Fig, shows a system of two concentric spheres of radii r_(1) and r_(2) and kept at temperature T_(1) and T_(2) (T_(1) gt T_(2)) respectively. Find the expression for radial rate of flow of heat through the substance.

Fig, shows a system of two concentric spheres of radii r_(1) and r_(2)

1.	Fig, shows a system of two concentric spheres of radii r_(1) and r_(2) and kept at temperature T_(1) and T_(2) (T_(1) gt T_(2)) respectively. Find the expression for radial rate of flow of heat through the substance.
Answer» Solution :Consider an elementary portion of TWO spheres between two concentric spherical sheels of radius x and `(x+dx)` fig. Let T and `(T-dT)` be the temperature of the inner and outer part of the elementary portion of the shells. Then RATE of FLOW of heat through the elementary portion is `(dQ)/(dt) = (-KA[T-(T-dT)])/(dx) = (-KA dT)/(dx)` Here -ve SIGN shown that the heat is lost by the spheres. Surface area of elementary portion of two spheres, `A = 4 PI x^(2)` `:. (dQ)/(dt) = -K 4 pi x^(2) (dT)/(dx)` or `dT = (-1)/(4 pi K) (dQ)/(dt) x^(-2) dx` Intergrating both the side within the proper limits, we have `int_(T_1)^(T_2) dt = -(1)/(4 pi K) (dQ)/(dt) int_(r_1)^(r_2) x^(-2) dx` `[T]_(T_1)^(T_2) = -(1)/(4 pi K) (dQ)/(dt)[-(1)/(x)]_(r_1)^(r_2)` `T_(2)-T_(1) = (1)/(4 pi K) (dQ)/(dt)[1/(r_2)-1/(r_2)]` or `T_(1)-T_(2) = (1)/(4 pi K) (dQ)/(dt)(1/(r_1)-1/(r_2))` `=(1)/(4 pi K) (dQ)/(dt)(((r_2)-(r_1))/(r_(1)r_(2)))` `(dQ)/(dt) = (4 pi K (T_(1)-T_(2))r_(1)r_(2))/((r_(2)-r_(1))`.