Appendix C — Acronyms, Notation, and Sign Conventions
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 |
| BVODE | Boundary value ordinary differential equation |
| CSTR | Continuous (flow) stirred-tank reactor |
| DAEs | Differential-algebraic equations |
| IVODE | Initial value ordinary differential 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 are used in Reaction Engineering Basics to 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.
A vertical bar with a subscript is used in Reaction Engineering Basics 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 asking for the value of \(x\) that causes some function, \(f\left(x\right)\), to equal \(y\) will be written as follows:
\[ x: f\left(x\right) = y \]
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.