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AE 4711

Mechanical Engineering

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AE 4711. Computational Assignment 1

C1. The specific heat capacity at constant pressure (as well as constant volume) of polyatomic molecules change with temperature (T). It is essential that the T dependence of CP is taken into account while calculating specific enthalpy changes of air that occurs in gas turbine engines. Write two subroutines (in Matlab or any programming language of your choice) which will calculate the change in specific (thermal) enthalpy of air (mixture of N2 and O2) between T1 and T2:

Subroutine 1: delta_h_con_cp - which assumes a constant value of Cp, that at 300 K.

Subroutine 2: delta_h_var_cp - which considers the variation of Cp with T.

The Cp dependence of gas phase fluids on T is usually expressed as polynomials. These polynomials can be found. Since properties are available only for pure species, you will have to use the Cps of N2 and O2 to calculate the Cp of air. Keep in mind that air is composed of 79% N2 and 21% O2 (mole or volume basis).

Use the subroutines 1 and 2 to calculate the change in specific enthalpy when a gas is heated from 300 K to 2000 K. Quantify the error that results from making the constant Cp assumption. (10 pts)

C2. Entropy of a species in a multi-component gas depends on its T and partial pressure. It is important in cycle analysis to figure out the final conditions at the end of an isentropic process (for modeling compressors and turbines). Write a subroutine to calculate final temperature of air, post an isentropic process, given the initial conditions (P1,T1) and final pressure (P2). Subroutine 3: isen_tf_var_cp - Taking into account the Cp variation with temperature

Similar to Cp dependence on temperature, polynomials are used to express the dependence of standard molar entropy (see lecture notes on multi-component gases) on temperature and can be found on the NIST database.

C3. Now, let’s try and model a compressor of a gas-turbine engine. Assume that the velocity of gas flow through the compressor is small enough that the kinetic energy of the gas is negligible compared to its thermal energy. The compressor is characterized by its pressure ratio, πc = Pout/Pin; and isentropic efficiency, ηc. Calculate work required to compress gas from an initial condition of (Tin, Pin) over a specified πc, using

a. a reversible compressor (Subroutine 4: comp_rev_low_mach) b. an irreversible compressor with ηc = 80% (Subroutine 5: comp_irrev_low_mach) Now, utilize the subroutines 4 and 5 to calculate the work required to compress gas from an initial condition of 300 K and 1 bar over a πc = 40. Comment on the result.

Note: These subroutines will be used for cycle analysis at a later stage. So the compressor subroutine for example should accept as arguments: inlet state properties, πc, and ηc.; and return as arguments the final state and required work input.

C4. In the previous problem C3, the work input to compress a gas across a specified pressure ratio was calculated, however the final temperature at the end of the process was not calculated. In order to calculate the final temperature, the change in temperature has to be calculated for a given change in enthalpy. Write a subroutine to calculate the change in temperature, given a change in thermal enthalpy.

Subroutine 6: delta_T_var_cp - which considers the variation of Cp with T. (5 pts)

[Hint: you have to use an iterative procedure]

C5. We have to develop a model for turbine operation. A turbine is usually characterized by the pressure ratio, πt = Pin/Pout, and its isentropic efficiency, ηT. Calculate the work output for:

a. a reversible turbine (Subroutine 7: turb_rev_low_mach)

b. an irreversible turbine with ηT = 85% (Subroutine 8: turb_irrev_low_mach)

Perform calculations using Subroutines 7 and 8 to expand a gas at 40 bar and 829 K in a turbine across πt = 40. The temperature at the exit can be calculated using Subroutine 6.

C6. Combustor modeling involves calculating the fuel-to-air ratio, given the inlet and exit (limited by turbine inlet) temperatures. Write a subroutine (Subroutine 9: comb_rev_low_mach) to calculate the fuel-to-air ratio and equivalence ratio at which the combustor will have to operate for a given inlet and exit temperature. Assume that the combustor uses dodecane (hf (298K) = -82.8 kcal/mol) as the fuel. The combustion properties of dodecane are similar to the combustion properties of real jet fuel, which is a mixture of many hydrocarbons.

C7. Now we have developed tools to model the operation of turboshaft engines, which are used to power helicopters and to generate electricity. Using Subroutines 1-9, write a program to simulate turboshaft operation. Prepare the following plots and explain the trends observed.

a. For a given turbine inlet temperature, T4 = 1600 K, plot the thermal efficiency, work per unit

mass flow, equivalence ratio, and fuel-to-air ratio as a function of compressor pressure ratio

for a reversible and irreversible (ηC = 85%; ηT = 90%) engine.

b. Plot thermal efficiency and work output as a function of pressure ratio for three values of

T4 = 1400 K, 1600 K, and 1800 K. Assume that engine is reversible and plot the three cases on

the same chart. From the plot, explain why there is a need to develop turbine blades that can

withstand higher temperatures.

c. The T-64-GE-100 turboshaft engine made by GE is used to power the Sikorsky MH-53T

helicopter. It has an overall pressure ratio of 14 and turbine inlet temperature of 827 C. It

generates a maximum power of ~4300 shp (shaft horsepower) and ~3800 shp under

continuous operation. GE has also revealed that the engine has a specific fuel consumption

of ~80 g/(MW.s) for an air flow 13.3 kg/s. Use the developed tool to analyze and then

comment on the design choices and also the accuracy of the developed tool.

Information to check if subroutines are working properly:

Subroutine	Inputs	Output
2: delta_h_var_cp	T1 = 400 K, T2 = 1400 K	Δh = 32.37 kJ/mol
	T1 = 800 K, T2 = 1900 K	Δh = 37.93 kJ/mo
3: isen_tf_var_cp	T1 = 400 K, P1 = 1e5 Pa, P2 = 1e6 Pa	Tf = 752.43 K
	T1 = 800 K, P1 = 1e5 Pa, P2 = 3e6 Pa	Tf =1833.17 K
9: comb_rev_low_mach	T1 = 300 K, T2 = 1800 K	φ = 0.604
	T1 = 600 K, T2 = 1800 K	φ = 0.458

 

The correctness of other subroutines can be checked using the above three subroutines or using values derived from constant Cp based results.