Heating Up

In this time of year, dead winter approaches. As it grows colder and colder, the urge to open my window at night with the fan on and the heater going slowly disappears as the window stays shut and the heater starts blasting. However, as we talked about the movement of heat in physics, I grew interested and noticed it all around; in our sweaters, our food, our bodies- right now, it's a very seasonal and on-point topic. While our current heaters blow air into our rooms, electric heaters use radiation to slowly heat up the air in the room. As the thermostat goes up and down these days, I wanted to see how long it would take for an electric heater to heat up an empty small dorm-sized room from about 2C to the heater's comfortable temperature of 23C.

A nifty little aluminum heater. (http://img.hisupplier.com/var/userImages/2008-11/11/hisunrays$011813759(s).jpg)

First of all, since the heater is using radiation, we will be looking at the equation for heat and radiation. To start off, here is the equation for the net heat flow in radiation. The second equation beneath it is changed in order to find time.

Now for our variables; we are assuming that the heater is made from aluminium. To obtain the surface area of the radiator, I assumed a 1mx0.3mx1m sized radiator.

 Now, we must find Q in this instance, as we have each variables besides it. Below is the equation for Q, or heat:

The mass of the air in the room is calculated below. I am assuming a small room that is 6mx5mx3m, (3m tall in consideration of objects present in a room that would normally be 4m tall) resulting in the volume of air to be 120m cubed. Now, as a cubic meter of air is about 1.3 kg, I calculated further to get a mass of 156 kg of air.

Furthermore, taking our change in temperature (23-2=21) and the specific heat of air:

Plugging it in gives us a heat of 3292.38J! Lastly, now that we have Q, we can plug everything back in...

Thus, we see that it takes 162.6s, almost 3 minutes, to fully heat a room to a nice and comfy 23C. So, next time you turn up the heater, don't fiddle with it until it fully heats the room up- you want your room to be heated not too hot, not too cold, but just right!