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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.

Geometry:
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.

Given:
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.

Problem:
Calculate the equilibrium temperature.

Solution:
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.

Results:
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

 

QuickField

Theory

Temperature, °C

20.93

20.99

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

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