Why are you concerned about closed refrigerant loops? Have you even started modelling this yet? Do you have any converged process unit ops? If you are forced to model this process for your class, then you should concentrate on getting some results that show you tried.
For the refrigerant loops themselves, you just specify completely the conditions of your stream 5 and stream 14 -- no loops in the model. The refrigeration compressors and condensers will be at the tail end of the refrigeration portion of your model and you just specify those units so that you match what you know forced stream 5 and stream 14 to be. Designate the outlet tail streams '5X' and '14X'. When your model is complete, stream 5 and stream 5X will be identical. Stream 14 and stream 14X will be identical. You don't need to use recycles in the simulation model.
You will need the flow rate of some stream, so make one up for the natural gas if it was not given to you. You also need a starting point for your refrigerant streams. Make composition estimates for your natural gas and refrigerant streams if they are not supplied. So beginning the model, you have complete definitions for streams 5, 14, 22. With these, you can solve HX-1. See my previous comments about keeping a large temperature approach with excess refrigerant flow. With HX-1 solved, you try to add HX-2 and HX-3. Save your work often. I expect this simulation to be very frustrating for you. How many months do you have to complete it?
I will tell you my approach to this model if I only had the info you provided. I would start with stream 5 miniscule flow! This leaves you with one mixed refrigerant loop! That is still not easy to model, but simplifies things. After solving for a single mixed refrigerant, then you can go back and add flow to your propane refrigerant loop if you want to improve overall efficiency. The advantage to this procedure is that you may get a converged model and an 'answer' even if it is not completely optimized for efficiency.