When you talk about perfectly competitive markets having no profit, you're probably thinking of the term "economic profit". The sort of profit everyone usually thinks of is revenue minus cost, which is called accounting profit by economists so as to distinguish it from economic profit. Also economists are really bad at naming things. Economic profit is revenue-costs-opportunity costs.

In perfect competition, firms do make accounting profit, but they don't make economic profit.

Thanks for posting your model here and getting involved in the discussion. It's always good to be able to discuss these things publicly because I'm sure many people are learning a lot from it.