Introduction:
Heat transfer is the process of energy transfer from a region of higher temperature to a region of lower temperature. In this case, we have a spherical shell with an inner radius R1 and an outer radius R2, and we want to determine the radial distance from the center of the shell where the temperature is halfway between the temperatures at the inner and outer surfaces of the shell.

Heat Transfer in Spherical Shells:
The rate of heat transfer through a spherical shell can be determined using the formula:

Q = 4πk(T1 - T2)/[1/(R2 - R1)]

Where Q is the rate of heat transfer, k is the thermal conductivity of the material, T1 is the temperature at the inner surface of the shell, T2 is the temperature at the outer surface of the shell, and R2 - R1 is the thickness of the shell.

Temperature Distribution:
The temperature distribution within the spherical shell can be assumed to be radially symmetric. This means that the temperature at any radial distance r from the center of the shell can be expressed as:

T(r) = T1 + (T2 - T1)(r - R1)/(R2 - R1)

Finding the Radial Distance:
To find the radial distance from the center of the shell where the temperature is halfway between T1 and T2, we set T(r) equal to the average of T1 and T2:

T(r) = (T1 + T2)/2

Substituting the expression for T(r) into the equation above, we get:

T1 + (T2 - T1)(r - R1)/(R2 - R1) = (T1 + T2)/2

Simplifying the equation, we find:

r - R1 = (R2 - R1)/2

r = (R2 - R1)/2 + R1

Therefore, the radial distance from the center of the shell where the temperature is halfway between T1 and T2 is given by (R2 - R1)/2 + R1.

Conclusion:
In this analysis, we have determined the radial distance from the center of a spherical shell where the temperature is halfway between the temperatures at the inner and outer surfaces. This is achieved by considering the heat transfer through the shell and the temperature distribution within the shell. The final result is given by (R2 - R1)/2 + R1, which provides the desired radial distance.