Finding Cv values for Hydrogen and Water at high temperatures

[ZA]

Currently, I am trying to find the optimal O/F ratio for H2 and O2 in terms of maximal gas escape velocities. I have researched Cp, and Cv values for H2O and H2 extensively, yet I can not find a function of their values depending on temperature. Does anyone know where I could find functions of Cp/Cv values for Hydrogen and water accurate to at least 3000 K? Also, in order to find the overall Cp/Cv value, should I just multiply the mole fraction of each product by it's Cp/Cv value and add all of them up? ((Mol frac. H2)*Cp/Cv(t)(H2)+(Mol frac. H2O)*Cp/Cv(t)(H2O))=Cp/Cv? Finally, I am interested in finding the combustion chamber tempreature. In order to do that, should I equate the enthalpy of a unit mass of reactants with the integral of the combined Cp function (per mass unit) of the products?
Tfinal
(Delta H)=Integral( Cp(T)(H2O+H2))dT
Tinitial
I would greatly appreciate any answers to these questions.
thank you for taking your time to read this.

 

[ZA]

One more thing: I have been given approximate Cp and Cv values twice already, but I need numbers and functions accurate to at least 2 zeros, thank you

 

[FredGarvin]

No offense, but I question your statement about accuaracy to 4 decimal places. In my experience with fuel/oxidizer systems, the worrying of being .0001 off on anything is not practical.

That being said, from my thermodynamics reference, Moran & Shapiro:

$$\frac{\overline{c_p}}{R} = \alpha = \beta T + \gamma T^2 + \delta T^3 + \epsilon T^4$$

For Hydrogen:
$$\alpha$$ = 3.057
$$\beta x 10^3$$ = 2.677
$$\gamma x 10^6$$ = -5.810
$$\delta x 10^9$$ = 5.521
$$\epsilon x 10^{12}$$ = -1.812

For water:
$$\alpha$$ = 4.070
$$\beta x 10^3$$ = -1.108
$$\gamma x 10^6$$ = 4.152
$$\delta x 10^9$$ = -2.964
$$\epsilon x 10^{12}$$ = .807

This is stated as being good from 300 to 2000K. How it varries after that I have no references available right now that cover it.

You may want to look into NASA SP-273 here:
http://www.openchannelsoftware.org/project/view_user_defined_page.php?user_defined_page_id=417&group_id=29

I doubt you'll want to spend the money for the source code, but it can give you some research options to follow up on.

 

[Clausius2]

Also, in order to find the overall Cp/Cv value, should I just multiply the mole fraction of each product by it's Cp/Cv value and add all of them up? ((Mol frac. H2)*Cp/Cv(t)(H2)+(Mol frac. H2O)*Cp/Cv(t)(H2O))=Cp/Cv? Finally, I am interested in finding the combustion chamber tempreature. In order to do that, should I equate the enthalpy of a unit mass of reactants with the integral of the combined Cp function (per mass unit) of the products?
Tfinal
(Delta H)=Integral( Cp(T)(H2O+H2))dT
Tinitial
I would greatly appreciate any answers to these questions.
thank you for taking your time to read this.

Fred has answered the first part. Now I have been thinking about the second. After writting some equations without any bibliography as a background, I propose you to calculate the adiabatic constant of a binary mixture on the next way: (subindex "j" refers to hydrogen, while "o" refers to oxygen):

i) The mixture thermal state (internal energy) must be the same than the sum of each component one:

$$U=\frac{PV}{\gamma-1}=\frac{P_jV}{\gamma_j-1}+\frac{P_oV}{\gamma_o-1}$$

using ideal gas equation and assuming an isothermic mixture, this last equation yields:

$$\gamma=\frac{1}{\frac{X}{\gamma_j-1}+\frac{1-X}{\gamma_o-1}}+1$$

where X is the hydrogen molar fraction. I have checked for X=1 then logically$$\gamma=\gamma_j$$. Try with this model or search in some book about thermodynamic properties of mixtures.

About the third part, assuming there is no heat losses:

$$H_{products}=H_{reactants}$$

where $$H_i=N_i\Big(\int_0^{T_a}cp_i(T)dT+h_{fi}\Big)$$ is the enthalpy of each component, being N_i the number of moles. The enthalpy is composed by two terms: one thermal enthalpy which is integrated to $$T_a$$, and another term of formation enthalpy. That temperature represents the "adiabatic flame" temperature, which will be of the same order than "the combustion chamber temperature". You must look at thermodynamic tables, where all these integrals are solved for each component and temperature, and iterate in $$T_a$$ because it is a non linear equation.

was it helpful?

EDIT: sorry. Hope you knew $$\gamma=C_p/C_v$$.

 

[Clausius2]

One more thing: I have been given approximate Cp and Cv values twice already, but I need EXACT numbers and functions accurate to at least 3-4 zeros, as this is literaly rocket science, and even the slightest mistake can result in a complete faliure!

It doesn't matter how many zeros you write in your calculator. The accuracy of your calculation is dominated by the accuracy of experimental data obtained for adiabatic constants, heat capacities, and enthalpies. I doubt too much any experimentalist is going to show you data with +/-0.0001 accuracy. They won't feel like :tongue2: . And I don't think Von Braun employed 4 decimals when calculating the Saturn V. He was better an old german wolf.

 

[ZA]

Thank you all very much for the information provided. It is very helpful. Sorry about asking for so many zeros. I guess I got carried away. I now have just one question left. Does anyone know a place where I could find the functions for Cv values? Thanks.

 

[Clausius2]

Thank you all very much for the information provided. It is very helpful. Sorry about asking for so many zeros. I guess I got carried away. I now have just one question left. Does anyone know a place where I could find the functions for Cv values? Thanks.

I don't know. Surely you could find yourself with different values in bibliography. Why do yo need it?. I hadn't needed any of these values when I calculated the adiabatic temperature, because every enthalpy is usually tabulated, you could find it in a chemics or combustion book (or see Jannaf tables). Anyway, if you are calculating it seriously, I recommend you the typical book of reference: "Molecular Theory of Gases and Liquids AKA Green Monster" of Hirschfelder. You will have fun with it :tongue2: .