INCORRECT: Disappointed almost to the point of tears, an empty test tube was the worst outcome possible.
In this example, what the participle phrase modifies is missing.
CORRECT: Disappointed almost to the point of tears, the professor founded that the test tube was empty.
INCORRECT: Sitting on the park bench, the sun disappeared behind the clouds.
Where is the noun that sitting on the park bench modifies? It is not modifying sun or clouds, but a person.
CORRECT: Sitting on the park bench, I watched the sun disappear behind the clouds.
INCORRECT: Speeding through the tunnel, the station came into view.
The station was not speeding through the tunnel, and there is no noun that speeding through the tunnel modifies.
CORRECT: Speeding through the tunnel, the train approached the station.