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When this happens, does it ever fail to start until it cools off? If so, when that happens, I need you to confirm fuel pressure and check for spark. Here are a couple links that may be helpful. I realize the pump is new, but there could be a blockage.
Next, what you described sounds like an issue with the crankshaft position sensor. When they start going bad, they will internally short when they get hot. The result is no spark, which can shut down the engine. Interestingly, they don't always set a code. Take a look through these links. The second one describes symptoms of a bad sensor.
I'm not sure what engine you have, so I went with the most common one, the 5.7L Here are the directions specific for sensor replacement. If you have a different engine, let me know. Also, the attached pictures correlate with the directions.
1. Remove the electrical connector.
2. Remove the sensor hold down bolt.
3. Remove the sensor from the timing cover.
NOTE: When installing or removing a crankshaft position (CKP) sensor make sure the sensor is fully seated and held stationary in the front cover before torquing the hold down bolt into the front cover. A sensor which is not seated may result in erratic operation and lead to the setting of false codes.
4. Inspect the sensor O-ring for wear, cracks or leakage. Replace if necessary. Lube the new O-ring with clean engine oil before installing.
NOTE: Make certain that the Crankshaft Position (CKP) sensor mounting surfaces are clean and free of burrs before installing the CKP sensor.
1. Install the sensor into the timing cover.
2. Install the sensor hold down bolt.
Tighten the hold down bolt to 8 Nm (71 lb in).
3. Install the electrical connector.
Let me know if this helps or if you have other questions.
Images (Click to enlarge)
Wednesday, June 19th, 2019 AT 8:59 PM