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Exhaust gas from a furnace is used to preheat the combustion air supplied to the furnace burners. The gas, which has a flow rate of 15 kg/s and an inlet temperature of 1100 K, passes through a bundle of tubes, while the air, which has a flow rate of 10 kg/s and an inlet temperature of 300 K, is in cross flow over the tubes. The tubes are unfinned, and the overall heat transfer coefficient is 90 W/m2·K.

Determine the total tube surface area, in m2, required to achieve an air outlet temperature of 850 K. The exhaust gas and the air may each be assumed to have a specific heat of 1075 J/kg·K.

Answer :

Answer:

The total tube surface area in m² required to achieve an air outlet temperature of 850 K is 192.3 m²

Explanation:

Here we have the heat Q given as follows;

Q = 15 × 1075 × (1100 - [tex]t_{A2}[/tex]) = 10 × 1075 × (850 - 300) = 5912500 J

∴ 1100 - [tex]t_{A2}[/tex] = 1100/3

[tex]t_{A2}[/tex]  = 733.33 K

[tex]\Delta \bar{t}_{a} =\frac{t_{A_{1}}+t_{A_{2}}}{2} - \frac{t_{B_{1}}+t_{B_{2}}}{2}[/tex]

Where

[tex]\Delta \bar{t}_{a}[/tex] = Arithmetic mean temperature difference

[tex]t_{A_{1}[/tex] = Inlet temperature of the gas = 1100 K

[tex]t_{A_{2}[/tex] = Outlet temperature of the gas = 733.33 K

[tex]t_{B_{1}[/tex] =  Inlet temperature of the air = 300 K

[tex]t_{B_{2}[/tex] = Outlet temperature of the air = 850 K

Hence, plugging in the values, we have;

[tex]\Delta \bar{t}_{a} =\frac{1100+733.33}{2} - \frac{300+850}{2} = 341\tfrac{2}{3} \, K = 341.67 \, K[/tex]

Hence, from;

[tex]\dot{Q} = UA\Delta \bar{t}_{a}[/tex], we have

5912500  = 90 × A × 341.67

[tex]A = \frac{5912500 }{90 \times 341.67} = 192.3 \, m^2[/tex]

Hence, the total tube surface area in m² required to achieve an air outlet temperature of 850 K = 192.3 m².

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