Calculate COP & VCC for Refrigerant - Urgent Help Needed

[alexisonsmith]

COP of a Refrigerant...Urgent Please Help!

How do you find the COP ( Coefficient of Performance ) of a Refrigerant if you know the basic details of the refrigerant such as:
Critical tempertaure
Critical Pressure
w
Cp

Also I am trying to calculate the VCC as well!? Any advice would be great ,

Thanks

 

[Topher925]

A refrigerant doesn't have a COP, but a refrigeration cycle does. The COP is defined was;

COP = Heat Transfer of Interest / Total Work Input

The heat transfer of interest is will either be the total heat transfer from the condenser or the evaporator.

I don't know what a VCC is? Can you elaborate?

 

[alexisonsmith]

VCC = Volumetric cooling capacity. I am tryign to calculate it using the Peng-Robinson Equation of State equations...I have no idea how to calculate it using the equation you have given me and what I already have...

 

[Yeti08]

Finding the COP should be basic thermodynamics:

$$COP=\frac{h_{1}-h_{4}}{h_{2}-h_{1}}$$

where h₁, h₂, and h₄ are the enthalpies at the compressor inlet, compressor oulet, and valve (or capillary) outlet, respectively. To determine the enthalpies from the PR EOS you should be able to use enthalpy departure calculations. As Topher925 said the fluid doesn't have an inherent COP - the enthalpies depend on the temperature and pressure at each point and are used in the reduced temperature an pressure in the PR EOS.

 

[alexisonsmith]

Yes, this I understand however I do not understand how I am meant to get the enthalpies by using the critical temperature and critical pressure from which I have been given...the paper is shown below. I am trying to calculate the 1st approach I have found all of the values but it is just the COP I am having problems with...

https://www.physicsforums.com/showthread.php?t=367342

 

[Yeti08]

As I implied before, you can't calculate COP (or VCC) from the information that you listed. If you read the paper that you posted in the other thread, you'll see that they used condenser and evaporater temperatures of 30C and -40C, respectively, with 5C of superheat at the evaporater and 5C subcool at the condenser. All other components are assumed ideal.

 

[alexisonsmith]

Ok so without sounding stupid, how am I meant to calculate the enthalpy of the given temperatures?

 

[Yeti08]

I think you can look at the change in entropy departure between two states and that would be the actual change in entropy since they are both using the same reference point, but I could be wrong on that.

 

[Yeti08]

Okay, I knew I left out a term. The change in enthalpy can be calculated through

$$\Delta H = \Delta H'_{1} + \Delta H'_{12} - \Delta H'_{2}$$


where $$\Delta H'_{1}$$ is the enthalpy departure (or residual enthalpy) at state 1, $$\Delta H'_{2}$$ is the enthapy departure at state 2, and

$$\Delta H'_{12} = \int C'_{p}dT$$

evaluated from T₁ to T₂, where C'_p is the ideal gas heat capacity.

The same format applies to entropy except

$$\Delta S'_{12} = \int C_{p}\frac{dT}{T}-R\ ln\frac{P_{2}}{P_{1}}$$

 

[alexisonsmith]

OK perfect, I can see this all coming together now! So delta h = h1 - h2 I am guessing.

Do I now have enough information to calculate H1 and H2 I know that H = U + PV however i think this could be the wrong form of the equation. or should I look the values for H1 and H2 in the steam tables!?

 

[Yeti08]

Yes, Delta H is H1-H2. Enthalpy is defined as U+PV, but if you already know what the internal energy is then you shouldn't have any problem. You could always use tables to get properties, but I was under the impression that you wanted to use Peng-Robinson EoS to do this. To use PR, you use the enthalpy departure (residual enthalpy) defined as

$$\frac{\Delta H'}{R\ T} = T\ \int_0^P \left(\frac{\partial Z}{\partial T}\right)_{P}\frac{dP}{P}$$

which for the Peng-Robinson compressibility factor becomes

$$\Delta H' = h_{T,P}-h_{T,P}^{ideal} = R\ T_{c}\left[T_{r}\left(Z-1\right)-2.078\left(1+\kappa\right)\sqrt{\alpha}\ ln\left(\frac{Z+2.414B}{Z-0.414B}\right)\right]$$

where

$$\kappa = 0.37464+1.54266\omega - 0.26992\omega^{2}$$

$$B = 0.07780\frac{P_{r}}{T_{r}}$$

 

[alexisonsmith]

By using the PR equations I am assuming the following 2 equations:

Tr=T/Tc
Pr=P/Pc

Am I assuming that T= the evaportaing temperature or the condensing temperature?

 

[Yeti08]

Yes, all the standard notation for Peng-Robinson - T_c is critical temp, P_r is reduced pressure, omega is acentric factor and so forth. You evaluate at the temperature and pressure for the state point in question.

 

[alexisonsmith]

So T is the state point in question which in this case is 30? But then what would be the state point in question for P?

 

[Yeti08]

I mean state point as in a fully defined state, so a state point would require knowing both temperature and pressure. I'm not sure exactly how you could get saturation pressure from the PR EoS. Using van der Waals you could use Maxwell equal area rule, or maybe you could calculate the chemical potentials of the liquid and gas since $$\mu_{G}=\mu_{L}$$ at phase equilibrium. To be honest, I have never had a use for this sort of calculation since I could write a program in a couple minutes that would calculate the COP for the stated case for 30-40 refrigerants, and it's been years since I've studied this particular concept. What, might I ask, are you trying to do this for?

 

[alexisonsmith]

I am currently trying to explain in an assigment different papers on refrigerants, one of the ways in which this is being done is by reproducing the results which were obtained in the paper, I am currently trying to explain the results which were found by calculating one set of results for each of theapproaches however I did not realize it was simple to produce a program for this! How could I go about producingsuch a program? Thank you for all of your help by the way it is really helping me understand what I find a difficult subject

 

[Yeti08]

I meant that I could write a paper that could quickly check and compare COPs of different refrigerants which seems to be the goal of that paper. Doing it specifically with the Peng-Robinson EoS is a different story, but I think I figured it out. I thought of this right after I posted before but I had a meeting to attend. The forumla for the compressability factor for PR is a cubic, so there can be one or three real roots. If there are three, two of them correspond to the vapor and liquid compressability factors - $$Z^{V}$$ and $$Z^{L}$$. The liquid will be the smallest non-zero root and the vapor will be the largest; a single root means single phase. You can then use the compressability factors to calculate the fugacity of the saturated gas and liquid. Basically, combine the standard Peng-Robinson equations, the equations that I already mentioned and these:

Find fugacity ($$f^{V}$$) of saturated vapor and liquid by using the respective compressibility factors:
$$ln\frac{f}{P}=\left(Z-1\right)-ln\left(Z-B\right)-\frac{A}{2\sqrt{2}B}ln\left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]$$

At phase equilibrium the fugacities are equal:

$$f^{V}=f^{L}$$

This should provide enough equations to determine the saturation pressure (the unknown in the fugacity equations), though it may take some iterations depending on how you choose to calculate.

 

[alexisonsmith]

Brilliant I will work on that, in the mean time I am also woking on another paper I have got very far however I cannot seem to calculate the reduced ideal density, I understand the formula however I do not understant where the range of densities come from? Here is the link to the paper
https://www.physicsforums.com/showthread.php?t=367554
Thanks again for your help

 

[Yeti08]

You get varying saturated liquid reduced ideal densities because the density is dependant on the reduced temperature (which is dependant on temperature). So you know the temperature from the reduced temperature and critical temperature of the fluid, you know the pressure from the saturation pressure, and you know the liquid compressilbility so you can find the saturated liquid density and therefore the saturated liquid reduced ideal density.

EDIT:
Just so you know, I was able to reproduce the $$log_{10}\left(\rho_{r}^{id}\right)^{S}$$ vs. $$\tau$$ graph for R134a using the PR equations.