I have a 3D printer and want to print an optimized port for my subwoofer. Let's assume the port has a circular cross-section at every point. x=0 is the front baffle, with x increasing as you go deeper into the port. What function for the radius, r(x) is optimal?
The goal is to avoid port compression and chuffing sounds.
The goal is to avoid port compression and chuffing sounds.
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No 3D printer needed, I use ABS pipe available at any plumbing shop for my ducts. They're sturdy as hell, and Oatey ABS glue fastens those onto MDF as strongly as anyone could wish. Since the outside diameter is a half-inch greater than the inside, hole saws of the appropriate sizes are readily found. My front baffles are double-thickness so the inside panel is cut for the outside diam. and the outer panel is cut for the I.D. Pop in the duct from the cabinet interior and you're golden.
I find a 3/4" radius roundover does a fine job on both the front panel and rear support panel.
No 3D printer needed, I use ABS pipe available at any plumbing shop for my ducts. They're sturdy as hell, and Oatey ABS glue fastens those onto MDF as strongly as anyone could wish. Since the outside diameter is a half-inch greater than the inside, hole saws of the appropriate sizes are readily found. My front baffles are double-thickness so the inside panel is cut for the outside diam. and the outer panel is cut for the I.D. Pop in the duct from the cabinet interior and you're golden.
I find a 3/4" radius roundover does a fine job on both the front panel and rear support panel.
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Since the air in the port needs to meet the flat front baffle I imagine that as x -> 0, r -> infinity. This boundary condition is met by any radius chamfer.
If port compression is caused by turbulence then we want laminar flow. From the Wikipedia article that suggests we want the Reynolds number of the fluid to stay below 1800.
Assume the narrowest part of the port occurs at x=m and that Re = 1800 at that point. How should the port's radius change to not increase Re while growing at the fastest rate possible?
Re = QD/(νA)
D is the hydraulic diameter of the pipe (m) (which for round completely full pipes is 2x the radius)
Q is the volumetric flow rate (m3/s)
A is the pipe's cross-sectional area (m2)
u is the mean speed of the fluid (SI units: m/s);
ν is the kinematic viscosity of the fluid
Re = 2Qr/(v*pi*r^2)
Group all the constant terms into a coefficient K
Re = K r / r^2 = K/r
So any increase in r will reduce Re. This implies any increasing are would meet the criteria. But clearly, in real life, it helps to have a more gradually expanding port. I tried to come up with an equation that considers both the axial and radial velocity of air particles, but no luck yet.
Regarding the use of a 3d printer and more complicated geometries, it is not necessary for sure. I find this to be part of the fun.
If port compression is caused by turbulence then we want laminar flow. From the Wikipedia article that suggests we want the Reynolds number of the fluid to stay below 1800.
Assume the narrowest part of the port occurs at x=m and that Re = 1800 at that point. How should the port's radius change to not increase Re while growing at the fastest rate possible?
Re = QD/(νA)
D is the hydraulic diameter of the pipe (m) (which for round completely full pipes is 2x the radius)
Q is the volumetric flow rate (m3/s)
A is the pipe's cross-sectional area (m2)
u is the mean speed of the fluid (SI units: m/s);
ν is the kinematic viscosity of the fluid
Re = 2Qr/(v*pi*r^2)
Group all the constant terms into a coefficient K
Re = K r / r^2 = K/r
So any increase in r will reduce Re. This implies any increasing are would meet the criteria. But clearly, in real life, it helps to have a more gradually expanding port. I tried to come up with an equation that considers both the axial and radial velocity of air particles, but no luck yet.
Regarding the use of a 3d printer and more complicated geometries, it is not necessary for sure. I find this to be part of the fun.
FYI/FWIW, I use the radius*0.732 pipe end correction on a flat baffle, so this becomes the flare radius while some others have claimed you can't go wrong with the radius of the pipe = the flare radius.
I was told by a speaker designer friend that the ideal port flare for minimum port noise was a tractrix. He said it was the subject of a US patent dispute involving JBL and one other maker possibly Bose. He was called as a prior art expert. I think his side won
The tractrix makes sense to me intuitively. If you add both the radial and axial dimensions of the flow velocity and want that to remain constant, that is what a tractrix is doing by having a fixed length pole sweep out a curve. It also meets the boundary condition that it is tangent to the front baffel.
The equation of a tractrix
"a" is defined by the radius of the port at the baffle. Then you simply trace backwards until you get to the minimum throat radius, r_min, before compression would begin. The closer r_min is to a the smaller the expanding portion of the port will be.
For non-round ports, I suppose the approach would need to be modified, or you would need non-round straight sections for the port which is a pain.
The equation of a tractrix
"a" is defined by the radius of the port at the baffle. Then you simply trace backwards until you get to the minimum throat radius, r_min, before compression would begin. The closer r_min is to a the smaller the expanding portion of the port will be.
For non-round ports, I suppose the approach would need to be modified, or you would need non-round straight sections for the port which is a pain.
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For non-round ports, perhaps you just compute the tractrix from the circular part of the port forward, and the tractrix from the actual outlet shape backwards and then smoothly interpolate between them.
So if f(a,x) defines a tractrix and x in range [0,1]
radius(x, phi) = f(a_circular_outlet_of_same_area, x)*x + f(a_actual_outlet(phi),x)*(1-x)
where
x is the axial location along the port
phi is the angle that rotates around the axis of the port
a_circular_outlet_of_same_area is the radius that a circle would have with the same area of the space you have for the port
a_actual_outlet(phi) is the distance form the axis to the outside of the shape in the phi direction
Then at x=1 the port will be circle-shaped (where the straight section of port begins) and at x -> 0 it will smoothly mesh with the actual shape of the port.
After thinking about the above more, I am not sure it's a good idea. You really want only a single "a" for everything to keep the inspiration related to constant air velocity.
Also related concept that might help derive the optimal shape: flow seperation is probably what causes chuffing in ports of expanding radius that should theoretically only allow the reynolds number to decrease
So if f(a,x) defines a tractrix and x in range [0,1]
radius(x, phi) = f(a_circular_outlet_of_same_area, x)*x + f(a_actual_outlet(phi),x)*(1-x)
where
x is the axial location along the port
phi is the angle that rotates around the axis of the port
a_circular_outlet_of_same_area is the radius that a circle would have with the same area of the space you have for the port
a_actual_outlet(phi) is the distance form the axis to the outside of the shape in the phi direction
Then at x=1 the port will be circle-shaped (where the straight section of port begins) and at x -> 0 it will smoothly mesh with the actual shape of the port.
After thinking about the above more, I am not sure it's a good idea. You really want only a single "a" for everything to keep the inspiration related to constant air velocity.
Also related concept that might help derive the optimal shape: flow seperation is probably what causes chuffing in ports of expanding radius that should theoretically only allow the reynolds number to decrease
Folks have used various horn flares in rectangular shapes by making two equal horn tapered ducts of half the total length and attaching them back to back for high power, very low vent mach systems.
Dual flare will compress then decompress so is ideal for high velocity.
Keep in mind chuffing is from the velocity of the port being to high.
And non ideal port shapes causing poor efficiency.
To reduce chuffing or compression. Use the highest efficiency shape.
A round port , which is large enough to keep port velocity close to ideal.
Often quoted to be around 5% the speed of sound or about 17 meters per second.
As you soon find out. The high efficiency of round ports tends to require the longest length compared to rectangle or square ports.
So often non ideal shapes are used, or smaller round ports are used.
Because Ideal ports can be extremely long and not fit well within a enclosure.
So that is why flaring comes into play. Flaring helps reduce chuffing with non ideal port shapes or sizes.
Ideal flare is dual flare or both ends of the port flared. Assuming the port is non ideal.
If the port is closer to Ideal and around a velocity of 17 to 18 ms. Flaring is not really necessary.
Otherwise with a dual flare, first flare is compressing the air into the small non ideal port. Then the exit flare is decompressing.
Actual chuffing is more a issue below port tuning. Even though ideal velocity is around 17 to 18 ms.
You can get away with up to 20 to 25 ms.
But when the port unloads its even higher.
So with subwoofers and using non ideal port velocity.
Most the chuffing can be solved using a over excursion filter, or highpass filter , or sub sonic filter.
Since most the extremely high velocity is below port tuning.
And at port tuning or above you can tend to get away with higher velocity.
I tend to shoot for 18 to 22 ms and there will be no chuffing.
Below port tuning might shoot up to 25 to 30 ms
but the filter will remove that.
Ideal flare is long slow taper. If your ideal port was say 5" but your stuck using 3" port
Then the flare could start out anywhere from ideal 5" and the compress slowly into your non ideal 3" port. Then same again decompress from 3" to another 5" flare.
The real magic isnt the flare itself, its ideal velocity.
Which can be achieved without a flare.
Again as mentioned flares come into play when your using non ideal high velocity.
Keep in mind chuffing is from the velocity of the port being to high.
And non ideal port shapes causing poor efficiency.
To reduce chuffing or compression. Use the highest efficiency shape.
A round port , which is large enough to keep port velocity close to ideal.
Often quoted to be around 5% the speed of sound or about 17 meters per second.
As you soon find out. The high efficiency of round ports tends to require the longest length compared to rectangle or square ports.
So often non ideal shapes are used, or smaller round ports are used.
Because Ideal ports can be extremely long and not fit well within a enclosure.
So that is why flaring comes into play. Flaring helps reduce chuffing with non ideal port shapes or sizes.
Ideal flare is dual flare or both ends of the port flared. Assuming the port is non ideal.
If the port is closer to Ideal and around a velocity of 17 to 18 ms. Flaring is not really necessary.
Otherwise with a dual flare, first flare is compressing the air into the small non ideal port. Then the exit flare is decompressing.
Actual chuffing is more a issue below port tuning. Even though ideal velocity is around 17 to 18 ms.
You can get away with up to 20 to 25 ms.
But when the port unloads its even higher.
So with subwoofers and using non ideal port velocity.
Most the chuffing can be solved using a over excursion filter, or highpass filter , or sub sonic filter.
Since most the extremely high velocity is below port tuning.
And at port tuning or above you can tend to get away with higher velocity.
I tend to shoot for 18 to 22 ms and there will be no chuffing.
Below port tuning might shoot up to 25 to 30 ms
but the filter will remove that.
Ideal flare is long slow taper. If your ideal port was say 5" but your stuck using 3" port
Then the flare could start out anywhere from ideal 5" and the compress slowly into your non ideal 3" port. Then same again decompress from 3" to another 5" flare.
The real magic isnt the flare itself, its ideal velocity.
Which can be achieved without a flare.
Again as mentioned flares come into play when your using non ideal high velocity.
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