The standard process is a stoichiometric, component-by-component mass balance built around a defined feed basis and reaction conversion. For a gas-solid thermal decomposition reaction, engineers first balance the reaction and identify the molecular weights, then calculate feed moles from the material composition. Stoichiometry and conversion determine the solid products and evolved gas, after which all streams are converted to mass flow rates and checked against an overall mass balance.

Core takeaway: Treat the reactor as a system of conserved components, usually chemical elements or independently reacting species. Account separately for reactive solids, inert solids, generated gases, unreacted feed, moisture, and measured losses so that the calculated outlet streams satisfy both stoichiometry and conservation of mass.

Define the Thermal Decomposition System

Identify the Feed and Outlet Streams

List every material entering and leaving the pilot plant reactor.

Typical feed streams include:

Raw mineral or chemical solid
Inert impurities or gangue
Carrier gas or purge gas
Moisture
Fuel or combustion air, when heating is integrated with the reactor

Typical outlet streams include:

Decomposed solid product
Unreacted feed
Evolved decomposition gas
Inert solids
Dust or entrained fines
Moisture and carrier gas

The balance boundary must be clear. A balance around only the reactor differs from one around the entire pilot plant, which may also include cyclones, condensers, gas scrubbers, or product collectors.

Determine Whether the Process Is Steady or Transient

For continuous pilot operation, balances are commonly written in mass flow units such as kg/h or molar flow units such as kmol/h.

For a batch experiment, use total quantities such as kilograms, kilomoles, and total gas evolved during the run. The same stoichiometric method applies, but accumulation inside the reactor must be considered during operation.

Establish the Reaction Basis

Write Balanced Decomposition Reactions

The first calculation step is to write a balanced equation for every relevant decomposition reaction.

For calcium carbonate calcination:

[ \mathrm{CaCO_3(s) \rightarrow CaO(s) + CO_2(g)} ]

For a general carbonate:

[ \mathrm{MCO_3(s) \rightarrow MO(s) + CO_2(g)} ]

If several minerals decompose, write each independent reaction separately. For example, a feed may contain multiple carbonate species that produce different solid oxides and carbon dioxide.

Obtain Molecular Weights

Use molecular weights consistent with the units used in the balance.

For calcium carbonate calcination:

[ M_{\mathrm{CaCO_3}} = 100.09\ \mathrm{kg/kmol} ]

[ M_{\mathrm{CaO}} = 56.08\ \mathrm{kg/kmol} ]

[ M_{\mathrm{CO_2}} = 44.01\ \mathrm{kg/kmol} ]

Thus, one kilomole of calcium carbonate produces one kilomole each of calcium oxide and carbon dioxide.

Select a Calculation Basis

Choose a basis that makes the feed composition easy to interpret.

Common choices are:

100 kg of raw feed
1,000 kg of raw feed
1 kmol of the key reactive component
A measured pilot-plant feed rate, such as 100 kg/h

A 100 kg basis is particularly useful when the feed contains mass percentages because each percentage directly represents kilograms of that component.

Calculate the Feed Composition

Separate Reactive and Inert Material

Suppose a 100 kg feed contains:

85 wt% calcium carbonate
10 wt% inert solids
5 wt% moisture

The component masses are:

[ m_{\mathrm{CaCO_3}} = 85\ \mathrm{kg} ]

[ m_{\mathrm{inert}} = 10\ \mathrm{kg} ]

[ m_{\mathrm{water}} = 5\ \mathrm{kg} ]

Only the calcium carbonate participates in the stated decomposition reaction. The inert solids pass through the reactor unless another reaction or separation mechanism is included.

Convert Reactive Feed to Moles

The reactive calcium carbonate amount is:

[ n_{\mathrm{CaCO_3}} = \frac{85\ \mathrm{kg}}{100.09\ \mathrm{kg/kmol}} = 0.849\ \mathrm{kmol} ]

This molar quantity is the basis for calculating theoretical product and gas generation.

Include Feed Purity and Mineralogical Composition

If the supplied material is not pure, calculate the amount of each reactive compound from its assay or mineral analysis.

A feed described as “90% carbonate” does not automatically mean 90% of one specific carbonate. The balance should identify whether the reactive fraction is calcium carbonate, magnesium carbonate, hydrated carbonate, or a mixture.

Apply Conversion and Stoichiometry

Calculate the Reaction Extent

If the fractional conversion of calcium carbonate is (X), the reacted amount is:

[ n_{\mathrm{reacted}} = X n_{\mathrm{CaCO_3},0} ]

For complete conversion, (X=1). For an 80% conversion:

[ n_{\mathrm{reacted}} = 0.80(0.849) = 0.679\ \mathrm{kmol} ]

The reaction extent is a convenient way to connect feed conversion directly to outlet quantities.

Determine Unreacted Solid

The unreacted calcium carbonate is:

[ n_{\mathrm{CaCO_3,out}} = n_{\mathrm{CaCO_3},0} - n_{\mathrm{reacted}} ]

For the example:

[ n_{\mathrm{CaCO_3,out}} = 0.849 - 0.679 = 0.170\ \mathrm{kmol} ]

Its mass is:

[ m_{\mathrm{CaCO_3,out}} = 0.170(100.09) \approx 17.0\ \mathrm{kg} ]

Determine Solid Product Formation

The stoichiometric coefficient of calcium oxide is one, so:

[ n_{\mathrm{CaO,out}} = n_{\mathrm{reacted}} ]

Therefore:

[ m_{\mathrm{CaO,out}} = 0.679(56.08) \approx 38.1\ \mathrm{kg} ]

Determine Gas Evolution

The stoichiometric coefficient of carbon dioxide is also one:

[ n_{\mathrm{CO_2,out}} = n_{\mathrm{reacted}} ]

Thus:

[ m_{\mathrm{CO_2,out}} = 0.679(44.01) \approx 29.9\ \mathrm{kg} ]

The gas generation rate is obtained in the same way when the feed is expressed as a continuous mass flow rate.

Complete the Overall Material Balance

Account for Inerts and Volatiles

The inert solids remain at approximately 10 kg on the selected basis, assuming no entrainment or loss.

The 5 kg of feed moisture may leave as water vapor unless it condenses or participates in another reaction. It should therefore be included in the gas outlet or in a separate condensed-water stream.

The calculated outlets are approximately:

17.0 kg unreacted calcium carbonate
38.1 kg calcium oxide
10.0 kg inert solids
29.9 kg carbon dioxide
5.0 kg water vapor

The total is:

[ 17.0 + 38.1 + 10.0 + 29.9 + 5.0 = 100.0\ \mathrm{kg} ]

This confirms the total material balance for the idealized system.

Use the General Material Balance Equation

For any component:

[ \text{Input} - \text{Output} + \text{Generation}

\text{Consumption} = \text{Accumulation} ]

For a steady-state reactor, accumulation is zero. For a nonreactive solid:

[ \dot{m}_{\mathrm{inert,in}}

\dot{m}_{\mathrm{inert,out}} ]

For a reactive species, the reaction terms are determined by stoichiometry and conversion.

Check Elemental Conservation

An overall mass check is necessary but may conceal incorrect species assignments. Elemental balances provide a stronger verification.

For calcium carbonate calcination, independently check calcium, carbon, and oxygen:

[ \text{Ca in} = \text{Ca in unreacted CaCO}_3

\text{Ca in CaO} ]

[ \text{C in} = \text{C in unreacted CaCO}_3

\text{C in CO}_2 ]

[ \text{O in} = \text{O in all outlet species} ]

For complex mineral feeds, elemental balances are often more reliable than attempting to track every mineral formula.

Extend the Method to Complex Pilot Plants

Track Multiple Independent Reactions

When several decomposition or reduction reactions occur, represent the outlet species using reaction extents:

[ N_i = N_{i,0} + \sum_j \nu_{ij}\xi_j ]

where:

(N_i) is the outlet amount of species (i)
(N_{i,0}) is its feed amount
(\nu_{ij}) is the stoichiometric coefficient of species (i) in reaction (j)
(\xi_j) is the extent of reaction (j)

Using independent reactions avoids redundant equations and provides a systematic method for multicomponent gas-solid systems.

Use Elemental Balances for Pyrometallurgical Systems

For ores, fluxes, coke, and complex oxide feeds, the unknown streams may be calculated by balancing elements such as:

Iron
Manganese
Silicon
Carbon
Oxygen
Nitrogen

Nonvolatile oxides may be assigned to slag, while carbon may partition among molten metal, carbon monoxide, carbon dioxide, and unreacted solid carbon. This approach remains valid even when the complete mineralogical structure is uncertain.

Reconcile Calculations with Pilot Data

The theoretical balance should be compared with measured inlet and outlet flow rates.

Useful measurements include:

Solid feed and product mass
Product moisture
Gas flow rate
Gas composition
Dust collection rate
Residual reactive material
Carbon dioxide concentration

Mass balance reconciliation can reveal gas leaks, solids entrainment, incomplete collection, moisture errors, or incorrect assumptions about conversion.

Understanding the Trade-offs

Theoretical Conversion Is Not Actual Conversion

A stoichiometric calculation gives the products expected at a specified conversion. It does not establish the conversion itself.

Actual conversion depends on temperature, residence time, particle size, heat transfer, gas composition, pressure, and reactor mixing. Pilot data or a kinetic model is required to determine whether the assumed conversion is realistic.

Gas Flow Depends on Basis and Composition

Carbon dioxide generated by decomposition is only part of the total outlet gas when purge gas, air, steam, or combustion products are present.

For example, total wet gas may include carbon dioxide, water vapor, nitrogen, oxygen, and unreacted steam. Gas recovery calculations must specify whether the reported flow is wet, dry, or corrected for a reference temperature and pressure.

Inerts Do Not Always Remain in the Reactor

The assumption that inert solids pass quantitatively to the solid product can fail when the reactor produces dust or fines.

A practical pilot-plant balance may therefore require separate outlet streams for product solids, collected dust, and unaccounted solids. The inert-component balance is especially useful for estimating these losses.

Mass Closure Does Not Prove Process Accuracy

A balance can close numerically while the chemistry is wrong if compensating measurement errors exist.

The final analysis should therefore combine total mass closure, elemental closure, individual species balances, and comparison with measured gas composition or solid assays.

Making the Right Choice for Your Goal

Use the balance method that matches the question being answered.

If your primary focus is theoretical product yield: Choose a defined feed basis, apply complete stoichiometry, and calculate solid and gas products from the assumed conversion.
If your primary focus is pilot-plant gas recovery: Include carrier gas, moisture, gas composition, gas losses, and the distinction between wet and dry gas flow.
If your primary focus is reactor performance: Use measured feed and outlet streams together with conversion, reaction extent, and component or elemental balances.
If your primary focus is a complex mineral feed: Balance conserved elements and use independent reaction extents rather than relying on a single overall decomposition equation.
If your primary focus is troubleshooting: Reconcile individual component balances to identify dust losses, leaks, unreacted solids, sampling errors, or unmeasured moisture.

A reliable gas-solid decomposition balance connects feed composition, reaction stoichiometry, conversion, measured pilot data, and elemental conservation in one consistent calculation.

Summary Table:

Balance step	Key calculation	Purpose
Define system	Identify feed, product, gas, inert, moisture, and loss streams	Establish the balance boundary
Select basis	Use 100 kg feed, 1 kmol reactant, or a measured flow rate	Set a consistent calculation basis
Balance reactions	Write equations and obtain molecular weights	Determine stoichiometric relationships
Calculate conversion	Apply reaction extent and conversion to reactive feed	Find reacted and unreacted material
Determine outlets	Calculate solid products, evolved gases, moisture, and inerts	Quantify every output stream
Verify closure	Perform total-mass and elemental balances	Detect losses, leaks, or incorrect assumptions
Reconcile data	Compare calculations with gas, solids, moisture, and dust measurements	Assess actual pilot-plant performance

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