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cooling and leaking process mathematical model

1. A heated object is allowed to cool in a room temperature which has a constant temperature of To.

a. Analyse the cooling process.
b. Formulate mathematical model for the cooling process.

2. At time t= 0 water begins to leak from a tank of constant cross-sectional area A. The rate of outflow is proportional to h, the depth of water in the tank at time t. Write the constant of proportion kA where k is constant.

a. Analyse the tank leaking process.
b. Formulate mathematical model for the leaking process.

Write conclusions based on your formulated mathematical model for leaking process.

Solution Preview

1. a. As the object emits heat energy, it is absorbed by the surrounding at To. Because of this process of heat exchange, the object temperature decreases and tends towards To (== Qo + 273: Qo in degree C and To in K).

The heat energy emitted by the object, depends on it's instantaneous temperature Tb. Because, of surrounding temperature To, surrounding also gives some heat energy to the object. As Tb > To, in net, the object temperature decreases.

The heat energy emitted by the object is based on Stefan's law, dH = s*T^4 * A*dt

Where, s: Stefan's constant, A: surface ares through which heat energy accepted or emitted, dt: time ...

Solution Summary

Two problems are solved. First: mathematical model of cooling process to room temperature. Second: Mathematical model of leaking process of water filled tank as function of depth.