Mathematical Foundations of Chemical Engineering


Joel Moxley, Erik Allen, Hal Alper


Coming out of our MIT Chem Eng PhD program, I always thought the curriculum could be made so much simpler by having a unifying glue of the mathematical derivation tree from first principles of every possible problem that could be asked in a PhD qualifying exam.  The key would be to define a consistent set of quantities (whose variable names and representations never change) that can be cross-indexed locally within a section and globally for the entire book, all labeled by their SI units.  To this end, Hal Alper and I wrote an outline.

Since then, we’ve added Erik Allen to the team and partnered with a LaTeX-in-the-cloud company to develop the sub-indexable quantity technology and translated the early work into a manuscript outline.

It’s not meant to replace the traditional chemical engineering textbook pillars (physics / stat mech, thermo, transport, and mathematical methods) but instead be the glue between them all.  For every undergrad and grad textbook, we can cross-reference section-by-section where we cover the material.


Please see a draft of the manuscript outline here: moxley-et-al-mathematical-foundations-of-chemical-engineering-2016-11-12.pdf


An excerpt of the index may be found below.

 

 


W $-$ Work $[J]$( 2, 4, 46, 52, 73, 74, 75, 76, 77, 80, 89, 103, 109, 142, 146, 155, 158, 185, 192, 205 )

F $-$ Force $[N]$( 2, 3, 14, 15, 16, 17, 74, 79, 80, 84, 92, 114, 116, 117, 118, 119, 120, 141, 142, 143, 145, 146, 148, 149, 153, 154, 155, 156, 157, 158, 159, 163, 175, 184, 185, 186, 187, 189, 191, 192, 203, 205, 208, 222, 223 )

$\underline {x}$ $-$ General displacement vector $[m]$( 2, 14, 15, 16, 17 )

P $-$ Thermodynamic Pressure $[Pa]$( 2, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 83, 84, 89, 91, 92, 93, 94, 96, 102, 103, 105, 106, 112, 114, 118, 119, 135, 140, 141, 142, 146, 147, 149, 157, 163, 164, 165, 172, 178, 179, 182, 183, 186, 187, 197, 202, 204, 205, 209, 212, 213, 215, 216, 217, 218, 219, 220, 221, 233, 234, 244 )

V $-$ Volume $\left [m^{3}\right ]$( 2, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 89, 90, 92, 97, 100, 101, 106, 107, 108, 111, 112, 120, 125, 128, 132, 133, 136, 148, 149, 153, 154, 155, 156, 157, 159, 160, 163, 166, 170, 184, 185, 186, 187, 189, 191, 193, 202, 215, 216, 222, 223, 248 )

$F_{f}$ $-$ Friction Force $[N]$( 2 )

$\sigma $ $-$ Surface Tension $\left [\frac {N}{m}\right ]$( 2, 14, 15, 16, 17, 43 )

$\underline {a}$ $-$ Total surface area $[m^2]$( 2, 14, 15, 16, 17 )

$\varepsilon $ $-$ Electric potential $\left [\frac {J}{Coulomb}\nobreakspace {}or\nobreakspace {}\frac {J}{A\cdot s}\nobreakspace {}or\nobreakspace {}V\right ]$( 2 )

$\underline {q}$ $-$ Electric charge $\left [Coulomb\nobreakspace {}or\nobreakspace {}A\cdot s\right ]$( 2, 252 )

E $-$ Electric Field Strength $\left [\frac {V}{m}\right ]$( 2, 117, 170 )

D $-$ Electric Displacement $\left [\frac {Coulombs}{m^{2}}\right ]$( 2, 47, 48, 78, 79, 86, 90, 91, 92, 93, 99, 105, 106, 110, 112, 114, 115, 116, 118, 119, 120, 123, 130, 131, 132, 135, 136, 138, 141, 148, 160, 161, 162, 173, 190, 192, 193, 195, 196, 198, 199, 202, 206, 207, 208, 209, 211, 215, 216, 217, 219, 220, 221, 222, 223, 248 )

H $-$ Magnetic Field Strength $\left [\frac {A}{m}\right ]$( 2, 140, 250 )

$\underline {B}$ $-$ Magnetic Induction $\left [\frac {V\cdot s}{m^{2}}\right ]$( 2, 126 )

$\underline {V}_{0}$ $-$ Original volume $[m^3]$( 2, 252 )

x $-$ Displacement $[m]$( 2, 35, 47, 80, 81, 82, 83, 84, 90, 93, 94, 95, 97, 98, 101, 102, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 126, 129, 130, 132, 133, 136, 142, 146, 148, 149, 150, 151, 152, 165, 197, 200, 205, 224, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 238, 239, 240, 241, 242, 243, 244, 245, 247 )

$\Omega $ $-$ One-dimensional strain (linear strain) $[none]$( 2, 252 )

0 $-$ Initial( 2, 17, 35, 36, 42, 43, 44, 50, 51, 52, 53, 54, 62, 63, 67, 68, 154, 155, 156, 158, 159, 163, 164, 165, 166, 167, 170, 173, 175, 179, 185, 186, 187, 188, 189, 190, 191, 192, 196, 197, 198, 199, 200, 203, 205, 213, 214, 215, 216, 217, 219, 220, 221, 222, 223, 228, 230, 233, 234, 235, 236, 243, 244, 245, 246, 247, 249 )

u $-$ Force Potential( 3, 76, 80, 83, 84, 85, 86, 89, 96, 102, 110, 129, 130, 135, 150, 190, 198, 199, 206, 207, 208, 209, 218, 219, 227, 234, 236, 237, 240 )

r $-$ Position $[m]$( 3, 4, 13, 18, 35, 37, 38, 44, 46, 78, 81, 84, 93, 95, 96, 99, 100, 101, 104, 105, 107, 110, 111, 113, 115, 125, 126, 135, 140, 142, 143, 144, 148, 152, 153, 154, 155, 156, 159, 160, 161, 162, 164, 172, 173, 174, 175, 176, 179, 184, 185, 188, 189, 190, 191, 193, 194, 195, 196, 200, 202, 203, 206, 209, 211, 212, 213, 214, 217, 220, 221, 233, 234 )

$q_{i}$ $-$ charge $[Coulombs]$( 3, 250 )

$\varepsilon $ $-$ Particle potential energy $[J]$( 3, 4, 252 )

$k_{B}$ $-$ Boltzmann Constant $\left [\frac {J}{K}\right ]$( 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 18, 150, 164, 165, 166, 170, 209 )

T $-$ Temperature $[K]$( 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 89, 91, 93, 99, 100, 101, 102, 105, 106, 107, 108, 109, 112, 114, 115, 116, 117, 118, 119, 120, 125, 126, 127, 128, 130, 131, 132, 135, 139, 143, 144, 145, 150, 152, 157, 158, 159, 163, 164, 165, 166, 167, 168, 169, 170, 172, 173, 174, 175, 176, 178, 179, 182, 186, 187, 190, 191, 192, 193, 197, 199, 200, 202, 203, 209, 211, 215, 216, 218, 227, 230, 231, 239, 247, 248, 249 )

$\sigma $ $-$ Molecular Radius $[m]$( 3, 4, 11, 12, 18, 57, 58, 73, 74, 76, 89, 100, 141, 143, 145, 147, 160, 167, 170, 172, 193, 197, 198, 199, 203, 205, 249 )

$\Phi $ $-$ Potential( 4, 13, 18, 35, 44, 94, 105, 231, 232, 233, 234, 235, 236 )

R $-$ Ideal Gas Constant( 4, 18, 35, 36, 37, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 74, 75, 79, 85, 86, 87, 88, 92, 93, 96, 105, 106, 107, 108, 110, 112, 114, 116, 117, 118, 119, 120, 125, 128, 135, 140, 142, 143, 145, 146, 148, 152, 153, 159, 160, 161, 162, 163, 164, 166, 167, 169, 173, 174, 175, 176, 178, 180, 181, 182, 183, 186, 187, 188, 190, 192, 193, 195, 196, 198, 199, 200, 202, 203, 205, 209, 216, 218, 220, 221, 225, 248 )

a $-$ Spherical core radius $[m]$( 4, 252 )

$\nu $ $-$ Frequency $\left [\frac {1}{s}\right ]$( 4, 150, 209 )

N $-$ Number of Moles $[mol]$( 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 36, 39, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 59, 65, 71, 72, 76, 79, 89, 90, 91, 93, 105, 112, 115, 116, 118, 119, 120, 128, 135, 138, 154, 156, 157, 163, 165, 166, 167, 168, 170, 171, 172, 173, 174, 179, 186, 187, 189, 197, 198, 202, 209, 220, 224, 225, 226, 227, 239, 240, 241, 249 )

E $-$ Energy $[J]$( 5, 6, 7, 8, 9, 10, 13, 47, 73, 77, 80, 83, 89, 91, 92, 99, 100, 101, 102, 103, 104, 120, 136, 158, 164, 167, 168, 172, 176, 178, 190, 191, 192, 197, 198, 209, 212, 213, 214, 248, 249 )

$\beta $ $-$ Boltzmann Normalization $\left [\frac {1}{J}\right ]$( 5, 7, 8, 18, 35, 249 )

B $-$ Generalized variable $[none]$( 5 )

$P_{i}$ $-$ Probability distribution of energy states $[none]$( 5, 250 )

i $-$ State or Level( 5, 9, 10, 14, 15, 16, 17, 18, 41, 42, 46, 47, 48, 49, 50, 51, 56, 57, 58, 59, 60, 61, 62, 63, 64, 71, 72, 73, 76, 77, 79, 80, 81, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 96, 105, 106, 107, 108, 111, 114, 115, 116, 117, 118, 119, 120, 126, 128, 129, 130, 131, 133, 135, 138, 142, 143, 145, 147, 149, 153, 157, 158, 159, 162, 163, 164, 165, 168, 170, 172, 174, 175, 177, 178, 183, 186, 187, 190, 191, 192, 194, 199, 200, 201, 203, 204, 205, 206, 215, 216, 218, 224, 225, 226, 227, 228, 229, 230, 233, 234, 241, 242, 243, 248, 249 )

$n_{i}$ $-$ Moles of species i( 6, 7, 252 )

$\tilde{N}$ $-$ Total number of moles( 6, 7, 252 )

$\Omega $ $-$ Microcanonical Partition Function( 6, 7, 9, 10, 137, 160, 162, 163, 164, 193, 194, 196 )

$\omega $ $-$ Microcanonical Partition Function( 6, 7, 8, 69, 70, 72 )

S $-$ Entropy $\left [\frac {J}{K}\right ]$( 6, 9, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 38, 39, 40, 41, 45, 46, 47, 52, 55, 56, 64, 65, 66, 73, 74, 75, 76, 77, 80, 82, 91, 93, 100, 112, 119, 120, 136, 140, 142, 143, 144, 145, 146, 148, 149, 158, 160, 161, 162, 164, 178, 180, 181, 182, 183, 187, 192, 193, 194, 195, 196, 202, 203, 204, 205, 206, 209, 212, 213, 214, 217, 221, 222, 223, 235, 238, 243, 248 )

$Q_{N}$ $-$ Canonical Partition Function( 7, 8, 9, 10 )

A $-$ Helmholtz Free Energy $[J]$( 7, 9, 10, 11, 12, 14, 17, 21, 22, 23, 24, 27, 28, 30, 31, 33, 39, 46, 48, 55, 64, 65, 80, 89, 90, 91, 92, 99, 109, 111, 112, 115, 116, 135, 136, 140, 141, 142, 143, 144, 146, 148, 152, 156, 157, 158, 159, 160, 161, 162, 164, 165, 166, 167, 168, 172, 173, 174, 175, 178, 179, 180, 181, 182, 183, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 198, 199, 202, 203, 204, 205, 206, 209, 213, 215, 218, 224, 225, 226, 227, 228, 236, 240 )

$\mu $ $-$ Chemical Potential $[J]$( 7, 9, 10, 14, 15, 16, 17, 37, 39, 41, 43, 44, 45, 47, 48, 49, 50, 55, 65, 66, 76, 80, 81, 82, 86, 89, 91, 92, 93, 94, 95, 96, 97, 98, 100, 105, 130, 135, 136, 137, 140, 148, 150, 166, 167, 174, 209, 221, 222, 223 )

$\Xi $ $-$ Grand Canonical Partition Function( 7, 9, 10, 43 )

$\Delta $ $-$ Isobaric/Isothermal Partition Function( 8, 9, 10, 13, 38, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 61, 62, 63, 64, 73, 74, 75, 76, 77, 80, 83, 85, 88, 89, 91, 92, 93, 94, 97, 102, 103, 105, 106, 107, 111, 115, 125, 126, 131, 132, 140, 150, 152, 158, 159, 173, 174, 175, 187, 190, 191, 192, 193, 199, 200, 209, 216, 218, 228, 229, 230, 231, 237, 238, 241, 242, 243 )

G $-$ Gibbs Energy $[J]$( 8, 9, 10, 11, 12, 16, 17, 21, 22, 23, 25, 27, 29, 31, 32, 33, 36, 39, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 61, 62, 65, 68, 148, 152, 159, 173, 174, 175, 191, 216, 218 )

U $-$ Internal Energy $[J]$( 9, 11, 12, 14, 15, 17, 21, 22, 23, 24, 26, 27, 28, 31, 33, 38, 39, 40, 41, 46, 54, 55, 56, 65, 66, 67, 73, 74, 80, 82, 88, 211 )

H $-$ Enthalpy $[J]$( 10, 11, 12, 15, 16, 17, 21, 22, 23, 24, 26, 27, 28, 29, 31, 33, 37, 38, 39, 40, 42, 46, 47, 52, 53, 54, 55, 56, 61, 62, 63, 64, 65, 66, 67, 73, 74, 76, 78, 79, 87, 88, 89, 91, 111, 115, 142, 143, 144, 145, 146, 147, 158, 159, 190, 191, 192, 193, 199, 200, 201, 203, 204, 205, 208, 214, 224, 227, 230 )

$\pi $ $-$ Constant( 11, 12, 13, 18, 37, 38, 44, 170, 249 )

m $-$ Mass $[Kg]$( 11, 12, 13, 38, 39, 40, 47, 49, 55, 56, 65, 66, 71, 72, 111, 133, 134, 136, 165, 166, 170, 194, 206, 214, 215, 229, 231, 236, 241, 242, 243 )

h $-$ Planck's Constant $\left [J\cdot s\right ]$( 11, 12, 13, 83, 108, 150, 170, 201, 209, 210, 219 )

e $-$ Base of natural ogarithm( 11, 12, 81, 89, 94, 95, 105, 111, 113, 114, 117, 118, 147, 149, 173, 174, 236, 240, 243, 244, 245, 249 )

$C_{V}$ $-$ Constant Volume Heat Capacity $\left [\frac {J}{Kg\cdot K}\right ]$( 11, 12, 19, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 40, 56, 66, 75, 89 )

I $-$ Moment of Inertia $\left [Kg\cdot m^{2}\right ]$( 11, 47, 48, 110, 111, 161, 170, 175, 196, 212, 218, 226, 239, 242 )

$\Theta _{\nu }$ $-$ Vibrational Frequency Factor( 11, 12, 13, 251 )

$D_{e}$ $-$ Energy of Atomization $\left [\frac {J}{mol}\right ]$( 11, 12, 13, 250 )

$\omega _{e}$ $-$ Electronic Degeneracy $[none]$( 11, 12, 13, 251 )

$\Theta $ $-$ Dimensionless Concentration or Temperature $[none]$( 12, 81, 94, 105, 106, 107, 108, 109, 110, 111, 113, 125, 126, 128, 129, 130, 132, 133, 134, 142, 143, 144, 157, 158, 159, 178, 186, 187, 190, 191, 192, 200, 201, 202, 203, 231, 235, 243, 244, 245 )

$D_{0}$ $-$ Measured Bond Dissociation Energy $\left [\frac {J}{mol}\right ]$( 12, 13 )

v $-$ Wavenumber $\left [\frac {1}{cm}\right ]$( 12, 13, 252 )

c $-$ Speed of Light( 12, 13, 80, 126, 199, 205, 206, 224 )

$\lambda $ $-$ Wavelength $[m]$( 12, 13, 111, 115, 117, 118, 126, 143, 144, 145, 201, 204, 224, 226, 232, 233, 234, 236, 240, 241, 242, 243 )

$q_{trans}$ $-$ Translational Molecular Partition Coefficient( 12, 168, 170 )

$q_{rot}$ $-$ Rotational Molecular Partition Coefficient( 12, 13, 168, 170 )

$q_{vib}$ $-$ Vibrational Molecular Partition Coefficient( 12, 13, 168, 170 )

$q_{elec}$ $-$ Electronic Molecular Partition Coefficient( 12, 13, 170 )

$Q_{cl}$ $-$ Classical Partition Coefficient( 13 )

$Q_{int}$ $-$ Internal Partition Coefficient( 13 )

$Q_{cm}$ $-$ Center-of-Mass Partition Coefficient( 13 )

Q $-$ Heat Flow $[W]$( 13, 46, 51, 52, 73, 74, 75, 76, 77, 89, 91, 99, 105, 158, 159, 168, 173, 191, 192, 225, 248 )

$Z^{*}$ $-$ Configurational Integral $[none]$( 13, 110 )

$\Lambda $ $-$ DeBroglie Wavelength $[m]$( 13, 46, 109, 224, 225, 226, 236, 248 )

$q_{int}$ $-$ Internal Molecular Partition Coefficient( 13, 46 )

$q_{nuc}$ $-$ Nuclear Molecular Partition Coefficient( 13, 170 )

$V_{f}$ $-$ Free Volume $\left [m^{3}\right ]$( 13, 250 )

$\rho $ $-$ Density $\left [\frac {Kg}{m^{3}}\right ]$( 13, 18, 35, 43, 67, 68, 73, 78, 79, 80, 81, 82, 83, 86, 87, 88, 91, 92, 93, 94, 96, 97, 102, 103, 105, 106, 107, 111, 112, 118, 126, 130, 131, 132, 139, 140, 143, 145, 148, 149, 150, 151, 160, 161, 162, 164, 193, 194, 195, 196, 199, 200, 201, 202, 204, 209 )

$\Phi _{ij}(r)$ $-$ Potential Function( 13, 14, 251 )

g(r) $-$ Radial Distribution Function $[none]$( 13, 14 )

$x_{i}$ $-$ $i^{th}$ Conjugate Coordinate( 14, 15, 16, 17, 18, 62, 63 )

$\xi $ $-$ Conjugate Coordinate( 14, 15, 16, 17, 18, 251 )

$B_{n}$ $-$ $n^{th}$ Virial Coefficient $\left [\left (\frac {m^{3}}{Kg}\right )^{n-1}\right ]$( 18, 35, 36, 41, 43, 44, 67, 99 )

$N_{A}$ $-$ Avogadro's number $\left [\frac {molecules}{ mol}\right ]$( 18, 35, 186, 208 )

$\epsilon $ $-$ Energy well depth $[J]$( 18, 252 )

$C_{P}$ $-$ Constant Pressure Heat Capacity $\left [\frac {J}{Kg\cdot K}\right ]$( 19, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 36, 40, 52, 56, 66, 67, 75, 78, 79, 86, 87, 105, 106, 107, 126, 139, 150, 158, 159, 190, 191, 192, 199, 200, 209, 215, 216 )

Z $-$ Compressibility $[none]$( 35, 48, 64, 67, 68, 109, 157, 166, 172, 174, 175, 177, 178, 186, 187 )

$\omega $ $-$ Accentricity Factor $[none]$( 35, 251 )

mix $-$ Mixing Function( 35, 36, 41, 42, 45, 46, 47, 64 )

$x_{i}$ $-$ Fraction of species $i$ $[none]$ ( 35, 37, 38, 41, 42, 43, 44, 45, 46, 47, 49, 52, 57, 58, 60, 61, 62, 64, 81, 84, 86, 87, 90, 93, 94, 107, 116, 126, 127, 128, 138, 141, 147, 148, 149, 200, 201, 249 )

$\phi $ $-$ Electric Potential( 35, 36, 46, 48, 57, 58, 59, 60, 61, 81, 93, 107, 109, 114, 116, 117, 118, 119, 120, 133, 136, 160, 161, 162, 164, 187, 190, 193, 194, 195, 196, 200, 209, 210, 211, 237, 238, 246, 251 )

$\phi _{i}$ $-$ Flory-Huggins Mole Fraction of species $i$ $[none]$( 35, 44, 45 )

$\kappa $ $-$ COME BACK( 36, 70, 72 )

$\alpha $ $-$ Thermal Diffusivity $\left [\frac {m^{2}}{s}\right ]$( 36, 37, 39, 40, 55, 56, 57, 58, 59, 62, 63, 66, 79, 106, 107, 110, 111, 113, 115, 125, 128, 130, 139, 150, 176, 199, 201, 209, 215, 223, 243, 244, 249 )

$\Im $ $-$ degrees of freedom $[none]$( 37, 38 )

$\nu _{i}$ $-$ Charge Valency of species $i$( 37, 38, 251 )

B $-$ Exergy $[J]$( 38, 39, 46, 47, 52, 55, 64, 65, 76, 77, 78, 79, 90, 91, 110, 111, 115, 116, 123, 125, 128, 135, 136, 140, 141, 142, 143, 145, 146, 158, 160, 166, 167, 168, 172, 173, 174, 175, 179, 180, 181, 182, 183, 190, 191, 192, 193, 203, 204, 205, 209, 214, 215, 218, 219, 227, 229, 235, 247 )

$\delta $ $-$ Boundary Layer Thickness $[m]$( 38, 54, 55, 65, 72, 73, 74, 75, 76, 77, 89, 91, 108, 125, 126, 129, 130, 133, 134, 137, 150, 151, 157, 163, 166, 167, 186, 187, 197, 201, 210, 231, 236, 249 )

C $-$ Concentration $\left [\frac {kg of i}{m^{3}}\right ]$( 39, 43, 46, 55, 57, 58, 65, 76, 79, 83, 84, 86, 87, 88, 90, 91, 92, 93, 96, 99, 102, 105, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 123, 126, 128, 131, 138, 143, 144, 145, 152, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 172, 174, 175, 176, 183, 184, 186, 187, 188, 189, 190, 192, 193, 194, 195, 196, 199, 200, 202, 203, 204, 206, 207, 208, 213, 215, 217, 219, 223, 227, 230, 231, 233, 235 )

$\xi $ $-$ Extent of Reaction $[none]$( 41, 52, 53, 54, 198 )

z $-$ Height $[m]$( 41, 73, 74, 238 )

$f$ $-$ Fugacity( 42, 46, 48, 49, 50, 51, 56, 57, 58, 59, 60, 61, 62, 63, 74, 78, 85, 88, 92, 101, 104, 105, 107, 111, 113, 142, 144, 147, 148, 152, 175, 176, 177, 187, 200, 202, 217, 220, 221, 228, 229, 230, 231, 232, 234, 235, 236, 238, 241, 242, 243, 247, 249 )

EX $-$ Excess property( 42, 44, 45, 49, 52, 62, 63 )

$\gamma _{i}$ $-$ Activity Coefficient for species $i$ $[none]$( 42, 49, 52, 57, 58, 60, 61, 62, 63, 175 )

$\Pi $ $-$ Osmotic Pressure $[Pa]$( 43, 44 )

k $-$ Thermal Conductivity $\left [\frac {W}{m K}\right ]$( 43, 44, 45, 46, 69, 72, 79, 87, 89, 90, 91, 93, 105, 107, 108, 109, 112, 114, 115, 125, 126, 135, 139, 143, 145, 150, 151, 152, 156, 160, 161, 162, 164, 166, 167, 168, 169, 172, 173, 174, 175, 177, 178, 179, 181, 188, 189, 192, 193, 194, 195, 196, 199, 200, 201, 202, 203, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 220, 221, 223, 225, 229, 242, 243, 247, 248, 249 )

$M_{W}$ $-$ Molecular Weight $\left [\frac {g}{mol}\right ]$( 43, 90, 112, 135, 202, 209 )

$a_{i}$ $-$ Activity coefficient of $i$( 44, 252 )

$N_{i}$ $-$ Moles of $i$( 44, 252 )

$\chi $ $-$ Flory-Huggins Parameter $[none]$( 44, 45, 128 )

$N_{i}$ $-$ Mass Species Flux of $i$ $\left [\frac {Kg}{m^{2} s}\right ]$( 44, 46, 47, 114, 115, 116, 117, 118, 119, 153, 154, 157, 184, 185 )

c $-$ Critical( 45, 87 )

$\Omega _{mix}$ $-$ Combinatorial Mixing Function $[none]$( 45, 251 )

$\underline {S}$ $-$ Stress Vector( 45, 75, 89, 140, 141, 146 )

atherm $-$ Athermal Function( 46, 252 )

Bi $-$ Biot Number $[none]$( 46, 47, 90, 107, 108, 109, 200, 201, 207, 210 )

a $-$ Activity( 49, 71, 72, 73, 74, 84, 128, 156, 157, 162, 175, 187, 189, 194, 206, 218, 219, 225, 227, 228, 233, 234, 235, 244, 246 )

K $-$ Dilation Factor( 49, 92, 93, 109, 117, 135, 137, 143, 144, 152, 167, 168, 173, 175, 176, 178, 179, 190, 203, 209, 212, 213, 214, 215, 216, 217, 218, 219, 221, 223, 236, 239, 241 )

v $-$ Velocity $\left [\frac {m}{s}\right ]$( 50, 51, 73, 74, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 110, 111, 114, 115, 116, 117, 118, 119, 125, 126, 128, 130, 131, 132, 136, 137, 138, 140, 141, 143, 145, 146, 149, 150, 151, 152, 153, 157, 163, 165, 166, 167, 168, 169, 170, 172, 175, 183, 186, 187, 190, 199, 200, 201, 204, 212, 213, 214, 215, 230, 231, 238 )

$K_{a}$ $-$ Equilibrium Constant $[varies]$( 50, 51, 52, 53, 54, 152, 175 )

$K_{\phi }$ $-$ Fugacity Coefficient Equilibrium Constant Contribution $[varies]$( 50, 250 )

$K_{y}$ $-$ Mole Fraction Equilibrium Constant Contribution $[varies]$( 50, 250 )

$K_{P}$ $-$ Pressure Equilibrium Constant Contribution $[varies]$( 50, 51 )

$\phi $ $-$ Fugacity Coefficient $[none]$( 50, 53, 54, 68 )

q $-$ heat flux $\left [\frac {W}{m^{2}}\right ]$( 50, 51, 73, 78, 86, 87, 91, 92, 97, 111, 112, 126, 128, 139, 140, 143, 144, 145, 149, 168, 201, 202, 203, 204, 225, 232 )

$K_{\gamma }$ $-$ Activity Coefficient Equilibrium Constant Contribution $[varies]$( 51, 250 )

$\beta $ $-$ Thermal Expansion Coefficient $\left [\frac {1}{K}\right ]$( 56, 57, 58, 59, 60, 62, 63, 112, 131, 132, 198, 202, 215, 247, 248 )

L $-$ Length Scale $[m]$( 60, 61, 64, 80, 82, 83, 85, 88, 90, 91, 92, 93, 94, 97, 100, 102, 107, 108, 109, 111, 116, 117, 118, 125, 126, 127, 128, 130, 131, 132, 150, 160, 161, 172, 177, 193, 195, 198, 199, 200, 201, 209, 210, 216, 217, 218, 219, 220, 224, 225, 232, 233, 235, 236 )

$\mathcal {K}_{i}$ $-$ Partition Coefficient $[none]$( 61, 90, 108, 201, 210, 218 )

t $-$ Time $[s]$( 73, 153, 154, 156, 158, 167, 172, 176, 177, 179, 183, 184, 185, 189, 190, 192, 197, 199, 201, 202, 204, 215, 216, 217, 220, 221, 222, 223, 231, 235, 238, 239, 240, 241, 242, 243, 244, 249 )

g $-$ Gravitational Acceleration( 73, 74, 76, 80, 83, 89, 92, 98, 102, 103, 114, 130, 131, 132, 140, 142, 144, 148, 149, 150, 164, 165, 192, 203, 205, 209, 234, 243 )

K $-$ Taylor Dispersion( 73, 81, 110 )

U $-$ Velocity $\left [\frac {m}{s}\right ]$( 75, 76, 80, 83, 84, 85, 86, 89, 91, 92, 97, 100, 102, 103, 105, 110, 125, 130, 131, 132, 148, 149, 150, 151, 158, 159, 190, 191, 192, 198, 209, 215, 221, 224, 225 )

$\eta $ $-$ heat cycle efficiency $[none]$( 76, 251 )

$\underline {\nabla }$ $-$ Gradient Operator( 78, 79, 80, 81, 82, 83, 90, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 110, 111, 112, 114, 115, 116, 117, 118, 125, 126, 128, 130, 131, 132, 137, 138, 139, 140, 142, 143, 144, 145, 146, 153, 160, 183, 202, 203, 204, 205, 206, 207, 208, 238 )

$\Phi $ $-$ Dissipation Function( 78, 85, 86, 93, 94 )

$\underline{\underline {\sigma }}$ $-$ Stress Tensor( 79, 82, 142, 146, 147, 148 )

$J_{i}$ $-$ Molar Diffusive species flux of $i$ $\left [\frac {mol}{s m^{2}}\right ]$( 79, 86, 88, 105, 114, 115, 138, 143, 144, 145, 153, 172, 183, 203, 204 )

h $-$ Heat Transfer Coefficient $\left [\frac {W}{m^{2}k}\right ]$( 80, 91, 94, 102, 103, 109, 125, 140, 143, 144, 145, 165, 168, 169, 172, 173, 192, 202, 203, 234 )

$\wp $ $-$ Dynamic Pressure $[Pa]$( 80, 81, 82, 83, 84, 85, 86, 88, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 118, 130, 131, 132, 133, 147, 148, 151 )

Re $-$ Reynolds Number $[none]$( 80, 82, 83, 85, 86, 87, 92, 97, 99, 100, 101, 102, 122, 123, 125, 129, 130, 132, 148, 150, 151, 209, 226, 240, 241 )

Sr $-$ Strouhal Number $[none]$( 80, 82, 83, 99, 100, 101, 102, 103, 104 )

$\tau $ $-$ Torque $[N m]$( 81, 84, 86, 87, 88, 94, 96, 105, 107, 114, 116, 118, 130, 131, 136, 137, 140, 141, 146, 148, 151, 156, 160, 188, 189, 193, 197, 198, 199, 204, 222 )

$\underline{\underline {\Gamma }}$ $-$ Rate-of-Strain Tensor( 81, 136, 137 )

$\Psi $ $-$ Stream Function $\left [\frac {m^{2}}{s}\right ]$( 83, 99, 100, 101, 102, 103, 104, 105, 117, 192 )

$\underline {w}$ $-$ Vorticity Vector( 83, 102, 103, 137, 224, 226, 227, 240 )

$R_{V}$ $-$ Volumetric Species Source $\left [\frac {mol}{s m^{3}}\right ]$( 87, 91, 93, 114, 118, 126, 160, 193 )

$D_{i}$ $-$ Pseudobinary diffusivity of $i$ $\left [\frac {m^{2}}{s}\right ]$( 90, 91, 106, 107, 108, 111, 114, 116, 117, 118, 125, 126, 128, 131, 138, 143, 145, 150, 172, 198, 199, 200, 201, 204, 209, 210 )

Sh $-$ Sherwood Number $[none]$( 90, 122, 126, 128, 129, 130, 133, 134, 135, 150, 209 )

Pr $-$ Prandtl Number $[none]$( 91, 122, 123, 124, 125, 128, 129, 130, 131, 132, 133, 134, 135, 150, 209 )

$\eta $ $-$ Intrinsic Viscosity $\left [\frac {m^{3}}{Kg}\right ]$( 94, 99, 113, 128, 160, 161, 162, 163, 194, 195, 196, 206, 207, 208, 211, 231, 233, 234, 243, 244, 245, 246 )

$\underline {e}$ $-$ Multicomponent Energy Flux ( 95, 98, 114, 115, 126, 132, 147 )

Br $-$ Brinkman Number $[none]$( 105, 125 )

Pe $-$ Peclet Number $[none]$( 106, 117, 122, 123, 125, 126, 127, 128, 129, 132, 150, 198, 199, 209 )

$H_{V}$ $-$ Volumetric Energy Source $\left [\frac {W}{m^{3}}\right ]$( 106, 107, 126 )

Da $-$ Damkohler Number $[none]$( 106, 108, 156, 189, 198, 199, 200, 207 )

$k_{c,i}$ $-$ Mass Transfer Coefficient for species $i$ $\left [\frac {mol}{s m^{2}conc.difference}\right ]$( 108, 150, 201, 209, 210 )

$\mathcal {R}$ $-$ Surface Reaction Rate( 111, 112, 202 )

$\mathcal {H}_{i}$ $-$ Henry's Law Constant for specie $i$ $\left [\frac {atm}{mol fraction}\right ]$( 114, 115, 250 )

$q^{(x)}$ $-$ Dufour Heat Flux( 114, 250 )

$\underline {n}$ $-$ Normal Vector $[none]$( 115, 140, 141, 142, 146, 147, 148 )

i $-$ Current Density $\left [\frac {A}{m^2}\right ]$( 117, 250 )

Sc $-$ Schmidt Number $[none]$( 122, 123, 124, 125, 128, 129, 130, 133, 134, 135, 150, 209 )

Nu $-$ Nusselt Number $[none]$( 122, 125, 128, 129, 130, 133, 134, 135, 150, 209 )

Gr $-$ Grashof Number $[none]$( 124, 131, 132, 133, 134, 135, 150, 209 )

Ra $-$ Rayleigh Number $[none]$( 131, 250 )

$\beta _{i}$ $-$ Solute Expansion Coefficient due to species $i$ $\left [\frac {1}{mol}\right ]$( 131, 250 )

$D_{ij}$ $-$ Diffusivity of component $i$ in fluid $j$ $\left [\frac {m^{2}}{s}\right ]$( 138 )

$\tau _{ij}$ $-$ Shear Rate in direction $i$ perpendicular to $j$ (CHECK FOR ONSISTENCY) $\left [\frac {N}{m^{2}}\right ]$( 147, 251 )

$\underline {e}_{i}$ $-$ unit vector in direction $i$ $[none]$( 147, 250 )

$c_{D}$ $-$ Drag Coefficient $[none]$( 148, 250 )

$F_{B}$ $-$ Body Force $[N]$( 148, 250 )

$\epsilon $ $-$ Dielectric Constant $\left [\frac {Coulombs^{2}}{J m}\right ]$( 156, 189, 207, 208 )

$\hat{f}$ $-$ Mixture Fugacity( 175, 250 )

$f_{\underline {e}}_{i}}}}$ $-$ Differential Force in direction of $\underline {e}}_{i}}$( 250 )

$\mathcal {H}$ $-$ Surface Mean Curvature( 250 )

$S_{A}$ $-$ Surface Area $\left [m^{2}\right ]$( 250 )

$\underline{\underline {w}}$ $-$ Vorticity Tensor( 250 )

$z_{i}$ $-$ Charge of species $i$ $[Coulombs]$( 250 )

$\epsilon _{M},\epsilon _{i},\epsilon _{E}$ $-$ Eddy Diffusivity for Momentum, Species and Energy $\left [\frac {m^{2}}{s}\right ]$( 250 )

$\Theta _{n}$ $-$ $n^{th}$ FFT Coefficients( 251 )

$\lambda $ $-$ Latent heat $\left [\frac {J}{mol}\right ]$( 251 )

$\lambda $ $-$ Debye Length $[m]$( 251 )

$\mu $ $-$ Viscosity $[Pa s]$( 251 )

$\nu $ $-$ Kinematic Viscosity $\left [\frac {m^{2}}{s}\right ]$( 251 )

$\nu _{ij}$ $-$ Stoichiometric Coefficient for species $i$ in reaction $j$( 251 )

$\sigma $ $-$ Symmetry Factor( 251 )

$\Phi _{n}$ $-$ FFT Basis Function( 251 )

$\Psi $ $-$ Dimensionless Electric Potential $[none]$( 251 )

$\mathcal {F}$ $-$ Faraday's Constant( 251 )

x $-$ Extensive Scalar( 251 )

$\underline {x}$ $-$ Extensive Vector( 251 )

$\underline{\underline {x}}$ $-$ Extensive Vector( 251 )

$\tilde{x}$ $-$ Extensive Vector( 251 )

$\textbf {X}$ $-$ Matrix( 252 )

$\hat{x}$ $-$ Intensive Scalar( 252 )

$\overline {x}$ $-$ Partial Molar( 252 )

$\dot{x}$ $-$ Time derivative( 252 )

$\underline {\nabla }_{s}$ $-$ Surface Gradient Operator( 252 )

$E^{4}$ $-$ Axisymmetric Biharmonic Operator( 252 )

$\sigma $ $-$ boundary( 252 )

id $-$ Ideal Gas( 252 )

sat $-$ Saturated Vapor( 252 )

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