Temperature change brought about by radiation

[Sai Maurice]

Homework Statement

A room 10mX10mX5m room is lighted by four 100 W lightbulbs. Assuming 73% of the energy is converted to heat, how much warmer will they room be in 6.9 hr. The room is insulated

Relevant Equations

dQ/dt=σAT⁴ radiation

I tried modeling the problem quite a few ways. one was to say that the difference between the heat emitted by the room and the heat emitted by the bulbs would equal the heat absorbed by the room, and that could allow us to calculate temperature. This did not work. I'd appreciate your help

 

[BvU]

Hello Sai,

Seems to me you didn't render the full problem statement -- I miss some things like c_p, $\rho$ for the air (and/or the walls). Were they given ?

quite a few ways. one was to say

No, you did not 'say', you wrote something down and worked it out.

How did you calculate ? We can't pick up what you did right/wrong by telepathy. Post your work

This did not work

How do you know ? Or do you already have the book answer and want PF to guess what it is ?

(for example, about four degrees...)

 

[Chestermiller]

Are you assuming that the light bulbs heat up the walls and ceiling of the room or just the air in the room? If they heat up the walls and ceiling, are the outsides of the these insulated? If not, then the insides of the walls must be considered insulated, so that no energy crosses. Where does the other 27% of the energy go?

 

[BvU]

Out the window ?

 

[Sai Maurice]

Hello Sai,

Seems to me you didn't render the full problem statement -- I miss some things like c_p, $\rho$ for the air (and/or the walls). Were they given ?
No, you did not 'say', you wrote something down and worked it out.

How did you calculate ? We can't pick up what you did right/wrong by telepathy. Post your work

How do you know ? Or do you already have the book answer and want PF to guess what it is ?

(for example, about four degrees...)

I assure you, that was all the info given. I can send you a picture if you'd like. The Reason it's hard to be specific is that this was a returned question, where the professor marked it as wrong without explanation, expecting us to return have a second attempt and get it right. The only problem is, I have no idea how to approach this problem with the info given. I set up the problem as the statement I laid out using the equation marked in the OP. I'll send more detail about my approaches when I have the opportunity.

 

[Sai Maurice]

Are you assuming that the light bulbs heat up the walls and ceiling of the room or just the air in the room? If they heat up the walls and ceiling, are the outsides of the these insulated? If not, then the insides of the walls must be considered insulated, so that no energy crosses. Where does the other 27% of the energy go?

The room is insulated

 

[BvU]

@Sai Maurice : further demo the problem statement needs clarification: are the lamps inside this huge room or outside ?

Correct my 'guess' for the answer to around 12 degrees (I read 15 m high instead of 5 )

 

[BvU]

that was all the info given. I can send you a picture if you'd like

So the problem statement included a picture ?
And you

I laid out using the equation marked in the OP

used what for T ?

 

[hutchphd]

Are you assuming that the light bulbs heat up the walls and ceiling of the room or just the air in the room? If they heat up the walls and ceiling, are the outsides of the these insulated? If not, then the insides of the walls must be considered insulated, so that no energy crosses. Where does the other 27% of the energy go?

I believe the intent of the problem is to consider only radiative losses after a new equilibrium is reached so the heat capacities don't matter. But I do wonder about the 27%... I guess those are conductive/convective losses. Also for an exact solution one needs the initial temperature...

 

[BvU]

But I do wonder about the 27%

So do I -- mayby the prof wants to confuse his students a bit by throwing this in ...

 

[hutchphd]

I'll send more detail about my approaches when I have the opportunity.

Please send your entire attempt. Then we can help

 

[Chestermiller]

I believe the intent of the problem is to consider only radiative losses after a new equilibrium is reached so the heat capacities don't matter. But I do wonder about the 27%... I guess those are conductive/convective losses. Also for an exact solution one needs the initial temperature...

I don't think so. The intent seemed to me to be to use the first law of thermodynamics to determine the internal energy rise (and temperature rise) of the air, given the heat input from the bulbs.

 

[hutchphd]

I feel a little like the blind men with the elephant here! You may be correct.

 

[Sai Maurice]

So the problem statement included a picture ?
And you
used what for T ?

dQ/dt=σAT⁴ I used this for the T, assuming the room was initially at room temperature. The result was that the walls of the room were radiating more heat to the lightbulbs than the other way around by a large factor.

 

[hutchphd]

So explicitly how did you get "12 degrees" ?? SHOW YOUR WORK PLEASE

 

[Sai Maurice]

Extra Info
I was just told we can assume the room has a specific heat capacity of 1000 J/kg, and the air has a density of 1.3 g/L. With this info, i can now solve the problem, thanks for taking the time to read my post.

 

[BvU]

I was just told we can assume

tooth fairy ?

No T^4 then, just $Q = m c_p \Delta T$, leading to $approx$ 12 degrees ...

 

[Chestermiller]

tooth fairy ?

No T^4 then, just $Q = m c_p \Delta T$, leading to $approx$ 12 degrees ...

It should be $C_v$, not Cp since the volume is constant. Anyway, I get about 17 C.

 

[BvU]

since the volume is constant

Stuffy atmosphere ...

 

[haruspex]

dQ/dt=σAT⁴ I used this for the T, assuming the room was initially at room temperature. The result was that the walls of the room were radiating more heat to the lightbulbs than the other way around by a large factor.

I don’t see how that equation is relevant. Heat energy radiated by the walls goes back to the walls. All you care about is the total energy consumed by the bulbs in the given time, less that 27% that presumably went out the window as light. You need to assume, though, that the walls have only a small thermal capacity and nearly all the energy gets transferred to the air by conduction.

It is unclear to me that C_v is appropriate. A room so well sealed is unrealistic.

 

[Chestermiller]

I don’t see how that equation is relevant. Heat energy radiated by the walls goes back to the walls. All you care about is the total energy consumed by the bulbs in the given time, less that 27% that presumably went out the window as light. You need to assume, though, that the walls have only a small thermal capacity and nearly all the energy gets transferred to the air by conduction.

It is unclear to me that C_v is appropriate. A room so well sealed is unrealistic.

If the room is not well sealed, the mass of gas in the room decreases. So, instead, that needs to be taken into account. As the escaping air leaves, there is less remaining air in the room receiving the same intensity of heat. The only reasonable assumption for this problem, in my judgment, is to assume constant gas volume and mass in the room.

 

[jbriggs444]

The only reasonable assumption for this problem, in my judgment, is to assume constant gas volume and mass in the room.

If one is going for this level of detail, a more reasonable assumption is that the room leaks and that the gas leaking out carries some thermal energy with it despite the insulation.

 

[Chestermiller]

If one is going for this level of detail, a more reasonable assumption is that the room leaks and that the gas leaking out carries some thermal energy with it despite the insulation.

That seems possible, but involves use of the open system version of the first law of thermodynamics which seems beyond the scope of the intent of this question.

 

[BvU]

Let's try to help the OP -- the quality of the exercise doesn't merit detailing...