Dear Mr. Ahmadreza Jamali

Greeting

The behavior of heat transfer through the tube or ball not like the behavior of heat transfer from the rectangle object, because the tube or ball has a critical radius, and that mean if you insulate the tube there is a heat transfer through the (wall+ insulation) , but if you neglect the value of critical radius and increase the insulation thick to approaching to the critical diameter at that time the heat transfer not reduced but increase and that normal behavior.

you need some assumptions to solve problem like this

One of the best reference you can follow ...

"Engineering Heat Transfer" by N. V. Suryanarayana

Always I try to find more references and send it to you

With my respect

Thank you for your valuable comment. This book is a good reference and helps so much.

Hello Ali D. Salman ,

I agree with what you say about a critical radius for thermal insulation around a central cylindrical source of heat such as a lagged hot pipe, but in the present situation we have two sources of heat: 1) the central conductor, and 2) the outer shield (screen). In the case of the central conductor, there is an initial increase in the thermal resistance due to the lower thermal conductivity of the surrounding insulation, but this effect is countered by the increase in the surface area of the insulation due to the further increased radius (thickness) of the insulation. However, at the outer shield (screen) part of the heat due to Joule heating flows inward, and, here, both the decrease of the thermal conductivity of the insulation and the decrease in the surface area as the radius decreases both increase the thermal resistance. It is not clear to me that the heat flows exactly cancel each other, but I suppose it is possible - the results of Ahmadreza Jamali seem to indicate such is the case.

Regards,

Tom Cuff

For most insulating material the thermal conductivity increases with temperature. The magnitude of the effect depends upon the heat flux, dimensions and materials. However, the increased thermal conductivity certainly affects the heat losses and the temperature field: suggest that you look at simple potential theory (e.g solutions to Laplaces equation for the spherical geometry), or perhaps simpler use the old Carlslaw and Jaeger book on transient heat transfer for analytical solutions to this.