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Equilibrium temperature

transient heat transfer simulation, transient heat transfer FEA analysis

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

Problem Type
Axisymmetric problem of heat transfer.

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

Task
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 (animation):
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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