Help with finite heat release analysis

[Jason Louison]

https://www.engr.colostate.edu/~allan/thermo/page8/page8.html

The link above takes you to a site I have found to be very helpful in my studies of the Otto cycle, but on this particular page, it depicts an equation for the rate of change of cylinder Pressure vs. crank angle, and in this particular equation, cylinder pressure itself is also a variable... or is it? They say that this equation will effectively solve the equation for cylinder pressure vs crank angle, but the first thing that perplexed me was that cylinder pressure was a variable in its own derivative, and second, how would someone go about deriving an actual equation for cylinder pressure vs crank angle from this?

 

[anorlunda]

That is called a differential equation. You solve it using integration to get pressure as a function of time. Do you have any background in calculus and differential equations?

 

[Jason Louison]

That is called a differential equation. You solve it using integration to get pressure as a function of time. Do you have any background in calculus and differential equations?

Yes, but how do I solve for pressure if pressure is its own variable in the differential equation?

 

[anorlunda]

Yes, but how do I solve for pressure if pressure is its own variable in the differential equation?

I sounds like you didn't understand my answer from #2. May I suggest the following link as an excellent source for you to teach yourself the answers to your questions.

https://www.khanacademy.org/math/differential-equations

 

[Jason Louison]

I sounds like you didn't understand my answer from #2. May I suggest the following link as an excellent source for you to teach yourself the answers to your questions.

https://www.khanacademy.org/math/differential-equations

I understand completely, what I DON'T understand is how are we going to integrate an equation for cylinder pressure if the differential equation is DEPENDENT ON CYLINDER PRESSURE??

 

[anorlunda]

View attachment 208254View attachment 208255

I understand completely, what I DON'T understand is how are we going to integrate an equation for cylinder pressure if the differential equation is DEPENDENT ON CYLINDER PRESSURE??

That is absolutely ordinary in differential equations. You need to study solution methods to solve equations involving both P and dP/dt.

There is also the digital simulation method. Start at time 0, given initial P, calculate dP/dt with the equation. Then set P=P+E*(dP/dt) where E is constant representing a very small time increment. That gives you the value for P at t=E seconds. Then repeat the whole procedure again and again until you get tired. You will have a series of numbers which are the values of P for each instant in time. Just be sure to choose E small enough. That's hard using pencil and paper, but it is trivially easy using a computer.

 

[Jason Louison]

That is absolutely ordinary in differential equations. You need to study solution methods to solve equations involving both P and dP/dt.

There is also the digital simulation method. Start at time 0, given initial P, calculate dP/dt with the equation. Then set P=P+E*(dP/dt) where E is constant representing a very small time increment. That gives you the value for P at t=E seconds. Then repeat the whole procedure again and again until you get tired. You will have a series of numbers which are the values of P for each instant in time. Just be sure to choose E small enough. That's hard using pencil and paper, but it is trivially easy using a computer.

I've been looking around and haven't gotten very far with the integration, but I'll try the computer technique.