When we enclosed our Lanatna Lanai we also added heat and AC. The county required our contractor to get a heat/AC survey to determine if our existing heat/ac units could handle the extra Square footage (approx 400sf). This survey was conducted by Mumms and it takes into consideration the type of windows and features (Low E, Insulated, heat gain etc) you are enclosing your room with, the extra SF to be cooled, the direction the added room faces to include the amount of full window space, and where the sun sets. After this survey we were able to add a heat/cooling duct directly to our existing furnace that provided EQUAL air flow to the added sf. The absolute worst senario would be to extend the duct work from your family area to the new lanai area. As you said you would be stealing heat/cooling air flow and it would just be uncomfortable. BE AWARE alot of contractors when adding heat ac to an enclosed lanai space will price it out as an extension from your family room so your contract needs to be specific as to how heat/ac will be piped into the new SF area.
If your heat/ac survey shows that your existing furnace/ac units cannot accomodate the new added sf...the ductless ac vent system is a great alternative. Miitisibuti Mr BIG is a very popular one and very effective and comes in diferent sizes. This feature looks good, and is a reasonably priced alternative.
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