# 1D_packbed.ipynb (Source)

Data files used in this example: Ammonia-Ru-Ba-YSZ.yaml

# One-dimensional packed-bed, catalytic-membrane reactor model without streamwise diffusion¶

The present model simulates heterogeneous catalytic processes inside packed-bed, catalytic membrane reactors. The gas-phase and surface-phase species conservation equations are derived and the system of differential-algebraic equations (DAE) is solved using the scikits.odes.dae IDA solver.

In :
# Import Cantera and scikits
import numpy as np
from scikits.odes import dae
import cantera as ct
import matplotlib.pyplot as plt

%matplotlib inline
%config InlineBackend.figure_formats = ["svg"]
print("Runnning Cantera version: " + ct.__version__)

Runnning Cantera version: 2.6.0


## Methodology¶

One-dimensional, steady-state catalytic-membrane reactor model with surface chemistry is developed to analyze species profiles along the length of a packed-bed, catalytic membrane reactor. The same model can further be simplified to simulate a simple packed-bed reactor by excluding the membrane. The example here demonstrates the one-dimensional reactor model explained by G. Kogekar, et al., "Efficient and robust computational models of heterogeneous catalysis in catalytic membrane reactors," (2022), In preparation .

### Governing equations¶

Assuming steady-state, one-dimensional flow within the packed bed, total-mass, species mass and energy conservation may be stated as 

$$\frac{d(\rho u)}{dz} = \sum_{k=1}^{K_{\mathrm{g}}} \dot {s}_k W_k A_{\mathrm{s}} + \frac{P_{\mathrm b}}{A_{\mathrm b}} j_{k_{\mathrm M}}, \\ \rho u \frac{dY_k}{dz} + A_{\mathrm{s}} Y_k \sum_{k=1}^{K_{\mathrm{g}}} \dot {s}_k W_k = A_{\mathrm s} \dot {s}_k W_k + \delta_{k, k_{\mathrm M}} \frac{P_{\mathrm b}}{A_{\mathrm b}} j_{k_{\mathrm M}}, \\ \rho u c_{\mathrm p} \frac{dT}{dz} + \sum_{k=1}^{K_{\mathrm g}} h_k (\phi_{\mathrm g} \dot {\omega}_k + A_{\mathrm s} \dot {s}_k) W_k = \hat h \frac{P_{\mathrm b}}{A_{\mathrm b}}(T_{\mathrm w} - T) + \delta_{k, k_{\mathrm M}} \frac{P_{\mathrm b}}{A_{\mathrm b}} h_{k_{\rm M}} j_{k_{\mathrm M}}.$$

The fractional coverages of the $K_{\rm s}$ surface adsorbates $\theta_k$ must satisfy $$\dot {s}_k = 0, {\ \ \ \ \ \ } (k = 1,\ldots, K_{\mathrm s}),$$ which, at steady state, requires no net production/consumption of surface species by the heterogeneous reactions .

The pressure within the bed is calculated as $$\frac{dp}{dz} = - \left( \frac{\phi_{\mathrm{g}} \mu}{\beta_{\mathrm{g}}} \right) u,$$ where $\mu$ is the gas-phase mixture viscosity. The packed-bed permeability $\beta_{\rm g}$ is evaluated using the Kozeny-Carman relationship as $$\beta_{\mathrm g} = \frac{\phi_{\mathrm{g}}^3 D_{\mathrm p}^2}{72 \tau_{\mathrm{g}}(1 - \phi_{\mathrm{g}})^2},$$ where $\phi_{\mathrm{g}}$, $\tau_{\mathrm{g}}$, and $D_{\mathrm {p}}$ are the bed porosity, tortuosity, and particle diameter, respectively.

The independent variable in these conservation equations is the position $z$ along the reactor length. The dependent variables include total mass flux $\rho u$, pressure $p$, temperature $T$, gas-phase mass fractions $Y_k$, and surfaces coverages $\theta_k$. Gas-phase fluxes through the membrane are represented as $j_{k, {\mathrm M}}$. Geometric parameters $A_{\mathrm s}$, $P_{\mathrm b}$, and $A_{\mathrm b}$ represent the catalyst specific surface area (dimensions of surface area per unit volume), reactor perimeter, and reactor cross-sectional flow area, respectively. Other parameters include bed porosity $\phi_{\mathrm g}$ and gas-phase species molecular weights $W_k$. The gas density $\rho$ is evaluated using the equation of state (ideal Eos, RK or PR EoS).

If a perm-selective membrane is present, then $j_{k_{\mathrm M}}$ represents the gas-phase flux through the membrane and ${k_{\mathrm M}}$ is the gas-phase species that permeates through the membrane. The Kronecker delta, $\delta_{k, k_{\mathrm M}} = 1$ for the membrane-permeable species and $\delta_{k, k_{\mathrm M}} = 0$ otherwise. The membrane flux is calculated using Sievert's law as $${j_{k_{\mathrm M}}}^{\text{Mem}} = \frac{B_{k_{\mathrm M}}}{t} \left ( p_{k_{\mathrm M} {\text{, mem}}}^\alpha - p_{k_{\mathrm M} \text{, sweep}}^\alpha \right ) W_{k_{\rm M}}$$ where $B_{k_{\mathrm M}}$ is the membrane permeability, $t$ is the membrane thickness. $p_{k_{\mathrm M} \text{, mem}}$ and $p_{k_{\mathrm M} \text{, sweep}}$ represent perm-selective species partial pressures within the packed-bed and the exterior sweep channel. The present model takes the pressure exponent $\alpha$ to be unity. The membrane flux for all other species ($k \neq k_{\mathrm M}$) is zero.

### Chemistry mechanism¶

This example uses a detailed 12-step elementary micro-kinetic reaction mechanism that describes ammonia formation and decomposition kinetics over the Ru/Ba-YSZ catalyst. The reaction mechanism is developed and validated using measured performance in a laboratory-scale packed-bed reactor . This example also incorporates a Pd-based H2 perm-selective membrane.

### Solver¶

Above governing equations represent a complete solution for a steady-state packed-bed, catalytic membrane reactor model. The dependent variables are the mass-flux $\rho u$, species mass-fractions $Y_k$, pressure $p$, temperature $T$, and surface coverages $\theta_k$. The equation of state is used to obtain the mass density, $\rho$.

The governing equations form an initial value problem (IVP) in a differential-algebraic (DAE) form as follows: $$f(z,{\bf{y}}, \bf {y'}, c) = 0,$$ where $\bf{y}$ and $\bf{y'}$ represent the solution vector and its derivative vector, respectively. All other constants such as reference temperature, chemical constants, etc. are incorporated in vector $c$ (Refer to  for more details). This type of DAE system in this example is solved using the scikits.odes.dae IDA solver.

1. G. Kogekar, H. Zhu, R.J. Kee, 'Efficient and robust computational models of heterogeneous catalysis in catalytic membrane reactors', (2022), In preparation
2. B. Kee, C. Karakaya, H. Zhu, S. DeCaluwe, and R.J. Kee, 'The Influence of Hydrogen-Permeable Membranes and Pressure on Methane Dehydroaromatization in Packed-Bed Catalytic Reactors', Industrial & Engineering Chemistry Research (2017) 56, 13:3551 - 3559
3. R.J. Kee, M.E. Coltrin, P. Glarborg, and H. Zhu, 'Chemically Reacting Flow: Theory, Modeling and Simulation', Wiley (2018)
4. Z. Zhang, C. Karakaya, R.J. Kee, J. Douglas Way, C. Wolden,, 'Barium-Promoted Ruthenium Catalysts on Yittria-Stabilized Zirconia Supports for Ammonia Synthesis', ACS Sustainable Chemistry & Engineering (2019) 7:18038 - 18047
5. G. Kogekar, 'Computationally efficient and robust models of non-ideal thermodynamics, gas-phase kinetics and heterogeneous catalysis in chemical reactors' (2021), Doctoral dissertation.

### Define gas-phase and surface-phase species¶

In :
# Import the reaction mechanism for Ammonia synthesis/decomposition on Ru-Ba/YSZ catalyst
mechfile = "data/Ammonia-Ru-Ba-YSZ.yaml"
# Import the models for gas-phase
gas = ct.Solution(mechfile, "gas")
# Import the model for surface-phase
surf = ct.Interface(mechfile, "Ru_surface", [gas])

# Other parameters
n_gas = gas.n_species  # number of gas species
n_surf = surf.n_species  # number of surface species
n_gas_reactions = gas.n_reactions  # number of gas-phase reactions

# Set offsets of dependent variables in the solution vector
offset_rhou = 0
offset_p = 1
offset_T = 2
offset_Y = 3
offset_Z = offset_Y + n_gas
n_var = offset_Z + n_surf  # total number of variables (rhou, P, T, Yk and Zk)

print("Number of gas-phase species = ", n_gas)
print("Number of surface-phase species = ", n_surf)
print("Number of variables = ", n_var)

Number of gas-phase species =  4
Number of surface-phase species =  6
Number of variables =  13


### Define reactor geometry and operating conditions¶

In :
# Reactor geometry
L = 5e-2  # length of the reactor (m)
R = 5e-3  # radius of the reactor channel (m)
phi = 0.5  # porosity of the bed (-)
tau = 2.0  # tortuosity of the bed (-)
dp = 3.37e-4  # particle diameter (m)
As = 3.5e6  # specific surface area (1/m)

# Energy (adiabatic or isothermal)
solve_energy = True  # True: Adiabatic, False: isothermal

# Membrane (True: membrane, False: no membrane)
membrane_present = True
membrane_perm = 1e-15  # membrane permeability (kmol*m3/s/Pa)
thickness = 3e-6  # membrane thickness (m)
membrane_sp_name = "H2"  # membrane-permeable species name
p_sweep = 1e5  # partial pressure of permeable species in the sweep channel (Pa)
permeance = membrane_perm / thickness  # permeance of the membrane (kmol*m2/s/Pa)

if membrane_present:
print("Modeling packed-bed, catalytic-membrane reactor...")
print(membrane_sp_name, "permeable membrane is present.")

# Get required properties based on the geometry and mechanism
W_g = gas.molecular_weights  # vector of molecular weight of gas species
vol_ratio = phi / (1 - phi)
eff_factor = phi / tau  # effective factor for permeability calculation
# permeability based on Kozeny-Carman equation
B_g = B_g = vol_ratio**2 * dp**2 * eff_factor / 72
area2vol = 2 / R  # area to volume ratio assuming a cylindrical reactor
D_h = 2 * R  # hydraulic diameter
membrane_sp_ind = gas.species_index(membrane_sp_name)

# Inlet operating conditions
T_in = 673  # inlet temperature [K]
p_in = 5e5  # inlet pressure [Pa]
v_in = 0.001  # inlet velocity [m/s]
T_wall = 723  # wall temperature [K]
h_coeff = 1e2  # heat transfer coefficient [W/m2/K]

# Set gas and surface states
gas.TPX = T_in, p_in, "NH3:0.99, AR:0.01"  # inlet composition
surf.TP = T_in, p_in
Yk_0 = gas.Y
rhou0 = gas.density * v_in

# Initial surface coverages
# advancing coverages over a long period of time to get the steady state.
Zk_0 = surf.coverages

Modeling packed-bed, catalytic-membrane reactor...
H2 permeable membrane is present.


### Define residual function required for IDA solver¶

In :
def residual(z, y, yPrime, res):
"""Solution vector for the model
y = [rho*u, p, T, Yk, Zk]
yPrime = [d(rho*u)dz, dpdz, dTdz, dYkdz, dZkdz]
"""
# Get current thermodynamic state from solution vector and save it to local variables.
rhou = y[offset_rhou]  # mass flux (density * velocity)
Y = y[offset_Y : offset_Y + n_gas]  # vector of mass fractions
Z = y[offset_Z : offset_Z + n_surf] # vector of site fractions
p = y[offset_p]  # pressure
T = y[offset_T]  # temperature

# Get derivatives of dependent variables
drhoudz = yPrime[offset_rhou]
dYdz = yPrime[offset_Y : offset_Y + n_gas]
dZdz = yPrime[offset_Z : offset_Z + n_surf]
dpdz = yPrime[offset_p]
dTdz = yPrime[offset_T]

# Set current thermodynamic state for the gas and surface phases
# Note: use unnormalized mass fractions and site fractions to avoid over-constraining the system
gas.set_unnormalized_mass_fractions(Y)
gas.TP = T, p
surf.set_unnormalized_coverages(Z)
surf.TP = T, p

# Calculate required variables based on the current state
coverages = surf.coverages  # surface site coverages
# heterogeneous production rate of gas species
sdot_g = surf.get_net_production_rates("gas")
# heterogeneous production rate of surface species
sdot_s = surf.get_net_production_rates("Ru_surface")
wdot_g = np.zeros(n_gas)
# specific heat of the mixture
cp = gas.cp_mass
# partial enthalpies of gas species
hk_g = gas.partial_molar_enthalpies

if n_gas_reactions > 0:
# homogeneous production rate of gas species
wdot_g = gas.net_production_rates
mu = gas.viscosity  # viscosity of the gas-phase

# Calculate density using equation of state
rho = gas.density

# Calculate flux term through the membrane
# partial pressure of membrane-permeable species
memsp_pres = (p * gas.X[membrane_sp_ind])
# negative sign indicates the flux going out
membrane_flux = (-permeance * (memsp_pres - p_sweep) / W_g[membrane_sp_ind])

# Conservation of total-mass
# temporary variable
sum_continuity = As * np.sum(sdot_g * W_g) + phi * np.sum(wdot_g * W_g)
res[offset_rhou] = (drhoudz - sum_continuity - area2vol * membrane_flux * membrane_present)

# Conservation of gas-phase species
res[offset_Y:offset_Y+ n_gas] = (dYdz + (Y * sum_continuity - phi * np.multiply(wdot_g,W_g)
- As * np.multiply(sdot_g,W_g)) / rhou)
res[offset_Y + membrane_sp_ind] -= area2vol * membrane_flux * membrane_present

# Conservation of site fractions (algebraic contraints in this example)
res[offset_Z : offset_Z + n_surf] = sdot_s

# For the species with largest site coverage (k_large), solve the constraint equation: sum(Zk) = 1
# The residual function for 'k_large' would be 'res[k_large] = 1 - sum(Zk)'
# Note that here sum(Zk) will be the sum of coverages for all surface species, including the 'k_large' species.
ind_large = np.argmax(coverages)
res[offset_Z + ind_large] = 1 - np.sum(coverages)

# Conservation of momentum
u = rhou / rho
res[offset_p] = dpdz + phi * mu * u / B_g

# Conservation of energy
res[offset_T] = dTdz - 0  # isothermal condition
# Note: One can just not solve the energy equation by keeping temperature constant.
# But since 'T' is used as the dependent variable, the residual is res[T] = dTdz - 0
# One can also write res[T] = 0 directly, but that leads to a solver failure due to singular jacobian

if solve_energy:
conv_term = (4 / D_h) * h_coeff * (T_wall - T) * (2 * np.pi * R)
chem_term = np.sum(hk_g * (phi * wdot_g + As * sdot_g))
res[offset_T] -= (conv_term - chem_term) / (rhou * cp)


### Calculate the spatial derivatives at the inlet that will be used as the initial conditions for the IDA solver¶

In :
# Initialize yPrime to 0 and call residual to get initial derivatives
y0 = np.hstack((rhou0, p_in, T_in, Yk_0, Zk_0))
yprime0 = np.zeros(n_var)
res = np.zeros(n_var)
residual(0, y0, yprime0, res)
yprime0 = -res


### Solve the system of DAEs using ida solver¶

In :
solver = dae(
"ida",
residual,
first_step_size=1e-15,
atol=1e-14,  # absolute tolerance for solution
rtol=1e-06,  # relative tolerance for solution
algebraic_vars_idx=[np.arange(offset_Y + n_gas, offset_Z + n_surf, 1)],
max_steps=8000,
one_step_compute=True,
old_api=False,  # forces use of new api (namedtuple)
)

distance = []
solution = []
state = solver.init_step(0.0, y0, yprime0)

# Note that here the variable t is an internal varible used in scikits. In this example, it represents
# the natural variable z, which corresponds to the axial distance inside the reactor.
while state.values.t < L:
distance.append(state.values.t)
solution.append(state.values.y)
state = solver.step(L)

distance = np.array(distance)
solution = np.array(solution)
print(state)

SolverReturn(flag=<StatusEnumIDA.SUCCESS: 0>, values=SolverVariables(t=0.05008188075110137, y=array([1.40862546e-03, 4.99997977e+05, 6.85053268e+02, 8.87695903e-02,
4.76156874e-01, 4.11794146e-01, 2.31457254e-02, 3.31389994e-03,
9.93036016e-01, 2.72895231e-03, 8.58521045e-04, 3.67135998e-08,
6.25735914e-05]), ydot=array([-1.00712547e-02, -4.63384892e+01,  2.35197310e+01,  1.93773529e+00,
-1.09694323e+01,  9.02163101e+00, -4.93684753e-06,  1.65140796e-01,
-3.26729842e-01,  1.54845328e-01,  5.55655962e-03,  4.81233014e-07,
1.18667692e-03])), errors=SolverVariables(t=None, y=None, ydot=None), roots=SolverVariables(t=None, y=None, ydot=None), tstop=SolverVariables(t=None, y=None, ydot=None), message='Successful function return.')


### Plot results¶

In :
plt.rcParams["font.size"] = 14
f, ax = plt.subplots(3, 2, figsize=(12, 12), dpi=96)

# Plot gas pressure profile along the flow direction
ax[0, 0].plot(distance, solution[:, offset_p], color="C0")
ax[0, 0].set_xlabel("Distance (m)")
ax[0, 0].set_ylabel("Pressure (Pa)")

# Plot gas temperature profile along the flow direction
ax[0, 1].plot(distance, solution[:, offset_T], color="C1")
ax[0, 1].set_xlabel("Distance (m)")
ax[0, 1].set_ylabel("Temperature (K)")

# Plot major and minor gas species separately
minor_idx = []
major_idx = []
for j, name in enumerate(gas.species_names):
mean = np.mean(solution[:, offset_Y + j])
if mean <= 0.1:
minor_idx.append(j)
else:
major_idx.append(j)

# Plot mass fractions of the gas-phase species along the flow direction

# Major gas-phase species
for j in major_idx:
ax[1, 0].plot(distance, solution[:, offset_Y + j], label=gas.species_names[j])
ax[1, 0].legend(fontsize=12, loc="best")
ax[1, 0].set_xlabel("Distance (m)")
ax[1, 0].set_ylabel("Mass Fraction")

# Minor gas-phase species
for j in minor_idx:
ax[1, 1].plot(distance, solution[:, offset_Y + j], label=gas.species_names[j])
ax[1, 1].legend(fontsize=12, loc="best")
ax[1, 1].set_xlabel("Distance (m)")
ax[1, 1].set_ylabel("Mass Fraction")

# Plot major and minor surface species separately
minor_idx = []
major_idx = []
for j, name in enumerate(surf.species_names):
mean = np.mean(solution[:, offset_Z + j])
if mean <= 0.1:
minor_idx.append(j)
else:
major_idx.append(j)

# Plot the site fraction of the surface-phase species along the flow direction
# Major surf-phase species
for j in major_idx:
ax[2, 0].plot(distance, solution[:, offset_Z + j], label=surf.species_names[j])
ax[2, 0].legend(fontsize=12, loc="best")
ax[2, 0].set_xlabel("Distance (m)")
ax[2, 0].set_ylabel("Site Fraction")

# Minor surf-phase species
for j in minor_idx:
ax[2, 1].plot(distance, solution[:, offset_Z + j], label=surf.species_names[j])
ax[2, 1].legend(fontsize=12, loc="best")
ax[2, 1].set_xlabel("Distance (m)")
ax[2, 1].set_ylabel("Site Fraction")
f.tight_layout()

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