Calculating the yard capacity

Yes, there is such a thing - check out my track plan analysis formulas.

These formulas will enable you to estimate the number of cars the layout can support, the number of cars you can reasonably move (taking into account your passing tracks and staging), and from there you can estimate the number of trains you can expect (assuming a typical train length as suggested by your passing sidings).

Once you know the number of trains you will be able to run per session on the plan and the typical size of a train the plan supports, you will be able to estimate the yard capacity you will need, The general rule (as you mention) is double the cars you expect to be in the yard.

But to know how many cars you expect in the yard, you need to know how many trains you will be running - and you can't know that until you know the staging capacity and the length of your passing sidings. The formulas help you evaluate all that, and adjust as necessary.

Just as important as the number of trains, is the types of trains and possible destinations from your yard (although the two are most definately directly related - except you could have a lot of through trains on the mainline that travel to the same destination, with only one or two actually tasked to drop/lift cars). How many destinations/directions/routes cars can take out of the yard on different trains will affect how much switching there is. At minimum, you could have local cars in, and out. Or if it's a branchline junction, you could have mainline east, mainline west and branchline. The large yard at our club layout supports several different branchline trains, plus local industries, plus 4 possible mainline directions, plus a little bit of car storage for local industry loadings.

You also always want to have at _least_ one or two clear tracks for runarounds and arrivals, or the yard can be completely jammed.

Do you expect every train to be broken up and remade at the yard, or will some/many pass through with all/some just dropping off/picking up a cut of cars?  That can make a big difference on the capacity requirements for the yard.  So, you need to estimate the proportion/number of cars that will be matriculated within the yard in determining the number of cars occupying the yard at a given time.