How can I start my Heatsink fan of Compaq Laptop

The laptop will need to be dismantled down to the motherboard level, use a can of compressed air attach straw and blow out all ports, heatsink and fan. It might be a good idea to remove both for closer inspection and several good blast of air for good measure, and check the thermal paste between the heatsink and processor ic has not dried out.

check the underneath of the laptop, there should be several access panels, remove these and look for fans, clear them of any dust you can see, the main fan should be blowing through a heatsink (lots of small fins) try to get to the part between the fan and heatsink as sometimes you can get a lot of dust build up there.

I have a feeling that it has been overheating for some time and this explains the shutting off. Overheating is the killer of laptops, caused by dust and fluff build up in the cooling fan and copper heatsink / radiator areas of the laptop.
(Assuming the battery is charging, the adaptor is sending current to the laptop (check this by getting someone to use a multimeter with the red probe set to the internal jack pin and the black to the external part of the jack and at a DC range of 20 volts.)
Your only option now is to have the laptop stripped to clean out the heatsink and fan assembly, not hard to do if you take your time and see if that works. You may be fortunate and the processor and graphics chip may not have been damaged by overheating, however, preapre for the possibility that the motherboard may need replacing

if the cooling fan is not working then the laptop will shut off due to heat issues. Than could just be clogged stopping it from spinning. I have seen the cooling fans go bad. The fan will def need to be repaired or replaced if not working. You can probably find one for it and replace it yourself if you have any technical experience

If the laptop shuts down and won't boot up until it has cooled down, then the CPU is overheating and goes into thermal shutdown. This is caused by either a faulty fan or the fan and heatsink assembly is clogged with dust.
If this is not fixed then eventually the CPU will fail completely.
Check the fan to see if it spins freely if not then replace the fan.
If the fan and heatsink is clogged with dust then remove the dust from the fan and heatsink fins.

“My Toshiba laptop suddenly shuts down by itself without any warning. Sometimes it works fine for hours, sometimes it shuts down in 10-15 minutes.” This complaint we hear from our customers over and over again. About 15-20% of all Toshiba laptops we get for repair, suffer from an overheating problem. Yep, OVERHEATING!
This is one of the most common problems with Toshiba laptops we deal with.
Indications of laptop overheating problem:

1. The keyboard and the bottom of your laptop are very hot when the laptop is working.
2. The CPU fans are working all the time at maximum rotation speed and operate much louder than before.
3. The laptop suddenly shuts down by itself without warning. When it just started, the laptop was shutting down after 1-2 hours and how it shuts down after 5-10 minutes of operation.
4. The laptop works fine when it runs idle, but shuts down as soon as you start using any memory demanding applications (DVD player, image editing software, video editing software, etc.).

Solution:
If the CPU heatsink is not clogged with dust and lint completely, you can use canned air and just blow it inside the laptop through the openings on the bottom and on the sides. It’s nice as a precaution measure, but it might not work if your laptop already has a problem and the heatsink is completely clogged.

1. Open the laptop case, so you can access the CPU fan and the heatsink. In some cases you can access the heatsink through the latch on the bottom of the laptop. Sometimes you have to open the laptop case all the way down.
2. Carefully disconnect the fan cables on the system board and remove the fan. If the fan makes unusual sound when it spins (grinding sound), I would recommend to replace the fan.
3. Clean the fan and the heatsink with compressed air.
4. I would also recommend removing old thermal grease from the CPU and applying new grease for better heat conductivity.

If the CPU heatsink is not clogged with dust and lint completely, you can use canned air and just blow it inside the laptop through the openings on the bottom and on the sides. It’s nice as a precaution measure, but it might not work if your laptop already has a problem and the heatsink is completely clogged.

1. Open the laptop case, so you can access the CPU fan and the heatsink. In some cases you can access the heatsink through the latch on the bottom of the laptop. Sometimes you have to open the laptop case all the way down.
2. Carefully disconnect the fan cables on the system board and remove the fan. If the fan makes unusual sound when it spins (grinding sound), I would recommend to replace the fan.
3. Clean the fan and the heatsink with compressed air.
4. I would also recommend removing old thermal grease from the CPU and applying new grease for better heat conductivity.

Remove your cpu/gpu heatsink, apply a ermal bonding, this will prevent shut off because of cpu/gpu overheating(chipset detects temperature faulty in the firsts seconds of operating your laptop), if this didnt resolve your problem, naybe and issue with your Mobo.

Thanks
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vcrtech

Try to clean heatsink and fan. Make sure fan is spinning(common problem).

It appears your CPU is overheating and is shutting down.
Does the fan spin at all when the laptop is switched on?
The fan could be faulty, if it does not spin freely.
To test the fan, try removing it and connect it to a 12 volt DC power supply, if the fan spins fast then it is OK and the fault is on the laptop, possibly the power supply to the fan. In this case you need professional help.
If the fan appears OK then the problem could be insufficient too old heat sink compound between the CPU and the heatsink. You will need to remove the fan and heatsink and the old compound and put some fresh heatsink compound on the CPU/heatsink and reassemble the fan/heatsink.