USING INTERIOR POINT OPTIMIZATION TECHNIQUE FOR
MUFFLERS COST REDUCTION
Waleed El-Sallamy
1and Ahmad Abosrea
21Production Engineering and Printing Technology, Akhbar El-Yom Academy, Giza, Egypt
2Group for Advanced Research in Dynamic Systems (ASU-GARDS), Ain Shams University, Elsarayat St., Abbaseya, Cairo, Egypt
E-Mail: [email protected]
ABSTRACT
Muffling devices are essential parts of any vehicle/machine that uses internal combustion engines. From automotive manufacturers point of view, they contribute massive sheet metal processing and production time, especially mufflers that are produced at high rates for commercial vehicles. This paper is concerned to reduce the production cost of any commercial automotive mufflers through theoretical studies and experimental measurements verification. Mufflers design parameters (transmission loss and pressure drop) were selected for performance judgment, parameters were simulated and experimentally measured. The transfer matrix method was used to calculate the propagation of sound waves along the muffler. Interior point optimization algorithm in MATLAB optimization tool box was used in this paper. A commercial automotive muffler with a hybrid combination of dissipation and reflection was selected for application and a new optimized design was proposed for this muffler with a reduced manufacturing cost. The manufacturing operation time was reduced by 92% and consequently the cost relative to the original muffler was reduced by 60%.
Keywords: commercial vehicle, vehicle economics, muffler optimization, exhausts system manufacturing.
1. INTRODUCTION
Mufflers are an important part in the automotive industry due to their impact upon environment noise legislation and the engine performance. Effective noise control can be accomplished by careful selection of the internal muffler elements. Perforated tubes are one of the most expensive parts in the manufacturing; it is used to damp the acoustic waves by dissipation through the holes. On the other hand, manufacturing perforate tube needs long sheet metal processing time and manufacturing steps that consequently raise the final product cost.
Perforated mufflers can be modeled by the segmentation approach [1] by modeling a perforated pipe passing through a muffler cavity as four port element. Elnady and Åbom [2] used the two-port transfer matrix method to improve the approach to be more flexible by dividing the perforated muffler element to be a network of pipes and perforates. The modeling was done using SIDLAB computer software [3] which uses the improved approach in calculations. Up to the plane wave limit Elnady et al. [4] verified this developed technique on 15 mufflers with different perforation configurations. The flow generated noise radiated from mufflers were studied by Abosrea [5] for different muffler configurations and verified by experimental measurements.
Along this track, many researchers investigated the theoretical modeling of mufflers aiming to optimize the muffler efficiency by increasing the noise attenuation or reducing the backpressure. The first shape optimization for a muffler cavity was done by Bernhard [6] without any volumetric constrains. Using Genetic Algorithm Yeh et al. [7]-[9] introduced constrains to the muffler optimization, the algorithm was applied on different cavities configurations. Elsaadany [10] used optimization barrier method to reach a target parameter value under while keeping the exhaust volume and pressure drop as constrains. Researches were focusing on noise attenuation
efficiency and pressure drop, ignoring the cost effect of the muffler internal design due to man power, material, machines used and production process. Abosrea et al. [11], [12] studied the procedures possible to reduce the muffler costs without using an optimization method. In this paper, a commercial muffler with internal design depending mainly on perforated tubes was selected. Production stages were proposed for the selected muffler, the internal shape was modified, accordingly the production steps and time were reduced and total cost reduction was achieved. The mufflers acoustic performance and pressure drop were simulated and experimentally measured. Several optimization trials were performed to reach a reduced cost muffler while maintaining the original muffler outer size (to keep interchange ability) and performance as a minimum optimization target.
Finally, the paper come up with a reduced cost muffler model while monitoring the performance parameters described by Munjal [13] for petrol vehicles, and keeping the outer shell size as a constrain. To assure the later industrial application of this work, all design and process changes were done with focusing on possible interactions and consistence with standard product realization in ISO/TS 16949:2002 [14] inside a manufacturing facility specialized in automotive feeding industry.
2. Theory
2.1 Transmission loss
= 22 21 12 11 T T T T T (1) ii
T are the transfer matrix coefficients while 1 represent the inlet side of the element and 2 represent the outlet side. If the transmission propagates from port 1 to port 2, thenT11is the reflection coefficient at the inlet, T22
is the reflection coefficient at the outlet, T12 is the propagation of the wave from inlet to outlet andT21 is the
reflection of the wave from outlet to inlet.
A c Z=
(2)
Where,
= fluid density.
c = speed of sound.
A = cross section area of the element.
The transmission loss TL is then calculated by:
(
)
(
)
+ + + + + = 2 2 1 22 21 1 2 12 11 1 2 2 2 2 1 10 4 1 1 log 10 Z Z T T Z Z T T Z M Z M TL (3) Where,M = Mach number calculated from the flow velocity/speed of sound,when solving with flow.
2.2 Pressure Drop
Similarly, the pressure drop of each element can be calculated [15] using the flow transfer matrix.
= 1 0 1 f flow R T (4) f
R = element flow resistance in the propagation
direction.
The 0 and 1 represent no resistance in the direction opposite to flow from outlet to inlet and the continuity of flow from inlet to outlet respectively.
Q A k
Rf e
2 2 = (5) Where, e
k = flow loss coefficient constant due to area change which leads to a pressure dropP.
Q = volume flow.
v P
ke
=2
(6)
Using Bernoulli’s equation;
e s
s v gh P v gh gh
P1+0.512+ 1= 2+0.522+ 2+ (7)
Where,
s
P = static pressure.
v = flow velocity.
g = gravitational acceleration.
h = fluid head.
e
h = head losses.
After ignoring the difference due to head change inside a muffler network, the total pressure will be;
2 1 2
1 P K 0.5 v
Ps = s + e (8)
From Equation (5):
1 2
1 P R Q
Ps = s + f
(9)
At a constant volume velocity Q for a single inlet and single outlet element, the flow properties can be represented now by total pressure and volume flow. The flow in the two-ports element can be solved by:
= 2 2 1 1 1 0 1 Q P R Q
Pflow f flow
(10)
Where,
flow
P = stagnation pressure, calculated from the sum of
static pressure and the velocity head.
2
5
.
0
v
P
P
flow=
s+
(11)
3. Optimization
The objective of the optimization is to reach the maximum muffler attenuation “Transmission loss” while keeping the pressure drop within the allowable limits. This is done by shape optimization for the muffler by defining the allowable lower and upper limits of dimension variations and any other shape constrains.
The objective function to reach a target transmission loss X-target for a current state of transmission loss X is
min
min min
maxG = −G (12)
) min( arg min X Xt et
G = −
(13)
minxf(x)such that
{
c(x) ≤ 0 ceq(x) = 0
A. x ≤ b Aeq. x = beq
lb ≤ ub
b and beq are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions. lb, and ub are the lower and upper bounds.
4. Simulation
Exhaust systems manufacturing are considered as one of the longest processes in the automotive feeding industry, in the same time the manufacturing rates should be high enough to keep up with the vehicle final assembly lines. The manufacturing of the exhaust systems includes but not limited to; sheet metal processing, pipe forming, machining and welding. Reducing the manufacturing processes were achieved by removing the perforated pipes which contributes the longest manufacturing steps.
4.1 Original Muffler
It is a hybrid muffler type relying on perforated tubes (resistive) and resonators (reactive) in the form of extended tubes (un-perforated sections) inside the muffler chambers. It has a triangular cross-section with inlet, outlet and internal perforated pipe as shown in Figures 1 and 2. The maximum allowable limits of the pressure drop was measured as 2660 Pa at 32 m/s.
The simulation of the muffler is based on two port technique where every geometrical element is connected to anther by nodes. The network is built to simulate the flow path through the muffler from inlet to outlet.
Figure-1. 3D model for the original muffler.
Figure-2. Original muffler main dimensions.
4.2 Proposed mufflers
The internal sub-assembly design of the original muffler was modified from production cost point of view and simplified by using two non-perforated pipes instead of the three ones. The new mufflers have the same outer dimensions of the original muffler.
Iteration A: Simple expansion chamber
The highest reduction in cost can be achieved by removing the whole internal assembly of the original muffler, and convert it to a reactive muffler with empty chamber.
Figure-3. 3D model for the simple expansion chamber “Iteration A”.
Iteration B: Expansion chamber with extended inlet and outlet
exceeded by the new iterations and the transmission loss curve of the original muffler was used as a target curve for optimization.
The optimum value of the extended inlet pipe is 185 mm and the optimum value of the extended outlet pipe is 92 mm.
Figure-4. 3D model for the Expansion chamber with extended inlet and outlet.
Iteration C: Expansion chamber with extended inlet and outlet and two baffles
Two inner baffles were returned to the model, this is to enhance the transmission loss. The baffles were kept unchanged to avoid any change in their manufacturing stamps that will raise the manufacturing process cost.
From Figure-5 the flow enters directly to the middle chamber to face the baffle wall, the middle chamber is pressurized to force the flow to enter the first chamber; flow is then directed through the outlet pipe. The length of the extended pipe length is 196 mm while the length of the extended outlet pipe is 194 mm.
Figure-5. 3D model for the two baffles model.
5.Experimental verification
Transmission loss was measured using the acquisition module in SIDLAB, the software uses the two sources method to measure the transfer function by the means of speakers for excitations and six microphones for
data capturing [17] [18]. The transmission loss was measured with and without air flow to simulate the real mufflers situations. This technique has previously been standardized by Elnady [19], the used test rig is illustrated in Figure-6.
Figure-6. Transmission loss measurement test rig.
The flow pressure-drop curves of the mufflers were measured by measuring the static pressure drop across the muffler and plotting it against the inlet flow at different velocities. The pressure drop was measured by two tubes that are connected to a digital manometer while the flow speed is measured using a flow meter before the muffler.
6. RESULTS
The properties of the original muffler were measured to verify the simulation of the original muffler. Next, the three iterations of the muffler were simulated to study the effect on its performance.
The original muffler was first simulated and measured with and without flow to verify the simulation model. Figures 7 and 8 show the transmission loss results without flow and with air flow.
Figure-7. Original muffler Transmission loss verification without flow.
0 100 200 300 400 500 600 700 800 900 1000 5
10 15 20 25 30 35 40 45 50 55
Frequency [Hz]
T
ra
n
s
m
is
s
io
n
L
o
s
s
[
d
B
]
Figure-8. Original muffler Transmission loss verification at 15m/s.
The transmission loss of the original muffler showed a constant value due to the impedance effect of perforated pipes while the peaks of the resonators are damped in the presence of flow.
The pressure drop of the original muffler is shown below to be 2660 Pa at 32 m/s (0.053 Kg/sec).
Figure-9. Original muffler pressure drop verification up to 32 m/sec.
After verifying the simulation model of the original muffler, three iterations of the muffler were simulated starting from the lowest in price.
The transmission loss of the original muffler was used a target curve for optimization, that the optimized model should exceed. Figure-10 show the optimized curves of all the iterations without flow and Figure-11 show the optimized curves at flow of 15 m/s.
Figure-10. Transmission loss of the orginal muffler compared with the three iterations made
without flow.
Figure-11. Transmission loss of the orginal muffler compared with the three iterations made at
15m/sec inlet flow speed.
The first iteration “Iteration A” with the lowest price is to optimize the original muffler by turning it into a simple expansion chamber. However, it showed lower values than the original, therefore, an introduction of extended inlet and outlet pipes were introduced. Iteration B with extended inlet and outlet pipes showed a better response of transmission loss except the region below 300 Hz in case of flow. The peak at 235 Hz in the original muffler curve was considered as a critical constrain, it should exist in the optimized muffler to account for any noise known harmonic from the engine. In Iteration C, the two baffles are used and three steel rods are used to hold the two baffles in the muffler shell. Two peaks were introduced at 195 Hz and 245 Hz to replace the resonance in the original muffler curve.
Iteration C was manufactured and compared with original muffler to show its advantage over the original muffler except the small drop at 550 Hz.
0 100 200 300 400 500 600 700 800 900 1000 5
10 15 20 25 30 35 40 45 50 55
Frequency [Hz]
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Simulation Measurement
0 0.01 0.02 0.03 0.04 0.05 0.06 0
500 1000 1500 2000 2500 3000
Inlet Mass Flow [Kg/s]
P
re
s
s
u
re
D
ro
p
[
P
a
]
Simulation Measurement
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10 20 30 40 50 60 70 80
Frequency [Hz]
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Original muffler Iteration A Iteration B Iteration C
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10 20 30 40 50 60 70 80
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Figure-12. Measured transmission loss comparison of the 3rd iteration and the original muffler without flow.
To overcome the drop in the middle range, rock wool with light density (14 kg/m3) was added into the third chamber. Figures 13 and 14 show that the 550 Hz drop was damped.
Figure-13. Iteration C after adding porous material compared with the measurement of the original
muffler without flow.
Figure-14. Measured transmission loss comparison of the 3rd iteration and the original muffler with flow.
Iteration C was manufactured to study its operation time, setting time and total cost. The production steps reduced from 18 to only 4 steps, the operation time of the internal assembly (perforated pipe and the baffles) takes 3 hrs. While the proposed model only takes 15 min. The use of pipes with the same production steps and identical supporting rods advances the new model with the ease of production flow and decreased the machines setup time. This optimization contributed 60% reduction in the muffler cost.
7. CONCLUSIONS
Material processing for vehicles produced for long periods is a concern for automotive feeding industry. Mufflers are an example of the vehicle’s parts that consume time and material due to the long sheet metal processing. A minimization for the cost was targeted by optimizing the muffler inner parts while keeping the performance the same. Manufacturing steps and applicable muffler design parameters (transmission loss and pressure drop) were studied, simulated and measured. Optimization through the objective function in MATLAB “fmincon” was used to reach the highest transmission loss and the minimum pressure drop. New design was proposed and compared to the original model to show 92% reduction in production time and consequently reduction of the entire cost.
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