Hot flue gases from a combustion furnace flow through a chimney, which is 7-m-tall and has a hollow cylindrical cross section with inner diameter di = 30 cm and outer diameter do = 50 cm. The flue gases flow with an average temperature of Tg = 300°C and convective heat transfer coefficient of hg = 75 W/(m2 K). The chimney is made of concrete, which has a thermal conductivity of k = 1.4 W/(m K). It is exposed to outside air that has an average temperature of Ta = 25°C and convective heat transfer coefficient of 15 W/(m2 K). For steady-state conditions, (a) determine the inner and outer wall temperatures, (b) plot the temperature distribution along the thickness of the chimney wall, and (c) determine the rate of heat loss to outside air from the chimney. Solve the problem by numerical

Question

analysis using a nodal network with ?r = 2 cm and ?? = 10°.



GIVEN

- Hot flue gas flow in a hollow cylindrical 7-m-tall chimney with inner and outer diameters of

di = 0.3 m and do = 0.5 m, and thermal conductivity k = 1.4 W/(m K).

- Average hot gas temperature Tg = 300°C and heat transfer coefficient



- Outside air average temperature Ta = 25?C and heat transfer coefficient





FIND

- The temperature distribution in chimney wall and the inside and outside wall temperatures.

- Rate of heat loss from gas to outside air through the chimney.



ASSUMPTIONS

- Steady state conditions, and there is no heat generation in the chimney wall and the conduction

along the height of the chimney is negligible.

Answer 1

(a) For steady-state conditions, the heat conduction equation is



which is subject to the following boundary conditions



Construct a nodal network with 2 cm spacing in the radial direction in the thickness of the chimney’s

hollow cylinder (6 nodal points total, including the nodal points on the inner and outer wall) and

10-degree spacing in the angular direction (36 nodal points). This would result in the following form

of discretized equation



Though, it may be noted that because of the circular symmetry, this can be solved as a one-

dimensional problem without using the angular nodes.



The numerical solution (carried out on MATLAB) yields the following temperature values at the six

(6) radial nodes along the wall thickness



266.1°C 235.7°C 208.5°C 183.7°C 160.9°C 126.8°



(b) This temperature distribution is depicted as an isotherm contour plot in the figure below.





(c) Also, the rate of heat loss from the outer wall of the chimney to air can be calculated as