Reactors and Catalysis

Recycle Stream in terms of Conversion

X_ai = R/(R+1) * X_af

Recycle Stream in terms of Concentration

C_ai = (C_a0 + R*C_af) / (R+1)

Average Height

Area under curve / width of the area

Iterative solution to X_A1 (recycle ratio)

If KL=PQ, guess is correct

If KL<PQ, R is too high and X_A1 should be larger

If KL>PQ R is too low, and X_A1 should be smaller

Concentration in terms of conversion

C_a = C_a0 (1 - X_a)

Optimum recycle ratio (conc. and spacetime relation)

spacetime = C_a0 (average height X_af)

Trapezoidal Rule

find width by b - a / number of trapezoids

T- Profile for Batch reactor

Batch: vary the whole-reactor temperature over time (heating/cooling duty)

T-profile for PFR

PFR: establish a temperature profile along the reactor length (e.g. via countercurrent heat exchange)

T- profile for MFR

MFR (series): different temperature in each stage

How to find Temperature Progression

Find maximums of the rates, join the dots and

Recycle Ratio, R

Ratio of recycled fluid volumetric flow to the flow leaving the overall system.

Recycle PFR

A PFR with a fraction of its product stream returned to the inlet.

X_A1

Conversion at the point where fresh feed and recycle stream mix (reactor inlet).

X_Af

Final/exit conversion of the reactor.

Average height

The mean value of 1/(−r_A) over an interval, used to convert an integral (area) into an equivalent rectangle.

Optimal recycle ratio

The R that minimizes reactor volume for a given final conversion.

Locus of maximum rates

The curve joining the points of maximum reaction rate across a family of r(C,T) curves.

Optimal temperature progression

The sequence of temperatures through a reactor (or reactor system) that minimizes total volume for a given conversion.

T_max

Maximum allowable operating temperature (limited by materials, safety, side reactions).

Equilibrium constant, K

Thermodynamic ratio describing the extent of reaction at equilibrium; depends only on T.

Equilibrium conversion, X_Ae

The theoretical maximum (or, for reversible reactions, the actual maximum) conversion attainable at a given T.

ΔH_r

Heat of reaction — heat transferred from surroundings to the reaction system.

ΔC_p

Difference in heat capacities between products and reactants (Σ ν_i C_p,i).

C_p′ / C_p′′

Heat capacity of unreacted feed (′) vs fully reacted product (′′), per mole of A.

Adiabatic operation

No heat exchange with surroundings; all reaction heat stays in the system.

Non-adiabatic operation

Heat is deliberately added/removed, or lost to surroundings (term Q).

Operating line

The X_A vs T relationship imposed by the energy balance for a given reactor/heat configuration.

Adiabatic operating line slope

C_p / (−ΔH_r) — relates conversion change to temperature change.

Endothermic reaction

ΔH_r > 0; absorbs heat; T decreases with conversion (adiabatically).

Exothermic reaction

ΔH_r < 0; releases heat; T increases with conversion (adiabatically).

Residence Time Distribution (RTD)

The distribution of times that different fluid elements spend inside the reactor

Mean residence time

t̄ Average time fluid spends in the vessel t̄ = V/ν

E curve

Exit age distribution — probability density function of residence times ∫E dt = 1

F curve

Cumulative form of E (dimensionless

θ (theta)

Dimensionless time θ = t/t̄

Eθ

E curve normalised by mean residence time: Eθ = t̄·E(t̄θ) ∫Eθ dθ = 1

Microfluid

Fluid state where individual molecules mix freely and lose their identity (e.g. gases

Macrofluid

Fluid state where molecules remain grouped in aggregates that act as individual mini-batch reactors (e.g. emulsions

Earliness of mixing

Whether streams mix early

affects contacting and conversion for two-stream systems

Closed vessel

Plug flow at entrance/exit

flat velocity profile no diffusion or eddies crossing the boundary more than once D = 0

Open vessel

Eddies/non-laminar flow and possible diffusion across the reacting-zone boundaries D > 0

Dispersion Coefficient

D Parameter [m²/s] describing the degree of back-mixing/spreading (developed further in Module 6)

Tracer

A trackable

Pulse experiment

Tracer test: an instantaneous injection of tracer (M units) output is the Cpulse curve normalised to give E

Step experiment

Tracer test: inlet stream switched entirely to tracer at constant concentration Cm output is the Cstep curve normalised to give F

Channelling / short-circuiting

Flow path that allows some fluid to bypass most of the vessel spending less time than average

Stagnant region

Region of the vessel with little/no flow causing some fluid to spend much longer than average (risk of fouling

Dirac Delta Function

δ(t−t0) Idealised pulse of zero width and infinite height with area = 1 represents the E curve of an ideal PFR

Variance σ²

A measure of the spread of the RTD σ² = ∫t²E dt − t̄²

Compartment Model

Models a real vessel as a combination of PFR and MFR regions (plus dead/bypass/recycle flow), matched against the measured E curve

Active volume

Volume where fluid actually flows and reacts — sum of plug flow (Vp) and mixed flow (VM) regions

Dead / stagnant volume, Vd

Volume with no flow and no conversion

Bypass flow, Vb

Fluid flow that does not enter the vessel at all

Recycle flow, Vr

Fluid recycled from the vessel exit back to the inlet

Axial (longitudinal) dispersion

Spreading of a tracer pulse along the flow direction due to local velocity differences

Dispersion Coefficient, D

Parameter [m²/s] describing the degree of back-mixing; larger D = more spreading; D=0 = pure plug flow

Vessel Dispersion Number, D/uL

Dimensionless group; D/uL ≈ 0 → PFR; 0 < D/uL < 0.01 → Dispersion Model valid; D/uL ≫ 0 → far from plug flow

Variance, σ²

Measure of RTD spread; σ² = ∫t²E dt − t̄²; additive across units in series

Δσ²

Increase in variance across a vessel; Δσ²/(Δt̄)² = Δσθ² = 2(D/uL)

Probability paper

Plotting tool: cumulative F values form a straight line if the RTD is Gaussian; used to extract σ from the 16%/84% points

Tank-in-Series (TIS) Model

Models a non-ideal reactor as N equal-sized ideal MFRs in series; N = t̄²/σ² = 1/σθ²

Independence

Fluid 'loses its memory' between tanks — behaviour in one tank doesn't affect behaviour in the next; satisfied by complete mixing

Convolution

Mathematical operation combining an input signal with a system's RTD (E curve) to give the output signal: Cout = Cin ∗ E