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Equilibrium temperature
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:
All dimensions are in meters.
Steel sphere volume V_{steel} = 65.4·cm^{3}, water volume V_{water} = 3600 m^{3}.
Given:
T_{steel}= 85°C, T_{water}=20°C;
Volume density of steel ρ_{steel} = 7800 kg/m^{3},
Volume density of water ρ_{water} = 1000 kg/m^{3},
Specific heat of steel C_{steel} = 460 J/kg·K,
Specific heat of water C_{water} = 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 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:

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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