What you are describing is a contract position, not consulting. For consulting work you bill them for hours worked, all hours. For contract work they pay you a fixed sum per week, month, or length of the contract. I've worked both types. Consulting work is somewhat better because they can't abuse your hours. If they do, they'll pay for that abuse. The downside is that you are responsible to pay your own payroll taxes, medical, personal time off, etc. For contract work the pay and associated taxes are often handled by a 3rd party, sometimes directly by the company, and there can also be benefits included like paid holidays, medical, and some even offer 401k.
If you take this position, make sure to read over the contract terms and ensure there are limits to the number of hours they can ask you to work per week without overtime pay. It is easy to get screwed on contracts. Be careful.