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I'm in a combustion lab and we are performing experiments on a turbojet engine. However, I am having some trouble calculating the exit velocity and compressor/turbine powers of the engine.

Consider the nozzle exit, where the exhaust gases go. Assuming 100% adiabatic efficiency of the nozzle and applying the 1st law of thermodynamics gives $$ \frac{u_e^2}{2} = h_{0e}-h_e = c_{p,N}(T_{0e}-T_e)=c_{p,N}T_{0e}[1-(p_a/p_{0e})^{(\gamma_n - 1)/\gamma_n}], $$ assuming $p_e = p_a$, the ambient pressure. In the lab, we've set up thermocouples that measure the stagnation temperature and pressure at the exit, as well as ambient conditions. My problem, however, is that I have no idea where to get $c_{p,N}$, the specific heat of the gases in the nozzle from. I've looked up in a reference book that the average specific heat ratio $\gamma_n = 1.36$, but this still doesn't give me the specific heat that I need because I don't know the specific gas constant $R$ for these gases. For reference, it is an SR-30 turbojet with Jet-A fuel.

In addition, I have the same problem when calculating the power outputs. Assuming that all of the power of the turbine is transmitted to the compressor, $$ \dot{W_T} = (\dot{m_a} + \dot{m_f})c_{p,T}(T_{04}-T_{05})=\dot{m_a}c_{p,C}(T_{03}-T_{02}) = \dot{W_C} $$ Again, I have no clue what the specific heats are, only their ratio. Is there some way I can calculate these quantities or look them up somewhere? Or even find the specific gas constants somewhere?

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4 Answers 4

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Do you have a copy of the book Gas Turbine Theory by H. Cohen, G.F.C. Rogers and H.I.H. Saravanamuttoo? You may well find the answers in there I think.

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  • $\begingroup$ Yes, they are on the back page. But, here’s a website with them: ohio.edu/mechanical/thermo/property_tables/air/air_cp_cv.html Walsh and Flecther’s book Gas Turbine Performance is also a great source for this sort of info. See amazon.com/Gas-Turbine-Performance-Philip-Walsh/dp/0791800679 $\endgroup$
    – Penguin
    Sep 6, 2018 at 10:21
  • $\begingroup$ Alright guys, I’ve found cp for the compressor and turbine. My next question is would the nozzle cp be the same as the turbine cp since the same gases are flowing through them? I can’t seem to find cp nozzle in any of the references you gave. $\endgroup$
    – Josh Pilipovsky
    Sep 6, 2018 at 16:38
  • $\begingroup$ @JoshPilipovsky. cp (and cv) is a function of total temperature, and fuel air ratio. In a civil engine, the FAR and total temp in the exhaust nozzle is the same at the low pressure turbine exit. But, it will be different to the value at the HP turbine entry, or LP turbine entry, as total temperature is decreasing through the turbine as energy is extracted. If your engine includes an afterburner which is operating, that will change the FAR and total temperature between the LPT exit and nozzle. Use the formula for cp and cv as a function of total temp and FAR as outlined in ... $\endgroup$
    – Penguin
    Sep 7, 2018 at 10:37
  • $\begingroup$ ...this paper here: citeseerx.ist.psu.edu/viewdoc/… $\endgroup$
    – Penguin
    Sep 7, 2018 at 10:38
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R or R* is the Universal Gas Constant, 8.3143 joules/K-mol. The gas constant r for a particular gas, or the specific gas constant is calculated as r = R/m where m is the molecular weight of the gas in question.

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Well known values used in manual gas turbine calculations for the specific heat for air are 1005 J/kg/K and 1150 J/kg/K for combustion gasses. However, note that these are chosen values that are convenient for the calculation, these have no value when considering the actual gases.

The specific heat (at constant pressure, $c_p$) of gases depends on the composition and the temperature. This implies that your calculations are always incorrect using the formulae expressed in the equation. You need to integrate over the compression or expansion process.

It is possible to calculate the specific heats, note that Gordon and Mc Bride from NASA Cleveland, Ohio have shown how this can be done through a computer program: Computer Program for Calculation of Complex Chemical Equilibrium Compositions and Applications, known as CAE. The calculation is based on polynomials calculating the individual contributions for the species in the gases (to make the polynomial dimensionless, it is given in the form of $\frac{C_p}{R}$).

$$\frac{c_p(T)}{R} = a_1T^{-2} + a_2T^{-1} + a_3 + a_4T + a_5T^{2} + a_6T^{3} + a_7T^{4} $$

In this equation the constants $a_1$ through $a_7$ are defined below and above 1000 K. The $c_p$ found using this equation is in (J/mole/K). To get $c_p$ in (J/kg/K), one can use the molar weight of the specie. The $c_p$ of a mixture is calculated by taking the weighted average between the values of $c_p$ of the species present in the mixture. In case of a $c_p$ in (J/kg/K) the mass fraction is to be used and in case of $c_p$ in (J/mole/K) the mole fraction:

$$ c_{pg} = \sum_{i=1}^{NS} ({m_i \cdot c_{p,i}}) = 1 $$

where:

  • $c_{pg}$ = specific heat at constant pressure of the (gaseous) medium (J/kg/K),
  • NS = number of species in mixture (-),
  • $m_i$ = mass fraction of specie $i$ (-),
  • $c_{p,i}$ = specific heat at constant pressure of specie $i$ (J/kg/K).

The specific heat used in gas turbine performance simulation software used to be tabulated based on the fuel to air ratio (FAR) of kerosene in air. Using interpolation, the values for the calculation are found. However, more complex and modern gas turbine simulation software tools have some sort of an implementation of NASA's CEA.

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Find and old Pratt & Whitney handbook. It list k values for combustion products of alkanes at 100% stoich, 200% stoich, and 400% stoich.

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