How to obtain the cycle length as a function of the loading pattern for a pressurized water reactor?
In a French REP, REP is the French acronym for PWR (Pressurized Water Reactor), the loading plan has no direct influence on the cycle time. The first goal of the loading plan is to try to obtain a power density as equal as possible over the whole core volume to avoid the appearance of hot spots. For example, new fuel elements are placed at the periphery of the core where the thermal neutron flux is lowest, and partially spent fuel elements are placed in the core center where the thermal neutron flux is highest. At each 12 to 18 month period, the reactor will be deliberately shut down for the installation of a new loading plan, and a fuel element will be moved 3 or 4 times during its lifespan of about 5 years. Without this voluntary shutdown, the reactor could operate for another 2 or 3 years, depending on the average composition of the average 235U content. The reactor remains a critical mass up to an average 235U content of about 0.7 to 1%. If you use the French language, you can join Futura forum physique .
The short answer is that you put the loading pattern into a core simulator and let the core simulator calculate the cycle length. It isn't something you can do by hand.
If you want to know how to design a loading pattern that will meet a certain cycle length, that is a lot broader question. You might want to start with a book on "Nuclear Fuel Cycles". There is quite a bit of research going on now to try to optimize the fuel cycle using different optimization techniques such as simulated annealing, neural networks, etc.