Archive Page 2

The Variable Valve Timing and Lift Electronic Control (VTEC) System

     We’ve all heard the acronym “VTEC” thrown around constantly in the world of cars.  So much so that “VTEC” has become massively misused over and over again so that it is used now with sarcasm.  But here I’m going to clear up all the misunderstanding and explain the world of valvetrain technology.

     For those a bit less invested in the world of auto technology, I’ll explain the valvetrain for you a bit.  The valvetrain is what controls what goes in and out of the cylinder (intake and exhust).  It consists of the camshaft, the rocker arms, and of course the valves.  The camshaft is a long rod rotating along with the engine running along the length of the engine.  It has a few protrusions (the cam profile) coming out of it which are intended to push a set of rocker arms which then push open the valves.  The valves are the “doors” that let the air and exhaust enter and exit the cylinder during  the four-stroke combustion cycle.

     For a small history lesson; Honda brought the VTEC system to the masses in Japan in 1989 with their Intergra and CRX SiR models.  The U.S. was introduced to it in 1991 with the arrival of their everyday supercar NSX which produced 290 horses with only 3.0 liters of displacement.  Honda then installed their new tech into the Type R versions of their B16 and B18 engines for an easy 100 plus hp per liter.  The B16B built specifically for the Civic Type R boasted an impressive 115.8 hp per liter in a relatively affordable car.  The VTEC system culminates in the S2000, which is the only car to use their infamous F20c engine.  It produces 240 horsepower with “only” 2.0 liters of displacement and no forced induction.  Introduced in 1999 to celebrate the new millenium, the S2000’s F20c still has the highest specific output for a naturally aspirated 2.0 liter car under $100,000.

     Let’s get down to the tech side here.  V-T-E-C stands for Variable valve Timing and lift Electronic Control.  In english, VTEC (and all other variable valve technology) modulates the valves of each cylinder with a computer (just like everything else in this world).  VTEC doesn’t modify the valves directly, but rather the camshafts that the valves are connected to.  As the engine turns, the camshafts turn in relation and they push the valves open for the cylinder.  What VTEC does is change both the duration and lift of the valves as engine speeds change by using different cam profiles.  For example, for maximum performance at high RPMs, both valve lift and duration should be relatively longer.  But for drivability, comfort, fuel consumption, and emissions, valve duration and lift should be relatively minimal.  So for a minivan, larger cam profile is not needed, but for a performance-oriented car that is driven both in traffic on the road and on the track, the best of both worlds are needed.  If you want shorter duration and lift, you want your profiles to be smaller, since they won’t push open the valves as far and for as long.  If you want longer duration and lift, you obviously want a larger profile.  How do we get both smaller AND larger profiles onto one camshaft?  For our example, let’s imagine two cam profiles pushing two rocker arms to open two valves.  Here’s a simple diagram:

“3_a1.gif.” Image. Honda Motor Co. “The VTEC Breakthrough: Solving a Century Old Dilemma.” Honda Worldwide. N.p., 2009. Web. 3 Dec. 2009. <

     What Honda decided to do was to build a camshaft with two cam profiles and the ability to switch between them.  When the time comes to switch from low-RPM comfort mode to high-RPM performance mode, the camshaft makes a small change and switches to its longer duration and lift cam profile.  And then the car slows back down to lower-RPM driving, the camshaft profile switches back.  So how does Honda change the cam profiles on the fly?  Quite simple actually.

     What VTEC does is install another rocker arm (let’s call it the VTEC rocker) between the two rocker arms, and a larger cam profile to operate this larger rocker arm.  This larger cam profile would obviously push the valves open for a longer duration and lift.  The larger cam profile and rocker doesn’t push anything open as there aren’t any valves underneath it, but are just situated between the smaller rockers.  Like here:

“3_a1.gif.” Image. Honda Motor Co. “The VTEC Breakthrough: Solving a Century Old Dilemma.” Honda Worldwide. N.p., 2009. Web. 3 Dec. 2009. <

     During low-RPM operation, the valvetrain works normally; ie. the smaller profiles push the smaller rocker arms which then push open the valves.  But when the engine enters its higher-RPM range, a hydraulic system pushes a pin that connects the VTEC rocker with the two rockers beside it.  When that happens, the larger VTEC cam profile pushes all three rockers even further due to the fact that the VTEC rocker arm is now connected to two non-VTEC rocker arms.  When the larger cam profile isn’t needed anymore, the pin disengages and unconnects the three rocker arms, reverting back to the smaller cam profile.  An animation for your entertainment:

“cam_a07.swf.” Animation. Honda Motor Co. “The VTEC Breakthrough: Solving a Century Old Dilemma.” Honda Worldwide. N.p., 2009. Web. 3 Dec. 2009. <

     It’s quite a simple mechanism actually, just a simple trigger (a certain RPM), a hydraulic system, and an extra set of cam profiles and rockers.  The problem with VTEC is that it is an “on/off” switch, where the cam profiles are either small or large.  The next step is to continuously modulate the cam profile for any and every driving condition.  BMW introduced its “Valvetronic” system in 2001, which Toyota followed with its “Valvematic” system and Nissann with its VVEL (Variable Valve Event and Lift) system.

     Well, that’s basically how the VTEC system (and most variable valve systems) work.  Leave a comment if you’d like.

     //Drive on.

Vroom = Math?


     As a society, we find assurance in math due to their binary nature.  It’s either right or wrong.  There is no “I prefer”, or “I believe” within math, and that makes it so much easier (to a certain extent).  Everything can be reduced to math (mostly).  And math (to me) makes everything easier to understand.  It gives us a certain way to relate, compare, and think.  When you look at a car on the road, I doubt the first thing that pops into your mind is “MATH”, or “I wonder what the exact numbers are behind it”.  But if you want to understand how something works, using math is one of the best ways to go about it.  Just like how being able to teach a subject means you truly understand it, its the same for math and physics/engineering/worldly dynamics.  We might be able to explain “why” a part works using words, but to understand it using math is completely different.  From the chassis to the engine to the suspension, every part is designed using math.  And if we can figure out how each of these parts was designed with math, we can pick it apart and eventually understand it better.  In a world of ebb and flow, of constant change and infinite unknowns, the basics of math keep us grounded in reality and truth. 

     This is why I ultimately delve into the math behind cars: to critically define, compare, and ultimately, understand them.  Because in a way, to understand something is to enjoy it.

     //Drive on.

Brake Horsepower vs Flywheel Horsepower

     Horsepower is measurable, meaning that there are numbers involved and can therefore be compared.  The problem with measuring horsepower is it isn’t always measured in the same way.  When we add two with seven, we know there is only one definition for “two” and “seven”.  But when talking about cars, there are infinite unknowns which, as variables, we can’t always eliminate.  While 500 ponies is greater than 499 ponies, the problem is how both horsepowers are measured.  Differentiating between flywheel horsepower (engine horsepower) and brake horsepower (power to the wheels) definately helps, but it’s still not perfect, but I’ll leave that to another post.  There is also the problem of not being able to accurately convert between the two (flywheel and brake horsepower).  Flywheel horsepower is usually between 10% to 20% greater than brake horsepower due to drivetrain losses, but what does that really mean?

     When we  change parts to make our car faster, we usually use horsepower to determine whether it was successful/useful/worth the money or not.  We say that we’ve “gained” or “added” 10 horsepower to our engine/car, but that’s not always correct.  What we measure when we dyno our cars to find the horsepower is the power that’s conveyed to the wheels.  So from releasing the energy from fuel via combustion to sending them to the wheels to increase our velocity, there’s a bunch of stuff in between.  What the dyno measures is just the end product (the energy sent to the wheels).  We know there are various ways to make our car “faster” or “quicker” (there is a difference), but sometimes the only way we measure that is through horsepower.  Let me explain.

     One of the easiest ways to make our car faster is to decrease the weight of our wheels, since unsprung weight (or mass) has a larger effect on acceleration than sprung weight (or mass).  It takes more energy to rotate a mass than to push it linearly.  So if we’ve installed lighter aftermarket wheels and take it to a dyno to measure the effects, let’s say that we see an increase in 2 horsepower.  Now that’s not alot, but it’s still measurable through the dynometer, and thus we can make comparisons.  Lighter wheels = more power, right?  Not exactly.  Nothing has changed in both the input and output of the engine whatsoever.  Your engine still produces the same amount of power it did before you replaced your wheels. 

     But how does that make sense if our dyno clearly shows an increase in horsepower?  And an exact 2 hp at that?  An increase in brake horsepower doesn’t always equal an increase in engine horsepower.  And the engine is the only place to produce horsepower, not the wheels.  This is a direct example of that.  We can’t always convert our brake horsepower to flywheel horsepower just to fluff our power numbers.  Reducing the weight of our wheels does NOT make our engine produce more power.  The same amount of air is fed into the cylinder and the same amount of fuel is burned through combustion.  There is no change within the engine, nor is there any change from the input and the output of the engine.  The flywheel horsepower stays the same, but what has changed is the rotational losses of when the energy of the fuel is being sent from  the engine to the pavement (the brake horsepower).  And this is what seperates flywheels horsepower from brake horsepower.  Brake horsepower measures the power being put to the ground, while flywheels horsepower measures the power the engine produces.

     So while our dyno says there’s an increase in horsepower the engine doesn’t see any change.  Thus, while we haven’t “gained” any power, we’ve “freed” or “unlocked” some power through weight reduction.  A quick example of rotational losses would be the driveshaft, clutch, flywheel, brakes, tires, wheels, etc, anything that the engine has to turn while transmitting the power to the ground.  Reducing the weight of any of those parts will reduce the amount of power sapped before the engine power reaches the ground, no matter how small or large the weight decrease.

     So the next time you dyno your car, take care when converting your brake horsepower into flywheel horsepower.  The best thing is to just leave your brake horsepower as is, and to not convert it.

     //Drive on.

How to Go Fast Faster: The Math Behind Turbocharging. Part 8b: The Result


Part 1 is here.

Part 2 is here.

Part 3 is here.

Part 3b is here.

Part 4 is here.

Part 4b is here.

Part 5 is here.

Part 5b is here.

Part 6 is here.

Part 7 is here.

Part 7b is here.

Part 8 is here.

     After doing all that math to piece together the most efficient turbo system for the C30A engine for the Acura NSX, Mr. Bell goes on to verify and test his math with actual data.  Keep in mind that a twin-turbo setup was used, with one intercooler for each turbo.  The aim was around 375 bhp, an increase of about 36% power over stock (275 bhp).  The engine’s already been tuned and broken in, so no worries about that.  Let’s take a look at the end result.

     First we’ll take a look at the compressor and the Compressor Efficiency to see how well it’s performaing.  If you remember from our previous post, to calculate our Compressor Efficiency we need the Ambient Temperature, Pressure Ratio, Compressor Inlet Temperature, and Compressor Outlet Temperature.  For our Pressure Ratio, we’ll use the equation involving the Boost Pressure since that’s what we’re given:

Pressure Ratio = Pressure Ratio

Pressure Ratio 1.41

Here’s the rest of the information in a table:


     We see three sets of runs at around 100 °F ambient using engine speeds of 4,000 and 6,000 RPMs at 6 psi of boost.  Using these numbers, we can calculate for the Compressor Efficiency and then compare that to the theoretical compressor efficiency of the compressor map.  Since there are three sets of runs, I’ll calculate the compressor efficiency for each set, and then find the average:

Compressor Efficiency (%) = Compressor Efficiency

Compressor Efficiency Run 1 (%) ≈ 77.4%

Compressor Efficiency Run 2 (%) ≈ 74.0%

Compressor Efficiency Run 3 (%) ≈ 74.3%

Average Compressor Efficiency (%) ≈ 75.2%

      As a reminder, the compressor map for the Model 128 ,which we installed, peaked at an efficiency of  76% when running at 60% load.  And I just calculated for an Average Compressor Efficiency of 75.2% over three seperate runs at both 4,000 and 6,000 RPMs.  This is very encouraging, as our end result was less than 1% off from the theoretical compressor efficiency.

     Looking at the table, we can see a small downward trend in efficiency from the first run to the last one due to the increase in heat.  Obviously, with the first run, the engine hasn’t been pushed yet, so is quite cool.  But by the 8th run, we can see that the efficiency has dropped from 78.4% and evened out around 74.4%.  With just this engine, from a change of 6°F, we get a decrease of about 5% or so in efficiency.  I thought there would be a noticible difference between the 4,000 and 6,000 RPM runs, but there’s not enough data to support this.  The first two sets of runs showed a decrease in efficiency when raising the engine load from 4,000 to 6,000 RPM of either 2.1% or 1.1%, but the third run actually showed an increase in efficiency of .3%.  By now, you can see I’m over-analyzing the results, but this is interesting nontheless.  Let’s move

      As for the turbine side, Mr. Bell attached a pressure gauge to a fitting on the turbine inlet and recorded a boost pressure of 15 psi, with a maximum of 15.5 psi.  An efficient turbine which produces minimal backpressure has a turbine inlet pressure anywhere between 2 to 3 times the boost pressure.  In our case that would be anywhere between 12 to 18 psi, so our 15 psi is good enough for good low-speed response at the cost of a tiny increase in exhaust backpressure.  He states that increasing the turbine size wouldn’t be worth the increase in turbo response, and is thus happy with his turbine selection.

     Now let’s take a look at the intercooler and how efficient its running.  We’re aiming for the maximum temperature drop while having a pressure loss of less than 1 psi.  Unfortunately, the pressure loss was recorded at a “tick over 1 psi at 7700 to 7800 RPM.”  Mr. Bell was a bit disappointed, but nevertheless stuck with his selection.  As for the temperatures, here’s a table of the data collected:


     Once again, these runs were done under 100 degree weather.  He made 4 runs here at 4,000 RPM.  Now let’s plug in the numbers to find our Intercooler Efficiency:

Intercooler Efficiency (%) =

Intercooler Efficiency Run 1 (%) ≈ 83.1%

Intercooler Efficiency Run 2 (%) ≈ 80.6%

Intercooler Efficiency Run 3 (%) ≈ 81.7%

Intercooler Efficiency Run 4 (%) ≈ 81.6%

Average Intercooler Efficiency (%) ≈ 81.8%


     This gives us an intercooler which operates at an efficiency of around 82%, around 3% less than the 85% Mr. Bell used earlier.  He seemed a bit dissapointed at the 1 psi loss of pressure through the intercooler, but he expected this from a street-application intercooler.

     And finally, the result of all our math.  First the logged stock performance data:

  • 0-60: 5.7 seconds
  • 1/4-mile time: 14.0 seconds
  • 1/4-mile speed: 101.0 mph
  • Power: 268 bhp

     And the end result:

  • 0-60: 4.7 seconds
  • 1/4-mile time: 13.0 seconds
  • 1/4-mile speed: 111.5 mph
  • Power: 390 bhp

     Shaved off one second from the average 0-60 time.  Not bad for a fully CARB legal turbo system for the streets.

     This concludes our section on the math behind turbocharging.  The next section will be about engine management tuning.  Stay tuned.

How to Go Fast Faster: The Math Behind Turbocharging. Part 8: An Example


Part 1 is here.

Part 2 is here.

Part 3 is here.

Part 3b is here.

Part 4 is here.

Part 4b is here.

Part 5 is here.

Part 5b is here.

Part 6 is here.

Part 7 is here.

Part 7b is here.

     By now we’ve covered most of the basics of designing your own turbocharger kit, as well as some other random stuff I decided was relevant using a whole lot of math and graphs.  This will probably be the last post of this section where I’ll provide a full example using most, if not all, of the steps using Corky Bell’s Maximum Boost: Designing, Testing and Installing Turbocharger Systems.  It’s in Chapter 17 and starts on page 196 if you want to double check/follow along.  On with the show.

The *< and >* denotes parts which were not included in the chapter.   I have taken the liberty to add them in myself and do the math to show an example for each of the equations I’ve covered.

     We’ll be using the Honda/Acura NSX as our base car, which uses the 3.0 liter C30A engine designed specifically for this supercar.  I’ll provide a full in-depth look for the NSX later in the Spotlight section, but for now all you need to know is the engine specs.  It produces around 275 horsepower and 210 pounds of torque to the crank in a compact V-6 form.  It was also the first car in the U.S. to use their revolutionary VTEC system, creating a very linear and flat torque curve for both low-RPM and high-RPM driving.  It maxes out at a stratospheric 8,000 RPM due to its lightweight internals, but the maximum power output was a bit disappointing for a supercar.  Here’s a rundown of the stock performance measured by Mr. Bell:

  • 0-60: 5.7 seconds
  • 1/4-mile time: 14.0 seconds
  • 1/4-mile speed: 101.0 mph

     And this is where a turbo comes into play.  Using a turbo, we will give it more juice yet keep it street-legal and CARB legal, meaning the original catalytic converters stay put and emissions must stay within CARB limits.  Let’s go.

     Mr. Bell decided that in order to keep the C30A’s flat torque, one huge turbo wouldn’t work out since it wouldn’t provide enough power down low.  So it came down to either a sequential or parallel twin turbo setup.  Sequential turbo systems are very complicated, so that was elimanted.  That leaves us with a twin-turbo setup, splitting the work between two smaller turbos and thus giving us the low down torque we want. Our end goal is around 375 ponies, so let the math start.

     First the we want to find the Performance Gain (%).  Our Desired Bhp is obviously 375, while our Stock Bhp is 275.  Let’s plug it in:

Performance Gain (%) = Performance Gain

Performance Gain (%) = 36%


     Using our Performance Gain, we can now solve for the Boost Required:

Boost Required (psi) = Boost Required

Boost Required (psi) ≈ 5.3 psi


     So we’re aiming for a 36% increase in horsepower.  First let’s make sure our Fuel System is up to the job of our increased power.  Our Maximum RPM for the C30a is 8,000 RPM.  Plug it in:

Time of One Revolution (msec) = Time of One Revolution

Time of One Revolution (msec) = 7.5 msec

     Time to see if we have enough headroom to increase our Injector Pulse Duration.  The Stock Pulse Duration was measured at 5.0 msec, so plug that in along with the 7.5 msec from our previous equation:

Available Increase (%) = Available Increase

Available Increase (%) = 50%

     Our injectors have the extra time to cope with the extra airflow, but let’s check to see if our fuel pump can handle the extra Fuel Pressure:

Fuel Pressure Required (psi) = Required Fuel Pressure

Fuel Pressure Required (psi) = 83 psi

     According to Mr. Bell, the stock fuel system can handle the increased Fuel Pressure.  So now that our fuel systems are sorted out, let’s move on to our actual turbo selection.

     First up is the Pressure Ratio, which is simple:

Pressure Ratio = Pressure Ratio2

Pressure Ratio = 1.36


     Now that we have that, let’s move onto our Airflow Rate.  We’re first going to need to convert our Displacement from Liters to Cubic Inches using this equation:

Displacement (in3) = Displacement Converter

Displacement (in3) = 183 in3

     With our converted Displacement, we can solve for our stock Airflow Rate while using a Volumetric Efficiency of 90%:

Airflow Rate (cfm) =

Theoretical Flow Rate

Airflow Rate (cfm)381 cfm


     Then plug in our 8,000 RPM, Pressure Ratio of 1.36, Volumetric Efficiency of 0.90, and our 183 in3 of Displacement into our Turbo Airflow Rate equation:

Turbo Airflow Rate (cfm) =

Turbo Airflow Rate2

Turbo Airflow Rate (cfm) ≈ 519 cfm

     Since we’re going with a twin-turbo setup, only 260 cfm will flow through each turbo at Maximum RPM (8,000 RPM).

     With our Pressure Ratio and Turbo Airflow Rate figured out, we can start looking at compressors and compressor maps.  Corky Bell used two examples here, the Aerocharger model 101 and model 128:

Model 101:


”Fig. 17-10.” Chart. Maximum Boost, Designing, Testing and Installing Turbocharger
Systems. By Corky Bell. Cambridge, MA: Bentley Publishers, 1997. 201.

Model 128:


”Fig. 17-11.” Chart. Maximum Boost, Designing, Testing and Installing Turbocharger
Systems. By Corky Bell. Cambridge, MA: Bentley Publishers, 1997. 201.

     Fortunately for us, Mr. Bell has already drawn the Pressure Ratio line.  All we have to do is read the map and decide between the two.  Model 101’s maximum efficiency is a nice 78%, but as our engine speeds increase, it drops off pretty quickly into the low 50s.  Model 128, on the other hand, although its maximum efficiency is less (at 76%), its maximum efficiency is closer to the middle of the torque band, and doesn’t drop off as badly in terms of efficiency.  Mr. Bell decided here to choose the Aerocharger compressor Model 128 over the 101 due to its greater efficiency at maximum load instead of the one with just the highest maximum efficiency.


     As for the turbine side, let’s solve for the Exducer Bore (in) diameter:


”Fig. 3-10.” Chart. Maximum Boost, Designing, Testing and Installing Turbocharger
Systems. By Corky Bell. Cambridge, MA: Bentley Publishers, 1997. 31.

     Using our Turbo Airflow Rate of around 260 cfm for each turbocharger, we should use an Exducer Bore between 1.5 and 2.0 inches.  While Mr. Bell doesn’t go into details about the Turbine Selection, he does settle for an Exducer Bore diameter of 2.0 inches.


     Now we’re going to calculate the Temperature Rise at Maximum Compressor Efficiency using the Compressor Efficiency equation moved around a bit.  We’ll be plugging in an Ambient Temperature of 80°F and a Compressor Efficiency of 76%:

Temperature Rise at Max Compressor Efficiency (°F) =

Temperature Rise at Maximum Compressor Efficiency

Temperature Rise at Max Compressor Efficiency (°F) = 63°F

     We have the Temperature Rise when the compressor is at its maximum efficiency, but we need to also calculate for the Temperature Rise at Maximum Load, which is 8,000 RPM in our case.  Calculating for the Maximum Load will always give you the highest Temperature Rise, since that’s when the engine is working its hardest.  We need the Temperature Rise from before, and the highest efficiency and lowest efficiency percentages which the Pressure Ratio line passes through (on the Compressor Map).  Here’s the equation:

Temperature Rise at Maximum Load (°F) = Temperature Rise at Maximum Load

Temperature Rise at Maximum Load (°F) = 79°F

     Now onto intercooling.  Using the last answer, we can actually solve for the Temperature Post-Intercooler (°F) due to the heat removal (use °F as your unit here, not absolute):

Intercooled Temperature (°F) =

Intercooled Temperature (°F) = 13°F

     We can then use that to find the Intercooler Gain (%) by using an Intercooler Efficiency of 85% (an assumption), the previous Ambient Temperature of 80°F, and the Intercooled Temperature of 13°F:

Intercooler Gain (%) =

Intercooler Gain

Intercooler Gain (%) = 11.9%

     Now to figure out the dimensions for our intercooler.  This starts with finding a suitable Internal Flow Area (in²) for our output.  We’ll use the equation instead of the graph here:

Internal Flow Area (in3) = Internal Flow Area

Internal Flow Area (in3) ≈ 23 in2

     With that, we can find the Area of the Charge Air Face, or the “top” of the intercooler:

Area of the Charge Air Face (in2) = Area of the Charge Face

Area of the Charge Air Face (in2 51 in2

     So if we use an intercooler with a 3 inch depth, our width would be around 17 inches.  With a 2 inch deep intercooler, it would be around 26 inches.  We then have the problem of finding a good place to put the intercooler so it gets fresh air.  Mr. Bell found a suitable area just behind the rear wheel well, with enough spacec for an intercooler core measuring 3.5 by 9 inches.  It was enough to split the intercooler into two cores, one behind each wheel well using a 3 inch deep intercooler while satisfying our specifications of a 3 by 17 inch intercooler.


     Moving onto the Intercooler Piping, let’s plug in our Turbo Airflow Rate of 260 cfm (since each turbo will have its own intercooler) and use a Velocity of 440 ft/sec, or Mach 0.4:

Pipe Diameter (in) = Pipe Diameter

Pipe Diameter (in)  0.90 in


     And there we go.  We’ve solved for the “best” or most efficient in our case, turbo setup for a C30A engine from the Honda/Acura NSX.  Now since this book was published a while ago (1997), I’m sure there are a slew of new and (maybe) better parts to turbocharge this engine.  This is just an example of what you can do.

How to Go Fast Faster: The Math Behind Turbocharging. Part 3b: Brake Specific Fuel Consumption (BSFC)


Part 1 is here.

Part 2 is here.

Part 3 is here.

*EDIT: Sorry this is out of order.  I pushed every post up since I had to squeeze this post in between Part 2 and Part 4.  I expanded the fuel system section quite a bit and decided it needed its own post. 

     In the last post I explained bits about the Electronic Fuel Injection (EFI) system and how to modify it to cope with the upcoming turbo power.  Near the end, I introduced the idea of the Brake Specific Fuel Consumption (BSFC) to estimate the amount of fuel the engine will need per hour.  Here I’ll expain a bit more on the BSFC and how it affects our cars.

     So I’ve already defined the BSFC as the amount of fuel the engine requires to produce one horsepower per hour, but I didn’t specify a unit.  We can figure that out by looking at how to calculate the BSFC.  You’ll only need two variables: the Fuel Rate (Lb/Hr) and the Power (Hp).  Here is the equation:

BSFC (Lb/Hp*Hr) = BSFC

Just divide the Fuel Rate by the Power.  And if you use only the units, you’ll see that the unit for BSFC is Lb/Hp*Hr.

     When you take a look at the equation, it’s very simple.  We’re just multiplying two numbers.  But if you start comparing the two variables, Fuel Rate and Power, you realize that you can never have an exact BSFC for an engine or car.  Why?  Because the BSFC is the rate of fuel consumption over one hour of operation.  What if for the first hour you drove mildly on the highway, and then the second  hour you were sitting in bumper to bumper traffic?  The two Fuel Rates would be completely different.  As would the Power as well.  When we’re sitting in traffic, we’re not using and power at all.  And when we’re cruising on the highway, I doubt you’re using all your engine’s power for a full hour.

     So the problem becomes the use of BSFC if we can’t calculate an exact number for an engine.  Well, we can get a range of BSFCs for engines.  For example, in our last post I used 0.65 as a safe number for a turbocharged engine.  Most turbo’d engines run between 0.6 and 0.65 BSFC while supercharged cars have a BSFC between 0.55 and 0.6, and naturally aspirated engines use only 0.45 to 0.5 Lbs/Hp*Hr.  These are only approximations, but you can clearly see the difference between naturally aspirated engines and turbocharged engines.  Turbocharged engines usually require more fuel to keep detonation at bay due to the increased temperature and pressure of the intake air.  This is why a turbocharged engine uses more fuel per horsepower per hour.  Now that we’ve established that the BSFC of an engine is in constant flux, let’s take a look at how and why it changes.

     First let’s take a look at how the BSFC changes in relation to the Engine’s Speed (RPM).  One would guess that at the car’s maximum RPM would be when the BSFC is the highest, and vice versa.  But take a look at this:


Edgar, Julian. “Brake Specific Fuel Consumption.” AutoSpeed. Web Publications
     Pty Limited, 10 Apr. 2008. Web. 27 Sept. 2009. <

     The red curve shows the Power (in kW for this graph) while the green curve represents the Torque (shown as the Brake Mean Effective Pressure here) so then the pink curve is obviously the engine’s BSFC (in g/kWh this time).  What’s interesting is that the pink curve isn’t linear or exponential and that the lowest point isn’t at the lowest RPM, but the curve begins around the 1,000 RPM point, and then drops to its lowest point at around the 2,500 RPM area.  It can’t be linked to either the highest Torque point or the highest Power point, so it seems that power can’t be directly linked to the BSFC. 

     There are a few reasons that I can think of for the BSFC being at neither the lowest nor the highest engine speeds:

  1.      At lower RPMs, time between the engine cycles lets the intake air cool down too much during the compression cycle.  We want the coolest air to fill up the cylinder to pack more air in there (The Pressure-Temperature Law), but once the valves close and the compression cycle starts, we want the pressure and temperature to increase to give us more torque.  Remember that heat is a form of energy, and the more there is in the compressed air, the more energy is converted when ignition occurs.
  2.      At high RPMs, there is an exponential increase in frictional loss within the engine, from cylinders to camshafts to belts.  Faster engine speeds = more friction.
  3.      Also at high RPMs, the speed the piston is descending on its intake stroke is faster than the air filling up the cylinder.  This is what creates a vacuum at higher RPMs, since the engine doesn’t get the air fast enough.  This vacuum creates extra work for the engine, thus reducing efficiency.
  4.      Most engines are tuned for mid-range torque, meaning all the geometry of the engine, the timing of the cams,camshafts etc are all optimized for the best efficiency in the middle of the RPM range, not the lowest or the highest point.

     But there’s another problem.  This is more or less a dyno graph, meaning the data logged is when the car is going full-throttle from 1,000 all the way to 7,000 RPMs.  Our engines rarely run full throttle while puttering around town.  So lets take a look at how Throttle Position (%) (or engine load) affects our BSFC.


 Edgar, Julian. “Brake Specific Fuel Consumption.” AutoSpeed. Web Publications
     Pty Limited, 10 Apr. 2008. Web. 27 Sept. 2009. <

     Going by this graph, by using 100% throttle, we’re actually getting the most efficienct BFSC, and we’re getting the worst by using “only” 25% throttle.  You can once again see that the lowest point on each curve is in the middle of the RPM range; between 2500 and 3500 RPM in this case. 

     The BSFC for full throttle (100%) is at most 0.50 Lb/Hp*Hr, with a 0.43 Lb/Hp*Hr best.  But when we look at the 50% throttle curve, it has a horrible 0.80 Lb/Hp*Hr best, but a decent 0.48 Lb/Hp*Hr best.  But when we take a look at the 25% throttle curve, it gets alot worse.  Its worst BSFC is a 1.50 Lb/Hp*Hr, with only a 0.70 Lb/Hp*Hr best.  That’s about three times the fuel consumption at its worst, and just under twice at its best.  So if we had to two engines running at the same RPM (lets use 3,000 RPM), one at 100% throttle, the other at 25% and producing the same amount of power, the engine using only 25% throttle would use up almost twice the amount of fuel.  It seems that while we linearly decrease our throttle, our BSFC increases exponentially.  Now please remember that BSFC does NOT equate to the fuel efficiency at a certain throttle or engine speed, but the fuel efficiency of a certain throttle or engine speed in comparison to the Power produced.  When we were comparing the 100% throttle and 25% throttle, we were saying that an engine using 100% throttle is using 0.43 Pounds of fuel per Horsepower per Hour

     In mathematical terms, the more power we’re using, the smaller our BSFC (if the Fuel Rate doesn’t change), since BSFC is equal to Fuel Rate divided by Power.  But to get more Power, we usually have to increase our Engine Speed, which in turns raises our Fuel Rate, and eventually our BSFC (you see the dilemma?).  This is why the BSFC of an engine isn’t as simple as just the lowest or highest RPM.  What if we ran our car at a constant 3,200 RPM with 100% throttle?  We’d obviously be getting the best BSFC possible.  But on the other hand, using 100% throttle means increasing the Engine Speed faster, and in turn increasing our Power faster.  And the larger our Power, we lower the BSFC (once again if the Fuel Rate doesn’t change).  Also, there’s no way we can keep the throttle open all the way yet keep it at 3,200 RPM.  Its like a dog chasing its own tail.

     On the last note, we’ll take a look at a graph of an engine’s actual BSFC in comparison to its Engine Speed (RPM) and Engine Load/Torque (BMEP):


Edgar, Julian. “Brake Specific Fuel Consumption.” AutoSpeed. Web Publications
     Pty Limited, 10 Apr. 2008. Web. 27 Sept. 2009. <

     As you can see, the best BSFC is a 0.42 Lb/Hp*Hr (the red island), at round 2,000 RPM while using 100 BMEP (Torque).  The black dots represent a car’s BSFC taken at 1 second intervals.  See how they rarely enter the 0.42 Lb/Hp*Hr island, and are mostly spread out between 0.50 and 1.70 Lb/Hp*Hr.  And at the worst BSFC is when the car is idling, where the engine consumes fuel, but doesn’t create any Power, and ultimately doesn’t get anywhere.

     This post is actually quite a diversion from the turbocharging process, but it interested me, so I decided to add it in here.  Mathematically I haven’t solved anything (I haven’t found how to achieve the “best” BSFC in a real world setting), but this was a fun research topic.  Stay tuned for the next post where I’ll get back on topic.

How to Go Fast Faster: The Math Behind Turbocharging. Part 3: Increasing Fuel Delivery


Part 1 is here.

Part 2 is here.

*EDIT: Sorry this is out of order.  I pushed every post up since I had to squeeze this post in between Part 2 and Part 4.  I expanded the fuel system section quite a bit and decided it needed its own post.

     Our next step will involve our fuel system.  We have to make sure our fuel system is up to the job for our soon-to-be turbo’d engine.  So if we’re increasing the performance of our engine by stuffing more air in the engine, then we have to also increase the fuel in the cylinder to balance it out.  We can’t just add more air without fuel since that won’t achieve anything.  So if we’re deciding to stuff more air, and thus more fuel, we have to make sure our fuel system can pump the necessary amount of fuel with the increased amount of air.  There are a number of ways in which we can supply the engine with more fuel, but each one has its own advantages and disadvantages. 

     The first option would be to increase the Pulse Duration (msec) of the injectors.  The Pulse Duration is how long each injector’s nozzle is open to mix the fuel with the air.  The problem is that the length of the duration is limited to the time for the engine to complete one whole cycle.  So the faster an engine speeds up, (increase in revolutions, or RPMs), the less time the injectors have to spray its fuel.  Obviously even at the highest RPMs of a stock engine, the stock injectors have enough time to do its job despite the fact that its limited to just milliseconds.  Here’s a simple graph to illustrate my point:


 “Fig. 7-3.” Chart. Maximum Boost, Designing, Testing and Installing Turbocharger
     Systems. By Corky Bell. Cambridge, MA: Bentley Publishers, 1997. 88.

     To increase  our Pulse Duration, we must first figure out how long it takes for one revolution of the engine.  We just need our engine’s maximum Revolutions (RPM) here.  Here’s the equation for the Time of one Revolution:

Time of one Revolution (msec) = Time of One Revolution

Here we’re just dividng 60,000 by our maximum RPM.  The 60,000 is to convert from minutes to milliseconds (msec).

     You then need to find out how long the stock pulse duration is for our fuel injector, since this is different for every engine/car.  This I can’t help you calculate.  You have to research that number on your own.  Once we find that, we can calculate how much we can push the stock fuel system before it maxes out.  We then need the Time for one Revolution (msec) and the Stock Pulse Duration (msec) to find out if our Available Increase:

Available Increase (%) = Available Increase 

Just divide your previous answer (Time of one Revolution) by the Stock Pulse Duration, subtracting by one, and then multiplying by 100% to convert to percentages.

     This will calculate how much room we have to increase our fuel supply for the cylinders.  Some engines might have alot of headroom, while others have none.  Typically, a low-boost (under 7 psi) turbo’d engine will have enough headroom for the stock injectors to satisfy the power demand, but do make sure before you install your turbo.  Then we need to compare this Available Increase percentage with our Performance Gain (%) from the last post.  If our Available Increase is more than our Performance Gain, then we don’t have to physically modify our fuel system for our turbocharging project.  We do, however, need to make changes to the Pulse Duration at the software end, meaning we still need to make changes to the ECU (once again, not covered here).  This is just one option to increase our fuel supply into the cylinders, but should only be used if your Boost Pressure (psi) is under 10 psi.  If this option doesn’t work (ie. the Available Increase isn’t enough), then keep on reading.

     The next option would be to increase the system’s Fuel Pressure by either installing a Fuel Pressure Regulator, or once again modifying the ECU to make the changes via the software.  A Fuel Pressure Regulator increases the Fuel Pressure linearly to the amount of Boost Pressure (psi) in the intake manifold as long as your stock fuel system can handle it.  To calculate how much extra Fuel Pressure you’re going to need, you’ll need your Performance Gain (%) and your car’s Stock Fuel Pressure (psi).  (If you don’t remember what that is, just divide your Desired Bhp by your Stock Bhp, subtract by one, and then multiply by 100%):

Fuel Pressure Required (psi) =

Required Fuel Pressure

What we’re doing here is approximating the required Fuel Pressure with a certain amount of Performance Gain.  The approximation is squaring the sum of the Performance Gain and one, and then multiplying that by your car’s stock Fuel Pressure.  Most engine’s use a stock Fuel Pressure of 43.5 psi, but don’t take my word for it, do some research.

     When you’ve found that out, check to make sure that your stock fuel system can handle that pressure/flow.   The problem with this solution is that although you might be able to squeeze out a few more psis of pressure, the stock pump will reach its limit quite quickly since it’s rated for the engine’s original power.  Once again, this option should only be considered if you’re running a low-boost turbo (10 psi or less).  You can also install a new Fuel Pump to cope with the stock pump’s limitations, but without upgrading the rest of the fuel system, this isn’t a recommended solution.

     If your Boost Pressure is over 10 psi, or your Available Increase isn’t enough, then your best option is to take a look at your whole fuel system and start looking to replace a few parts.  A typical fuel system includes a Fuel Pump, a Fuel Pressure Regulator, the Fuel Lines, and the Fuel Injectors.  This isn’t the most cost effective way, but it is the safest for your engine (if done right).  Let’s start by looking at the Fuel Injectors.  This is the tail end of the fuel system, and is what can actually restrict the amount of fuel you want from mixing with the air.  A Fuel Injector is rated by its Fuel Flow, which can be represented in either pounds per hour (lb/hr) or cubic centimeters per minute (cc/min).  If you’re looking to convert between them, here’s the simple equation:

Fuel Flow Unit Conversion (cc/min) = Fuel Flow

Just multiply your lb/hour Fuel Flow by 10.5 to change your units to cc/min.  Or divide  your cc/min Fuel Flow by 10.5 to get lb/hour.

     To figure out your desired Injector Fuel Flow is quite simple too.  You just need your Desired Bhp (Hp) and the Number of Injectors you have (usually 1 per cylinder, but double check in case):

Injector Fuel Flow (lb/hr) = Injector Flow Rate

We’re just multiplying our Desired Bhp by 0.65 and then dividing by the number of injectors for equal fuel distribution among each injector.

     The 0.65  is the Brake Specific Fuel Consumption (BSFC), a number estimated to be the amount of fuel needed to produce 1 hp for 1 hour for turbocharged engines.  I will go into the BSFC in another post (Part 3b) in more detail.  This tells us how much fuel our injectors need to push out, but you should always choose the next larger size to be on the safe side.  You don’t want your engine running lean do you or have to replace your injectors if you want just a small increase in power?

     Now that you’ve figured out the size of your injectors, we can take a look at the Fuel Pump.  The pump must be able to supply the amount of fuel demanded by the engine so make sure you don’t starve the engine of fuel on this end either.  A Fuel Pump is rated by three variables:  the Voltage it requires, the Fuel Flow, and its Fuel Pressure capabilities.  The Voltage is simple, as most pumps are rated at either 12 or 13.5 volts.  Double check the voltage your engine supplies, and the amount your pump requires and make sure they match.  The Fuel Flow is simple too since it was basically just calculated.  Here it is again:

Fuel Flow (gal/hr) = Flow Rate

We’re just multiplying our Desired Bhp by the BSFC (Brake Specific Fuel Consumption) and then dividing by 6.34 to convert the lb/hr to gal/hr since 6.34 is the weight of fuel per pound in a gallon.

     Fuel Pressure isn’t that hard to solve for either.  Your Base Fuel Pressure (psi) should be around 43.5 psi, and you just have to add your Boost Pressure (psi) to it.  I’ll also be adding in another 10 psi for pumping losses due to hydrodynamic losses (friction, bends etc.) since there is always a pumping loss (usually around 5 psi, but I’ll be using 10 psi just in case).

Fuel Pressure (Psi) = Fuel Pressure

Just add the 43.5 to your desired Boost Pressure, and then add another 10 to that.

     This should give us a safe estimate for the amount of Fuel Pressure our pump needs to be able to handle.  We just then need to search for a suitable Fuel Pump using those numbers.  You should always buy “up”, meaning buy a part with a little bit of headroom in case the engine is more thirsty than we’ve calculated for.

     After taking a look at our Fuel Pump and Fuel Injector, make sure your Fuel Lines and Fuel Pressure Regulator is also up to par, and you’re basically done with this part.

     The next post will be about the Brake Specific Fuel Consumption (BSFC).  Stay tuned.

June 2017
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