For simplicity we assume that the cooling jacket temperature can
be directly manipulated, so that an energy balance around the
jacket is not required. We also make the following assumptions

ïPerfect mixing (product stream values are the same as the
bulk reactor fluid)

ïConstant volume

ïConstant parameter values

The constant volume and parameter value assumptions can easily
be relaxed by the reader, forfurther study.

**2.1 Parameters and Variables
**

The parameters and variables that will appear in the modeling
equations are listed below for convenience.

A | Area for heat exchange |

CA | Concentration of A in reactor |

CAf | Concentration of A in feed stream |

cp | Heat capacity (energy/mass*temperature) |

F | Volumetric flowrate (volume/time) |

k0 | Pre-exponential factor (time-1) |

R | Ideal gas constant (energy/mol*temperature) |

r | Rate of reaction per unit volume (mol/volume*time) |

t | Time |

T | Reactor temperature |

Tf | Feed temperature |

Tj | Jacket temperature |

Tref | Reference temperature |

U | Overall heat transfer coefficient (energy/(time*area*temperature)) |

V | Reactor volume |

DE | Activation energy (energy/mol) |

(-DH) | Heat of reaction (energy/mol) |

r | Density (mass/volume) |

**2.2 Overall material balance
**

The rate of accumulation of material in the reactor is equal to
the rate of material in by flow - the
material out by flow.

Assuming a constant amount of material in the reactor (= 0), we find that

If we also assume that the density remains constant, then

and

**2.3 Balance on Component A
**

The balance on component A is

(1)

where r is the rate of reaction per unit volume.

**2.4 Energy Balance
**

The energy balance is

(2)

where Tref represents an arbitrary reference temperature for enthalpy.

**2.5 State Variable form of Dynamic Equations
**

We can write (1) and (2) in the following state variable form
(since dV/dt = 0)

(1a)

(2a)

where we have assumed that the volume is constant. The reaction
rate per unit volume (Arrhenius expression) is

\b(3)

where we have assumed that the reaction is first-order.