Battle of Eknomus

Roman formation
The plan of the 2 consuls was to row to the neighborhood of Carthago. They were rowing with Sicily on their right, coming from Messana. If they divided their infantry over the battle ships with 120 legionnaires on each ship, like Polybius tells us, there were 35 ships needed for a legion. This means there were approximately 150 battle ships plus a number of towed boats for the horses. One can argue there were also ships needed for the auxiliares. Possibly but they are (as often is the case) not mentioned and Polybius reports the best men were hand picked. Above that there had to stay garrisons to guard Sicily.

The fleet was, as always in the Roman army, in an orderly formation. Each ship was to protect the ship in front of him,and therefor not straight behind her, but beside her. Thus forming a triangle. At the bottom of the triangle a 3rd squadron towing boats with the horses. Behind this line the 4th squadron called the triarii to protect the rear side of the formation. A really strong formation.

The Carthaginian formation
Polybius reports there were 150,000 men and 350 battle ships on the side of Carthago. These numbers are more than likely exaggerated. Feeding 150.000 men would have been a logistical hell. Gathering 100,000kg grain every day to begin with, while the men were at sea and couldn't help foraging. The aria would soon be cleared of food and the distance between place collect and delivery would expand rapidly. I think a formation of approximately 150 war ships would be quite impressive.

The Punic fleet assembled in Heraclea and moved to the Romans. The left wing was under command of Hamilcar and the right wing under command of Hanno. The most left part of Hamilcar's wing sailed under an angle and close to the coast, to prevent the enemy to circumvent the Carthaginian formation.

Phase I
The first part of the Carthaginians was a good one. When the Romans attacked their center, the center turned around and fled, seemingly. Now the Romans made a mistake that easily could have cost them the victory. In the hurry to keep up with the much faster Carthaginian ships the 1st and 2nd squadron lost contact with the other two, much slower squadrons.

Phase II
Now the two parts of the Roman fleet were separated, the way to victory for Carthago was wide open. There were 2 options: Either attack the one part or the other but not both at the same time. Hamilcar's center should have been keeping busy the first 2 squadrons while the others attacked the slower part or almost all together the 1st and 2nd Roman squadron.

Both parties sofar ignored (or just hadn't invented yet) an important and basic principle: concentration of your means and creating (locally) superiority in numbers. The Romans should have kept their strong formation together and the Carthaginians should have attacked only half of the Roman fleet.

Phase III
Hamilcar's center turned around and attacked the Romans. They were short in numbers and still didn't know how to deal with the corvus. Once Hamilcar was beaten the two Romans squadrons turned and hastily came to the aid of the other two squadrons that were having a hard time. The third squadron had to disconnect the horse boats in order to be able to defend themselves. Now the Romans were superior in numbers and quality and this part of the Carthaginian fleet was defeated.