Why do you want a textarea to submit when you hit enter?
A "text" input will submit by default when you press enter. It is a single line input.
<input type="text" value="...">
A "textarea" will not, as it benefits from multi-line capabilities. Submitting on enter takes away some of this benefit.
<textarea name="area"></textarea>
You can add JavaScript code to detect the enter keypress and auto-submit, but you may be better off using a text input.