This document summarizes the principal equations used for engineering diagnosis of desulfurization systems.
The equations support:
- identification of reaction-, transport-, diffusion-, adsorption-, hydrodynamic-, and separation-limited behavior;
- comparison of intrinsic and apparent rates;
- screening of porous-catalyst or adsorbent limitations;
- assessment of hydrodynamic process intensification;
- sulfur-balance evaluation;
- energy-normalized comparison with a defined reference case;
- planning of system-specific validation.
Evidence status
- The individual equations are established engineering relationships when used within their stated assumptions.
- Their integration, parameterization, threshold selection, and use in this repository constitute an E3 engineering screening framework and executable research prototype.
- The equations are not a universally validated industrial design model. Real applications require traceable data, dimensional consistency, uncertainty analysis, sulfur closure, and independent validation.
Unless otherwise stated:
- concentrations are expressed on a consistent molar or mass basis;
- rates are expressed as positive removal rates;
- time is expressed in seconds or another explicitly stated unit;
- absolute temperature is used in kinetic expressions;
- absolute pressure is used in vapor-pressure and cavitation calculations;
- all quantities in a dimensionless ratio must use compatible units and bases.
A symbol such as
- a first-order volumetric rate coefficient,
$\mathrm{s^{-1}}$ ; - a surface-based coefficient;
- a catalyst-mass-based coefficient;
- a higher-order kinetic coefficient with concentration-dependent units;
- an overall apparent coefficient incorporating reaction and transport effects.
Coefficients on different bases must not be combined without conversion.
| Symbol | Meaning | Representative unit |
|---|---|---|
| sulfur-species concentration |
|
|
| oxidant concentration | same concentration basis used in the rate law | |
| volumetric sulfur-removal rate | concentration per time | |
| apparent pseudo-first-order coefficient | ||
| liquid-side mass-transfer coefficient | ||
| interfacial area per reactor volume | ||
| volumetric mass-transfer coefficient | ||
| porous-particle radius | ||
| effective diffusivity | ||
| Thiele modulus | dimensionless | |
| internal effectiveness factor | dimensionless | |
| energy |
|
|
| sulfur mass |
|
For a constant-volume batch system, sulfur conversion may be written as:
where:
-
$C_{S,0}$ is the initial sulfur concentration; -
$C_S$ is the sulfur concentration at the specified time; -
$X_S$ is dimensionless.
For a continuous system or a process with changing phase volume, sulfur molar or mass flow should be used rather than concentration alone:
An equivalent mass-flow expression may also be used.
A decrease in sulfur concentration in one measured phase can result from:
- chemical conversion;
- transfer to another phase;
- adsorption;
- precipitation;
- sampling bias;
- dilution;
- analytical loss.
A concentration decrease in one phase is not automatically equivalent to complete sulfur removal.
For many screening calculations, sulfur disappearance is represented by:
where:
-
$r_S$ is the volumetric sulfur-species removal rate; -
$k_{\mathrm{obs}}$ is an observed or apparent first-order coefficient; -
$C_S$ is the sulfur-species concentration.
Dimensional consistency requires:
For a constant-volume batch system with constant
Integration gives:
The concentration profile is:
The corresponding sulfur conversion is:
This integrated expression assumes:
- constant system volume;
- no sulfur inflow or outflow during the batch interval;
- constant apparent rate coefficient;
- no unmodeled phase transfer;
- no significant sampling-volume correction;
- a consistent analytical basis.
The fitted
- intrinsic reaction kinetics;
- external mass transfer;
- internal diffusion;
- mixing;
- catalyst accessibility;
- adsorption;
- oxidant activation;
- hydrodynamic intensification;
- simultaneous downstream removal.
It must not automatically be interpreted as an intrinsic kinetic coefficient.
A generalized oxidative-desulfurization rate may be represented as:
where:
-
$k$ is the intrinsic or apparent rate coefficient; -
$C_S$ is sulfur-compound concentration; -
$C_{\mathrm{ox}}$ is oxidant concentration; -
$n$ is the apparent order with respect to sulfur; -
$m$ is the apparent order with respect to oxidant.
If
When oxidant concentration remains approximately constant:
Only when
The apparent coefficient may still depend on:
- catalyst concentration;
- phase ratio;
- interfacial area;
- temperature;
- mixing;
- oxidant decomposition;
- oxidant utilization;
- sulfur speciation.
The pseudo-first-order reduction is not justified when oxidant concentration changes materially, catalyst activity changes, mass transfer evolves, or the reaction order varies during the fitted interval.
The temperature dependence of a kinetic coefficient may be represented by:
where:
-
$A$ is the pre-exponential factor; -
$E_a$ is the apparent or intrinsic activation energy; -
$R$ is the universal gas constant; -
$T$ is absolute temperature in kelvin.
The linearized form is:
The units of
An apparent activation energy can be affected by:
- external mass transfer;
- internal diffusion;
- adsorption equilibrium;
- catalyst deactivation;
- changing phase behavior;
- temperature-dependent viscosity;
- temperature-dependent mixing;
- analytical and fitting limitations.
A low apparent activation energy may indicate transport influence, but it is not conclusive proof of a specific limitation.
For gas–liquid or liquid–liquid transfer on a liquid concentration basis:
where:
-
$r_{\mathrm{mt}}$ is the volumetric transfer rate; -
$k_L$ is the liquid-side mass-transfer coefficient; -
$a$ is interfacial area per reactor volume; -
$k_La$ is the volumetric mass-transfer coefficient; -
$C^*$ is the equilibrium concentration corresponding to the interfacial condition; -
$C$ is the bulk-phase concentration.
Dimensional consistency requires:
This relationship is relevant to:
- hydrogen-sulfide absorption;
- oxygen or oxidant transfer;
- oxidative desulfurization;
- multiphase catalytic oxidation;
- liquid–liquid extraction-assisted systems.
The volumetric coefficient combines two contributions:
An observed change in
For a pressure-based gas-side driving force:
where:
-
$N_{H_2S}$ is the volumetric hydrogen-sulfide transfer rate; -
$K_Ga$ is an overall volumetric gas-side transfer coefficient; -
$P_{H_2S}$ is the bulk gas-phase hydrogen-sulfide partial pressure; -
$P_{H_2S}^*$ is the equilibrium partial pressure corresponding to the liquid-side condition.
When the rate is expressed in
Observed performance may depend on:
- solvent chemistry;
- chemical enhancement;
- interfacial area;
- circulation rate;
- gas velocity;
- temperature;
- total pressure;
- contactor geometry;
- equilibrium representation.
Pressure-based and concentration-based transfer coefficients must not be combined without an explicit equilibrium relation and unit conversion.
When reaction and mass transfer can both be represented by compatible first-order coefficients:
Approximate screening interpretation:
-
$Da_{\mathrm{diag}}\ll1$ : reaction-controlled tendency; -
$Da_{\mathrm{diag}}\approx1$ : coupled reaction–transport tendency; -
$Da_{\mathrm{diag}}\gg1$ : mass-transfer-controlled tendency.
The characteristic reaction time is:
The characteristic mass-transfer time is:
Therefore:
The ratio is dimensionless only when
For higher-order kinetics, complex reactor models, reactive absorption, or changing interfacial area, the Damköhler number must be constructed from consistent characteristic rates or timescales.
Its exact definition depends on:
- reactor type;
- rate law;
- characteristic length;
- phase system;
- geometry;
- selected concentration basis.
This repository therefore uses the ratio as an illustrative diagnostic quantity, not as a universal Damköhler-number definition.
For an isothermal spherical porous particle with a first-order volumetric reaction:
where:
-
$\phi$ is the Thiele modulus; -
$R_p$ is particle radius; -
$k_v$ is a first-order coefficient on a compatible volumetric catalyst basis; -
$D_{\mathrm{eff}}$ is effective diffusivity.
Dimensional consistency is:
If the rate coefficient is expressed per catalyst mass, external surface area, or active-site inventory, it must be converted to a compatible volumetric basis before use.
-
$\phi\ll1$ : weak internal diffusion influence; -
$\phi\approx1$ : reaction and diffusion are coupled; -
$\phi\gg1$ : substantial internal diffusion influence.
The expression assumes:
- spherical particle geometry;
- first-order reaction;
- constant effective diffusivity;
- isothermal behavior;
- uniform porous structure;
- a defined external particle concentration.
The internal effectiveness factor is defined as:
For a first-order reaction in an isothermal spherical particle:
An equivalent computational form is:
As
A numerically stable small-$\phi$ expansion is:
For large
The leading large-$\phi$ behavior is:
-
$\eta\approx1$ : most of the particle is effectively utilized; -
$0<\eta<1$ : internal gradients reduce catalyst utilization; -
$\eta\ll1$ : substantial internal diffusion resistance is present.
A computational implementation should:
- reject negative
$\phi$ ; - return
$\eta=1$ at$\phi=0$ ; - use the series expansion for sufficiently small
$\phi$ ; - avoid subtracting nearly equal floating-point quantities;
- verify that
$0<\eta\leq1$ for the stated model.
For idealized finite-capacity adsorption:
where:
-
$q_e$ is equilibrium adsorption capacity; -
$q_{\max}$ is maximum monolayer capacity; -
$K_L$ is the Langmuir affinity coefficient; -
$C_e$ is equilibrium sulfur concentration.
Dimensional consistency requires:
Therefore,
When
When
The Langmuir model assumes:
- a homogeneous adsorption surface;
- finite equivalent adsorption sites;
- monolayer adsorption;
- no interaction among adsorbed species.
Agreement with the equation does not prove that these assumptions describe the actual adsorption mechanism.
An empirical heterogeneous-surface relation is:
where:
-
$K_F$ is the Freundlich capacity coefficient; -
$n_F$ is an empirical heterogeneity parameter.
The units of
- the units of
$q_e$ ; - the units of
$C_e$ ; - the value of
$1/n_F$ .
Values of
The Freundlich equation has no finite saturation capacity. Extrapolation outside the fitted concentration range may therefore become physically unrealistic.
Agreement with the model is not proof of a specific adsorption mechanism.
A commonly used empirical adsorption-kinetics expression is:
where:
-
$q$ is loading at time$t$ ; -
$q_e$ is the fitted equilibrium loading; -
$k_2$ is the pseudo-second-order coefficient.
If
For
Good agreement with this model is not, by itself, proof of chemisorption.
Mechanistic interpretation requires complementary evidence such as:
- equilibrium measurements;
- temperature dependence;
- spectroscopy;
- diffusion analysis;
- competitive adsorption;
- regeneration behavior.
For two linear sequential resistances expressed on a common basis:
The equivalent overall coefficient is:
When reaction is much slower than mass transfer:
When mass transfer is much slower than reaction:
The additive-resistance expression is appropriate only when:
- the resistances are sequential;
- the model is linear or suitably linearized;
- coefficients use compatible units;
- driving forces are consistently defined;
- strong coupling does not invalidate independent resistance addition.
It is a conceptual screening relationship, not a universal reactor equation.
For hydrodynamic-cavitation-assisted desulfurization:
where:
-
$k_{\mathrm{app,HC}}$ is the apparent coefficient under cavitation-assisted conditions; -
$k_{\mathrm{app,ref}}$ is the apparent coefficient for a defined reference case.
Interpretation:
-
$F_{\mathrm{app}}>1$ : higher apparent rate than the reference; -
$F_{\mathrm{app}}=1$ : no apparent rate difference; -
$F_{\mathrm{app}}<1$ : lower apparent rate than the reference.
The two coefficients should be obtained using comparable:
- feeds and sulfur speciation;
- temperature;
- pressure;
- oxidant or hydrogen dose;
- catalyst or adsorbent loading;
- phase ratio;
- analytical method;
- fitting interval;
- product and separation boundary.
An enhancement factor greater than one does not prove industrial usefulness. The assessment must also consider:
- additional energy;
- pressure drop;
- oxidant consumption;
- erosion;
- corrosion;
- fouling;
- emulsion formation;
- downstream separation;
- product quality;
- reliability;
- maintenance;
- scale-up transferability.
A commonly used hydrodynamic cavitation number is:
where:
-
$p_{\mathrm{ref}}$ is a stated downstream, recovery, or reference absolute pressure; -
$p_v$ is vapor pressure at the local fluid temperature; -
$\rho$ is fluid density; -
$v$ is the stated characteristic velocity.
Dimensional consistency is:
Every reported value should identify:
- the pressure-measurement location;
- whether pressure is absolute;
- the velocity definition;
- the flow area;
- fluid temperature;
- density;
- vapor-pressure basis;
- device geometry;
- flow rate.
A lower cavitation number may indicate stronger cavitation tendency within a defined setup, but equal values do not guarantee equal:
- cavity dynamics;
- collapse intensity;
- residence pattern;
- erosion risk;
- chemical effect;
- scale-up performance.
The cavitation number is a hydrodynamic descriptor, not a universal process-performance correlation.
For comparison with a defined reference case:
Interpretation:
-
$\Delta m_S>0$ : positive incremental removal; -
$\Delta m_S=0$ : no incremental removal; -
$\Delta m_S<0$ : poorer removal than the reference.
Both removal quantities must use the same:
- feed basis;
- throughput;
- treatment duration;
- product boundary;
- analytical method.
Gross removal in the intensified case must not be substituted for incremental removal when evaluating the benefit attributable to intensification.
The incremental energy associated with intensification may be defined as:
The energy boundary should specify, as applicable:
- pump energy;
- measured motor input;
- heating;
- cooling;
- oxidant generation;
- recirculation;
- separation;
- auxiliary equipment.
The following quantities are not interchangeable:
- nameplate power;
- measured electrical input;
- shaft power;
- hydraulic power.
For positive incremental energy input:
Representative units include:
or:
The reciprocal specific energy demand is:
Representative units include:
or:
For positive finite quantities expressed in reciprocal units:
- If
$\Delta E=0$ ,$EN_S$ is undefined. - If
$\Delta m_S=0$ ,$SEC_S$ is undefined. - If
$\Delta m_S<0$ , the intensified case provides negative incremental benefit. - If
$\Delta E<0$ , the comparison requires explicit interpretation. - Uncertainty should be reported when either numerator is calculated by subtracting similar measured values.
A high value of
A stoichiometric oxidant-utilization efficiency may be defined as:
The stoichiometric oxidant requirement is:
where:
-
$n_{\mathrm{ox,supplied}}$ is the supplied oxidant quantity; -
$n_{S,\mathrm{converted,target}}$ is the quantity of target sulfur converted; -
$\nu_{\mathrm{ox}}$ is the stoichiometric oxidant requirement per mole of target sulfur.
A rigorous definition must specify:
- oxidant identity;
- oxidant purity;
- stoichiometric basis;
- target sulfur species;
- target oxidation state;
- side reactions;
- decomposition;
- residual oxidant;
- analytical uncertainty.
For a consistent closed basis:
A calculated value above one indicates that the stoichiometric basis, sulfur analysis, oxidant analysis, or assumed reaction pathway requires review.
For transformed sulfur entering a downstream separation step:
For a consistent nonnegative basis:
The numerator represents sulfur actually removed from the final product phase.
The denominator represents transformed sulfur entering the defined separation boundary.
The final product sulfur concentration should be measured after the complete reaction and separation sequence.
Sulfur conversion is not necessarily equivalent to sulfur removal.
For a process evaluated over a defined accounting interval:
For a closed batch system:
A relative sulfur-balance closure error may be expressed as:
The sulfur balance should include, where relevant:
- treated hydrocarbon or gas;
- aqueous phase;
- extractant or solvent;
- adsorbent;
- catalyst deposits;
- precipitated solids;
- purge and vent streams;
- sampling losses;
- identified sulfur products.
A closed sulfur balance demonstrates consistency of sulfur accounting within the defined boundary and analytical uncertainty. It does not establish a reaction mechanism.
For a ratio:
with independent standard uncertainties
This approximation may be used for:
- apparent enhancement factors;
- energy-normalized removal;
- specific energy demand;
- oxidant-utilization efficiency;
- separation efficiency.
The linearized relation may become unreliable when:
- the denominator approaches zero;
- uncertainty is large;
- variables are correlated;
- distributions are strongly non-normal;
- the numerator or denominator is obtained from a small difference between large quantities.
In such cases, a more appropriate propagation method, such as Monte Carlo analysis, should be considered.
No single equation should be used in isolation to claim process superiority.
A technically defensible assessment should combine, where relevant:
- sulfur speciation;
- sulfur-balance closure;
- intrinsic and apparent kinetics;
- interphase mass transfer;
- internal diffusion;
- adsorption and surface phenomena;
- hydrodynamics;
- oxidant or hydrogen utilization;
- downstream separation;
- energy consumption;
- catalyst or adsorbent durability;
- erosion;
- corrosion;
- fouling;
- product quality;
- uncertainty;
- reliability;
- scale-up transferability.
The analysis should determine:
- which mechanism controls overall performance;
- which secondary limitations remain important;
- whether the proposed intervention addresses the controlling limitation;
- whether the complete process improves relative to a defined reference.
Executable implementations should:
- validate that required inputs are finite;
- reject negative concentrations, diffusivities, radii, energies, and times;
- reject zero denominators before evaluating ratios;
- document every unit and coefficient basis;
- use numerically stable limiting expressions;
- distinguish warnings from fatal input errors;
- preserve significant digits consistent with data uncertainty;
- test representative, boundary, and failure cases;
- reproduce published demonstration outputs from version-controlled inputs.
The repository examples are screening demonstrations. They should not silently replace missing pilot or industrial data with assumed values.
| Element | Status | Interpretation |
|---|---|---|
| Individual classical equations | Established within stated assumptions | Require correct geometry, units, rate law, and boundary conditions |
| Repository integration | E3 — Engineering screening framework | Research prototype for transparent diagnosis and comparison |
| Python examples | E3 — Executable demonstrations | Illustrative calculations, not validated industrial models |
| Default parameter values | Illustrative | Not measured pilot or industrial design data |
| Regime thresholds | Screening guidance | Must be justified for the specific system |
| Industrial prediction | Not established | Requires independent validation and scale-up evidence |
Use the equations for:
- transparent engineering screening;
- dimensional checks;
- sensitivity analysis;
- experimental planning;
- regime diagnosis;
- reference-case comparison;
- identification of missing evidence.
Do not use them as:
- universal kinetic correlations;
- final reactor-design calculations;
- guaranteed catalyst or adsorbent performance;
- process-safety calculations;
- emissions-compliance predictions;
- industrial performance warranties.
- Repository README
- Framework summary
- Regime classification
- Validation guidelines
- HDS internal-diffusion example
- ODS reaction–mass-transfer example
- Cavitation-intensification example
- Illustrative parameter table
The equations in this document provide a structured basis for diagnosing coupled desulfurization phenomena.
Their responsible use requires:
- compatible units and coefficient bases;
- explicit assumptions;
- defined system boundaries;
- defined energy boundaries;
- sulfur-balance closure;
- uncertainty assessment;
- comparison with an appropriate reference;
- system-specific validation.
The repository implements these relationships as an E3 engineering screening framework and executable research prototype, not as a universally validated industrial design model.