###############################################################################
# The Institute for the Design of Advanced Energy Systems Integrated Platform
# Framework (IDAES IP) was produced under the DOE Institute for the
# Design of Advanced Energy Systems (IDAES).
#
# Copyright (c) 2018-2023 by the software owners: The Regents of the
# University of California, through Lawrence Berkeley National Laboratory,
# National Technology & Engineering Solutions of Sandia, LLC, Carnegie Mellon
# University, West Virginia University Research Corporation, et al.
###############################################################################


# Flowsheet Gibbs Reactor Simulation and Optimization of Steam Methane Reforming#

Author: Brandon Paul
Maintainer: Brandon Paul
Updated: 2023-06-01

## Learning Outcomes#

• Call and implement the IDAES GibbsReactor unit model

• Construct a steady-state flowsheet using the IDAES unit model library

• Connecting unit models in a flowsheet using Arcs

• Fomulate and solve an optimization problem

• Defining an objective function

• Setting variable bounds

## Problem Statement#

Following the previous example of Steam Methane Reformation in an Equilibrium Reactor, this example solves the flowsheet using a Gibbs Reactor instead. The steam methane reformation example is adapted from S.Z. Abbas, V. Dupont, T. Mahmud, Kinetics study and modelling of steam methane reforming process over a NiO/Al2O3 catalyst in an adiabatic packed bed reactor. Int. J. Hydrogen Energy, 42 (2017), pp. 2889-2903. Typically, the process follows the chemical equations below:

CH4 + H2O → CO + 3H2
CO + H2O → CO2 + H2

However, the GibbsReactor unit model solves the equilibrium by minimizing Gibbs free energy. Conveniently, this eliminates the need for a reaction package although a thermophysical package is still required.

The flowsheet that we will be using for this module is shown below with the stream conditions. As in the prior example, we will be processing natural gas and steam feeds of fixed composition to produce hydrogen. The process consists of a mixer M101 for the two inlet streams, a compressor to compress the feed to the reaction pressure, a heater H101 to heat the feed to the reaction temperature, and a GibbsReactor unit R101. We will use thermophysical properties following the Peng-Robinsion cubic equation of state for this flowsheet.

The state variables chosen for the property package are total molar flows of each stream, temperature of each stream and pressure of each stream, and mole fractions of each component in each stream. The components considered are: CH4, H2O, CO, CO2, and H2 and the process occurs in vapor phase only. Therefore, every stream has 1 flow variable, 5 mole fraction variables, 1 temperature and 1 pressure variable. ## Importing Required Pyomo and IDAES Components#

To construct a flowsheet, we will need several components from the Pyomo and IDAES packages, as well as some utility tools to build the flowsheet. For further details on these components, please refer to the Pyomo documentation.

From IDAES, we will be needing the FlowsheetBlock and the following unit models:

• Feed

• Mixer

• Compressor

• Heater

• GibbsReactor

• Product

from pyomo.environ import (
Constraint,
Var,
ConcreteModel,
Expression,
Objective,
TransformationFactory,
value,
units as pyunits,
)
from pyomo.network import Arc

from idaes.core import FlowsheetBlock
from idaes.models.properties.modular_properties import GenericParameterBlock
from idaes.models.unit_models import (
Feed,
Mixer,
Compressor,
Heater,
GibbsReactor,
Product,
)

from idaes.core.solvers import get_solver
from idaes.core.util.model_statistics import degrees_of_freedom
from idaes.core.util.initialization import propagate_state


## Importing Required Thermophysical Package#

As mentioned earlier, the GibbsReactor does not require a reaction package.

Let us import the following module from the IDAES library:

• natural_gas_PR as get_prop (method to get configuration dictionary)

from idaes.models_extra.power_generation.properties.natural_gas_PR import get_prop


## Constructing the Flowsheet#

Let us create a ConcreteModel and add the flowsheet block.

m = ConcreteModel()
m.fs = FlowsheetBlock(dynamic=False)


We now need to add the property packages to the flowsheet. Unlike the previous example, we do not need to add a reaction package for the reactor model to calculate results. We will use the Modular Property Framework to build a state block for the parameter dictionary.

thermo_props_config_dict = get_prop(components=["CH4", "H2O", "H2", "CO", "CO2"])
m.fs.thermo_params = GenericParameterBlock(**thermo_props_config_dict)


Let us start adding the unit models we have imported to the flowsheet:

m.fs.CH4 = Feed(property_package=m.fs.thermo_params)
m.fs.H2O = Feed(property_package=m.fs.thermo_params)
m.fs.PROD = Product(property_package=m.fs.thermo_params)
m.fs.M101 = Mixer(
property_package=m.fs.thermo_params, inlet_list=["methane_feed", "steam_feed"]
)
m.fs.H101 = Heater(
property_package=m.fs.thermo_params,
has_pressure_change=False,
has_phase_equilibrium=False,
)
m.fs.C101 = Compressor(property_package=m.fs.thermo_params)

m.fs.R101 = GibbsReactor(
property_package=m.fs.thermo_params,
has_heat_transfer=True,
has_pressure_change=False,
)


## Connecting Unit Models Using Arcs#

We have now added all the unit models we need to the flowsheet. Let us connect the unit models by defining and building each Arc:

m.fs.s01 = Arc(source=m.fs.CH4.outlet, destination=m.fs.M101.methane_feed)
m.fs.s02 = Arc(source=m.fs.H2O.outlet, destination=m.fs.M101.steam_feed)
m.fs.s03 = Arc(source=m.fs.M101.outlet, destination=m.fs.C101.inlet)
m.fs.s04 = Arc(source=m.fs.C101.outlet, destination=m.fs.H101.inlet)
m.fs.s05 = Arc(source=m.fs.H101.outlet, destination=m.fs.R101.inlet)
m.fs.s06 = Arc(source=m.fs.R101.outlet, destination=m.fs.PROD.inlet)


Finally, we can use Pyomo’s TransformationFactory to write the equality constraints on each Arc:

TransformationFactory("network.expand_arcs").apply_to(m)


## Adding Expressions to Compute Operating Costs#

For this flowsheet, we are interested in computing hydrogen production in millions of pounds per year, as well as the total costs due to pressurizing, cooling, and heating utilities:


operating cost = $39.958 million per year  For this operating cost, what conversion did we achieve of methane to hydrogen? m.fs.R101.report() print() print(f"Conversion achieved = {value(m.fs.R101.conversion):.1%}")  ==================================================================================== Unit : fs.R101 Time: 0.0 ------------------------------------------------------------------------------------ Unit Performance Variables: Key : Value : Units : Fixed : Bounds Heat Duty : 1.7819e+07 : watt : False : (None, None) ------------------------------------------------------------------------------------ Stream Table Units Inlet Outlet Total Molar Flowrate mole / second 309.01 429.02 Total Mole Fraction CH4 dimensionless 0.24272 0.034965 Total Mole Fraction H2O dimensionless 0.75725 0.32532 Total Mole Fraction H2 dimensionless 9.9996e-06 0.49984 Total Mole Fraction CO dimensionless 9.9996e-06 0.059609 Total Mole Fraction CO2 dimensionless 9.9996e-06 0.080265 Temperature kelvin 500.00 920.80 Pressure pascal 2.0000e+05 2.0000e+05 ==================================================================================== Conversion achieved = 80.0%  ## Optimizing Hydrogen Production# Now that the flowsheet has been squared and solved, we can run a small optimization problem to determine optimal conditions for producing hydrogen. Suppose we wish to find ideal conditions for the competing reactions. The GibbsReactor does not drive equilibrium forward based on temperature, so we will see small amounts of intermediate components present in the product stream. We will allow for variable reactor temperature and pressure by freeing our heater and compressor specifications, and minimize cost to achieve 90% methane conversion. Since we assume an isentopic compressor, allowing compression will heat our feed stream and reduce or eliminate the required heater duty. Let us declare our objective function for this problem. m.fs.objective = Objective(expr=m.fs.operating_cost)  Now, we need to add the design constraints and unfix the decision variables as we had solved a square problem until now, as well as set bounds for the design variables (reactor outlet temperature is set by state variable bounds in property package): m.fs.R101.conversion.fix(0.90) m.fs.C101.outlet.pressure.unfix() m.fs.C101.outlet.pressure.setlb( pyunits.convert(1 * pyunits.bar, to_units=pyunits.Pa) ) # equals inlet pressure m.fs.C101.outlet.pressure.setlb( pyunits.convert(10 * pyunits.bar, to_units=pyunits.Pa) ) # at most, pressurize to 1 bar m.fs.H101.outlet.temperature.unfix() m.fs.H101.heat_duty.setlb( 0 * pyunits.J / pyunits.s ) # ensures outlet is equal to or greater than inlet temperature m.fs.H101.outlet.temperature.setub(1000 * pyunits.K) # at most, heat to 1000 K  We have now defined the optimization problem and we are now ready to solve this problem. results = solver.solve(m, tee=True)  Ipopt 3.13.2: nlp_scaling_method=gradient-based tol=1e-06 max_iter=200 ****************************************************************************** This program contains Ipopt, a library for large-scale nonlinear optimization. Ipopt is released as open source code under the Eclipse Public License (EPL). For more information visit http://projects.coin-or.org/Ipopt This version of Ipopt was compiled from source code available at https://github.com/IDAES/Ipopt as part of the Institute for the Design of Advanced Energy Systems Process Systems Engineering Framework (IDAES PSE Framework) Copyright (c) 2018-2019. See https://github.com/IDAES/idaes-pse. This version of Ipopt was compiled using HSL, a collection of Fortran codes for large-scale scientific computation. All technical papers, sales and publicity material resulting from use of the HSL codes within IPOPT must contain the following acknowledgement: HSL, a collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk. ****************************************************************************** This is Ipopt version 3.13.2, running with linear solver ma27. Number of nonzeros in equality constraint Jacobian...: 598 Number of nonzeros in inequality constraint Jacobian.: 0 Number of nonzeros in Lagrangian Hessian.............: 506 Total number of variables............................: 205 variables with only lower bounds: 14 variables with lower and upper bounds: 181 variables with only upper bounds: 0 Total number of equality constraints.................: 203 Total number of inequality constraints...............: 0 inequality constraints with only lower bounds: 0 inequality constraints with lower and upper bounds: 0 inequality constraints with only upper bounds: 0 iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls 0 3.9958388e+07 1.49e+06 3.46e+01 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0 1 3.8920063e+07 1.48e+06 1.52e+03 -1.0 7.19e+06 - 3.91e-01 6.43e-03f 1 2 7.0948609e+07 1.15e+06 1.86e+06 -1.0 4.83e+06 - 1.51e-01 2.26e-01h 1 3 1.0553921e+08 5.23e+05 1.04e+07 -1.0 2.42e+06 - 3.41e-01 5.67e-01h 1 4 1.0874890e+08 1.58e+05 7.64e+06 -1.0 8.45e+05 - 7.09e-01 7.11e-01h 1 5 1.0751027e+08 1.51e+04 1.67e+06 -1.0 2.97e+05 - 9.49e-01 9.09e-01f 1 6 1.0721898e+08 5.95e+00 9.98e+03 -1.0 3.47e+04 - 9.90e-01 1.00e+00f 1 7 1.0721794e+08 3.43e-05 8.84e+01 -1.0 1.59e+02 - 9.90e-01 1.00e+00f 1 8 1.0721794e+08 8.85e-09 7.14e-01 -1.0 1.43e-02 - 9.92e-01 1.00e+00h 1 9 1.0721794e+08 1.50e-08 1.92e-06 -2.5 1.72e-02 - 1.00e+00 1.00e+00f 1 iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls 10 1.0721794e+08 1.02e-08 2.70e-06 -3.8 4.73e-04 - 1.00e+00 1.00e+00f 1 11 1.0721794e+08 1.02e-08 2.31e-06 -5.7 2.63e-05 - 1.00e+00 1.00e+00f 1 12 1.0721794e+08 1.49e-08 1.00e-06 -7.0 3.06e-07 - 1.00e+00 1.00e+00h 1 Number of Iterations....: 12 (scaled) (unscaled) Objective...............: 1.0721793780338201e+08 1.0721793780338201e+08 Dual infeasibility......: 1.0011035875619890e-06 1.0011035875619890e-06 Constraint violation....: 1.5205920451933321e-12 1.4901161193847656e-08 Complementarity.........: 9.0909090914354020e-08 9.0909090914354020e-08 Overall NLP error.......: 9.0909090914354020e-08 1.0011035875619890e-06 Number of objective function evaluations = 13 Number of objective gradient evaluations = 13 Number of equality constraint evaluations = 13 Number of inequality constraint evaluations = 0 Number of equality constraint Jacobian evaluations = 13 Number of inequality constraint Jacobian evaluations = 0 Number of Lagrangian Hessian evaluations = 12 Total CPU secs in IPOPT (w/o function evaluations) = 0.004 Total CPU secs in NLP function evaluations = 0.004 EXIT: Optimal Solution Found.  print(f"operating cost =${value(m.fs.operating_cost)/1e6:0.3f} million per year")

print()
print("Compressor results")

m.fs.C101.report()

print()
print("Heater results")

m.fs.H101.report()

print()
print("Gibbs reactor results")

m.fs.R101.report()

operating cost = \$107.218 million per year

Compressor results

====================================================================================
Unit : fs.C101                                                             Time: 0.0
------------------------------------------------------------------------------------
Unit Performance

Variables:

Key                   : Value      : Units         : Fixed : Bounds
Isentropic Efficiency :    0.90000 : dimensionless :  True : (None, None)
Mechanical Work : 3.0334e+06 :          watt : False : (None, None)
Pressure Change : 9.0000e+05 :        pascal : False : (None, None)
Pressure Ratio :     10.000 : dimensionless : False : (None, None)

------------------------------------------------------------------------------------
Stream Table
Units         Inlet     Outlet
Total Molar Flowrate     mole / second     309.01     309.01
Total Mole Fraction CH4  dimensionless    0.24272    0.24272
Total Mole Fraction H2O  dimensionless    0.75725    0.75725
Total Mole Fraction H2   dimensionless 9.9996e-06 9.9996e-06
Total Mole Fraction CO   dimensionless 9.9996e-06 9.9996e-06
Total Mole Fraction CO2  dimensionless 9.9996e-06 9.9996e-06
Temperature                     kelvin     353.80     619.25
Pressure                        pascal 1.0000e+05 1.0000e+06
====================================================================================

Heater results

====================================================================================
Unit : fs.H101                                                             Time: 0.0
------------------------------------------------------------------------------------
Unit Performance

Variables:

Key       : Value      : Units : Fixed : Bounds
Heat Duty : 5.8781e-09 :  watt : False : (0.0, None)

------------------------------------------------------------------------------------
Stream Table
Units         Inlet     Outlet
Total Molar Flowrate     mole / second     309.01     309.01
Total Mole Fraction CH4  dimensionless    0.24272    0.24272
Total Mole Fraction H2O  dimensionless    0.75725    0.75725
Total Mole Fraction H2   dimensionless 9.9996e-06 9.9996e-06
Total Mole Fraction CO   dimensionless 9.9996e-06 9.9996e-06
Total Mole Fraction CO2  dimensionless 9.9996e-06 9.9996e-06
Temperature                     kelvin     619.25     619.25
Pressure                        pascal 1.0000e+06 1.0000e+06
====================================================================================

Gibbs reactor results

====================================================================================
Unit : fs.R101                                                             Time: 0.0
------------------------------------------------------------------------------------
Unit Performance

Variables:

Key       : Value      : Units : Fixed : Bounds
Heat Duty : 2.1076e+07 :  watt : False : (None, None)

------------------------------------------------------------------------------------
Stream Table
Units         Inlet     Outlet
Total Molar Flowrate     mole / second     309.01     444.02
Total Mole Fraction CH4  dimensionless    0.24272   0.016892
Total Mole Fraction H2O  dimensionless    0.75725    0.31609
Total Mole Fraction H2   dimensionless 9.9996e-06    0.51498
Total Mole Fraction CO   dimensionless 9.9996e-06   0.093140
Total Mole Fraction CO2  dimensionless 9.9996e-06   0.058900
Temperature                     kelvin     619.25     1087.4
Pressure                        pascal 1.0000e+06 1.0000e+06
====================================================================================


Display optimal values for the decision variables and design variables:

print("Optimal Values")
print()

print(f"C101 outlet pressure = {value(m.fs.C101.outlet.pressure)/1E6:0.3f} MPa")
print()

print(f"C101 outlet temperature = {value(m.fs.C101.outlet.temperature):0.3f} K")
print()

print(f"H101 outlet temperature = {value(m.fs.H101.outlet.temperature):0.3f} K")

print()
print(f"R101 outlet temperature = {value(m.fs.R101.outlet.temperature):0.3f} K")

print()
print(f"Hydrogen produced = {value(m.fs.hyd_prod):0.3f} MM lb/year")

print()
print(f"Conversion achieved = {value(m.fs.R101.conversion):.1%}")

Optimal Values

C101 outlet pressure = 1.000 MPa

C101 outlet temperature = 619.248 K

H101 outlet temperature = 619.248 K

R101 outlet temperature = 1087.385 K

Hydrogen produced = 32.070 MM lb/year

Conversion achieved = 90.0%