How to estimating the heating cost of an open bathroom window?

Question

I forgot to close the bathroom window in the morning and the heating was on. I'm wondering if it's possible to roughly estimate the costs?

Unfortuantely I have not much physics knowledge. I just need a rough estimate, but searching the internet I did not find a satisfactory answer to my question, but similar questions, so I thought it might be interesting for others too.

The scenario:

The window (60x60cm) was open for 11 hours. Outside temperature approx -10°C. The window is next to the wall mounted heater, the window pane covers the heater when opened. Especially it covers the electric heat cost allocator. During that time the heater thermostat (located near the floor) was on 1 (Defrost), but the heater itself was very hot when I returned. The heater is a wall mounted bathroom heater with an estimated capacity of 960W (EN442) (for more details see page 22 in this brochure) The bathroom door was closed. The bathroom is approx 13m³ large. The energy supplier accounts 6ct/kwh.

My calculation:

I assume the heater was heating all the time at full power, since the thermostat was in the cold air falling down the window, opening the hot water flow to the max.

1) Can I just calculate 11h*0,96w*6ct/kWh = 63,36ct ?

2) I think EN442 defines the heat capacity at a certain temperature (75C/65C/20C, which is water temp in/out/room temp). So is it ok to use 960W for my scenario? Is that the maximum capacity for a heater like that or could it get much higher?

3) I'm not quite sure how the heat cost allocator (sample) calculates costs. I think it measures the difference between the room temperature and the heater. Does it make a huge difference that the window pane (double glass) was in front of it? Could the pane have shielded the allocator a bit from the cold air? Basically is it still accurate under such extreme conditions?

Answer 1

Your assumption that the room heater was on continuously is reasonable. It seems you have a heater of the hot water type - but instead of the temperature difference being roughly 40°C (midway between 75 and 65 °C for the temperature inside the heater, and 20°C for the room temperature) you have a temperature difference closer to 70°C (since the window is wide open, the room might be almost as cold as the outside - really hard to estimate without knowing how much the air could circulate; especially, if the window pane is partly in front of the heater, this may reduce the circulation and this would reduce the amount of heat transferred).

The heat transfer is roughly proportional to the temperature difference - if you get 960 W at 40°C difference, you will get 70/40*960 = 1680 W if it gets really cold in the vicinity of the heater (see comment above about the window pane). The actual average energy output will be a bit lower.

Using that upper estimate of the heat, you could put the cost of the heating at approximately 120 cents ( 11 hrs * 1.68 kW * 6.6 c/hr = 121.96) - that's a maximum.

Now it's not clear to me that your heating system is electrical - usually circulating hot water systems are heated with gas, in which case using the electrical rate will give you the wrong answer. In the US, gas is priced in BTU (British Thermal Units); on December 8, the whole sale price was $2.772/MBTU according to WolframAlpha, and a MBTU if 293 kWh. That's obviously a lot lower than the residential price, but I don't know what the markup in your region is. At any rater, the price of heating with gas is typically a little lower than heating with electricity (which is why people heat with gas... it saves all the losses from conversion). Without knowing more about your heating system, I can't give you a better answer.