Chemical Reactors’ Design Equations
1. Reactor Mole Balances.
General Mole Balance Equation (for the species “i”): Eq. 1.1
Table 1.1 Reactor Specific Mole Balances (moles and molar flowrates form)a.
Type Diagram Unsteady State Operation Steady State Operation
Batch
Eq. 1.2a Not Applicable -
Reactor (BR)
Semi-Batch
Reactor Eq. 1.3a Not Applicable -
(SBR)b
Continuous
Stirred Tank
Eq. 1.4a Eq. 1.4b
Reactor
(CSTR)
Plug Flow
Reactor Eq. 1.5a Eq. 1.5b
(PFR)
Packed Bed
Reactor Eq. 1.6a Eq. 1.6b
(PBR)
Well-Mixed
Fluidized Bed
Eq. 1.7a Eq. 1.7b
Reactor
(WM-FBR)
a
This set of reactor design equations is applicable to both multiple and single reaction systems, for both liquid phase reactions and gas phase reactions. It is not applicable to those systems in which reacting phase goes on phase
change (means multiphase flow on process side cannot be handled but multiphase flow on utility side can be handled). For multiple reaction systems there is need to make mole balances on all chemical species involved, for single
reaction systems only molar balance is needed for the limiting reactant.
b
For semi-batch reactors, mole balance must be adapted in function to the feeding and discharge pattern for each chemical species.
Table 1.2 Reactor Specific Mole Balances (conversion form)c.
Type Diagram Unsteady State Operation Steady State Operation
Batch
Eq. 1.2b Not Applicable -
Reactor (BR)
Semi-Batch
Reactor Eq. 1.3b Not Applicable -
(SBR)d
Continuous
Stirred Tank
Eq. 1.4c Eq. 1.4d
Reactor
(CSTR)
Plug Flow
Reactor Eq. 1.5c Eq. 1.5d
(PFR)
Packed Bed
Reactor Eq. 1.6c Eq. 1.6d
(PBR)
Well-Mixed
Fluidized
Eq. 1.7c Eq. 1.7d
Bed Reactor
(WM-FBR)
c
This set of reactor design equations is applicable only to single reaction systems, it is perfectly applicable to liquid phase reactions and also applicable to gas phase reactions for which volume expansion of accumulated fluid during
the course of reaction can be ignored (or considered negligible). It is not applicable to those systems in which reacting phase goes on phase change (means multiphase flow on process side cannot be handled but multiphase flow on
utility side can be handled). For these systems there is no need to make mole balances on all chemical species involved, mole balance only in limiting reactant (denoted by L) is needed to be solved in combination with appropriate
stoichiometric and kinetic equations.
d
For semi-batch reactors, the feeding and discharge pattern for each chemical species must be considered.
2. Reactor Energy Balances.
General Energy Balance Equation:
Eq. 2.1a
Eq. 2.1b
Table 2.1 Reactor Specific Energy Balances (moles and molar flowrates form)a.
Type Diagram Energy balance equation
Unsteady state operation:
Batch Reactor
Eq. 2.2a
(BR)
Unsteady state operation:
Semi-Batch
Eq. 2.3a
Reactor (SBR)b
Unsteady state operation:
Eq. 2.4a
Continuous
Stirred Tank
Reactor
(CSTR) Steady state operation:
Eq. 2.4b
Table 2.1 Reactor Specific Energy Balances (moles and molar flowrates form, continue)a.
Type Diagram Energy balance equation
Unsteady state operation:
Eq. 2.5a
Eq. 2.5b
Plug Flow
Reactor (PFR) Steady state operation:
Eq. 2.5c
Eq. 2.5d
Packed Bed Eq. 2.6a
Unsteady state operation:
Reactor (PBR)
Eq. 2.6b
Steady state operation:
Unsteady state operation:
Well-Mixed
Fluidized Bed
Eq. 2.7a
Reactor (WM-
FBR)
Table 2.1 Reactor Specific Energy Balances (moles and molar flowrates form, continue)a.
Type Diagram Energy balance equation
Steady state operation:
Well-Mixed
Fluidized Bed
Eq. 2.7b
Reactor (WM-
FBR)
a
This set of reactor design equations is applicable to both multiple and single reaction systems, for both liquid phase reactions and gas phase reactions. It is not applicable to those systems in which reacting phase goes on phase
change (means multiphase flow on process side cannot be handled but multiphase flow on utility side can be handled). Fluid expansion during chemical reaction is considered. Some variables follow certain sign conventions:
stoichiometric coefficients are positive for products and negative for reagents, heat transfer rate is positive when heat is supplied by the utility fluid and negative when it is released by reaction medium, work transfer rate is positive
when work is done over the reaction system and negative if work is done by the reaction medium. Heat capacities are considered to vary with temperature so polynomial or hyperbolic function must be used for calculation.
b
For semi-batch reactors, energy balance must be adapted in function to the feeding and discharge pattern for each chemical species.
Table 2.2 Reactor Specific Energy Balances (moles and molar flowrates reduced form) c.
Type Diagram Energy balance equation
Unsteady state operation:
Batch Reactor
Eq. 2.2b
(BR)
Unsteady state operation:
Semi-Batch
Eq. 2.3b
Reactor (SBR)d
Unsteady state operation:
Eq. 2.4c
Continuous
Stirred Tank
Reactor Steady state operation:
(CSTR)
Eq. 2.4d
Plug Flow Eq. 2.5c
Unsteady state operation:
Reactor (PFR)
Table 2.2 Reactor Specific Energy Balances (moles and molar flowrates reduced form, continue) c.
Type Diagram Energy balance equation
Steady state operation:
Plug Flow
Eq. 2.5d
Reactor (PFR)
Unsteady state operation:
Eq. 2.6c
Fixed Bed
Reactor (FBR) Steady state operation:
Eq. 2.6d
Unsteady state operation:
Eq. 2.7c
Well-Mixed
Fluidized Bed
Reactor (WM- Steady state operation:
FBR)
Eq. 2.7d
c
This set of reactor design equations is applicable to both multiple and single reaction systems, for both liquid phase reactions and gas phase reactions, it is also applicable to gas phase systems but volume expansion of accumulated
fluid during the reactions is ignored. It is not applicable to those systems in which reacting phase goes on phase change (means multiphase flow on process side cannot be handled but multiphase flow on utility side can be handled).
Fluid expansion during chemical reaction is considered. Some variables follow certain sign conventions: stoichiometric coefficients are positive for products and negative for reagents, heat transfer rate is positive when heat is
supplied by the utility fluid and negative when it is released by reaction medium, work transfer rate is positive when work is done over the reaction system and negative if work is done by the reaction medium. Heat capacities are
considered constant at their mean values over the applicable temperature ranges.
d
For semi-batch reactors, energy balance must be adapted in function to the feeding and discharge pattern for each chemical species.
Table 2.3 Reactor Specific Energy Balances (conversion form)e.
Type Diagram Energy balance equation
Unsteady state operation:
Batch Reactor
Eq. 2.2c
(BR)
Unsteady state operation:
Semi-Batch
Eq. 2.3c
Reactor (SBR)f
Unsteady state operation:
Eq. 2.4e
Continuous
Stirred Tank
Reactor Steady state operation:
(CSTR)
Eq. 2.4f
Plug Flow Eq. 2.5e
Unsteady state operation:
Reactor (PFR)
Table 2.2 Reactor Specific Energy Balances (conversion form, continue)e.
Type Diagram Energy balance equation
Steady state operation:
Plug Flow
Eq. 2.5f
Reactor (PFR)
Unsteady state operation:
Eq. 2.6e
Fixed Bed
Reactor (FBR) Steady state operation:
Eq. 2.6f
Unsteady state operation:
Eq. 2.7e
Well-Mixed
Fluidized Bed
Reactor (WM- Steady state operation:
FBR)
Eq. 2.7f
e
This set of reactor design equations is applicable only to single reaction systems, it is perfectly applicable to liquid phase reactions and also applicable to gas phase reactions for which volume expansion of accumulated fluid during
the course of reaction can be ignored (or considered negligible). It is not applicable to those systems in which reacting phase goes on phase change (means multiphase flow on process side cannot be handled but multiphase flow on
utility side can be handled). Some variables follow certain sign conventions: stoichiometric coefficients are positive for products and negative for reagents, heat transfer rate is positive when heat is supplied by the utility fluid and
negative when it is released by reaction medium, work transfer rate is positive when work is done over the reaction system and negative if work is done by the reaction medium. Heat capacities are considered constant at their mean
values over the applicable temperature ranges.
f
For semi-batch reactors, the feeding and discharge pattern for each chemical species must be considered.
1.1 Miscellaneous equations for adiabatic operation.
1.2 Design equations for chemical reactors.
1.3 Miscellaneous equations for catalytic reactors.
Particle Density of Bulk Density of Bulk vs Catalyst Density Void Fraction of
Catalyst Catalyst Catalyst Bed
Eq. 1.14 Eq. 1.15 Eq. 1.16 Eq. 1.17
Integral Mass of Eq. 1.18 Differential Mass of Eq. 1.19 Relation between Homogeneous Eq. 1.20 Solid Fraction of Eq. 1.21
Catalyst Catalyst and Heterogeneous Rate of Catalyst Bed
Reaction
2. Reactor Heat Transfer Rate Equations.
Table 3.1 Reactor Specific Energy Balances (moles and molar flowrates form)a.
Type Diagram Energy balance equation
Unsteady state operation:
Batch Reactor
Eq. 2.2a
(BR)
Unsteady state operation:
Semi-Batch
Eq. 2.3a
Reactor (SBR)b
Unsteady state operation:
Eq. 2.4a
Continuous
Stirred Tank
Reactor
(CSTR) Steady state operation:
Eq. 2.4b
Table 2.1 Reactor Specific Energy Balances (moles and molar flowrates form, continue)a.
Type Diagram Energy balance equation
Plug Flow Eq. 2.5a
Unsteady state operation:
Reactor (PFR)
Steady state operation: Eq. 2.5b
Unsteady state operation:
Eq. 2.6a
Fixed Bed
Reactor (FBR) Steady state operation:
Eq. 2.6b
Unsteady state operation:
Well-Mixed
Fluidized Bed
Eq. 2.7a
Reactor (WM-
FBR)
Table 2.1 Reactor Specific Energy Balances (moles and molar flowrates form, continue)a.
Type Diagram Energy balance equation
Steady state operation:
Well-Mixed
Fluidized Bed
Eq. 2.7b
Reactor (WM-
FBR)
3. Stoichiometric equations.
3.1 Single reaction systems.
Table 3.1 Stoichiometric variables
Concentration Component Global
Concentration stoichiometric stoichiometric Feed molar ratio
(differential
(integral form) Eq. ratio ratio Eq.
Eq. 3.1a form) Eq. 3.2a Eq. 3.2b
3.1b 3.3
Table 3.2 Reaction coordinates
Molar extent of
Fractional conversion Molar extent of reaction (for
reaction (for batch
continuous processing)
processing)
Eq. 3.4 Eq. 3.5a Eq. 3.5b
Liquid hourly space Gas hourly space
Space velocity Weight hourly space velocity
velocitya velocitya
Eq. 3.6a Eq. 3.6b Eq. 3.6c Eq. 3.6d
Heterogeneous Instantaneous
catalytic reaction Overall selectivity of
Space time selectivity of
effectiveness factor reaction
reaction
Eq. 3.7 Eq. 3.8 Eq. 3.9a Eq. 3.9b
Mole fraction of Mass fraction of Instantaneous Overall yield of reaction
chemical species “i” chemical species “i” yield of reaction
Eq. 3.10a Eq. 3.10b Eq. 3.11a Eq. 3.11b
a
Volumetric flowrates must be measured for liquid at 60 °F for liquid phase and at standard temperature and pressure for gas phase; for homogeneous systems (BR, CSTR and PFR) the volume term
refers to the volume of the reactor, while for heterogeneous systems (PBR, FBR, TBR and SR) it refers to the catalyst volume.
Table 3.3 Stoichiometric relations for ideal gases.
Mole fraction Gas phase concentration
Eq. Partial pressure
Eq. 3.13 Eq. 3.14
3.12
Mole difference Total molar flowrate
Total molar flowrate (based on
of reaction Eq. (based on molar extent of
Eq. 3.23b fractional conversion) Eq. 3.22b
3.15 reaction)
Table 3.4 Stoichiometric relations for systems with constant density
Variable Batch processing Continuous processing
Conversion Eq. 3.4a Eq. 3.4b Eq. 3.4c
Molar extent of
Eq. 3.5a Eq. 3.5b Eq. 3.5c
reaction
Correlation between
conversion and molar Eq. 3.16a Eq. 3.16b Eq. 3.16c
extent of reaction
Limiting reactant Eq. 3.17a Eq. 3.17b Eq. 3.17c
Total amount of
chemical species
before occurrence of Eq. 3.18a Eq. 3.18b Eq. 3.18c
chemical reaction (or
at reactor inlet)
Total amount of
chemical species at Eq. 3.19a Eq. 3.19b Eq. 3.19c
any instant
Amount of chemical
Eq. 3.20a Eq. 3.20b Eq. 3.20c
species at any instant
Table 3.4 Stoichiometric relations for systems with constant density (continue)
Variable Batch processing Continuous processing
Reactor final (or
outlet) amount of each
Eq. 3.21a Eq. 3.21b Eq. 3.21c
chemical species as
function of conversion
Reactor final (or
outlet) amount of all
Eq. 3.22a Eq. 3.22b Eq. 3.22c
chemical species as
function of conversion
Reactor final (or
outlet) amount of each
chemical species as Eq. 3.23a Eq. 3.23b Eq. 3.23c
function of molar
extent of reaction
Reactor final (or
outlet) amount of all
chemical species as Eq. 3.24a Eq. 3.24b Eq. 3.24c
function of molar
extent of reaction
Maximum molar
Eq. 3.25a Eq. 3.25b Eq. 3.25c
extent of reaction
Equilibrium molar
extent of reaction and Eq. 3.26a Eq. 3.26b Eq. 3.26c
fractional conversion
Table 3.5 Stoichiometric relations for systems with variable density
Variable Conversion Molar extent of reaction
Reaction Batch system Eq. 3.4a Continuous flow Eq. 3.4b Batch Eq. 3.5a Continuous Eq. 3.5b
coordinate system system flow system
Molar fluid
Eq. 3.27a Eq. 3.27b
expansion factor
Batch system
Batch and continuous flow system Eq. 3.28b
Concentration Eq. 3.28a
Continuous flow system
Eq. 3.28c
Volume Eq. 2.29a Eq. 2.29b
Volumetric
Eq. 2.30a Eq. 2.30b
flowrate
3.2 Multiple reaction systems.
Mole consumption/generation of the “ith” chemical species in the “jth” reaction
Batch processing Continuous processing
Partial molar
extent of reaction
Eq. 2.31a
of the “jth”
chemical reaction
Reactor final (or Eq. 2.32a
outlet) amount of
each chemical
species as function
of molar extent of
reaction
Partial fractional
conversion of the
Eq. 2.33a
“ith” chemical
species
Global fractional
conversion of the
Eq. 2.34a
“ith” chemical
species
Correlation
between partial
fractional
Eq. 2.35a
conversion and
partial molar
extent of reaction
Correlation
between global
fractional
Eq. 2.36a
conversion and
partial molar
extent of reaction
Flujo molar total
(solo gases):
Eq. 2.37a No aplica
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