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Equilibrium temperature

Hot steel sphere is submerged into the cup of cold water. Calculate the equilibrium temperature.

Problem Type:
Axisymmetric problem of heat transfer.

Slot of electric machine In this tutorial we will analyze the skin effect occurring at the industrial frequency 50 Hz. The resistance of the conductor, carrying 500 A sinusoidal current, in the slot of the electric machine will be calculated. Steel Water Air Ø 5 cm Ø 20 cm Ø 10 cm

Steel sphere volume Vsteel = 65.4·cm3, water volume Vwater = 3600 cm3.

Tsteel= 85°C, Twater=20°C;
Volume density of steel ρsteel = 7800 kg/m3,
Volume density of water ρwater = 1000 kg/m3,
Specific heat of steel Csteel = 460 J/kg·K,
Specific heat of water Cwater = 4200 J/kg·K.

Calculate the equilibrium temperature.

We have to divide two bodies in order to prevent heat exchange between them at initial stage. In the transient problem the bodies are connected through the periodic boundary condition (T1=T2).

According to the 1st law of thermodynamics, the energy of the isolated system is preserved*:
(CρV)water·(Twater - T) + (CρV)steel·(Tsteel - T) = 0, where
T is the equilibrium temperature.

Analytical solution
(CρV)water = 4200·1000·3600·10-6 = 15120 [J/K]
(CρV)steel = 460·7800·65.4·10-6 = 235 [J/K]
15120·(20 - T) + 235·(85 - T) = 0
T = 20.99°C

Temperature distribution calculated in QuickField:

heat exchange simulation




Temperature, °C



*A textbook of Engineering Thermodynamics, R.K.Rajput, Luxmi publication (P) Ltd, Fourth edition, 2010.

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