Explained: Specific Heats (Cp & Cv)
ฝัง
- เผยแพร่เมื่อ 31 ต.ค. 2016
- In this video we will derive the expressions for the specific heats at constant volume and constant pressure from the combined first and second law of thermodynamics (video [1] below).
The expressions become simpler when we assume a thermally perfect gas, where the energy and enthalpy are only a function of temperature (video [2] below).
===== RELEVANT VIDEOS =====
[1] : goo.gl/soiSr9
[2] : goo.gl/pdL6A8
===== REFERENCES =====
:: Notes by Matt MacLean
:: Modern Compressible Flow, Anderson
:: Elements of Gasdynamics, Liepmann and Roshko
:: Gas Dynamics, Zucrow and Hoffman - วิทยาศาสตร์และเทคโนโลยี
Thank you Josh. I am taking one thermodynamics course, and your video makes these confusing concepts straightforward.
You're welcome! I'm glad they're helpful!
you are an absolute legend
Thanks for the refresher. In the control volume analysis of heat convection cases (external or internal) I notice that for incompressible flow, they use h=CpT to compute enthalpy in many textbooks. I have a question about the validity of this. I can see how if the fluid were ideal we could have h=u+pv=CvT+TR/v*v=(Cv+R)T=CpT. However, when the fluid is modeled as incompressible how can the same relation be used, Is there a relation between enthaly and temperature when the fluid is modeled as incompressible? What about the internal energy? Is CvT still valid ?
I'm going to assume that the fluid you're mentioning here is any gas, not a liquid. Liquids are very nearly incompressible, but they also clearly don't follow the ideal gas law.
For gases, there are a few things we can assume to make our lives easier when analyzing different problems. The first assumption is that our gas follows the ideal gas law. This is the case when we assume we have a thermally perfect gas (TPG), or a calorically perfect gas (CPG). This is a reasonable assumption when we are not at extremely low or high temperatures or pressures (in general). For almost all cases that you'll see in fluids and heat transfer, assuming an ideal gas is reasonable.
Now that we have stated that our fluid behaves like an ideal gas, we can specify whether it behaves like a TPG or a CPG. If it behaves like a TPG, then the internal energy and the enthalpy are functions of temperature (see my video on thermally perfect gases). This means that the equation h = CpT and e = CvT are valid, but note that Cp and Cv are actually functions of temperature (so it might be better to write h(T) = Cp(T)*T and e(T) = Cv(T)*T). If we make a more limiting assumption that we have a CPG, then by definition, the specific heats are constant, and the equations h = CpT and e = CvT are still valid, but Cp and Cv now do not depend on temperature. This makes things easier because we don't need to worry about how these values change when solving for things such as entropy (see my isentropic relations video, where these values come out of the integral).
The last assumption we can make is whether we are dealing with an incompressible or compressible flow. The key is to note that for both incompressible and compressible flows, we can assume the gas is ideal. For both of these types of flows, we can even assume the gas is calorically perfect, our most limiting assumption. I'm not saying that these assumptions are always reasonable though. It will be up to you to decide whether you can assume a CPG, or a TPG, or whether you can't assume an ideal gas at all.
So to sum everything up, whether your flow is incompressible or compressible, you can still assume that it behaves like an ideal gas, and the relationships h = CpT and e = CvT will be valid. However, you now need to know whether you are assuming a TPG or a CPG in order to see the final form of these equations. For a TPG, Cp and Cv are still functions of temperature, while for a CPG, they are constant (but not equal).
Thanks for your comprehensive response. So it's just a matter of assumption. My confusion arises with the fact that if the flow is incompressible then the ideal gas law which is the basis for the derivation of these relations wont hold. But from what I understand you're saying that enthalpy and internal energy can still be linked to temperature thru those equations.
Ok I think I see where the problem lies. When we make the assumption that a gas is incompressible, the only thing we're saying is that the density of the gas does not change with position (x, y, z) or time (t). We haven't specified anything about the type of gas. We can certainly still assume that the gas is ideal. When we have a compressible flow, now we're saying that the density can change with position or time. For compressible flow, we can also certainly make the assumption that we have an ideal gas. The assumptions of incompressible/compressible and ideal/non-ideal are not mutually exclusive.
So for both incompressible and compressible flow, if we decide to assume that the gas is ideal, then the expression h = CpT and e = CvT will hold. Within the assumption of an ideal gas, we can specify even further, which is where the distinction between thermally perfect and calorically perfect comes into play.
Hey! the video was awesome! but can you explain how did you write dh = del q +VdP? We define h as e +pv right??
Differentiate h=e+Pv, using the chain rule on the Pv term to get Pdv+vdP, swap e for dq-dw, swap dw for -Pdv.
very useful, thanks
Glad I could help!
But could u tell me why you used a three lined equal to rather than '='? Pls help asap
Pls help
Pls help
The three-lined equal sign just means "is defined as", instead of "is equal to".
@@JoshTheEngineer thanks your video made my concept krystal clear.
What does the 'e' and gamma stand for?
The variable 'e' is the specific internal energy, and 'gamma' is the ratio of specific heats, or Cp/Cv.
@@JoshTheEngineer thanks
Great. Maybe slow down a little..
Thanks for the feedback!
Useful video but man you speak fast!
Thanks, and yea I know I speak fast, but it's easier to have people slow down or re-watch the video instead of me having to re-film it.
lol i still watched at 2x
@@yashmagarwal2715 God tier brain genetics