In a hydrogen-generation water-gas shift (WGS) pilot plant, conversion, limiting reactant, and extent of reaction are evaluated from inlet and outlet molar flow rates. For the reaction

[ \mathrm{CO + H_2O \rightleftharpoons CO_2 + H_2} ]

the stoichiometric coefficients are (\nu_{\mathrm{CO}}=-1), (\nu_{\mathrm{H_2O}}=-1), (\nu_{\mathrm{CO_2}}=+1), and (\nu_{\mathrm{H_2}}=+1). The limiting reactant is identified from the feed ratio, conversion is the fraction of that reactant consumed, and the extent of reaction measures the absolute molar progress of the reaction.

Core takeaway: Determine the limiting reactant using the smallest feed amount divided by its stoichiometric requirement, calculate conversion from the consumed limiting-reactant flow, and calculate extent from the molar change of any species divided by its stoichiometric coefficient. In a real pilot plant, reliable results depend on consistent gas-flow measurements, water accounting, and a validated outlet material balance.

Start With a Consistent Material-Balance Basis

Use molar flow rates for continuous operation

For a steady-state pilot plant, use molar flow rates rather than total moles:

[ F_{i,\text{in}} \quad \text{and} \quad F_{i,\text{out}} ]

where (F_i) is the molar flow rate of species (i), commonly expressed in mol/s or kmol/h.

For a batch or finite-duration experiment, the equivalent quantities are total inlet and outlet moles:

[ n_{i,\text{in}} \quad \text{and} \quad n_{i,\text{out}} ]

The equations are identical after replacing (F_i) with (n_i).

Correct the analytical basis first

Gas analyzers often report dry-basis concentrations, while steam is present on a wet basis. Convert all measurements to a common basis before calculating conversion or extent.

Account for:

Steam condensed in downstream equipment
Water remaining in the outlet gas
Carrier or inert gases
Hydrogen or carbon monoxide already present in the feed
Differences between standard and actual volumetric flow conditions
Gas leaks, sampling losses, and analyzer calibration errors

A material balance based on inconsistent wet and dry flow rates can produce apparently impossible conversions.

Identify the Limiting Reactant

Compare stoichiometric feed capacities

For WGS, one mole of CO reacts with one mole of water. The maximum extent permitted by each reactant is therefore

[ \dot{\xi}_{\max,\mathrm{CO}}

\frac{F_{\mathrm{CO,in}}}{1} ]

and

[ \dot{\xi}_{\max,\mathrm{H_2O}}

\frac{F_{\mathrm{H_2O,in}}}{1} ]

The limiting reactant is the species with the smaller maximum extent:

[ \dot{\xi}_{\max}

\min\left( F_{\mathrm{CO,in}}, F_{\mathrm{H_2O,in}} \right) ]

Thus, if the steam-to-CO molar ratio is below one, steam is stoichiometrically limiting. If it is above one, CO is limiting.

Distinguish limiting from excess

A reactant is limiting because it has the smaller stoichiometric feed capacity, not because it must be completely consumed in the reactor.

For example, CO may be the limiting reactant even when the catalyst converts only 70% of the inlet CO. The remaining 30% simply exits unreacted.

Use feed ratios carefully

The steam-to-CO ratio is commonly written as

[ S/C

\frac{F_{\mathrm{H_2O,in}}}{F_{\mathrm{CO,in}}} ]

For the WGS reaction:

(S/C=1): stoichiometric feed
(S/C>1): steam excess
(S/C<1): CO excess and steam stoichiometrically limiting

In practice, steam is frequently operated in excess to reduce the likelihood of carbon deposition and to support high CO conversion. However, excess steam increases heating and condensation requirements.

Calculate the Extent of Reaction

Define extent from species changes

For a single WGS reaction, the extent rate is calculated from any species:

[ \dot{\xi}

\frac{F_{i,\text{out}}-F_{i,\text{in}}}{\nu_i} ]

For the WGS stoichiometry, this gives:

[ \dot{\xi}

F_{\mathrm{CO,in}}-F_{\mathrm{CO,out}} ]

[ \dot{\xi}

F_{\mathrm{H_2O,in}}-F_{\mathrm{H_2O,out}} ]

[ \dot{\xi}

F_{\mathrm{CO_2,out}}-F_{\mathrm{CO_2,in}} ]

[ \dot{\xi}

F_{\mathrm{H_2,out}}-F_{\mathrm{H_2,in}} ]

The extent rate has units of molar flow, such as mol/s. For a fixed quantity rather than a continuous flow, use:

[ \xi

\frac{n_{i,\text{out}}-n_{i,\text{in}}}{\nu_i} ]

Use multiple species as a consistency check

For an ideal single-reaction system, all four calculations should produce approximately the same extent.

For example:

[ F_{\mathrm{CO,in}}-F_{\mathrm{CO,out}} \approx F_{\mathrm{CO_2,out}}-F_{\mathrm{CO_2,in}} ]

and

[ F_{\mathrm{CO,in}}-F_{\mathrm{CO,out}} \approx F_{\mathrm{H_2,out}}-F_{\mathrm{H_2,in}} ]

Significant disagreement indicates measurement error, unaccounted condensation, an unsteady operating condition, or additional reactions.

Interpret extent physically

The extent represents the number of reaction-equivalent moles that have occurred. One mole of extent corresponds to:

One mole of CO consumed
One mole of water consumed
One mole of CO₂ formed
One mole of H₂ formed

This makes extent especially useful for checking the complete reaction stoichiometry rather than relying on only one analyzer measurement.

Calculate Conversion

Conversion based on the limiting reactant

The fractional conversion of species (i) is

[ X_i

\frac{F_{i,\text{in}}-F_{i,\text{out}}} {F_{i,\text{in}}} ]

For the limiting reactant, this can also be written as

[ X_{\text{lim}}

\frac{\dot{\xi}} {\dot{\xi}_{\max}} ]

For CO as the limiting reactant:

[ X_{\mathrm{CO}}

\frac{F_{\mathrm{CO,in}}-F_{\mathrm{CO,out}}} {F_{\mathrm{CO,in}}}

\frac{\dot{\xi}}{F_{\mathrm{CO,in}}} ]

For steam as the limiting reactant:

[ X_{\mathrm{H_2O}}

\frac{F_{\mathrm{H_2O,in}}-F_{\mathrm{H_2O,out}}} {F_{\mathrm{H_2O,in}}}

\frac{\dot{\xi}}{F_{\mathrm{H_2O,in}}} ]

Percentage conversion is simply:

[ X_{\text{lim}}(%)

100X_{\text{lim}} ]

Conversion depends on the chosen basis

CO conversion and steam conversion are not necessarily equal when the feed is not stoichiometric.

For example, with excess steam, the same reaction extent is divided by a larger inlet steam flow. Therefore, steam conversion will be lower than CO conversion even though both are describing the same WGS reaction.

Always report the basis explicitly, such as CO conversion or limiting-reactant conversion.

Do not confuse conversion with hydrogen yield

Hydrogen production is related to extent, but hydrogen yield may use a different denominator. For example, hydrogen formed per mole of inlet CO is

[ Y_{\mathrm{H_2/CO}}

\frac{F_{\mathrm{H_2,out}}-F_{\mathrm{H_2,in}}} {F_{\mathrm{CO,in}}} ]

If no hydrogen enters with the feed and only WGS occurs, this value is equal to the CO conversion. In a real plant with hydrogen in the feed or side reactions, they are not necessarily equal.

Apply the Equations to Pilot-Plant Measurements

Example calculation

Suppose the measured inlet flows are:

[ F_{\mathrm{CO,in}}=10\ \mathrm{mol/s} ]

[ F_{\mathrm{H_2O,in}}=15\ \mathrm{mol/s} ]

The steam-to-CO ratio is 1.5, so CO is the stoichiometrically limiting reactant.

If the outlet CO flow is (2\ \mathrm{mol/s}), then:

[ \dot{\xi}

10-2

8\ \mathrm{mol/s} ]

The CO conversion is:

[ X_{\mathrm{CO}}

\frac{10-2}{10}

0.80 ]

or:

[ X_{\mathrm{CO}}=80% ]

The ideal single-reaction prediction would be:

[ F_{\mathrm{CO_2,out}}=F_{\mathrm{CO_2,in}}+8 ]

[ F_{\mathrm{H_2,out}}=F_{\mathrm{H_2,in}}+8 ]

[ F_{\mathrm{H_2O,out}}=15-8=7\ \mathrm{mol/s} ]

These predicted values should be compared with the measured outlet flows.

Use an overall reaction balance

For a steady-state reactor without accumulation:

[ F_{i,\text{out}}

F_{i,\text{in}}+\nu_i\dot{\xi} ]

For each WGS species:

[ F_{\mathrm{CO,out}}

F_{\mathrm{CO,in}}-\dot{\xi} ]

[ F_{\mathrm{H_2O,out}}

F_{\mathrm{H_2O,in}}-\dot{\xi} ]

[ F_{\mathrm{CO_2,out}}

F_{\mathrm{CO_2,in}}+\dot{\xi} ]

[ F_{\mathrm{H_2,out}}

F_{\mathrm{H_2,in}}+\dot{\xi} ]

This relationship allows the operator to calculate an extent from each measured component and diagnose inconsistencies.

Compare performance with equilibrium separately

The measured conversion is the actual reactor conversion. It reflects catalyst activity, temperature, residence time, pressure, feed ratio, heat transfer, and mass-transfer effects.

For the reversible WGS reaction, equilibrium conversion provides a thermodynamic benchmark. It should not be confused with the limiting-reactant calculation or with the measured extent.

At a specified temperature and composition, the equilibrium constant can be expressed using species activities or partial pressures:

[ K

\frac{a_{\mathrm{CO_2}}a_{\mathrm{H_2}}} {a_{\mathrm{CO}}a_{\mathrm{H_2O}}} ]

For an ideal-gas analysis, activities may be approximated using partial pressures. The equilibrium comparison is meaningful only when temperature, pressure, feed composition, and water basis are defined consistently.

Understanding the Trade-offs

A high conversion is not automatically the best operating point

Higher conversion can require greater catalyst volume, longer residence time, higher steam ratios, or more severe operating conditions.

The best operating point may instead balance conversion against hydrogen productivity, energy use, pressure drop, catalyst life, and downstream separation requirements.

Excess steam improves some objectives but increases burden

Steam excess can help drive the reaction toward hydrogen and reduce carbon-formation risk. It also increases the duty required to generate and heat steam, enlarges the wet gas flow, and may create condensation problems during sampling.

The steam ratio should therefore be reported alongside conversion.

A single-reaction extent can be misleading

The WGS equations assume that the measured CO and hydrogen changes are caused only by WGS. In a practical catalyst system, methanation, methane reforming, carbon deposition, oxidation, or other reactions may occur.

If side reactions are present, one extent cannot represent the entire reactor. Use separate reaction extents, selectivity calculations, and a complete elemental balance.

Measurement error can dominate the result

Small errors in steam measurement or dry-basis gas analysis can produce large errors in calculated water conversion and extent.

Use calibrated analyzers, stable flow meters, condensate collection, inert-tracer checks where appropriate, and repeated steady-state measurements before accepting the result.

How to Apply This to Your Pilot Plant

The following workflow provides a defensible evaluation:

If your primary focus is identifying the limiting reactant: Calculate (F_{i,\mathrm{in}}/|\nu_i|) for CO and H₂O; the smaller value identifies the stoichiometrically limiting species.
If your primary focus is conversion: Measure inlet and outlet flows of the selected basis species and calculate (X_i=(F_{i,\mathrm{in}}-F_{i,\mathrm{out}})/F_{i,\mathrm{in}}).
If your primary focus is reaction extent: Calculate (\dot{\xi}=(F_{i,\mathrm{out}}-F_{i,\mathrm{in}})/\nu_i) independently from CO, H₂O, CO₂, and H₂, then compare the results.
If your primary focus is hydrogen generation: Use the net hydrogen increase, (F_{\mathrm{H_2,out}}-F_{\mathrm{H_2,in}}), and verify that it agrees with the WGS extent.
If your primary focus is scale-up or optimization: Compare measured conversion with equilibrium conversion while also tracking steam ratio, temperature, residence time, catalyst condition, energy demand, and material-balance closure.

A sound pilot-plant evaluation uses limiting-reactant identification, conversion, extent, and material-balance closure together, rather than treating any single value as a complete measure of reactor performance.

Summary Table:

Evaluation	Formula	Key interpretation
Limiting reactant	(\dot{\xi}{max}=\min(F{CO,in},F_{H_2O,in}))	The reactant with the smaller stoichiometric feed capacity
Steam-to-CO ratio	(S/C=F_{H_2O,in}/F_{CO,in})	(S/C>1): steam excess; (S/C<1): steam-limited
Reaction extent	(\dot{\xi}=(F_{i,out}-F_{i,in})/\nu_i)	Reaction progress calculated from any WGS species
Reactant conversion	(X_i=(F_{i,in}-F_{i,out})/F_{i,in})	Fraction of a selected reactant consumed
Hydrogen generation	(\Delta F_{H_2}=F_{H_2,out}-F_{H_2,in})	Net hydrogen formed; compare with WGS extent
Balance validation	Compare extents from CO, H₂O, CO₂, and H₂	Differences may indicate measurement error, condensation, or side reactions

Improve Your WGS Pilot-Plant Evaluation

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