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) =

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. <http://autospeed.com/cms/
     title_Brake-Specific-Fuel-Consumption/A_110216/article.html> 

     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. <http://autospeed.com/cms/
     title_Brake-Specific-Fuel-Consumption/A_110216/article.html>

     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. <http://autospeed.com/cms/
     title_Brake-Specific-Fuel-Consumption/A_110216/article.html>

     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.
     Print.

     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) =

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 (%) =  

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) =

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) =

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) =

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) =

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) =

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.

How to Go Fast Faster: The Math Behind Turbocharging. Part 7b: Intercooler Efficiency

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.

     So now that we have an intercooler, we can calculate its Internal Flow Area (in³), Intercooler Lag (sec), Intercooler Gain (%), Power Loss (%), and ultimately, the Intercooler Efficiency (%) based on some logged data and intercooler/engine specs.

     If we do not know the exact Internal Flow Area, we can always figure it out by some measuring.  We just need the Number of Channels, the Channel Width (in), and the Channel Length (in).  Here’s the easy equation:

Internal Flow Area (in³) =

This equation is really simple.  Just multiply the three dimensions of the intercooler channels, and we get our Internal Flow Area, or basically the volume.

     With all the extra piping of an intercooler, the air takes a longer route from the throttle body all the way into the engine.  This can cause a small lag between the time you actually open up the throttle and the time the intercooled air reaches the engine.  This can be called Intercooler Lag.  To calculate that, we need the Volume/Internal Flow Area (in³) of the intercooler and the Flow Rate (cfm or ft³/min) at a certain rpm.  Then we can approximate the time by a simple equation:

Intercooler Lag (sec) =

All we’re doing is diving the Volume of the air by the Flow Rate to get the Intercooler Lag.  But we also have to multiply that by 60 to convert the minutes to seconds, and then divide by 1728 to convert the feet cubed to inches cubed.  Then its all just multiplied by 2, which is a result of the increased airflow going through the engine when going from cruise to boost.

     We can also calculate how much “more” air we’re stuffing into the cylinder by cooling and making the air more dense via the intercooler.  This is called Intercooler Gain, and to calculate that, all we need is the Absolute Temperature (^o) of the ambient air, the Temperature before the intercooler (^oF), and the Temperature after the intercooler(^oF).  We then plug all that into this equation to find our gain:

Intercooler Gain (%) =

So here we’re just dividing the temperature of the air before entering the intercooler by the air exiting the intercooler, ie. the change in temp caused by the intercooler.  We then just subtract by one and then multiply by 100% to get an answer in percent form.

     Unfortunately while the Intercooler Gain does indicate how much extra air should be entering the combustion chamber with the addition of the intercooler, the Intercooler Gain does not equal the same increase in power due to aerodynamic loses inside the intercooler and the piping.  This would be called Power Loss.  We can calculate the Power Loss by using  another formula here, using the Boost-Pressure (psi) of the system, and the Boost-Pressure (psi) after the installation of the intercooler:

Power Loss (%) = 

We’re adding the air pressure (more commonly called boost) to the atmospheric pressure (14.7) to find the absolute pressure.  We do this twice, as the boost pressure before the installation of the intercooler will be higher because of the aerodynamic drag in the intercooler.  We then divide the absolute pressures, subtract one by all that, and then multiply all that by 100% to find our Power Loss in percentage form.  Hopefully the intercooler won’t have lowered our boost pressure too much, resulting in a minimal Power Loss.

      The last equation in this sequence would then be the calculation of the actual Intercooler Efficiency. The Intercooler Efficiency is found by the temperature removed by the intercooler divided by the temperature rise due to the compressor.  This is similar to the Intercooler Gain, but this number is more applicable to the exact intercooler without influences from the turbo/engine/piping/etc.  This time we need the:

• Temperature Pre-Intercooler/Post-Compressor (^oF)
• Temperature Post-Intercooler (^oF)
• Ambient Temperature (^oF)

     Here’s the equation:

Intercooler Efficiency (%) =

EQUATIONPLACEHOLDER

We’re basically finding the  temperature removed divided by the temperature rise.  As our intercooler has a higher efficiency, it’ll remove more heat, and thus we’ll have a higher nominator.  We want a small as possible denominator, but we obviously can’t change that with the intercooler.  Just muliply the dividend by 100% to change the efficiency to a percentage.

     And that’s your Intercooler Efficiency.  Obviously this can only be calculated with your intercooler installed and with certain equipment, but this is a very accurate way of finding out how your intercooler is performing (considering everything else works perfectly).