where:

big subscript X = fuelcycle X

subscript i = stage of fuelcycle X (all stages except end use by vehicles or power plants)

subscript f = process fuel f

GX,i = g/BTU CO2-equivalent emissions from stage i of fuelcycle X

ENX,i,f = use of process fuel f (e.g., electricity, diesel fuel ) at stage i of fuelcycle X: BTUs of process fuel f per 1.0 BTU of fuel X made available to end users outside of fuelcycle X.

EMf = emission factor: grams of CO2-equivalent emissions per BTU of process fuel used

 

 

 

Note that in this representation, ENX,i,f is BTUs of process fuel f per 1.0 BTU of fuel X made available to end users outside of fuelcycle X, not BTUs of process fuel per BTU of energy produced by stage i. Generally, 1.0 BTU of energy out of stage i might not end up as one BTU of fuel X made available to end users outside of fuelcycle X, because some of the energy output from stage i might be lost in stages downstream (for example, methanol production requires about 1.5 BTUs of natural-gas input to produce 1.0 BTU of methanol), and some might be used internally within the fuelcycle as a process fuel, and hence be unavailable outside of the fuelcycle. Therefore, given data on process fuel use at a particular stage of the fuelcycle, and energy output of the stage, the parameter ENX,i,f will be shown to be:

 

where:

ENX,i,f is as defined above

PX,i,f = BTUs of process fuel f used at stage i of fuelcycle X, per 1.0 BTU of energy of energy out of stage i (estimated from primary data)

K*X,i = the stage i energy-conversion or energy-loss factor: BTUs of fuel or feedstock energy out of stage i of fuelcycle X per 1 BTU of fuel energy output from the final stage

UX = total internal use (own-use) of fuel X as a process fuel, in fuelcycle X: BTUs of own-use per 1.0 BTU of X output from the final stage of the fuelcycle X (estimated as a fraction of Pi, at each stage)

Note that in the definition of UX,, the 1.0 BTU of X output from the final stage includes the amount that is recycled internally, so that the amount available outside of the fuelcycle is 1.0 - UX.

Note too that the lost energy represented by the factor K*X,i just as well could be counted as internal or own use, and so be incorporated into UX. Consider, for example, fuel lost to evaporation or leakage during the fuelcycle. If 5% of the fuel evaporates or leaks during the distribution and station stage, then the K* factor, as defined above, is 1.00/(1.00-0.05) = 1.053. But the 5% also can be viewed as a sort of own use U (non-combustion own use, in this case), such that the 1-Ux denominator of the expression for ENX,i,f is 1-0.05, which gives 1.053 in the numerator -- the same as the K* factor.

 

Estimation of own-use

The following diagram shows energy input and output for a simple four-stage fuelcycle (recovery, transmission, production, and distribution). Pi is the total amount of process energy (from all process fuels f) used in stage i per 1.0 BTU of energy output from stage i, and Ki (not K*i) is the number of BTUs from stage i needed as input to stage i+1 in order to produce 1.0 BTU from stage i+1.

P1 Ø			P2 Ø			P3 Ø			P4 Ø
REC	Æ1.0	K1 Æ	TRAN	Æ1.0	K2 Æ	PROD	Æ1.0	K3 Æ	DIST	Æ1

 

The process energy factors, Pi, are estimated from primary data on process energy use and fuel or feedstock output, at each stage. The conversion/loss factors, Ki, are estimated from energy-in/energy-out data for each stage, and typically are close to 1.0 for all stages except for fuel production. Note that Ki is expressed relative to 1.0 BTU output from stage i+1, whereas K*i is expressed relative to 1.0 BTU output from the final stage, such that K*i is the product of the Ki from stage i to the penultimate stage:

K*i = Ki . Ki+1 . ... . Kfinal-1

 

there being no Kfinal because K is expressed relative to 1.0 BTU output from stage i+1 and by definition there is no stage after the final stage.

Recall that the overall objective is to express process energy inputs per BTU of final product delivered to consumers outside of the fuelcycle. In order to do this, we must account for the multiplicative effect of the Ki factors, and for own-use of final fuel. First I account for the multiplicative effect of the Ki factors, by representing the four separate stages as one system, the output of which is one BTU of fuel product.

 

P1 . K1 . K2 . K3 Ø				P2 . K2 . K3 Ø			P3 . K3 Ø		P4 Ø
K1 . K2 . K3 Æ	REC	Æ	TRANS		Æ	PROD	Æ	DIST	Æ1

 

 

Next, I account for own use. The 1.0 BTU of fuel output from the final stage includes some fuel that is recycled back to the stages of the fuelcycle, as process fuel. Hence, the amount of fuel available to end users outside of the fuelcycle is less than 1.0. Let Fi be the fraction of Pi that is the end use fuel x that comes out of the fuelcycle X. We now have:

 

F1 . P1 . K1 . K2 . K3 Ø +				F2 . P2 . K2 . K3 +			F3 . P3 . K3 +		F4 . P4 +	¨ U
(1-F1) . P1 . K1 . K2 . K3 Ø				(1-F2) . P2 . K2 . K3 Ø			(1-F3) . P3 . K3 Ø		(1-F4) . P4 Ø	Æ 1-U
K1 . K2 . K3 Æ	REC	Æ	TRANS		Æ	PROD	Æ	DIST	Æ 1

 

ADD IN THE LINES AT THE END

 

This diagram shows that, given inputs Pi, the whole fuelcycle produces 1-U BTUs of x for end users outside of the fuelcycle itself, where U is the total amount of own use at all stages of the fuelcycle (equal to F1 . P1 . K1 . K2 . K3 + F2 . P2 . K2 . K3 + F3 . P3 . K3 + F4 . P4). Thus, to end up with 1 BTU of X for end users outside of the fuelcycle, we must scale all inputs by 1/1-U:

TABLE IS MISSING: WOULD NOT MOVE INTO HTML FORMAT: SEE ORIGINAL DOCUMENT

 

The foregoing shows energy flows, ENX,i.The final step is to incorporate these expressions for ENX,i into the expression for CO2 - equivalent emissions (GX,i), by multiplying them by the appropriate emission factors EM. (Recall from above that .) For any fuelcycle X, the emission factor EMf for any process fuel f that is not the output x of X -- i.e., for any non- "own-use" process fuel -- is the full fuelcycle emission factor, where the full fuelcycle includes emissions from production, distribution, etc., as well as from final end-use of the process fuel in fuelcycle X. This should be intuitively clear: for those process fuels outside of the fuelcycle in question, the entire fuelcycle emission must be counted. I designate such a full fuelcycle emission factor as EMFC.

However, with this method, the emission factor EM for own-use fuel x in fuelcycle X is just the emission factor for final or direct use of the own-fuel as a process fuel within its fuelcycle. For example, in the method presented above, the appropriate emission factor for diesel fuel used by tanker trucks in the diesel fuelcycle is the emission factor for diesel end-use use by trucks -- not the full fuelcycle emission factor for diesel fuel. This is because, in this method, the emissions attributable to the making of the own-use fuel already are accounted for by virtue of the own-use fuel being subtracted from net output. I designate such an end-use-only emission factor as EMEU.

Combining this with the derivation for EN, above, we now can derive the following expression for complete fuelcycle emissions of CO2-equivalent GHGs (Gx):

where:

little subscript x = fuel x produced by fuelcycle X

GX,i, ENX,i,f , EMf, PX,i, KX,i, and UX are as defined above

GX = complete fuelcycle CO2-equivalent emissions of greenhouse gases from the entire fuelcycle X, except end use, per BTU of fuel output

Fx,i = the fraction of Pi that is the end use fuel x that comes out of the fuelcycle X

Ff,i = the fraction of Pi that is process fuel f

EMx,EU = emission factor for end use of own-fuel x (CO2-equivalent g/BTU)

EMf,FC = emission factor for full fuelcycle production and use (including end use) of process fuel f (CO2-equivalent g/BTU)

 

 

Development of an equivalent, simpler method

The method just developed is appealing because it is derived from a clear, general representation of a fuelcycle. It does, however, have two minor disadvantages. First, it requires that own-use Ux be estimated for the entire fuelcycle. Second, it requires two different kinds of emission factors: EMX,EU for own-use fuel, and EMf,FC for other fuels.

Because of these disadvantages, I have derived from the method above a simpler but less intuitive method that does not require the estimation of own use Ux, or the use of different kinds of emission factors. The method is:

where all the terms are as defined previously, and the summation over f process fuels includes the own-use fuel x. The advantages of this method are that it does not require the estimation of Ux per se, or the designation of separate kinds of emission factors for own-use fuel. It is notationally and programatically simpler than the original method.

This new method can be shown to be equivalent to the original method. First, expand the expression for GX,i into terms for own use fuel x and other fuels f:

where:

EMx,FC = emission factor for full fuelcycle production and use, including end use , of own fuel x (CO2-equivalent g/BTU)

Now substitute this expression for GX,i into the expression for GX:

where EMx,FC* = complete fuelcycle emission factor for fuel x, except end-use emissions. Note, though, that EMx,FC* is just GX: complete fuelcycle CO2-equivalent emissions of greenhouse gases from the entire fuelcycle X, except end use. Hence, we have:

 

From this point, we can make an infinite number of substitutions of for GX. After two more such substitutions we have:

With an infinite number of substitutions for GX, we have:

The second term in this expression is the binomial expansion of (1-UX)-1. Hence, the alternative expression for GX is exactly equal to:

which is precisely the original expression derived earlier.

 

Application of the new method

The model now uses the new method -- -- to calculate g/BTU CO2-equivalent emissions of GHGs from all stages of the fuelcycle except end use. It is evident from the demonstration above that this new method is circular, or recursive: emissions at each stage (GX,i) depend on total fuelcycle emissions (GX), which is the sum of emissions from each stage: . The spreadsheet handles this circularity by iterative calculations, and converges on a solution after 20 or so iterations (as revealed by comparing the results of the new method with the results of the old method, which is not circular in the same way). Thus, the new method in effect transfers some of the work of estimating fuelcycle GHG emissions from me to the spreadsheet.

In the new method, the factor K is used to account for energy lost by evaporation or leakage, and for energy lost in feedstock-to-fuel conversion processes. For example, in the conversion of natural gas to methanol, about 1.5 BTUs of natural gas are required to produce 1.0 BTU of methanol. Although it would be possible to treat 0.5 BTUs of natural gas as an additional fuel input used to "process" the 1.0 BTU of natural gas that emerges as 1.0 BTU of methanol, it would be awkward to do so.

 

Related changes

Where an energy source X is used to recover energy source X (e.g., coal used at the mine site as a source of energy), the fuelcycle emissions for such "own use" should not include emissions from a feedstock transmission stage. I have adjusted the model accordingly.