This NASCAR Cup Series engine...
This NASCAR Cup Series engine has some pretty advanced designs in the intake and heads, designs that when you understand the theory behind them can benefit your weekly racing program.
This month, we'll discuss cylinder heads and intake manifolds, including some of their relationship with valve events. The specifics of valvetrain and valvetrain components will be covered in some detail in the next installment.
To begin, we should discuss some theoretical concepts before getting into the practical applications of heads and manifolds. The purpose of this approach is to help create a mental picture of what is taking place in the combustion space.
An internal combustion engine oxidizes fuel in a specific volume (combustion chamber). Oxidation is a chemical reaction which adds oxygen to create combustion (burning). This reaction is exothermic which means that the reactants (fuel and oxygen) produce heat and a byproduct--exhaust. The gases created from the combustion process are at high temperature and pressure, expanding against all exposed surfaces, but most notably the piston. The defining feature of an internal combustion engine is that useful work is performed by the expanding gases acting directly on the piston(s). It is this force that is transferred into actual work being produced (torque).
Internal combustion engines are heat engines. In other words, the heat released by burning (oxidizing) the fuel is converted into mechanical energy (torque and power). To increase the power output of an engine, we can increase the amount of useable heat transferred, improve the efficiency of the transfer process, or reduce losses associated with the system. To increase the amount of useable heat transferred to mechanical energy, we can increase the displacement of the engine. This has the affect of passing a larger amount of fuel and oxygen into the system (engine). Another possibility is to make the engine "think" the displacement is larger by increasing the mass of combustible products which pass through the system by making the ducting (intake and exhaust passages) more efficient.
Given the freedom to design...
Given the freedom to design an intake path, this is the approach you'd like to take. Line-of-sight passages minimize flow losses and maximize volumetric efficiency, particularly at high engine speeds.
An engine's ability to make power is directly related to how efficiently it is able to induct fuel and oxygen and rid itself of the by-products of combustion. The magnitude and balance of these fluid flows affect the amount and efficiency of the heat released by the fuel, during the combustion process. For this reason, assuming the engine is operating properly and within reasonable limits, more mass flow equates to more power.
Oxygen is needed to burn fuel. But, during the inlet cycle, air is what the engine inducts. Dry air contains 21 percent O2 (oxygen) and 79 percent inert gases. Typically, for a racing gasoline, the leanest air/fuel ratio for best torque is approximately 13.5:1. Therefore, for every pound of fuel consumed, 13.5 pounds of air must be ingested, although only 2.8 pounds of this supports combustion.
One cubic-foot of air at standard temperature and pressure (STP), assuming average composition, weighs approximately 0.0807 pounds. At STP, 13.5 pounds of air is approximately 167 cubic feet of air. If a Cup engine uses 170 lbs/hr of fuel at peak power (at 13.5:1 air/fuel ratio), that would equate to 2,295 lbs/hr of air. So, at standard temperature and pressure, 2,295 pounds of air would be 28,439 cubic feet per hour. This would be an astounding 474 standard cubic feet per minute. In this particular case, STP would be 60 degrees F and 14.696 psia. If you heated this air to a more reasonable (typical) inlet temperature, the volume would increase by the ratio of the absolute temperatures. In reality, this is an incredible amount of air to flow through a port.
Now let's get to something practical. No matter what you race, from a lawnmower to a Cup engine (for a given fuel type and if you want to make more power from a four-stroke spark-ignited engine and have limited resources), you want to work on improving the mass flow rate of fuel and oxygen (air) through the engine. The three areas to concentrate on are:
While these valves and head...
While these valves and head are off the NASCAR Spec engine you can see how they have been enlarged.
* Cylinder heads
* Intake manifolds
* Valve event timing (valvetrain)
From the previous theory section, it should now be apparent why gas exchange is important. It should also be evident that improving the efficiency of that exchange should help increase power.
Essentially, air is moved into the cylinder by the difference between pressure in the cylinder (during the induction stroke) and atmospheric pressure. When compared to the pressure in the cylinder during combustion, the negative pressure created by the piston moving down is not very large. Any losses in the ducts (ports) leading to the combustion space become significant. Because virtually any loss is important, considerable effort should be placed on improving the efficiency of intake ports.
Clearly, this F1 intake port...
Clearly, this F1 intake port displays the current trend in line-of-sight flow path layout. Considering the engine speed range in which these engines operate (in excess of 17,000 rpm), this approach to port design is virtually mandatory. Weekly racing engines utilizing smaller ports and lower rpm can benefit from the same practice.
In Cup racing today, very large intake valves are common. Most people think of valve sizes in terms of diameter, but a more important characteristic is valve area. Top Cup engines today have an intake valve which is between 26 percent and 29 percent of the area of the engine's cylinder bore. In these same engines, typical exhaust valve areas are between 50 percent and 59 percent of the intake valve area.
All Cup engines have intake valves that have a compound angle. This means the valve is "rolled" from vertical and canted toward the side. This is the arrangement pioneered by big-block Chevrolet ("porcupine") cylinder heads. It allows the valve to un-shroud itself from the cylinder wall, during the valve opening cycle. The intake and exhaust valves are also rotated (from a common line), and are therefore not aligned.
NASCAR mandates that all Cup heads do not have canted exhaust valves. If canted exhaust valves were legal, all manufactures would use this method. When the rules in a given series do not specify a cylinder head configuration, canted-valve heads should be considered.
The regulations for most circle track racers, outside of NASCAR's top series, require the use of small-block engines with traditionally inline valves and no side-canting. However, this valve layout may restrict the size (area) of the valves that can be packaged. Remember to consider the top ring position in the pistons, with relation to the valve pockets, when changing valve size.
Valve lifts of 0.900-inch are being raced in Cup today. As you suspect, large cylinder bores allow increased valve areas. Many times, it is not possible to run large valves, but is it possible to increase valve lift? Increasing lift only provides a benefit if the heads continue to be flow-efficient at high valve lifts. Don't overlook the fact some port configurations actually begin to flow less at very high lifts (see the baseline port in Chart 1). In this case, increasing valve lift may not have much effect on either flow or power.
A chart of this type is an...
A chart of this type is an excellent tool to compare different cylinder heads with existing valve events or to relate how different cam profiles work with a baseline head. The integrated area (area under the curve) is a good metric to compare different combinations. More area equates to more performance potential.
One way to understand the relationship between the valve events and air flow potential of the cylinder head is to plot air flow vs. cam or crankshaft angle. To generate this type of graph, make a table of airflow vs. valve lift. Next, create a table of cam or crank angle and valve lift. Table A shows an example of this type of table. Finally, calculate the airflow at each valve lift. Chart 1 is an example.
|Cam Angle||Net Valve Lift||Air Flow|
Valve Head and Throat Size
Intake air or exhaust gases see the valve head as a restriction. That portion of the port under the valve seat should have the smallest cross-sectional area in the port. This area is called the "throat." One trend in Cup that may improve the performance of your cylinder heads is to increase the throat diameter until the valve seat has just enough area to seat the valve. Increasing the back-angle on the valve, in conjunction with this technique, has proven to be effective.
This airflow "box" of an F1...
This airflow "box" of an F1 port package clearly displays line-of-sight design. While many circle track classes do not allow the freedom to modify otherwise "production" aftermarket racing parts to this extent, searching for ways to modify in this direction (where allowed) can lead to beneficial power gains.
Trends in Intake Port Design or Modification
With big valves and extremely high lifts, it is not surprising that the ports of today's Cup engines are quite large (in volume). These ports might be confused with drag racing-style ports from just a few years ago. In general, circle track rules tend to prevent ports from being raised to the extent that is legal in drag racing. However, one of the current trends has been to use a "line-of-sight" philosophy when creating the intake system. If you've been around the development of racing engines for very many years, this almost amounts to a "duh." Ed Winfield would be proud of this approach.
Line-of-sight, as it sounds, can be depicted by drawing the straightest line possible from the carburetor (or fuel injector) throttle plates to the intake valve. An intake manifold of this type is usually combined with large-volume ports that emphasize power at its peak and beyond. This is an excellent technique, when combined with a high rpm engine package that includes the appropriate camshaft. Typically, a line-of-sight porting technique produces shorter runner lengths and sacrifices a bit of torque at torque peak and below, unless the camshaft's valve events are adjusted to compensate.
Traditionally, circle track engine builders prefer smaller-volume ports to improve performance at lower engine speeds, enhancing throttle response. Today, a new technique is being used which has a slightly different philosophy. This technique is to make the intake valve large in area and the ports large in volume, to minimize any ducting losses, and uses a line-of-sight strategy for the intake track. In turn, this can improve low rpm performance when used with shorter and more aggressive cam profiles.
Such an approach achieves good low-speed torque, provided by the valve event timing and aggressive lobe profiles, and good power that results from the extremely efficient and large-volume ports that combine with the line-of-sight induction system. The traditional approach uses longer intake runners and smaller port volumes and longer valve events that are intended to build lower speed performance and improve peak power.
As viewed through the plenum...
As viewed through the plenum and toward its end, this small-block RO7 intake manifold includes a raised plenum floor that minimizes flow path angle differences between the cylinder head and manifold runners, thus improving the wet-flow approach to the head.
While both techniques achieve similar results, the new line-of-sight technique seems to generate a better torque curve and still make more peak power, while carrying the power past its peak. This type of intake system minimizes ducting losses in the intake system and minimizes air/fuel charge separation in the inlet track, when compared to the smaller-volume and bigger-cam approach. This seemingly small gain in efficiency makes a difference, because of the incredible flow rates required to support the power of today's Cup engines. It can work for your Saturday night applications, too.
The majority of the mass flow through the exhaust port comes at high pressure ratios (cylinder pressure vs. atmospheric pressure) at or near sonic flow conditions. Under these conditions, the exhaust gases flow off the back of the valve in a line (as straight as possible) to the exhaust port roof and then out of the cylinder head.
Like the intake, the throat of an exhaust port is the key to improving exhaust port performance. Funda-mentally, the exhaust port throat must be as large as possible for a given valve size. Exhaust port development (and flow) must be kept in balance with port design/modifications on the intake side.
Intake manifolds are the primary tuning component for four-stroke, spark-ignited engines. Inlet runners function as organ pipes. The optimized length of a runner will provide wave reinforcement (energy) at the correct moment of the engine cycle to increase the pressure in the cylinder, before the valve is shut. This "wave reinforcement" improves the filling of the cylinder (volumetric efficiency) and thus increases torque and power. Wave reinforcement occurs over a very narrow engine speed range. Longer runners tune at lower frequencies, like large organ pipes, and shorter runners tune at higher frequencies (or engine speed). The frequency of the tuning point directly relates to engine speed as follows:
* Longer runners "tune" at low engine speed
* Shorter runners "tune" at higher engine speed
Looking toward the side of...
Looking toward the side of this RO7 intake, you can see how the manifold's runners (the "short" or "inside" pairs) are elevated to come more into alignment with the cylinder head's intake ports. In such cases, it's desirable to have manifold runners that essentially become "extensions" of the intake ports...where possible.
Where rules allow fuel injection and open intake manifold design, the "trumpets" of the individual runners clearly resemble organ pipes whose lengths can be adjusted to provide desired power gains at the intended rpm.
Unfortunately, the geometry of common single-plane, single-carburetor inlet manifolds, and most circle track regulations, do not allow for radical changes in runner lengths. Depending on the rules for your particular series, modifications to the intake manifold can have a profound effect on the shape of the volumetric efficiency vs. engine speed curve.
With a common-plenum, single-carburetor manifold, the overall volume of the manifold correlates inversely with runner length. A larger volume manifold, when modified efficiently, will have shorter runners. A smaller volume manifold tends to have longer runners. Cup teams traditionally built one type of "open racing" engine and tuned the engine to the track with the intake manifold only!
If you don't believe the importance of a circle track engine's intake manifold, today's top teams may run a completely different valvetrain and cylinder head at short tracks than at tracks 1.5 miles and longer. Until very recently, the only difference between a Cup engine raced at Martinsville and one raced at Michigan was the intake manifold.
This fire breathing 410 sprint...
This fire breathing 410 sprint car engine runs on alcohol. Notice the organ pipe style injectors.
If your racing takes you to tracks you have not been to before, try a few different manifolds to see if one suits the track or track conditions better. Check the volume of your intake manifolds by filling them with water and try the smaller volume manifolds when you need to move the power down and larger ones when you need more high-speed power.
Here's one more point on intake manifolds. Airflow in an intake manifold is neither one-directional nor continuous. Remember this last statement; not continuous. Each time an inlet cycle begins, there is residual "energy activity" in the manifold. In fact, this is a subject unto itself. The point is that an engine's volumetric efficiency (as it affects torque output) is a function of how quickly and efficiently each succeeding cylinder can be filled, in the firing order. As the pressure differential builds between cylinder pressure and atmospheric (as influenced by engine displacement and rpm), the smaller the intake manifold's volume, the quicker it can contribute to v.e. Keep in mind that smaller volumes associated with this condition can become flow restrictions (and contributors to the mechanical separation of air and fuel) at higher engine speeds. You simply need to determine the specific rpm range in which the engine will be run and select intake manifolds accordingly.
Today, cylinder heads, intake manifolds and valve event timing are developed as a system. The intake track starts at the carburetor (or fuel injector manifold's base) and ends at the inlet valve. Cylinder heads and intake manifolds have almost become one word. The tuning of the engine is done by bringing port design, manifold design and valve event timing together. The extraordinary levels of performance of today's Cup engines can be directly related to the development of these separate systems as one whole--which, in fact, is an approach that can be applied directly to the weekly racer.
Next month, we will discuss push rod valvetrains, component by component, in addition to some applicable fundamentals about problems controlling valve motion and what happens when it isn't. We'll see you next month and--don't spin out.
Superflow's SF-1020 SuperBench...
Superflow's SF-1020 SuperBench is a top of the line flowbench used by some of the best teams and engine builders in the industry.
Some Thoughts on Airflow Testing
Since we devoted some space this month to comments on cylinder heads and intake manifolds, it seemed appropriate to provide a few thoughts on airflow testing, not focused on the more exotic computer-aided means by which such evaluations can be performed. Rather, we figured the Saturday night racer or engine builder could benefit from some "back room" perspectives on airflow bench use. Following are some suggestions to consider.
When flow benches first emerged in the performance parts and engine building community, longer ago than we'd like to admit, the axiom that "more was better" seemed the goal. However, it wasn't long until testers concluded that there's a definable link between an engine's pumping capacity (volumetric efficiency) and the volume of air it can ingest, based on a given piston displacement and engine speed. Concurrent to that realization was the importance of air flow quality, not just quantity.
Underscoring the importance of this notion was the fact air (in both intake manifolds and cylinder heads) is the means by which fuel is admitted to the combustion space. So, the ability of providing combustible air/fuel charges in the most efficient fashion was a function of airflow quality. In addition, air is compressible and (for comparative purposes) fuel is not. And, because they are of substantially different mass (in this case weight), air can change both direction and velocity much more easily. Sometimes this is good, sometimes not. Typical inlet paths can become very circuitous, even with the best efforts of either designing or modifying them otherwise.
Further adding to the complexity of "air as a means of fuel conveyance" is the fact inlet flow is both unsteady and interrupted during a given inlet cycle. Flow benches, obviously, cannot replicate these conditions. Adding a fluid to the system (wet-flow benches) can provide additional information and patterns with respect to how fuel is conveyed or altered from air inlet to combustion space, but the reality of all this is it's a compromise to actual engine running conditions. So the object is to design test methods that minimize the most inconsistent conditions first and then, at the very least, identifying those less inconsistent ones.
So what practical application does this entire dialog have to performing flow bench tests? Let's attempt some answers to that by trying to examine what you can and cannot get from such efforts. At the risk of oversimplification, suppose we try something to make a point. Visualize a given intake path from where air would enter the induction system to where the intake valve stem joins the valve head. Imagine a string positioned at the center of the air inlet, connected to this valve stem-point and pulled taut. In theory, this is the shortest flow path between these two points.
Keeping in mind that air tends to follow the shortest path, its movement will more closely approximate this distance than fuel. Our visualization is intended to emphasize that there are multiple opportunities for fuel and air to become separated between air inlet and combustion space. In a running engine, the object is to minimize the potential differences in actual (dynamic) flow paths between air and fuel. So the obvious question becomes, "How can you attempt to resolve this problem by using a flow bench?" This leads us directly to the flow quality issue.
By at least one definition, flow quality relates to the distribution of pressure (or pressure profile) at any given location in a flow path. Although basic physics utilizes certain Bernoulli principles to describe pressure distribution, let's simply state that this relates to where pressure resides throughout a flow path. In a wet-flow environment, the absence of a low-flow pressure condition is one circumstance that promotes the separation of air and fuel. Another circumstance that encourages mechanical separation of air/fuel charges is a sudden change in airflow direction or when eddies are created.
If you'll think back to our "taut string" example for a moment, you can visualize that in areas most remote from the string, you can expect low pressures to encourage puddling and further variations in air/fuel ratios along the flow path. Of course, none of these conditions are in support of delivering proper and consistent mixtures to the combustion space.
On a flow bench, the use of two types of probes can help identify improper or unwanted pressure conditions. One is a "velocity probe" that utilizes a small, open-ended tube (on the order of 0.030-inch i.d.), connected to a manometer. When pointed in the direction of flow (parallel to the flow), it can indicate flow rate. So-called "mapping" of port cross-sections, along the flow path, can be performed with such a probe. The other is a "J-probe" that consists of another small tube bent in such a way the hook in the "J" is pointed against the direction of flow. In this way, since atmospheric pressure is acting on the open ends of the manometer and probe, only pressures less than atmospheric will deflect the manometer. This type of probe is valuable in performing mapping similar to the velocity probe, differing in the fact its readings indicate areas of boundary flow separation and negative influence on mixture suspension.
You'll note that none of these last few paragraphs have mentioned flow quantity, only quality. Again, keep in mind that aside from its participation in the combustion process, air is the means by which fuel is either conveyed into the combustion space, helps qualify mixtures for combustion, or both-depending on the method of introducing fuel to an engine (carbureted or fuel injected). The point of this discussion is to not become locked into the need for simply increasing an engine's inlet airflow without a measure of concern for flow quality. Whether you're choosing or modifying intake manifolds or cylinder heads, keep an eye on the need for making certain air/fuel ratios delivered to the combustion space have been properly conditioned. And if you're on the dyno and curious about how much raw fuel is actually passing directly through the engine, try sampling unburned hydrocarbons (HC) using a rapid-response emissions instrument. Your observations could be quite revealing-and point you back toward the flow bench for some additional airflow quality measurements. Relying on brake specific fuel consumption data isn't the only way to get a reading on combustion efficiency.
It's pretty easy to tell from...
It's pretty easy to tell from the excessive carbon deposits that this engine was running too rich.
On Reading Combustion Patterns
Since cylinder heads were on the discussion list this month, it seems appropriate to devote some space to the value of reading combustion patterns. In reality, it is from these tell-tale signs that you can determine much about an engine's combustion efficiency-or a lack thereof.
It has previously been suggested that besides participating in the combustion process, air is the means by which fuel enters the combustion space. Poorly conditioned, airflow quality can lead to a wide range in air/fuel ratios when the burn begins. Aside from the fact an advancing combustion flame can alter ratios ahead of its progress, variations often occur prior to ignition. It is these that an engine builder (or parts designer) can address to improve net power. But first, you need to identify where the problems lie, and then take corrective steps.
Two fundamental surfaces can provide clues to such problems. One is the combustion chamber, the other on piston crowns. With respect to the former and more than 60 years ago, Sir Harry Ricardo advanced the importance of maintaining "turbulence set up by gases during their entry before combustion." While this concept is important in a high-speed racing engine, the value of keeping fuel properly atomized and suspended, prior to combustion, can be overdone by allowing excessive velocity or turbulence to mechanically separate fuel from air. This leads to unwanted extremes in air/fuel ratios and lost power. The evidence of such separation can be "read" on both combustion chamber walls and piston tops. You know of these as "washed" or clean areas which suggest combustion was not present on or around these surfaces. One quick look at the backside of a small-block Chevy combustion chamber (the intake side of the spark plug location) will verify what we're suggesting.
Sooty or heavily-carboned surfaces, particularly in gasoline-fueled engines, is an indication of fuel-rich combustion, often associated with locations of lesser airflow activity prior to combustion. You may also find this type of combustion residue located on the down-side of valve notches or protrusions on piston crowns and often linked to sharp edges that create vortices (low pressure points) and collected fuel.
When you consider peak instantaneous inlet flow velocity in a 5.7L engine turning 7,000 rpm can exceed 400 ft/sec (when flow directional changes occur), there's ample opportunity for fuel to separate from its "carrier" air. As a result, any steps you can take to help ensure fuel is maintained in suspension clearly points to the potential for increased power.
Admitting the approach is more subjective than scientific, you can actually gain insight into an engine's efficiency by studying combustion patterns. Look for extremes in residue color. These can indicate a range of combustion activity from overly-rich mixture "zones" (dark) to no appreciable combustion at all (clean or light in color). When accompanied by an engine's apparent need to require an inordinate amount of spark timing (to optimize power) or what appears to be a requirement for excessive fuel, you can also look for air/fuel mixture quality problems.
Further adding to this problem, data suggests an engine's air/fuel mixtures will often vary in ratio from the point of fuel admission all the way into the burn space, whether carbureted or fuel injected. Both these methods of delivery require that attention be given to the ability of air to transport fuel in its most efficiently combustible form, up to and during the burn. At no point in the process can you assume that initially creating the proper air/fuel ratio will ensure it'll remain at that proportion until combusted.
In the end, it's a matter of the amount and distribution of kinetic energy present and characterized in the inlet stream and combustion space. Making certain that lows and highs in that energy content are minimized, abrupt changes in flow direction are eliminated or reduced, wet-flow surfaces are not made too smooth ("dead" boundary layers can lead to separated air and fuel), working toward increased power from reduced spark timing, exploring best power from minimal fuel flow and spending time examining the "evidence" from combustion can all lead to improved on-track performance. Sometimes, it's proper reading of the evidence that leads to successful completion of all the other objectives mentioned.
Torque Peak vs. Flow Passage Size
Essentially, this is a subject almost unto itself. However, since we're discussing port size fundamentals and their relative effect on engine performance (torque output), a brief but general discussion seems appropriate.
Here's the deal. Even though intake and exhaust flow is considered "unsteady" and "bi-directional" (at times), it is possible to determine what we'll call "mean flow velocity" (m.f.v.) at a given time (rpm). Simply stated, there is a mean flow velocity that's measurable at any rpm point, specifically at peak torque. Further, it turns out this m.f.v. is virtually the same for any engine at its peak torque rpm. It's a combination of conditions involving fluid dynamics and acoustics.
From a parts selection or component design standpoint, peak torque rpm can be influenced (sometimes significantly) by passage size or cross-sectional area. If you subscribe to the concept that a common m.f.v. will exist among engines at their peak torque rpm, then increasing passage area raises this rpm point and decreasing area lowers the point. While it's true that passage length affects torque output (long vs. short), this dimensional change typically tends to "rock" or see-saw a given torque curve about the rpm point that we'll call the "critical" flow velocity at peak torque. You've seen this characteristic when lengthening or shortening header pipe lengths. Short ones tend to add torque above the peak, longer ones increase torque below peak.
Rather than detract from the core text of this month's Series segment, suffice to say that the "critical" m.f.v. is on the order of 245 ft/sec and is largely governed by rpm, piston displacement and passage cross-section area (intake or exhaust). If you want to test this theory from a practical standpoint, do this: Study a dyno sheet to determine the peak rpm torque point. Even though this point is influenced by factors other than (for example) header pipe section area, the exhaust and intake passages are each major contributors to the determination of peak torque rpm.
Calculate the primary header pipe section area, using pipe i.d. in the calculation for area. For the sake of discussion, if you happen to be using 1.75-inch primary pipes of 0.040-inch wall thickness, this computes to an area of 2.19 square inches.
Next, determine the volume of one cylinder. If the engine displaces 350 cubic inches, one cylinder's volume (for purposes of this calculation) would be 43.75 cubic inches. Now, multiply 2.19 x 88,200 (a mathematical constant) and divide the answer by 43.75, and you'll have determined that these headers should reach their "critical" flow velocity at 4,416 rpm. Installing headers of 2.0 inches o.d. (still of 0.040-inch wall thickness) and making no other changes would yield a cross-section area of 2.895 and, using the same method of calculation, shifts the torque peak rpm upward to 5,837. Again, lengthening or shortening the primary pipe length simply adds (or removes) torque below or above this rpm point.
Now, those of you mathematically inclined have already seen that these dimensional and results relationships can be transformed into a rather simple algebraic equation that looks like the following:
Torque Peak (rpm) = (88,200) x (Passage Area) / Displacement of One Cylinder
Although not as accurate as some computer models that take more functional elements into account (including valve motion characteristics, mechanical compression ratio, cylinder pressure histories, etc.), this simplistic approach can reveal why an engine's torque output may vary (in both quantity and placement) throughout a given rpm range, when contemplating or changing exhaust systems. Given this equation and through simple transposition of terms, you can select an rpm at which you'd like a torque (volumetric efficiency) boost and solve for the passage area required to produce that increase (at that point). It's a pretty useful tool.
And, interestingly, you'll find that this approach can also be applied to intake systems (manifolds in particular) when you're either selecting parts or evaluating dyno (or track) performance of a particular engine package. If personal experience has any measure of value, I've included this method in both the initial and in-progress design of intake and exhaust systems for a number of years. It's a pretty good way to perform evaluative and directional steps, quickly.