250W lamp vs. hot water bottle

[lemur]

I'm trying to compare the heat output between a 250W lamp and a 1 liter hot water bottle filled with boiling water. I think this involves specific heat of water but I'm not sure how to set up the math. This is what I think so far:

 

4.187 kJ/kgK

so does that mean 1kg of water produces @4200 joules per degree temperature change?

Then I suppose I would have to know how fast the hot water bottle is cooling.

Is there some short cut to compare the two without knowing the exact cooling rate of the water bottle? It doesn't have to be exact. I'm just trying to get a general idea of how they compare as heat sources.

[swansont]

I'm trying to compare the heat output between a 250W lamp and a 1 liter hot water bottle filled with boiling water. I think this involves specific heat of water but I'm not sure how to set up the math. This is what I think so far:

 

4.187 kJ/kgK

so does that mean 1kg of water produces @4200 joules per degree temperature change?

Then I suppose I would have to know how fast the hot water bottle is cooling.

Is there some short cut to compare the two without knowing the exact cooling rate of the water bottle? It doesn't have to be exact. I'm just trying to get a general idea of how they compare as heat sources.

 

 

The power output of the bottle depends on the conditions. Conduction is proportional to the temperature difference, but if there's nothing in direct contact then it's mostly radiation with some convection. Radiation varies as the difference in T^4 and the surface area.

 

Let's assume radiation. This is given by the Stefan–Boltzmann law. The water is at 373 K, and we'll assume the surroundings are at 280K (~ 7ºC). The water container is 1000 cm^2 (or .1 m^2) (a sphere would be 482, a cube would be 600) and we'll assume a perfect radiator.

 

That gives you 75 Watts. You could increase this by having the container have even more surface area. Of course, if the surroundings are warmer or cooler, that changes the number.

[lemur]

The power output of the bottle depends on the conditions. Conduction is proportional to the temperature difference, but if there's nothing in direct contact then it's mostly radiation with some convection. Radiation varies as the difference in T^4 and the surface area.

 

Let's assume radiation. This is given by the Stefan–Boltzmann law. The water is at 373 K, and we'll assume the surroundings are at 280K (~ 7ºC). The water container is 1000 cm^2 (or .1 m^2) (a sphere would be 482, a cube would be 600) and we'll assume a perfect radiator.

 

That gives you 75 Watts. You could increase this by having the container have even more surface area. Of course, if the surroundings are warmer or cooler, that changes the number.

Thanks. I knew that black-body radiation means that things at the same temperature mostly emit the same level of radiation, but I didn't know the equation (and I'd probably mess up applying it). Now I'm wondering how it changes things to wrap the water bottle in a towel. Would that reduce the emissivity or would it be treated as itself an absorber and re-emitter with its own characteristics? The more I think about, the whole comparison falls apart when you start dealing with insulation. The heat lamp is designed to provide heat with little if any insulation while the water bottle works well by allowing it to slowly cool down in a well-insulated situation.

[swansont]

The insulation slows the cooling by slowing the rate of conductive heat transfer; the temperature on the outside is lower than on the inside. So even if it was a perfect radiator, T has gone down, and the radiated power drops very fast.

[alibaba441]

yes the thermal output is indeed ~ 4200J/K for a 1kg bottle.

 

The heat is going to be emitted in a mixture of conduction (through the skin etc) convection(heat dissipation to the air) and radiation.

 

Since the temperature of the water bottle never reaches above 80C and drops quite quickly initialy, the radiation aspect is probably small. so the majority of the heat output will be through conduction and convection.

 

to calculate this we make some assumptions:

 

the temperature of the water bottle starts at 80C and drops to 20C (room temperature) in the space of 2 hours.

 

hence we have

 

Q = M*Cp*dT

 

where q = TOTAL thermal output (Joules)

M = mass in kg

Cp = heat capacity (J/kg C)

dT = total change in temperature

 

so we have M = 1kg, Cp = 4200J/Kg C

dT = (80-20) = 60 C

 

so Q = 4200*60*1 = 252,000 J or 252kJ of thermal energy released over 2 hours

 

if you want the power you want to know the heat ouput in Joules/s ( Power = energy/time)

 

P = 252kJ/(2 hours)

P = 252kj/(2*60*60)

P = 35Watts

 

so a hot water bottle is nowhere near comparable to a 250W incandescent bulb. However, dont forget that not all the energy from a light bubl goes to heat (some will go to light) hence the thermal output of the bulb is slightly lower than its 250W rating.

 

Its always good to check if your results make sense. the warmth we feel from a water bottle outputting 35 Watts is significant but can we check that it is realistic? Googling caring after a baby squirrel:

 

http://blogs.fingerlakes1.com/mumsy/raising-orphaned-eastern-grey-squirrels/

 

reveals that it si suggested they be kept warm by way of a 7 Watt bulb. this is a fifth of the hot water bottle thermal output (approx). this makes sense, a hot water bottle would probably output way too much heat for a small animal to handle so our answer seems to be in the right order of magnitude

 

 

 

 

[robomont]

if im correct then one horse could keep a hundred squirrels warm?

Edited May 25, 2013 by robomont

[swansont]

yes the thermal output is indeed ~ 4200J/K for a 1kg bottle.

 

The heat is going to be emitted in a mixture of conduction (through the skin etc) convection(heat dissipation to the air) and radiation.

 

Since the temperature of the water bottle never reaches above 80C and drops quite quickly initialy, the radiation aspect is probably small. so the majority of the heat output will be through conduction and convection.

I calculated a radiation number, and as it's comparable in magnitude to your answer (in fact it's larger), that tells us that radiation can't be ignored.

 

so a hot water bottle is nowhere near comparable to a 250W incandescent bulb. However, dont forget that not all the energy from a light bubl goes to heat (some will go to light) hence the thermal output of the bulb is slightly lower than its 250W rating.

No, this is wrong. Visible light still counts as heat. Visible light being absorbed will warm things up.