Mass and Energy Balances
Extrusion is a system with a lot of mass and energy interactions going on.  While it may be more complex than many systems, performing a mass and energy balance is a matter of being thorough enough to account for the inputs and outputs.

Items needed to make a relatively complete mass and energy balance include:
• Proximate analysis of the dry blend
• Dry feed rate into the preconditioner
• Water rate into the preconditioner
• Steam rate into the preconditioner
• Extruder motor energy input into the extrudate
• Water rate into the extruder
• Steam rate into the extruder
Complete mass and energy balances will also include heat exchanges across system boundaries (such as a jacketed cooling system) and account for energy in protein denaturation and starch gelatinization, as well as other sources/sinks of energy.

Calculation of Heat Capacity

While it may be possible to locate or measure specific heat capacities for the ingredients used in the extruder, it may be more practical to calculate it from the following equation:
cp = 1.424 Xcarbohydrate + 1.549 Xprotein    + 1.675 Xfat + 0.837 Xash + 4.187 Xwater

This is from "Food Process Engineering, 2nd Edition" by Dennis R. Heldman and R. Paul Singh, Published by Van Nostrand Reinhold.  Copyright 1981.  This is equation 3.33 found on page 101 of the book.

cp is the heat capacity (kJ/kg °C)

Here is a spreadsheet (OpenOffice format) for calculation of heat capacity: heat capacity spreadsheet

X is the weight fraction (expressed as a decimal) of the component, the component is listed in subscript.  There are 2 possible different values for fat.  As fat in extruded products tend to be low, the issue of using the wrong value is minimal.

It is possible to perform a mass and energy balance on the combined preconditioner/extruder system, but it is frequently useful to perform 2 mass and energy balances.  The first will be performed on the preconditioner and the second on the extruder.  This allows the calculated temperature out of the preconditioner to be compared to the value being measured, as a check of accuracy.

Mass and Energy Balance for the Extrusion System

The reader is owed an apology for the current state of the mass and energy balance here.  This document is being written over time.  As with any detailed document, even when the details are all present, it may still be difficult reading.  It is a given that not all the details are here yet.

For the mass and energy balance on the preconditioner, here is the data needed:

• Mass flow of the dry feed
• Composition (carbohydrate, protein, fat, ash, water) of the dry feed
• Temperature of the dry feed
• Mass flow of water added
• Temperature of the water added
• Mass flow of the steam added
• Quality of the steam added

This first pass at a mass and energy balance will ignore:

• Energy input from the preconditioner motor
• Heat exchange across the preconditioner wall
• Heat loss or gain due to any reactions

We know that:

the sum of energy going in = sum of energy going out

Using degrees Celsius directly is an arbitrary decision, but as the energy balance is about heat loss and gain, any arbitrary reference temperature will work.  In this case, we will use 0°C as that arbitrary reference.

The energy going in is:
Input for the dry feed = Edf = ṁdf cp,df Tdf,inlet
Input for the liquid water = Epcw = ṁw cp,w Tw,inlet
Input for the steam is the sum of the heat of vaporization and the thermal energy of water at 100°C = Epcs = Q ṁs h + ṁs cp,w 100°C

The energy coming out is:
(ṁdf cp,df + ṁw cp,w + ṁs cp,w) TPC_exit

This allows the exit temperature to be solved for:
TPC_exit = (ṁdf cp,df Tdf,inlet + ṁw cp,w Tw,inlet + Q ṁs h + ṁs cp,w 100)/(ṁdf cp,df + ṁw cp,w + ṁs cp,w)
= (Edf + Epcw + Epcs)/(ṁdf cp,df + ṁw cp,w + ṁs cp,w)

ṁ = mass flow rate of the component
Q = steam quality, expressed as a decimal
h = heat of condensation for steam
T = temperature
Key to subscripts:
df = dry feed
w = water
s = steam
PC_exit = Preconditioner exit

The sum of energy exiting the system is expressed in a factored form since all components are at the same temperature or will come to equilibrium at this temperature.

Here is a spreadsheet (OpenOffice format) for calculation of temperatures related to extrusion (both preconditioner and extrudate before the die): preconditioner and extruder temperatures spreadsheet

The preconditioned material then drops into the extruder.  The derivation of the temperature to a point just before exiting the die is similar to the preconditioner above.  The energy input from the motor cannot be ignored in this case, however.  So we have the energy coming into the system through the preconditioner (as above):

The energy going into the preconditioner is:
Input for the dry feed = ṁdf cp,df Tdf,inlet
Input for the liquid water = ṁw cp,w Tw,inlet
Input for the steam is the sum of the heat of vaporization and the thermal energy of water at 100°C = Q ṁs h + ṁs cp,w 100°C

And the additional energy going into the extruder:
Input for the liquid water = Ebw = ṁw cp,w Tw,inlet
Input for the steam is the sum of the heat of vaporization and the thermal energy of water at 100°C = Ebs = Q ṁs h + ṁs cp,w 100°C
Input for the extruder motor: Specific Mechanical Energy (SME)

So the energy going into the extruder is the sum of the energy going into the preconditioner plus the energy going into the extruder.

This allows us to solve for the temperature right before the extrudate exits the die:

Textruder_exit = (Energy going into the preconditioner + Energy going into the extruder)/(sum of (mass flow rate * heat capacity))
=  (Edf + Epcw + Epcs + Ebw + Ebs + SME)/{(ṁdf cp,df + ṁw cp,w + ṁs cp,w)preconditioner + (ṁw cp,w + ṁs cp,w)barrel}

Special Cases and Considerations

When the extrudate exits the die, there is a release of pressure.  If the extrudate is hot enough (greater than 100°C), then some portion of the water in the extrudate will flash off as steam.  As a rough estimate, the temperature after steam flashes will be about 100°C.  This is due to the water in the system wanting to come to an equilibrium state at 1 Atmosphere at the boiling point of water.

In the event the extruder is vented, then the water in the system will want to come to an equilibrium condition based on the pressure at the vent.  Work done by the screw beyond that point will again heat the extrudate.