24 kg of water are heated by a thermostatically controlled 4Kw electric heater. The thermostat switches the heater on at 60 degree centigrade and off at 65 degree centigrade. If the time between switching off and switching on is 3.0 min, what will be the time between switching on and switching off. Assume that the rate of cooling of the water uniform over the range 60 degree centigrade to 65 degree centigrade.
The time between switching off and on (the cooling time) depends upon the rate of heat loss. It is the time it takes to cool 5 C. You knoww that to be 3.0 minutes
The time it takes between switching on and off (the heating time) depends upon the time it takes to heat up 5 C, while the the heat loss is still going on.
You can get the heat loss rate from the 3 minute cooldown time, and the rate of heat addition from 4 kW figure.
Qloss = 24 kg * 4184 J/kg C * 5 C/180 s
= 2.79 kW
Qin = 4.00 kW
Qin - Qloss = 1.21 kW
Energy is lost at a rate that is 2.3 times faster than it can be added when the heat loss is still going on.
The heating cycle time will be 2.3 times 3 minutes, or 6.9 minutes
To find the time between switching on and switching off, we need to determine the rate of cooling of the water.
First, let's calculate the heat absorbed by the water when the heater is on. We can use the formula:
Heat = Power x Time
Given that the power of the electric heater is 4 kW and the time between switching off and switching on is 3.0 minutes, we have:
Heat (on) = 4 kW x 3.0 min
Next, let's calculate the heat lost by the water during the time between switching off and switching on. We can assume that the rate of cooling is uniform over the temperature range of 60 to 65 degrees Celsius. Therefore, the average temperature change during this time is:
Temperature Change = (65°C - 60°C) / 2
Now, we can use the specific heat capacity of water to calculate the heat lost:
Heat (off) = Mass x Specific Heat Capacity x Temperature Change
Given that the mass of water is 24 kg and the specific heat capacity of water is approximately 4,186 J/kg°C, we have:
Heat (off) = 24 kg x 4,186 J/kg°C x Temperature Change
Since the heat absorbed during heating is equal to the heat lost during cooling, we can equate the two expressions:
4 kW x 3.0 min = 24 kg x 4,186 J/kg°C x Temperature Change
Simplifying the equation, we get:
12,000 J/min = 100,464 J/°C x Temperature Change
Now, we can solve for the temperature change:
Temperature Change = 12,000 J/min / 100,464 J/°C
Temperature Change ≈ 0.1196 °C
Finally, to find the time between switching on and switching off, we need to determine how long it takes for the temperature to change by 0.1196 degrees Celsius.
We know that the rate of cooling is uniform, so we can assume that it will take the same amount of time to cool from 60°C to 59.8804°C as it did to heat from 59.8804°C to 60°C. Therefore, the time between switching on and switching off is double the time of 0.1196 minutes (or approximately 7.176 seconds).
Therefore, the time between switching on and switching off is approximately 0.2392 minutes (or approximately 14.352 seconds).