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It sounds like they want to replace the heater core. Here is what I need you to do: Allow the engine to warm up to operating temperature. Turn the heat on high. Locate the heater core hoses that will run to the firewall under the hood. Both hoses should be hot. If only one is hot, then the core may be plugged. If both are hot, then I suspect a blend air door actuator has failed. The actuator is what controls the temperature entering the interior of the vehicle.
Now, the problem I'm having is that in the US, the i-30 isn't offered. However, I believe it is the same vehicle as our Elantra. So, I attached a picture below that shows where the heater core hoses attach at the firewall. I highlighted the hose connection points.
Do this test and let me know what you find. Also, here is a link you may find helpful.
Take care,
Joe
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Monday, August 26th, 2019 AT 7:21 PM