## Annex 3A1 First Order Decay Model

3A1.1 INTRODUCTION

The first order decay (FOD) model introduced in Chapter 3 is the default method for calculating methane (CH4) emissions from solid waste disposal sites (SWDS). This Annex provides the supplementary information on this model:

• mathematical basis for the FOD model (see Section 3A1.2),

• key issues in the model, such as the estimation of the mass of degradable organic carbon available for anaerobic decomposition at SWDS (DDOCm) (Section 3A1.2) and the delay time from disposal of waste in the SWDS till the decomposition starts (Section 3A1.3),

• introduction of the spreadsheet model developed to facilitate the use of the FOD method (3A1.4),

• how to estimate the long-term storage of carbon in SWDS (Section 3A1.5),

• different approaches to the FOD model, including an explanation of the differences between the current and earlier IPCC methods (Section 3A1.6).

3A1.2 FIRST ORDER DECAY (FOD) MODEL - BASIC THEORY

The basis for a first order decay reaction is that the reaction rate is proportional to the amount of reactant remaining (Barrow and Gordon, 1996), in this case the mass of degradable organic carbon decomposable under anaerobic conditions (DDOCm). The DDOCm reacted over a period of time dt is described by the differential equation 3A.1.1:

Equation 3A1.1 Differential equation for first order decay d(DDOCm) = -k • DDOCm • dt

Where:

DDOCm = mass of degradable organic carbon (DOC) in the disposal site at time t k = decay rate constant in y-1

The solution to this equation is the basic FOD equation.

Where:

DDOCm = mass of degradable organic carbon that will decompose under anaerobic conditions in disposal site at time t

DDOCmo = mass of DDOC in the disposal site at time 0, when the reaction starts k = decay rate constant in y-1

Substituting t =1 into Equation 3A1.2 shows that at the end of year 1 (the year after disposal), the amount of DDOCm remaining in the disposal site is:

Equation 3A1.3 DDOCm remaining after 1 year of decay

The DDOCm decomposed into CH4 and CO2 at the end of year 1 (DDOCm decomp) will then be:

Equation 3A1.4 DDOCm DECOMPOSED AFTER 1 YEAR OF DECAY

The equation for the general case, for DDOCm decomposed in period T8 between time (t-1) and t, will be:

The equation for the general case, for DDOCm decomposed in period T8 between time (t-1) and t, will be: Equations 3A1.4 and 3A1.5 are based on the mass balance over the year.

In Section 3.2.3, the parameter half-life time of the decay is discussed. Half-life is the time it takes for the amount of reaction to be reduced by 50 percent. The relationship between half-life time and the reaction rate constant k is found by substituting DDOCm in Equation 3A1.2 with 1/2DDOCm0, and t with t1/2: 3A1.3 CHANGING THE TIME DELAY IN THE FOD EQUATION

In most SWDS, waste is disposed continuously throughout the year, usually on a daily basis. However, there is evidence that production of CH4 does not begin immediately after disposal of the waste (see Section 3.2.3 in Chapter 3).

Equations 3A1.3 and 3A1.4 assume that the decay reaction starts on January 1 in the year after disposal, i.e., an average six month delay before the reaction commences.

The equations can easily be transformed to model an earlier start to the decay reaction, i.e., start of the decay reaction in the year of disposal. This is done by moving the ekt curve backwards along the time axis. For example, to model a reaction start on the first of October in the year of disposal (i.e., an average time delay of three months before the decay reaction commences, instead of six months), Equation 3A1.2 will be transformed into the following:

Then there will be two solutions, one for the year of disposal and one for the rest of the years:

Then there will be two solutions, one for the year of disposal and one for the rest of the years:

Equation 3A1.8 DDOCm DECOMPOSED IN YEAR OF DISPOSAL (3 MONTH DELAY) 8 T denotes the year for which the estimate is done in relation to deposition year.

Where:

DDOCm decompY = DDOCm decomposed in year of disposal

DDOCm decompT = DDOCm decomposed in year T (from point t-1 to point t on time axis) T = year from point t-1 to t on the time axis, where year 1 is the year after disposal. Y = disposal year The same can be done to find the equations for reaction start within the year after disposal.

### 3A1.3.1 Disposal profile

The method presented here assumes that CH4 production from all of the waste disposed during the first year (Year Y) begins on the 1st of January on the year after disposal. Year 1 is defined as the year after disposal.

Some inaccuracy will be introduced by the fact that, in reality, waste disposed at the beginning of the year will begin to produce CH4 earlier, and waste disposed at the end of the year will begin to produce CH4 later. Comparison of results calculated with the simple FOD method presented here and the exact day-by-day method, which is presented in Section 3A1.6.3, has been used to evaluate this error. With a half-life time of 10 years, evaluating CH4 emissions with the exact method gives a decay profile only 1 day difference from the simplified version of the method. With a half-life time of 3 years, the simple method gives 3.5 days difference from the exact method. Even with a half-life time of 1 year, the difference between the exact and simple methods is just 10 days. The error introduced by the assumption in this simple method is very small in comparison with other uncertainties in the parameters, especially given that the uncertainty in delay time is at least two months.

In order to estimate CH4 emissions for all solid waste disposal sites in a country, one method is to model the emissions from the waste disposed in each year as a separate row in a spreadsheet. In the IPCC Waste Model, CH4 formation is calculated separately for each year of disposal, and the total amount of CH4 generated is found by a summation at the end. A typical example, for six years of disposal of 100 units of DDOCm each year, with a decay rate constant of 0.1 (half-life time of 6.9 years), and CH4 generation beginning in the year after disposal, is shown in the table below. The figures in the table are the DDOCm decomposed from that waste each year, from which the CH4 emissions can be calculated.

When considered over a period of 50 years, which is necessary for the FOD method, this leads to a rather large calculation matrix. The spreadsheet uses a more compact and elegant approach to the calculations. This is done by adding the DDOCm disposed into the disposal site in one year to the DDOCm left over from the previous years. The CH4 emission for the next year is then calculated from this 'running total' of the DDOCm remaining in the site. In this way, the full calculation for one year can be done in only three columns, instead of having one column for each year (see Table 3A1.1).

The basis for this approach lies in the first order reaction. With a first order reaction the amount of product (here DDOCm decomposed) is always proportional to the amount of reactant (here DDOCm). This means that the time of disposal of the DDOCm is irrelevant to the amount of CH4 generated each year - it is just the total DDOCm remaining in the site that matters.

This also means that when we know the amount of DDOCm in the SWDS at the start of the year, every year can be regarded as year number 1 in the estimation method, and all calculation can be done by these two simple equations:

Equation 3A1.10

Mass of degradable organic carbon accumulated at the end of year T

Equation 3A1.11 Mass of degradable organic carbon decomposed in year T

Where:

the decay reaction begins on the 1st of January the year after disposal.

DDOCmax = DDOCm accumulated in the SWDS at the end of year T

DDOCmdx = mass of DDOC disposed in the SWDS in year T

DDOCmai.j = DDOCm accumulated in the SWDS at the end of year (T-l)

DDOCm decompT = DDOCm decomposed in year T

 Table 3A1.1 New FOD calculating method year DDOCm disposed DDOCm accumulated DDOCm decomposed 0 100 100 0 1 100 190.5 9.5 2 100 272.4 18.1 3 100 346.4 25.9 4 100 413.5 33.0 5 100 474.1 39.3 6 100 529.0 45.1

3A1.4.1 Introducing a different time delay into the spreadsheet model

The table and equations presented above assume that anaerobic decomposition of DDOCm to CH4 begins on 1st of January in the year after disposal (an average delay of 6 months before the decay reaction begins).

If the anaerobic decomposition is set to start earlier than this, i.e., in the year of disposal, separate calculations will have to be made for the year of disposal. As the mathematics of every waste category or waste type/fraction is the same, only parameters are different, indexing for different waste categories and types/fractions are omitted in the equations 3A1.12-17, and 3A1.19:

Equation 3A1.12 DDOCm REMAINING AT END OF YEAR OF DISPOSAL

DDOCm remT = DDOCmdT

(Column F in CH4 calculating sheets in the spreadsheet model)

Equation 3A1.13 DDOCm DECOMPOSED DURING YEAR OF DISPOSAL

DDOCm decT = DDOCmdT

(Column G in the CH4 calculating sheets in the spreadsheet model)

Where:

DDOCm remT = DDOCm disposed in year T which still remains at the end of year T (Gg) DDOCmdT = DDOCm disposed in year T (Gg)

DDOCm decT = DDOCm disposed in year T which has decomposed by the end of year T (Gg) T = year T (inventory year)

M = month when reaction is set to start, equal to the average delay time + 7 (month) k = rate of reaction constant (y-1)

Equations 3A1.10 and 3A1.il will then become: Where:

DDOCmax = DDOCm accumulated in the SWDS at the end of year T, Gg

DDOCmax-1 = DDOCm accumulated in the SWDS at the end of year (T-1), Gg

DDOCm decompT = DDOCm decomposed in year T, Gg

The spreadsheets are based on Equations 3A1.12 to 3A1.15. If the reaction start is set to the first of January the year after disposal, this is equivalent to an average time delay of 6 months (month 13). Equations 3A1.14 and 3A1.15 will then be identical to Equations 3A1.10 and 3A1.11.

3A1.4.2 Calculating DDOCm from amount of waste disposed

Data on waste disposal is entered into the spreadsheet. The data can be given by waste type (waste composition option) or as bulk waste. In the waste composition option, waste is split by waste type/material (paper and cardboard, food garden and park waste, wood, textiles and other waste). In the bulk waste option, waste is split only by main waste category (MSW and industrial waste). Not all DOCm entering the site will decompose under the anaerobic conditions in the SWDS. The parameter DOCf is the fraction of DOCm which will actually degrade in the SWDS (see Section 3.2.3 in Chapter 3). The decomposable DOCm (DDOCm) entering the SWDS is calculated as follows:

Equation 3A1.16 Calculation of decomposable DOCm from waste disposal data

(Column D in the CH4 calculating sheet in the spreadsheet model)

Where:

DDOCmdT = DDOCm disposed in year T, Gg

WT = mass of waste disposed in year T, Gg

DOC = Degradable organic carbon in disposal year (fraction), Gg C/Gg waste

DOCf = fraction of DOC that can decompose in the anaerobic conditions in the SWDS (fraction)

MCF = CH4 correction factor for year of disposal (fraction) (see Section 3.2.3)

3A1.4.3 Calculating CH4 generation from DDOCm decomposed

The amount of CH4 generated from the DDOCm which decomposes is calculated as follows:

Equation 3A1.17 CH4 generated from decomposed DDOCm

CH4 generatedT = DDOCm decompT • F • 16/12 (Column J in the CH4 calculating sheets in the spreadsheet model)

Where:

CH4 generatedT = amount of CH4 generated from the DDOCm which decomposes

DDOCm decompT = DDOCm decomposed in year T, Gg

F = fraction of CH4, by volume, in generated landfill gas

16/12 = molecular weight ratio CH4/C (ratio).

The CH4 generated by each category of waste disposed is added to get total CH4 generated in each year. Finally, emissions of CH4 are calculated by subtracting first the CH4 gas recovered from the disposal site, and then CH4 oxidised to carbon dioxide in the cover layer. Where:

CH4 emittedT = CH4 emitted in year T, Gg x = waste type/material or waste category Rt = CH4 recovered in year T, Gg OXT = Oxidation factor in year T, (fraction)

### 3A1.5 CARBON STORED IN SWDS

Only part of the DOCm in waste disposed in SWDS will decay into both CH4 and CO2. An MCF value lower than 1, means that part of the DOCm will decompose aerobically to CO2, but not to CH4. The DOCm available for anaerobic decay will not decompose completely either. The decomposing part of this DOCm (DDOCmd) is given in Equation 3A1.16. The part of DOCm that will not decompose will be stored long-term in the SWDS, which will then be:

Using the default value for DOCf = 0.5, 50 percent of the disposed DOCm will remain there for long term. Equation 19 describes the annual increase in the stock of long-term stored carbon in the SWDS. The long-term stored carbon in harvested wood products (HWP) disposed in SWDS (see Chapter 12 in the AFOLU volume) can be estimated using this equation. For the waste composition option, the amount of DOCm which is long-term stored in HWP waste disposed in SWDS is calculated directly from material information in the Activity sheet. Using the bulk waste option, the fraction of waste originating from HWP needs to be estimated first. If this is not known, the regional or country-specific default fractions for paper and cardboard, garden and park and wood waste can be used (see Section 2.3). The calculations are performed in the spreadsheet model in the sheet called 'Stored C' and 'HWP'.

### 3A1.6 DIFFERENT FOD APPROACHES

Different FOD approaches have been used to estimate the CH4 emissions from SWDS. The differences between the approach used in these Guidelines, previous IPCC approaches and the so-called exact FOD method are discussed below. The approach used in this Volume has been chosen mainly for the following reasons:

• the method describes the FOD reaction mathematically more accurately than the previous IPCC approaches,

• it is easy to use in a spreadsheet model,

• it gives, as a by-product, an estimate of changes in carbon stored in SWDS (annual changes in carbon stock, for both long-term and short-term storage as the mass-balance of conversions of carbon into CH4 and CO2 in the SWDS are maintained by the approach).

3A1.6.1 1996 Guidelines - The rate of reaction approach

In the Revised 1996IPCC guidelines (1996 Guidelines, (IPCC, 1997)) the estimation of the CH4 emissions from SWDS was based on the rate of reaction equation. This is a common way of looking at the mass transformation in a chemical reaction. This is obtained by differentiating Equation 3A1.2 with respect to time:

Equation 3A1.20 First order rate of reaction equation

DDOCm reaction rate = - d(DDOCm)/ dt = k • DDOCm0 • e~kt

The rate of reaction equation shows the rate of reaction at any time, and the rate of reaction moves along a curve. Therefore it has to be integrated to find the amount of reacted DDOCm over a period of time.

We want to find the DDOCm decomposed into CH4 and CO2 per calendar year. The start is year number 1 going from point 0 to point 1 on the time axis. Year number 1 is associated to point 1 on the time axis. Therefore the integration has to be performed from t-1 to t, which leads to an equation identical to Equation 3A1.5.

However, the equation presented in the 1996 Guidelines (Equation 4, Chapter 6) is:

Equation 3A1.21 IPCC 1996 Guidelines equation for DOC reacting in year T

In fact, this is the rate of reaction equation. Effectively this means that the yearly CH4 production is calculated from the rate of reaction at the end of the year. This is an approximation which involves summing a series of rectangles under the rate of reaction curve, instead of accurately integrating the whole area under the curve. An error is introduced by the approximation; the small triangles shown on the top of the columns in Figure 3A1.1 are neglected, and mass balance over the year is not obtained. The method based on the equation in the 1996 Guidelines using a half-life time of 10 years would give results 3.5 percent lower than the full mass balance calculations used in these Guidelines (see equations 3A.1.4-5).

However, where the method in the 1996 Guidelines is used with half life times developed specifically for this method, calculations will be correct.

Figure 3A1.1 Error introduced by not fully integrating the rate of reaction curve

Figure 3A1.1 Error introduced by not fully integrating the rate of reaction curve 3A1.6.2 IPCC 2GGG Good Practice Guidance

In the Good Practice Guidance and Uncertainty Management in National Greenhouse Gas Inventories (GPG2000, IPCC, 2000), Equation 5.1, a normalisation factor A is introduced into the rate of reaction equation. When this 'normalisation factor' is multiplied into Equation 5.1 the result is a solved integral:

 Equation 3A1.22 IPCC 2000GPG FOD equation for DDOCm reacting in year T DDOCm decompT = DDOCm0 • kt _ e-k(t+1)

This is equivalent to the correct equation (Equation 3A1.5) as it integrates the decay curve. However, for year 1 it integrates from point 1 to point 2 on the time axis, and therefore the CH4 formed in the first year of reaction is not counted (see Figure 3A1.2). This means that with a half life time of 10 years the GPG2000 equation calculates results that are 7 percent lower than those calculated with approach taking the full mass balance into account.

Figure 3A1.2 Effect of error in the GPG2000 equation

Figure 3A1.2 Effect of error in the GPG2000 equation 12 3 4 5 years

The intention of the normalisation factor has obviously been to fill in the small triangles on top of the columns in Figure 3A1.1. It fails because the normalisation factor used is equivalent to an integration going from point t to (t +1) on the time axis. As the integration using year number as a basis has to go from t-1 to 1, the normalisation factor filling in the whole area under the rate of reaction curve would be A = ((1/e"k) _ 1)/k.

3A1.6.3 Mathematically Exact First-Order Decay Model

The First Order Decay (FOD) model as described above can be shown to be mathematically equivalent to a model for which the total amount of DOC is assumed to be disposed at a single point in time in each disposal year, i.e., on a single date. If there is no delay in the commencement of the decay process, this date would be the middle of the year, i.e., 1st of July, with a delay of 6 months the assumed reaction start with the full amount of material is 31st December/1st January. This assumption, though counter-intuitive, leads to numerical errors that are small compared to the uncertainty in the understanding of the chemical processes, activity data, emission factors and other parameters of the emission calculation.

An alternative formulation of the FOD method is presented here for completeness. The delay in the commencement of the decay process can be represented, and simple recursive formulations can be given.

Equation 3A1.23 represents the formulation of the FOD with disposal rate D(t). The first term in the bracket represents the inflow into the carbon pool in the SWDS (disposal), the second term represents the outflow from the site (carbon in form of CH4); the sum of the two terms represents the overall change in carbon stock in the SWDS.

Equation 3A1.23 FOD with disposal rate D(t)

Where:

dDDOCm(t) = change in DDOCm at time t D(t) = DDOCm disposal rate at time t

DDOCm(t) = DDOCm available at time t for decay

If there is a delay of A years in the commencement of the decay process after the DDOCm has been disposed, it will be necessary to distinguish the part of the stock that is available for decay, to which Equation 3A1.23

applies, and the inert part of the stock. For a disposal rate D(t) that is constant during each disposal year (and equal to the amount of DDOC disposed during that year divided by one year) it can be shown that the carbon stocks at the end of year i can be expressed in terms of the carbon stocks at the end of year i-1 and the amounts of disposal in year i and year i-1 (Pingoud and Wagner, 2006):

Equation 3A1.24 Degradable organic carbon accumulated during a year

DDOCma( i +1) = a • DDOCma( i) + b • DDOCmd(i -1) + c • DDOCmd(i)

Where:

DDOCma (i) = DDOCm stock in the SWDS at the beginning of year i, Gg C

DDOCmd (i) = DDOCm disposed during year i, Gg C

A = delay constant, in years (between 0 and 1 years)

For an immediately starting decay (A=0), the constant b is equal to zero, so that Equation 3A1.24 reduces to an equation that relates the carbon pool in a given year i to the carbon pool in the previous year i-1 and the amount of DOC being deposited during year i.

It can further be shown (Pingoud and Wagner, 2006) that this form can be used to calculate recursively the corresponding CH4 produced in a given year:

Equation 3A1.25 CH4 generated during a year

CH4 gen( i) = q •[a' • DDOCma( i) - b • DDOCmd (i -1) + c' • DDOCmd (i)]

Where:

CH4 gen (i) = CH4 generated during year i, Gg C

DDOCma(i) = DDOC stock in the SWDS at the beginning of year i, Gg C

DDOCmd(i) = DDOC disposed during year i, Gg C

b' = 1/k • (e-k(1-A)-e-k) - A • e-k = b (constant)

c' = 1- A - 1/k • (1-e-k(1-A)) = 1 - c (constant)

### References

Pingoud, K. and Wagner, F. (2006). Methane emissions from landfills and decay of harvested wood products: the first order decay revisited. IIASA Interim Report IR-06-004.

Barrow, Gordon M. (1996). Physical Chemistry, Mc Graw Hill, NewYork, 6th ed.

IPCC (2000). Good Practice Guidance and Uncertianty Management in National Greenhouse Gas Inventories. Penman, J., Kruger D., Galbally, I., Hiraishi, T., Nyenzi, B., Enmanuel, S., Buendia, L., Hoppaus, R., Martinsen, T., Meijer, J., Miwa, K. and Tanabe, K. (Eds). Intergovernmental Panel on Climate Change (IPCC), IPCC/OECD/IEA/IGES, Hayama, Japan.

IPCC (1997). Revised 1996 IPCC Guidelines for National Greenhouse Inventories. Houghton, J.T., Meira Filho, L.G., Lim, B., Tréanton, K., Mamaty, I., Bonduki, Y., Griggs, D.J. and Callander, B.A. (Eds). Intergovernmental Panel on Climate Change (IPCC), IPCC/OECD/IEA, Paris, France.

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### Responses

• giovanna
What is a first order decay model?
11 days ago