I would add a couple of suggestions for the design:
You are using 741 OP-AMP, which is not rail-to-rail, and you're using it for driving the base of a transistor: what happens is that when the output of the 741 is high, it will be at about Vcc - 1V, that is enough to keep the transistor on. I would suggest using a rail-to-rail OPAMP or adding a small resistance to the emitter of the transistor to limit the current when the input is high (could be even better because you mantain the fan at a slower speed but still cooling).
When designing with sensors, such as photoresistors or thermistors, it's better to - first know the value at room temperature of these sensors - and then picking a potentiometer just bigger to simulate the behavior of this sensor, and check that the circuit is working.
UPDATE: from the datasheet, the typical voltage swing is 13-14 V (you can measure the exact maximum value just measuring the positive saturation voltage), and by design the lose in the range tends to be more in the upper rail, because the output stage has a \$ {V_{CE}}^{sat} + {V_{BE}^{ON}} \simeq 0.2 + 0.6 \simeq 0.8 V \$.
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UPDATE 2: Now I see that you are powering your circuit at +12V / 0V, that is NOT the exact supply voltage specified for the 741 OPAMP: it requires a dual-rail, \$ \pm 15V \$ fix this as the first thing.
You can see as your OPAMP is outputting 10 V instead of 12, and 1.2V instead of 0; the first, with the drop over the resistor, makes the transistor always on, as you can see that the base voltage is 11V, enough to keeping it on.
And...why did you use a diode to simulate a fan??? Seems a quite different load.
UPDATE TO THE UPDATE:
I'm glad that it works, at least the simulation: however, you are still using a single rail supply (+12:0, +15:0). The 741 wants +15:-15, so the best thing to do is CHANGING THE OPAMP. It's not expensive at all and you can use a rail-to-rail (again), that is better for single supply applications, down to 3.3V if you need that; or, for your case, +12 or +5.
This is an option, here there is plenty, you have only to choose, based primarily on availability for your purpose. For the simulator, you can also find many options.
Yes, it is possible to charge an NiMH battery with that circuit. Just not reliably, and no, it would not be safe.
∆V charge termination is fiddly and unreliable at best. Worse, it is not a state, but a single event, and events can be missed. This is an easily missed event, so there is the definite possibility of uncontrolled overcharging, and it is even fairly likely.
And of course, cells will often exhibit false voltage depressions at various points during a charge, especially fresh NiMH cells. So its just as likely your microcontroller will terminate the charge much too soon.
dT/dt is the most reliable, safest, and preferred method for charge termination of both NiCd and NiMH cells when you are charging at a relatively high rate like 1A. This requires a temperature sensor, which you will need anyway if you want to build a safe charger. You cannot make a charger that is safe without it being able to sense the battery's temperature. If a charger can't do that, it cannot be safe. It's as simple as that.
Temperature sensors are a dime a dozen and cheap as anything though. Any thermistor will do as long as you know its resistance tables and have them programmed on your µC. You simply charge at a constant current, with a hard voltage limit as well. I.e., it will charge at 1A unless the voltage needed to maintain 1A rises above some set level. If it does, the charger will simply stay at that maximum voltage and the current will fall. This is because the chemical reactions in the cell are not purely current driven, there are different (and unwanted) reactions that normally don't occur, but can if enough voltage is applied to the cell. Most manufacturers seem to agree that this cutoff is about 1.8V per cell for NiMH batteries. So you want to charge at 1A, but only if you can do so using less than 1.8V.
You do that while constantly doing a running calculation on how fast the battery's temperature is rising. As soon as it starts increasing faster than 1°C per minute, terminate the charge.
Even this alone is not enough however, because a cell with high internal resistance will charge at a lower current (due to the 1.8V maximum) and may not ever actually be dissipating enough power to heat that quickly. It will instead just heat up until it vents (explodes in a controlled manner) from pressure. A safe charger requires a failsafe, which is a temperature charge termination failsafe. Terminate the charge due to temperature rise, but also terminate it if the battery reaches a certain temperature. 50°C is really the maximum you should ever aim for, but 45°C is going to be a little safer and will probably help with cell longevity.
Finally, while this is neither required for effectiveness or safety, it is good to have a timer termination as well. Yes, a third charge termination system. This is literally just a timer, set for the maximum reasonable amount of time you feel like a cell could ever take to charge (remembering that old cells with higher internal resistance could take quite a while longer to charge than brand new cells) and always terminate the charge due to the timer if the other termination conditions fail to ever occur first. This prevents the charger from cooking a weak cell for hours or days, keeping it at 40 degrees C or something but the cell is at equilibrium thermally, and just sits like that. NiMH cells do not like heat, and that will artificially age the battery very quickly, and have a very negative impact on its current condition and long term usable life.
Best Answer
Here's a poor man's 2-BJT version. POT+Q1 set the voltage at the base of Q2.
(V(R2)-Vbe(Q2))/R1 sets the current.
This circuit, however, is too sensitive to V1 variations. Furthermore (depending on the resistor values), there could be an excess current, for heavier loads (smaller YOUR_LOAD values in Ohm).
simulate this circuit – Schematic created using CircuitLab
A better solution is the "not so poor man's" 6 components solution (capacitors excluded). Some remarks:
simulate this circuit