This was titled a Simple Heat Transfer Solution. It is in fact, an impossible question to answer well, because it is not well enough defined.
Let me suppose that I can replace this scenario with a solid conductive rod at a known temperature, externally supported by a ceramic bush of width x inch and thickness y inches mounted on an an insulated surface. The ceramic's thermal conductivity is ? and the insulating mounting surface has a thermal conductivity of ?? and temperature ???
I was "hoping" for a simple conduction solution. Using Fouriers formula. I don't understand how to apply the "q" from the heater. Ideally we are trying to use the bushings as the insulators. Your simplified solution will work I think. In other words, the shaft is hot (known temp) and the q is known, using known k values for the bushings. What bushing thickness (y inches) gives what temp on the outside surface of the bushing.
An electrical analogy will help you, I hope. You have a rod at a known potential, fixed to a substrate by two resistors of known value. What is the potential of the substrate at the point of connection?
You don't know, because you have not specified the effective resistance of the substrate. So the potential could be ground, or the rod potential, or somewhere in between. To put it in temperature terms: If the place that the bushings touch is at ambient temperature, then the bushings could be at ambient temperature (if the mounting surface is conductive, or the bushings could be at rod temperature if the mounting surface is very much more insulating than the bushings.
It is known that too much insulation on slender pipes can increase the heat lost.
This could be analogous, if the bushings are sinking heat all around their circumference. Are they? If they only touch the substrate at one radius, then greater thickness represents lowered heat loss.
But what is ideal for you?
1) Minimizing heat lost through ceramic bushes to the substrate?
2) Minimizing heat lost from the cylindrical surface at the air boundary?
3) minimizing the sum of these losses?
4) Minimizing trhe capital cost of the device?
5) Minimizing the running cost? Minimizing the sum of these two?
Don't have enough grasp of your configuration and constraints to be useful, I'm afraid.
I thought this would be simple.... The substrate is aluminum. The shaft is heated internally to a measured outside (O.D) temperature. The shaft is mounted/supported by two ceramic (or other material) insulators. The insulators are flat on the bottom and bolted down to a flat aluminum plate. Just like a horizontal hot water tank, resting on two concrete blocks.
I have got a deal of pleasure out of this heat transfer puzzle. Mostly because you keep telling us how simple and basic it is....
Anyway, a 300 watt heater in a tube could put the surface temperature somewhere between 40 degC and 300 degC depending.
You mentioned you want to minimize temperature rise of the substrate, so here's the answers you were looking for:
1) Make the ceramic standoffs as long as possible.
2) Make their transverse cross section as small as possible, consistent with the strength needed.
3) Radiation will heat the substrate - somewhere between 3 and 30 degrees, possibly, so there's no sense in reducing the conducted heat flux at the standoff's base to less than the radiative flux heating the surface away from the ceramic standoff.
4) If radiative heating is excessive, make the opposed surfaces reflective or white. Failing an acceptable reduction in sub strate temperature, increase the standoff distance.
5) Maintain good airpaths round the tube and around the standoffs for convective cooling. Perforate the ceramic for enhanced air cooling as needed.