Manifold Design in a PEM Fuel Cell Stack to Improve Flow Distribution Uniformity

Abstract

In this paper, a numerical study was performed to investigate the flow distribution in a 52-cell proton exchange membrane (PEM) fuel cell stack. The non-uniformity factor and standard deviation parameters were used to determine the flow distribution uniformity. Flow channels of each bipolar plate were replaced with straight parallel channels filled with porous media to reduce computational costs. The effect of external and integrated humidifiers on the gas distribution among the channels was investigated. Using integrated humidifiers improved the non-uniformity factor and standard deviation by 35% and 19%, respectively. Two methods were employed to improve the flow distribution: gradual reduction of the manifold height, and installing a bump at the bottom wall of the inlet manifold. Reducing the height of the inlet manifold in the stack with integrated and external humidifiers decreased the non-uniformity factor by 62% and 44%, respectively. The installation of the bump on the manifold wall enhanced flow distribution in the stack with the external humidifier. The results show that by using an integrated humidifier in this method, the flow distribution became more non-uniform. The best flow distribution in the stack was obtained with an integrated humidifier and a 90% reduction in manifold height. In this case, the flow rate passing through each channel was more than 99% of the average mass flow rate passing through the entire channel.

1. Introduction

Fuel cells are devices that convert the chemical energy of a fuel directly into electrical energy. In heat engines and thermal power plants, fuel combustion energy will be available as useful work through some processes and mechanical tools. Due to the absence of moving components and the ability to convert chemical energy directly into electrical, the efficiency of fuel cells is greater than that of common thermal energy generation systems. Among the other types of fuel cells, the PEM fuel cell has obtained much attention due to its low-temperature operation, high power density, and solid electrolyte, and can be an alternative for internal combustion engines [1].

A single PEM fuel cell can only produce a voltage in the range of 0.6 to 1 V. Several fuel cells are connected in series, forming a fuel cell stack to increase the output voltage. The fuel and oxidant enter the inlet manifold from the feeding source in a fuel cell stack. The inlet manifold is responsible for distributing the gas between the fuel cells. Because fuel cells in a stack are connected in series, the output voltage and efficiency of a stack are limited by the performance of each individual cell [2]. In the ideal condition, all fuel cells should receive an equal amount of reactant gas. Thus, the gas distribution in a manifold is more important than that in the gas flow channel of a bipolar plate [3]. In fuel cell stacks, the reactant gases are not distributed uniformly between the cells, and the flow maldistribution increases as the number of cells in a stack increases. The destructive disadvantages of maldistribution such as fuel cell flooding, membrane drying, and non-uniform temperature distribution reduce the stack efficiency [4,5].

There are few experimental studies carried out on the flow maldistribution in fuel cell stacks. Luo et al. [6] experimentally studied the flow and temperature distribution in a 40-cell PEM fuel cell stack. The voltage variation of cells showed that the effect of the temperature distribution on the performance of cells is more than that of flow distribution. Kim and Kim [7] experimentally measured the instantaneous velocity of airflow in a 10-cell PEM fuel cell stack. Their results showed that flow separation occurs in a manifold with 90° turns, which causes an increase in flow maldistribution. They observed that using a manifold with smooth turns improves the maximum power output of the PEM fuel cell stack by 10.3%. Lebeak et al. [3,8] modeled the cathode side of a 70-cell PEM fuel cell stack by locating a pressure plate between the straight parallel channels. They showed that the flow transition from the circular geometry of the inlet tube to the rectangular geometry of the manifold leads to forming a jet flow at the entrance of the manifold, resulting in a more uneven flow distribution. They added a diffuser to the inlet manifold, and the flow distribution was significantly improved. Anbumeenakshi and Thansekhar [9] experimentally investigated the water flow distribution in the parallel microchannels with two types of inlet configurations. They observed that a more uniform flow distribution is obtained when the fluid enters the manifold in a direction perpendicular to its length. They realized that the velocity and momentum of the fluid decreased after colliding with the manifold wall, which improves the flow distribution. Tanii et al. [10] measured the voltage of cells in a 9-cell fuel cell stack with serial and parallel flow patterns. Experimental data showed that the cell voltage variations in the parallel flow pattern were less than in the serial flow pattern.

The analytical methods reduce computational costs using the hydraulic network approach. Maharudrayya et al. [11] performed a 1D analysis to study the pressure and flow distribution in a fuel cell stack. Their results showed that the flow maldistribution was strongly affected by the rib width, manifold, and channels dimensions. Hossein et al. [12] used CFD and the analytical relations proposed by Maharudrayya et al. [11] to investigate the flow uniformity and distribution in parallel channels. The analytical results showed that most of the flow passed through the end 20% of channels. The CFD simulations indicated that the primary flow channels received less flow due to the high flow velocity at the header inlet. Their results showed that the gradual reduction in the height of the header led to a better flow distribution. Park and Li [13] developed a non-isothermal model to study the flow and temperature distribution in a 61-cell PEM fuel cell stack. They found that the flow distribution improved by reducing the hydraulic diameter of the flow channels and increasing the manifold dimensions. Qin et al. [14] investigated the effect of flow channel depth and width on the cell voltages of a PEM fuel cell stack. Their results showed that the average cell voltage increased from 0.62 V to 0.66 V when the channel width was reduced from 0.6 mm to 0.2 mm, while the pressure drop in the flow channels increased from about 0.5 kPa to 6.7 kPa. Mustata et al. [15] showed that the pipe network approximations cannot provide accurate results for gas flow distribution in PEM fuel cell stacks due to a large number of flow outputs from the manifold to the flow channels of fuel cells.

Analytical models have been developed to simplify the governing equations based on assumptions and approximations such as laminar flow, one dimensional flow, hydraulic network approach, etc. Therefore, researchers have used CFD to achieve more accurate results, as well as to model phenomena such as turbulence, separation, etc. Chen et al. [16] used the CFD to simulate flow distribution and pressure variation within a 72-cell fuel cell stack. Their results showed that a significant pressure drop in the flow channels, larger manifold width, and lower gas feeding rate improved the flow distribution. Jackson et al. [17] proposed a relationship for the width of the inlet and outlet manifolds in terms of the number of fuel cells. Their optimized design dramatically improved the flow distribution in the stack. However, the manifold width in the optimized design increased by 931% for a 26-cell stack. Lim et al. [18] studied the velocity and pressure distribution in a PEM fuel cell stack and investigated the effect of manifold configurations on the flow maldistribution. They realized that flow distribution in the stack with a double inlet/double outlet design is more uniform than the conventional single inlet/outlet configurations. Chen et al. [19] used the Taguchi method to investigate the effect of the inlet tube diameter, the inlet tube to intermediate zone length ratio, and the intermediate zone width on the pressure distribution in a 30-cell PEM fuel cell stack. They found that the inlet tube diameter is the most influential factor in the uniform flow distribution in the stack.

The performance of a PEM fuel cell is influenced by the relative humidity of the oxidant and fuel gases [20]. Because the proton conductivity of Nafion membranes depends on relative humidity, the reactant gases undergo a humidification process to increase the efficiency of PEM fuel cells. Wilberforce et al. [21] numerically and experimentally showed that the humidification of reactant gases influences the overall performance of a PEM fuel cell. The humidification process can be done outside the stack using an external humidifier [22]. The external humidifier increases the relative humidity of the fuel and oxidant gases, then gases with high relative humidity are transferred to the stack. The integrated humidifier can also be used to increase the relative humidity of the reactant gases [23]. In this method, the humidifier is connected to the stack, and the dry gases from suppliers are directly transferred to the stack.

The flow maldistribution in the cathode manifold is more severe [24]. Thus, the flow distribution improvement in the cathode manifold is more important. In some reviewed analytical studies [12,25], the gas flow channels of the bipolar plates were chosen hypothetically, and the pressure drop in the flow channels was calculated using the hydraulic network approximation. In some other studies [11,14], the pressure drop in the flow channels was assumed as a variable, and researchers tried to obtain the optimal flow distribution in the stack by changing the pressure drop coefficient. In CFD studies, the flow and pressure distribution in fuel cell stacks is usually simulated by two approaches. In the first approach, the fuel cells are replaced by rectangular channels, and the pressure drop in the flow channels of bipolar plates is ignored [12,18]. In these studies, the flow maldistribution is more severe due to the lack of simulating pressure drop. In the second approach, the parallel channels filled with a porous medium are used for considering the pressure drop in individual fuel cells [26]. In this approach, increasing the pressure drop in the channels reduces the flow maldistribution, which is more in line with the actual condition of fuel cell stacks.

In this paper, the flow distribution in the cathode side of a 52-cell PEM fuel cell stack is simulated. For simplification, the individual fuel cells are replaced by parallel rectangular channels filled with porous medium. The pressure drop obtained from the simulation of the flow channels of a single PEM fuel cell [27] is used to calculate the permeability coefficient of the porous medium. Then, the flow distribution within the stack with the external and integrated humidifiers is investigated. In the reviewed studies, the effect of the integrated humidifier on the flow distribution has not been investigated. This paper compares the flow maldistribution in a PEM fuel cell stack with external and integral humidifiers for the first time. A tapered manifold has been mentioned as a solution to reduce flow maldistribution. Therefore, three tapered manifolds with different inclination angles are used to investigate the effect of reducing the height of the manifold on the flow distribution. Also, a new method is presented to eliminate the effects of jet flow in the stack with the external humidifier. The effect of installing an arc bump at the bottom wall of the inlet manifold and gradual reduction in the height of the manifold on flow distribution is studied for stack with external and integrated humidifiers.

2. Governing Equation

The three-dimensional version of the incompressible Navier–Stokes equations is used to simulate the flow distribution in the PEM fuel cell stack. The following assumptions are applied to simplify the stack model:

• Heat, mass, and electrochemistry transport phenomena and the gravity force are ignored.
• The incompressible saturated oxygen is used as working flow.
• Straight channels replace the individual fuel cells.
• The channels are filled with porous media.

Based on these assumptions, the governing equations are given as below:

Continuity and momentum equations:

$$\nabla .(\rho U)=0,$$

(1)

$$\nabla .(\rho UU)=-\nabla p+\nabla .\tau -\frac{\mu }{\alpha }U,$$

(2)

where ρ and μ are the density and viscosity of saturated oxygen, U and p are the flow velocity and pressure, τ is the viscous stress tensor and α is the permeability. The Reynolds number in the manifold inlet is above 7000, and the standard k-ε model is used to simulate turbulent flow.

Turbulent kinetic energy and turbulent dissipation rate equations:

$$\nabla .(\rho kU)=\nabla .((\mu +\frac{{\mu }_t}{{\delta }_k})\nabla k)+P_k-\rho \epsilon +S_k,$$

(3)

$$\nabla .(\rho \epsilon U)=\nabla .((\mu +\frac{{\mu }_t}{{\delta }_{\epsilon }})\nabla \epsilon )+C_{1\epsilon }\frac{\epsilon }{k}{P}_k-\rho C_{2\epsilon }\frac{{\epsilon }^2}{k}+S_{\epsilon },$$

(4)

where k is turbulent kinetic energy, ε is the turbulent dissipation rate, μ_t is the turbulent viscosity, and P_k is the rate of the generation of the turbulence kinetic energy. δ_k = 1, δ_ε = 1.3, C_1ε = 1.44 and C_2ε = 1.92 are empirical constants. S_k and Sε are source terms.

3. Computational Domain and Boundary Conditions

The three dimensional computational domains of the 52-cell stack with external and integrated humidifiers are shown in Figure 1. The dimensions for this PEM fuel cell stack are obtained from a 12-kW PEM fuel cell stack and are based on an active area of 500 cm². The height and width of the input and output manifolds are 10 mm and 20 mm. The pressure drop in a single PEM fuel cell at the stack operating condition is 1142 Pa, which is obtained from the simulation of the flow channels of a single fuel cell in Ref. [27]. The flow field of the cathode section of a single fuel cell are simulated in Ref. [27] and the pressure drop is calculated. The simulated single fuel cell has a large active area (500 cm²) and requires 2.5 million meshes to achieve a grid-independent solution. As a result, for a stack containing 52 cells, more than 130 million meshes are required for simulation. A straight channel filled with porous media creates an equivalent pressure drop to reduce the calculation time. A rectangular channel filled with porous medium with length, width and height of 0.8, 20 and 56.6 mm is simulated. The length and width of the channel are equal to the depth of the flow channels of the fuel cell and the width of the manifold, respectively. The height of the channel is chosen as desired. The flow through a single fuel cell is passed through the channel. The permeability coefficient of the porous medium is calculated by trial and error so that the pressure drop in the channel is equal to the pressure drop in a fuel cell. The permeability coefficient is set to 9.22 $\times$ 10⁻⁸ for applying the equivalent pressure drop. In this paper, humidifiers are not simulated, and it is assumed that saturated oxygen enters the stack.

In the case of using an external humidifier (Figure 1a), a tube with a diameter of 8 mm is used to deliver the saturated oxygen flow from the humidifier to the stack. Due to end-plates and connections, the distance from the connection tube to the first channel is 50 mm. In the case of using integrated humidifiers (Figure 1b), the flow enters the manifold after passing six planar membrane humidifiers. Since the anode side humidifiers are first connected to the stack, the distance between the humidifiers and the first channel is 58.4 mm. A two-dimensional, cross-sectional schematic of the stack with integrated humidifiers is displayed in Figure 2. The length of the channels and ribs are 0.8 mm and 6.6 mm, respectively.

At the stack inlet, the mass flow rate is obtained from the number of PEM fuel cells (N = 52), current density (i = 0.6 A/cm²), fuel cell active area (A_active = 500 cm²), oxygen stoichiometric ratio (St = 1.4), operating temperature (T = 70 °C), operating pressure (P = 2 atm) and relative humidity (RH = 100%). The no-slip condition is applied to all walls. A constant pressure equal to the operating pressure is imposed at the stack outlet, and the normal gradients of other unknowns are set to zero. The governing equations and boundary conditions are implemented in the computational fluid dynamics software OpenFOAM. A second order scheme is used to discretize the convection and diffusion terms. The temporal terms are discretized with the first order Euler scheme. The flowchart of the simulation process is displayed in Figure 3.

4. Validation and Mesh Independency

The experimental data acquired by Kim and Kim [7] is used to check the accuracy of the numerical results obtained in this paper. In the experiment of Kim and Kim [7], air at atmospheric conditions enters the manifold of a 10-cell fuel cell stack, and instantaneous air velocity was measured as it passed through parallel channels. In Figure 4, the normalized air velocity obtained from the present study is compared with the numerical results and experimental data obtained by Kim and Kim [7]. Figure 4 shows that the numerical results have a good agreement with experimental data.

The flow distribution in the stack with integrated humidifiers is simulated with five different meshes, and the instantaneous flow velocity at the center of the channels is plotted in Figure 5. The horizontal axis demonstrates the channel number in the stack, where channel no.1 is the nearest channel to the stack inlet. According to Figure 5, the maximum difference between the results of simulations performed by 1.1 and 1.6 million elements is less than 0.65%. Thus, the mesh with 1.1 million elements is selected for further simulations.

5. Results

This study uses two parameters to measure the severity of flow maldistribution: non-uniformity factor and standard deviation. Non-uniformity factor [11,17] calculates the severity of maldistribution using the maximum and minimum mass flow rates passing through the channels:

$$F_1=\frac{\max ({\dot{m}}_1\cdots {\dot{m}}_{52})-\mathrm{m}\mathrm{i}\mathrm{n}({\dot{m}}_1\cdots {\dot{m}}_{52})}{\max ({\dot{m}}_1\cdots {\dot{m}}_{52})},$$

(5)

The standard deviation of the mass flow rate through the channels is given by:

$$F_2=\sqrt{\frac{{{\displaystyle \sum ({\dot{m}}_i-{\dot{m}}_{ave})}}^2}{N-1}},$$

(6)

where N is the total number of fuel cells in the stack, ${\dot{m}}_i$ is the mass flow rate in each individual cell, and ${\dot{m}}_{ave}$ is the average mass flow rate of 52 fuel cells. The lower values of F₁ and F₂ show a more uniform flow distribution.

5.1. The Effect of Humidifiers on Flow Distribution

Figure 6 shows the flow distribution among the parallel channels for the stack with the integrated and external humidifiers. The mass flow rate through each channel is normalized using the average mass flow rate to show the severity of the maldistribution. Due to the flow maldistribution in the stack manifold, some channels near the stack inlet receive a flow less than the average. The normalized mass flow rate for these channels is less than 1. The other channels receive more flow than the average to balance the total mass flow rate. For this reason, the normalized mass flow rate in downstream channels is greater than 1. According to Figure 6, the first channel in the stack receives 93.4% and 97% of the average flow passing through the channels for the external and integrated humidifiers

In the stack with the external humidifier, an insulation tube is usually used to supply the humidified oxidants to the manifold. The entry of gas from the tube into the manifold causes the formation of a jet flow, which increases the axial velocity at the inlet region of the initial channels. The velocity contour at the middle plane of the stack and inlet manifold is shown in Figure 7. As shown in Figure 7a, the jet formed at the entrance of the inlet manifold significantly impacts the first four channels. Figure 7b shows that the jet is asymmetric, which is consistent with the results obtained by Lebeak et al. [8].

Figure 8 shows the velocity contour in the case of using integrated humidifiers. As shown in Figure 8a, the high speed flow from the outlet of the humidifiers vertically enters the manifold and hits the upper wall. As a result, the flow velocity near the upper wall is higher than near the entrance of the channels. Also, the width of the outlet of the integrated humidifiers is equal to the manifold width. Therefore, the flow velocity across the manifold width is almost the same (Figure 8b), unlike the use of an external humidifier, where the flow velocity in the center of the manifold is higher than near the walls (Figure 7b). By comparing Figure 7 and Figure 8, it can be seen that spreading the inflow across the width of the manifold and shifting the high velocity region from the manifold center to near the manifold upper wall can improve flow distribution.

5.2. Geometry Optimization

Researchers have proposed various methods to reduce the flow maldistribution. The advantages and disadvantages of each method should be considered in the design of the stack. Increasing the pressure drop in the flow channels of the bipolar plates improves the flow distribution in the fuel cell stack. The reactant gases are transported to the fuel cell stack via the electric compressor. The input power of the compressor increases with pressure loss; thus, the electric efficiency of the system decreases as the pressure drop in the flow channel increases. Increasing the dimensions of the manifold is another approach to improve flow distribution. This method increases the volume and weight of the fuel cell stack. As a result, increasing the manifold dimensions reduces the power density (ratio of output power to stack volume) and specific power (ratio of output power to stack weight) of the stack.

In this paper, two approaches are examined to reduce the flow maldistribution. In the first approach, the height of the manifold is gradually reduced from the beginning of the inlet to its end. Three cases are considered for the gradual reduction of the manifold height, in which the manifold height (h) is gradually reduced by 70, 80, and 90%. In the second approach, an arc bump is placed on the lower wall of the manifold, aiming to reduce the effects of the jet flow. Three bumps with different heights are considered. The length of the bumps is fixed and equal to 28.2 mm, and the height of the bumps (r) is 50, 60, and 70% of the manifold height, respectively. Figure 9 shows the geometry of the proposed designs.

Simulated cases and their different geometrical parameters are listed in

Table 1. Simulated cases and their different geometrical parameters.

Case Name	H (mm)	h (mm)	r (mm)
Design A (default)	10	0	0
Design B	10	7	0
Design C	10	8	0
Design D	10	9	0
Design E	10	0	5
Design F	10	0	6
Design G	10	0	7

.

5.2.1. The Effect of Manifold Height Reduction on Flow Uniformity (Designs B–D)

The normalized mass flow rate through the channels for the stack with integrated humidifiers is plotted in Figure 10. The numerical results show that the normalized mass flow rate in the first channel increased from 0.97 for design A to 0.994 for design D. In design D, the normalized mass flow rates in individual channels were more than 0.99, and a uniform flow distribution was obtained.

The impact of manifold height reduction on the flow distribution for the stack with the external humidifier is displayed in Figure 11. The mass flow rate in the first half channels of the stack increased with gradually decreasing the manifold height. The mass flow rate through the second half channels of the stack decreased to maintain the constant total mass flow rate. As shown in Figure 11, the normalized mass flow rate in the first channel from 0.934 for design A increased to 0.96 for design D.

Figure 12 shows the static pressure and axial velocity of the flow inside the inlet manifold at 0.5 mm above the inlet of the channel. In design A, the cross-sectional area of the manifold is constant, and the local mass flow rate decreases along the length of the manifold. As a result, the axial flow velocity progressively decreases along the manifold length, which increases static pressure. Downstream channels receive more mass flow rate due to increased static pressure along the manifold. As seen in Figure 12, reducing the manifold height along the manifold length increased the axial velocity of the flow. Thus, the static pressure increased in the first half of the manifold and decreased in the second half. For this reason, the primary channels in designs B–D received more flow than design A, and a more uniform flow distribution was obtained.

Generally, a bigger manifold decreases the flow maldistribution, and a uniform flow distribution increases the electric efficiency of the stack. However, the power density and specific power of the stack decrease with increasing manifold size. Therefore, the uniform flow distribution in the fuel cell stack with a smaller manifold improves the technical specifications. This section shows that the flow distribution is more uniform in the stack with integrated humidifiers and tapered manifold. In other words, a smaller manifold can be used in this approach.

To compare the impact of the proposed designs on the gas flow distribution, the flow uniformity parameters for designs A–D are tabulated in

Table 2. Flow distribution parameters for designs A–D.

Design	Humidifier	Non-Uniformity Factor (F1)	Standard Deviation (F2) (10−7)
Design A	External	0.106	9.58
	Integrated	0.068	7.77
Design B	External	0.085	6.98
	Integrated	0.053	6.19
Design C	External	0.076	5.76
	Integrated	0.047	5.25
Design D	External	0.058	3.92
	Integrated	0.025	2.79

. The flow distribution parameters for design A show that integrated humidifiers improved the non-uniformity factor and standard deviation by 35.8% and 18.9%, respectively. In the case of using the external humidifier, the non-uniformity factor and standard deviation for design D were 45.3% and 59% lower than for design A. These values improved to 63% and 64% in the case of using integrated humidifiers. In calculating the non-uniformity factor, the maximum and minimum mass flow rates through the individual channels were used. In other words, only the mass flow rate through two channels was considered. In comparison, the mass flow rate in all channels is important in calculating the standard deviation. As shown in Figure 10 and Figure 11, with a tapered manifold, the mass flow rate changes in all channels, which causes the standard deviation to decrease more than the non-uniform factor.

The influence of the tapered manifold on the flow distribution between parallel channels in the laminar flow was investigated [12,28]. They assumed the flow at the manifold inlet to be fully developed. Tong et al. [28] showed that the effect of a tapered manifold on flow distribution increased with the Reynolds number. Hossain et al. [12] found that using a tapered manifold can improve gas flow distribution in the flow field of a fuel cell. This paper studied the effect of a tapered manifold in a stack with integrated and external humidifiers. The flow was turbulent and the Reynolds number range was not the same as studies [12,28]. It can be concluded that the results obtained from this paper are consistent with the results of [12,28].

5.2.2. The Effect of the Bump on Flow Uniformity (Designs E–G)

The numerical results show that the flow distribution in the stack with integrated humidifiers is more uniform because the flow velocity across the manifold width is approximately equal. Using the external humidifier, a jet is formed in the manifold, affecting the mass flow rate through primary channels. To overcome this problem, an arc bump is placed on the bottom wall of the inlet manifold. Figure 13 shows the velocity contour obtained from the numerical simulation of flow distribution for design G. The high velocity jet flow hits the bump. It deflects toward the upper wall of the manifold. The high velocity region is shifted from the manifold center to the top of the bump. Figure 13a,b show that the flow is spread across the manifold width after the jet hits the bump. By comparing Figure 7b and Figure 13b, it can be concluded that in the presence of a bump, the flow velocity increases near the sidewall of the manifold. The numerical results of Section 5.2.1 show that shifting the high velocity region from the center of the manifold to near the top wall and spreading the flow across the manifold width can improve the flow uniformity in the stack. Therefore, installing a bump is favorable for improving the flow distribution in a stack with an external humidifier. The numerical results show that the flow velocity above the bump increases significantly with a further increase in the bump height, and the flow distribution becomes more uneven. As a result, there is an optimal value for the bump height. Lebæk et al. [3] and Chen et al. [19] used a diffuser at the inlet of the manifold to reduce the effects of the jet flow on the flow distribution. Their results showed that the diffuser reduces the effects of jet flow on the flow maldistribution but does not remove the jet flow from the manifold. It is shown in Figure 13 that when the jet hits the arc bump, its structure is destroyed, and the velocity of the flow becomes uniform across the manifold.

Figure 14 shows the normalized mass flow rate in the stack channels with the external humidifier for designs A and E–G. The numerical results indicate the normalized mass flow rate in the first channel increases from 0.934 for design A to 0.973 for design G. Figure 14 indicates that the effects of the jet on the primary channels are significantly reduced for designs E–G. It is clear by comparing Figure 11 and Figure 14 that the mass flow rate in the primary channels for designs E–G increases more than for designs B–D, while the mass flow rate in the other channels changes slightly. However, in designs B–D, the mass flow rate in all channels is closer to the average value, and a more uniform flow distribution is obtained

The normalized mass flow rate in the individual channels of the stack with integrated humidifiers is plotted in Figure 15 for designs E–G. Placing the bump on the lower wall of the manifold decreases the mass flow rate in the primary channels and increases the flow maldistribution, unlike the case of using the external humidifier. It can be explained that the bump reduces the cross-sectional area of the manifold. As a result, the axial flow velocity above the bump increases by up to 40%, and the mass flow rate in the primary channels decreases for designs E–G. Therefore, placing the bump on the manifold bottom wall for the stack with the external humidifier improves the flow distribution due to neutralizing the effect of the jet flow. While in the case of using integrated humidifiers, the significant increase in the axial flow velocity causes an increase in the flow maldistribution.

The flow distribution parameters for designs E–D are calculated and tabulated in

Table 3. Flow distribution parameters for designs A, E–G.

Design	Humidifier	Non-Uniformity Factor (F1)	Standard Deviation (F2) (10−7)
Design A	External	0.106	9.85
	Integrated	0.0679	7.77
Design E	External	0.077	8.54
	Integrated	0.076	8.34
Design F	External	0.071	8.23
	Integrated	0.083	8.77
Design G	External	0.069	8.14
	Integrated	0.093	8.82

. In the case of using the external humidifier, the non-uniformity factor and standard deviation for design G are reduced by 34.9% and 17.4% compared to design A. In contrast, these values increase for integrated humidifiers by 37% and 13.5%. In designs B–D, the mass flow passing through all channels is affected by reducing the manifold height. Therefore, the standard deviation changes more than the non-uniformity factor. In designs E–G, only the mass flow rates through the primary channels are modified; therefore, the changes in the non-uniformity factor are more significant.

6. Conclusions

This paper investigates the flow maldistribution in the cathode side of a 52-cell PEM fuel cell stack. The non-uniformity factor and standard deviation determine the flow maldistribution in the stack with external and integrated humidifiers. In the stack with the external humidifier, a jet flow forms in the inlet manifold entrance, which reduces the mass flow rate in the primary channels. Using integrated humidifiers leads to a more uniform flow distribution, and the non-uniformity factor and standard deviation improve by 35.8% and 18.9%, respectively, compared to using the external humidifier. Then, two approaches are proposed to reduce flow maldistribution. Reducing the manifold height improves the flow distribution in the stack with both integrated and external humidifiers. The non-uniformity factor and standard deviation decreased by 45.3% and 59% for the external humidifier. These values enhance to 63% and 64% for integrated humidifiers. Placing an arc bump on the manifold bottom wall improves the flow distribution in the stack with the external humidifier. The non-uniformity factor and standard deviation decreased by 34.9% and 17.4%. This approach increases the flow maldistribution in the stack with integrated humidifiers.