Conventions
ID_tube1 is inside dia. of smallest tube
OD_tube1 is outside dia. of smallest tube
ID_tube2 is inside dia. of middle tube
OD_tube2 is outside dia. of middle tube
ID_tube3 is inside dia. of biggest tube
OD_tube3 is outside dia. of biggest tube
CSA is the Cross-sectional area presented to the airflow. Ideally constant throughout the device
Calculating tube diameters
Keeping the area presented to the airflow constant;
area inside middle tube - area occupied by smallest tube = area inside smallest tube
pi * (ID_tube2 / 2)^2 - pi * (OD_tube1 / 2)^2 =pi * (ID_tube1 / 2)^2
multiplying both sides by 4/pi
ID_tube2^2 - OD_tube1 ^2 =ID_tube1)^2
rearranging terms
ID_tube2^2 =ID_tube1^2 + OD_tube1^2
taking square root of both sides
Inside diameter of middle tube = sqr(ID_tube1^2 + OD_tube1^2 )
area inside biggest tube - area occupied by middle tube = area inside smallest tube
pi * (ID_tube3 / 2)^2 - pi * (OD_tube2 / 2)^2 =pi * (ID_tube1 / 2)^2
multiplying both sides by 4/pi
ID_tube3 ^2 -OD_tube2 ^2 = ID_tube1 ^2
rearranging terms
ID_tube3 ^2 = ID_tube1 ^2 + OD_tube2 ^2
taking square root of both sides
Inside diameter of biggest tube = sqr(ID_tube1 ^2 + OD_tube2^2)
Calculating end gaps
Assume for a moment that you have an inner tube with a hole as shown
The area of the hole will be the length of the arc segment times the height of the hole.
If we increase the arc segment to encompass the entire circumference of the tube, we will define the area presented to the air as it flows around the end of the tube. This area must equal the CSA of our small tube.
Firstly, the end gap of the small tube
CSA=circumference_small_tube * endgap_small_tube
re-arranging terms
endgap_small_tube = CSA / circumference_small_tube
substituting for CSA
endgap_small_tube=( pi * (ID_tube1 / 2 ) ^ 2 ) / circumference_small_tube
For strict accuracy, we need to use the average circumference
endgap_small_tube=( pi * (ID_tube1 / 2 ) ^ 2 ) / ( pi*( ID_tube1 + OD_tube1)/2)
simplifying
endgap_small_tube= ( ID_tube1 ^ 2 ) / ( 2 * ( ID_tube1 + OD_tube1))
Second, the end gap of the middle tube
CSA=circumference_tube2 * endgap_tube2
endgap_tube2=CSA/average circumference_tube2
endgap_tube2=( pi * (ID_tube1 / 2)^2 ) / ( pi * (ID_ tube2 + OD_tube2) / 2)
endgap_tube2= (ID_tube1 ^ 2 ) / ( 2* (ID_ tube2 + OD_tube2))
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Classic format , using a graphic
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As a reality check, if you set ID = OD, these equations reduce to those in the simple proofs
If one was insane enough to construct a 5 tube port, similar reasoning would give:
Inside diameter of tube2 = sqr(ID_tube1^2 + OD_tube1 ^2)
Inside diameter of tube3 = sqr(ID_tube1 ^2 + OD_tube2 ^2)
Inside diameter of tube4 = sqr(ID_tube1 ^2 + OD_tube3 ^2)
Inside diameter of tube5 = sqr(ID_tube1 ^2 + OD_tube4 ^2)
endgap_tube1= (ID_tube1 ^ 2 ) / ( 2 * (ID_tube1 + OD_tube1))
endgap_tube2= (ID_tube1 ^ 2 ) / ( 2* (ID_ tube2 + OD_tube2))
endgap_tube3= (ID_tube1 ^ 2 ) / ( 2* (ID_ tube3 + OD_tube3))
endgap_tube4= (ID_tube1 ^ 2 ) / ( 2* (ID_ tube4 + OD_tube4))
The next page details the Acoustic length vs Physical length relationship
