A block of aluminum is heated to a temperature of 100 °C. It is then dropped into a flowing stream of water which is held at a constant temperature of 10°C. After 10 seconds the temperature of the block is 60°C. (Aluminum is such a good heat conductor that its temperature is essentially uniform throughout its volume during the cooling process.) The rate of cooling is proportional to the temperature difference between the block and the water.
(a) Write a differential equation for this system with the temperature of the water as the excitation and the temperature of the block as the response.
(b) Compute the time constant of the system.
(c) If the same block is cooled to 0 °C and dropped into a flowing stream of water at 80 °C, at time at what time will the temperature of the block reach 75°C?
a) The controlling differential equation equates the rate of change of the aluminum temperature to the difference between the water and aluminum temperatures, times a rate constant ? ,
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