Scale-up of geometrically similar extruders is
relatively straight-forward:

- Feed rates if the various ingredients need to be
proportional to the diameter of the screw cubed (D
^{3})

- The screw profile needs to be identical between scales
- The screw speed is identical between scales
- Heating or cooling of the extruder barrel needs to be
minimized. Heat transfer does not scale with
geometrically similar extruders.

- Approach to the die holes needs to be as similar as possible between scales
- There are a number of manufacturers who have vastly different die geometries for different scales of extruders. This can cause great difficulty in scaling up.
- Flow rate per die hole needs to be identical between scales.

This sounds deceptively easy. It really can be this
easy. The largest caution in scaling up the die is a
significant change in geometry for the die between pilot plant
and production plant. More than one manufacturer uses a
pilot plant set-up that puts the die opening(s) directly in
front of the screw with no obstructions, but a much more
restrictive design on a commercial scale. When a page is
added on this subject, a link will be added here.

Below is a simple description of why geometrically similar
extruders scale-up in the manner described above. More
comprehensive and and rigorous descriptions are possible, but
will not be covered here.

Extruders designed to be geometrically similar are a case where direct scale-up is possible if some basic assumptions are not violated. If two extruders are geometrically similar, it means that they look exactly alike except for the size of the extruders. Every dimension of the extruder is multiplied by some constant. One way to think of this is with scale models in the same way there are scale models of trains, planes, and cars. These scale models are models of the real item that have been shrunk in every dimension. Alternately, if you are somewhat familiar with CAD software, it is similar to zooming in on an object: what shows on the screen has the same relative dimensions, with all dimensions being increased by the same amount.

The major basic assumptions to be aware of include:

- L:D of the extruders is the same
- This is not always the case, and can be overcome in some
cases

- Gap between the screws and the barrels are scaled

- This is not always the case, measure to make sure
- Heating or cooling of the barrel sections is minimal
- Many companies choose to actively heat or cool extruder barrel sections. Heat transfer through the barrel wall will not scale, but minor amounts of heating or cooling can be dealt with

If two extruders are geometrically similar, then we can do some
basic analysis of the system to demonstrate how the throughput of
the extruder scales up with diameter of the extruder and why
extruders should not be actively heated or cooled if there is a
desire to scale the process.

First we need to look at the geometry of the screw in a
scale-independent (dimensionless) manner. The screw element
has the following characteristics:

- Diameter (D), which we will scale everything else to
- This is also the outer diameter (D
_{o})

- Root Diameter (inner diameter), D
_{i}

- Flight thickness (t), which will typically vary based on
distance from D
_{i}. - Length of the extruder (L),

Dimension |
Symbol indicating this variable |
Notes on the variable |

Diameter of the screw |
D | This is the basis for 1 unit of length
measurement for the extruder. All other dimension will
be converted in terms of D. This measurement is also referred to as the outer diameter (D _{o}). |

Inner diameter of the screw |
D_{i} |
This is the narrowest dimension on the screw,
basically the diameter of the screw (D) minus double the
depth of the channel defined by the flights.
This is fixed on typical twin-screw extruders, but may
change along the length of the extruder for a single-screw
extruder or for twin-screw extruders where the 2 screws are
not parallel. |

Flight thickness |
t |
This can be a constant thickness from the
root diameter to the outer diameter, but is more typically
variable and defined by the elliptical channel. |

Length |
L |
This is the length of the extruder.
Extruder lengths are often expressed in terms of
diameters. |

Pitch |
P |
This is the distance forward material in the
screw would travel per revolution of the screw in the area
of this pitch. A full pitch screw would move material
forward 1 D, 3/4 pitch would move material forward 0.75 D,
etc, assuming there are no inefficiencies in the conveying
of the screw. |

Gap between the barrel wall and the diameter
of the screw part |
δ |
This is 1/2 the difference in diameter for
the barrel and the screw. |

Fraction of the barrel cross-sectional area
occupied by the screw elemnt |
C_{1} |
This is a mathematical constant across scales
for geometrically similar extruders |

Ratio of the barrel diameter to screw
diameter |
C_{2} |
The barrel diameter is slightly larger than
the screw. Using a ratio of barrel diameter to screw
diameter will make the math more easily understood. |

Created Variable |
C_{3} |
A constant that is a combination of a
function of C_{1} and C_{2}. Use of C_{3}
will make some later calculations be easier to work
with. |

Diagram of a screw element (side view):

Looking at the side view of a screw element, we can see that for
1 full revolution of a screw, the screw will convey the extrudate
forward equal to the pitch of the screw. This ignores any
inefficiencies in pumping ability in the screw. These
inefficiencies in pumping will also scale, and that will be
covered a bit later in this page. So: the volume of
extrudate conveyed forward is proportional to the diameter of the
extruder.

One revolution of a screw brings the extrudate forward by the
pitch of the screw (a full pitch screw for this example):

Diagram of a screw element (end view):

The blue in the image above indicates the cross-section of the
metal at the end of the screw. This is the area of the
cross-section in the barrel that is not extrudate. The
barrel is a slightly larger diameter than the screw, so if we show
a cross-section of the extruder with both the extrudate (red) and
the metal of the screw (blue), the full cross-section of the
barrel can be visualized:

If the cross-section of the metal is removed, we are left with
just the cross-section of the extrudate:

The portion of the cross-sectional area of the barrel contains the screw element, not extrudate. This can be calculated:

This area is proportional to some set of constants multiplied by D

The cross-sectional area of the barrel can be calculated:

Now the volume of extrudate being conveyed forward can be calculated:

The constant C3 was created to make the equation show a little
more clearly for later calculations.

So now we have the volume of extrudate carried forward per revolution of the screw by multiplying the cross-sectional area of the extrudate by the length the extruate is conveyed forward:

This tells us that the volume conveyed forward in an extruder screw is proportional to the diameter of the screw cubed. A similar calculation can be done to show the same relationship applies for twin-screw extruders. Since the residence time is proportional to diameter of the extruder for geometrically similar extruders, residence time is independent of extruder diameter for geometrically similar extruders being fed scaled feed rates and at equal screw speeds.

Heat generation in extruders occur
through viscous dissipation of energy being delivered through the
extruder screw(s). Energy input is a function of shear rate of the
extrudate. It can be shown that for geometrically similar
extruders fed at scaled feed rate, the energy input is a function
of shear rate and time in the extruder. The shear rates can be
calculated for any point in the extrudate. For the sake of an easy
geometry, a cross-section of the screw (in blue) and the extrudate
(in red) will be used for the example. This is shown in the
image below. The diameter of the barrel is D+2δ where δ is
the gap between the screw element and the barrel wall at the
narrowest point. For different areas in the extrudate, the
distance between the moving screw element and the barrrel
wall will be different.

For this example, we use the area of highest shear for the calculation, but the shear at any point could be used. The gap between the screw tip and the wall is the point of highest shear and can be expressed mathematically as:

For geometrically similar extruder being fed at scaled rates and run at a given RPM, the feed rate increases at a rate proportional do D3. Since energy input per unit mass is constant and the mass flow rate is proportional to D3, the energy input to the extruder is a function of D3. The area available for heat exchange is only proportional to D2: surface area available per unit mass is proportional to 1/D.

The easier example is when scaling from a shorter extruder to a longer one. In this case, the shorter extruder can be scaled to the longer one by adding conveying elements in the early part of the extruder screw profile. The added conveying elements would not be filled, so would add minimal extra energy to the extrudate. The one area to be a little cautious may be in the location of the first set of paddles (kneading blocks). In many cases, the first set of paddles are used as a way to seal the screws and prevent injected steam from exiting through the extruder inlet. In cases like that, the conveying screws should be added after the first set of paddles, again it should have minimal impact on the energy input to the extrudate.

The more difficult situation is going from a longer extruder to a shorter one. In a screw profile with a lot of paddles or other work elements, it may not be possible to remove conveying elements due to there not being enough remaining conveying elements force all the extrudate to be conveyed through the die - the extruder will back up.