Critical radius of insulation refers to the radius at which the heat transfer through the insulation is minimized. In this case, we need to calculate the critical radius of insulation for a steel steam pipe with given dimensions and properties.

Given:
Inner diameter of the pipe (D_i) = 10 cm
Outer diameter of the pipe (D_o) = 11 cm
Thermal conductivity of insulation (k) = 1 W/mK
Convective heat transfer coefficient (h) = 8 W/m^2K

To calculate the critical radius of insulation, we need to consider the heat transfer through the insulation and the heat transfer by convection from the outer surface of the insulation to the surrounding air.

First, let's calculate the radius of the pipe:
Radius of the pipe (r) = Inner diameter / 2 = 10 cm / 2 = 5 cm = 0.05 m

Now, let's calculate the radius of the outer surface of the insulation:
Radius of the outer surface of insulation (R) = Outer diameter / 2 = 11 cm / 2 = 5.5 cm = 0.055 m

Now, let's calculate the critical radius of insulation (r_c) using the formula:

r_c = (k / h) * (R - r)

Substituting the given values:

r_c = (1 W/mK / 8 W/m^2K) * (0.055 m - 0.05 m)
= 0.125 m

Therefore, the critical radius of insulation is 0.125 m.

Explanation:

The heat transfer through the insulation is given by Fourier's law of heat conduction:

q = (k / L) * A * (T1 - T2)

Where:
q = heat transfer rate (W)
k = thermal conductivity of insulation (W/mK)
L = thickness of insulation (m)
A = surface area perpendicular to heat flow (m^2)
T1 = temperature on one side of the insulation (K)
T2 = temperature on the other side of the insulation (K)

In this case, we assume steady-state heat transfer and neglect the thermal resistance of the steel pipe since the conductivity of steel is much higher than that of insulation.

The heat transfer by convection from the outer surface of the insulation to the surrounding air is given by Newton's law of cooling:

q = h * A * (T_s - T_inf)

Where:
h = convective heat transfer coefficient (W/m^2K)
T_s = temperature of the surface of insulation (K)
T_inf = temperature of the surrounding air (K)

To minimize the heat transfer through the insulation, we need to minimize both the heat transfer through the insulation and the heat transfer by convection from the outer surface of the insulation. The critical radius of insulation is the radius at which these two heat transfer mechanisms are balanced, resulting in minimum heat transfer.