Chemical Engineering formulas — 33 solvers

Absorption Factor

A = \frac{L}{m V}

Chemical EngineeringThe single number that decides whether an absorber can do its job: the liquid's capacity to carry the solute away divided by the gas's capacity to deliver it. Below one, no number of stages will finish the separation.

Boilup Ratio

V_B = \frac{V}{B}

Chemical EngineeringThe vapour the reboiler raises divided by the bottoms drawn off beneath it. The stripping section's answer to the reflux ratio, and the term the steam bill is really written in.

Chilton–Colburn Analogy for Mass Transfer

\mathrm{Sh} = \frac{f}{2} \, \mathrm{Re} \, \mathrm{Sc}^{1/3}

Chemical EngineeringMass transfer inferred from pressure drop. Chilton and Colburn found in 1934 that the j-factor for mass transfer equals half the Fanning friction factor, which lets a Sherwood number be predicted from nothing but a Reynolds number and a Schmidt number.

Column Material Balance (Distillate and Bottoms Split)

D = F\,\frac{z_F - x_B}{x_D - x_B}

Chemical EngineeringHow much of the feed leaves overhead and how much out the bottom, from nothing but the three compositions. Two balances — total flow and light key — solved together, and the answer is a lever rule.

Conversion in n Equal CSTRs in Series

X = 1 - \frac{1}{\left(1 + \mathrm{Da}\right)^{n}}

Chemical EngineeringConversion through a cascade of n identical stirred tanks, where Da is the Damköhler number of a single tank. Splitting one vessel into several buys conversion for nothing but partition walls, and as n grows the cascade converges on plug flow.

CSTR Conversion (First Order)

X = \frac{\mathrm{Da}}{1 + \mathrm{Da}}

Chemical EngineeringHow far a first-order reaction gets in a single perfectly mixed tank, as a function of the Damköhler number alone. The curve rises steeply and then flattens: it passes 50% at Da = 1, needs Da = 9 for 90%, and needs an infinite tank for 100%.

Damköhler Number (First Order)

\mathrm{Da} = k\,\tau

Chemical EngineeringThe single dimensionless group that decides how far a first-order reaction gets: rate constant times residence time. It compares the speed of the chemistry with the speed of the flow, and once it is fixed the conversion is fixed — whatever the vessel's actual size.

Feed Line (q-Line)

y = \frac{q}{q - 1}\,x - \frac{z_F}{q - 1}

Chemical EngineeringThe line the feed's thermal condition draws on a McCabe–Thiele diagram. Both operating lines must cross on it, and where they cross is the optimum feed stage — so q decides where the feed nozzle goes.

Fractional Conversion from Concentration

X = \frac{C_{A0} - C_A}{C_{A0}}

Chemical EngineeringThe fraction of the limiting reactant that has been consumed, read straight off an inlet and an outlet analysis. Every reactor design equation on this site is written in terms of this one number, so it is worth getting the basis right before anything else.

Gilliland Correlation (Actual Stages)

\frac{N - N_{min}}{N + 1} = 1 - \exp\!\left[\left(\frac{1 + 54.4X}{11 + 117.2X}\right)\!\left(\frac{X - 1}{\sqrt{X}}\right)\right],\quad X = \frac{R - R_{min}}{R + 1}

Chemical EngineeringThe empirical bridge across the middle of column design: given the two unbuildable limits — Fenske's fewest stages and Underwood's least reflux — it returns the stage count a column actually needs at the reflux you intend to run.

Half-Life of a Second-Order Reaction

t_{1/2} = \frac{1}{k\,C_{A0}}

Chemical EngineeringHow long a second-order reaction takes to consume half its reactant. Unlike first-order decay, this time depends on the starting concentration — and each successive half-life is twice as long as the one before it.

Instantaneous Selectivity

S_{D/U} = \frac{r_D}{r_U}

Chemical EngineeringThe rate of the reaction you want divided by the rate of the one you do not, at the conditions inside the reactor right now. When two reactions compete for the same feed, this ratio decides which vessel to build and where to run it.

Interphase Mass Flux

N_A = K_y \left( y - y^{*} \right)

Chemical EngineeringThe point of the whole apparatus: a coefficient multiplied by a departure from equilibrium gives the moles crossing each square metre of interface every second. Everything else in mass transfer exists to estimate one of the two terms on the right.

Kremser Equation for Absorption Stages

N = \frac{\ln \left[ \dfrac{y_1}{y_2} \left( 1 - \dfrac{1}{A} \right) + \dfrac{1}{A} \right]}{\ln A}

Chemical EngineeringHow many theoretical stages a tray absorber needs, from the recovery demanded and the absorption factor available. Kremser's 1930 shortcut, which replaced stepping off a McCabe–Thiele diagram by hand and has outlived the diagram.

Mean Residence Time from a Tracer

\bar{t} = \frac{\int_0^{\infty} t\,C\,dt}{\int_0^{\infty} C\,dt} = \frac{M_1}{M_0}

Chemical EngineeringThe average time fluid actually spends in a vessel, taken as the first moment of a tracer curve divided by its zeroth moment. Compared with the nominal V/v₀, it is the single most useful diagnostic a reactor engineer has: it finds dead volume and short-circuiting that no drawing will show.

Minimum Reflux Ratio (Underwood, Binary)

R_{min} = \frac{1}{\alpha - 1}\left(\frac{x_D}{z_F} - \frac{\alpha\left(1 - x_D\right)}{1 - z_F}\right)

Chemical EngineeringThe least reflux that could ever achieve a given split, at the price of an infinite number of stages. The lower limit of the design, opposite Fenske's upper one, and the number every real reflux ratio is quoted as a multiple of.

Overall Column Efficiency

E_o = \frac{N_t}{N_a}

Chemical EngineeringThe bridge from the calculation to the hardware: theoretical stages divided by the actual trays needed to do their work. Real trays never reach equilibrium, so there are always more of them than the theory asked for.

Packed Column Height from HTU and NTU

Z = H_{OG} \, N_{OG}

Chemical EngineeringA packed column's height split into the two things that set it: how hard the separation is, and how good the packing is at it. Chilton and Colburn's transfer-unit idea, and the reason packing is specified in metres of HTU rather than in stages.

Péclet Number for Mass Transfer

\mathrm{Pe} = \frac{u L}{D}

Chemical EngineeringWhether the stream carries the solute along faster than diffusion can spread it out: bulk advection divided by molecular diffusion. The number that decides if a reactor behaves like a plug or like a stirred tank.

PFR Conversion (First Order)

X = 1 - e^{-\mathrm{Da}}

Chemical EngineeringHow far a first-order reaction gets in a plug-flow tube, as a function of the Damköhler number alone. Reactant decays exponentially along the length, so conversion closes on 100% far faster than a stirred tank can: Da = 2.3 gives 90%, Da = 4.6 gives 99%.

Power-Law Reaction Rate

r = k\,C_A^{\,n}

Chemical EngineeringThe rate of a reaction as a power of the reactant concentration. The reaction order n is fitted from data, not read off the balanced equation — and because the units of k depend on n, this is the equation where careless unit handling does the most damage.

Rayleigh Equation (Simple Batch Distillation)

\ln\frac{B_0}{B_1} = \frac{1}{\alpha - 1}\ln\!\left[\frac{x_0}{x_1}\left(\frac{1 - x_1}{1 - x_0}\right)^{\!\alpha}\right]

Chemical EngineeringA pot still with no column and no reflux: vapour is drawn off as fast as it forms, so what is left behind gets steadily poorer in the light component. This says how much charge remains once the pot has fallen to a given composition.

Reaction Yield

Y = \frac{n_D}{n_{A0} - n_A}

Chemical EngineeringMoles of the desired product formed per mole of limiting reactant actually consumed. Yield is the honest partner to conversion: a reactor can consume all its feed and still make very little of what you wanted.

Rectifying Operating Line (McCabe–Thiele)

y = \frac{R}{R + 1}\,x + \frac{x_D}{R + 1}

Chemical EngineeringThe straight line above the feed that ties the vapour rising off a stage to the liquid falling onto it. Slope R/(R+1), and it always passes through the point (x_D, x_D) on the 45° line — which is how it gets drawn.

Reflux Ratio

R = \frac{L}{D}

Chemical EngineeringThe liquid sent back down the column divided by the product drawn off the top. One number that sets the slope of the rectifying line, the height of the tower, and most of the energy bill.

Schmidt Number

\mathrm{Sc} = \frac{\mu}{\rho D}

Chemical EngineeringThe ratio of momentum diffusivity to mass diffusivity — how readily a fluid spreads motion compared with how readily it spreads molecules. The mass-transfer twin of the Prandtl number, and the fluid property every mass-transfer correlation is written around.

Sherwood Number

\mathrm{Sh} = \frac{k_c L}{D}

Chemical EngineeringThe convective mass transfer coefficient made dimensionless: transfer with the flow divided by transfer by diffusion alone. Named for Thomas Sherwood, whose 1934 wetted-wall work with Gilliland set the pattern every correlation since has followed.

Space Velocity

\mathrm{SV} = \frac{1}{\tau}

Chemical EngineeringHow many reactor volumes of feed pass through per unit time — the reciprocal of space time, and the number catalyst vendors quote on a datasheet. Reported in reciprocal hours: an LHSV of 2 h⁻¹ means two vessel volumes of liquid feed every hour.

Stanton Number for Mass Transfer

\mathrm{St}_D = \frac{k_c}{u}

Chemical EngineeringThe mass transfer coefficient measured against the velocity that produced it: what fraction of the oncoming stream is actually delivered to the surface. The group the Chilton–Colburn j-factor is built from.

Stripping Operating Line (McCabe–Thiele)

y = \frac{V_B + 1}{V_B}\,x - \frac{x_B}{V_B}

Chemical EngineeringThe straight line below the feed, written in terms of the boilup ratio. Slope greater than 1, and it passes through (x_B, x_B) on the 45° line — the mirror of the rectifying line, anchored on the bottoms instead of the distillate.

Transfer Units for Dilute Absorption

N_{OG} = \frac{\ln \left[ \dfrac{y_1}{y_2} \left( 1 - \dfrac{1}{A} \right) + \dfrac{1}{A} \right]}{1 - \dfrac{1}{A}}

Chemical EngineeringThe number of transfer units a dilute absorber demands — the packed-column counterpart of a stage count. Colburn's 1939 integration of the design equation for a straight equilibrium line, and the number that multiplies HTU to give a height.

Two-Film Overall Mass Transfer Coefficient

\frac{1}{K_{OG}} = \frac{1}{k_G} + \frac{m}{k_L}

Chemical EngineeringTwo films in series, and the resistances add. Lewis and Whitman's 1924 picture of a gas film and a liquid film meeting at an interface in equilibrium is still the working model for every absorber and stripper built.

Wilke–Chang Liquid Diffusivity

D_{AB} = 7.4 \times 10^{-8} \, \frac{(\phi M_B)^{1/2} \, T}{\mu_B V_A^{0.6}}

Chemical EngineeringAn estimate of how fast a solute diffuses through a liquid, from the solvent's molar mass, viscosity and association behaviour plus the solute's molar volume. Wilke and Chang's 1955 correlation, still the default when nobody has measured the real thing.