Numerical Analysis for Real-time Nonlinear Model Predictive Control of Ethanol Steam Reformers

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Numerical Analysis for Real-time Nonlinear

Model Predictive Control of Ethanol Steam

Reformers

An undergraduate thesis presented by

Robert Joseph George

Advised by: Xinwei Yu

Supervised by: Paul Buckingham

Abstract

The utilization of renewable energy technologies, particularly hydrogen, has seen a boom in interest and has spread

throughout the world. Ethanol steam reformation is one of the primary methods capable of producing hydrogen

eﬃciently and reliably. This paper provides an in-depth study of the reformulated system, both theoretically and

numerically, as well as a plan to explore the possibility of converting the system into its conservation form. Lastly, we

oﬀer an overview of several numerical approaches for solving the general ﬁrst-order quasi-linear hyperbolic equation

to the particular model for ethanol steam reforming (ESR). We conclude by presenting some results that would

enable the usage of these ODE/PDE solvers to be used in non-linear model predictive control (NMPC) algorithms

and discuss the limitations of our approach and directions for future work.

Submitted in partial fulﬁllment of the Honors requirements for the degree

of Bachelor of Honors in Applied Mathematics and Computer Science.

Department of Mathematics and Statistics

Edmonton, Canada

arXiv:2210.13745v3 [math.AP] 26 Apr 2023

Contents

1 Introduction................................................... 2

2 ModelDescription ............................................... 3

2.1 ChemicalReactions .......................................... 3

3 FirstOrderQuasiLinearPDESystem.................................... 5

4 ConservationLaw................................................ 7

5 SolutionoftheSystem............................................. 11

5.1 MethodofCharacteristics....................................... 11

5.2 Solution................................................. 12

6 UniquenessandExistenceTheorem...................................... 12

6.1 OurModel ............................................... 12

6.2 Uniqueness Theorem for Linear Systems of First Order . . . . . . . . . . . . . . . . . . . . . . 13

7 Uniqueness Theorem for Quasi Linear Systems of First Order . . . . . . . . . . . . . . . . . . . . . . . 16

8 Existence Theorem for Quasi Linear Systems of First Order . . . . . . . . . . . . . . . . . . . . . . . . 16

8.1 Theoretical Analysis of our System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

9 NumericalAnalysis............................................... 18

9.1 Ordinary Diﬀerential Equation Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

9.2 EulerMethods ............................................. 20

9.3 Runge-Kutta .............................................. 21

9.4 Partial Diﬀerential Equation Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

9.5 FiniteElementMethod ........................................ 22

9.6 Tools .................................................. 22

10 Discussion.................................................... 22

11 Limitations ................................................... 25

12 FutureWork .................................................. 28

13 Conclusion ................................................... 28

Appendices ...................................................... 30

A EigenvaluesandEigenvectors..................................... 30

B ODESolver............................................... 30

C Simpliﬁedsystem............................................ 31

1 Introduction

The growth in population has resulted in a considerable increase in global energy consumption. This is due to the

fact that energy is required for practically all activities like transportation, heating etc. Nowadays, hydrogen has

emerged as an attractive energy source that may be utilized to store energy from renewable sources. The time has

come to capitalize on hydrogen’s potential to play a signiﬁcant role in addressing critical energy challenges. Recent

breakthroughs in renewable energy technology have proved that technical innovation has the potential to build

worldwide clean energy ﬁrms that do not rely on hydrocarbon-based fuels, therefore helping to reduce pollution

levels [3].

Because it possesses one of the highest energy densities per unit mass, hydrogen is appealing as a fuel. According to

[5], energy density is deﬁned as the quantity of energy stored per unit volume in a speciﬁc system or region of space.

According to this statistic, hydrogen has an energy density of 35,000 watts per kilogram (W/kg), but lithium-ion

batteries have an energy density of just 200 watts per kilogram (W/kg), and because hydrogen has a tenfold higher

energy-to-weight ratio than lithium-ion batteries, it can deliver a greater range while being substantially lighter.

Fuel cells, when pure hydrogen is used, can be turned into electricity at high eﬃciency, durability, and eﬃciency as

needed.

Because of the physical features of hydrogen, it is diﬃcult to transport and store, prompting us to study safe ways

to produce hydrogen locally eﬀectively. One conceivable method for safe hydrogen transportation that appears to

be the most eﬃcient is to store the fuel as a low-pressure liquid and then use an onboard ethanol steam reformer

(ESR) to produce hydrogen as needed [1]. This is owing to its mobility, renewable nature, and low toxicity.

The development of real-time eﬃcient and trustworthy control systems to assure device eﬃciency while reducing the

impact of interruptions such as signiﬁcant variations in interior and external temperatures during transportation, as

mentioned in [1], are crucial for the design of ethanol steam reformers. There have been few mathematically modelled

studies on the best design of control techniques for ethanol steam reformers. Previous research has examined the

steady-state behaviour of ethanol steam reformers in order to build proportional-integral-derivative (PID)-based

decoupled control loops while disregarding physical and operational restrictions and needs [6] and [7]. Another study

[4] presented the use of a model which manages a mass ﬂow control of ethanol/water and temperature regulation of

a 1 kWe thermal plasma reformer. Although these work, they have certain limitations, such as neglecting physical

and operational constraints and not using the system’s accessible information. Some of the research relies on linear

process models, which might be inaccurate in general, as in [4]. Given the complexity of nonlinear models for

ethanol steam reformers, research has been done to employ nonlinear process models, which have a relatively high

computational cost for computing the control rule.

Model predictive control (MPC) is a sophisticated process control method that has been utilized extensively in

industrial and chemical processes since the 1980s. The MPC approach is a collection of control techniques that employ

a mathematical model of an investigated system to get control actions by minimizing a cost function connected to

speciﬁed control objectives while taking desired system performance into account [2].

The authors of [2] conducted the ﬁrst research, which was a rigorous examination of the extent of nonlinear model

predictive control, in which they analyze a mechanistic model that is a single distributed parameter system which,

in fact, makes this physics-based distributed parameter system for ethanol steam reformer a unique system. In

addition, the same paper presented a method for constructing an approximation reduced-order nonlinear dynamics

model with lower computing cost while preserving the same structure of the original equations and physical model

parameters.

Further extending the study [1] introduced a zero error approach in the model reformulation. Zero error means

that no error is introduced when reformulating the model, and no additional assumptions are made. When putting

the model into a nonlinear model predictive control algorithm, this new formulation preserves the same advantages

of being dependent on physical model parameters and considerably lowering the real-time computations required.

Figure 1shows the ESR, which is a nonlinear dynamical system comprised of a catalytic ethanol steam reactor

in series with a separation stage that includes a selective membrane for hydrogen removal as described in [1]. The

mechanistic model described is a function of time and axial direction only (only one spatial dimension) for the system

of partial diﬀerential equations.

Our report will provide the ﬁrst in-depth examination of the aforementioned reformed system, with the goal of

attempting to turn it into some type of conservation law or similar standard method ( examples include convection

Page 2 of 32

form, diﬀusion form etc.). We will prove the uniqueness and existence of our system’s solutions by understanding

their characteristic system, which is usually done by the Riemann method. Expanding on this, we look at any

potential singularities and how to mitigate this during numerical analysis. This is particularly crucial since any

numerical approach we wish to use requires us to discretize the domain (i.e. discretize both the spatial and temporal

parts of the domain and consider a bounded domain). This is signiﬁcant because it allows us to avoid using numerical

approaches that might cause the system to blow up and instead utilize more eﬃcient numerical methods created

expressly for 1D conservation rules (examples include The ﬁnite volume method, Approximate Riemann Solvers etc).

Our main contribution with this paper is to perform a numerical comparison of the methods applied to our system

based on dependability and eﬃciency, as deﬁned by computation time and the number of analyses required to

achieve a speciﬁc level of accuracy, as well as to guarantee that the algorithm is stable, converges and is consistent

[3]. This would be accomplished by converting the systems of partial diﬀerential equations to a system of ordinary

diﬀerential equations, for which there are already various techniques for solving this system of ODEs (examples

include Euler Methods, Adaptive methods etc). This may also be compared to a direct approach to solving PDE

problems utilizing numerical techniques such as ﬁnite diﬀerence methods, ﬁnite element methods, ﬁnite volume

methods, spectral methods etc. Finally, we take this further by determining the best system formulation and reﬁning

the accompanying numerical approach even more to reduce computing costs.

Aligned with [3], the goals of the study are to enable a mechanistic model to be employed in real-time control

calculations while explicitly accounting for input, state and output constraints with minimal computation cost. This

would open up a new ﬁeld of research into nonlinear distributed parameter systems with additional features such as

mass, and particle number, all of which are common in other areas of research (examples include thermodynamics,

ﬂuid dynamics etc ). This research may be extended to other reactor systems that are frequently encountered in

chemical process control applications etc. Lastly, this would bring us closer to our goal of manufacturing hydrogen

safely, which in turn could be used as green energy.

The remainder of this article is organized as follows: Section 3 describes the ethanol steam reformer model under

consideration, as well as the chemical processes and assumptions. Section 4 discusses the original system of partial

diﬀerential equations as well as the reformed system. Section 5 deﬁnes the conservation law and the convection form

of the system. Section 6 summarizes the eﬀorts made to convert the system into conservation legislation. Section 6

illustrates how to solve a ﬁrst-order quasi-linear PDE problem. The Uniqueness and Existence Theorem for First-

Order Linear Systems is covered in Section 7. Sections 8 and 9 explore the Uniqueness and Existence theorem for

First Order Quasi Linear Systems and how it cannot be directly applied to our model. Section 10 provides a brief

explanation of the numerical methods that are applied to our model. Section 11 summarizes the results as well as the

ﬁndings and programming methods employed, whereas Section 12 highlights the limits of some of the work done in

this research. Finally, Sections 13 and 14 discuss future work and the main conclusions, followed by the bibliography

and appendix.

2 Model Description

We now proceed to describe the Ethanol Steam Reformer(ESR) as a nonlinear dynamical system that combines in

series two process unit operations:

1. a reformer stage comprised of a catalytic ethanol steam reactor

2. a separation stage comprised of a selective membrane through which hydrogen can penetrate.

In the reactor, ethanol is reformed with water to generate a gas mixture from which hydrogen is extracted. The

entire process takes place within a single integrated module known as a staged membrane reactor which is described

in 1.

2.1 Chemical Reactions

Chemical reactions take place in a tubular packed-bed reactor with a single intake and output. There are 4 primary

chemical reactions over cobalt-based catalysts that take place in the staged membrane reactor are as follows [1]

C2H5OH −→ CH3CHO + H2,(1a)

C2H5OH −→ CO + CH4+ H2,(1b)

CO + H2OCO2+ H2,(1c)

CH3CHO + 3H2O−→ 2CO2+ 5H2.(1d)

Page 3 of 32

Figure 1: Staged membrane reactor scheme

These four reactions are occurring in the same location and under the same circumstances at the same time. To begin,

ethanol dehydrogenates to produce hydrogen and formaldehyde (1a), which is then reformed with water to produce

carbon dioxide (1d). Furthermore, at normal working circumstances, cobalt catalysts are active for the Water Gas

Shift (WGS) process (1c). The unfavourable process is ethanol breakdown, which produces carbon monoxide and

methane (1b). The Pd-Ag membrane absorbs only hydrogen during the membrane separation step, leaving waste

gases on the retentate side [2].

For the mathematical modelling of the ethanol steam reformer, various plausible and slightly overlapping assumptions

are made which are as follows [1]

1. The concentrations in the ethanol steam reformer are easily characterized as two plug-ﬂow units connected in

sequence.

2. The radial dependency of pressure and temperature in the tubular reactor and membrane separator is insignif-

icant.

3. In the tubular reactor and membrane separator, the ﬂuid is entirely radially mixed at each axial location.

4. In comparison to convection in the axial direction, the impact of molecular diﬀusion in the axial direction is

modest.

5. In the operating pressure range, the ideal gas assumption is true.

These assumptions ensure that the mechanistic model for the two process units is a function of time and axial

direction, implying that the partial diﬀerential equations describing the system’s nonlinear dynamics have just one

spatial dimension. This leads to the model being a system of ﬁrst-order quasi-linear hyperbolic equations.

Page 4 of 32

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NumericalAnalysisforReal-timeNonlinearModelPredictiveControlofEthanolSteamReformersAnundergraduatethesispresentedbyRobertJosephGeorgeAdvisedby:XinweiYuSupervisedby:PaulBuckinghamAbstractTheutilizationofrenewableenergytechnologies,particularlyhydrogen,hasseenaboomininterestandhasspreadthroughoutthewo...

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Numerical Analysis for Real-time Nonlinear Model Predictive Control of Ethanol Steam Reformers

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