2. A simple turbojet cycle
3. The goal of this project
Ref. 1. Useful constants
Ref. 2. Useful equations
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There are various means of producing power. In many respects gas turbines are the most satisfactory. Gas turbines are simple. They have no rubbing and reciprocating components. This simplifies balancing, increases reliability and reduces lubrication consumption. The advantages were first realized with steam driven turbines used to generate electricity. (Steam driven turbines are still predominant in generating electricity.) It is not uncommon to find steam turbine power plants producing over 500 MW electricity at efficiencies up to 40%.
In spite of it's predominance in power generation, steam turbines do have a disadvantage. Generating high-pressures high-temperature steam requires expensive and bulky equipment. The turbines are also not driven from the high temperature gasses directly. The gasses are used to heat water to generate steam. This has further efficiency impacts. Using the gasses directly would therefore be much more efficient.
Serious development of the gas turbine started just before the Second World War. These turbines were meant to generate shaft power, but the attention was soon transferred to the development of the turbojet engine as a method for propelling aircraft. It was only by the mid nineteen fifties that gas turbines started competing in other fields. Today gas turbine technology has a profound impact on a great variety of applications.
One of the fields where gas turbine technology is fairly new, is that of radio controlled model aircraft. Radio modelers have been making use of ducted fans and pulse jets to propel scale jet aircraft for years.
Ducted fans are essentially shrouded fans driven by high speed two-stroke engines. This method produces very high noise levels, bad throttle response and a relatively low thrust to weight ratio. The high speed two-stroke engine requires expensive fuel. The high speed reciprocating piston vibrates the engine thereby reducing reliability. These two-stroke engines also have low efficiencies.
Pulse jets are rockets with a one way valve at the front of the combustion chamber. The air fills the chamber from the front due to forward velocity. Fuel is injected into the combustion chamber and ignited. This produces a forward thrust pulse. After the pressure in the combustion chamber has dropped under the stagnation values, the one way valve opens to repeat the process. Pulse jets require high velocities to operate and generate very high noise levels. Pulse jets have expensive and exotic fuel requirements. A low efficiency is also a common problem.
Substituting the above means of propulsion with a small gas turbine would be highly beneficial. Noise levels can be reduced with careful exhaust nozzle design. Throttle response can be improved by using light weight materials in the construction of rotating parts. The overall thrust to weight ratio can easily be improved. A carefully designed gas turbine will also have a higher efficiency, mainly because it operates at higher temperatures. Gas turbines can be designed to run on almost any fuel. This makes it possible to run a propulsive system using diesel instead of exotic fuels.
2. A simple turbojet cycle
A turbojet engine is made up of an intake, a compressor, a combustion chamber, a turbine and a propulsive nozzle. The intake accelerates or decelerates ambient air to subsonic conditions. The compressor then compresses the air, thereby increasing the stagnation pressure and temperature of the air. The air is fed into the combustion chamber where a fuel is injected into it with an atomizer. The fuel/air mixture is ignited. This increases the stagnation temperature. There is a slight drop in stagnation pressure through the combustion chamber. The combustion gas is then allowed to expand through a turbine that transfers some of the energy contained in the combustion gasses back to the compressor. The remaining energy in the gasses is then used constructively by the propulsive nozzle. If the nozzle is unchoked the gasses are allowed to expand to the point where they are at ambient pressure before they are allowed to leave the nozzle. If the nozzle is choked the gasses are allowed to expand until it reaches a velocity equal to the velocity of sound at the specific temperature before it is allowed to leave the nozzle. A schematic of a small turbojet is shown in Figure 2.1. The intake is located between a and 1. The compressor with it's diffuser is located between points 1 and 2. The combustion chamber is located between 2 and 3. The turbine is located between 3 and 4. The propulsive nozzle is located between 4 and 5.
Most modern compressors are axial compressors because of their higher efficiency. When it comes to small gas turbines though, centrifugal compressors appear to be much more efficient and easier to manufacture than axial compressors. If centrifugal compressor can be compared to moving water up a slope using big, infrequent stroke, the axial compressor can be compared to moving the water using small, very frequent strokes. From this image it can be seen that an axial compressor will be more efficient than a centrifugal compressor. If the compressor is sufficiently small, this however is not the case. An axial compressor is made of small aerodynamically shaped blades that accelerate the air to increase the stagnation conditions. The air is then decelerated by the diffuser (another set of blades.) This brings it's static conditions closer to the new stagnation conditions. Incorporating such blades in a sufficiently small compressor is close to impossible. A centrifugal compressor inhales air at the front, and then using centrifugal action accelerates the air outwards with vanes. The air is picked up by simple diffuser vanes. The diffuser feeds the air to the combustion chamber. The uncomplicated working and fewer components of a centrifugal compressor makes it ideal for use in a small gas turbine.
3. The goal of this project
The main goal of this project is to develop a small inexpensive centrifugal compressor. It will later be used in the development of a gas turbine aimed at propelling radio control aircraft.
3.1 Design requirements
The end product has to meet the following standards:
3.2 Design characteristics
The following characteristics have been set for the final design:
3.2.1. Assumed component efficiencies:
Gas turbine compressors have to be designed for a certain mass flow. It is therefore necessary to characterize the complete gas turbine before the compressor can be designed. The reasonably accurate idea of the mass flow can be deduced by analysing the overall performance of a gas turbine. The overall analysis usually gives a specific thrust. The mass flow can then be calculated from the specific thrust because the thrust requirements are known. (Specific thrust is the amount of thrust a gas turbine would produce for a given mass flow.) To do this, a good knowledge of one dimensional compressible fluid dynamics is required. The equations used in calculating the overall performance of a theoretical gas turbine are include in Ref. 2.
4.1 Analytical prediction of performance
4.1.1 Stagnation conditions after intake:
The ambient conditions (pressure and temperature) are known. Radio controlled aircraft have relatively low velocities. The stagnation conditions at the intake can therefore be assumed to be roughly equal to the ambient conditions.
4.1.2 Stagnation conditions after compressor
The impeller rotates at 85 000 rpm and therefore does work on the air. This changes the stagnation conditions of the air. The compressor efficiency is known. The changes in the stagnation temperature can thus be calculated using Eq. 2.1.2 in Ref. 2.
The change in stagnation temperature can be used to calculate the change in stagnation pressure using Eq. 2.1.1. in Ref. 2.
4.1.3 Change in stagnation conditions over turbine
The turbine inlet temperature is known. The amount of work absorbed by the compressor can be calculated. The efficiency of the mechanical power transfer is also known. This makes it possible to calculate the change in stagnation temperature over the turbine.
The pressure loss over the combustion chamber is known. This makes it possible to calculate the stagnation pressure at the turbine inlet.
The turbine efficiency is known. The availability of the stagnation pressure at the turbine inlet makes it possible to calculate the stagnation pressure at the turbine outlet using Eq. 2.1.1. in Ref. 2.
4.1.4. Is the nozzle choked or unchoked?
The nozzle pressure ratio is given as the ratio between the turbine outlet stagnation pressure and the ambient pressure. The critical nozzle pressure ratio is given by Eq. 2.3.1. in Ref. 2. Because the nozzle pressure ratio is smaller than the critical pressure ratio, the nozzle is unchoked.
4.1.5. Static conditions after the nozzle
The efficiency of the nozzle is known. All ambient conditions are known. All stagnation conditions at the turbine outlet are known. The static pressure of the nozzle at the exit is equal to the ambient pressure. (An unchoked nozzle gives maximum thrust when the static pressure at the exit is equal to the ambient pressure.)
The static temperature at the exit can be calculated using Eq. 2.1.4. in Ref. 2.
The static density at the exit can be calculated using Eq. 2.1.3. in Ref. 2.
The outlet velocity can be calculated using Eq. 2.1.2. in Ref. 2.
4.1.6. Specific thrust
The ratio of area to mass flow can easily be calculated using Eq. 2.4.3 in Ref. 2.
All the terms in Eq. 2.3.1. are known. The specific thrust can then be calculated.
Because the thrust requirements are known, the mass flow can easily be calculated from the specific thrust.
4.2. Results of the calculations done in 4.1
The results of the method discussed in 4.1 are given in Table 4.2.1. All the calculations are in the counter book included with this report. The results were verified using a spread sheet. The spread sheet is named "SPECTHR.XLS" and is in MS Excel format.
The resulting mass flow is given as 0.0656 kg/s. The theoretical turbine will have a specific thrust of 381.1 N/(kg/s).
5. Design approach
As discussed before, gas turbine compressors are divided in to three groups: axial, radial and hybrid compressors. The choice of a radial compressor is the best for reasons previously discussed. From all the known requirements and dimensions the compressor can be designed. It should also be kept in mind that exceeding a mach number of 0.8 might cause shocks inside the compressor. This would have a profound impact on the efficiency. It is therefore a further requirement that all mach numbers should be kept under 0.8.
5.1. Analytical design of impeller
5.1.1. Calculation of impeller diameter
Because the required pressure ratio and maximum rotational speed of the impeller are known, it is simple to determine the impeller diameter using Eq. 2.1.2 in Ref. 2. All that is required is the slip factor. The impeller slip factor is defined by Eq. 2.5.1 in Ref. 2 where n is the number of impeller vanes.
5.1.2 Calculation of impeller power requirements
The impeller power requirements can be calculated using the product of mass flow and velocity where the velocity is defined by Eq. 2.1.2 in Ref. 2.
5.1.2 Calculation of the impeller vane angles at the annulus root and tip
The axial velocity can be calculated at the impeller inlet by iteration. The iteration will have the following criteria: The difference between the static and stagnation conditions due to the velocity gives rise to a mass flow of 0.0656 kg/s. The axial velocity of the air and the tangential velocity of the annulus tip and root can then be combined in a velocity triangle to solve the vane angles.
5.1.3. Calculation of the impeller channel depth at the tip
The channel depth can be calculated by assuming that half of the losses in the compressor occur in the impeller. A radial velocity is assumed for the air, and the tangential velocity is calculated from the impeller tip speed and slip factor. This makes it possible to define the static conditions of the air at the impeller tip. Requiring that the mass flow in the radial direction stays 0.0656 kg/s makes it possible to calculate the impeller channel depth.
5.1.4. Calculation of the mach number at the impeller tip
Finally from the static conditions the mach number can be calculated.
5.2. Results of the calculations done in 5.1.
An impeller diameter of 70 mm is required to generate a 2:1 pressure ratio. The air entering the impeller should have a velocity of 56.96 m/s. The impeller annulus vane angles should be 18° at the tip and 47° at the root. The annulus tip diameter is 40 mm and the root diameter is 12 mm. An impeller channel depth of 8 mm will give rise to a radial velocity of around 27.1 m/s. Curving the vanes back at about 30° will improve component efficiencies without too big an impact on component performance. The compressor should have a 5.58 kW power requirement when operating at maximum performance. All the calculations in 5.1 are in the included counter book. The calculations were also verified using a spread sheet: "IMPELLER.XLS"
5.3. Analytical design of diffuser
5.3.1. Calculation of new impeller tip air velocities
Because the impeller will have curved vanes the velocity triangle at the tip of the impeller will change. The new velocities can simply be calculated using this velocity triangle.
5.3.2. Calculation of diffuser inlet velocity triangle and vane angle
Because the air has a rotational (tangential) component, the tangential velocity at the diffuser inlet can be calculated by requiring constant angular momentum. Assuming that the diffuser channel depth is the same as the impeller channel depth makes it possible to calculate the cross sectional area of the radial flow at the diffuser inlet. The radial velocity can be calculated by iterating and specifying that the mass flow remains constant. The velocity triangle deduced from the tangential and radial velocities at the diffuser inlet can then be used to define the diffuser inlet angle.
5.3.3 Calculation of the diffuser outlet velocity and mach number
The required stagnation conditions at the diffuser outlet are known. Specifying inner and outer diameters for the axial outlet of the diffuser makes it possible to calculate the cross sectional area for air leaving the compressor. Specifying constant mass flow makes it possible to calculate the axial flow velocity by iterating. The speed of sound can be calculated from the air's static conditions at the outlet. This makes the calculation of the mach number straight forward.
5.4. Results of calculations performed in 5.3
To ensure even flow from the 5 mm vane less space into the diffuser with the assumed channel depth of 8 mm, the diffuser requires a vane angle of approximately 5.8°. With the assumed axial outlet dimensions (90 mm inner diameter, 106 mm outer diameter) the diffuser will have an outlet velocity of 16.86 m/s. Because of the vane less space, the minimum diffuser diameter is 80 mm. Theoretically the mach number will never exceed 0.8.
5.5. Conclusion of theoretical compressor performance
Table 5.5.1 gives a list of all the stagnation and static conditions at various points in the compressor. Note that 1 is at the impeller intake, 2 is at the impeller tip, 2' is at the diffuser intake, and 3 is at the diffuser outlet.
5.6 Possible manufacturing techniques
5.6.1 Manufacturing the impeller
The cost of composite materials is offset by the ease with which it can be worked, it's very high strength, and it's ease of machining. The impeller can be made of a combination of epoxy impregnated carbon fibre and glass fibre. All unfilled areas which may cause stress concentrations can be filled with low density epoxy foam. (Epoxy filled with small hollow glass beads)
A possible manufacturing technique:
5.6.2 Manufacturing the diffuser and shroud.
The manufacturing of the diffuser/impeller shroud combination and vanes can be done in precisely the same way as for the impeller.
All the theoretical calculations are included in the two zipped files for "SPECTHR.XLS" and "IMPELLER.XLS".
6. Conclusion of theoretical results
6.1. The Impeller
Annulus tip diameter: 40 mm
Annulus root diameter: 12 mm
Overall impeller diameter: 70 mm
Impeller channel depth at tip: 8 mm
Annulus tip angle: 18°
Annulus root angle: 47°
Number of vanes: 13
Vane sweep: 30°
Rotational speed: 85 000 rpm
Power requirements: 5.58 kW
Stagnation pressure ratio: 2:1
Isentropic efficiency: 0.875
Highest component mach number: 0.717
Carbon-fibre and glass-fibre composite structure.
6.2. The diffuser
Outer axial exit diameter: 106 mm
Inner axial exit diameter: 90 mm
Vane less space: 5 mm
Minimum vane diameter: 80 mm
Diffuser channel depth: 8 mm
Vane angle: 6°
Isentropic efficiency: 0.875
Highest component mach number: 0.580
Carbon fibre and glass-fibre composite structure.
6.3. Technical Sketches
The technical sketches are included in the back of the report.
According to the theoretical analysis the criteria in section 3 can be met. The only criteria that can not been met is the availability criteria. Most of the construction is based on carbon-fibre. Carbon-fibre may not always be readily available. Constructing these components from carbon fibre is simple and cheap compared to investment alloy casting or precision milling which is the standard method of making centrifugal compressors.
For air: For combustion gas:
Reference 2: Useful equations
2.1 Useful isentropic one dimensional properties of a compressible fluid:
2.2 Propulsive duct specific thrust:
2.3 Useful propulsive nozzle equations:
2.3. Other useful fluid equations:
2.4. Useful empirical equations:
The design sketches are currently hand drawings. On scanning the required
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Pieter E S Smith