Along cylindrical rod of diameter 240 mm with thermal conductivity of 0.6 w/m ⋅ k experiences uniform volumetric heat generation of 24,000 w/m3 . the rod is encapsulated by a circular sleeve having an outer diameter of 440 mm and a thermal conductivity of 6 w/m ⋅ k. the outer surface of the sleeve is exposed to cross flow of air at 27°c with a convection coefficient of 25 w/m2 ⋅ k. (a) find the temperature at the interface between the rod and sleeve and on the outer surface. (b) what is the temperature at the center of the rod?

temperature at the interface between the rod and sleeve: 75.87 °C

temperature on sleeve's  the outer surface: 58.42 °C

temperature at the center of the rod: 219.87 °C

Explanation:

Data

rod radius,

rod thermal conductivity,

heat generation,

sleeve outer radius,

sleeve thermal conductivity,

air temperature,

convection coefficient,

From figure attached, the unknowns are

temperature at the interface between the rod and sleeve,

temperature at the outer surface,

temperature at the center of the rod,

Assumptions: steady state, heat flux in radius direction only

(a)

Heat flux from the inner face of the sleeve to air

Thermal resistance (R_t) combines conduction through sleeve and convection to air (L is the length of the sleeve and the rod)

Rod's Volume is calculated as

Energy balance for the rod gives

where from steady state assumption, also, for this system. So

Energy balance for the sleeve gives

where from steady state assumption, also, for this system. So

Heat flux is computed as

For the case inner face of the sleeve to air, the equation is

Heat flux from sleeve's outer face to air has a thermal resistance called R_{conv}

For the case outer face of the sleeve to air, heat flux the equation is

(b)

Rod's temperature as function of radius can be computed as

For r = 0 (the center)

oil flow is a highly convective flow compared to water flow at normal temperature and pressure.

answer:

put more,