Physics (thermo) question

[TheOmegaCode]

I have a problem that's just kicking my ass.
A 6-cylinder gasoline engine has a displacement volume of 3 liters, a compression ration of 9.5 and operates at 4000rpm. The air-fuel mixture enters a cylinder at p=1atm and an ambient temperature of 27celsius. During combustion, the mixture reaches a temp of 1,350celsius. Calculate the power delivered by this engine.

So, in a PV diagram, we have a path from A-> B-> C-> D-> A. Since the process is cyclic, the internal energy goes to zero for the cycle. We know from A->B and C->D the process is adiabatic and the process from B->C and D->A is isometric.

The heat enters the system from B->C and heat is expelled from D->A.
So Qh is the process from B->C and Qc is the process from D->A.

W=Qh-Qc

Qh=Cv(Tc-Tb)

Qc=Cv(Ta-Td)

Is it wrong to assume Ta=Tb and Tc=Td since between these respective processes there is no heat exchanged? Also, how do we find the heat capacity at constant volume for an air-fuel mixture? Do we even need it? Also, should we treat the mixture as an ideal gas and use the given initial pressure to our advantage?

We know that for a single cylinder, the non-compressed volume is .5L*((10^-3*m^3)/L) and the compressed volume is .05263L*((10^-3*m^3)/L).

Any tips or points to another resource would be greatly appreciated. I just can't seem to figure this one out.

Once we find the work, the problem is just about done. We know that power is delivered every other revolution of the crankshaft. So this work, multiplied by the six cylinders and then solve P=dW/dT. The power is 'delivered' every 2/4000 minutes * 60 seconds/ minute or 3/100 seconds. So the power would be the Work of one cylinder, W1c, divided by (3/100)s, times six cylinders or 200W1c/second.

I've even tried applying the fact that for an adiabat, T1V1^(cp/cv -1)=T2V2^(cp/cv -1), to use, but I wasn't clear on what temps and points to use. Using points A and C respectively, I got a gamma = 1.75, which doesnt make any sense. That's a higher yield than a monotomic gas.

 

[DrPizza]

The memories you just brought back of thermo from way back when. Sorry, been way too many years for me.

 

[aesthetics]

Physics makes me hate life.

 

[Toastedlightly]

Originally posted by: TheOmegaCode
I have a problem that's just kicking my ass.
A 6-cylinder gasoline engine has a displacement volume of 3 liters, a compression ration of 9.5 and operates at 4000rpm. The air-fuel mixture enters a cylinder at p=1atm and an ambient temperature of 27celsius. During combustion, the mixture reaches a temp of 1,350celsius. Calculate the power delivered by this engine.

So, in a PV diagram, we have a path from A-> B-> C-> D-> A. Since the process is cyclic, the internal energy goes to zero for the cycle. We know from A->B and C->D the process is adiabatic and the process from B->C and D->A is isometric.

The heat enters the system from B->C and heat is expelled from D->A.
So Qh is the process from B->C and Qc is the process from D->A.

W=Qh-Qc

Qh=Cv(Tc-Tb)

Qc=Cv(Ta-Td)

Is it wrong to assume Ta=Tb and Tc=Td since between these respective processes there is no heat exchanged? Also, how do we find the heat capacity at constant volume for an air-fuel mixture? Do we even need it? Also, should we treat the mixture as an ideal gas and use the given initial pressure to our advantage?

We know that for a single cylinder, the non-compressed volume is .5L*((10^-3*m^3)/L) and the compressed volume is .05263L*((10^-3*m^3)/L).

Any tips or points to another resource would be greatly appreciated. I just can't seem to figure this one out.

Once we find the work, the problem is just about done. We know that power is delivered every other revolution of the crankshaft. So this work, multiplied by the six cylinders and then solve P=dW/dT. The power is 'delivered' every 2/4000 minutes * 60 seconds/ minute or 3/100 seconds. So the power would be the Work of one cylinder, W1c, divided by (3/100)s, times six cylinders or 200W1c/second.

I've even tried applying the fact that for an adiabat, T1V1^(cp/cv -1)=T2V2^(cp/cv -1), to use, but I wasn't clear on what temps and points to use. Using points A and C respectively, I got a gamma = 1.75, which doesnt make any sense. That's a higher yield than a monotomic gas.

just reading, I would assume that Ta and Tb are NOT equal, same goes with Tc and Td. The adiabatic compression just means that no heat was lost. Work is still being done on the system (in the form of compression).

To start you down a path that I woudl do.. I'd use enthaly instead of internal energy for this calculation (enthalpy deals with a pressure term, while internal energy doesn't, IIRC).

Let me think about it a bit more and see if I can come up w/ anything.

EDIT: Do you have a PV diagram to go along w/ this problem (including isotherms, adiabatic equivilence lines, ethalpy lines and the like)? I know it is a long shot, but that seems to be the best situation with the non-ideal mixture.

 

[CraKaJaX]

Originally posted by: aesthetics
Physics makes me hate life.

'Physics' isn't THAT bad. However, when it's e-mag fields and waves, it can get sheisty REAL fast.

On a side note, my roommate is always bitching about his thermo exams and hw's (Chem Eng)

 

[TheOmegaCode]

Nothing but the italicized text was given. I'm relying on the PV Diagram for an ideal otto cycle. i.e. http://www.grc.nasa.gov/WWW/K-12/airplane/otto.html, where 2/6, 3, 4, 5 are my a, b, c, d respectively. The only thing I know about enthalpy is what I've seen from some cambridge videos on thermo. The way our text approaches the material for our level is, zeroth, 1st, and 2nd law of thermo. Where for the 2nd law we only cover heat engines, refrigeration, carnot engines and entropy. That's all I have to work with from the curriculum. Though, it's a problem from outside the text, so I'm not above using outside material/resources.

 

[Born2bwire]

Originally posted by: CraKaJaX

Originally posted by: aesthetics
Physics makes me hate life.

'Physics' isn't THAT bad. However, when it's e-mag fields and waves, it can get sheisty REAL fast.

/me looks down at the Maxwell's Equations T-shirt he is wearing.

 

[Fenixgoon]

i'd have to crack open my thermo book and refresh.. it's been 2 years.

 

[Eli]

Cool.

I wish I knew how to figure this out.

 

[Howard]

Originally posted by: Eli
Cool.

I wish I knew how to figure this out.

Borrow a thermodynamics textbook.

 

[CycloWizard]

It's hard to say for sure how to solve this without knowing what you've learned thus far in the course. There are many different approaches that will work, but if we tell you one that is far beyond what you actually know/understand, I doubt your prof will believe that you did it (and, of course, he'd be right).

As others have said, adiabatic processes are not generally isothermal. If I isochorically combust gasoline in the engine without transferring heat, I have increased the temperature and pressure of the fluid in the chamber without transferring heat to or from the surroundings. However, if I know how much gasoline I combusted, I can compute the change in internal energy of the fluid and can then calculate its final temperature and pressure. The key to solving this problem is to understand the process to the point where you can identify and solve individual steps. As this is a standard ICE problem, there are literally entire books and websites dedicated to it, so I'm not going to do the whole thing for you, but hopefully the hints will get you going in the right direction.

 

[TheOmegaCode]

Yeah, I've come to the conclusion that the temperatures cannot be the same. Moreover, Ta=Tb(1/9.5)^(cp/cr -1) and Tc=Td(9.5)^(cp/cv -1), since the process between a->b and c->d are adiabatic. So the temp increases from A to B to C and decreases from C to D to A. And I could arrive at the work two different ways, either finding the work done on the system in the adiabatic phases and adding them together, or find the heat entering the system and subtracting heat leaving the system. The problem is, I don't know the degrees of freedom of the air/gasoline mixture or the specific heat capacity. That's the hurdle I'm trying to get over.

Also, I'm treating the mixture as an ideal gas, the problem states that it is initially at near STP.

What I'm trying to do now is come up with some sort of system of equations to, hopefully, sub in values and get a definitive value.

 

[Paperdoc]

This is a classic problem involving the CARNOT cycle. Look that up to calculate the energy released in one cycle of one cylinder. (Remember the compression ratio in calculating the pressure in the cylinder prior to ignition.) Then remember that a 4-cycle gas engine only fires each cylinder every SECOND revolution. Then you have the rpm value, with which you can calculate the number of cylinder explosions (and energy releases) per minute. That gives you a rate of energy release, which is power; convert to whatever units you need.

 

[TheOmegaCode]

What they wanted, apparently, was just to make an assumption on the Cv. I just used 7/2R for polyatomic molecules. So gamma was approximately 1.28.

and for the work, Qh=nCv(Tc-Tb) and Qc=nCv(Ta-Td) you use n=PV/RT

So the work for a single cylender ends up being PV/T*7/2*(Tc-Tb+Ta-Td). Multiply that by 200 and it gets you the power.