I'll trust you on Q3 being NPN, I can't read the marking from your picture. That said, your schematic likely has collector and emitter swapped. No longer a Darlington stage, but still a form of cascaded transistor pair.

Keep the pot on short leads, the added capacitance could make for a nice oscillator when you don't want one. Be careful of getting the pot too low in value, the thing might go a bit nuts.

 

Stating the obvious, we don’t know what is inside of U1. It is probably a propriety ASIC, which is why you can’t find a datasheet. The SM5651 datasheet indicates the sensor likes to be driven with a constant current source of 1.5 mA. The sensor is configured as a bridge circuit, whose output is “corrected” for various errors (i.e. temperature and linearity) and scaled. In your case, the output of the sensor is 45mV when a differential pressure of 1 Pa is applied.

U1 will have a few functions; the main one is to generate a constant current source of 1.5 mA. (It could be DC or at some AC frequency.) Another is to “amplify” the 45 mV to 10 volts at the output. There could be other functions too which may further compensate for SOME the remaining errors of the sensor or provide additional features, but without its datasheet we will never know.

I think RPLaJeunesse (Post #19) may be on the right track – R8 and R9 do seem to have the configuration of gain resistors. There are still too many questions regarding the external transistors to speculate on, but you may not really need to know in order to scale the overall circuit to 250 Pa.

The easiest path to follow is to try changing the value of R8 as RPLaJeunesse suggests. There could be negative effects in doing this though such as making U1 oscillate or affecting other performance factors than what is advertised for the board. Be on the lookout for anything that may affect YOUR requirements.

If changing R8 does not give the results you need, then I would suggest adding an external gain of 4 amplifier as sghioto suggests (Post #12). The 2nd half of the LM358 isn’t needed. Rather than ‘just using it’, a better solution would be to just ground pins 5 and 6 and take the output directly from pin 1. (Why add the errors of another op-amp that you don’t need?) If the LM358 doesn’t respond fast enough for your requirements, you can substitute a faster op-amp such as a TL072. Keep in mind that a gain of 4 amplifier at the output will increase sensor gain, offset and thermal errors and similar errors from the U1 circuit by a factor of 4 also. If these prove to be a problem for your application, then you have no choice but to order a 250 Pa sensor.

 

The IC has pins 5 & 6 as differential inputs. Pin 4 is gnd (or V-). Pin 13 (across from pin 4) might be a power pin (V+?)
getting power from the "transistor mess"? The "sure thing" is that it's 16 pin...

 

I appreciate the responses very much, I'm going to grab all my solder gear and put a pot on R8, it's been a while since I even powered my soldering station.

As for powering, there are V+ and V- additionally in the row with the sensor wires, and those are connected directly to two pins on the IC as well, but I didn't draw them, since they are NC in the circuit.

The power pin that is in use is Pin 2. The 24VDC goes directly to that pin through a protection diode.

As for making the precision worse and so on, this is all fine. This is going into a ventilation system and it is used to compensate for filter degradation. There is a very low gain PI controller, controlling the fans closed loop on the value of the sensor, the precision is honestly fine even if it's 2%, as long as the signal is linear and the transient response is also pretty much irrelevant - everything is happening very slowly.

Since the setpoint values are found empirically by measuring flow vs pressure with an adjusted system and then entered as the very last thing, the actual measured values are also irrelevant, the actual and setpoint values just need to be within a certain voltage range and the actual value has to respond to pressure change in a linear fashion. That's it. Not a whole lot of requirements here.

I have around 35-120pa pressure depending on intensity setting and the 35 pa is out of the commercial controllers acceptable voltage range, so I can't set the setpoint low enough. I did not have any equipment to measure this before ordering the sensors, and due to my inexperience, I got the 0-1 kpa ones.

P.S.
The output clamp is done with a zener diode. It's not on the diagram but you can see it in the picture. It's marked WH or WA, that'd put it at around 10 volts.

 

I included possible errors to watch for in case your application was a precision one; most pressure applications are not fast, but may be precise. Do be on the lookout for oscillation at what you have labeled "Vout" on R9 and at the sensor drive (I+ and I-). Use an oscilloscope for this measurement as most meters will not respond to a high frequency oscillation. If there is oscillation, that will have to be dealt with also.

 

Well, I am afraid this one is going to be very anticlimatic.

Before soldering the pot in, I checked the sensor one last time on the bench, to make sure it's working. Well it isn't. No output anymore.
Nothing apparent wrong with it either, all the wires seem ok. So no idea what happened there - something got somehow damaged when measuring continuity with the meter. Other sensor still works that I have.

Since troubleshooting that really isn't on the cards, I'm giving up and ordering a pair of new sensors with the correct measurement range, since I have to now wait anyway for at least one sensor to arrive.

 

Ouch, that hurts, but we have all been there. Order the correct sensor, but in the meantime. . . The sensor may be OK, but U1 fried. Isolate the sensor from the circuit and apply 1.5 mA of excitation current to the sensor. Use a resistor and DC supply, with the resistor setting the current (start at a high resistance and work your way down until you get 1.5 mA). With zero differential pressure you should measure about 0 mV across the sensor output leads. With 1000 Pa differential pressure you should measure about 45 mV. If not, the sensor is bad. If you do measure 45 mV, you still have a 1000 Pa sensor for the next project.