# Calculate output pressure of a syringe pump

I've built this syringe pump following those instructions. Now I would need how do I calculate the output pressure given the step rate of the stepper motor, and vice-versa.

I don't know if it is helpful anyway but I know that the relationship between the torque exerted by the motor and the linear force upon the syringe plunger is given by:

$$M = F r_a \tan(\alpha), \; \tan(\alpha) = \frac{p}{2 \pi r_a}$$

where $$r_a$$ is the radius of the threaded rod, $$\alpha$$ the thread angle and $$p$$ the pitch.

Thus we have:

$$M = \frac{F p}{2 \pi} = \frac{P r^2 p}{2}$$

where $$r$$ is the syringe radius. Is it right until now? Moreover, is it any useful? I found myself unwillingly stuck in this project but I never studied such cases.

The problem here is that the exerted torque depends on the load so I wouldn't know how to link output liquid pressure with motor step rate.

• Why not use the rate of volume change and apply the conservation of mass? Apr 21, 2019 at 18:27
• It depends on motor current limit, full-steps low-max-acceleration and higher-velocity and desired flow rate. Apr 22, 2019 at 17:24

The question asks how to derive output pressure from step-rate.
The answer does not perform all the calculations but attempts to define the variables.

Minimum required specs:

• s = Step incr. [m] (after gear reduction)
• (°/step * mm/rev(wormgear) * 360°/1000[mm/m])
• F = minimum stepper force [kg] ( that must exceed load to prevent skipping at speed)
• Torque * gear Radius, r [m] = F [kg] (converted from N-m)
• The minimum stepper force, F must exceed syringe input load force
• F=Pi/Area to avoid skipping

Pi = inlet pressure
Po = outlet pressure
L = length of tube
η = viscosity
V = volume of the fluid at outlet pressure
v = velocity of the fluid at outlet pressure

repeat for each concentric stage

$$Flowrate = \dfrac{π\cdot r^4 \cdot (P-Po) }{ 8\cdot η\cdot L} [m^3/s]$$

If below output is into air, can we assume P1=P2? NO.

It may be better to measure Flow rate with Force input for target syringe then define desired flow rate it is capable of then determine gear ratio and motor torque and step rate needed to perform task.. Yet considering the optimization of stepper speed is a controlled acceleration to prevent skipping.

If you have the open source GRBL Panel software (S/W) with these variables in an Arduino CNC shield, you can maximize velocity then control a ratio of this for flow rate, with this S/W better than anything else, that I have come across.

• On my first attempts I had tried like you said, but once I calculate the plunger pressure, then the needed force and finally the torque, how do I relate it with the step rate? Datasheets only provides maximum torque and some steppers (not this one) are also provided with the maximum torque as a function of the step rate. This does not tell the step rate to have that pressure at the plunger Apr 23, 2019 at 8:59
• You must do some R&D , get a variety of needle sizes , measure flow rates. The formulae don't cover initial stiction or stepper static force. Flow rate requires more force at some velocity where stepper forces reduce. A reducing step rate acceleration increases the stepper velocity possible significantly, therefore, the dynamic force at speed determined by step rate may be optimized by design. So I have no idea what you can achieve but it could be different from what I could achieve. At least the formulae helps you understand tradeoffs. Apr 23, 2019 at 12:24
• In other words define range of flows needed, velocity of flow, nozzle area, barrel area, linear step* area= step volume then ramp step rate ramp down and stop for a controlled volume. The fluid velocity will have some limit, so work back from this. Apr 23, 2019 at 12:34
• is this for a 3D printer? youtube.com/watch?v=YU5SdzRVtv0 Apr 23, 2019 at 12:55