Appendix C — Acronyms, Notation, Sign Conventions, and Computing
This appendix describes acronyms, notation, and sign conventions that are used throughout Reaction Engineering Basics.
C.1 Acronyms
The primary use of acronyms in Reaction Engineering Basics is to identify reactor types and mathematical equation types. The first time each of the acronyms listed here is used, its meaning is stated, but after that these common acronyms are used without explanation.
| Acronym | Meaning |
|---|---|
| ATE | Algebraic-transcendental equation |
| BSTR | Batch stirred-tank reactor |
| CI | Confidence interval. |
| CSTR | Continuous (flow) stirred-tank reactor |
| CVODE | Coupled value ordinary differential equation |
| IVODE | Initial value ordinary differential equation |
| MVODE | Missing value ordinary deiferential equation |
| PFR | Plug-flow reactor |
| SBSTR | Semi-batch stirred tank reactor |
C.2 Notation
This section describes notation that is used with variable symbols and in equations throughout Reaction Engineering Basics.
Dots over variable symbols designate flow variables.
- \(n_i\) represents the moles of \(i\) whereas \(\dot n_i\) represents the molar flow rate of \(i\).
- \(W\) represents work (energy) whereas \(\dot W\) represents the rate of doing work (power).
Carats, breves, and tildes over heat capacities are used to differentiate between molar, volumetric and gravimetric heat capacities.
- \(\hat{C}_{p,i}\) represents the molar heat capacity of reagent \(i\) and has units of energy per mol per degree of temperature.
- \(\breve{C}_p\) represents the volumetric heat capacity of a fluid and has units of energy per volume per degree of temperature.
- \(\tilde{C}_p\) represents the gravimetric heat capacity of a fluid and has units of energy per mass per degree of temperature.
Lines under variable symbols are used to indicate a set of values of the quantity represented by the variable symbol.
- \(\underline{f}_A\) might represent a set of experimental measurements of the fractional conversion of A.
A vertical bar with a subscript is used to specify the conditions at which a quantity is evaluated.
- \(\Delta H_j^0 \Bigr\rvert_{T=\text{300 K}}\) represents the standard enthalpy change for reaction \(j\) evaluated at 300 K.
- \(n_A \Bigr\rvert_{t=0}\) represents the molar amount of A at \(t=0\).
Summations and continuous products will occastionally indicate a lower value and an upper value of the index. More commonly only the index variable will be shown. This means that all possible values of the index should be included. For example, in the following definition of the mole fraction of reagent A, the summation in the denominator includes the molar amount \(n_i\) of every reagent, \(i\), in the system.
\[ y_A = \frac{n_A}{\displaystyle \sum_i n_i} \]
In some instances the sum or product of some sub-set of the indexed quantity is desired. In these cases in Reaction Engineering Basics the index variable will typically have a prime (\(^\prime\)) and the items to be included in the sum or product will be indicated immediately following the equation. As an example, in the following equation, only the reactions in a complete, mathematically independent subset of the reactions occurring in the system are included in the summation.
\[ \dot n_i = \dot n_{i,in} + \sum_{j^\prime} \nu_{i,j^\prime} \dot \xi_{j^\prime} \]
\(\qquad \qquad\) independent subset of the reactions occurring in the system.
Confidence intervals for estimated parameters are reported as
value, 95% CI [lower_limit, upper_limit].
For example, \(m\) = 10 g, 95% CI [9.3, 11.2] means that the estimated value of \(m\) is 10 g with a 95% confidence interval between 9.3 and 11.2 g.
Implicit equations for the value of a variable \(x\) that causes some function, \(f\left(x\right)\), to equal \(y\) are written as
\[ x: f\left(x\right) = y \]
The maximum and minimum of a variable or function, \(y\), are designated as follows:
\[ \max \left(y\right) \qquad \qquad \min \left(y\right) \]
The maximum or minimum of \(y\) with respect to variable \(x\) is indicated as follows:
\[ \underset{x}{\max} \left(y\right) \qquad \qquad \underset{x}{\min} \left(y\right) \]
The value of \(x\) at which \(y\) is maximized or minimized is written as follows:
\[ \underset{x}{\arg\max} \left(y\right) \qquad \qquad \underset{x}{\arg\min} \left(y\right) \]
C.3 Sign Conventions
During the derivation of some of the equations presented in Reaction Engineering Basics a mathematical sign (i. e. positive vs. negative) must be assigned to some quantities. The resulting sign conventions for Reaction Engineering Basics are as follow:
- Stoichiometric coefficients of reactants are negative.
- Stoichiometric coefficients of products are positive.
- If a reagent is neither a reactant nor a product in a given reaction , its stoichiometric coefficient in that reaction is zero.
- Heat added to a system is positive.
- Work done by a system is positive.
C.4 Computing
Reaction Engineering Basics does not present any computer code. This is intentional because different readers will prefer to use different mathematics software packages. Instead, the numerical implementation of calculations is described in a general way that (hopefully) is sufficient for understanding the examples in the book and for writing the necessary code using any software package. However, it is necessary to make a few assumptions in order to describe the numerical implementation of calculations.
Functions
In Reaction Engineering Basics, “function” is used to refer to a unit of computer code. Some software packages may use “subroutine,” “script,” “module,” etc. instead. It is further assumed that functions can contain other functions. The numerical implementation of the calculations for completing any one example assignments can then be described in terms of two kinds of functions: utility functions and example calculations functions.
Given and Known Constants
Reaction Engineering Basics Examples typicaly begin with an Assignment Summary that lists the values of variables that are provided in the assignment narrative. It is assumed that these constants are converted to consistent units and are available in all functions associated with that example.
Making Variables Globally Available
In some situations the arguments that are provided to a function and the quantities that it returns are specified by the software package being used. When it is necessary to provide additional inputs to such functions, Reaction Engineering Basics assumes that some other function will make that additional input “globally available,” meaning that the quantity is available in all functions associated with the example.