Simulation & Optimization of Pressure Swing Adsorption

Yajun Wang & Alexander W. Dowling Lorenz T. Biegler

Department of Chemical Engineering Carnegie Mellon University

March 2014

Motivation

• Objective

-- Modify PSA simulation

-- Diagnose convergence problems in PSA optimization

• Pressure Swing Adsorption (PSA)

-- Gas separation (e.g. H2 – CO2)

-- In IGCC power plant

2 [1] Alexander W. Dowling, Sree R. R. Vetukuri, and Lorenz T. Biegler. Large-scale optimization strategies for pressure swing adsorption cycle synthesis.

Pressure Swing Adsorption (PSA) Superstructure[2]

State variables: CCO2 CH2 qCO2 qH2 T

5 operating steps (time slots): α β φ Pads Pdes

Additional control variables: ts Lbed

Cyclic steady state

3 [2] Anshul Agarwal. Advanced Strategies for Optimal Design and Operation of Pressure Swing Adsorption Processes. PhD thesis, Carnegie Mellon University, 2010.

PSA modeling[4]

• Mass balance 𝜖𝑏𝜕𝐶𝑖

𝜕𝑡+ 1 − 𝜖𝑏 𝜌𝑠

𝜕𝑞𝑖

𝜕𝑡+𝜕 𝑣𝐶𝑖

𝜕𝑥= 0

• Mass transfer 𝜕𝑞𝑖𝜕𝑡= 𝑘𝑖(𝑞𝑖

∗ − 𝑞𝑖)

• Energy balance

𝜖𝑡 𝐶𝑖𝑖

𝐶𝑝𝑔𝑖 − 𝑅 + 𝜌𝑠𝐶𝑝𝑠

𝜕𝑇

𝜕𝑡− 𝜌𝑠 ∆𝐻𝑖

𝑎𝑑𝑠

𝑖

𝜕𝑞𝑖𝜕𝑡+ 𝜕 𝑣ℎ

𝜕𝑥+ 𝑈𝐴 𝑇 − 𝑇𝑤 = 0

• Momentum balance

• Adsorption Isotherm

• Ideal gas equation

• Check valve equation, Bed connection equations, Auxiliary equations

PDEs

DAEs

4 [4] Ling Jiang, Lorenz T Biegler, and V Grant Fox. Simulation and optimization of pressure-swing adsorption systems for air separation.

PSA Simulation

System discretization Discretize space domain[1]

Finite volume method

Flux limiter

• Keep sharpness of steep fronts • Prevent oscillations

convert PDEs to ODEs

Smearing Oscillation 5

Two types of flux limiters

Van Leer[4]

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Non-differentiable

7 [3] Reza Haghpanah, et al. Multiobjective optimization of a 4-step adsorption process for post-combustion CO2 capture using nite volume technique.

WENO[3]

Differentiable

Two types of flux limiters

WENO performs similarly as van Leer to maintain accurate fronts

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Without flux limiter, the simulation is poor near fronts.

No flux limiter VS van Leer WENO VS van Leer

Difference in CO2 concentration

Mole/m3

Difference in CO2 concentration

Mole/m3

WENO is computationally more efficient

Time to reach cyclic steady state

WENO Reduce 17.84% computational time

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Optimization formulation[1]

Separation specific energy

Bed equations

Quality requirements

CO2 purity ≥92%

CO2 recovery ≥90%

Variable bounds

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Periodic Boundary Condition Formulation[1]

• Add cyclic steady state equations

• Optimal variables: initial states and control variables

CCO2 CH2 qCO2 qH2 T

Step 1 Step 2 Step 3 Step 4 Step 5

Switch Beds and Repeat

Step 1

……

α β φ Pads Pdes ts Lbed

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Convergence troubles[1]

• Sensitivity calculation -- Expensive (more than 95% computational time) -- Approximate within integrator tolerances, O(10-6 to 10-10)

• Hard to converge

-- Large number of iterations, computational time -- Infeasibility

• Sensitive to initial guess

MATLAB fmincon

SQP

Active-set

Interior-point

12 *First derivatives from direct sensitivity

020

4060

80100

120140

0

50

100

150

-2

-1

0

1

2

3

x 105

Notscaled Hessian

Optimization Diagnostics

Optimization ends up with no feasible solution

Final iteration 92

Objective value 85.08

Constraints violation 0.989

KKT violation 29.69

Large KKT violation is caused by φ

Check approximate Hessian matrix at final point, large elements also correspond to φ

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Importance of scaling

Not scaled Scaled by 100 Scaled by 1000

Objective value 86.24 86.26 85.8

Constraints violation 0.289 9.78×10-4 2.32×10-3

KKT violation 29.42 3.338 8.598

Optimization data at iteration 80

Scale φ

With scaled φ, both primal and dual feasibility can be improved largely.

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Redo optimization with all control variables except φ

Final iteration 232

Objective value 84.71

Constraints violation 7.68×10-7

KKT violation 4.39×10-3

Reduce problem dimension

Fix control variables optimization problem nonlinear equations solving problem # bed state variables = # bed equations

Optimization with only one type of control variables α β φ Pads Pdes ts Lbed

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Conclusion & Future work

WENO flux limiter is more suitable for PSA optimization – high efficiency and better differentiability

Scaling improves optimizer performance

Simplified problems converge to optimum

o Continue optimization diagnostics

o Consider optimization with reduced order models

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Acknowledgement

• Professor Larry Biegler

• Alexander W. Dowling

• The Department of Chemical Engineering at Carnegie Mellon University

• National Energy Technology Laboratory (NETL) (initial funding)

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