A gas-fired industrial furnace is used to generate steam. The furnace is a 3 m cubic structure and the interior surfaces are completely covered with boiler tubes transporting pressurized wet steam at 150°C. It is desired to keep the furnace losses to 1% of the total heat input of 1 MW. The outside of the furnace can be insulated with a blanket-type mineral wool insulation [k = 0.13 W/(m °C)], which is protected by a polished metal sheet outer shell. Assume the floor of the furnace is insulated. What is the temperature of the metal shell sides? What thickness of insulation is required?

Question

GIVEN

? An insulated cubic furnace with steam filled tubes on the inner walls

? Steam temperature (Tst) = 150°C

? Length of a side of the furnace (L) = 3 m

? Thermal conductivity of mineral wool insulation (ki) = 0.13 W/(m°C)

? Insulation is protected by metal sheet outer shell

? Furnace losses (qc) = 1% of total heat input

? Total heat input (qin) = 1 MW = 106 W

FIND

(a) Temperature of the metal sheel sides (Ts)

(b) The thickness of insulation (s) required

ASSUMPTIONS

? Steady state operation

? Thermal resistance of the convection within the steam pipes, the steam pipe walls, the furnace

walls, and the metal shell negligible compared to that of the insulation

? Air outside the furnace is still

? The floor is well insulated —heat loss is negligible

? Temperature of the metal shell is uniform

? Ambient temperature (T?) = 20°C (293 K)

? Edge effects are negligible

? The emissivity of the polished metal shell (E) = 0.05 (see Table 9.2)

SKETCH



PROPERTIES AND CONSTANTS

From Appendix 1, Table 5, the Stephan-Boltzmann Constant

Answer 1

(a) Since the Grashof number on the outside of the metal shell will depend on the temperature of the metal shell, an iterative procedure is required. For the first iteration, let Ts = 100°C (373 K).

From Appendix 2, Table 28, for dry air at the mean temperature of 60°C



The Grashof number for the four sides of the furnace, assuming the insulation thickness is small compared to 3 m, is



The heat transfer from the furnace will be calculated by treating the sides as vertical flat plates and the top as a horizontal flat plate facing upward. From Equation (8.13), the heat transfer coefficient for the sides is



The characteristic dimension for the top of the furnace (Lc) is



The Grashof and Rayleigh numbers based on this dimension are



The average Nusselt number is given by Equation (8.16)



The rate of convection and radiation must be 1% of the total heat input



By trial and error: Ts = 327 K = 54°C

Following the same procedure for other iterations





Therefore, the surface temperature (Ts) ? 61°C

(b) The rate of conductive heat transfer through the insulation must also be 1% of the input heat



Solving for the insulation thickness