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Introduction to Modeling

Modeling in engineering and science transcends the glitz of fashion runways and the meticulous craftsmanship of hobbyists. It embodies a rigorous approach to understanding, analyzing, and optimizing systems, processes, and operations crucial to society's advancement.

Mathematical modeling, a cornerstone of applied mathematics, is more than a mere problem-solving tool - it is a language deeply rooted in mathematical physics. Engineers and scientists leverage mathematical models to conceptualize system behaviors, rigorously testing these hypotheses through meticulous validation and verification.

Engineering modeling entails:

• Crafting mathematical representations of physical reality, expressed through equations and algorithms.
• Employing simulations of these mathematical models as proxies for real-world experimentation.
• Elevating the value of models through rigorous validation processes.

While engineering modeling shares some traits with fashion and hobby modeling, its essence lies in its ability to distill complex phenomena into actionable insights. Like fashion modeling:

• Models serve as idealized depictions of reality, facilitating understanding and analysis.
• They establish relationships between different system states and parameters.

Resonating with hobby modeling:

• Careful construction is paramount to ensure fidelity to the real-world system.
• Models amalgamate empirical laws and constitutive relations to capture system dynamics.
• The distinction between system states and parameters is pivotal in model formulation.

Historically, engineering relied heavily on empirical methods, with designs forged through trial and error experimentation. However, the advent of mathematical modeling, coupled with modern computing tools, has revolutionized this paradigm. Mathematical models serve as powerful guides for equipment and process design, streamlining experimentation, and mitigating risks associated with empirical approaches.

Equation-based models, derived from fundamental principles like mass and energy conservation, thermodynamics, and kinetics, empower chemical engineers to address a myriad of questions, such as:

• Will our concepts translate into practical solutions?
• What measurements are essential for validation?
• Which factors significantly influence system behavior?
• Is the proposed solution economically viable and safe?
• Can we effectively control and optimize the system?
• What scale can we feasibly implement?

Engineers and scientists navigate complex technical challenges with precision and confidence by strategically integrating mathematical modeling, propelling innovation, and societal progress. In navigating this terrain, we must balance simplicity and utility. Levenspiel's (2002) concept of "the US$10, US$100, and US$1000 models" serves as a reminder: while complex models may incur significant costs, they do not always offer proportional increases in understanding. Expensive experiments yield to relatively inexpensive computer simulations, offering a cost-effective alternative. Moreover, as depicted in Figure 1.1, the decreasing cost of computing contrasts with the rising costs of traditional experimentation over time.

Models vary in complexity based on their intended purposes. Simple models can often provide sufficient insight to guide decisions on whether to proceed with further development, pause for further investigation, or alter engineering directions. However, the demand for increased model detail grows as the need for accuracy and precision intensifies in design and decision-making processes.

The cost associated with developing a mathematical model hinges on the level of uncertainty deemed acceptable in the model predictions. As depicted in Figure 1.2, generally, the higher the tolerance for uncertainty in the calculated results, the lower the overall cost of the model.

The escalating costs of achieving greater certainty in model predictions often stem from the endeavor to solve the model equations. In preliminary engineering models, there is a propensity toward higher uncertainty - lower precision - particularly when efforts to mitigate uncertainty fail to justify the exponentially increasing modeling expenses.

Contrastingly, scientific models tend to gravitate toward minimizing uncertainty - higher precision - aiming for a comprehensive comprehension of natural phenomena. To navigate the uncertainty, engineers often incorporate design safety factors. For instance, a chemical engineer might design a distillation column by calculating the minimum number of ideal stages, then doubling this figure and adjusting the reflux to attain the desired separation (Walas 1990).

Figure 1.1 Computing costs are decreasing as the cost of experimentation increases over time.

Figure 1.2 Cost of mathematical modeling as a function of the degree of uncertainty in the model predictions.

1.1 Numerical Methods

Numerical methods are computational techniques employed to approximate solutions to mathematical models that may prove inconvenient, challenging, or even impossible to solve through standard analytical methods. Analytical solutions yield symbolic representations, often involving explicit rearrangements of equations to isolate variables. However, when analytical solutions are unattainable, numerical methods offer practical approximations.

Consider calculating the molar volume of a gas using models that correlate molar volume with temperature and pressure. The ideal gas law and the Redlich-Kwong equation of state for nonideal behavior are two potential candidates. We can rearrange the ideal gas law to solve for the molar volume:

where Rg is the ideal gas constant, T is the temperature, and P is the pressure. However, we cannot rearrange the following nonideal Redlich-Kwong model explicitly for either molar volume or temperature1:

The parameters a and b are functions of the critical temperature and gas pressure. There is no way to rearrange Eq. (1.2) with the molar volume only appearing on one side of the equation. Instead, we determine the Redlich-Kwong equation's molar volume from numerical approximation methods for specific cases of T and P.

Many numerical methods rely on iterative calculations, often facilitated by computers. Compared to analytical solutions, numerical approaches can be easier to implement, saving time and effort that can be directed toward addressing other challenges.

In chemical engineering, typical problems involve large systems of linear equations arising from applying fundamental principles like the conservation of momentum, mass, and energy. Additionally, nonlinear equations abound due to the inherent complexities of physical and chemical properties, transport phenomena, thermodynamics, and chemical reactions.

Some common examples of nonlinear functions encountered in chemical engineering include the temperature-dependent Antoine equation for vapor pressure, the modified Arrhenius function for reaction rate constants, and the modified Henri function for enzyme reaction kinetics with substrate inhibition.

While linear equations possess analytical solutions, determining the solution of a system larger than three or four equations can often prove tedious. Analytical solutions for such large systems entail extensive algebraic manipulations and symbolic bookkeeping. Moreover, many nonlinear equations lack analytical solutions altogether. Hence, methods for obtaining practical approximations of nonlinear functions and large systems of linear equations are essential.

To underscore the necessity of numerical methods, let us consider a simple model from chemical reaction engineering.

The following reaction mechanism is elementary, irreversible, and first order in species A:

Consider the steady-state mole balance for product species B around a perfectly stirred chemical reactor, as illustrated in Figure 1.3, with feed and effluent concentrations of CA0 and unreacted CA and product CB, respectively.

We start with species conservation to mathematically model this reactor. With no B in the feed, the concentration of B leaving the reactor equals the product of the residence time, t, with the rate of generation of B per unit volume in the reactor, rB:

For elementary, first-order reactions, the rate of production of B is a linear function of concentration:

where (CA0-CB) is equivalent to the concentration of unreacted A in the reactor in terms of the initial concentration of reactant A, and k is the first-order reaction rate constant. We can rearrange Eqs. (1.7) and (1.8) explicitly for the concentration of product CB exiting the reactor:

Figure 1.3 Well-mixed, steady-state...