Exhaust System Performance Math

Choosing exhaust pipe diameter and header primary-tube diameter and length can be confusing. Generally, people tend to copy what others have done, often choosing too-large tube and pipe diameters with the belief that bigger is always better. Not so. Choosing diameters that are too large can rob the engine of torque and horsepower at lower engine speeds; diameters that are too small can choke the engine.

 


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Sizing an exhaust system doesn’t need to be a guessing game. A few formulas will aid you in determining the optimum pipe diameter and length.

Sizing an exhaust system doesn’t need to be a guessing game. A few formulas will aid you in determining the optimum pipe diameter and length.

 

The engine needs to breathe, but the exhaust path needs to be matched to the engine’s operating speed (the RPM at which you want maximum torque and/or horsepower) and the engine’s VE. Selecting the proper header primary tube diameter and length affect where maximum torque and/or horsepower are achieved in a specific powerband range. Matching header tube size to your engine is a major factor in maximizing power at low-end, mid-range, or top-end.

 

Finding Swept Volume

As a basis for several calculations, you first need to know how to determine the swept volume of one cylinder and total engine displacement. Either of the following two formulas can be used to determine the swept volume of each cylinder:

Swept Volume = Pi x (bore ÷ 2) x (bore ÷ 2) x stroke

Swept Volume = bore x bore x stroke x .7854

Where:

Pi (p) = 3.14159

.7854 = math constant

As an example, let’s say that the engine features a bore diameter of 4.030 inches and a stroke of 4.000 inches. Using the first formula:

3.14159 x (4.030 ÷ 2) x (4.030 ÷ 2) x 4.000

3.14159 x 2.015 x 2.015 x 4.000 = 51.022

 

Exhaust pipe is sized based on OD. Shown here is a 3-inch pipe.

Exhaust pipe is sized based on OD. Shown here is a 3-inch pipe.

 

The ID of this 3-inch exhaust pipe measures 2.88 inches. This is typical for pipe that features about a .060-inch wall thickness

The ID of this 3-inch exhaust pipe measures 2.88 inches. This is typical for pipe that features about a .060-inch wall thickness

 

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This is an example of measuring pipe wall thickness. This pipe features .060-inch wall thickness.

This is an example of measuring pipe wall thickness. This pipe features .060-inch wall thickness.

 

Measuring pipe diameter should be done with a caliper or micrometer, rather than eye-balling with a tape measure.

Measuring pipe diameter should be done with a caliper or micrometer, rather than eye-balling with a tape measure.

 

Using the second formula:

4.030 x 4.030 x 4.000 x .7854 = 51.022

The resulting figure of 51.022 ci indicates the displacement of a single cylinder. If the engine features eight cylinders, simply multiply by 8. In this example, total engine displacement is 408 ci.

 

Calculating Primary Tube Size

Wave tuning is realistically the realm of high-level race engine builders, where variables such as valve opening timing, connecting rod length, and overlap are considered in relation to tube diameter and length. For instance, peak horsepower is usually about 1,500 to 1,700 rpm above the engine’s torque peak. This RPM spread between torque and horsepower is influenced by the engine’s rod-to-stroke ratio. With stroke remaining the same, using a shorter connecting rod should broaden the separation between peak torque and horsepower. A longer rod should lessen this spread, with torque and horsepower peak RPM being closer together.

Rather than driving yourself nuts trying to understand the complexities involved in wave tuning, you can use the following formulas as a guide in the right direction.

 
Cylinder Volume

Because primary tube diameter affects engine torque, you also want to consider the engine’s peak-torque RPM. To calculate cylinder volume, you can use this formula:

Cylinder Volume = displacement ÷ number of cylinders

For example, a 408-ci engine with 8 cylinders has a volume of 51 cc per cylinder (408 ÷ 8).

 

Cross-Section Area

If you already know (or anticipate) the engine’s peak-torque RPM, you can use this formula to deter-mine the primary tube’s cross-section area, which is also known as the primary pipe area (PPA):

Cross-Section Area = (cylinder volume x RPM) ÷ 88,200

Where:

88,200 = mathematical constant

RPM = here, RPM at peak torque

For example, a 408-ci engine at 4,200 rpm has a cross-section area of 2.428 ci [(51 x 4,200) ÷ 88,200].

You want to select the best tube diameter for your application, with a specific peak-torque RPM in mind. As another example, let’s use a 350-ci engine that features a single-cylinder displacement of 43.75. Using your planned peak torque at 5,000 rpm, the cross-section area is 2.480 [(5,000 ÷ 88,200) x 43.75]. You can also find the cross-section area by first determining the pipe’s inside diameter. As you recall, tube sizes are designated by outside diameter, so you need to subtract the total wall thickness of the pipe (wall thickness x 2). This formula for finding the cross-section area is:

Cross-Section Area = pR2

Where:

p = 3.14159

R = the tube’s inside radius, which is half of the ID

If a tube measures 1.75 inches in outside diameter and features a wall thickness of .040 inch, the inside diameter is 1.67 inch. The radius is half of this, .835 inch. So using the formula above, the cross-section area is 2.19039 (3.14159 x .8352 = 3.14159 x .697225. This is a bit small for the intended peak-torque RPM. If you bump up to a primary tube OD of 1.875 inches with a .040-inch wall thickness, the cross-section area increases to 2.530 (3.14159 x .89752 = 3.14159 x .8055).

Given that the theoretical goal was to obtain a cross-section area of 2.480, you can compromise by choosing the 1.875-inch primary tube size, which theoretically allows peak RPM to occur around 5,100 rpm, which is pretty close to the initial target. You can further tailor where the peak torque occurs by changing the length of the primary tubes. By lengthening the tubes, you emphasize torque below the 5,100-rpm range. If you go with shorter tubes, torque is emphasized at higher RPM.

 
Inside Diameter

Once the cross-section area has been determined, you can then calculate the primary tube inside diameter using this formula:

Inside Diameter = √cross-section area x 1.273

Continuing the example above, you get a primary tube ID of 1.98 inches [√2.428 x 1.273)]. Considering commonly available tube diameters, this indicates a choice of a tube with a 1.5-inch inside diameter. Depending on the tube wall thickness, you can also calculate ID based on the tube’s OD and its wall thickness. For instance, if you purchase a set of headers labeled as having 1.75-inch primaries, this means that the OD is 1.75 inches. The industry standard uses OD to refer to tube diameter and ID to refer to pipe diameter. Header manufacturers typically refer to OD when identifying primary tube diameters.

 

5A

 

For example, headers labeled as having 1.5-inch primaries actually have an ID of around 1.42 to 1.37 inch, depending on wall thickness of the material. If you buy headers labeled as having 1.75-inch primaries, the inside tube diameter is slightly smaller, but that doesn’t mean that you’re being cheated.

Tubing is simply sized according to its OD. Here’s another formula for calculating inside diameter:

Inside Diameter = OD – (2 x wall thickness)

For example, if the tube OD measures 1.75 inches and the wall thickness is .047 inch (18 gage), the ID is 1.656 inch [1.75 – (2 x .047) = 1.75 – .094].

Here’s another example: If the tube OD measures 1.75 inches and wall thickness is .0625 (16 gauge), the ID is 1.625 inch [1.75 – (2 x .0625) = 1.75 – .125].

 

Pipe CFM

A rule of thumb is to determine engine intake volume and then approximately match this volume for the exhaust. To calculate it you can use this formula:

Exhaust Volume = (RPM x .001) x displacement ÷ 2

 

5B

 

For instance, if your target is to obtain optimum performance at, say, 4,000 rpm, and the engine displacement is 350 ci, the formula works out to 700 cfm [(4,000 x .001) x 350 ÷2].

The exhaust pipe(s) total CFM should be in the same range.

 

Exhaust Waves

Most street vehicle owners who intend to install a set of exhaust headers for improved performance are more concerned with installation and fit rather than the nuances involved in tuning for peak power and torque. Generally speaking, smaller primary tube diameters and longer primary tubes with collectors provide better low-end torque, which is best suited for the street. However, compromises are commonly made simply because of available space and fit in the vehicle. Serious racers are more inclined to “tune” the exhaust to maximize power and torque to suit their engine’s operating parameters on the track.

 

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Exhaust system math can be difficult to comprehend. I explain here how wave pulses affect the exhaust system in understandable terms. When the engine is running, pressure waves run in both directions through the primary tubes of the headers. These include compression waves and expansion waves that tend to move faster than the actual exhaust gas particles. Compression waves (or pulses) are positive waves that push exhaust particles from the cylinder head, through the remainder of the system. Expansion waves are negative waves that push gases in the opposite direction. The expansion wave is created at the end of the exhaust outlet, where negative pressure reflects back up into the tube, which creates a pressure drop when it reaches the combustion chamber. This can create a pressure differential (or vacuum) at the cylinder area. This pressure drop then allows the atmospheric (or forced-induction) air to increase, packing more air and fuel into the cylinder.

 

5C

 

In other words, a combination of a positive pressure wave and negative pressure wave can improve exhaust scavenging, resulting in an increase in power output. As far as exhaust tuning is concerned, this pressure/ expansion wave phenomenon is influenced by engine RPM, primary tube diameter, and length. Smaller diameter primary tubes create more pressure and better scavenging, but if sized too small, the primary tubes can cause flow restriction and can make the engine work harder to pump the charge. Larger diameter tubes result in lower pressures that can be less efficient at scavenging.

Connecting rod length (crank-to-rod ratio) is yet another aspect to consider in terms of how close in the RPM range that peak torque and peak horsepower are generated. With the same crank stroke, shorter rods tend to widen the RPM difference between peak torque and peak power while longer rods tend to narrow this gap (moving peak torque and peak horsepower closer together in terms of engine speed). Altering rod length doesn’t make a difference in peak values, but it can broaden or tighten the RPM range between torque and power.

Again, remember the basics: Smaller-diameter tubes cater to low-end torque, while larger-diameter tubes tend to favor peak torque at a higher RPM band. In terms of primary tube length, shorter tubes favor higher engine speeds while longer tubes favor lower engine speeds.

Although all of this theory is interesting to note, the reality is that most street applications benefit from smaller diameters and longer tubes (considering the variable of practical fitment to the vehicle at hand). Racers experiment with diameters and lengths to suit specific engine requirements based on where they need the most torque and power at a given RPM.

 

Primary Tube Length

Here is the formula for determining primary tube length:

Primary Tube Length = [850 (360 – EVO) ÷ RPM] – 3

Where:

EVO = exhaust valve opening point (refer to your cam card)

For example, let’s say that your engine’s cam has an exhaust valve opening point of 82 degrees and your desired peak torque hits at about 4,200 rpm. Using the formula, the length is 53.26 inches ([850 (360 – 82) ÷ 4,200] – 3).

In general, longer primary tubes move the powerband below the engine’s peak torque point and shorter primary tubes tend to move the torque value to above the peak-torque RPM.

 

Collector Diameter and Length

A collector (where the primary tubes terminate as a group) is most beneficial at or below peak-torque RPM. In theory, as you add collector volume, you increase torque. As you reduce collector volume, you slightly reduce torque. If you’re more concerned about operating the engine at peak horsepower, a collector may not be needed. If you’re more concerned about running the engine at or near its peak-torque RPM, a collector makes sense. Here are the applicable formulas for estimating collector diameter and length:

Collector Diameter = 1.9 x primary tube diameter

Collector Length = .5 x primary tube length

 

Where:

1.9 = math constant

.5 = math constant

For example, let’s say that your primary pipes are 1.75 inches in outside diameter, and the primaries are 38 inches in length. Using the formulas you get these results:

Collector Diameter = 1.9 x 1.75 = 3.325 inches

Collector Length = .5 x 38 = 19 inches

In general, a collector with a smaller volume should benefit mid-range power, while a larger volume collector should benefit top-end power.

 

Written by Mike Mavrigian and Posted with Permission of CarTechBooks

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