TGS4
Mechanical
- Nov 8, 2004
- 3,915
In our most recent "climate change" discussion, I made a point about temperature averaging that seemed to be misunderstood. Because of its importance (in my opinion), I wanted to make its own thread.
It is clear to me that there is significant confusion about the cp of air and a global average temperature. Perhaps an example will clear things up:
I have two temperatures: -30°C (for argument's sake, let's say that the RH is 50%) and +30°C with an RH of 50%. The mathematical average of the temperatures is 0°C. However, based on the specific heat capacities:
[ul]
[li]the energy in 1 kg of the -30°C (243.15K) air is 1.005 [kJ/kg*K] * 1 [kg] * 243.15 [K] = 244.36575 kJ[/li]
[li]the energy in 1 kg of the +30°C (303.15K) air is 1.03 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 312.2445 kJ.[/li]
[/ul]
Now, average the energies to get 278.305125 kJ. If you wanted an average temperature based on average energy (something that can actually be averaged), you are stuck in the backwards calculation of which value of specific heat capacity to use.
[ul]
[li]If you use 1.005 [kJ/kg*K], you get an "average temperature" of 276.92K or 3.77°C.[/li]
[li]If you use 1.03 [kJ/kg*K], you get an "average temperature" of 273.20K or -2.95°C.[/li]
[li]If you use the "average" of the specific heat capacities (1.005+1.03)/2=1.0175 [kJ/kg*K], then you get an "average temperature" of 273.52K or 0.37°C.[/li]
[/ul] (Note that these calculations are all at sea-level).
So, which "averaging" calculation is correct? What is the true average temperature? Even IF you know the instantaneous humidity coincident with the temperature reading, you still have a problem in the averaging. But, here's another problem to highlight it even more:
I have two temperatures: +30°C (for argument's sake, let's say that the RH is 10% because it's in Denver) and +30°C with an RH of 100% (Miami). The average of the temperatures is, obviously, 30°C, right? However, based on the specific heat capacities:
[ul]
[li]the energy in 1 kg of the Denver air at 30°C (303.15K) air is 1.01 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 306.1512 kJ[/li]
[li]the energy in 1 kg of the Miami air at 30°C (303.15K) air is 1.056 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 320.1264 kJ.[/li]
[/ul]
Now, average the energies to get 313.1388 kJ. If you wanted an average temperature based on average energy (something that can actually be averaged), you are still stuck in the backwards calculation of which value of specific heat capacity to use.
[ul]
[li]If you use 1.01 [kJ/kg*K], you get an "average temperature" of 310.04K or 36.89°C.[/li]
[li]If you use 1.056 [kJ/kg*K], you get an "average temperature" of 296.53K or 23.38°C.[/li]
[li]If you use the "average" of the specific heat capacities (1.01+1.056)/2=1.033 [kJ/kg*K], then you get an "average temperature" of 303.14K or 29.99°C.[/li]
[/ul]
Again, what's the real metric here. It takes 4.5% more "energy" to heat the Miami air up an additional degree as compared to the Denver air. If we are truly "worried" about energies and fluxes, then why are we even using temperatures at all?
I am most certainly NOT talking about trends or anything else (yet). I am talking about the most basic of metrics used. Why are we using an average temperature?
It is clear to me that there is significant confusion about the cp of air and a global average temperature. Perhaps an example will clear things up:
I have two temperatures: -30°C (for argument's sake, let's say that the RH is 50%) and +30°C with an RH of 50%. The mathematical average of the temperatures is 0°C. However, based on the specific heat capacities:
[ul]
[li]the energy in 1 kg of the -30°C (243.15K) air is 1.005 [kJ/kg*K] * 1 [kg] * 243.15 [K] = 244.36575 kJ[/li]
[li]the energy in 1 kg of the +30°C (303.15K) air is 1.03 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 312.2445 kJ.[/li]
[/ul]
Now, average the energies to get 278.305125 kJ. If you wanted an average temperature based on average energy (something that can actually be averaged), you are stuck in the backwards calculation of which value of specific heat capacity to use.
[ul]
[li]If you use 1.005 [kJ/kg*K], you get an "average temperature" of 276.92K or 3.77°C.[/li]
[li]If you use 1.03 [kJ/kg*K], you get an "average temperature" of 273.20K or -2.95°C.[/li]
[li]If you use the "average" of the specific heat capacities (1.005+1.03)/2=1.0175 [kJ/kg*K], then you get an "average temperature" of 273.52K or 0.37°C.[/li]
[/ul] (Note that these calculations are all at sea-level).
So, which "averaging" calculation is correct? What is the true average temperature? Even IF you know the instantaneous humidity coincident with the temperature reading, you still have a problem in the averaging. But, here's another problem to highlight it even more:
I have two temperatures: +30°C (for argument's sake, let's say that the RH is 10% because it's in Denver) and +30°C with an RH of 100% (Miami). The average of the temperatures is, obviously, 30°C, right? However, based on the specific heat capacities:
[ul]
[li]the energy in 1 kg of the Denver air at 30°C (303.15K) air is 1.01 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 306.1512 kJ[/li]
[li]the energy in 1 kg of the Miami air at 30°C (303.15K) air is 1.056 [kJ/kg*K] * 1 [kg] * 303.15 [K] = 320.1264 kJ.[/li]
[/ul]
Now, average the energies to get 313.1388 kJ. If you wanted an average temperature based on average energy (something that can actually be averaged), you are still stuck in the backwards calculation of which value of specific heat capacity to use.
[ul]
[li]If you use 1.01 [kJ/kg*K], you get an "average temperature" of 310.04K or 36.89°C.[/li]
[li]If you use 1.056 [kJ/kg*K], you get an "average temperature" of 296.53K or 23.38°C.[/li]
[li]If you use the "average" of the specific heat capacities (1.01+1.056)/2=1.033 [kJ/kg*K], then you get an "average temperature" of 303.14K or 29.99°C.[/li]
[/ul]
Again, what's the real metric here. It takes 4.5% more "energy" to heat the Miami air up an additional degree as compared to the Denver air. If we are truly "worried" about energies and fluxes, then why are we even using temperatures at all?
I am most certainly NOT talking about trends or anything else (yet). I am talking about the most basic of metrics used. Why are we using an average temperature?
