Question

A furnace of 250 MW rating is used to melt and raise the temperature of aluminium from 25\(^{\circ}\)C to 900\(^{\circ}\)C. Aluminium has a solid-state specific heat, latent heat, and liquid-state specific heat of 0.9 kJ/kg-K, 390 kJ/kg, and 1.108 kJ/kg-K, respectively, and the furnace has 70% efficiency. The melting point of aluminium is 660\(^{\circ}\)C. The amount of aluminium that can be processed per hour is ________ kg (rounded off to 1 decimal place).

Hint:

Add up the heat needed per kg for solid heating, melting, and liquid heating, then divide the furnace's hourly useful output (rating times efficiency times 3600 s) by that value.

Answer 1

Correct Answer: 513271.7

Step 1: Work in a per-second rate instead of a per-hour total.
Find the useful thermal power actually delivered to the aluminium, then divide by the energy needed per kg to get a processing rate in kg/s, and finally scale up to an hourly figure.

Step 2: Useful power.
$P_{useful} = 0.70 \times 250{,}000$ kW $= 175{,}000$ kW $= 175{,}000$ kJ/s.

Step 3: Specific energy demand per kg (same three-stage sum as before).
$q = 0.9(660-25) + 390 + 1.108(900-660) = 571.5+390+265.92 = 1227.42$ kJ/kg.

Step 4: Mass processing rate.
\[ \dot{m} = \frac{P_{useful}}{q} = \frac{175{,}000}{1227.42} \approx 142.575\ \text{kg/s} \]

Step 5: Scale to one hour.
$m_{hour} = \dot{m} \times 3600 = 142.575 \times 3600 \approx 513{,}270\ \text{kg}$.

Final Answer:
This rate-based route confirms the amount of aluminium processed per hour is about 513,271.7 kg.
\[ \boxed{m \approx 513271.7\ \text{kg/hr}} \]