A spherical stainless steel vessel at 93°C contains 45 kg of water initially at the same temperature. If the entire system is suddenly immersed in ice water, determine (a) the time required for the water in the vessel to cool to 16°C, and (b) the temperature of the walls of the vessel at that time. Assume that the heat transfer coefficient at the inner surface is 17 W/(m2 K), the heat transfer coefficient at the outer surface is 22.7 W/(m2 K), and the wall of the vessel is 2.5-cm-thick.
GIVEN
• A spherical stainless steel vessel of water is suddenly immersed in ice water
• Initial temperature of vessel and water (Ti) = 93°C
• Mass of water in the vessel (m) = 45 kg
• The inner heat transfer coefficient ci h = 17 W/(m2 K)
• The outer heat transfer coefficient co h = 22.7 W/(m2 K)
• The vessel wall thickness (L) = 2.5 cm = 0.025 m
FIND
(a) The time required for the water in the vessel to cool to 16°C (b) The temperature of the walls of the vessel at that time (Tsf)
ASSUMPTIONS
• The water in the vessel is well mixed, therefore its temperature is uniform
• The vessel is completely filled with water
SKETCH
If the vessel is completely filled with water
The internal resistance of the water can be neglected since the water is assumed to be well mixed. The importance of the internal resistance of the vessel wall is indicated by the Biot number of the vessel wall. The characteristic length for the vessel wall is
Therefore, the vessel and its contents can be treated as a lumped capacitance and the system approximated two lumped capacitances as covered in Section 2.6.1 of the text.
(a) The temperature-time history of the water in the vessel is given
e temperature-time history of the water is
For the water to cool to 16°C
y trial and error: t = 55,870 s = 15.5 hours
(b) The energy balance for the fluid is
Differentiating the temperature-time history
Substituting this into the energy balance for the fluid
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