The Experimental Repeatability

h convection heat transfer coefficient at thermocouple wire surface

Now, the analysis method of uncertainty can be utilized to obtain the uncertainty in the flame temperature from the correlation associated with h, T_t, and ε. The relationship between temperature and error is shown in Fig. 2.6.

2.6.2 Uncertainty Analysis of mass flow controller

The apparatuses must be corrected by other standard instruments to make sure that they can normally operate and let the inaccuracy of the experimental results reduce to minimum. In this study, the major sensor in the experiment was the mass flow controller (MFC). The measurement range of the TC-1350 MFC adopted in this study was 0.6-100L/min±0.2%. The author also used different type of MFC, series TC-3100, which had wider measurement range as the standard correction apparatus to correct the TC-1350 MFC. All the uncertainties in different flow rate were between -0.05% and 0.12%.

2.6.3 The Experimental Repeatability

To verify experimental accuracy, perform one test using the specified mixed fuel at the specified pressure and flow rate; perform each test three times to ensure experimental repeatability. The following examples demonstrate experimental repeatability. The volume flow rates for four fuels with LHVs and

28 

RPMs are chosen to demonstrate experimental repeatability. The evaluation included three measurements for volume flow rate; the average value for each test was used. Standard deviation is defined as the absolute difference among the three volume flow rates. Table 2.2 and Fig. 2.7 show the coefficients of variation (CV) and experimental error bars. The CV is defined as the ratio of standard deviation s to mean X, where s is derived by

∑

=

= ^N

1 i

2 i -X) N (X

s 1

(2.3) The CV is a dimensionless number that can be used to specify the variation of data points in a data series around the mean. As the experiments were conducted outdoors, environmental conditions were difficult to control, for safety reason. As a consequence, the errors (<3%) in these experiments were expected to be higher than general experiment errors, but they should be acceptable. Experimental repeatability was apparently very high (Table 2.2).

29 

Table 2.2: Experimental repeatability for volume flow rate for various fuels.

60% Methane

RPM 1st(L/min) 2nd(L/min) 3rd(L/min) average s CV(%) 40000 45.9382 45.6556 44.3682 45.32 0.68 1.51

RPM 1st(L/min) 2nd(L/min) 3rd(L/min) average S CV 45000 42.0912 41.9728 40.8332 41.63 0.57 1.36 50000 46.8568 47.9964 44.8588 46.57 1.30 2.78 55000 51.874 51.2672 51.2228 51.45 0.30 0.58 60000 54.7452 51.4448 54.1828 53.46 1.44 2.70

80% Methane

RPM 1st(L/min) 2nd(L/min) 3rd(L/min) average S CV 45000 35.0908 35.7124 35.3572 35.39 0.25 0.72

RPM 1st(L/min) 2nd(L/min) 3rd(L/min) average s CV

45000 31.5388 31.302 30.906 31.25 0.26 0.84

30 

Chapter 3

Numerical analyses

3.1 Domain descriptions

The domain of combustion chamber consists of three major components.

They are the solid annulus dual liners (inner and outer), which have several discharge holes, the vaporizing tube, which plays the role of fuel transportation, and the fluids, such as gas fuel and air, filling with the remaining spaces. The annulus combustor has twelve fuel tubes and the liner holes and relative components are in a periodic arrangement shown as Fig 3.1(a) and (b).

Therefore, the combustor can be divided into one-twenty fourth sub-chambers geometrically as shown in Fig 3.1(c) and (d), which also represent the configuration of the full-size, three dimensional model domain employed for simulation. The components configuration of sub-chambers is shown as Fig. 3.2, and the size specification of sub-chamber is shown as Fig. 3.3.

3.2 Governing equations

In order to make the physical problem more tractable, several assumptions are made as follows:

1. One-twenty fourth annular zone of actual physical domain is considered due to symmetry, as shown in Fig. 3.1(c).

2. All gaseous mixtures are regarded as the ideal gases.

3. The flow is steady, compressible, and turbulent.

4. Properties in solid are constant.

5. Neglect the turbulence-combustion interaction and radiation heat transfer.

31 

6. One step global reaction is adopted to represent the chemical reaction of methane gas combustion.

7. Soret (or thermo) diffusion, accounting for the mass diffusion resulting from temperature gradients, is neglected [41].

Based on the assumptions mentioned above, the governing equations are given as follows [42]:

Mass conservation:

( )

=0

•

∇ ρVv (3.1)

which describes the net mass flow across the control volume’s boundaries is zero.

Momentum conservation:

Based on the general Newton’s second law and particular viscous stress law, the momentum equations are developed as Navier-Stokes equations:

( )

In Newtonian fluid, the viscous stresses are proportional to the deformation rates of the fluid element. The nine viscous stress components can be related to velocity gradients to produce the following shear stress terms:

'

32  assume their ensemble averaged values. Get back to Eq. 3.3, the rightmost term represents the additional Reynolds stresses due to turbulent motion. These are linked to the mean velocity field via the turbulence models. u^' is fluctuation about the ensemble average velocity and the overbar denotes the ensemble averaging process.

Energy conservation:

Heat transfer processes are computed by energy equation in the form known as the total enthalpy equation:

( )

( ) ( )

i ij h

where iis internal energy. k_eff is the effective thermal conductivity of the material. In laminar flow, this will be the thermal conductivity of the fluid, k; In turbulent flow,

33  molecular weight of i-th species. ω is global one-step reaction rate and in the Arrhenius form for methane combustion is computed as [43]:

2 where ^k is the reaction rate constant for methane combustion reaction:

Here four-step global reaction mechanism is adopted, the pre-exponential factor, temperature exponent and activation temperature are described as follows:

(1)Four-step global reaction mechanism

2

with the reaction rate expression as [43]:

[ ]

2 ⁴⁵

34  The dynamic viscosity for each species is calculated using Sutherland’s law

[44]:

where a and b are constants, they are specific for different species. The selected values for species which we need are tabulated in Table 3.1.

The mean specific heat at constant pressure, Cp, can be written as JANNAF thermodynamic polynomials are used to estimate the specific enthalpy,

hi, and specific heat, C_pi, for each species, [45]

Gas thermal conductivity, k, is derived from Prandtl number:

k C_p

= μ

Pr (3.18) in this work, Pr=0.707 is selected.

The diffusion coefficient, D, is calculated by Schmidt number:

D

= μ

Sc (3.19) in this work, Sc=0.7 is selected. The constant properties of solid are tabulate in Table 3.3. [46]

35 

Turbulent model

The Reynolds averaged Navier-Stokes simulation adopt k−ε model which involves solutions of transport equations for turbulent kinetic energy and its rate of dissipation. The model adopted here is based on Launder and Spalding (1974).

In the model, the turbulent viscosity is expressed as:

υ ^με with the production term P defined as:

m

the five constants used in this model are [36]:

3 used in the near-wall regions where viscous effects dominate the effects of turbulence. Instead, wall functions are used in cells adjacent to walls. Adjacent to a wall the non-dimensional wall parallel velocity is obtained from

+

36 

υ^τ yu y⁺ = ,

uτ

u⁺ = u (u_τ =C_μ¹⁴k¹² and κ =0.4), and E =9.0 for smooth walls [36].

3.3 CFD-ACE+ software

This work uses a commercial package software, CFD-ACE+, to carry out the numerical computation. CFD-ACE+ is an Advanced CFD and Multiphysics software for simulations of fluid, thermal, chemical, biological, electrical and mechanical phenomena [47]. The solver, based on a finite-volume method [48], divides the physical domain into many control volumes invoked by computational grids and then discretizs differential equation into algebraic one for numerical computation on each grid. [49, 50]

3.4 Numerical method

The CFD-ACE+ employs the finite-volume method to discrete the partial differential equations and then utilize SIMPLEC scheme to obtain the pressure and velocity fields by solving mass and momentum conservation equations.

Then, substitute the obtained pressure and velocity fields into energy, species equations, etc. in sequence.

3.5 Boundary conditions

In the model domain, it consists of two inlets, one outlet, two physical symmetric surfaces (including one cyclic symmetric surface), interfaces between two different phases (solid and gas), and the remaining wall. The non-slip boundary conditions are applied on the solid walls. The outlet locates at the rear

37 

of combustion chamber whose function is to guide the hot gas to generate thrust power. The outlet conditions of combustion chamber mainly determined by what is happened upstream. The parameters, such as temperature, velocity, and pressure, at outlet boundary are assumed zero gradients there, and the flow rate in outlet boundary must satisfy with mass balance. The air and fuel inlet boundary conditions are summarized in Table 3.4.

What follows give an example involved the calculation methods and steps for setting of boundary conditions.

Operating conditions:

The compressor wheel works at the rotational speed 80000 rpm, and compression ratio (CR) 1.35.

3.5.1. The inlet boundary conditions

There are two inlet boundaries. One supplied compressed air from the end of diffuser, and the other one supplied mixed fuel gas.

3.5.1.1 Air inlet boundary conditions of combustor

The boundary conditions here are demanded for temperature, velocity, turbulent kinetic energy, turbulent dissipation rate, and spicy mass fraction.

Step1. Modified compression ratio:

The user manual of MGT MW54 accounts for the compression ratio of compressor produced by Garrett is needed to be modified when it is assembled on this engine type. The empirical formula of modified compression ratio is

π

38 

Step2. Look up the corresponding values for mass flow rates

•

m and compression efficiency ^η at the specified operating conditions on compressor map for a rotational speed 80000 rpm, and compression ratio 1.35.

)

Step3. Decide the temperature rises after compressing process:

The temperature after compressor was decided by experimental data.

K

Step4. Calculate the air flow speed velocity at compressor outlet:

According to state equation and continuity equation, the air flow velocity for outlet of compressor is derived as

RT

where A is the outlet cross-section of compressor

⎟⎠

Step5. Decide turbulent kinetic energy and dissipation rate Set turbulence intensity by empirical formula [48]:

)

where D_his the hydraulic diameter of compressor outlet

04 .

=0

⇒ I

Set dissipation rate in terms of hydraulic diameter:

39 

Step6. Set species compositions and mass fraction of air at compressor outlet Mass fraction: N₂ =0.768,O₂ =0.232

3.5.1.2 Fuel inlet boundary conditions of combustor

The boundary conditions here are demanded for temperature, mass flow rate, turbulent kinetic energy, turbulent dissipation rate, and species mass fraction.

Step1. The volume flow rate of the fuel

Total fuel consumption is 56.8(L/min) at P=10.083bar.

Step2. Set species compositions of mixed fuel at fuel tube inlet

Set different ratio CO2 gas mixed with methane. For example, volume percentage as 90% methane, 10% CO2 possess mixed molecular weight:

4

Step3. Set boundary temperature T_ambient =298K

Step4. Set turbulence intensity I =0.0564, dissipation rate in terms of hydraulic diameter D_h =0.001(mm).

Because the practical operating conditions of the compressor could afford range from critical rotating speed 80000 rpm to idle 45000 rpm according to compressor map, in addition to implement the case of operating condition 80000 rpm to simulate, the study provides the other three operating conditions of rpm 70000, 60000, and 50000 to simulate.

3.5.2 The outlet boundary conditions

40 

The only one outlet locates at the rear of combustor chamber whose function is to exhaust the hot gas to generate thrust power. The outlet conditions of combustion chamber are mainly determined by what is happening upstream.

The pressure at outlet boundary is defined as casing pressure P2 which is derived from Eq. (3.29). The parameters such as temperature, velocity, pressure, at outlet boundary are assumed zero gradients, and the flow rate in outlet boundary must satisfy with mass balance.

3.5.3 The symmetry boundary conditions

The component of velocity vector which perpendicular to symmetric surfaces are set zero.

3.5.4 The interface boundary conditions

The interfaces between two different phases, solid and gas, are the no-slip boundary conditions (u, v, w = 0).

3.5.5 Wall boundary conditions

Except the inlet, outlet, symmetry, and interface boundaries, the remaining geometry is all wall boundaries, which are the no-slip boundary conditions (u, v, w = 0) for velocity and adiabatic for temperature.

3.6 Computational procedure of simulation

In solving the Navier-Stoke Equations, the Semi-Implicit Pressure Linked Equation Correlation (SIMPLEC) algorithm was a widely used numerical method. Different from other algorithms adopting finite difference method, this

41 

SIMPLEC adopted finite volume method. Figure 3.4 (a) shows the model of annular combustion chamber used in the CFD-ACE+. It was developed by Solid-Works; then, imported to the CFD-ACE+ as IGS file. Afterwards, the mesh arrangement was accomplished and shown in Figure 3.4 (b) by the CFD-ACE+ pre-processing, also called CFD-GEOM.

3.7 Grid-independence test

For obtaining the acceptable numerical solution, this study applied the unstructured grids produced from geometry models to carry out grid-independence test. The grid-independence test comprised combustion case.

Because the over-heating problem of liners is one of concerns in this research, the grid densities is increased especially near the liners, whose thicknesses are only 0.4mm, and the fuel tube, which is about 0.5mm in thickness. Owing to adoption of upwind numerical difference scheme, the grid amount should be dense enough to avoid the false diffusion phenomenon.

For grid test, the operating conditions are specified as that the air inflow of

m& =0.0044 kg/s is generated by compressor with a compression ratio 1.35 under the rotation rate 80000 rpm, and the fuel tube supplies a mixing fuel of methane (mass fraction 0.766) and CO2 (mass fraction 0.234) under a fixed mass flow rate (m& =3.325E-05kg/s). From the above discussion, the mass flow rate, velocity and temperature of gas mixture ejected from the rear of combustion chamber are the emphasized data

The grid numbers adopted for the grid-independence tests in this case are 477181, 717974, and 1484378. The test results are listed in Table 3.5. From the information given by the table, it can be seen that the maximum relative errors

42 

of various physical quantities are all less than 3%. Under such circumstance, it is naturally to select the grid number of 477181 to compromise the computational time. Pentium 4 with CPU 3.0 GHz, 2GB RAM is used to carry out the computation, and select convergence criterion as 10^-3. Then the computational time for a typical simulation in the c case needs about 60 hours.

43 

Table 3.1: Sutherland's law coefficients of dynamic viscosity

Species Sutherland's

Law Coefficients

CH4 O2 CO2 H2O CO H2 H N2

A 1.25E-06 1.78E-06 1.50E-06 1.86E-06 1.50E-006 6.89E-007 6.89E-007 1.40E-06

B 197.4 156 222.26 708 136.35 96.69 96.69 111.5

44 

Table 3.2: JANNAF coefficients of gas specific heat

Species

※ These coefficients will be used at temperatures between the lower limet and the break point

a1 7.79E-01 3.21E+00 2.28E+00 3.39E+00 3.26E+00 3.30E+03 2.5 3.30E+00 a2 1.75E-02 1.13E-03 9.92E-03 3.47E-03 0.15E-02 0.82E-03 0 1.41E-03 a3 -2.78E-05 -5.76E-07 -1.04E-05 -6.35E-06 -3.88E-006 -8.14E-07 0 -3.96E-06 a4 3.05E-08 1.31E-09 6.87E-09 6.97E-09 5.58E-09 -9.48E-11 0 5.64E-09 a5 -1.22E-11 -8.77E-13 -2.12E-12 -2.51E-12 -2.47E-12 4.13E-13 0 -2.44E-12 a6 -9.83E+03 -1.01E+03 -4.84E+04 -3.02E+04 -1.43E+04 -1.01E+03 2.55E+04 -1.02E+0

3 a7 1.37E+01 6.03E+00 1.02E+01 2.59E+00 4.85E+00 -3.29E+00 -0.46E+0

0 3.95E+00

Coefficients at upper

limit

※ These coefficients will be used at temperatures between the break point and the upper limit

a1 1.68E+00 3.70E+00 4.45E+00 2.67E+00 3.02E+00 2.99 E+00

2.5 2.93E+0 0

a2 1.02E-02 6.14E-04 3.14E-03 3.06E-03 0.14E-02 0.70E-03 0 1.49E-03

a3 -3.88E-06 -1.26E-07 -1.28E-06 -8.73E-07 -5.63E-07 -5.63E-08

0 -5.68E-0 7

a4 6.79E-10 1.78E-11 2.39E-10 1.20E-10 1.02E-10 -9.23E-12 0 1.01E-10

a5 -4.50E-14 -1.14E-15 -1.67E-14 -6.39E-15 -6.91E-15 1.58E-15

0 -6.75E-1 5

a6 -1.01E+04 -1.23E+03 -4.90E+04 -2.99E+04 -1.43+04 -8.35E+02

2.55E+04 -9.23E+0 2

a7 9.62E+00 3.19E+00 -9.55E-01 6.86E+00 6.11E+00 -1.36E+00 -0.46E+0 0

5.98E+0 0

45 

Table 3.3: Properties of solid liner Material Density(kg/m³) Specific

heat(J/kg-K) Thermal conductivity(W/m-K)

Steel_AISI_1020 7900 470 48

Table 3.4: Boundary conditions of combustion chamber

Air Inlet Fuel Inlet

Conditions

Hydraulic diameter(D_h)=0.012 m Turbulence intensity (I )=0.04

YN2=0.768 YO2=0.232

45000 1.081 0.001765 307.8 94.80 73.60 62.58 55.30

46000 1.088 0.001800 308.5 96.20      

46 

Table 3.5: Grid test results of different grid densities for numerical simulation 477181 grids 717974 grids 1484378 grids Maximum Relative

Error (%)

Outlet mass flow rates

(kg/s) 4.18×10-3 4.17×10-3 4.25×10-3 1.9 Outlet max.

temperature (K) 645 641 655.2 2.2

Outlet max. velocity

(z-direction) (m/s) 108.9 106 106.4 2.3

Average outlet velocity (z-direction) (m/s)

80.6 81.5 82.9 2.5

47 

Chapter 4

Dynamic Model of MGT for Control Strategy

This section describes the development of a dynamic model for the proposed MGT system. According to the thermal process of the MGT, different models have been developed to predict the dynamic response behaviors of gas turbine systems. The MGT’s dynamic model used in this study is adapted from that developed by Rowen [51]. This model is commonly used due to its simplicity and flexibility in adjusting to turbines with different characteristics.

An outline of the structure, including the control and fuel systems, was generated using Matlab/Simulink (Fig. 4.1); the relevant equations are

[ ]

of the fuel, T_amb is ambient temperature, Tr is the maximum exhaust temperature, Tx is predicted exhaust temperature, Torqueis predicted mechanic torque produced by the MGT, a, y and zare correlation factors, and N (p.u) is compressor rotational speed. These equations can calculate turbine torque and exhaust temperature algebraically.

Signal Vfuel, which represent the volume flow rate of fuel, is the most

48 

important variable when operating the MGT. In numerical simulation of the dynamic model of the MGT, compressor rotational speed, exhaust temperature of the MGT, and MGT loading were functions of Vfuel. Notably, Vfuel was utilized to calculate MGT mechanic torque and exhaust temperature after the simulation signal was modulated by a time delay block generated by combustion reactions.

49 

Chapter 5

Results and Discussions

5.1 Experimental results and discussions

The original MGT, which was powered by liquid fuel, was modified to run on gaseous fuel. Therefore, the original pipes were changed accordingly.

Additionally, the lubricant for bearings in the original MGT was pre-mixed with oil fuel. The lubricant supply system was separated from the fuel pipes after modification. In the experiments, stable rotational speed, 45,000 RPM, was reached. To increase fuel efficiency, data were obtained at each step as the rotational speed was increased by 5000 RPM increments. Each step was maintained for approximately 10 seconds to ensure that the engine reached a stable condition such that measured output data at each step were meaningful;

otherwise, stop-and-run experiments would consume a substantial amount of gas and the time required would be excessive. Additionally, maximum exhaust temperature was set at 800°C for safety reasons.

5.1.1 Tests using various fuels with no load

The proposed MGT was tested without loading. Various fuels with LHV with various mixture ratios of methane (CH4) to carbon dioxide (CO2) were used.

The concentrations of CH4 in the fuel were in the range of 50–90%, and those of CO2 increased from 10% to 50%. The temperatures of the main components of the MGT were measured at a specified fuel and fuel flow rate against cylinder gauge pressure. Thermal efficiency was calculated accordingly. Finally, the performances of the MGT under the various test conditions were compared.

50 

5.1.1.1 Pressures and volume flow rates of fuel in the MGT

Fuel pressure was the pressure exerted by the cylinder (bottle); this pressure was adjusted using a control valve. Fuel pressure was determined by fuel volume flow rate. The volume flow rate of each fuel was measured as a function of pressure, and calculated from mass flow rate. (CH4 density is 0.7168 g/L at 0°C, 1 atm) The volume flow rate of each fuel was roughly linearly proportional to fuel pressure. Figure 5.1 plot rotational speeds as functions of volume flow rates for the various fuels with LHV.

In the test using fuel comprised of 90% CH4 and 10% CO2, the rotational rate approached 85,000 RPM as fuel pressure approached a maximum of 12bar.

Under the same pressure, a rate of only 47,500 rpm was reached with 60% CH4

and 40% CO2. The condition of 50% methane could not generate power when loading was applied. More specifically, the power of MGT at 40,000 RPM while applying 50% methane fuel was not able to drive the generator. The author present the results of 50% methane here were tending to offer more information for future research interest. Clearly, the combustible fuel concentration in the LHV fuels influenced the performance of MGT. According to Fig. 5.1, as expected, the fuel with the higher heating value performs better because it can supply more energy to MGT. Also, a higher pressure must be applied to the lower heating-value fuel to approach idleness at 45,000 RPM.

Since MGT performance was proportional to the CH4 concentration in fuel and volume flow rate, increasing volume flow rate increased MGT performance.

Furthermore, using a fuel with a LHV in the MGT originally designed for oil fuel may result in choking of both the turbine wheel and fuel supply pipe. This choking can occur in the nozzle throat or in the annulus at the turbine outlet. As rotational speed increases to >85,000 rpm, choking may markedly limit the

51 

rotational rate of the turbine wheel because the rotational speed increases as flow rate increases; however, the maximum fuel pressure reached at 85,000 rpm was 12 bar. Therefore, turbine choking did not occur during tests in this study.

Compared to the turbine choking, the choking at fuel supply pipe needs more attention. The original pipe which only have 0.5mm diameter was designed to deliver 0.05L liquid fuel per minute at 80,000 rpm. However, the experiment results showed that the volume of the fuel was 58L and the pressure was 12 bar when the maximum power was reached. Thus, in order to avoid choking

Compared to the turbine choking, the choking at fuel supply pipe needs more attention. The original pipe which only have 0.5mm diameter was designed to deliver 0.05L liquid fuel per minute at 80,000 rpm. However, the experiment results showed that the volume of the fuel was 58L and the pressure was 12 bar when the maximum power was reached. Thus, in order to avoid choking

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