# Space time (chemical engineering)

Space time (or space-time) is the time necessary to process one reactor volume of fluid, given a particular set of entrance conditions. Commonly represented by the Greek letter τ, it is obtained by dividing the reactor volume by the volumetric flow rate entering the reactor. It is the equivalent to the hydraulic retention time (HRT=V/Q).[1] The space time is frequently used in calculation involving chemical reactors and is often a more useful quantity to work with than the average residence time. The average residence time and the space are equal if all of the following conditions are met:[2]

1. Temperature and pressure are constant
2. The reaction proceeds with constant density
3. The volumetric flow rate is evaluated at reactor inlet conditions

## Design Equations

Design Equations are equations relating the space time to the fractional conversion and other properties of the reactor. Different design equations have been derived for different types of the reactor and depending on the reactor the equation more or less resemble that describing the average residence time. Often design equations are used to minimize the reactor volume or volumetric flow rate required to operate a reactor.[2]

### Batch Reactor

Batch reactors are reactors in which the reactants are put in the reactor at time 0 and react until the reaction is stopped. Consequently, the space time is the same as the average residence time in a batch reactor.

${\displaystyle \tau =N_{AO}\int {\frac {1}{(-r_{A})V_{R}}}\,df_{A}}$

### PFR

Plug flow reactors (PFRs) are reactors in which the reactants enter the reactor at one end and react as they move down the reactor. Consequently, the reaction rate is dependent on the concentrations which vary along the reactor requiring the inverse of the reaction rate to be integrated over the fractional conversion.

${\displaystyle \tau =C_{AO}\int {\frac {1}{(-r_{A})}}\,df_{A}}$

### CSTR

Continuously stirred tank reactors (CSTRs) are reactors in which the reactants continuously enter and leave a tank where they are mixed. Consequently, the reaction proceeds at a rate dependent on the outlet concentration.

${\displaystyle \tau ={\frac {C_{Ain}-C_{Aout}}{(-r_{AF})}}\ }$

### Recycle Reactors

Recycle reactors are PFRs with a recycle loop. Consequently, they behave like a hybrid between PFRs and CSTRs.

${\displaystyle \tau =C_{AO}(R+1)\int {\frac {1}{(-r_{A})}}\,df_{A}}$

In all of these equations :${\displaystyle -r_{A}}$ is the consumption rate of A, a reactant. This is equal to the rate expression A is involved in. The rate expression is often related to the fractional conversion both through the consumption of A and through any k changes through temperature changes that are dependent on conversion.[2]

## Variable Volume Reactions

In some reactions the reactants and the products have significantly different densities. Consequently, as the reaction proceeds the volume of the reaction changes. This variable volume adds terms to the design equations. Taking this volume change into consideration the volume of the reaction becomes:

${\displaystyle V_{R}=V_{Rinitial}(1-\delta _{A}f_{A})}$

Plugging this into the design equations above results in the following equations:

### Batch

${\displaystyle \tau =N_{AO}\int {\frac {1}{(-r_{A})V_{R}(1-\delta _{A}f_{A})}}\,df_{A}}$

### PFR

${\displaystyle \tau =C_{AO}\int {\frac {1}{(-r_{A})(1-\delta _{A}f_{A})}}\,df_{A}}$

### CSTR

${\displaystyle \tau ={\frac {C_{Ain}-C_{Aout}}{(-r_{AF})(1-\delta _{A}f_{A})}}\ }$

Generally, when reactions take place in the liquid and solid phases the change in volume due to reaction is not significant enough that it needs to be taken into account. Reactions in the gas phase often have significant changes in volume and in these cases one should use these modified equations.[2]

## References

1. ^ Elements of Chemical Reaction Engineering (4th Edition) by H. Scott Fogler, Prentice Hall PTR, 2005. ISBN 0-13-047394-4
2. ^ a b c d Chemical Engineering Kinetics and Reactor Design by Charles G. Hill, Jr. John Wiley & Sons Inc, 1977. ISBN 978-0471396093