Pressure drop and flow distribution in parallel-channel configurations of fuel cells: Z-type arrangement

international journal of hydrogen energy 35 (2010) 5498–5509

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Pressure drop and flow distribution in parallel-channel configurations of fuel cells: Z-type arrangement Junye Wang* North Wyke Research, Okehampton, Devon EX20 2SB, UK

article info

abstract

Article history:

A general theoretical model based on mass and momentum conservation has been

Received 23 September 2009

developed to solve the flow distribution and the pressure drop in Z-type configurations of

Received in revised form

fuel cells. While existing models neglected either friction term or inertial term, the present

26 February 2010

model takes both of them into account. The governing equation of the Z-type arrangement

Accepted 27 February 2010

was formulated to an inhomogeneous version of the U-type one. Thus, main existing

Available online 26 March 2010

models have been unified to one theoretical framework. The analytical solutions are fully explicit that they are easily used to predict pressure drop and flow distribution for Z-type

Keywords:

layers or stacks and provide easy-to-use design guidance under a wide variety of combi-

Fuel cell stack

nation of flow conditions and geometrical parameters to investigate the interactions

Flow distribution

among structures, operating conditions and manufacturing tolerance and to minimize the

Manifold

impact on stack operability. The results can also be used for the design guidance of flow

Parallel channels

distribution and pressure drop in other manifold systems, such as plate heat exchanges,

Pressure drop

plate solar collectors, distributors of fluidised bed and boiler headers.

Maldistribution

Crown Copyright ª 2010 Published by Elsevier Ltd on behalf of Professor T. Nejat Veziroglu. All rights reserved.

1.

Introduction

Fuel cells have the potential to revolutionize the way we power our world, offering cleaner, more-efficient alternatives to the combustion of gasoline and other fossil fuels to replace the internal-combustion engine in vehicles and the gas turbine at power station. Therefore, fuel cells are an important part of the comprehensive and balanced technology portfolio needed to address two most important energy challengesdsignificantly reducing carbon dioxide emissions and ending our dependence on fossil fuels. However, the widespread adoption of fuel cells has yet to take off. An essential prerequiresite is the production of compact, low cost and high performance stacks since fuel cells must be combined into fuel cell stack to obtain enough electricity for many power applications. Therefore, the stack is one of the most important components for the fuel cell commercialisation.

Under ideal conditions, the performance of a fuel cell stack is simply the linear sum of the performance of unit cells. However, this linear correlation is not achieved in practice because of non-uniform flow distribution. The parallelchannel configurations are of greatest interests due to their clear advantages of simplicity and less pressure drop over others. Thus, this type of configurations has the greatest potential to reduce development cost and accelerate design and manufacturing cycle times. This is especially true to achieve commercialised sized stacks, where a significant portion of development costs is at manufacturing, assembly, and testing. However, using the parallel-channel configurations, there is a possibility of the severe flow maldistribution problems. Some channels may be starved of reactants, while others may have them in excess, which reduces fuel cell performance. It is a key to predict the performance of various parallel-channel configurations of fuel cell stacks. Thus, flow

* Tel.: þ44 1837883552; fax: þ44 183782139. E-mail address: [email protected] 0360-3199/$ – see front matter Crown Copyright ª 2010 Published by Elsevier Ltd on behalf of Professor T. Nejat Veziroglu. All rights reserved. doi:10.1016/j.ijhydene.2010.02.131

international journal of hydrogen energy 35 (2010) 5498–5509

Nomenclature A1 A2 B B1 C Cf dc D f F J lc L M n p P Q r

constant in Eq. (22), defined by Eq. (24) constant in Eq. (22), defined by Eq. (25) constant in Eq. (30), defined by Eq. (31) constant in Eq. (23), defined by Eq. (26) constant in Eq. (31), (39) and (45) coefficients of turning losses diameter of the channels (m) diameter of header (m) fanning friction factor cross-sectional area of headers or channels (m2) constant in Eq. (30), defined by Eq. (32) length of the channels length of header (m) constant defined by Eq. (28) numbers of channels in a stack dimensionless pressure pressure in header coefficient in Eq. (29), defined by Eq. (22) root of characteristic equation

performance can be explored under various geometries of the stack, resulting in a higher efficiency fuel cell stacks and an optimal geometrical structure and cost reduction. Many attempts have been published for designs and developments of configurations of fuel cells [1–4]. There are three approaches to study pressure drop and flow distribution in parallel-channel configurations of fuel cells: computational fluid dynamics (CFD) [5–10], channel network models [11–14] and analytical models [10,15]. The CFD is a somewhat detailed approach in which modelling has potential to resolve real-world 3-D engineering structures. The pressure drop and flow distribution can be predicted using this approach without the knowledge of flow coefficients, such as the friction and pressure recovery coefficients. However, the CFD is unsuitable for optimizing fuel cell stack geometry and preliminary designs since it is too expensive to regenerate the computational geometry and mesh for each new configuration. Furthermore, there were still the difficulties for the turbulence models and the boundary models associated with solving swirling or curvature flows because the eddy-viscosity models failed to capture the anisotropy of strain and Reynolds stresses under the action of Coriolis and centrifugal forces [16–19]. In the recent years, lattice Boltzmann models (LBM) have been employed to study transports in porous media and fuel cells [20–23]. Unlike conventional numerical schemes based on discretisations of macroscopic continuum equations, the LBM is based on microscale, in which the physics and chemistries associated with the molecular level interaction can be incorporated more easily into mesoscopic kinetic equations. This feature gives the LBM the advantage of studying nonequilibrium dynamics, especially in multiphase interactions of complex flows involving micro and meso-scale interfacial dynamics and complex boundaries, such as porous media and fuel cells. However, the LBM needs more computer resources than the traditional CFD.

R u U vc w W x X

5499

coefficient in Eq. (29), defined by Eq. (23) dimensionless channel velocity channel velocity (m/s) dimensionless volume flow rate in the channels dimensionless velocity in header velocity in header (m/s) dimensionless axial coordinate in header axial coordinate in header (m)

Greek symbols b average velocity ratio in header (Wc/W ) 3 coefficient in Eqs. (23) and (29), defined by Eq. (27) r fluid density (kg/m3) s wall shear stress (N/m2) z average total head loss coefficient for channel flow n kinematic viscosity (m2/s) Subscripts c channel i intake header e exhaust header

Channel network model is also called discrete model in the field of flow in manifolds. In a channel network model, the stack structure is represented as a network of multiplejunctions traversed by the fluid flow. Then, mass and momentum conservation equations can be built at each junction. Finally, a set of difference equations is solved using an iteration program. Thus, a designer cannot use its results directly and it is still inconvenient for the preliminary design and optimisation of the stack structures. Analytical model is also called the continuous model in which flow is considered to be continuously branched along intake and exhaust manifolds. It has been shown intuitively, as well as mathematically, that the continuous manifolds are but limiting cases of the more interesting, discrete manifolds [24,25]. In mathematical viewpoint, the fluid mechanical principles in a continuous manifold lead to a differential rather than a difference equation in a discrete one. For the sake of this, the continuous models are also fundamentals of various discrete models. A main advantage of analytical models over the CFD and the channel network models is that it is simple and flexible to stack designers since results of computations based on the differential equation can be represented more simply and compactly than is the case for calculations using the nonlinear difference equations. A general analytical model has unique possibility to correlate explicitly the performance, such as flow distribution and pressure drop, and stack structures, such as duct and channel diameters and shape, pitch and duct lengths. Particularly, for preliminary design, we have less information if a geometrical structure is optimal but major decisions should be made in this stage, such as cost, durability, maintenance and performance. Thus, flow performance can be explored under various geometries of the stacks. The analytical models are traditionally based on Bernoulli theorem. This approach has received fairly good attentions. However, many researchers [24–38] have indicated that there are two main difficulties with applying a Bernoulli equation to

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the varying mass flow process. Firstly, it is assumed in Bernoulli equation for a dividing flow manifold that the fluid leaving through the branch has lost all its axial component of velocity before it crosses the control volume; the pressure change is related to the velocity change and friction loss. Since the velocity of the side stream is not exactly in the perpendicular direction when the stream flows through the branch, an extra pressure change caused by the flow branching cannot be captured which has been observed by the experiment. Secondly, because the lower energy fluid in the boundary layer branches through the channels the higher energy fluid in the pipe centre remains in the pipe. So the average specific energies in a cross-section will be higher in the downstream than in the upstream. Since energy balance is based on the average value of the cross-section, these higher specific energies cannot be corrected and lead to an error. Hence, according to the First Law of Thermodynamics, when the specific mechanical energies are multiplied by the relevant mass flow rate terms, the mechanical energy after branching for the manifold can apparently be greater than the approaching energy. To avoid the above problems, a more general theoretical approach has been developed based on mass and momentum conservation for flow in manifold system. For example, flow in dividing manifolds is usually formulated as a second order nonlinear ordinary differential equation (Eq. (4) in Refs. [32,35]): 1 dPi fi dWi þ ¼0 W2 þ ð2  bi ÞWi dX r dX 2Di i

(1)

Eq. (1) can be rearranged to the below equations: 1 dPi fi 2  bi dWi2 þ ¼0 Wi2 þ 2 dX r dX 2Di

(2a)

or to a discrete equation: DPi þ

rfi 2 ð2  bi Þr W DX þ DWi2 ¼ 0 2Di i 2

(2b)

This equation can be simplified to Bernoulli equation without the potential term when bi ¼ 1. Therefore, Bernoulli equation is a special case of the flow models in manifolds. For U-type arrangement, we can combine the dividing flow equation (Eq. (1)) with a combining flow one (e.g. Eq. (8) of Ref. [35]) to derive a governing equation (Eq. (14) in Ref. [35]) which is identical to Eq. (1) mathematically:    2    2   d pi  pe L fi fe Fi Fi þ w2i  ð2  be Þ þ ð2  bi Þ dx Fe 2 Di De Fe dwi wi ¼0 dx

ð3Þ

It has been a well known challenge in the field of flow in manifold systems to have a complete analytical solution of Eqs. (1) or (3) for a half century because of complexity of coupling pressure and velocity. In the past years, researchers had to simplify the model and neglected part of the models although this might cause a significant error or employ a numerical approach. Bajura and Jones [27,28] derived a similar equation for headers of power plant boiler. Bassiouny and Martin [29,30] extended Bajura and Jones’ model into plate heat exchangers. Their models have been used for designs of fuel cell stacks [10,12,15] in the recent years as well. However, these models were solved by neglecting friction

effect. Thus, they only validated for a short manifold. A study by Shen [31] showed that even for short manifolds the friction effects on the flow distribution are still not negligible. The present author [24,32–36] also indicated that the friction effects on the flow distribution are important. Kee et al. [14] and Maharudrayya et al. [15], on the other hand, retained the frictional term in their model equation but the inertial term was neglected totally. In practice, Eqs. (2a) and (2b) can be simplified to Eq. (5) of Ref. [15] when bi ¼ 0. Wang et al. [24,32] had analysed the effect of manifold structure on the friction and the pressure recovery factors. The friction factor will be dependent on three ratios: the channel diameter to the header diameter, the spacing length to the header diameter, and the sum of the areas of all the channels to the cross-sectional area of header. The pressure recovery factor in the intake manifold, bi, was at of order of 0.5–1. The value bi ¼ 0 implies that the flow leaves the manifold at right angles (i.e., axial component of the velocity is equal to zero) and represents the maximum possible static pressure recovery. The value bi ¼ 1 implies that the fluid would leave the manifold without losing any axial momentum. The analysis is applicable to a combination of manifolds, such as U- and Z-type arrangement. Thus, it is clear that Maharudrayya et al.’s model function is at an extreme case of Eqs. (2a) and (2b) while Bernoulli equation at another end case. Furthermore, whether bi and be are 0 or 1, the third term on the left hand in Eq. (3) will be eliminated when Fi/Fe ¼ 1. Thus because the second term on the left hand in Eq. (3) is always positive there is only one solution for Bernoulli equation or Maharudrayya et al.’s model [15], compared with the three possible solutions of Eq. (3) as a second order nonlinear ordinary differential equation which depends on the polynomial discriminate of the coefficients of the friction and inertial terms. Wang et al. [24,34] analysed flows in manifold where flows were divided for three regions for the first time: momentum dominant, momentum-friction reciprocity and friction dominant regions. Both the friction and momentum effects on the flow distribution are important. Wang [35] carried out successfully the first attempt to solve analytically Eq. (3). He proved that there were three solutions for Eq. (3) rather than one solution. These three solutions have also been proved experimentally [24,32–36]. It is clear that the neglecting of the inertial terms has lost two solutions with the triangular function and the exponent function. It is not surprising that the phenomena of pressure rise cannot be captured by their model because there is no flow branching effect. The neglecting of inertial term may cause an error of 20–50% since bi was at of order of 0.5–1 and be was at of order of 0.5 to 0.6. The analysis for the Z-type configurations is similar to that of the U-type done by Wang [35]. However, it will be more difficult to derive a general control equation of Z-type arrangement. Unlike U-type, there is no general control equation for Z-type which is ready for a solution. Following the previous work, this paper will focus on the parallelchannel configuration with Z-type arrangement, as shown in Fig. 1 to solve a complete theoretical model without any neglecting of friction and inertial effects. The unique feature of this model is that an analytical solution is possible in a generalized form with both the frictional and inertial terms. The present paper is arranged in the following order. In

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international journal of hydrogen energy 35 (2010) 5498–5509

We

Uc

W e, Pe

Wi(X)

τw

Pi(X)

Uc

Wc

Wi(X+ΔX) Pi(X+ΔX)

ΔX

Wi W i, Pi

Fig. 2 – Control volume for the intake header.

x L 2.1.2. Fig. 1 – Schematic diagram of Z-type arrangement.

Section 2, the mathematical model is presented. Its analytical solution is given in Section 3. The results and discussion are arranged in Section 4. In the last section, some concluding remarks are given.

2.

Mathematical model

Since momentum change in intake and exhaust headers is as a result of the flow branching and friction, both the inertial and the friction effects should be taken into account. The development of the present theoretical model is based on the following assumptions: 1) the manifold header is of constant cross-sectional area and has equally spaced channels of uniform size; 2) the temperature influence on fuel cells is directly through chemical reaction and phase change and indirectly through thermal buoyancy, and the thermal buoyancy from the fluid density is very small compared to the pressure drop and the effect of the fluid density on flow distribution can be neglected since most types of fuel cells are made of manifold systems with small channels and operate in a low range of temperatures.

2.1.

Intake manifold

The control volume in an intake manifold is shown in Fig. 2. The mass and momentum balances can be written as follows:

Mass conservation

  dWi DX þ rFc Uc rFi Wi ¼ rFi Wi þ dX

1 dPi fi dWi Fc n þ þ Uc Wc ¼ 0 W2 þ 2Wi dX r dX 2Di i Fi L

Fi L dWi Uc ¼  Fc n dX

(5)

(7)

After inserting swi, and Wc into Eq. (6) and neglecting the higher orders of DX, Eq. (6) can be rearranged as follows: 1 dPi fi dWi þ ¼0 W2 þ ð2  bi ÞWi dX r dX 2Di i

2.2.

(8)

Exhaust manifold

The control volume in the exhaust manifold is shown in Fig. 3. The mass and momentum balances can be written as follows:

2.2.1.

Mass conservation

  dWe DX þ rFc Uc rFe We ¼ rFe We þ dX

(9)

where Fe is the cross-sectional areas of the exhaust header, and We the axial velocity in the exhaust manifold. Fe L dWe Fc n dX

(10)

ΔX

(4)

where Fi and Fc are the cross-sectional areas of the intake header and the channel, respectively, Wi the axial velocity in intake header, Uc the channel velocity, X axial coordinate in the intake header, and r fluid density. Setting DX ¼ L=n; n is the number of channels and L length of the header.

ð6Þ

where Pi is pressure in the intake header, Di diameter of intake header, swi is given by Darcy–Weisbach formula swi ¼ f rðWi2 =8Þ; and Wc ¼ bi Wi . It is noted that pDi can be replaced for a rectangular header or other shapes using its wetted perimeter (the exhaust header and channels are same as the intake header). After inserting swi, and Wc into Eq. (6) and neglecting the higher orders of DX, Eq. (6) can be rearranged as follows:

Uc ¼

2.1.1.

Momentum conservation

  dPi DX Fi  swi p Di DX Pi Fi  Pi þ dX  2 dWi ¼ rFi Wi þ DX rFi Wi2 þ rFc Uc Wc dX

We (X) Pe (X)

τwe Wce

We (X+ΔX) Pe (X+ΔX)

Uc Fig. 3 – Control volume for the exhaust header.

5502

2.2.2.

international journal of hydrogen energy 35 (2010) 5498–5509



Momentum conservation

  dPe DX Fe  sw p De DX Pe Fe  Pe þ dX  2 dWe ¼ rFe We þ DX rFe We2 þ rFc Uc Wec dX

z

ð11Þ

where Pe is pressure in the exhaust header, De diameter of exhaust header, swe is given by Darcy–Weisbach formula swe ¼ f rðWe2 =6Þ; and Wec ¼ be We : After inserting swe, and Wec into Eq. (11) and neglecting the higher orders of DX, Eq. (11) can be rearranged as follows: 1 dPe fe dWe þ ¼0 W2 þ ð2  be ÞWe dX r dX 2De e

(12)

From Eqs. (5) and (10), we have a relationship between velocities in intake and exhaust manifolds for the Z-type arrangement:   Fi (13) We ¼ ðW0  Wi Þ Fe

2.3.

Control equation for Z-type arrangement

Subtracting Eq. (12) from Eq. (8), we have: 1 dðPi  Pe Þ fi fe dWi W2  W2 þ ð2  bi ÞWi þ 2Di i 2De e dX r dX dWe  ð2  be ÞWe ¼0 dX

ð14Þ

The flow in the channels can be described by Bernoulli’s equation with a consideration of flow turning loss. Hence, the velocity in a channel, Uc is correlated to the pressure difference between the intake and the exhaust manifolds as follows:   lc U2c U2 ¼ rz c (15) Pi  Pe ¼ r 1 þ Cfi þ Cfe þ fc dc 2 2 where Cfi is coefficient of turning loss from the intake header into the channels and Cfe that of turning loss from the channels into the exhaust header, and fc is average friction coefficient for the channel flow. Inserting Eq. (5) into Eq. (15), gives: 2  2  1 Fi L dWi Pi  Pe ¼ rz dX 2 Fc n

(16)

Eqs. (13), (14) and (16) can be reduced to dimensionless form using the following dimensionless groups: Pi Wi We Uc X ; wi ¼ ; we ¼ ; uc ¼ ;x ¼ : W0 W0 L rW02 W 0 we ¼ ð1  wi Þ FFei   d pi pe fi L 2 fe L 2 dwi dwe þ ð2be Þwe ¼0 w  w þð2bi Þwi dx dx dx 2Di i 2De e 2  2  1 Fi dwi pi  pe ¼ z dx 2 Fc n pi ¼

Inserting Eqs. (17) and (19) into Eq. (18), we have    2 d pi  pe fi L 2 fe L Fi dwi ð1  wi Þ2 þð2  bi Þwi þ wi  dx dx 2Di 2De Fe  2 Fi dð1  wi Þ ð1  wi Þ ¼0  ð2  be Þ Fe dx

 2  2 2 dwi d wi fi L 2 fe L Fi fe L Fi þ w  þ 2wi dx dx2 2Di i 2De Fe 2De Fe  2  2 fe L Fi dwi Fi dwi þ ð2  be Þ  w2i þ ð2  bi Þwi ð1  wi Þ dx Fe dx 2De Fe

Fi Fc n

2

¼0

ð20bÞ

Eqs. (20a) and (20b) is a general control equation for Z-type arrangement of fuel cell layers or stacks. To the best of my knowledge, there is no analytical solution of Eqs. (20a) and (20b). Bajura and Jones [28], and Bassiouny and Martin [30] solved the same equation after neglected the frictional terms. Maharudrayya et al. [15], retained the frictional terms but neglected the inertial term. In order to solve Eqs. (20a) and (20b), we have to carry out further derivation of Eqs. (20a) and (20b) as follows.  2  2  2 dw d2 w f L fe L Fi fe LRei fi Fi i i i 2 þ w  þ 2wi z FFcin dx dx2 2Di i 2De Fe 2De Rei fi Fe  2  2 fe L Fi dwi Fi dwi ð2be Þ  w2i þð2bi Þwi wi dx Fe dx 2De Fe  2 Re f dw i i i þð2be Þ FFei ¼0 Rei fi dx  2   2   2 dw d2 w L f fe Fi fe fi LDi F i i i   i þ w2i þ  z FFcin 2 dx dx 2 Di De Fe De y Rei fi Fe o dw f L  F 2 n h  2  2 i e i i fi Di e Fi w ¼ þ ð2bi Þ 1 2b þð2be Þ FFei y Re 2bi Fe ð i fi Þ i dx 2De Fe  2  2

 2 dwi d wi ð2bi Þ 2be Fi Fc n þ 1 2 dx dx 2bi Fe z Fi  2  2 

  2 ð2be Þ Fc n fi Di dwi L fi fe Fi Fc n   wi þ þ  dx z Fe y Rei fi 2z Di De Fe Fi  2  2  2fi Di fe L Fc n fe L Fc n  w2i ¼ þ  ð21aÞ 2zDe Fe y Rei fi 2De z Fe   2 

 2  2 dwi d wi ð2  bi Þ Fc n fi Di 2  be Fc n dwi   þ  1  þ wi dx dx2 z dx z Fi Fe y Rei fi  2   2  

L fi Fc n 2fi Di fe L Fc n  w2i  1  þ 2z Di Fi y Rei fi 2zDe Fe  2 fe L Fc n ¼ ð21bÞ 2zDe Fe Eqs. (21a) and (21b) is a new general control equation of Z-type arrangement. Its all functions are identical to those of Eqs. (20a) and (20b) for Z-type fuel cell layers or stacks. To the best of my knowledge, it is the first time that a complete control equation, Eqs. (21a) and (21b), has been derived for modelling of Z-type arrangement.

(17)

3.

Analytical solution

(18) (19)

ð20aÞ

Eqs. (21a) and (21b) is a second order inhomogeneous nonlinear ordinary differential equation for flow distribution in Z-type fuel cell layers or stack. This equation is inhomogeneous version of Eq. (16) in Ref. [35]. The second term in the left hand of Eqs. (21a) and (21b) represents the momentum contribution known as the momentum term, and the third term does a friction contribution as the friction term. However, it can be seen that there are obvious differences. Besides the equation is inhomogeneous, there is an additional

international journal of hydrogen energy 35 (2010) 5498–5509

term in both momentum and friction term. Particularly, there is effect of friction in the coefficient of the momentum term. We define three constants: Q¼

  2  2  ð2  bi Þ Fc n fi Di 2  be Fc n   1  3z 3z Fi Fe y Rei fi

¼ A1  ð1  MÞA2  2   2  L fi Fc n 2fi Di fe L Fc n  R¼ þ 1  ¼ B1 þð12MÞ3 4z Di Fi y Rei fi 4zDe Fe

(22)

(23)

 2 ð2  bi Þ Fc n A1 ¼ 3z Fi

(24)

 2 ð2  be Þ Fc n 3z Fe

(25)

 2 L fi Fc n 4z Di Fi

(26)

B1 ¼ 

 2 fe L Fc n 3¼ 4zDe Fe

(27)

f i Di  M¼  y Rei fi

(28)

2

(29)

Although it was derived based on headers and channels of circular shape, Eq. (29) is also the governing equation for cases with rectangular or other shapes when the diameters and the perimeter in intake and exhaust headers and channels are replaced using their hydraulic diameters and wetted perimeters. It should be noted that there is a little difference of constants Q, R and 3 between circular and other shapes. Therefore, Eq. (29) can be used for both circular and rectangular headers and channels when readers choose properly coefficients. Eq. (29) is inhomogeneous version of Eq. (19) in Ref. [35]. The general solution to an inhomogeneous nonlinear differential equation is the sum of the general solution of the related homogeneous equation and its particular solution. We have obtained a general solution of its homogeneous version in Ref. [35]. What we need to do is to find a particular solution to Eq. (29). We reproduce the characteristic roots and its general solution of the homogeneous version of Eq. (29) as follows:

8 >
1 1 pffiffiffi r1 ¼  B þ i 3J 2 2 > : r2 ¼ 1B  1ipffiffiffi3J 2 2

pffiffiffi k.

w0 ¼ 0

Inserting the particular solution and its derivatives into Eq. (29), we have: 2Rk ¼ 3 k ¼ 3=2R pffiffiffiffiffiffiffiffiffiffi w ¼ 3=2R

ð33Þ

Similar to the analysis in Ref. [35], determining which roots of Eq. (30) are real and which are complex can be accomplished by noting that if the polynomial discriminant Q 3 þ R2 > 0; one root is real and two are complex conjugates; if Q 3 þ R2 ¼ 0; all roots are real and at least two are equal; and if Q 3 þ R2 < 0; all roots are real and unequal. Thus, we have two sets of solutions of Eq. (29). One solution is r ¼ B which represents a case of no fluid flow in the stack. Another set is two conjugated solutions, r1 and r2. Let’s take the two conjugated solutions into account which will depend on the sign of R2 þ Q 3 . Case 1: R2 þ Q 3 < 0 R Defining q ¼ cos1 ðpffiffiffiffiffiffiffi Þ, and substituting it into Eq. (30), Q 3

Thus, Eqs. (21a) and (21b) is reduced as follows: dwi d wi dwi  2Rw2i ¼ 3 þ 3Qwi dx dx2 dx

Let’s assuming a particular solution to Eq. (29), w ¼ Thus, its first and second derivatives are as follows:

w00 ¼ 0

where

A2 ¼

5503

Eq. (30) can be reduced to:   pffiffiffiffiffiffiffiffi q r1 ¼ 2 Q cos 3   pffiffiffiffiffiffiffiffi q þ 2p r2 ¼ 2 Q cos 3 Thus, the general solution of Eq. (29) and boundary conditions can be written as follows: wi ¼ C1 er1 x þ C2 er2 x þ wi ¼ 0; at x ¼ 1 wi ¼ 1; at x ¼ 0

pffiffiffiffiffiffiffiffiffiffi 3=2R

Axial velocity in the intake and exhaust manifolds:

er1  er2 ðer1 þr2 x  er2 þr1 x Þ þ er1  er2

rffiffiffiffiffiffi  3 er1 þr2 x  er2 þr1 x er1 x  er2 x  1 er1  er2 er1  er2 2R

wi ¼

wi ¼

rffiffiffiffiffiffi! 3 ðer1 þr2 x  er2 þr1 x Þ 2R

rffiffiffiffiffiffi 3 r1 x ðe  er2 x Þ rffiffiffiffiffiffi 3 2R  þ er1  er2 2R

1

" we ¼ 1

1

(34a)

(34b)

rffiffiffiffiffiffi! rffiffiffiffiffiffi 3 3 r1 x r2 x ðer1 þr2 x er2 þr1 x Þ ðe e Þ rffiffiffiffiffiffi# 2R 3 Fi 2R  þ er1 er2 er1 er2 2R Fe (35)

ð30Þ Channel velocity through substituting Eq. (34a) into Eq. (8):

where  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 þ R  Q 3 þ R2 B ¼ R þ Q 3 þ R2

(31)

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=3 J ¼ R þ Q 3 þ R2  R  Q 3 þ R2

(32)

rffiffiffiffiffiffi! 3 ( 1 ðr2 er1 þr2 x  r1 er2 þr1 x Þ 2R Fi dwi Fi uc ¼  ¼ Fc n dx Fc n er1  er2 rffiffiffiffiffiffi 3 ðr1 er1 x  r2 er2 x Þ) 2R  er1  er2

ð36Þ

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international journal of hydrogen energy 35 (2010) 5498–5509

Case 3: R2 þ Q 3 > 0

Flow distribution through Eq. (36):

vc ¼

Fc Uc nFc ¼ uc ¼  Fi W0 =n Fi rffiffiffiffiffiffi 3 ðr1 er1 x  r2 er2 x Þ 2R þ er1  er2

rffiffiffiffiffiffi! 3 1 ðr2 er1 þr2 x  r1 er2 þr1 x Þ 2R

Two conjugated solutions retain same as Eq. (30): 1 1 pffiffiffi r1 ¼  B þ i 3J 2 2 1 1 pffiffiffi r2 ¼  B  i 3J 2 2

er1  er2 ð37Þ

Pressure drop in the channels after substituting Eq. (34a) into Eq. (15): rffiffiffiffiffiffi ( 3  2 ð1  ðr2 er1 þr2 x  r1 er2 þr1 x Þ 1 Fi 2R pi  pe ¼ z er1  er2 2 Fc n

)2

pffiffiffiffi 3 ðr1 er1 x  r2 er2 x Þ  2R r e 1  er2

ð38Þ

Case 2: R2 þ Q 3 ¼ 0 Two conjugated solutions of the characteristic equation can be reduced as follows: 1 r1 ¼ r2 ¼  R1=3 ¼ r 2 Thus, the general solution of Eq. (29) and its boundary conditions can be written as follows: pffiffiffiffiffiffiffiffiffiffi wi ¼ ðC1 þ C2 xÞerx þ 3=2R wi ¼ 0; at x ¼ 1 wi ¼ 1; at x ¼ 0

(39)

Axial velocity in the intake and exhaust header: rffiffiffiffiffiffi  rffiffiffiffiffiffi rffiffiffiffiffiffi   rffiffiffiffiffiffi  3 3 3 1 3 x erx þ wi ¼ 1   1 þ r 2R 2R 2Re 2R

we ¼

Axial velocity in the intake and exhaust header: ( rffiffiffiffiffiffi pffiffiffi ! 3 eBx=2 3 B=2 wi ¼ pffiffi3  2Re sin 2 Jx sin 2 J !) rffiffiffiffiffiffi pffiffiffi rffiffiffiffiffiffi  3 3 3 Jð1  xÞ þ sin þ 1 2 2R 2R

(40a)

(40b)

rffiffiffiffiffiffi

   Fi 3 1  ð1  xÞerx þ 1  erx þ xerx  xerðx1Þ Fe 2R

(41)

pffiffi rffiffiffiffiffiffi( Jð1  xÞ sin 23Jx 3 Bx=2 pffiffi e pffiffi eBð1xÞ=2 þ 1 wi ¼ 2R sin 23J sin 23J pffiffi ) sin 23Jð1  xÞ pffiffi eBx=2  sin 23J sin

ð42Þ

Flow distribution through Eq. (42): rffiffiffiffiffiffi  Fc Uc nFc 3 ¼ uc ¼  1  rerx Fi W0 =n Fi 2R rffiffiffiffiffiffi rffiffiffiffiffiffi   3 3 1 rx e þ ð1 þ rxÞ 1  þ 2R 2Rer

ð43Þ

Pressure drop in the channels after substituting Eqs. (40a) and (40b) into Eq. (15): rffiffiffiffiffiffi  2  1 Fi 3 1 rerx pi  pe ¼ z 2 Fc n 2R rffiffiffiffiffiffi rffiffiffiffiffiffi  2  3 3 1 rx e  ð1 þ rxÞ 1  þ 2R 2Rer

2

ð46bÞ

" rffiffiffiffiffiffi pffiffiffi ! eBx=2 3 B=2 3 we ¼ 1  pffiffi3  2Re sin 2 Jx sin 2 J !# rffiffiffiffiffiffi) pffiffiffi rffiffiffiffiffiffi  3 3 Fi 3 Jð1  xÞ  þ 1 sin 2 2R 2R Fe

ð47Þ

Fi dwi eBx=2 Fi pffiffi ¼ uc ¼  Fc n dx sin 23J Fc n ! pffiffiffi ! pffiffiffi rffiffiffiffiffiffi  rffiffiffiffiffiffi B 3 B=2 B 3 3 3 Jx  1 Jð1xÞ e sin sin 2 2 2 2R 2 2R ! ð48Þ pffiffiffi pffiffiffi !  rffiffiffiffiffiffipffiffiffi pffiffiffi pffiffiffiffi 3 3 3 3 3 B=2 3 Jcos Jx  1 Jcos Jð1xÞ  2R e 2 2 2 2R 2

(

)

Flow distribution through Eq. (48): Fc n eBx=2 pffiffi uc ¼ Fi sin 23J ! pffiffiffi !  rffiffiffiffiffiffi pffiffiffi rffiffiffiffiffiffi B 3 B=2 B 3 3 3 Jx  1 Jð1xÞ e sin sin 2 2 2 2R 2 2R ! ð49Þ pffiffiffi pffiffiffi !  rffiffiffiffiffiffipffiffiffi pffiffiffi rffiffiffiffiffiffi 3 B=2 3 3 3 3 3 Jcos Jx  1 Jcos Jð1xÞ  e 2 2 2 2R 2R 2

vc ¼

uc ¼ 

vc ¼

pffiffi3

(

Channel velocity through substituting Eqs. (40a) and (40b) into Eq. (5): rffiffiffiffiffiffi

 Fi dwi Fi 3 ¼ 1 rerx Fc n dx Fc n 2R rffiffiffiffiffiffi rffiffiffiffiffiffi    3 3 1 rx e  ð1 þ rxÞ 1  þ 2R 2Rer

ð46aÞ

Channel velocity through substituting Eq. (47) into Eq. (5):

rffiffiffiffiffiffi  3 1  erx þ xerx  xerðx1Þ 2R

wi ¼ ð1  xÞerx þ

Thus, the general solution of Eq. (29) and boundary conditions can be written as follows: pffiffiffi  pffiffiffi  pffiffiffiffiffiffiffiffiffiffi

wi ¼ eBx=2 C1 cos 3Jx=2 þ C2 sin 3Jx=2 þ 3=2R (45) wi ¼ 0; at x ¼ 1 wi ¼ 1; at x ¼ 0

(

)

Pressure drop in the channels after substituting Eqs. (46a) and (46b) into Eq. (15):  2  2 dwi pi  pe ¼ 12z FFcin dx  2 1 Fi eBx pffiffi ¼ z 2 Fc n sin2 3J 2 2 ð50Þ rffiffiffiffiffiffi pffiffi  pffiffi pffiffiffiffi 3 B=2 B 3 B 3 3 e sin 2 Jx  2 1  2R sin 2 Jð1  xÞ 2 2R rffiffiffiffiffiffi rffiffiffiffiffiffi pffiffi pffiffi  pffiffi 3 B=2 pffiffi3 3 3 Jcos 23Jð1  xÞ e 2 Jcos 23Jx  1   2 2R 2R

(

ð44Þ

)

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4.

Results and discussions

4.1.

Comparison with existing analytical solutions

It is clear that both the friction and inertial term of the second order inhomogeneous nonlinear ordinary differential equation (Eqs. (21a) and (21b)) are kept in the present solutions. This gives a three complete analytical solutions for the Z-type fuel cell configurations, R2 þ Q 3 < 0; R2 þ Q 3 ¼ 0 and R2 þ Q 3 > 0: It can be seen that the present solutions of axial velocity are sum of those solutions in the U-type arrangement and pffiffiffiffiffiffiffiffiffiffi additional terms with 3=2R. When 3 ¼ 0, all the present solutions are the same as those in the U-type arrangement. However, it should be noted that there is an additional parameter, M, in Eqs. (22) and (23). Only when M ¼ 0, Q and R in the Z-type arrangement are the same as those in the U-type. Furthermore, the governing equation of the Z-type arrangement (Eq. (29)) is inhomogeneous with a parameter, 3. Therefore, they are still not the same even if M ¼ 0 or Q and R are the same. It can also be found that there are similar solutions of axial velocity to those in Ref. [30] when R2 þ Q 3  0. For R2 þ Q 3 < 0, Eq. (34a) can also be simplified to Eq. (2b) of Ref. [30] when B1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 3 i fi Þ 3 and M z 1/2. 2R 1=2Mz1. Thus, Eq. (34a) is ¼ yðRe 2fi Di ¼ r x

5505

It can be found that the general governing equation of the Z-Type arrangement (Eqs. (20a) and (20b)) can be simplified to those based on Bernoulli equation when both bi and be are equal to one or to Maharudrayya, et al.’s [15] and Kee et al.’s [14] model when both bi and be are equal to zero. Therefore, both of them are a special case of the present solutions. Furthermore, it can be clearly seen that two dimensionless constants of K1 and K2 in Maharudrayya, et al.’s [15] and Kee et al.’s [14] model are always positive. Therefore, the control equation of their models has only one solution and lost two solutions with the triangular function and the exponential function. Thus, their models can’t capture the phenomena of the pressure change caused by flow branching. It can be found that there are the terms with bi and be which represent the inertial effects on the flow performance in Eqs. (34a) (34b), (40a) (40b) and (46a) (46b). The inertial term does make a contribution to the flow momentum change. The previous research [24,30–34,37] indicated that bi and be play an important role in flow distribution and pressure drop in a manifold. Based on Eqs. (21a) and (21b) and the present solutions, we can estimate the effect of bi and be on flow distribution. Therefore, it may cause a significant error about an order of magnitude of

r x

2 simplified to wi z1  ee1r1 e er2 . This equation is the same as Eq.

(2b) of Ref. [30]. Therefore, all the present solutions can be simplified to those obtained by Bassiouny and Martin and all the results in Ref. [30] can be reproduced when B1 ¼ 3. Hence, Bassiouny and Martin’s solution is a special case of the present solutions neglecting friction effect. Because of neglecting friction effects in Bassiouny and Martin’s solutions their model may cause a significant error for a longer header or a big ratio of length and diameter. Fig. 4 shows a comparison of axial velocity profiles between the U- and the Z-type arrangement with the same Q and R when R2 þ Q 3 < 0. The results are comparable to those in Fig. 3 of Ref. [30] by Bassiouny and Martin. Here a little of difference can be explained for the contribution from the friction to flow distribution.

Fig. 4 – A comparison of axial velocity distribution between Z-type and U-type at Case 1.

Fig. 5 – Axial velocity distribution in the intake header at Case 1: (a) effect of M, (b) effect of A2.

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20%–50% if the inertial term is neglected, particularly for a short manifold or a small ratio of length and diameter.

4.2.

Case 1 ðR2 þ Q 3 < 0Þ

Fig. 5 shows influence of the parameters, M and A2 on axial velocity in the intake header for R2 þ Q 3 < 0. It can be seen in Fig. 5a that the velocity patterns become more linear as M increases. This can be explained that the portion of 3 in R (Eq. (23)) decreases as M increases. Similarly, the portion of A2 in Q (Eq. (22)) decreases as M increases. This means that the effect of the exhaust manifold structure on flow in the intake manifold will decrease as well. Fig. 5b shows the effect of A2 on the axial velocity. A small A2 is more closed to A1. The effect of friction can balance that of momentum. However, the velocity profiles become more nonlinear as A2 increases. This means that Fe becomes smaller and smaller. The effect of the momentum in the intake manifold becomes dominant. It should be noted that Case 1 occurs usually at some extreme conditions, such as Fi/Fe [ 1. Fig. 6 shows the effect of parameter changes on flow distribution in the channels when R2 þ Q 3 < 0. It can be seen in Fig. 6a that a bigger M has a better flow distribution. This is

Fig. 7 – Effect of M on pressure drop at Case 1 when A1 [ 1, A2 [ 10, B1 [ 1, 3 [ 1.

in agreement with that in Fig. 5a. However, a variety of combination of parameters, A1, A2, B1 and 3 does not improve the uniformity of the flow distribution at a fixed M. Fig. 7 shows the effect of changing parameter, M, on pressure drop in the channels when R2 þ Q 3 < 0. It can be seen that when A1, A2, B1 and 3 are fixed, there is a more uniform pressure drop in the channels for a bigger M. Of course, as M increases, a uniform pressure drop can be expected.

4.3.

Case 2 ðR2 þ Q 3 ¼ 0Þ

Fig. 8 represents axial velocity profiles in the intake header when R2 þ Q 3 ¼ 0. For Case 2, the two roots of the characteristic equation were reduced to r1 ¼ r2 ¼ 12R1=3 ¼ r in which the solution of the momentum equation (Eqs. (21a) and (21b)) is only dependent on R. Hence, there is no influence of Q on the axial velocity distribution. It can be seen that the axial velocity profile approaches linear as M decreases. Unlike Case 1, there is a smaller influence of M on axial velocity profile. A linear

Fig. 6 – Flow distributions in channels at Case 1, (a) effect of M at A1 [ 1, A2 [ 10, B1 [ 1, 3 [ 1; (b) effects of friction and momentum factors at a fixed M.

Fig. 8 – Effect of M on axial velocity distribution at Case 2 when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1.

international journal of hydrogen energy 35 (2010) 5498–5509

Fig. 9 – Effect of M on flow distributions in channels at Case 2 when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1.

Fig. 11 – Effect of M on axial velocity distribution at Case 3 when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1.

4.4. profile can be found when M approaches zero. It is because both Q and R are equal to zero when M ¼ 0 since B1 ¼ 3 and A1 ¼ A2 in this figure. Thus, Eqs. (21a) and (21b) reproduces a linear velocity profile of Ref. [35]. Fig. 9 shows the effect of M on the axial velocity profile. Increasing M will result in more and more uneven. This can be explained that the effect of the exhaust header on the flow distribution increases as the M increases. That means that the flow distribution in the intake header for the Z-type arrangement will be more dependent on the exhaust structure than for the U-type one. The pressure drop in the channels is illustrated in Fig. 10 when R2 þ Q 3 ¼ 0. Since the solution of the momentum equation (Eqs. (21a) and (21b)) is only dependent on the friction coefficient, R, there is no influence of Q on the pressure drop. It can be seen that the non-uniform of the pressure drop increases as M increases. A uniform pressure drop can be found at M z 0. This is in agreement with those in Figs. 8 and 9.

Fig. 10 – Effect of M on pressure drop at Case 2 when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1.

5507

Case 3 ðR2 þ Q 3 > 0Þ

Influence of the structure parameter M on the axial velocity in the intake header is shown in Fig. 11 when R2 þ Q 3 > 0. A change in M does affect the axial velocity profile. The linear

Fig. 12 – Flow distributions in channels at Case 3, (a) effect of M when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1; (b) effect of friction and momentum factors at a fixed M.

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5.

Fig. 13 – Effect of M on pressure drop at Case 3 when A1 [ 1, A2 [ 1, B1 [ 1, 3 [ 1.

axial velocity can be approached as M decreases. For a smaller M, both Q and R approach zero in Eqs. (22) and (23) since A1 ¼ A2 and B1 ¼ 3. That is a linear axial velocity profile can be expected when both Q and R are equal to zero. Fig. 12 shows the influence of the structure parameters on flow distribution in the channels for R2 þ Q 3 > 0. The uniform distribution can be found only when M approaches zero, as shown in Fig. 12a. A change in either A1, A2, B1 or 3 results in non-uniformity as shown in Fig. 12b. A proper selection of a set of these parameters does improve flow distribution for a fixed M since they represent either momentum effect or friction. This is because a certain momentum always needs a corresponding friction balance. Therefore, it is possible to have a uniform flow distribution design when a set of proper structure parameters is selected. Fig. 13 shows influence of the structure parameter M on pressure drop in the channels when R2 þ Q 3 > 0. The uniform pressure drop can be found again only when both M approaches zero. Any change of M will result in nonuniformity more or less. However, a proper selection of M as well as a combination of parameters, A1, A2, B1 and 3 does improve uniformity of the pressure drop. It should be noted that the present model is completely applicable to the rectangular or other shapes when some small modifications are made for Eqs. (6), (11) and (15) using the hydraulic diameter and wetted perimeter. It is also applicable to Z-type arrangements with interdigitated channels, such as those in Ref. [39] when all the turning losses in curve channels are included in Eq. (15). All the present results demonstrate capability of the present model used for the Z-type fuel cell configurations. As expected, the present analytical solution can explore a wider variety of combination of structures and flow parameters than those proposed by Bajura and Jones [27,28], by Bassiouny and Martin [30], by Kee et al. [14] and Maharudrayya et al. [15] as well as Bernoulli equation since the model includes both inertial and friction term. The present results also demonstrate the possibility for preliminary design optimisation. Therefore, this work provides a flexible and direct tool for the designers of Z-type fuel cell configurations.

Conclusions

A general theoretical model based on mass and momentum conservation has been developed to solve the flow and pressure distribution of Z-type fuel cell configurations. While existing models neglected either friction term or inertial term, the present model takes both of them into account. Thus, the present model includes main existing theoretical models as its special cases. As the analytical solutions are fully explicit, they are easily used to predict pressure drop and flow distribution for Z-type layers or stacks and provide easy-to-use design guidance under a wide variety of combination of flow conditions and geometrical parameters to investigate the interactions among structures, operating conditions and manufacturing tolerance and to minimize the impact on stack operability. Parameter Sensitivity is analysed through five general characteristic parameters representing geometrical structures and flow conditions for the flow performance of the fuel cell stack. It is found that there is a general polynomial discriminant, R2 þ Q 3 to determine the flow distribution and pressure drop. The proper adjustment of geometrical structure and flow conditions can improve uniformity and achieve an optimal design of the Z-type fuel cell stacks. Bernoulli equation in manifolds is a special case of the present flow models. It is also obvious that the existing solution by Bajura and Jones [27,28] and by Bassiouny and Martin [30] is a special case of the present solutions without the friction effect and those by Kee et al. [14] and Maharudrayya et al. [15] are another special case without inertial effect. Therefore, the present model includes almost all the main existing models. The results are applicable to a wide variety of combination of fuel cell stack dimensions, channel geometries, and fluid velocity. It also provides a fundamental theory for network models. Furthermore, the model can be used for the design guidance of other manifold systems, such as plate heat exchanges, Z-type arrangements with interdigitated channels, plate solar collector, distributor of fluidised bed and boiler headers.

Acknowledgements North Wyke Research is supported by Biotechnology and Biological Sciences Research Council (BBSRC). The Author would like to thank editors and reviewers for their comments and suggestions.

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