Uniform Reaction Rates and Optimal Porosity Design for Hydrogen Fuel Cells

Transcription

1 Uniform Reaction Rates and Optimal Porosity Design for Hydrogen Fuel Cells Jamal Hussain Al-Smail Department of athematics and Statistics King Fahd University of Petroleum and inerals (KFUP) October 6, 2016 Page 1 of 20

2 1. Consider the 2d cross-section of hydrogen fuel cell along the gas channels Chemical reactions and Electricity production Page 2 of 20 Anode: 2H 2 4H + + 4e Cathode: 4H + + 4e + O 2 2H 2 O + Heat The electrons travel from the anode to the cathode through an external circuit generating electrical power.

3 2. A gentle walk through literature review for fuel cells optimization: Kermani, et all: 2004, Novruzi, et all: Feb, 2004: reaction rate is not uniform on CL Novruzi, et all: Jul, 2004: water accumulation occurs where reaction is low Secanel, et all: 2007: maximizing the current density on the cathode CL by optimizing the platinum loading and gas diffusion layer porosity Secanel, et all: : optimizing the cathode and anode assembly to maximize the current density awardi, et all: 2005: optimizing the operating conditions to maximize the current density Page 3 of 20

4 Song: 2004, et all: optimizing the cathode CL thickness to maximize the current density rujicic: 2004, et all: optimizing the cathode dimensions and inlet pressure to maximize the current density Kumar: 2003, et all: testing rectangular, triangular and hemispherical cathode air channels to maximize the current density Jamekhorshid: 2011, et all: the importance of uniform current density Santis: 2006, et all: optimizing the catalyst loading to have the current density even on the cathode CL Page 4 of 20

5 2.1. Experimental Findings: The current density or the reaction rate is not uniform on the cathode catalyst, which results in the following problems drying out of the membrane in regions with hight reaction rates water accumulation in regions with low water transport non-optimum usage of the cathode catalyst Page 5 of 20

6 3. Optimal Porosity Design of the DL Objective: to find an optimal porosity function 0.4 ε(x) 0.74 that minimizes the efficiency cost functional ( ) 2 E(ε) := ĉ(ε) ĉ(ε) dx, (1) subject to the state equations describing the fluid dynamics in the DL. Here, a and b are given nonnegative parameters. Take ε(x) = Σ N i=1ε i f i (x), 0.4 ε i 0.8 Page 6 of 20

7 3.1. athematical odeling Assumptions: steady state, isothermal, single gas phase Let ĉ o (or simply ĉ), ĉ n, ĉ w denote the mass fractions of oxygen, nitrogen and water vapor, and û g, ˆp g, ρ g denote the velocity, the pressure and the density of the mixture In the DL: Using the method of volume averaging, the state of the system in is modeled by Conservation of total mass where û g is the superficial or extrinsic velocity. Conservation of momentum (Darcy equation) (ρ g û g ) = 0, (2) û g = k(ε) ˆp g, (3) where ˆp g is the intrinsic pressure of the mixture, and K(ε) is the permeability K(ε) of the DL divided by the dynamic viscosity µ of the mixture. Hence, (ρ g k(ε) ˆp g ) = 0. (4) Page 7 of 20

8 ass conservation of oxygen and nitrogen gives N o = ( D o (ε)ρ g ĉ o + ρ g ĉ o û g ) = 0, N n = ( D n (ε)ρ g ĉ n + ρ g ĉ n û g ) = 0, where ĉ o and ĉ n are the extrinsic mass fractions of oxygen and nitrogen, and D(ε) is the effective diffusivity, which is a function of ε. Previous Findings: the gas density ρ g and the nitrogen mass fraction ĉ n can be assumed constant. This simplifies calculations as well. Page 8 of 20

9 3.2. DL athematical odel The physics in the DL is the described by then reduced Darry low (k(ε) p g ) = 0 and the advection-diffusion equation for oxygen ( D(ε) c + cu) = 0, where u = k(ε) p g Boundary Conditions: Let s assume that are both given on Σ. On the walls of the DL, p g = p Σ and c = c Σ Page 9 of 20 Since u 1 = k(ε) 1 p g, u 1 = 0 and N o ν = 0. k(ε) 1 p g = 0 on Γ w

10 Catalytic Reaction: On the catalyst layer, we have the Reaction Boundary Conditions: ρ g u 2 = N o ν + N w ν Also, the current density 1 N o ν = 1 N w ν o 2 w 2H O 2 + 2e H 2 O + Heat ( ) 1 i(x) = 4F N o ν o and from Butler-Volmer equation i(x) = 2i ( ) oc αc F c ref sinh o RT η Hence and N o ν = H m c, u 2 = K(ε) 2 p g = β m c Page 10 of 20

11 3.3. Weak Formulation Let q be a test function such that q = 0 on Σ. Then, the reduced Darcy is written as 0 = k(ε) p g q (k(ε) νp g )q g = k(ε) p g q (β m c)q Find p HΣ 1 () such that k(ε) p q (β m c)q = 0, for all q H 1 Σ(). Similarly, a weak form for the advection-diffusion equation reads: Find c HΣ 1 () such that [D(ε) ĉ K(ε) p g ĉ] ϕ + for all ϕ H 1 Σ (). H m ĉϕ = 0, Page 11 of 20

12 3.4. Porosity Optimization Recall that and the optimal porosity E(ε) := ( ) 2 c(ε) c(ε) dx, ε = (ε 1, ε 2,..., ε N ) is found by the radient Descent ethod ε n+1 i = ε n i εi E(ε n ), where i E(ε n ) requires εi c for all i = 1, 2,..., N. Page 12 of 20

13 But εi c =: c satisfies k p q + [D c k p c] ϕ k(ε) p q (β m c )q [D(ε) c k(ε) p c k(ε) p c ] ϕ H m c ϕ = 0, for all ϕ, q H 1 Σ (). Remark: It is computationally very expensive to solve this system N times for each iteration. Page 13 of 20

14 3.5. Adjoint System Now we choose particular ϕ, q HΣ 1 () making the terms involving p and c equal to zero, integrate by parts and use ν ϕ = ν q = 0 on Γ w : k p q + [D c k p c] ϕ + [k( q c ϕ)]p [ (D ϕ) + k p ϕ]c k( ν q c ν ϕ)p + (D ν ϕ + H m ϕ β m q)c = 0 Since εi E(ε) = g( c)c we set D ν ϕ + H m ϕ β m q = g( c) on. Since also p is unknown on, we set k( ν q c ν ϕ) = 0 on. Page 14 of 20

15 The adjoint system defined as I) (k q) = ( c ϕ) in Σ : q = 0, Γ w : ν q = 0 : k( ν q c ν ϕ) = 0 Coupled with II) (D ϕ) k p ϕ = 0 Σ : ϕ = 0, Γ w : ν ϕ = 0 : D ν ϕ + H m ϕ β m q = g Page 15 of 20

16 Then εi E(ε) = = gc [k p q + (D c k p c) ϕ] where c, p are the solutions of the state equations, and ϕ, q are the solutions of the adjoint equations. Remark: For each ε iteration, we only need to solve the state and adjoint equation to obtain on optimal porosity by means of ε n+1 i = ε n i εi E(ε n ). Page 16 of 20

17 4. Page 17 of 20

18 Thank You Page 18 of 20

19 4.1. Effect of the eometric Design of Γ on the FC Designing the shape of Γ so that the oxygen mass flux N o,y is as uniform and as large as possible on the catalyst layer (CL) and the in/out pressure drop p in p out along the channel as small as possible. Or equivalently, find Γ that minimizes the following cost functional: E(Γ) = ( N o,y 1 ) 2 N o,y, where a, b and e are some given nonnegative constants. Page 19 of 20 The variables N o,y and p in are obtained through a state problem solved on A and.

20 4.2. Optimal Design of the Air Channel Oxygen mass fraction Oxygen mass flux on the CL This shape design of Γ improves the FC s efficiency as the catalyst layer is entirely used by the reactants accumulation of water and heat is reduced Page 20 of 20