## 3. Calculation Procedure

The model used in this study, the Volume Correlation Model (VCM), has been described in detail in other studies [

19,

20,

21,

22,

23]. The model was implemented in Excel® spreadsheets in combination with MATLAB® [

36,

37]. The VCM can be used to evaluate the time difference between reported data on steel consumption and scrap collection. The model calculates the scrapped and apparent lifetimes of steel. The scrapped lifetime of steel is the (estimated) actual service lifetime of steel; the apparent lifetime is defined below.

In a previous study, the scrapped and apparent lifetimes of steel were calculated for Sweden and the world [

19,

20]. In this study, the lifetimes of steel were recalculated with Swedish and global steel data for 1900 to 2010, for direct comparison with the calculations for the U.S. as presented here.

The apparent lifetime is predominantly used to calculate moving averages of scrap recovery rates, as described below. The apparent lifetime is calculated by assuming full recovery (as scrap) of all steel in final products, i.e., every ton of steel that entered service as a finished product is assumed to be eventually recoverable as scrap. Clearly, this is not always the case, and incomplete recovery is considered when calculating the scrapped lifetime.

The apparent lifetime of steel (

λ_{app}) has been found as the time difference between a given year

t_{x} (with a known cumulative tonnage of scrap collected up to this time: the left term in the equation below) and the number of years (in the past) required to consume the same tonnage of steel:

In this expression, Δm_{scrap}(t) is the mass of scrap collected during year t (years since 1900), Δm_{consumed}(t) is the total steel consumption in year t, and λ_{app} is the apparent lifetime of steel. This expression is used to find the apparent lifetime (λ_{app}).

Not all consumed steel is recycled. The apparent recovery rate

η(

t) is found as the ratio of steel recycled in a given year to the moving average steel consumption over a longer period, taken to be 2

λ_{app}.

At the beginning of the time period considered in the calculations, the time period (for averaging the recovery rate) could be longer than the input data available. For calculating the recovery rate, the moving average was instead calculated over the time period zero to t, in cases where 2λ_{app} > t. For the input data used here, the shorter integration period was used for the first seven years of data for the automotive sector, four years for consumer goods, one year for the construction sector, and the first six years of industrial goods.

Given the long lifetime of steel, the non-recirculated amount of steel is a reserve that is potentially available for future collection. This reserve may become a loss if the steel is not collected after a significant time period. Since the annual apparent recovery rate is based on the moving average of steel consumption, it would not necessarily be equivalent to the recycling rate in any given year; however, the longer-term averages (weighted by tonnage of steel) of the apparent recovery rate and recycling rate would be equal. Using the (time-varying) recovery rate, the amount of potentially recyclable steel added from each year’s consumption is estimated as follows:

Here, potentially recyclable steel refers to previously consumed steel that has been recycled within an apparent lifetime of steel. The tonnage of recyclable steel depends not solely on the availability of a reserve of steel but also strongly on scrap price; this formulation helps to highlight this effect.

The scrapped lifetime of steel is calculated as the time difference (

λ_{scrapped}) between a given year

t_{x} (with a known cumulative tonnage of scrap collected) and time required to have accumulated the same tonnage of potentially recyclable steel in products:

In this expression, λ_{scrapped} is the scrapped lifetime of steel. As Equation (4) shows, the principle suggests that the cumulative mass of scrap collected up to a given year equals a fraction (the recovery rate) of the steel used in products up to a specific date λ_{scrapped} years earlier. Use of an annual recovery rate (rather than a constant ratio) helps to account for some of the variations in scrap collection.

The integration period used to calculate the moving average steel consumption rate affects the results. The effect of the integration period arises, in part, from fluctuations in steel consumption and recycling (in response to economic cycles) as well as the increase in steel consumption over the period considered. To test the effect of the integration period, the scrapped lifetime for total steel in the U.S. for the years 1900 to 2016 was calculated based on three different integration periods:

λ_{app}, 2

λ_{app}, and 3

λ_{app} (as used in Equation (2)). The resulting calculated scrapped lifetimes of steel for the different integration periods are shown in

Figure 3. The difference between the apparent lifetime and scrapped lifetime was larger with a shorter integration period. If this large difference were real, it would imply that a large proportion of the steel would be lost and never recycled, which does not appear to be realistic, given reported recycling rates. The results were also compared with weighted averages of the reported lifetime of steel (the box in

Figure 3) for 2010–2015. The reported sector lifetimes (listed in

Table 1) were weighted using either the relative amount of steel consumed by each sector (yielding an average lifetime of 29 years) or the relative amount of scrap recovered from each sector (yielding an average lifetime of 25 years). The proportions of steel consumed were as given in

Figure 2 and scrap ratios as in

Table 2. The calculated average lifetime was close to the scrapped lifetime of steel, calculated with 2

λ_{app} as the integration period. On the basis of these results, the integration period was chosen to be 2

λ_{app}.