In order to do conventional least-squares fitting to you need to determine .

Step response of NIH 3T3 fibroblasts to ionomycin

For instance, the figure above shows the fluorescent intensity of some 3T3 cells to ionomycin.   The traces are normalized to the initial values.  Notice that the cells don’t recover to the levels at which they began.  I wanted to know the time constant of the decay on the tail of the trace.

I solved this problem by performing the fit inside an iterative minimum-finding function.  First I identified the point of fastest decline and selected that as the point.  Then I iterated to find the value for which minimized the norm of the error:

for best-fit A and .

y0_fit = fminbnd( @(y0) ...
                 norm( y-y0 - exp( polyval( polyfit( x, log(y-y0), 1), x))) ,...
                 -10*y(1), y(1) );

This, along with other analysis:

For a population of cells (Showing mean and standard deviation for each test point. numbers in parentheses indicate number of cells in the sample.):

The US environmental protection agency are in the process of developing a fuel economy standard for hybrid and plug-in electric vehicles, and according to a recent draft version of the standard, General Motors’ flagship entry into the “range-extended electric vehicle” market would clock in at 230 miles per gallon.

Recently I entered a debate in defense of the EPA’s rating.  It was a retaliation against the claim that GM and the EPA have entered into a conspiracy in which the standards are designed to the specific product available and politically palatable, and the net result would be consumers’ further disillusionment at the very notion of “clean energy” when they discovered that their gleefully acquired products failed to perform to their expectations.

It was a Master’s student, and so naturally I assumed that he hadn’t thought out his arguments.  Comically, I then undertook the very error I was presuming, and eventually after working through the question, I came to an understanding of the physical conditions of the problem which I now share with you. The essential innovation of GM’s car is its all-electric drivetrain.  The motors which turn the wheels are all electric, and they’re powered by an on-board generator which is attached to a traditional internal-combustion engine which burns gasoline.  This is more efficient for two reasons.

First, you dispense with the transmission.  Instead of having to amplify the relatively narrow dynamic operating range of a combustion engine to match the realities of vehicle speed, we make immediate use of the full flexibility of the direct-current motor.  Ultra-high torque at low speeds, minuscule power consumption at high speeds, and a direct connection to the wheels.  Electronic speed control. Second, the gas motor which still supplies all the energy (when off-grid) has gone from the center of the vehicle and the Source of All Things to a specialized component.  It has exactly one purpose– to run a generator– which means it can be highly tuned to a specific optimal operating point and switched on when needed.

My misstep in the debate was in overestimating the energy in a gallon of gas.  Gasoline has a higher heating value of about 47 megajoules per kilogram [MJ/kg], which comes out to roughly 130 MJ or 36 kilowatt-hours per gallon.  The conventional four-stroke gas engine implements the Otto cycle, a well-characterized and carefully orchestrated alternation of isochoric and adiabatic processes.  Isochoric means “constant volume” (i.e. top dead center and bottom dead center of the piston stroke) and adiabatic means “no heat transfer” (i.e. compression and power strokes).  Maybe I should write another post about that sometime.

The thermodynamic efficiency of the Otto cycle is known; it’s a function of the working fluid (in this case a mixture of gasoline and air) and the compression ratio of the engine, and its upper limit is defined by the detonation temperature of the fuel (think anti-knock, octane rating).  In the case of modern gasoline engines, it’s about 47%.  This means that if you’re burning gasoline, you give up 18 kWh/gal “when you drive it off the showroom floor.”  Add to that mechanical losses and imperfect assumptions and you take off another (say) 22-23%, which puts the gas-tank-to-flywheel efficiency of the well-tuned engine at a practical maximum of about 36%. In the case of the Volt-style transmission-free hybrids in which the combustion engine drives a special purpose alternator, something close to that efficiency might be achievable.  Perhaps an optimistic assumption would be that 13 kWh of electric power can be harnessed from the combustion of a gallon of gasoline, where it can be stored into a battery and then run through copper windings and onto the pavement and into the winds on the open highway.

According to Chevy’s engineers, their car takes 25 kWh of electric power to go 100 miles, which comes out to 52 miles per gallon of gas at 36% efficiency.  In order to get the 230-mpg figure they must have assumed it drives 40 miles ( on a / free of ) charge and then another 12 miles at 52 mpg.  This bears resemblance to a hypothetical test in which the fuel consumption of a vehicle is measured over a 52-mile course. The interesting thing to note here is the bias in measuring gasoline consumption but ignoring electric grid power as “free.”  For me, the marginal cost of 25 kWh is about $3.50, compared to $6.50 for two gallons of gas, but if I were charging my car every day that would go up to $6-7, because Southern California Edison uses tiered pricing (which I heartily endorse).

As far as greenhouse gas emissions are concerned, it’s hard to do worse than coal.  If you get your grid power from a coal plant you’re putting out about 1 kg of CO2 per kWh.  US average is more like 0.62 kg/kWh; in California we’re down around 0.45 kg/kWh, due largely to the fact that we have no coal plants, plus we’ve got all that nuclear and hydro power.  The internal combustion engine, running at its “practical maximum” efficiency of 36%, comes in at around 0.86 kg/kWh.

So any move to plug-in vehicles is an improvement, as long as it’s coupled with a move away from coal.  That’s fortunate, because the people who love coal power are the same people who are unlikely to buy plug-in hybrids, so advocacy for these two things together won’t anger any additional people.

The more interesting question surrounds the 25 kWh/100 miles figure.  What speed of travel does that represent? I wonder what theoretical lower bounds can be produced on the amount of energy required to move a car 100 miles?  Because the 13 kWh/gallon of gas applies to any power plant burning gas in a 4-stroke engine, the only way to improve energy efficiency is to reduce the energy needed to go the distance.  Or to run off the grid.

Pity how badly I botched this topic in my public discussion.  Frankly, I’m embarrassed.  The whole thing can’t go away fast enough.