Carnot Cycle Refrigerator - Thermal Battery Problem

So to find the amount of ice and steam created, you would need to use the latent heat formulas to determine the amount of energy needed to create the ice and steam, and then use the specific heat capacities to convert that into moles of ice and steam. In summary, the problem involves using a Carnot cycle to transfer heat from a colder reservoir to a hotter one in order to store energy in the form of ice and steam. The temperatures of the reservoirs are given as the ice point and steam point at 1 atm. The energy stored is given as 1 kW*hr, which is equivalent to 3600 kJ. To find the amount of ice and steam created, the latent heat formulas and specific heat capacities must be
  • #1
Kelsi_Jade
59
0
The question is:
Consider a thermal battery - a device for storing mechanical energy. Use two large containers as heat reservoirs where one contains ice and water in equilibrium and the other contains steam and water in equilibrium. Both are held at pressure=1atm. Energy is stored by using a carnot cycle to transfer heat from the colder reservoir to the hotter - how much ice is created and how much vapor is created when 1kW*hr of energy is stored?

I know that a refrigerator is essentially a heat pump in reverse so we can consider a Carnot engine where heat is transferred from the cold reservoir. Work is done in order to do this, and the work for the refrigerator is negative.

Since I don't know the temperatures, and n=only the pressure of the systems and the energy stored, I know I can't use the COP or efficiency in the problem but I am unsure where to go next.

Any help is appreciated.
 
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  • #2
Kelsi_Jade said:
Since I don't know the temperatures
Really? Reread the problem carefully, they're hiding in there somewhere...
 
  • #3
OK...so is it correct to say that if the ice and water are in equilibrium, T1 would be the temp where ice and water can exist together at 1 atm and T2 would be the temperature where water and steam exist in equilibrium at 1 atm?
So, after a little research: T1= ice point= 273K
T2= steam point= 373K

Next:
Is the 1 kW*hour (3600kJ) energy given the internal energy of the system if it is the "energy stored"?
which for an isobaric process would be: ΔU=ncpΔT
which can be found using the temperatures I just defined and where:
cpice=38.09 J/mol*K
cpsteam=37.47 J/mol*K

When the question asks for how much ice/steam, does that mean that I am to find the mols of ice/steam? Which I can find by rearranging the above equation?
 
  • #4
Kelsi_Jade said:
OK...so is it correct to say that if the ice and water are in equilibrium, T1 would be the temp where ice and water can exist together at 1 atm and T2 would be the temperature where water and steam exist in equilibrium at 1 atm?
So, after a little research: T1= ice point= 273K
T2= steam point= 373K
Correct.

Kelsi_Jade said:
Next:
Is the 1 kW*hour (3600kJ) energy given the internal energy of the system if it is the "energy stored"?
I am not sure how to interpret this, by my take is that "energy stored" means energy that went into the hot reservoir, as this is where it can be kept for future use in a thermal engine.

Kelsi_Jade said:
When the question asks for how much ice/steam, does that mean that I am to find the mols of ice/steam?
Yes.

Kelsi_Jade said:
Which I can find by rearranging the above equation?
Yes, but you are missing one element.
 
  • #5
Hmm...so, energy going into the hot reservoir would be the Heat energy Q? Where,
Q =mCpΔT =ncpΔT...

If that's true, how is the equation for ΔU= the equation for Q?
 
  • #6
Kelsi_Jade said:
which for an isobaric process would be: ΔU=ncpΔT
I hadn't paid enough attention to what you wrote there. That equation for ΔU is not valid for a phase transformation. Look up "latent heat".

And I'll make more precise my previous hint: you have to use the fact that it is a Carnot engine that is transfering the heat.
 
  • #7
OK, latent heat is the amount of energy in the form of heat (Q) required to completely effect a phase change of a unit of mass.
Latent heat=Q/m
Latent heat for water vapor:
L(T) = (2500.8 - 2.36 T + 0.0016 T^2 - 0.00006 T^3)J/g

Latent heat for ice:
L(T) = (2834.1 - 0.29 T - 0.004 T^2)J/g

On the right track?
 
  • #8
Kelsi_Jade said:
OK, latent heat is the amount of energy in the form of heat (Q) required to completely effect a phase change of a unit of mass.
Latent heat=Q/m
Latent heat for water vapor:
L(T) = (2500.8 - 2.36 T + 0.0016 T^2 - 0.00006 T^3)J/g

Latent heat for ice:
L(T) = (2834.1 - 0.29 T - 0.004 T^2)J/g

On the right track?
That first formula is not valid at 100 °C. Wikipedia has a table with values you can use.

Otherwise, yes, you are on the right track.
 
  • #9
DrClaude said:
That first formula is not valid at 100 °C.

Oh, I see that now. Thanks!
 
  • #10
Ok, I may have confused myself a little going over the last bit. So, is the 3600kJ Q, or L(T)?
 
  • #11
Kelsi_Jade said:
Ok, I may have confused myself a little going over the last bit. So, is the 3600kJ Q, or L(T)?
That would be Q going into the hot reservoir.
 

FAQ: Carnot Cycle Refrigerator - Thermal Battery Problem

What is a Carnot Cycle Refrigerator and how does it work?

A Carnot Cycle Refrigerator is a type of refrigeration system that uses a reversible Carnot cycle to transfer heat from a cold reservoir to a hot reservoir. It works by compressing a gas at the cold reservoir to increase its temperature, then transferring the heat to the hot reservoir through a heat exchanger. The gas then expands at the hot reservoir, absorbing heat and cooling the cold reservoir.

What is the purpose of a Thermal Battery in a Carnot Cycle Refrigerator?

A Thermal Battery in a Carnot Cycle Refrigerator is used to store and release heat energy. It acts as a buffer between the hot and cold reservoirs, allowing the system to operate more efficiently by reducing temperature fluctuations.

What are the main challenges in designing a Carnot Cycle Refrigerator?

The main challenges in designing a Carnot Cycle Refrigerator include minimizing heat losses, maximizing heat transfer efficiency, and finding suitable materials for the thermal battery to store and release heat at the desired temperatures.

How does the efficiency of a Carnot Cycle Refrigerator compare to other refrigeration systems?

The efficiency of a Carnot Cycle Refrigerator is theoretically the maximum possible efficiency for a refrigeration system. However, in practice, it is difficult to achieve this theoretical limit due to various factors such as heat losses and irreversibilities. Therefore, the efficiency of a Carnot Cycle Refrigerator may be lower than other refrigeration systems, but it is still considered one of the most efficient techniques.

What are the real-world applications of Carnot Cycle Refrigerators?

Carnot Cycle Refrigerators are commonly used in industrial and scientific applications where precise temperature control is required, such as in semiconductor manufacturing, cryogenics, and medical refrigeration. They are also being researched for potential use in sustainable energy systems and as a way to store and transport renewable energy.

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