<p>If you have followed that guide, you have extruded the 100&nbsp;mm through the heated nozzle.
This results in your steps/mm value to also be influenced by hot end temperature, speed during the extrusion test, and potentially pressure on the extruder grip gear.</p>

<p>Try recalibrating your extruder, but this time, disconnect the Bowden tube either at the hot end, or at the extruder.
Once you have the steps/mm set to something <em>only</em> related to the mechanics of the extruder, you can calibrate your flow rate - which you should usually do per filament type and even spool individually.</p>

CC BY-SA 4.0
2019-09-18T08:55:17.543Z
Note that the speed in the guide is very low `G1 E100 F100`, so the filament extrusion is not calibrated to print speed/pressure of the filament!
0scar
CC BY-SA 4.0
2019-09-18T09:00:50.427Z
Thanks for the suggestion of removing the bowden tube... I really like the idea of being able to totally eliminate any resistance at the hotend, like @0scar says though, it does extrude it very slowly... it takes about a minute to extrude 100 mm. However, it doesn't skip when calibrating.... only subsequent prints following the calibration.
BG100
CC BY-SA 4.0
2019-09-18T06:10:40.497Z
2019-09-18T08:49:54.263Z
2019-09-18T08:49:54.263Z
towe
<p>Let's set up the basics of extrusion and then go over what has effects on the results of calibration.</p>
<p>Let's discuss the extruder as an item on itself first, then look into how the nozzle and filaments impact it.</p>
<h2>Extruder</h2>
<p>The extruder basically is a spinning motor that pushes along filament at some extrusion rate <span class="math-container">$e_r [\frac{\text{mm}}{\text{step}}]$</span> and it's related factor <span class="math-container">$s_e [\frac{\text{step}}{\text{mm}}]=\frac 1 {e_r}$</span> via a hobbed gear - or in the case of this extruder via a pair of synchronous hobbed gears. The motor in an Ender 3 is a typical NEMA17 with <span class="math-container">$s=1.8\ \frac{\text{deg}}{\text{step}}$</span> (and up to 16 micosteps<sup><a href="https://3dprinting.stackexchange.com/a/10988/8884">see here</a></sup>). It spins a hobbed gear, which has an outer diameter <span class="math-container">$d_o$</span>, and the teeth are cut to a depth that generates the inner diameter <span class="math-container">$d_i$</span>. Somewhere between these diameters is the <em>effective</em> diameter <span class="math-container">$d_e$</span>. So, using basic geometry we get:</p>
<p><span class="math-container">$e_r=\frac{C_e\times s}{360°}=C_e\times0.005\frac 1 {\text{step}}$</span> wherein <span class="math-container">$C_e=\pi d_e$</span></p>
<p><span class="math-container">$s_e=\frac {0.005\ \text{steps}}{C_e}$</span></p>
<h2>Filament effects</h2>
<p>Now, that we know the theoretical setting of mm/step or steps/mm (<span class="math-container">$e_r$</span> and <span class="math-container">$s_e$</span>) for the firmware, we need to discuss how the filament impacts this. First of all, the calculation above holds only true for pushing even thickness filament that the teeth bite in evenly. If the filament does change in thickness, the effective diameter of our gear changes, and as a result, the extrusion changes. A thicker diameter filament does not get dug in as deep, the effective diameter goes up, and thus the circumference <span class="math-container">$C_e$</span>.</p>
<p>Similarly, different filaments do have different hardnesses.<sup><a href="https://3dprinting.stackexchange.com/a/8396/8884">see also here</a></sup> A test on hardness was done for <a href="https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1026&amp;context=aseeil-insectionconference" rel="nofollow noreferrer">this paper</a>, exploring Type A and Type D Shore hardness. Type A is a flat tip, while D is a sharp tip. The last is somewhat similar in the effect to the hobbed gear. The difference means that the teeth don't bite the same, affecting the effective diameter of the hobbed gear <span class="math-container">$d_e$</span>.</p>
<p>This behavior is <strong>filament dependant</strong></p>
<h2>Nozzle behavior</h2>
<p>The next part we have to look at is the behavior in the hotend and nozzle. In perfect conditions, the heat zone would melt up the filament fully and ensure a stable, laminar flow through the nozzle as it necks down the material from its starting diameter to the extrusion width.</p>
<p>Assuming our filament stays laminar, then the flow of material in is exactly the same as the flow out, nothing stutters. But if the factors in the pipe are off, then we get turbulent flow.<sup><a href="https://en.wikipedia.org/wiki/Laminar_flow" rel="nofollow noreferrer">further reading</a></sup></p>
<p>So, the flow is not necessarily linear, and we can easily identify several factors that impact the behavior in the nozzle. Let's do a quick rundown of different factors:</p>
<ul>
<li>Materials do expand differently (with factor <span class="math-container">$\alpha$</span>) under heating, thus impacting the volume of the material in the nozzle, which in turn impacts the volumetric flow rate and pressure in the nozzle. This is <strong>temperature</strong> and <strong>material dependant</strong>.<sup><a href="https://3dprinting.stackexchange.com/a/8384/8884">further on this</a></sup></li>
<li>The viscosity of the material has a huge impact on the flow behavior. Most plastics viscosities are <strong>temperature dependant</strong> but also <strong>material dependant</strong></li>
<li>The nozzle shape can have an impact on the flow rate to a small degree (mainly if it is uneven or rough). The most influential factor, however, is the <strong>nozzle diameter</strong>, which directly impacts flow rate.</li>
</ul>
<h3>How all that effects flow in a nozzle</h3>
<p>Let's assume flow is ensured and that we can ignore friction of the nozzle on the material. Then we get the Freeman formula for Flow:</p>
<p><span class="math-container">$Q=Av$</span> where Q is the Volumetric Flow in 0.001 m³/s, A the area of the nozzle in m², v is the velocity at the exit in m/s.</p>
<p><span class="math-container">$A=r^2\pi$</span> is the well known circle formula, r is the nozzle radius, which is half the nozzle diameter<br>
<sup><span class="math-container">$\ \ \ A_{0.4\text{ mm Nozzle}}=1.256\times10^{-7}\text{ m}^2$</span></sup></p>
<p><span class="math-container">$v=\sqrt{2P}$</span> is the Bernoulli's Equation, telling us that the flow velocity is pressure dependant (see Nozzle Behavior point 1). As a result, we get that the volumetric flow through our nozzle depends on the pressure like this:</p>
<p><span class="math-container">$Q=\sqrt 2 \pi\times r\times\sqrt P$</span><br>
<sup><span class="math-container">$\ \ \ Q_{0.4\text{ mm Nozzle}}=1.777\times10^{-7}\text{ m}^2\times\sqrt{P}$</span></sup></p>
<p>The pressure in the nozzle is the sum of the pressure generated by the force with which the material is pushed in by the extruder (<span class="math-container">$P_F=F/A'$</span>) and the material expansion in the nozzle (<span class="math-container">$P_e$</span>).</p>
<p>As we established up in the Extruder part, the extrusion rate is somewhat dependent on the effective diameter of the hobbed gear <span class="math-container">$d_e$</span>. The effective diameter of the hobbed gear also has an effect on the pressure in the nozzle: how deep the teeth cut into the filament determines the force they transmit. The other factor that impacts the force transmitted via the filament is the extrusion speed <span class="math-container">$v_e$</span>, thus we write <span class="math-container">$F(d_e,v_e)$</span>. Atop that, the actual filament diameter <span class="math-container">$A'$</span> plays another factor, as explored under filament effects. The thermal expansion, which is dependant on the material coefficient <span class="math-container">$\alpha$</span> and the Temperature increase <span class="math-container">$\Delta T$</span> adds to the pressure in the nozzle, thus we write <span class="math-container">$P_e(\alpha,\Delta T)$</span>. So the expression for the volumetric flow out of the nozzle is</p>
<p><span class="math-container">$Q=\sqrt 2 \times r \pi \times\sqrt {\frac{F(d_e,v_e)}{A'}+P_e(\alpha,\Delta T)}$</span><br></p>
<p>If this flow is laminar and non-turbulent could be read from the accompanying Reynolds number <span class="math-container">$Re$</span>, which is dependant on the (dynamic/kinematic) viscosity <span class="math-container">$\mu=\frac \nu \rho$</span> (rho is the density). Viscosity is temperature dependant, thus we write <span class="math-container">$\nu(T)$</span>. Last factor in the formula for the Reynolds number is the hydraulic diameter <span class="math-container">$D_H$</span>, which for our case boils down to the diameter of the nozzle, so <span class="math-container">$D_H=2r$</span>. For our case this gives:</p>
<p><span class="math-container">$Re=\frac{2r\times Q}{2 r^2 \pi \times \nu(T)}=\frac Q {r \pi \times \nu(T)}=\frac {\sqrt 2}{\nu(T)} \sqrt {\frac{F(d_e,v_e)}{A'}+P_e(\alpha,\Delta T)}$</span></p>
<h2>tl;dr / Conclusion</h2>
<p>Ok, the theory is all nice, but how does all that affect our print?!</p>
<ol>
<li>It's a good idea to calibrate for one material and use that as your benchmark and then create profiles for your other materials measured against this benchmark</li>
<li>If our temperature is lower or higher than in calibration, the flow is different and, as a result, the extruder-steps don't match up. Instead of altering the steps, we should <strong>adjust</strong> the volumetric flow value up or down to match to the particular material. Usually, it is called either <code>Flow</code> or <code>extrusion multiplier</code> in the software. In the calculations above it is <span class="math-container">$Q$</span>.</li>
<li>A different material or material mix (like a new color or different brand) has a different thermal expansion, thus we <strong>adjust</strong> the <code>flow</code>/<code>extrusion multiplier</code> in our profile as needed.</li>
<li>If we swap the nozzle, we get a different flow characteristic. Most of the flow calculations for such change is already done in the slicer, we usually don't need to alter the profile further, but tiny changes could be done by adjusting the <code>flow</code>/<code>extrusion multiplier</code> value.</li>
<li>extruder clicking can show over extrusion, but it <strong>also</strong> can show that the print height is not achieved as it is calculated for and point to errors in <strong>bed leveling</strong> if happening in the first layer(s) and to errors in the <strong>z-Axis</strong> if it happens later in the print.</li>
</ol>

CC BY-SA 4.0
2019-09-18T15:32:55.543Z
2019-09-18T16:10:25.003Z
2020-06-18T08:29:41.667Z
Trish
<p>I have this same extruder on my Voxelab Aquila.
Default E-steps are 93. Using the printer menu to feed 100 mm of eSun black PETG filament I measured a feed length of 66 mm, so 34 mm under extruded.</p>
<p>Using the formula 100 / 66 * 93 gave me a new E-step value of 140.9.
Upon re-testing, I found I was now over extruding by 34mm.</p>
<p>This formula is missing something. I've chased E-step values up and down before so decided to try something different.</p>
<p>Since the new E-step value was over extruding by the same as the original E-step under extruding value, I added the original E-step value to the new E-step value and took the average. 93 + 140.9 / 2 = 116.9.
I re-tested this value and found it was over extruding by 0.4 mm. Since I was still over extruding slightly I calculated a new E-step, 100 / 100.4 * 116.9 = 116.4. Took the average between the 2 again and got a final E-step value of 116.6.</p>
<p>Hope this helps anyone else having issues calibrating E-steps.</p>

CC BY-SA 4.0
2021-07-30T23:54:58.740Z
2021-07-31T00:04:39.043Z
2021-07-31T00:04:39.043Z
Ash
<p>Consider that the extruder is skipping because it is unable to push filament at the rate you are requesting. By reducing the steps to ninety percent, you are reducing the rate by that much as well.</p>

<p>Typically, a skipping extruder is an indication of clogging, but it does not have to be clogging caused by particulates jamming the nozzle. At higher rates of filament travel, one needs higher temperatures to compensate for the cooling at those higher rates.</p>

<p>Consider to reduce the print speed to ninety percent of the current figure, or raise the nozzle temperature by five to ten degrees (in steps) to see if you'll get rid of the cold blocking that may be causing this problem.</p>

CC BY-SA 4.0
2019-09-18T08:56:10.240Z
I'm using the same filament at the same temp as I used when it was stock (PLA at 200 °C), not really had a problem before the new extruder, but that maybe because the new extruder is doing a better job of pushing the filament through... so thanks for the info, I will experiment with slightly higher temp and slower speeds.
BG100
CC BY-SA 4.0
2019-09-18T08:59:45.493Z
This answer contains information on the relation of print speed and filament pressure and as such skipping steps! This should be incorporated in all those guides that tune filament extrusion length! I even forget that myself sometimes...
0scar
CC BY-SA 4.0
2019-09-18T09:38:07.967Z
Technically, it is not *clogging* if the nozzle can't handle the flow *at this temperature*. Changing the temperature changes the flow parameters (especially viscosity) in the nozzle.
Trish
CC BY-SA 4.0
2019-09-17T21:23:09.387Z
2019-09-17T21:23:09.387Z
fred_dot_u
<p>I have replaced the stock extruder on my Ender 3 with one of these:</p>

<p><a href="https://i.sstatic.net/wDmXL.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/wDmXL.jpg" alt="New Extruder"></a></p>

<p>The grip gear has a smaller diameter, so I calibrated the esteps as per the top google search: <a href="https://all3dp.com/2/extruder-calibration-6-easy-steps-2/" rel="nofollow noreferrer">Extruder Calibration – 6 Easy Steps to Calibrate Your Extruder...</a></p>

<p>If I set the esteps so that it's spot on with 100&nbsp;mm of filament is used up when I ask it to extrude 100&nbsp;mm, then during a print I get the occasional skip on the extruder.</p>

<p>If I dial it back a bit and set it so that it extrudes 90&nbsp;mm of filament when I ask it to extrude 100&nbsp;mm, then I don't get the skips.</p>

<p>In both cases the print looks normal.</p>

<p>I've tried changing the nozzle as well in case there was some blockage, but it doesn't make a difference.</p>

<p>Should I just go with the under extrusion? or is these likely to be some other problem that isn't apparent?</p>

<p>I didn't notice any issues with the stock extruder and the stock estep setting, but I didn't think to check the calibration.</p>

CC BY-SA 4.0
2019-09-18T03:28:05.827Z
Why did you replace it? Likely this one is just worse than the stock one - incapable of producing the same pressure reliably.
R.. GitHub STOP HELPING ICE
CC BY-SA 4.0
2019-09-18T03:29:25.207Z
If you don't notice serious quality issues from a 10% change in extrusion amount (which is huge), it's probably not a good idea to make random changes to your printer.
R.. GitHub STOP HELPING ICE
CC BY-SA 4.0
2019-09-18T08:51:52.320Z
I replaced it because the stock one made of plastic was starting to wear at the point the filament was being pulled in, also the spring was too weak and didn't grip the filament strongly enough. There is nothing wrong with the new one... much better than the old one.
BG100
CC BY-SA 4.0
2019-09-18T09:08:01.203Z
If filament can slip in it (probably because the small gear has too little contact surface) it's not better.
R.. GitHub STOP HELPING ICE
CC BY-SA 4.0
2019-09-18T09:09:07.970Z
The filament isn't slipping in the gear.... the stepper motor is skipping steps.
BG100
CC BY-SA 4.0
2019-09-18T09:46:46.107Z
@R.. I had to replace mine after the plastic one broke. I chose the aluminium design of the original extruder though. This design is "double drive", pushing th filament from both sides. This has a better pushing characteristic but also is more prone to shredding the filament.
Trish
CC BY-SA 4.0
2019-09-18T10:20:08.647Z
Increasing stepper motor current can also help, if it is not already set to the maximum for that motor.
jpa
CC BY-SA 4.0
2019-09-18T11:51:58.233Z
@BG100: Ah, I misunderstood what you meant then. I think my initial tone may have been a bit hostile - sorry about that. What are the final steps per mm compared to the original extruder?
R.. GitHub STOP HELPING ICE
CC BY-SA 4.0
2019-09-18T13:04:01.183Z
@R.. No worries :) Original extruder was 93 steps per mm, and new one is 143.4 steps per mm. I'm using Klipper firmware and annoyingly it uses mm per step, so the actual setting is 0.00697.
BG100
CC BY-SA 4.0
2019-09-18T13:17:03.697Z
@BF100: More steps per mm should decrease the motor torque requirements so odd that it's missing steps... BTW I'd compute it exactly from mechanical properties rather than experimentally.
R.. GitHub STOP HELPING ICE
<extruder><creality-ender-3><calibration>
Calibrated Esteps Causes Extruder Skipping
BG100
# Calibrated Esteps Causes Extruder Skipping

(Asked by: username_0 on 2019-09-17)

I have replaced the stock extruder on my Ender 3 with one of these:

[![New Extruder](https://i.sstatic.net/wDmXL.jpg)](https://i.sstatic.net/wDmXL.jpg)

The grip gear has a smaller diameter, so I calibrated the esteps as per the top google search: [Extruder Calibration – 6 Easy Steps to Calibrate Your Extruder...](https://all3dp.com/2/extruder-calibration-6-easy-steps-2/)

If I set the esteps so that it's spot on with 100 mm of filament is used up when I ask it to extrude 100 mm, then during a print I get the occasional skip on the extruder.

If I dial it back a bit and set it so that it extrudes 90 mm of filament when I ask it to extrude 100 mm, then I don't get the skips.

In both cases the print looks normal.

I've tried changing the nozzle as well in case there was some blockage, but it doesn't make a difference.

Should I just go with the under extrusion? or is these likely to be some other problem that isn't apparent?

I didn't notice any issues with the stock extruder and the stock estep setting, but I didn't think to check the calibration.

**Tags:** extruder, creality-ender-3, calibration
> **Comments:**
> **username_1**: Why did you replace it? Likely this one is just worse than the stock one - incapable of producing the same pressure reliably.  
> **username_1**: If you don't notice serious quality issues from a 10% change in extrusion amount (which is huge), it's probably not a good idea to make random changes to your printer.  
> **username_0**: I replaced it because the stock one made of plastic was starting to wear at the point the filament was being pulled in, also the spring was too weak and didn't grip the filament strongly enough. There is nothing wrong with the new one... much better than the old one.  
> **username_1**: If filament can slip in it (probably because the small gear has too little contact surface) it's not better.  
> **username_0**: The filament isn't slipping in the gear.... the stepper motor is skipping steps.  
> **username_2**: @R.. I had to replace mine after the plastic one broke. I chose the aluminium design of the original extruder though. This design is "double drive", pushing th filament from both sides. This has a better pushing characteristic but also is more prone to shredding the filament.  
> **username_3**: Increasing stepper motor current can also help, if it is not already set to the maximum for that motor.  
> **username_1**: @BG100: Ah, I misunderstood what you meant then. I think my initial tone may have been a bit hostile - sorry about that. What are the final steps per mm compared to the original extruder?  
> **username_0**: @R.. No worries :) Original extruder was 93 steps per mm, and new one is 143.4 steps per mm. I'm using Klipper firmware and annoyingly it uses mm per step, so the actual setting is 0.00697.  
> **username_1**: @BF100: More steps per mm should decrease the motor torque requirements so odd that it's missing steps... BTW I'd compute it exactly from mechanical properties rather than experimentally.  

---

## 4 answers

### Answer by username_7 on 2019-09-17. Score: 6 | Accepted answer

Consider that the extruder is skipping because it is unable to push filament at the rate you are requesting. By reducing the steps to ninety percent, you are reducing the rate by that much as well.

Typically, a skipping extruder is an indication of clogging, but it does not have to be clogging caused by particulates jamming the nozzle. At higher rates of filament travel, one needs higher temperatures to compensate for the cooling at those higher rates.

Consider to reduce the print speed to ninety percent of the current figure, or raise the nozzle temperature by five to ten degrees (in steps) to see if you'll get rid of the cold blocking that may be causing this problem.

> **Comments:**
> **username_0**: I'm using the same filament at the same temp as I used when it was stock (PLA at 200 °C), not really had a problem before the new extruder, but that maybe because the new extruder is doing a better job of pushing the filament through... so thanks for the info, I will experiment with slightly higher temp and slower speeds.  
> **username_5**: This answer contains information on the relation of print speed and filament pressure and as such skipping steps! This should be incorporated in all those guides that tune filament extrusion length! I even forget that myself sometimes...  
> **username_2**: Technically, it is not *clogging* if the nozzle can't handle the flow *at this temperature*. Changing the temperature changes the flow parameters (especially viscosity) in the nozzle.  

---

### Answer by username_4 on 2019-09-18. Score: 3

If you have followed that guide, you have extruded the 100 mm through the heated nozzle.
This results in your steps/mm value to also be influenced by hot end temperature, speed during the extrusion test, and potentially pressure on the extruder grip gear.

Try recalibrating your extruder, but this time, disconnect the Bowden tube either at the hot end, or at the extruder.
Once you have the steps/mm set to something *only* related to the mechanics of the extruder, you can calibrate your flow rate - which you should usually do per filament type and even spool individually.

> **Comments:**
> **username_5**: Note that the speed in the guide is very low `G1 E100 F100`, so the filament extrusion is not calibrated to print speed/pressure of the filament!  
> **username_0**: Thanks for the suggestion of removing the bowden tube... I really like the idea of being able to totally eliminate any resistance at the hotend, like @0scar says though, it does extrude it very slowly... it takes about a minute to extrude 100 mm. However, it doesn't skip when calibrating.... only subsequent prints following the calibration.  

---

### Answer by username_2 on 2019-09-18. Score: 2

Let's set up the basics of extrusion and then go over what has effects on the results of calibration.

Let's discuss the extruder as an item on itself first, then look into how the nozzle and filaments impact it.

## Extruder

The extruder basically is a spinning motor that pushes along filament at some extrusion rate $e\_r [\frac{\text{mm}}{\text{step}}]$ and it's related factor $s\_e [\frac{\text{step}}{\text{mm}}]=\frac 1 {e\_r}$ via a hobbed gear - or in the case of this extruder via a pair of synchronous hobbed gears. The motor in an Ender 3 is a typical NEMA17 with $s=1.8\ \frac{\text{deg}}{\text{step}}$ (and up to 16 micosteps[see here](https://3dprinting.stackexchange.com/a/10988/8884)). It spins a hobbed gear, which has an outer diameter $d\_o$, and the teeth are cut to a depth that generates the inner diameter $d\_i$. Somewhere between these diameters is the *effective* diameter $d\_e$. So, using basic geometry we get:

$e\_r=\frac{C\_e\times s}{360°}=C\_e\times0.005\frac 1 {\text{step}}$ wherein $C\_e=\pi d\_e$

$s\_e=\frac {0.005\ \text{steps}}{C\_e}$

## Filament effects

Now, that we know the theoretical setting of mm/step or steps/mm ($e\_r$ and $s\_e$) for the firmware, we need to discuss how the filament impacts this. First of all, the calculation above holds only true for pushing even thickness filament that the teeth bite in evenly. If the filament does change in thickness, the effective diameter of our gear changes, and as a result, the extrusion changes. A thicker diameter filament does not get dug in as deep, the effective diameter goes up, and thus the circumference $C\_e$.

Similarly, different filaments do have different hardnesses.[see also here](https://3dprinting.stackexchange.com/a/8396/8884) A test on hardness was done for [this paper](https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1026&context=aseeil-insectionconference), exploring Type A and Type D Shore hardness. Type A is a flat tip, while D is a sharp tip. The last is somewhat similar in the effect to the hobbed gear. The difference means that the teeth don't bite the same, affecting the effective diameter of the hobbed gear $d\_e$.

This behavior is **filament dependant**

## Nozzle behavior

The next part we have to look at is the behavior in the hotend and nozzle. In perfect conditions, the heat zone would melt up the filament fully and ensure a stable, laminar flow through the nozzle as it necks down the material from its starting diameter to the extrusion width.

Assuming our filament stays laminar, then the flow of material in is exactly the same as the flow out, nothing stutters. But if the factors in the pipe are off, then we get turbulent flow.[further reading](https://en.wikipedia.org/wiki/Laminar_flow)

So, the flow is not necessarily linear, and we can easily identify several factors that impact the behavior in the nozzle. Let's do a quick rundown of different factors:

* Materials do expand differently (with factor $\alpha$) under heating, thus impacting the volume of the material in the nozzle, which in turn impacts the volumetric flow rate and pressure in the nozzle. This is **temperature** and **material dependant**.[further on this](https://3dprinting.stackexchange.com/a/8384/8884)
* The viscosity of the material has a huge impact on the flow behavior. Most plastics viscosities are **temperature dependant** but also **material dependant**
* The nozzle shape can have an impact on the flow rate to a small degree (mainly if it is uneven or rough). The most influential factor, however, is the **nozzle diameter**, which directly impacts flow rate.

### How all that effects flow in a nozzle

Let's assume flow is ensured and that we can ignore friction of the nozzle on the material. Then we get the Freeman formula for Flow:

$Q=Av$ where Q is the Volumetric Flow in 0.001 m³/s, A the area of the nozzle in m², v is the velocity at the exit in m/s.

$A=r^2\pi$ is the well known circle formula, r is the nozzle radius, which is half the nozzle diameter  
$\ \ \ A\_{0.4\text{ mm Nozzle}}=1.256\times10^{-7}\text{ m}^2$

$v=\sqrt{2P}$ is the Bernoulli's Equation, telling us that the flow velocity is pressure dependant (see Nozzle Behavior point 1). As a result, we get that the volumetric flow through our nozzle depends on the pressure like this:

$Q=\sqrt 2 \pi\times r\times\sqrt P$  
$\ \ \ Q\_{0.4\text{ mm Nozzle}}=1.777\times10^{-7}\text{ m}^2\times\sqrt{P}$

The pressure in the nozzle is the sum of the pressure generated by the force with which the material is pushed in by the extruder ($P\_F=F/A'$) and the material expansion in the nozzle ($P\_e$).

As we established up in the Extruder part, the extrusion rate is somewhat dependent on the effective diameter of the hobbed gear $d\_e$. The effective diameter of the hobbed gear also has an effect on the pressure in the nozzle: how deep the teeth cut into the filament determines the force they transmit. The other factor that impacts the force transmitted via the filament is the extrusion speed $v\_e$, thus we write $F(d\_e,v\_e)$. Atop that, the actual filament diameter $A'$ plays another factor, as explored under filament effects. The thermal expansion, which is dependant on the material coefficient $\alpha$ and the Temperature increase $\Delta T$ adds to the pressure in the nozzle, thus we write $P\_e(\alpha,\Delta T)$. So the expression for the volumetric flow out of the nozzle is

$Q=\sqrt 2 \times r \pi \times\sqrt {\frac{F(d\_e,v\_e)}{A'}+P\_e(\alpha,\Delta T)}$

If this flow is laminar and non-turbulent could be read from the accompanying Reynolds number $Re$, which is dependant on the (dynamic/kinematic) viscosity $\mu=\frac \nu \rho$ (rho is the density). Viscosity is temperature dependant, thus we write $\nu(T)$. Last factor in the formula for the Reynolds number is the hydraulic diameter $D\_H$, which for our case boils down to the diameter of the nozzle, so $D\_H=2r$. For our case this gives:

$Re=\frac{2r\times Q}{2 r^2 \pi \times \nu(T)}=\frac Q {r \pi \times \nu(T)}=\frac {\sqrt 2}{\nu(T)} \sqrt {\frac{F(d\_e,v\_e)}{A'}+P\_e(\alpha,\Delta T)}$

## tl;dr / Conclusion

Ok, the theory is all nice, but how does all that affect our print?!

1. It's a good idea to calibrate for one material and use that as your benchmark and then create profiles for your other materials measured against this benchmark
2. If our temperature is lower or higher than in calibration, the flow is different and, as a result, the extruder-steps don't match up. Instead of altering the steps, we should **adjust** the volumetric flow value up or down to match to the particular material. Usually, it is called either `Flow` or `extrusion multiplier` in the software. In the calculations above it is $Q$.
3. A different material or material mix (like a new color or different brand) has a different thermal expansion, thus we **adjust** the `flow`/`extrusion multiplier` in our profile as needed.
4. If we swap the nozzle, we get a different flow characteristic. Most of the flow calculations for such change is already done in the slicer, we usually don't need to alter the profile further, but tiny changes could be done by adjusting the `flow`/`extrusion multiplier` value.
5. extruder clicking can show over extrusion, but it **also** can show that the print height is not achieved as it is calculated for and point to errors in **bed leveling** if happening in the first layer(s) and to errors in the **z-Axis** if it happens later in the print.


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### Answer by username_6 on 2021-07-30. Score: 0

I have this same extruder on my Voxelab Aquila.
Default E-steps are 93. Using the printer menu to feed 100 mm of eSun black PETG filament I measured a feed length of 66 mm, so 34 mm under extruded.

Using the formula 100 / 66 \* 93 gave me a new E-step value of 140.9.
Upon re-testing, I found I was now over extruding by 34mm.

This formula is missing something. I've chased E-step values up and down before so decided to try something different.

Since the new E-step value was over extruding by the same as the original E-step under extruding value, I added the original E-step value to the new E-step value and took the average. 93 + 140.9 / 2 = 116.9.
I re-tested this value and found it was over extruding by 0.4 mm. Since I was still over extruding slightly I calculated a new E-step, 100 / 100.4 \* 116.9 = 116.4. Took the average between the 2 again and got a final E-step value of 116.6.

Hope this helps anyone else having issues calibrating E-steps.


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