Question

A manufacturer of helicopter blades heat-treats a part as one of the steps of the
processing procedure. Heat-treating involves heating the metal up to a highly specific
temperature and then quenching, or cooling, it down quickly. The blades are made from
an aluminum alloy and have a mass of 273 kg. Water is normally used for the
quenching phase. If the water is at a temperature of 22.0oC, how much water (in kg)
would be needed to cool the blade from 300.0oC to at least 50.0oC?
External Data
The Cp of water is 4,180 J/kg oC, and the Cp of aluminum alloy is 900 J/kg oC. The
equation for calculating heat gained or lost by a material is q = mCpΔT.
Prompt
Write a scientific explanation that identifies mathematical relationships that can be used
to answer the question. Include a calculation in your evidence.

Answer 1

Scientific explanation and mathematical relationships

When the hot aluminum-alloy blade is quenched in water, energy is conserved: the heat lost by the blade equals the heat gained by the water (neglecting losses to the surroundings and to vaporization). Using the specific-heat relation q = m c ΔT for each material:

- Heat lost by blade: q_blade = m_blade · c_al · (T_final − T_initial_blade)
- Heat gained by water: q_water = m_water · c_w · (T_final − T_initial_water)

Setting heat lost = heat gained (with signs handled via temperature differences):
m_blade c_al (T_initial_blade − T_final) = m_water c_w (T_final − T_initial_water)

Given data
- m_blade = 273 kg
- c_al = 900 J·kg⁻¹·°C⁻¹
- T_initial_blade = 300.0 °C
- T_final = 50.0 °C (minimum required)
- T_initial_water = 22.0 °C
- c_w = 4180 J·kg⁻¹·°C⁻¹

Calculation (solve for m_water)
q_blade = 273 kg · 900 J/(kg·°C) · (300.0 − 50.0) °C
= 273 · 900 · 250 J
= 61,425,000 J

Temperature rise of water: ΔT_water = 50.0 − 22.0 = 28.0 °C

m_water = q_blade / (c_w · ΔT_water)
= 61,425,000 J / (4180 J/(kg·°C) · 28.0 °C)
≈ 61,425,000 / 117,040
≈ 524.8 kg

Conclusion
About 5.25 × 10^2 kg (≈ 525 kg) of water is required to cool the 273 kg blade from 300.0 °C to 50.0 °C under the assumption of no heat losses and that the water warms to 50.0 °C without boiling. This value is the minimum mass of water required under those assumptions.