I live in Seattle where we have a relatively small number of days where it gets hot enough to need AC. I live in a two story house where the two floors, each being 1200 sq ft in area, are connected by a large opening such that it’s almost like having one large room with a high ceiling. I’m wondering if a single zone split mini heat pump system installed on the top floor will also keep the lower floor cool enough. I don’t need it to be really cool, say 75 degrees, just cool enough to get through a few days of hot days each summer. I’m thinking of a system capable of providing 24000 to 36000 btu cooling. I don’t need heat from the system so I don’t really need a heat pump but almost all split mini systems seem to be based on heat pumps. I should add that I can use my gas furnace blower to help mix air via the existing ducts. Thanks!
Certainly 36 kBTU/h from a single air handler would make a huge difference in a 2-story house (1200 ft² each story) in Seattle, but you can get minisplits with one condensing unit supplying more than one air handler, and the air handlers are individually controllable. If you get a minisplit, you would want at least one air handler on each floor.
Since hot air rises and cold air sinks, yes, one head unit upstairs will affect both levels of the home, to the extent that you describe wanting.
Hard to know sizing without knowing your needs (both climate [recently hellish in your area as I write this answer] and insulation level of the building.) I manage just fine with a 12K btu (nominal) head in a building slightly smaller (1060 sq ft/floor) but very well insulated - I do have one head per floor, as I do use them primarily for heating and only occasionally for cooling, but for cooling the single head upstairs is often plenty. The "apparent effective output" seems to be considerably better than a similarly-nominal-sized window unit, perhaps in part from not being installed in a leaky window and poorly insulated from the hot part of itself, perhaps also due to more efficient fan/coil design (also much quieter than a window unit, even a much smaller output one.)
I chose the one head per compressor configuration as it's more efficient than the multi-head form, and cost differential was quite small. But not all mini-splits are equal on that basis, so shop carefully if operating cost/efficiency is a concern, as it is for me.
With a "more typically insulated building" 36K might be needed, or even be not quite enough.
My own shopping experience was that number of heads strongly influenced the cost, and BTU capacity of the heads/compressors was a much smaller impact than I expected (larger cost more, but twice the capacity was less than 10% cost difference, IIRC - but oversizing also makes it inefficient, so bigger is only better up to a point.) I had intended to get more smaller heads before I understood the cost model I found, so I fully understand using a single head located upstairs if you are only looking for cooling. Adding a downstairs head might nearly double your cost, at least based on what I found in my market.