In your estimate of cold gap, you introduce a factor of 6.35/3.14mm (2mm) as an "extra gap". i would like to know the origin of this factor.
This is simply a "safety factor". The residual gap, when the kiln is in full operation, is always dependent on the temperature difference between the shell and the tyre. This temperature difference varies with the process, fuel and feed chemistry and other factors such as remaining refractory thickness, seasonal changes and so on. The "extra gap" is there to manage these variations without pinching off the shell, which could otherwise result in crushing refractory.