Appendix B — Prerequisite Knowledge

Reaction Engineering Basics assumes that students have acquired certain knowledge and skills previously and have the ability to do the things listed here.

B.1 General Problem-Solving

  • Identify intensive quantities. These are quantities that do not fix the size of the system. Examples inlcude pressure, partial pressure, mole fraction, concentration, temperature, etc.
  • Identify extensive quantities. These are quantities that fix the size of the system. Examples include moles, molar flow rate, mass, mass flow rate, volume, volumetric flow rate, etc.
  • Know that if a problem provides only intensive quantities, the value of one extensive variable may be choses as a basis for the calculations.

B.2 Algebra

  • Rearrange equations using the basic rules and properties of algebra.
  • Factor, reduce and add algebraic fractions.
  • Manipulate logarithms and exponentials.
  • Combine two equations by elimination of a variable common to both.
  • When it is possible, analytically solve \(N\) equations to obtain expressions for \(N\) unknowns.
  • Numerically solve sets of algebraic-transcendental equations.
    • A brief overview of the numerical solution of sets of algebraic-transcendental equations is presented in Appendix J.

B.3 Calculus

  • Evaluate definite and indefinite integrals analytically and numerically (see below for a brief overview of numerical integration).
  • Differentiate a function analytically and numerically (see below for a brief overview of numerical differentiation).

B.3.1 Numerical Integration

  • Given a vector containing \(N\) values of the independent variable, \(x\)
  • The trapezoid rule is one of several ways to numerically integrate a function, \(f\left(x\right)\), over the interval from \(x_1\) to \(x_N\).

\[\int_{x_1}^{x_N} f(x)dx \approx \sum_{k=1}^{N-1} \left[ \frac{1}{2} \left( f_k + f_{k+1} \right) \left( x_{k+1} - x_k \right) \right]\]

B.3.2 Numerical Differentiation

  • Given
    • A vector containing \(N\) values of the independent variable, \(x\)
    • A vector containing the corresponding \(N\) values of the dependent variable, \(y\)
  • The derivative, \(\frac{dy}{dx}\) corresponding to the \(k^{th}\) element in the vectors can be approximated numerically using
    • Backward differences: \(\frac{dy}{dx}\Bigr\rvert_{k} \approx \frac{y_{k} - y_{k-1}}{x_{k} - x_{k-1}}\)
    • Forward differences: \(\frac{dy}{dx}\Bigr\rvert_{k} \approx \frac{y_{k+1} - y_{k}}{x_{k+1} - x_{k}}\)
    • Central differences: \(\frac{dy}{dx}\Bigr\rvert_{k} \approx \frac{1}{2} \left( \frac{y_{k+1} - y_{k}}{x_{k+1} - x_{k}} + \frac{y_{k} - y_{k-1}}{x_{k} - x_{k-1}} \right)\)

B.4 Differential Equations

  • Understand what it means to “solve” a differential equation.
  • Understand what initial conditions and boundary conditions are.
  • Analytically solve a first order, initial-value ODE by separation of variables.
  • Numerically solve a set of coupled, initial-value ODEs.
    • A brief overview of the numerical solution of sets of coupled, initial-value ODEs is presented in Appendix K.

B.5 Linear Algebra

  • Know the difference between linear and non-linear equations.
  • Write equations in vector and matrix form.
  • Multiply scalars, vectors and matrices.
  • Understand what the rank of a matrix is.
  • Given a set of reactions or linear equations, identify a complete mathematically independent subset of those reactions or equations (see below for a brief overview of identifying mathematically independent linear equations).

B.5.1 Identifying Mathematically Independent Linear Equations.

In a mathematically independent set of linear equations, none of the equations in the set can be expressed as a linear combination of the other equations in the set.

Given a set of equations (or chemical reactions), one or more complete, mathematically independent subsets can be identified such that none of the equations/reactions in the subset can be expressed as a linear combination of the other equations in the subset, and all of the equations not in the subset can be expressed as a linear combination of the equations/reactions in the subset.

A complete, mathematically independent subset of a set of chemical reactions can be found as follows.

  • Create an empty working matrix where each row will represents a reagent that appears either as a reactant or as a product in the reaction set.
  • Add the first reaction in the full set of reactions to the working matrix by creating a column for that reaction. In each row in that column, insert the stoichiometric coefficient of the species corresponding to that row.
  • Successively for each remaining reaction:
    • Create a test matrix by adding an additional column for the next remaining reaction to the working matrix.
    • Calculate the rank of the test matrix
    • If the rank of the test matrix equals the number of reactions (columns) in the test matrix, use the test matrix as the new working matrix, otherwise discard the test matrix.
  • After all reactions have been processed, the columns in the resulting working matrix will correspond to a complete, mathematically independent subset.
    • There could be other complete, mathematically independent subsets.

B.6 Statistics and Probability

  • Understand what means, standard deviations and 95% confidence intervals are and be able to calculate them.
  • Understand what probabilities are.
  • Calculate expectation values using a probability density function (see below for a brief overview).

B.6.1 Probability Density Functions

  • A probability density function, \(p_X\left(z\right)\), for a random variable, \(X\), spans the range of possible values of that variable.
  • At any value of \(z\), the value of the probability density function, \(p_X \left(z\right)\), when multiplied by a differential portion of the range, \(dz\), equals the probability that a single observation of \(X\) will fall in the range between \(z\) and \(z + dz\).
  • A probability density function must satisfy two conditions \[p_X\left(z\right) >0 \quad \text{for} \quad -\infty \lt z \lt \infty\] \[\int_{-\infty }^{\infty } p_X(z)dz = 1\]
  • The average (expectation) value of any quantity, \(a\), that is functionally dependent upon the random variable, \(X\), can be calculated using the probability density function. \[a=f(X) \text{ and } X\sim p_X(z) \quad \Rightarrow \quad \left\langle a \right\rangle = \int_{-\infty}^{\infty}f(z)p_X(z)dz\]

B.7 Fluids and Fluid Mixtures

  • Understand and use the ideal gas and ideal liquid equations of state.
  • Understand and use composition variables such as mole fractions, concentrations and partial pressures.
  • Understand fluid flow and pressure drop in pipes.

B.8 Reaction Thermodynamics

  • Understand the meaning of and difference between latent and sensible heats and be able to calculate them.
  • Understand heat of reaction and be able to generate and use an equation to calculate it.
    • A brief overview of calculating heats of reaction is presented in Appendix D.
  • Understand chemical reaction equilibrium and be able to generate and use equations to calculate equilibrium constants and equilibrium composition at any temperature.
    • A brief overview of equilibrium calculations is presented in Appendix D.

B.9 Heat Transfer

  • Understand heat transfer coefficients and heat transfer area and be able to use them to calculate the rate of heat transfer.
    • A brief overview of selected ways to transfer heat to or from chemical reactors is presented in Appendix E.

B.10 Chemical Processes

  • Know how to write and use mass and energy balances on components of simple chemical processes including
    • flow stream splitting points
    • flow stream mixing points
    • heat exchangers
  • Know how to use mass and energy balance equations on the components of a simple chemical process to model that process.

B.10.1 Classification of Chemical Processes

There are a number of different ways to classify or describe chemical processes. Students should be familiar with the following terms used to describe chemical processes.

  • Adiabatic - no energy is added to or removed from the system while the process occurs, other than that associated with input and output material flow streams.
  • Isothermal - the temperature throughout the system is uniform (the same everywhere) and constant while the process occurs.
  • Isobaric - the pressure throughout the system is uniform and constant while the process occurs.
  • Isochoric - the system volume is constant while the process occurs.
  • Steady-state - the temperature, pressure and chemical composition are all constant over time at each location in the system.
  • Transient - the temperature, pressure or chemical composition vary with time at one or more locations in the system.

B.11 Computation

Students using Reaction Engineering Basics should have access to and be able to use software for numerical computation to perform the tasks listed below.

  • Routine calculations.
  • Reading data from files or entering data manually.
  • Displaying data on screen as text, printing it or saving it to a file.
  • Plotting data.
  • Performing parameter estimation with non-linear models.
    • A brief overview of parameter estimation and model assessment is presented in Appendix N.
  • Solving algebraic-transcendental equations.
    • A brief overview of the numerical solution of sets of algebraic-transcendental equations is presented in Appendix J.
  • Solving initial value ordinary differential equations.
    • A brief overview of the numerical solution of sets of coupled, initial-value ODEs is presented in Appendix K.

B.12 Symbols Used in this Appendix

With one exception, all of the symbols used in this appendix have no special meaning and won’t be listed here. For example, in this chapter \(f\left( x \right)\) is used to denote some generic function, \(f\), that is functionally dependent on the generic variable, \(x\). Neither \(f\) nor \(x\) has any special meaning attached to it. The one exception is the probability density function.

Symbol Meaning
\(p_X\left(z\right)\) Probability density function for the random variable, \(X\).