The theory governing the design of the pulsejet engine in this project was adapted from C.E. Tharratt’s “The Propulsive Duct”. Tharratt developed his equations in the 1960s with the imperial system of units in mind. The equations were modified to work with SI units before being used to design the project engine.
5.1.1. Tailpipe
Tharratt proposed the following basic equation governed the basic design of the pulsejet engine duct:
= 0.00316 5.1
Where:
V = Engine Volume (cu. ft.)
L = Effective acoustic length of engine (ft.) F = Thrust (lbf)
Manipulating this equation to take SI unit inputs produces equation 5.2:
= 0.000066 5.2
The simplest form of pulsejet is simply a straight tube of constant cross-section. It was decided to use this as a starting point.
19 Since for a straight pipe,
=
If this relationship is substituted into equation 5.2 and simplified, then a direct relationship between thrust and cross-sectional area is established. This cross-section area will be used as the tailpipe area and will be referred to as Ae (exhaust area) from here on.
= 0.000066 5.3
It was decided to make use of standard seamless pipe sizes available on the market to make the tailpipe. This would reduce the complexity of having a long welded seam running the entire length of the pipe.
Inputting the maximum desired thrust of 90N into equation 5.3 returned a tailpipe diameter of 87mm. The next smallest seamless mild steel pipe available on the market was 3” Sch40 pipe. This gave an internal diameter of 78mm.
Using 3” Sch40 pipe as the basis for the design, the expected thrust was calculated by substituting the area of the pipe back into equation 5.3. The expected thrust returned was 72.6N.
The total length of a pulsejet engine is what determines the frequency at which it operates according to another of Tharratts basic equations, 5.4.
= 4
5.4
There is still quite a bit of debate regarding the correct operating frequency for a given size of jet. Therefore it was decided to look at some existing designs, their operating frequencies and their length to diameter (L/D) ratios to determine a suitable length for this engine. The known properties of some existing pulsejet designs are shown in table 5.1.
20
Table 5.1 Known Pulsejet Properties
Engine Static Thrust Output (N) Frequency (Hz) L/D ratio
Argus AS-014 2,200 46 9.6
Dynajet 20 260 15
By assessing the above data and considering the intended thrust output of the project engine, it was determined that an L/D ratio of 14 would be a good starting point for the engine. This would allow trimming of the tailpipe later on during testing if needed.
Using this ratio and the internal diameter of 3” Sch40 pipe, a total engine length of 1.1m was calculated.
This length was substituted into equation 5.4. The operating temperature of the engine was estimated to be 1000K approx.
= 4 ≈√1.36 ∗ 287 ∗ 1000
4 ∗ 1.1 ≈ 142
This frequency falls within the expected range for a pulsejet of this size.
5.1.2. Valve Plate
Another important relationship which Tharratt developed was that which related the intake valve area to the exhaust area. He proposed that:
= 0.23 5.5
This equation does not take into account the inefficiencies associated with different valve layouts. It is generally assumed that a petal valve layout has an efficiency of 0.5.
Therefore the equation must be modified to allow for this. To make calculation simpler, equation 5.5 can be modified to allow for the efficiency factor and to take an input of tailpipe diameter rather than area:
= 0.115 5.6
= 2205
21 A “combustion chamber” is not necessary in a pulsejet engine. However, due to the layout of a petal valve system, it is usually necessary to include a wider section which resembles a combustion chamber at the front of the tailpipe. This wider section will be referred to as a combustion chamber for convenience. The valve plate layout must be designed before the dimensions of this section can be determined.
The valve plate was designed in ProEngineer by observing the following criteria and attempting meet the required valve area while keeping the outer diameter of the orifices as low as possible.
It was desired to keep the number of petals in the valve as low as possible so that the probability of failure due to fatigue could be kept low.
It was also observed by studying previous designs that the maximum diameter intake hole was 12mm to reduce deformation of the valve during the positive pressure cycle of the engine. To try and solve this, the valve orifices had to have a minimum distance of 12mm in one direction.
The valve plate needed to allow for 2mm valve overlap minimum around each orifice.
Due to machining constraints, the smallest radius included in the design could be no smaller than 3mm.
The final design, shown in figure 5.1, had 10 intake orifices and the outer diameter of the intake orifice ring was 90mm.
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23 5.7
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24
Figure 5.3 Petal Valve Geometry
According to Singiresu S. Rao’s text, “Mechanical Vibrations”,
=3
Where I is the second moment of area and for a simple beam:
= 12 Therefore:
= 4 5.8
If all variables in equation 5.8 are kept constant and only b is allowed to change, then it can be shown that, as b increases, so does k. Therefore k increases with distance from the root of the petal valve. If the petal were to be deflected through a small distance, then the majority of bending would occur at the root where k has its smallest value.
To simplify the problem of varying k, the petal valve was modelled as a cantilever beam of constant cross-section equal to that at the root of the petal, with a lumped mass at the end which would represent the extra mass of the side lobes of the valve.
The value of that extra mass was found by:
Finding the mass of the side lobes
Finding the centre of gravity of that mass
Calculating the moments produced by this mass about the root
25
Then calculating an equivalent mass which would produce the same moment about the root if it were placed at the tip of the valve.
The result is a constant-section beam with a lumped mass at its end as shown in figure 5.4 for which the natural frequency of vibration can be easily calculated.
Figure 5.4 Simplified Petal Valve Model
The equivalent mass of this model can be found using Rao’s equation:
= + 0.23 5.9
The natural frequency of vibration can now easily be calculated using
=
5.10 Where k is obtained using equation 5.8 and m is calculated from equation 5.9.
In order to find the position of the centroid of the side lobes, a simple 2D CAD program called QCad was used. The geometry of the valve was drawn and then split into the main “beam” and up to five other simple shapes as shown in figure 5.5.
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of the “beam
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Valve Geometry
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26 values input id using the
d using:
27 In order to tune the petal valves to the required natural frequency of vibration, it was necessary to investigate the relationship between natural frequency ω, spring stiffness k, length L and material thickness t. By combining equations 5.8 – 5.12 and simplifying, the following equation 5.13 was obtained.
= 4 ( + 0.23 )
5.13 By varying t and L in equations 5.8 and 5.13, it can be shown that an increase in t will increase the natural frequency of vibration but will also increase the static stiffness by a larger factor.
∝ ∝
If L is reduced instead, there is a smaller increase in static stiffness for the same increase in natural frequency.
∝ 1
∝ 1
Therefore it is more desirable to tune the valve frequency by reducing the effective length of the valve than by increasing the thickness. Keeping the static stiffness as low as possible is also necessary to allow the engine to produce static thrust as the valves do not have the benefit of ram-air pressure to help open them.