Chemistry · Lesson

Reactor concepts, catalysis and separation integration

Compare ideal reaction models and integrate reaction, separation and energy accounts within explicit boundaries.

By the end, you can

Compare ideal first-order CSTR and plug-flow balances.
Propagate remaining fractions through a finite equal-tank series.
Distinguish instantaneous selectivity from an integrated yield.
Close an isolated-product ledger and its specific energy denominator.

A reactor model connects rates with inventories and flow

A batch model follows a changing inventory in a defined vessel. Continuous stirred-tank and plug-flow idealizations connect incoming and outgoing streams with different assumptions about mixing and progression. Specify the volume and concentration basis used by the rate law.

A residence-time parameter from volume divided by volumetric flow belongs to the stated flow model. Real residence times can form a distribution. A mean time alone does not establish every molecule’s history.

Different concentration histories give different ideal predictions

A perfectly mixed steady CSTR has its outlet composition throughout the mixed volume. An ideal plug-flow model follows changing composition along its flow coordinate. Under the same irreversible first-order constant-density assumptions, these histories lead to different conversion expressions. The comparison below keeps volume, flow and assigned rate fixed.

A series of equal CSTRs divides the total space time among stages. Each outlet becomes the next inlet. Multiplying remaining fractions gives the overall conversion; adding stage conversions would repeatedly use an inappropriate feed basis. As the number tends to infinity under these assumptions, the account approaches the plug-flow expression.

These idealizations do not include every reaction, thermal effect, catalyst behaviour, transport limit or mixing distribution. Higher conversion in the chosen toy model does not establish a universally preferable physical reactor.

Different mixing assumptions can give different conversion relations

For a defined rate law, an ideal well-mixed continuous vessel uses its outgoing concentration throughout the reacting volume. An ideal plug-flow description follows composition along the flow direction. A batch description instead follows time for its closed or specified inventory.

Under a simple irreversible first-order model, these assumptions lead to different conversion relations. Compare them only with compatible temperature, coefficient, volume and flow conventions. The mathematical comparison does not predict every real reactor.

A local rate comparison is only one part of a process

Parallel channels with different assigned rate orders can change their relative contributions as concentration changes. In the original example, the desired rate rises with concentration while its fraction of total consumption falls. A local kinetic comparison needs integration over the actual composition history to predict an overall reaction output.

A separation then changes the isolated share of formed product. The complete external ledger includes unreacted A, desired P, unisolated P and competing W on the conserved symbolic-unit basis. Product recovery and reaction selectivity should remain separately named.

Specific energy divides the declared energy input by the chosen output amount. Improved recovery can coincide with more energy per isolated unit in an assigned scenario. Heat, mass and environmental comparisons require compatible units and boundaries rather than one unlabeled efficiency number.

Greater desired rate can accompany a smaller desired fraction

Instantaneous desired fraction

Tracked-A concentration / mol L⁻¹

Assigned concentration cases
— Assigned parallel-rate fraction

Underlying data

Tracked-A concentration / mol L⁻¹	Instantaneous desired fraction
0.5	0.7954545454545454 · Assigned concentration cases
2	0.49295774647887325 · Assigned concentration cases

Original instantaneous parallel-rate model r_P=0.0350C and r_W=0.0180C² in compatible tracked-unit concentration units. Circles mark C=0.500 and 2.00 mol L⁻¹; the dashed curve gives r_P/(r_P+r_W) for C>0. No integrated reactor yield is shown.

Heat and mass transport can change observed performance

Reaction heat, heat removal and incoming-stream conditions contribute to an energy balance. Temperature can affect kinetics and equilibrium, creating feedback that a constant-temperature calculation excludes. Mixing and transport can also limit access to reactive sites.

A catalyst’s measured apparent activity can include transport effects and changing catalyst state. A rate measured per mass of catalyst is different from a turnover based on active sites. Define the basis before comparing materials or models.

Reaction and separation can be coupled in a process

A separation can change concentrations entering a reacting unit or return unconverted material. The combined system needs a balance that includes both functions, with their respective yields and energy demands.

The original idealizations below teach model differences and conservation. They do not establish a safe operating envelope, reactor design or instructions for controlling a hazardous reaction.

Worked examples

Compare two ideal models under the same assumptions

Assign isothermal constant-density symbolic A→P, one conserved tracked unit per entity, first-order k=0.0420 min⁻¹, V=63.0 L, q=1.80 L min⁻¹ and inlet C_A=0.770 mol L⁻¹. Compare a perfectly mixed CSTR with ideal plug flow.

Space time τ=V/q=35.0 min and Da=kτ=1.470. For a CSTR, q(C_in−C_out)=kVC_out gives X=Da/(1+Da)=0.595141700 and C_out=0.311740891 mol L⁻¹.
For ideal plug flow, dC/dτ=−kC gives X=1−exp(−Da)=0.770074515 and C_out=0.177042624 mol L⁻¹.
A+P retains the assigned input tracked-unit amount in each model. These comparisons hold only for the stipulated irreversible first-order, constant-density, isothermal account.
CSTR space time is not every molecule’s residence time. No universal reactor preference, actual molecular mechanism, heat-transfer design or equipment performance is established.

At the same assigned Da, X_CSTR=0.595141700 and X_PFR=0.770074515.

Equal tanks approach a model limit

Keep a fixed total space time 48.0 min and first-order k=0.0270 min⁻¹ for the same ideal irreversible one-unit model. Compare one and two equal CSTRs with the ideal plug-flow limit.

Total Da=1.296. For n equal tanks, each stage retains fraction 1/(1+Da/n), so the final A fraction is (1+Da/n)⁻ⁿ. Multiply stage fractions rather than add stage conversions.
One tank retains 0.435540070, conversion 0.564459930. Two retain 0.368201527, conversion 0.631798473.
The n→∞ logarithmic limit is exp(−Da)=0.273624103, giving ideal plug-flow conversion 0.726375897. A finite series is not exactly this limit.
Total volume/flow, temperature and assigned rate stay fixed. Costs, heat effects, mass transfer and equipment design have not been supplied.

Two equal ideal tanks give conversion 0.631798473; ideal plug flow is the separate limiting model.

A faster desired rate can have a smaller desired fraction

Assign parallel symbolic one-unit A→P and A→W channels, with empirical toy rates r_P=k_PC_A and r_W=k_WC_A². Let k_P=0.0350 min⁻¹,k_W=0.0180 L mol⁻¹ min⁻¹ and free tracked-A concentration 0.500 mol L⁻¹.

r_P=0.0175 and r_W=0.00450 mol tracked unit L⁻¹ min⁻¹. The total A consumption is 0.0220 in the same units.
Instantaneous desired fraction is r_P/(r_P+r_W)=35/44=0.795454545. The desired/undesired rate ratio is 3.88888889.
The different assigned rate orders make the desired fraction concentration-dependent. Orders are model inputs, not inferred from the symbolic net coefficients. Instantaneous fraction does not give integrated reactor yield or isolated output.

Desired instantaneous fraction 0.795454545 under the assigned rate model.

Isolation and energy have their own denominators

For a steady symbolic one-unit process, assign fresh A=12.0, consumed A=11.0, formed P=7.20 and formed W=3.80 kmol tracked unit h⁻¹. P recovery is 0.900; assigned reactor/separation power is 5.00/14.0 kW. Calculate the isolated yield and energy per isolated unit.

Reaction closure is 12.0=1.00 remaining A+7.20 P+3.80 W. Reactor desired selectivity on consumed A is 7.20/11.0=0.654545455.
Isolation gives 6.48 P and 0.720 unisolated P. The external tracked-unit ledger is 12.0=1.00+6.48+0.720+3.80. Overall isolated yield is 6.48/12.0=0.540.
Total assigned power is 19.0 kW. At this steady isolated flow, specific assigned energy is 19.0/6.48=2.93209877 kWh per kmol isolated tracked unit, since kW/(kmol h⁻¹)=kWh kmol⁻¹.
Recovery and power are scenario assignments, without an actual separator or a universal relation between them. The symbolic tracked unit is not a molecular mass balance.

Isolated yield 0.540; assigned energy 2.93209877 kWh per kmol isolated tracked unit.

Try it yourself

Halve only the volume to 31.5 L with the same assigned rate constant, inlet concentration and volumetric flow.
Show the answer and reasoning
τ=17.5 min, Da=0.735; X_CSTR=0.423631124, X_PFR=0.520494541; exit A concentrations 0.443804035/0.369219203 mol L⁻¹.
The space time and Da halve; substitute into each declared ideal model.
Replace the two tanks with three equal ideal CSTRs, retaining the same total space time. Find the final remaining fraction and conversion.
Show the answer and reasoning
Each stage Da=0.432; remaining fraction 0.340542207 and conversion 0.659457793.
Multiply three remaining fractions 1/1.432; the finite series still differs from the plug-flow limit.
Change only free tracked-A concentration to 2 mol/L. Recompute the two instantaneous rates and their desired fraction.
Show the answer and reasoning
r_P=0.0700, r_W=0.0720 mol tracked unit L⁻¹ min⁻¹; desired fraction 35/71=0.492957746.
The desired rate rises fourfold and the undesired rate sixteenfold, so the desired fraction falls.
Change recovery to 0.95 and assigned separation power to 20 kW, keeping all reaction flows and reactor power fixed. Compare isolated yield and specific energy.
Show the answer and reasoning
Isolated P=6.84, unisolated P=0.360 kmol tracked unit h⁻¹; isolated yield 0.570; total power 25.0 kW and energy 3.65497076 kWh per kmol isolated unit.
Recovery improves, but power grows enough to increase the energy per isolated unit; reaction selectivity stays fixed.

What to carry forward

Ideal reactor comparisons depend on kinetics and mixing assumptions.
Multiply remaining fractions through a series rather than add conversions.
Instantaneous selectivity does not by itself give isolated yield.
A specific energy needs a declared input boundary and output denominator.

References

Teaching resources and reference material supporting this lesson’s explanations and examples.