A furnace wall is to be constructed of brick having standard dimensions 22.5 cm* 11 cm * 7.5 cm. Two kinds of material are available. One has a maximum usable temperature of 1040°C and a thermal conductivity of 1.7 W/(m K), and the other has a maximum temperature limit of 8700C and a thermal conductivity of 0.85 W/(m K). The bricks have the same cost and are laid in any manner, but we wish to design the most economical wall for a furnace with a temperature of 10400C the hot side and 2000C on the cold side. If the maximum amount of heat transfer permissible is 950 W/m2 , determine the most economical arrangement using the available bricks.

Question

GIVEN

Furnace wall made of 22.5 ? 11 ? 7.5 cm bricks of two types

Type 1 bricks Maximum useful temperature (T1,max) = 1040°C=1313 K

Thermal conductivity (k1) = 1.7 W/(m K)

Type 2 bricks Maximum useful temperature (T2,max) = 870°C= 1143 K

Thermal conductivity (k2) = 0.85 W/(m K)

Bricks cost the same

Wall hot side (Thot) = 1040°C=1313 K and cold side (Tcold) = 200°C=473 K

Maximum heat transfer permissible (qmax/A) = 950 W/m2

FIND

The most economical arrangement for the bricks

ASSUMPTIONS

One dimensional, steady state heat transfer conditions

Constant thermal conductivities

The contact resistance between the bricks is negligible

SKETCH

Answer 1

Since the type 1 bricks have a higher thermal conductivity at the same cost as the type 2 bricks, the

most economical wall would use as few type 1 bricks as possible. However, there must be a thick

enough layer of type 1 bricks to keep the type 2 bricks at 1040°C or less.

For one dimensional conduction through the type 1 bricks,





where L1 = the minimum thickness of the type 1 bricks

Solving for L1



This thickness can be achieved with 4 layers of type 1 bricks using the 3 in. dimension.

Similarly, for one dimensional conduction through the type 2 bricks



This thickness can be achieved with 8 layers of type 2 brick using the 3 in. dimension. Therefore, the

most economical wall would be built using 4 layers of type 1 bricks and 8 layers of type 2 bricks with

three inches dimension of the bricks used as the thickness.