Thermodynamic homework about 1st and 2nd Law of Thermodynamics

[alpyurtsever]

Hi, I need help in my Thermodynamics Homework. The question is that,

 

A rectangular steel tank having an internal volume of 1 m3 contains air at 2.5MPa and 20°C. A relief valve is opened slightly allowing air to escape to the atmosphere. The valve is closed when the pressure in the tank reaches 350 kPa.

a) Calculate the amount of heat that must be added so as to keep the tank contents at 20°Cthroughout the process.

B) Calculate the final temperature if the process takes place adiabatically.

 

--

 

I have done the a part, using the first law of thermodynamics.

I have posted it. I also tried to solve b with the same analysis but it failed because the equation at the end does not have a real solution.

I thought that the 2nd Law can be used also, but I couldn't solve it anyway, because it is an unsteady state system and I don't know entropy of the mass leaving the system.

Can you help me?

[cypress]

Hi, I need help in my Thermodynamics Homework. The question is that,

 

A rectangular steel tank having an internal volume of 1 m3 contains air at 2.5MPa and 20°C. A relief valve is opened slightly allowing air to escape to the atmosphere. The valve is closed when the pressure in the tank reaches 350 kPa.

a) Calculate the amount of heat that must be added so as to keep the tank contents at 20°Cthroughout the process.

B) Calculate the final temperature if the process takes place adiabatically.

 

--

 

I have done the a part, using the first law of thermodynamics.

I have posted it. I also tried to solve b with the same analysis but it failed because the equation at the end does not have a real solution.

I thought that the 2nd Law can be used also, but I couldn't solve it anyway, because it is an unsteady state system and I don't know entropy of the mass leaving the system.

Can you help me?

 

This can be solved two ways. You can continue to use the ideal gas law and as you note you will have to integrate since temperature is not constant. Or you can use the thermodynamic tables for air which are empirical but they do the work of integrating for you. Are you familiar with the tables and are you allowed to use them for this problem or must you do the math by hand? If you must do it by hand, I can help you find the error in your attempt. Just let me know.

 

A third and easier solution would be to note that the system can be modeled as simpler case. Since the system is to be treated as adiabatic, how is use of the relief valve different thermodynamically from expansion by say a piston? After noting the difference (if any) can you model this by simple adiabatic expansion? Where is the work performed in the relief valve case?

[alpyurtsever]

We are using the tables for steam in general, but I think that wouldn't be a big problem if I use it for air too. Instructor would accept it. However we won't have the table for air in exams, so it would also be good if I learn how to integrate it. On the other side, I was thinking about taking the temperature of the air leaving the system as constant at (Ti+Tf)/2 . Is it a bad approximation? Or does it work?

 

I didn't understand your third approach. How can I model it with another system. I will be happy if you can show me any example of it.

 

Thank you

Edited November 21, 2010 by alpyurtsever

[cypress]

We are using the tables for steam in general, but I think that wouldn't be a big problem if I use it for air too. Instructor would accept it. However we won't have the table for air in exams, so it would also be good if I learn how to integrate it. On the other side, I was thinking about taking the temperature of the air leaving the system as constant at (Ti+Tf)/2 . Is it a bad approximation? Or does it work?

 

I agree use of the tables would not be useful for understanding in the long run. Your approximation is not a good approach because it will give you an approximate answer and it does not contribute to understanding the concept being explored by the problem. I would mark you down for it. Let's explore the third approach because I believe it does illustrate the concept.

 

I didn't understand your third approach. How can I model it with another system. I will be happy if you can show me any example of it.

 

Thank you

 

Let's replace the relief valve with a turbo expander which extracts work from the expanding gas. How would that change the final temperature, pressure and volume state of the gas in the tank? Based on the answer to that question, tell me how the system is similar and different from a chamber as described only with a movable piston held by a pin that is allowed to move from one point to another where the final pressure is 350kPa without any heat transfer? What about a container that is separated by a membrane from an evacuated chamber and then the membrane is removed such that at the end, the pressure is 350kPa?

 

Does this give you any ideas of how you might set up the formulas.

[alpyurtsever]

I don't know what turbo expander is? Is it a turbine, a compressor or something different? But I understand that I can use the state variables to solve the question. Do you mean it? For example for the piston case, the initial and final pressures are the same. If I can find one more same state variable, than all the others will be same.?

 

Evacuated chamber can be a good approach, system can be selected as the all volume, it can be easier.

 

But I didn't understand it totally I think. Maybe I have to research more. Is it about only the first Law of thermodynamics, will I use the entropy balance equation or, is it only about the conservation of energy?

[cypress]

I don't know what turbo expander is? Is it a turbine, a compressor or something different? But I understand that I can use the state variables to solve the question. Do you mean it? For example for the piston case, the initial and final pressures are the same. If I can find one more same state variable, than all the others will be same.?

 

Evacuated chamber can be a good approach, system can be selected as the all volume, it can be easier.

 

But I didn't understand it totally I think. Maybe I have to research more. Is it about only the first Law of thermodynamics, will I use the entropy balance equation or, is it only about the conservation of energy?

 

 

It is a conservation of energy problem, but it is about recognizing the conditions under which the system is changed. I reviewed your work and it seems the error you made was in properly characterizing the system. I was trying to get you to make a distinctions about the system based on the various scenarios I suggested but perhaps we should return to the initial problem and review the conditions specified by the problem statement.

 

You have already noted that dU = dW + dQ and I suspect you know or have derived that dW = -PdV. Finally you are aware that enthalpy, the total energy in a thermodynamic system, H = U + PV which represents U, the amount of energy required to create the system plus PV, the amount of energy required to make room for the system in the environment that surrounds the system. Thus dH = dU + PdV + VdP which is equal to the heat capacity (at constant pressure) of the system times the temperature change nC_pdT.

 

Let's review each of these components in light of the problem statement again.

 

First carefully define your system and identify the boundary of the system. What is the system?

 

Next change in heat energy dQ, from the problem statement is heat energy being added or removed?

 

Third work energy dW, is the volume of the system changing?

 

Does this help?

[alpyurtsever]

I am posting my solution as a whole with part a and b. I know and understand these concepts but how can I use them? I have pointed where I am unable to continue solution in my answer sheet.

[cypress]

I am posting my solution as a whole with part a and b. I know and understand these concepts but how can I use them? I have pointed where I am unable to continue solution in my answer sheet.

 

Ok, on the second page you state that there is no shaft work but is it true that there is no work done on the environment surrounding your system? In your setup you defined the system boundary as the volume of the tank and by that definition no work is performed within the tank but there is mass and energy leaving the tank. This definition makes the equations much more difficult because you have mass and energy fluxes that are variable making the equations difficult to solve. You are missing the mass and energy balance equations which is why you describe it as indefinite. I was trying to get you to redefine the system to make mass constant. Try defining your system as the gas regardless of where it resides. When you do this you have the gas expanding into and displacing volume in the space surrounding the tank thus there is work being done.

[alpyurtsever]

So you suggest me to think that the tank is connected to an evacuated tank, and take the system boundries as the whole volume of two tanks? So there wouldn't be a mass flow out of the system, but I will have work term instead which is easier to compute?

[cypress]

Yes, that would be a good approach.

[alpyurtsever]

I will work on it and send it tomorrow again than. Thanks

[alpyurtsever]

I have tried to do that for part a of the question, but I have failed. I found something different, and I think it is because I calculate the heat given to all system, so it is greater than the heat given to part of gas which is still inside the volume A. I will ask it to course assistant, thanks anyway. If I will fail to understand it, I will repost here a message

[cypress]

Yes your previous method for A was fine. I am sorry I was not more clear about your part A. Only part B required a different approach due to the problem statement.

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