As explained above, the governing equations of the gas mixture that include continuity, momentum and mass fraction of the components, are solved at first. Transient conservation equations are as follow:
Mass conservation
(1)
(∂(ε(1-s)ρ_g))/∂t+∇.(ρ_g u ⃗_g )=S_m
In this equation u ⃗_g is superficial velocity vector of gas mixture, which related to intrinsic fluid velocity U as following:
(2)
(u_g ) ⃗=ε(1-s)U ⃗
In the gas transfer channel, the porosity is equal one and the liquid water volume fraction is zero; so in this area, the superficial velocity will be equal to the intrinsic velocity.
Momentum conservation
(3)
1/ε(1-s) ∂(ρ_g u ⃗_g )/∂t+1/(ε^2 (1-s)^2 ) ∇.(ρ_g u ⃗_g u ⃗_g )= -∇p_g+1/ε(1-s) ∇.(μ_g ∇u ⃗_g )-μ_g/K u ⃗_g
Species conservation: …show more content…
Now, the liquid flow governing equations are presented to determine the volume fraction of the liquid and its velocity. The capillary pressure at porous media is defined as the difference between pressure in the liquid and vapor phase.
Mass conservation for liquid phase is as follow:
(18)
(∂(εsρ_l))/∂t+∇.(ρ_l u ⃗_l )=S_(g-l)
Darcy law can be used for liquid momentum equation in porous media; therefore, momentum equation is reduced and can be expressed by:
(19)
u ⃗_l=-K_l/μ_l ∇p_l
Also, the liquid pressure can be extracted from definition of capillary pressure.
(20)
p_c=p_g-p_l=f(s)
Permeability for each phase is defined as below:
(21)
K_l=K_rl K K_rl=s^n
(22)
K_g=K_rg K K_rg=〖(1-s)〗^n
Although the relative permeability depends on material and other factors, n can be between 1 and 5. However, n has been considered 3 in the most of investigations. Mass and momentum conservation can be combined and then substitute liquid pressure by equation (22) to achieve a partial differential equation for liquid water …show more content…
Then, by calculating the liquid phase pressure, also obtain the liquid phase velocity.
The amount of liquid water that was in the catalyst layer, will be effected on current density and active area for chemical reaction. In this situation, Butler-Volmer equation can be written as follow:
(27)
i=(1-s) i_0 〖x_o〗_2/〖x_o〗_(2,ref) exp(αFη/(RT) ̅ )
The activation loss in equation (27) is as below:
(28)
η=V_oc-V_cell-σi
In order to account the diffusion resistant and porosity effect in catalyst layer, have been suggested to correct the current density with agglomerate method. For this purpose, current density must be multiplied by effective factor θ as below:
(29)
i ́=θi θ is between 0 and 1. The following relations can be used to calculate this coefficient:
(30)
c_(o_2)^m=Hc_(o_2 )
(31)
k(c_(o_2)^m )^0.5=i/2F
(32)
M_T=L_ct √(k/(D_i^m ))
(33)
θ=tanh〖M_T 〗/M_T
So, current density is corrected by using previous
Mass conservation
(1)
(∂(ε(1-s)ρ_g))/∂t+∇.(ρ_g u ⃗_g )=S_m
In this equation u ⃗_g is superficial velocity vector of gas mixture, which related to intrinsic fluid velocity U as following:
(2)
(u_g ) ⃗=ε(1-s)U ⃗
In the gas transfer channel, the porosity is equal one and the liquid water volume fraction is zero; so in this area, the superficial velocity will be equal to the intrinsic velocity.
Momentum conservation
(3)
1/ε(1-s) ∂(ρ_g u ⃗_g )/∂t+1/(ε^2 (1-s)^2 ) ∇.(ρ_g u ⃗_g u ⃗_g )= -∇p_g+1/ε(1-s) ∇.(μ_g ∇u ⃗_g )-μ_g/K u ⃗_g
Species conservation: …show more content…
Now, the liquid flow governing equations are presented to determine the volume fraction of the liquid and its velocity. The capillary pressure at porous media is defined as the difference between pressure in the liquid and vapor phase.
Mass conservation for liquid phase is as follow:
(18)
(∂(εsρ_l))/∂t+∇.(ρ_l u ⃗_l )=S_(g-l)
Darcy law can be used for liquid momentum equation in porous media; therefore, momentum equation is reduced and can be expressed by:
(19)
u ⃗_l=-K_l/μ_l ∇p_l
Also, the liquid pressure can be extracted from definition of capillary pressure.
(20)
p_c=p_g-p_l=f(s)
Permeability for each phase is defined as below:
(21)
K_l=K_rl K K_rl=s^n
(22)
K_g=K_rg K K_rg=〖(1-s)〗^n
Although the relative permeability depends on material and other factors, n can be between 1 and 5. However, n has been considered 3 in the most of investigations. Mass and momentum conservation can be combined and then substitute liquid pressure by equation (22) to achieve a partial differential equation for liquid water …show more content…
Then, by calculating the liquid phase pressure, also obtain the liquid phase velocity.
The amount of liquid water that was in the catalyst layer, will be effected on current density and active area for chemical reaction. In this situation, Butler-Volmer equation can be written as follow:
(27)
i=(1-s) i_0 〖x_o〗_2/〖x_o〗_(2,ref) exp(αFη/(RT) ̅ )
The activation loss in equation (27) is as below:
(28)
η=V_oc-V_cell-σi
In order to account the diffusion resistant and porosity effect in catalyst layer, have been suggested to correct the current density with agglomerate method. For this purpose, current density must be multiplied by effective factor θ as below:
(29)
i ́=θi θ is between 0 and 1. The following relations can be used to calculate this coefficient:
(30)
c_(o_2)^m=Hc_(o_2 )
(31)
k(c_(o_2)^m )^0.5=i/2F
(32)
M_T=L_ct √(k/(D_i^m ))
(33)
θ=tanh〖M_T 〗/M_T
So, current density is corrected by using previous