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Here's the LTspice simulation of a circuit to convert the 0Ω to 5.2Ω rheostat (U2) rotation into a 5V to 1v signal for a PWM input module.
The output range can be adjusted by the value of the U4 pot and the offset by the U3 pot.

upload_2017-8-27_14-29-29.png
 
The standard resistor method does use low, med and high, but the low speed resistor is selected to the motor will start.
It would be easy enough to arrange the adapter to generate a voltage in the range, say, 1V-3.3V rather than 0V-3.3V, to ensure the motor would start.
 
It would be easy enough to arrange the adapter to generate a voltage in the range, say, 1V-3.3V rather than 0V-3.3V, to ensure the motor would start.

My point was the OEM did not use PWM. So to get over "stiction", they started at high speed first. Typical resistor resistors to select speeds went in the more natural low medium and high, although ceiling fans start out on high first too.

The suggested PWM module was more complicated than I expected. It seems like a reasonable cost although there may be "over the big pond" transportation costs. Flipping looks like it can be done in the configuration or as crutschow did it.

The manual said that the inputs can't exceed 3.3 V which probably means they can't exceed the supply voltage unless it's current limited and the supply sequencing could be an issue. Probbably the specs are not in the manual of exactly what's required. One of my pet peave's is when manufacturer's don;t show what the equivalent circuit of any outside world inputs are.

It might be too complicated for the TS/OP too.

Being critical of the design (not that it's bad), but one caveot is it looks like he assumes a 13 V battery and it may need a regulated source. I think he's got the minimum current covered through he rheostat's wiper. 10 mA is the absolute minimum I'd go for for the OEM rheostat.
 
Here is my suggestion. I haven't worked out component values, but the idea is that you still use the rheostat current to run the motor, but add to that current in a fixed ratio to allow for the larger motor.

R1 and R2 are low-value current sensing resistors, and their ratio will be the ratio of rheostat current to added current. For instance, if you need three times the current, so you need to add twice as much current as the rheostat delivers, you could have R1 at 0.2 Ω and R2 at 0.1 Ω.

Design considerations:-
The Op-Amp needs to be a rail-to-rail op amp.
The MOSFET is a P-type and needs to take all the additional current, and will generate a lot of heat.
The resistor and the capacitor connected to the inverting input are to reduce high-frequency gain and stop oscillation. 10 kΩ and 1 μF would be a good start.
R1 and R2 need to take the rheostat current and the additional current respectively, so they need to be suitable power resistors.
 

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it may need a regulated source
I doubt it. This is a car heater fan we're talking about. Will slight speed variation with battery voltage really matter? It might even be an advantage, since fan speed would tend to rise somewhat in line with engine output or fall if the battery were stressed.
 
The MOSFET is a P-type and needs to take all the additional current, and will generate a lot of heat.
The P-MOSFET is shown upside-down.
The source terminal goes to the plus supply (assuming plus is on the top of the schematic).
 
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Heat dissipation is certainly a problem with the linear approach. I'm surprised the original rheostat, given its small physical size judging by the pic, can handle it.
 
The P-MOSFET is shown upside-down.
The source terminal goes to the plus supply (assuming plus is on the top of the schematic).
Ooops. Sorry.
You are correct on both counts, the +ve is the top of the schematic and the MOSFET should have the source on the +ve and the drain feeding current to the motor (via a sensing resistor)
 
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