the dew point temperature and the relative humidity at the dry side outlet with the dry side inlet temperature. It is observed that the dew point temperature at the dry side outlet increases slightly with an increase in the temperature at the dry side inlet. This is in good agreement with the experimental trends observed by Yu et al. [20], who observed an increase in the dew point temperature of the dry side outlet with an increase in the dry side inlet temperature. However, the relative humidity at the dry side outlet decreases with an increase in the dry side inlet temperature. With an increase in the dry side inlet temperature, the dry side outlet temperature increases. This results in an increase in the saturation pressure of water vapour at the dry side outlet. The partial pressure of water vapour at the dry side outlet is almost constant because of a negligible change in the water transfer rate with the dry side inlet temperature. This results in a decrease in the relative humidity at the dry side outlet. The fact that the relative humidity and the dew point temperature exhibit opposite trends shows that a single performance metric is not sufficient and one should choose suitable performance metrics to analyze the humidifier.

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0 1.1

1.2

1.3 1.4 1.5 1.6 Pressure at wet side inlet / bara

1.7

1.8

Fig. 12. Variation of the relative humidity at the dry side outlet and the water recovery ratio with the wet side inlet pressure.

The wet side temperature is dependent on the dry side temperature because the dry side stream after passing through the stack enters the wet side. Even though not shown here, the effect of the wet side temperature on the performance metrics was found to be similar to the effect of the dry side temperature. With an increase in the wet side temperature, no significant change in the water transfer rate was observed. However, the relative humidity at the dry side outlet was found to decrease. This is because of the heat transfer from the wet side to the dry side, which results in an increase in the saturation pressure at the dry side outlet. The partial pressure of water vapour at the dry side outlet did not change significantly with the wet side inlet temperature because of a nearly constant water transfer rate, thus resulting in a decrease in the relative humidity. 4.7. Effect of number of plates The number of plates in a humidifier is an important design parameter as it determines the flow characteristics and the cost

of the humidifier. A humidifier with a high number of plates can operate at high flow rates. Further, for the same inlet flow rates, increasing the number of plates would affect the water transfer rate because of a reduction in the gas velocities and an increase in the effective area for heat and mass transfer. The model developed assumes that there is no interaction between different plates, i.e. one dry plate interacts only with one wet side plate. The plate design is made in such a way that there is interaction between only one wet and one dry region. A sensitivity analysis on the number of plates was carried out using the operating conditions given in Table 3. The total molar flow rate of the gases at the wet and dry sides was maintained constant and the number of plates was varied between 20 and 60. Fig. 15 shows the variation of the water transfer rate and the pressure drops across the dry and wet sides with the number of plates. It is observed that the water transfer rate increases with an increase in the number of plates. With an increase in the number of plates, the overall effective area for mass transfer increases. However, the

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o

Temperature at dry side inlet / C Fig. 13. Variation of water transfer rate and the water recovery ratio with the dry side inlet temperature.

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Water transfer rate / mol s-1

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Dew point at dry outlet / C

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Temperature at dry side inlet / oC Fig. 14. Variation of dew point temperature and relative humidity at the dry side outlet with the dry side inlet temperature.

velocities at the dry and wet sides decrease because the flow is distributed in more number of plates. As explained in Sections 4.1 and 4.2, a decrease in the dry and wet side velocities reduces the water transfer rate. However, we observed that the effect of increase in the overall effective area for mass transfer dominates over the effect of the decrease in the velocities. It is also observed from Fig. 15 that the pressure drops decrease with the number of plates because of a decrease in the velocity with an increase in the number of plates. Even though not shown here, the relative humidity at the dry side outlet increases with an increase in the number of plates. This is because of an increase in the water transfer rate and a reduction in the pressure drop, which results in a higher partial pressure of water vapour at the dry side outlet. Hence, an increase in the number of plates results in an improved performance of the humidifier. However, it must be noted that increasing the number of plates adds to the cost and size of the humidifier. For a fuel cell system designed for automotive applications, the humidifier size can become a major concern

because of space limitations. The cost of the humidifier is another major factor governing the design. Therefore, a design trade-off must be made to obtain the optimum number of plates for the humidifier, taking into consideration the performance, size and the associated costs. 4.8. System simulation studies The analysis so far assumes that the inlet conditions at the wet side are independent of the dry side conditions. However, in a practical fuel cell system, the humidifier and the fuel cell stack are connected to each other such that the outlet of the cathode enters the wet side of the humidifier. Hence, the flow rates, temperature and pressure at the wet side inlet are dependent on the dry side conditions. In addition, the temperature and water flow rates at the dry side are a function of the wet side conditions. Thus, there is an interrelationship between the conditions at the dry and wet side of the humidifier. To study this interdependence, we simulated a

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Water transfer rate Dry side pressure drop Wet side pressure drop

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Water transfer rate / mol s-1

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50

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Number of plates Fig. 15. Variation of water transfer rate and the pressure drops across the dry and wet sides with the number of plates.

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Cathode outlet

Cathode inlet

Stack Dry side inlet

Dry side outlet

Humidifier Wet side outlet

Wet side inlet

Fig. 16. Schematic of the system with the interconnected stack and humidifier.

system consisting of a humidifier and a fuel cell stack. A schematic of the system is shown in Fig. 16. In this system, the dry side outlet of the humidifier is connected to the cathode of the fuel cell stack, whereas the outlet of the cathode is connected to the wet side inlet of the humidifier. The gas at the outlet of the fuel cell stack has a high moisture content due to the production of water in the fuel cell stack. Thus, this stream is fed to the humidifier and is used to humidify the gases which enter the cathode. In order to perform system simulation studies, we define some additional variables for the stack. The cell current density (j) of the fuel cell stack governs the amount of hydrogen that must be fed to the stack, and the stoichiometry for the reaction between hydrogen and oxygen is used to estimate the amount of oxygen required for the reaction. For a given cell current density, the molar flow rate of hydrogen input to the stack is given by N˙ H2 =

jA 2F

(30)

where A is the total stack area and F is the Faraday’s constant. Based on the stoichiometry, the molar flow rate of oxygen at the stack inlet is given by N˙ O2 =

jA a 4F

(31)

Here, a is the stoichiometric ratio and represents the ratio of the actual molar flow rate of air and the molar flow rate required to achieve a given current density. The base values of operating parameters for performing system simulations are given in Table 4. Since air was fed to the stack, the molar flow rate of nitrogen was calculated by considering air to be a mixture of 21% oxygen and 79% nitrogen. The molar flow rates of oxygen and nitrogen entering the dry side of the humidifier are equal to the molar flow rates at the inlet of the stack. It is assumed that water vapour is not present in the dry side inlet stream of the humidifier. The water flow rate at the stack inlet (same as humidifier dry side outlet) depends on the water transfer rate across the humidifier, which in turn depends on the dry and wet side conditions. Hence, the water flow rate at the stack inlet was obtained using iterations. The pressure drop across the stack is assumed to be a linear function of the velocity at the stack inlet, whereas the pressure Table 4 Base operating conditions used for the system simulations. Parameter

Numerical value

j a pdi Tdi Twi di

1.8 A cm−2 1.9 1.8 bar 85 ◦ C 85 ◦ C 0%

drops across the dry and wet side of the humidifier were calculated using Eqs. (5) and (6), respectively. The velocity at the stack inlet depends on the water transfer rate and heat transfer rate across the humidifier, which in turn depend on the dry and wet side conditions. Hence, similar to the calculations of the water flow rate, the pressure at the wet side inlet was calculated iteratively. 4.8.1. Combined effect of pressure at dry and wet sides As explained in Section 4.3, the wet side pressure is dependent on the dry side pressure. To study the combined effect of the dry and wet side pressure on the performance of the humidifier, the dry side pressure was varied between 1.6 and 2.1 bar, whereas, the pressure and water flow rate at the wet side inlet were calculated iteratively. The other operating conditions are the same as those given in Table 4. Fig. 17 shows the variation of the water transfer rate and the water recovery ratio with the dry side inlet pressure. It is found that with an increase in the dry side inlet pressure, both the water transfer rate and the water recovery ratio increase. It was shown earlier in Figs. 10–12 that an increase in the dry and wet side pressure has a negative and positive effect, respectively, on the water transfer rate and the water recovery ratio. However, we observe in Fig. 17 that the effect of the wet side pressure is dominant, thus resulting in an increase in the water transfer rate and water recovery ratio with an increase in the dry side pressure (and a corresponding increase in the wet side pressure). It was also shown in Figs. 9 and 12 that the relative humidity at the dry side outlet increases with an increase in the dry and wet side inlet pressures. Therefore, although not shown here, the combined effect of an increase in the dry and wet side pressure results in an increase in the relative humidity at the dry side outlet. From these results, it can be concluded that maintaining a high operating pressure in the fuel cell system improves the performance of the humidifier. Also, high pressures in the fuel cell stack result in lower voltage losses. However, with an increase in the operating pressure, the parasitic losses associated with the compressor increase. Hence, the fuel cell system should be operated at an optimum pressure, considering the performance of the humidifier and stack, and the power associated with pumping the gases. 4.8.2. Effect of current density For analyzing the stack and humidifier as a closed system, the current density was varied between 1.6 and 2.1 A cm−2 . The temperature at the dry and wet side inlets was maintained at 85 ◦ C. The pressure at the dry side inlet was kept constant at 1.8 bar, whereas, as explained earlier, the pressure at the wet side inlet was calculated iteratively. The effect of stoichiometric ratio (a ) was studied for three different cases, shown in Fig. 18: (i) case A: constant stoichiometric ratio with increasing current density, (ii) case B: increasing stoichiometric ratio with increasing current

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Pressure at dry side inlet / bara Fig. 17. Variation of the water transfer rate and water recovery ratio with the dry side inlet pressure for system simulation studies.

density, and (iii) case C: decreasing stoichiometric ratio with increasing current density. Fig. 19 shows the variation of water transfer rate with the cell current density for the three different cases of constant, increasing and decreasing stoichiometric ratios. It is observed that the water transfer rate does not vary significantly with the current density for case A. It is also observed that the water transfer rate increases with an increase in the cell current density for case C, whereas a decrease is observed for case B. An increase in the current density for a constant stoichiometric ratio (case A) results in an increase in the molar flow rates at the dry and wet sides. It was discussed in Sections 4.1 and 4.2 and shown in Figs. 6 and 7 that an increase in the dry and wet side molar flow rates results in an increase in the water transfer rate. However, these results are valid only when the

water concentration at the wet side inlet is maintained constant. With an increase in the current density for a constant stoichiometric ratio, the molar flow rates increase resulting in an increase in the pressure drop across the stack. This results in a decrease in the water concentration at the inlet of the humidifier. For case A, the negative effect of a decrease in the water concentration at the wet side inlet balances the positive effect of an increase in the total molar flow rates. This results in an insignificant change in the water transfer rate with an increase in the current density. For the case of decreasing stoichiometric ratio with an increase in the current density (case C), the increase in the water transfer rates can be explained by analyzing the water concentration at the stack outlet. An increase in the current density results in an increase

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Current density / A cm

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-2

Fig. 18. Variation of stoichiometric ratio with current density for three different cases.

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1 0.9

Water transfer rate / mol s

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Case A

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Case C

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Current density / A cm

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2.2

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Fig. 19. Variation of water transfer rate with current density for various stoichiometric ratios.

in the molar flow rates of nitrogen, water, and oxygen at the stack outlet. However, a decrease in the stoichiometric ratio results in a decrease in the molar flow rates of N2 and O2 , without significantly affecting the water flow rates. This results in an increase in the water concentration at the stack outlet, hence increasing the water transfer rates in the humidifier. For the case of increasing stoichiometric ratio with an increase in the current density (case B), the concentration of water at the stack outlet decreases due to two factors discussed earlier: (i) an increase in the molar flow rates of O2 and N2 at the stack outlet, and (ii) an increase in the pressure drop with an increase in the molar flow rates. Thus, low water concentrations at the humidifier wet side inlet result in a decrease in the water transfer rates, despite the increase in the total molar rates. Hence, the negative effect of a decreasing water concentration at the wet side inlet is dominant over the positive effect of an increase in the total molar flow rates. This results in a net decrease

of the water transfer rates with an increase in the current density for case B. Fig. 20 shows the variation of water recovery ratio with the cell current density for the three different cases of constant, increasing and decreasing stoichiometric ratios. It is observed that the water recovery ratio monotonically decreases with an increase in the cell current density for all the three cases. It is also observed from Figs. 19 and 20 that for a given current density, the water transfer rate and water recovery ratio decrease with an increase in the stoichiometric ratio. Hence, low current densities with low stoichiometric ratios would result in high water transfer rates and a high water recovery ratio. However, operating a fuel cell stack at current densities lower than the design limit would result in inefficient utilization of the stack. Also, a low stoichiometric ratio would result in an increase in the voltage losses, thus resulting in a decrease in the power output. These aspects should be

0.7

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Water recovery ratio

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Current density / A cm

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-2

Fig. 20. Variation of water recovery ratio with current density for various stoichiometric ratios.

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Case A

20

Case B Case C

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1.7

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Current density / A cm

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2.1

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-2

Fig. 21. Variation of relative humidity at dry side outlet with current density for various stoichiometric ratios.

considered while choosing the current densities and stoichiometric ratios in order to maximize the water transfer rates and efficiency of the humidifier. Fig. 21 shows the variation of relative humidity at the dry side outlet with the cell current density for the three different cases of constant, increasing and decreasing stoichiometric ratios. It is observed that the relative humidity at the dry side outlet decreases with an increase in the cell current density for cases A and B, whereas an increase is observed for case C. An increase in the cell current density for cases A and B results in an increase in the molar flow rates at the dry and wet side inlets. It was shown in Fig. 8 and explained in Section 4.2 that an increase in the wet side molar flow rate results in an increase in the relative humidity at the dry side outlet. However, this result is valid only when the concentration of H2 O at the wet side inlet is maintained constant. With an increase in the current density for cases A and B, the concentration of water at the wet side inlet decreases, as explained earlier. Thus, the positive effect of high molar flow rate at the wet side on the relative humidity at the dry side outlet is diminished. Nevertheless, it was shown in Fig. 5 and explained in Section 4.1 that an increase in the dry side molar flow rate results in a decrease in the relative humidity at the dry side outlet. Thus, the combined effect of an increase in the dry and wet side flow rates is a decrease in the relative humidity at the dry side outlet. As explained earlier, an increase in the current density for the case of decreasing stoichiometric ratio (case C) results in an increase in the water concentration at the stack outlet. This results in an increase in the water transfer rate, and hence, an increase in the relative humidity at the dry side outlet. From Fig. 21, it is also observed that for a fixed current density, the relative humidity at the dry side outlet decreases with an increase in the stoichiometric ratio. This is because as discussed earlier, an increase in the stoichiometric ratio for a fixed current density results in an increase in the molar flow rates of nitrogen and oxygen, whereas, the molar flow rate of water at the stack outlet does not change significantly. This results in a decrease in the water concentration at the humidifier wet side inlet, thus resulting in a lower relative humidity at the dry side outlet. It is observed from Figs. 19–21 that various performance metrics like the water transfer rate, water recovery ratio and relative humidity at the dry side outlet exhibit different trends with an increase in the current density. For the case of

constant stoichiometric ratio, the relative humidity at the dry side outlet and the water recovery ratio decrease even though the water transfer rate does not vary significantly with an increase in the current density. Similarly, for the case of decreasing stoichiometric ratio, the relative humidity at the dry side outlet and water transfer rate increase, whereas, the water recovery ratio decreases with an increase in the current density. Thus, a single performance metric is not suitable to measure the effectiveness of a humidifier. The design phase should consider multiple performance metrics and the choice of the performance metric should depend on the specific requirements of the system.

5. Conclusions A two-dimensional steady state model is developed for a cross flow plate-and-frame membrane humidifier for a fuel cell system. It is shown that the water transfer rate increases with an increase in the velocities at the dry and wet sides, wet side inlet pressure, wet side inlet relative humidity, dry side inlet temperature and the number of plates. The relative humidity at the dry side outlet is found to increase with an increase in the wet side velocity, pressure at the dry and wet sides, wet side inlet relative humidity and the number of plates. The pressure drop across the humidifier decreases with an increase in the number of plates in the humidifier. However, an increased number of plates would result in a higher cost and size, which might not be suitable for the packaging of the humidifier into the fuel cell system. Simulations are performed by integrating the humidifier with the fuel cell stack. The results show that a high operating pressure improves the performance of a humidifier, but increases the parasitic losses in the fuel cell system. The effect of current density for various stoichiometric ratios shows that a low current density and low stoichiometric ratio results in a high water transfer rate, high relative humidity at stack inlet and a high water recovery ratio. However, a low stoichiometric ratio could result in high voltage losses in the fuel cell stack. It is shown that any single performance metric is not sufficient to analyze the performance of the humidifier. Therefore, while designing the humidifier, it is essential to identify the performance parameter(s) of interest.

M. Sabharwal et al. / Journal of Membrane Science 409–410 (2012) 285–301

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Acknowledgements The authors would like to acknowledge Mercedes Benz Research and Development India and Daimler AG Kirchheim/Teck-Nabern, Germany for the financial support.

Nomenclature am A A c cair Cp d d Dg Dm F hf H Hf Hl j k kc km Kc Kp L Li N˙ k Np Nu p P Petr R Sh T Tamb Tmem u wc wl W Wi WRR x y

membrane relative humidity fuel cell stack area (cm2 ) cross-sectional area for flow (m2 ) molar concentration of H2 O (mol m−3 ) molar concentration of air (mol m−3 ) specific heat capacity (J mol−1 K−1 ) channel depth (m) hydraulic diameter (m) diffusivity of water in air (m2 s−1 ) diffusivity of water in the membrane (m2 s−1 ) Faraday’s constant (C mol−1 ) convective heat transfer coefficient (W m−2 K−1 ) height of the flow channel (m) overall heat transfer coefficient (W m−2 K−1 ) heat loss coefficient (W m−2 K−1 ) cell current density (A cm−2 ) thermal conductivity of gas (W m−1 K−1 ) convective mass transfer coefficient (m s−1 ) thermal conductivity of membrane (W m−1 K−1 ) overall mass transfer coefficient (m s−1 ) constant for pressure drop calculation (Pa s m−2 ) effective length of the plate (m) length of the plate (m) molar flow rate of species ‘k’ (mol s−1 ) number of plates Nusselt number pressure (Pa) wetted perimeter (m) transverse Peclet number universal gas constant (J mol−1 K−1 ) Sherwood number temperature of the gas phase (K) ambient temperature (K) membrane temperature (K) velocity (m s−1 ) channel width (m) land width (m) effective width of the plate (m) width of the plate (m) water recovery ratio position coordinate (m) position coordinate (m)

Subscripts d dry side w wet side inlet i o outlet

di wi

dry side inlet wet side inlet

Greek letters ım membrane thickness (m) a stoichiometric ratio membrane water content m  relative humidity (%)

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