Equilibrium temperature

QuickField simulation example

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

Problem Type
Axisymmetric problem of heat transfer.

Geometry

Steel sphere volume V_steel = 65.4·cm³, water volume V_water = 3600 cm³.

Given
T_steel= 85°C, T_water=20°C;
Volume density of steel ρ_steel = 7800 kg/m³,
Volume density of water ρ_water = 1000 kg/m³,
Specific heat of steel C_steel = 460 J/kg·K,
Specific heat of water C_water = 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 1^st law of thermodynamics, the energy of the isolated system is preserved^*:
(CρV)_water·(T_water - T) + (CρV)_steel·(T_steel - 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):

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

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