The inside wall of a furnace needs to be designed such that it transmits a
maximum heat flux of 600 W/m2
. The inside wall temperature will be 1100°C and the
outside temperature will be 150°C. We have two different types of bricks from which we
can make the wall:
Type 1: 24 x 12 x 8 cm, k = 0.7 W/m·K, maximum allowable temperature = 850°C
Type 2: 24 x 12 x 8 cm, k = 1.6 W/m·K, maximum allowable temperature = 2200°C
If the two bricks cost the same amount of money, what is the most economical design for
the brick wall?
The bricks can be oriented in any direction and as many bricks as necessary can be used.
Assume perfect heat transfer between adjacent bricks.
To determine the most economical design for the brick wall, we need to calculate the heat flux for each type of brick and compare it to the required maximum heat flux of 600 W/m².
First, let's calculate the heat flux for Type 1 brick:
1. Calculate the temperature difference between the inside and outside of the furnace:
ΔT1 = 1100°C - 150°C = 950°C
2. Calculate the thermal conductivity of Type 1 brick:
k1 = 0.7 W/m·K
3. Calculate the heat flux using Fourier's Law of Heat Conduction:
q1 = (k1 * ΔT1) / (thickness1)
Since the thickness of Type 1 brick is given as 8 cm (or 0.08 m), substituting the values:
q1 = (0.7 W/m·K * 950°C) / (0.08 m)
= 8,275 W/m²
Next, let's calculate the heat flux for Type 2 brick:
1. Calculate the temperature difference between the inside and outside of the furnace:
ΔT2 = 1100°C - 150°C = 950°C
2. Calculate the thermal conductivity of Type 2 brick:
k2 = 1.6 W/m·K
3. Calculate the heat flux using Fourier's Law of Heat Conduction:
q2 = (k2 * ΔT2) / (thickness2)
Since the thickness of Type 2 brick is also given as 8 cm (or 0.08 m), substituting the values:
q2 = (1.6 W/m·K * 950°C) / (0.08 m)
= 19,000 W/m²
Now, let's compare the calculated heat fluxes for each brick with the maximum allowed heat flux of 600 W/m²:
For Type 1 brick: q1 = 8,275 W/m²
For Type 2 brick: q2 = 19,000 W/m²
Since the maximum allowed heat flux is 600 W/m², Type 1 brick satisfies this requirement, while Type 2 brick exceeds it significantly. Therefore, the most economical design for the brick wall is to use Type 1 bricks.