Mastering Exhaust System Math to Build High-Performance Engines

Exhaust systems don’t generally invoke the glamour and gut level appeal associated with many high-performance induction systems, but they certainly play a critical role in the performance of every engine. Through decades of experience we have learned most of the ideal shapes, sizes, and configurations for high-performance exhaust systems, and we are able to calculate primary pipe sizes and lengths with a good deal of accuracy. It hasn’t come easy, as the necessity for catalytic converters introduced detrimental restrictions that took a long time to overcome. Today we enjoy smaller, high-flow cat systems that are not the performance hindrance they once presented.

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Of course, dual exhaust systems have long been recognized as a performance perk and interconnecting H-pipes and X-pipes have become the norm for highperformance exhaust systems. Three-inch and larger fulllength exhaust systems have become prevalent and a broad range of appropriately sized high-performance mufflers accommodate them. So we have acquired a pretty good understanding of performance exhaust systems, but there are still ways to further optimize them through careful header selection based on engine displacement and RPM, volumetric efficiency, and final application. This is particularly true for applications unhindered by mufflers or those that run certain types of racing mufflers that have become more prevalent.

One drawback to the broad variety of header types and sizes is the potential for misapplication based on prevailing trends or which way the wind blows. Many enthusiasts still favor the “bigger is better” approach, which completely discards the purpose of building headers sized to suit specific applications. If the only purpose of headers were simply to look good, manufacturers would just build 2-inch Pro Stock–style headers for all small-blocks and 21⁄2-inch headers for all big-blocks. Everyone would be happy, but performance would suffer dramatically. In racing applications, exhaust systems are usually less restricted. Most often they involve a muffler requirement, stock exhaust manifolds or a possibly a spec header. Builders often consider mixed rocker arm selections optimized to provide appropriate exhaust event timing to aid cylinder blow down. Spec mufflers may be required and testing with back pressure readings can help pinpoint the most beneficial exhaust pipe cross-sectional area, effective primary pipe lengths, and collector sizes if headers are permitted. Modeling this on a PC simulator like PipeMax is a good idea if you want to pinpoint the best overall dimensions.

PipeMax Exhaust System Calculations

For those who want to get deeper into header design there is an inexpensive header-pipe-design program available on the internet called Pipe Max. It’s offered by Meaux Racing Heads and you can download it from their website at www.maxracesoftware.com for a reasonable price. It is primarily a header design program, but it incorporates enough user data to function as an engine simulation program and it does calculate torque and horsepower output based on VE and the recommended header specs. It provides a wealth of good information about header dimensions and the proper design of headers to accommodate wave tuning. You input all the usual engine information along with details about your cylinder heads and camshaft and it calculates the optimum primary pipe diameters and cross-sectional areas, primary tube lengths and collector specifications for optimum performance. The results screen displays dimensions for a single primary pipe or 2- and 3-step header designs. It provides recommended pipe diameters and lengths for the primary pipes and the collectors. It also displays the best and worst specifications so you can avoid them. It calculates primary pipe harmonics for the first through eighth reflected wave and collector harmonics with mufflers and tailpipes. All of these specs are delivered in U.S. units and metric units automatically. The program is based on projected VE and it has a function to estimate VE for your particular engine specs. It allows you to select a dyno acceleration rate for the simulation and you can specify the mean flow velocity to be used in crosssectional area calculations. While quite inexpensive, it is very thorough and makes a perfect complement to other simulators you might be using. It pinpoints the ideal header dimensions for your engine and simplifies header design. If you study it closely it teaches you a lot about the effects of pressure wave activity in the engine’s exhaust system. I’m very comfortable in recommending it. Source: www.maxracesoftware.com.

Simulators are front loaded with all the mathematical fundamentals of engine performance. They can’t always predict absolute power and torque, but they can illuminate trends according to the physics and that amounts to a pretty good road map for serious engine builders. As long as the requirement is clearly identified in the rules, savvy engine builders always find ways to optimize within those rules. If a spec header is involved and you’re not comfortable with discreetly modifying pipe dimensions, consider the possibility of different cam timing in cylinders whose intake and exhaust timing peaks differ due to unequal port or pipe cross section or length. Like other components, any restriction on the exhaust system should be carefully evaluated for its effect on torque production and how it affects other parts of the engine package. There are usually ways to refine and assemble a more robust package even within restrictive rules.

I once did a dyno test with a client whose big-block Chevy arrived with 21⁄4-inch headers, oval port cylinder heads, a conservative 218-degree camshaft, and a dualplane intake manifold sporting an 850-cfm Holley 4-barrel. When it didn’t perform as expected, he was thoroughly disappointed and immediately surmised that something was wrong with the dyno. I suggested trying a set of 13⁄4-inch headers that I had in my testing inventory, but he insisted that his high-performance big-block needed the big-tube headers that he had seen on everyone else’s car at the track. The fact that he had a street engine and that everyone else was running big-tube headers regardless of application simply reinforced the depth of the problem. I finally convinced him to try the smaller headers, and he was astonished when they delivered a solid 38–ft-lbs gain at the torque peak and raised the power curve by about 20 hp across the board.

I knew from personal experience that this was the right move and any other good dyno shop would have done the same thing. The point is simple: Just because the catalog lists big-tube headers doesn’t necessarily mean that you need them to achieve the best performance from your particular application.

Calculating Primary Tube Cross Section

So, how do you determine the correct header size for any given engine combination? Copying the other guy amounts to a shot in the dark and trial and error is pretty expensive. Wave tuning theory has long been a popular method, but it is typically beyond the reach of the average enthusiast. Moreover, it can be complicated and is probably less effective when saddled with full-length exhaust systems that include catalytic converters, X-pipes, supersonic master mufflers, and all the latest “gotta have it” exhaust goodies. Most of these pieces serve a legitimate function, but their presence complicates wave tuning. In short, wave tuning seeks to take advantage of oscillating pressure pulses within the exhaust stream to aid cylinder scavenging. The theory is to time the pulses so that a reflecting pulse arrives at the cylinder at just the right time to lend its energy to carrying exhaust gases out of the cylinder. The timing is a function of primary tube length, and it is affected by the point where the pulse reaches atmospheric pressure at the end of the tube.

It is most effective at one particular engine speed, and its value is well illustrated in top-level engine simulation packages such as Motion Software’s Dynomation 5, or Performance Trend’s Engine Analyzer Pro. But it is not something the average enthusiast can jump right into on a one-off basis. This is not to suggest that simulators are not a valuable tool for more serious enthusiasts with real racing applications that can take advantage of their powerful capabilities. Both are top-rated simulators that deliver quality results while providing a solid learning experience neatly disguised as pure fun. I highly recommend both of them.

and a hand-held calculator, there is still a way of calculating primary header pipe diameter for effective performance based on engine displacement, engine speed, and volumetric efficiency. In the interest of raising awareness and furthering everyone’s performance goals, leading performance authority Jim McFarland has provided a simple formula for calculating the optimum crosssectional area (AC/S) of a header primary tube or pipe. This method optimizes for the engine’s torque peak or point of maximum volumetric efficiency and thus the point of maximum exhaust volume. At engine speeds above the torque peak, cylinder filling decays proportionately with RPM (time) because there is progressively less time available to fill the cylinder on each intake event. There are more power strokes per minute, but proportionately less exhaust volume to evacuate due to declining VE. Optimizing for the torque peak provides the ideal primary pipe diameter, and since peak power is usually no more than 1,500 to 1,750 rpm above the torque peak, the selected pipe size has no trouble accommodating an exhaust volume that essentially flatlines, and then begins to fade rapidly.

The RPM spread between the torque peak and the power peak is influenced by the engine’s rod-to-stroke (R/S) ratio (see Chapter 2). Given a constant stroke length, a shorter rod tends to broaden the separation between the torque peak and the power peak. Similarly, a longer rod tends to move the peaks closer together. The previously stated range applies for most of the stroke and rod lengths commonly used by performance enthusiasts. The formula can be written two ways: one to solve for primary pipe c/s area when the torque peak RPM is known or anticipated, and the other to predict torque peak engine speed based on a primary pipe cross section that is being considered. To accommodate specific cylinder volumes, the formula considers a single cylinder only, so you’ll have to divide your known or anticipated displacement by the number of cylinders to determine single cylinder volume or displacement.

Cylinder Volume = displacement ÷ number of cylinders
Ac/s = (cylinder volume x RPM) ÷ 88,200
Or RPM = (Ac/s x 88,200) ÷ cylinder volume
Where:
Ac/s = primary pipe c/s area
Cylinder volume = volume of a single cylinder
88,200 = mathematical constant
RPM = RPM at torque peak

Once we know the calculated c/s area we can calculate the corresponding primary pipe diameter to the nearest available pipe size.

If Area = diameter2 x 0.7854

Then pipe size equals the square root of the previously calculated c/s area times the reciprocal of the constant 0.7854.

Pipe Size = √[A x (1/0.7854)] Or
Pipe Size = √(A x 1.273)

If you’re solving for a torque peak RPM based on contemplated primary pipe size, calculate the c/s area by squaring the inside diameter (ID) of the pipe and multiplying by 0.7854. To determine the true inside diameter of a pipe for the purpose of calculating c/s area, use the measured outside diameter (OD) minus twice the wall thickness.

ID = OD – (2 x wall thickness)
Example: 1.75-inch OD pipe with a wall thickness of 0.040 inch
ID = 1.75 – (2 x 0.040) = 1.67 inches

As performance exhaust systems encounter various restrictions such as catalytic converters and some types of high-performance mufflers, peak power can be eroded. Accordingly, header pipe size selection based on manipulating peak torque RPM points outweighs the pursuit of absolute peak power. This suggests erring on the small side of primary pipe selection to preserve velocity with the minimum c/s required to service cylinder volume at peak torque; hence the formula for c/s area based on engine speed, cylinder volume, and VE. Additionally, it should be noted that tuned induction systems generate their own independent torque curves that contribute proportionately to an engine’s “net” torque peak RPM. The default header size for most small-block engines seems to be 13⁄4, or 1.75, inch. Whether or not this is really the best choice depends on individual cylinder displacement and the exhaust volume it generates at peak torque.

The accompanying chart shows some calculated examples based on popular muscle car engines. Note the advertised torque and the RPM at which it occurs, the individual cylinder volume, calculated c/s area, calculated pipe size, and closest available pipe size. Note further that all of these performance engines except perhaps the largerdisplacement L99 Camaro and the earlier 350 SS call for a primary pipe diameter smaller than 13⁄4 inch, yet it is extraordinarily difficult to convince people to run smaller appropriately-sized headers on street cars. All major header manufacturers offer correctly-sized headers for these cars, but many enthusiasts ignore the recommended size and choose the next larger size because their gut tells them it is necessary. In most cases, it’s not.

Referring back to the previously cited production engines, we note that the 302-ci Z28 calls for a 1.79- square-inch c/s and 1.51-inch ID primary pipe to complement its exhaust volume at the 4,200-rpm torque peak. Consulting the chart, we check the c/s area and corresponding ID’s for a header with 16-gauge primary pipes (most common). The closest match is a 1.625-inch OD pipe that offers a 1.507-inch ID with 1.783-square-inch c/s area. In a catalog that would be listed as a 15⁄8-inch header with 16-gauge primary pipes and it is a near perfect match for the 302 engine.

Some heavy-duty headers use 14-gauge primary pipes. In this case you should still select the 15⁄8-inch (1.625) primary pipes even though the c/s area is a bit smaller. The next-larger common pipe size is 1.750, but the 14-gauge c/s area of 2.010 inches would be excessive for the stock 302’s exhaust volume. It would tend to reduce torque and shift the torque peak to a higher RPM. Also recognize that even though you might rev your 302 to 6,500 rpm, VE is proportionately reduced at that engine speed and the declining exhaust volume will not overtax the indicated primary pipe. Again, the key to primary pipe selection is the inside (ID) diameter c/s area indicated for the engine’s torque peak.

To cite a further example, we can look at the results of the 2009 Engine Master’s Challenge dyno shootout produced by Popular Hot Rodding magazine. Four-time winner John Kaase captured first place with a 403-ci small-block Ford that delivered 677 hp at 6,400 rpm and 597 ft-lbs of torque at 5,400 rpm through the mufflers. He ran Hedman headers with 2-inch-diameter primaries. Now let’s see how well the formula matches up to this selection.

403 ci ÷ 8 = 50.37 (rounded to 50.4 cid per cylinder)
Ac/s = 50.4 x 5,400 ÷ 88,200 = 3.085 square inches
Therefore:
Primary Pipe Size = √3.085 x 1.273 = 1.98-inch ID

Pretty close, with minor accommodation for tube thickness. Note that the object of the Engine Master’s Challenge is to maximize torque, and thereby horsepower, across the broadest possible range of engine speed. An experienced engine builder like Kaase leaves nothing on the table, and our formula neatly confirms his chosen header size. Moreover, he entered a second engine in the challenge, a 511-ci monster based on a 429 Ford. It happily thumped out 856 hp at 6,400 rpm and 751 ft-lbs of torque at 5,500 rpm. Again he selected a Hedman header, but this set had 2.25-inch primaries.

511 ÷ 8 = 63.875 ci per cylinder
Ac/s = 63.875 x 5,500 ÷ 88,200 = 3.983 square inches
Therefore:
Primary pipe size = √3.983 x 1.273 = 2.251 inches

Right on the money. Note that a key component of this formula is individual cylinder volume and the number of times you have to process (evacuate) it per minute. Because the torque peak represents maximum VE, we know that it is the point of greatest cylinder filling, highest charge density, and maximum exhaust volume. By optimizing header capability to match peak VE we achieve optimum system resonance at the point of maximum efficiency. Max torque follows accordingly.

My thanks to Jim McFarland for providing this simple formula and kudos to Mr. Kaase for proving it so handily. Because it addresses exhaust volume at peak VE, it calculates the optimum primary pipe diameter independently of primary tube length. The calculated area delivers optimum exhaust efficiency because it is ideally sized to move exhaust volume based on cylinder displacement and processing speed (RPM). Within reason, the calculated pipe diameter will deliver top performance regardless of length as long as the individual lengths can also be juggled to help crutch deficient inlet flow paths as found on good-port/bad-port big-block Chevys.

Once a peak torque RPM has been established, changing primary pipe length will not change the RPM point, but it will cause torque values to pivot back and forth about that point. Lengthening the primary pipe tends to inflate the torque curve below peak torque, while shortening the primary pipe typically shifts torque value from below the peak and adds it above the peak. Again, it is the change in primary pipe diameter and thus c/s area that moves the torque peak up or down the RPM scale.

For your convenience, the Primary Pipe Selection Guide (see page 96) is a list of commonly used primary pipe diameters and their corresponding c/s areas based on available pipe diameters and wall thickness. Once you calculate your ideal c/s area and corresponding ID, locate them on this chart to determine the most appropriate primary pipe size for your header selection.

Calculating Primary Tube Length Despite the previous discussion, there is a formula for calculating a preferred primary tube length. In his book, Performance Tuning in Theory and Practice, A. Graham Bell cites empirical formulas for primary pipe length and diameter that yield ballpark results surprisingly close to the previously examined exhaust volume and VE formula.

Lin. = [850 (360 – EVO) ÷ rpm] – 3
Dia.in. = [(vol. x 16.38) ÷ (L + 3)25] x 2.1
Where:
L = primary pipe length
EVO = exhaust valve opening point from cam card
or internet
vol. = volume of a single cylinder
Dia. = calculated primary tube diameter

Using these formulas with the same information from John Kaase’s 403-ci Engine Master’s Challenge engine, we calculate a primary pipe length of 39.8 inches and a primary tube diameter of 2.06 inches. Since we don’t know his cam specs other than 246 degrees at 0.050-inch lift and 0.750-inch total lift, we can speculate a pretty racy camshaft with EVO at 88 degrees BBDC. If we’re wrong and we drop it down to 77 degrees BBDC, we calculate a primary pipe length of 41 inches. The former is more likely correct.

Calculating Collector Diameter and Length

Generally speaking, header collectors do their best work at or below peak torque RPM. Adding collector volume typically increases torque in this range while reducing collector volume tends to diminish torque values to a degree. In discussions with Jim McFarland, he pointed out the sprint car trend of running headers with virtually no collectors to limit low-end torque, which can be detrimental on dirt tracks. Similarly, Pro Stock headers merely merge the primary tubes and don’t use a collector since they never see operation below peak torque.

Collector dimensions are, at best, difficult to pinpoint and there are no reliable formulas to accommodate all the variables, including large area changes and the amplitudes of reflected waves as they enter the collector. Motion Software’s Larry Atherton suggests that smaller collectors offer less area change, thus reducing the amplitude of the first reflected wave and encouraging mid-range power. Larger collectors generate stronger waves at the primary/collector interface and since this is closer to the cylinder it tends to increase top-end power. For the ballpark crowd, Atherton suggests the following formulas for estimating collector length and diameter:

Collector Diameter = 1.9 x primary pipe diameter
Collector Length = 0.5 x primary pipe length

These are ballpark formulas, but they work pretty well. We previously calculated the primary pipe diameter of John Kaase’s 403-ci Ford at 2.00 inches. Using this formula we calculate 1.9 x 2 = 3.8-inch diameter for his collectors. We also calculate a primary pipe length of 39.8 inches, which yields a 20-inch collector according to our formula.

Collector Diameter = 1.9 x 2 = 3.8 inches
Collector Length = 0.5 x 38.9 = 19.9 inches

In practice it appears that Kaase used a slightly smaller merge collector fitted to performance mufflers, so we can’t be certain. Still, you can follow this example and use the formulas to select collectors based on your particular input.

Food for Thought

Another point relative to primary pipe section area and peak torque RPM suggests the possibility of further broadening a torque curve by constructing headers with two different primary pipe diameters alternating in size according to firing order. Jim McFarland patented a header and intake manifold system using just such an arrangement. By arranging the smaller intake runners to correspond with the smaller c/s area primary pipes and similarly for the larger runners and primary pipes, it is possible to effect two torque peaks in the net curve, thus further exploiting the relationship between c/s area and peak volumetric efficiency (torque).

Exhaust System Wave Dynamics

It may be useful to some readers to contemplate the function of wave dynamics within an exhaust system. This brief explanation barely scratches the surface of a complex discipline, but its intent is to illuminate the prevailing thought as it applies to the action of finite amplitude waves operating within the exhaust flow. Finite amplitude waves are invisible pressure waves moving back and forth through an exhaust primary pipe completely independent of the flow of exhaust gas particles. There are two types: compression waves and expansion waves.

Compression waves are positive pressure disturbances with a pressure ratio greater than one relevant to ambient pressure within the primary exhaust pipe. Expansion waves have a strong negative pressure (less than 1).

These waves are a physical phenomenon traveling much faster than the gas particles in the exhaust stream. Because the exhaust system operates at much higher pressures than the induction system, these waves (or pulses) can be very strong, and they can exert considerable influence on the flow of exhaust gases. Positive pressure “compression” waves help push gas particles in the same direction that the wave is traveling, while negative pressure “expansion waves” help push gases in the opposite direction of their travel. This means that both waves can assist in the evacuation of exhaust gases from the cylinder.

Near the completion of a combustion event, the exhaust valve opens and a compression wave moves through the exhaust port into the header primary pipe. This high-pressure wave lends its energy to the outgoing gas flow and drives it toward the end of the pipe, which may or may not end at a header collector. It is the positive wave moving gas particles in the same direction of travel. When it reaches the end of the pipe, a negative pressure expansion wave of near equal intensity is reflected back up the pipe toward the cylinder. Since negative pressure waves push gases in the opposite direction of wave travel, the expansion wave also helps move gases toward the primary pipe exit. When this wave reaches the cylinder, it arrives with a significant drop in pressure. If, through careful tuning of the primary pipe length, it can be timed to arrive at the beginning of the valve overlap period (both valves slightly open at the same time), the low pressure will create a pressure differential (vacuum) in the cylinder which allows atmospheric pressure to push additional air and fuel into the cylinder.

This effect is called scavenging and it accomplishes three things. It assists the flow of outbound exhaust gases, it tugs on the incoming fuel charge to help initiate cylinder filling, and it helps purge the cylinder of residual exhaust gases. The timing of these events is critical and is largely dictated by engine speed and length of the primary pipe. The timing and intensity of wave action is controlled by primary pipe length and diameter. In the application of wave dynamics, primary pipe size is reflective of overall cylinder volume and the relationship of pipe size to the intensity of pressure waves. Large-diameter pipes generate lower pressures. Correspondingly lower positive pressure waves reflect lower amplitude suction waves that are less effective at scavenging residual exhaust and encouraging cylinder filling. Smaller pipes create higher pressures, which generate stronger scavenging waves but increase restriction and pumping effort. The correct balance can be effective depending on cylinder displacement (and corresponding exhaust volume), engine speed, and camshaft timing with regard to exhaust valve opening point and overlap period.

Primary pipe length dictates the timing of the suction wave’s arrival at the cylinder. With shorter tubes the scavenging wave arrives sooner and achieves tuning resonance at a higher RPM. The tradeoff is reduced cylinder blowdown due to the decrease in available time. This retains higher cylinder pressure and increases pumping work against the piston trying to empty the cylinder. Opening the exhaust valve earlier helps, but it forfeits cylinder pressure that could still be driving the piston. With early exhaust valve opening, larger pipes are required to optimize scavenging wave arrival time. Since these wave changes occur many times during a single exhaust event, they also become progressively weaker. There are counteracting forces at work here. Higher engine speeds require shorter pipes and earlier exhaust valve opening, but early opening requires longer pipes to optimize scavenging. A delicate balance is required.

The addition of collectors on headers further complicates the matter, but with potentially beneficial results. With no collector, the individual tubes create stronger suction waves that tend to peak within a narrow range of engine speed. Collectors improve performance and broaden the effective range by extending the width of the returning suction wave. Larger diameter collectors present a more direct path to atmospheric pressure and strengthen the front of the suction wave while smaller collectors tend to strengthen the end of the suction wave. The length of the collector controls the timing between the leading and trailing edge of the suction wave and can thus affect overall tuning across all of the pipes connected to it.

Because there are overlapping tradeoffs to wave tuning, other factors come into play. Equal-length primary pipes are well known to boost power by equalizing the work performed by each cylinder. But according to wave dynamics, they function best at or near one optimum speed depending on their length. Equal lengths often require more bends, which may increase restriction in some pipes, but wave dynamics infers that unequal-length tubes with minimal bending can effectively broaden the power range.

Of course this does not account for the presence of mufflers, catalytic converters, and convoluted tailpipes. Hence it more likely applies to racing applications where these elements are not present. Most race engine builders do not agree with this theory, but it further suggests that wave tuning action might be useful in crutching “bad” intake ports like those found on big-block Chevys. If the “poor” ports can be wave tuned (via the exhaust) to match the “good” ports, overall performance improves. There may be more of this type of tuning going on than is readily apparent, and whatever success it might provide is, for the most part, unreported because nobody’s talking.

Due to the complexity of tuning elements involved, wave dynamics is probably best left to professional racing efforts that are equipped and funded to pursue it. Still, the subject is fascinating, and armchair tuners willing to go the extra mile can study and apply it through the miracle of PC simulation. Motion Software’s Dynomation 5 Wave Action Simulator and Comp Cams’ DeskTop Dyno 5 are highly effective simulators that can lead the casual tuner to a higher level of expertise with regard to wave action tuning. Performance Trends’ Engine Analyzer Pro also incorporates wave tuning simulation to predict engine performance. If you’re a street enthusiast who just wants to make a good decision on a set of off-the-shelf headers, you should focus on the relationship between cylinder volume and torque peak RPM to choose your primary pipe size. But once you get a feel for wave action simulators you’ll really enjoy working with them.

Written by John Baechtel and Posted with Permission of CarTechBooks

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