Tau-X acceleration enrichment is a fuel adjustment method used to calculate engine fuel needs during transient, or fast changing conditions. The model accounts for most of the "lean stumble" that happens when a throttle is opened quickly at idle. On a carburator, the accelerator pump is mostly used to realize the needs of this model.
This method is a break from the past in that it does not rely on TPS, MAP or MAF to indicate throttle transients, it is based strictly on the changing fuel requirements of the engine. As such, it can only be fully applied by a system that can recalculate engine fuel demands on a real-time basis, at every cylinder firing. Most aftermarket systems don't do this, which gives this system an advantage in developing a fully realized tau-X acceleration enrichment model.
The basic theory behind Tau-X is that much of the fuel entering the engine does not come directly from the fuel injector (or carburator) but rather is evaporated from a puddle of fuel in the manifold that has accumulated there over time. At idle speeds, over half the fuel injected will hit the manifold walls and puddle there without vaporizing. And over half the fuel burned by the engine each cycle will come from the puddle. The puddle size can easily exceed 30 full fuel injection pulses, so it can take awhile to build up. Most of this behavior is determined by the chemical and physical properties of gasoline.
Liquid gasoline cannot burn by itself, of course, it has to mix with air first. In addition, small drops or particles of fuel aren't good enough, gasoline must evaporate (vaporize) in order to mix with air. Then, the vapor must mix with air in the right amount, between about 8 and 20 parts air to one part gasoline vapor. Another factor to consider is that gasoline is a mixture of different chemicals, all of which evaporate at different temperatures. Winter gas is a different mixture of chemicals than summer gas. Another factor is time; gasoline does not evaporate instantly, even if the temp is high enough, it takes time to evaporate.
As the name inplies, Tau-X is primarily concerned with the effects of two independant variables. The X term is used to account for the percent fuel going into the manifold puddle, and the Tau term is used to account for the time needed to vaporize the fuel in the puddle. Because tau is time related, most of the coding effort in realizing the model involves calculation of the tau related term. The tau term is usually modeled as a first order decay function, which means that it can be specified in either half-life units (time to 50% decay) or time constant units (time over e, or about 63% decay). The use of time constant units for tau is the more usual case, so that the % decay of the fuel puddle, during a specified time, can be found using the formula:
puddle_decay_factor = 1-[1/e^(tau_time_constants*seconds)]
Here are graphs of the above formula for typical tau values at various engine rpm. The decay factor calculated for 720 degrees crankshaft rotation at the rpm shown. Notice that it's possible to keep a fairly linear relationship, if tau values aren't too small, and RPM is kept to within about 500 to 3,000 RPM:
Here's a linear formula that approximates the decay factor to within about 1%, from 700rpm to 2,500rpm, for a tau of one. Time is expressed in 2us counts for 720 degrees of crankshaft rotation:
puddle_decay_factor = (0.1168 * 2us_count) + 373.4724
For a value of tau other than one, the result of this calc will be seen to vary linearly with 1/tau, again to within about 1% if tau is restricted to +-20%.
So what is the result, or usefulness, of this analysis of decay factor? Why not just create a lookup table for it and be done with it? Well, one issue is that most ECU's that claim to use tau-x, use a lookup table for tau, not decay factor. As can be seen, a fair amount of calculation must still be done to get from tau to decay factor, so why not build that calculation into the table and just lookup decay factor directly? I think the answer to that question is these ECU's are not calculating decay factor correctly, they are assuming that tau is related to decay by a simple factor, which is clearly not the case.
How should tables be setup for an accurate tau-x system? The tables should provide the data needed to calculate an injection amount. Since the injection amount is calculated and performed every 720 deg, then the tau-x calculations must coincide with that timing requirement. The lookup tables for X need to coincide with the same assumptions for tau. When that's done correctly, the following relationships for required pulse width will apply:
fuel_inj = [ fuel_req - (curr_puddle * evap_%) ] / [ 1 - X ]
proj_puddle_accel = ( fuel_inj * X ) + curr_puddle - ( curr_puddle * evap_% )
proj_puddle_steady = fuel_inj * X / evap_%
Where:
fuel_req = The total fuel needed by the engine this 720 deg cycle.
fuel_inj = The actual fuel to be injected this 720 deg cycle. Same as fuel_req if the engine has been running at a constant speed and load (no acceleration enrichment needed).
evap_% = decay factor, the percent of puddle mass projected to be lost in the next 720 deg cycle.
X = X factor, the percent of fuel_inj that will enter the puddle mass in the next 720 deg cycle.
proj_puddle = Calculation of puddle mass (expressed as cummulative pulse width counts) available at beginning of next 720 deg cycle.
curr_puddle = puddle at beginning of next 720 deg cycle, same as proj_puddle_accel from last cycle. If there is no result available from last cycle, can use calculated value of proj_puddle_steady as an initial, or seed estimate.
(to be continued)
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