Copper flows in buildings, infrastructure and mobiles: a dynamic model and its application to Switzerland

Abstract

During the last century, the consumption of materials for human needs increased by several orders of magnitude, even for non-renewable materials such as metals. Some data on annual consumption (input) and recycling/waste (output) can often be found in the federal statistics, but a clear picture of the main flows is missing. A dynamic material flow model is developed for the example of copper in Switzerland in order to simulate the relevant copper flows and stocks over the last 150 years. The model is calibrated using data from statistical and published sources as well as from interviews and measurements. A simulation of the current state (2000) is compared with data from other studies. The results show that Swiss consumption and losses are both high, at a level of about 8 and 2 kg/(cap year), respectively, or about three times higher than the world average. The model gives an understanding of the flows and stocks and their interdependencies as a function of time. This is crucial for materials whose consumption dynamics are characterised by long lifetimes and hence for relating the current output to the input of the whole past. The model allows a comprehensive discussion of possible measures to reduce resource use and losses to the environment. While increasing the recycling reduces losses to landfill, only copper substitution can reduce the different losses to the environment, although with a time delay of the order of a lifetime.

Appendices

Appendix A1: Model equations for buildings

For the explanation of the equations below, see “Mathematical model” section.

Buildings

Balance equation for stocks:

$$ \dot{M}^{(3)} (t) = A_{13} (t) + A_{23} (t) - A_{36} (t) - A_{37} (t) $$

(3)

Abrasions/corrosion:

$$ A_{37} (t) = a_{Build} (t) \cdot M^{(3)} (t) $$

(4)

Input into buildings:

$$ A_{13} (t) + A_{23} (t) = I_{\text{Inner}} (t) + I_{\text{Outer}} (t) $$

(5)

Copper stock in buildings:

$$ M^{(3)} (t) = M_{\text{Inner}} (t) + M_{\text{Outer}} (t) $$

(6)

Copper import into buildings:

$$ A_{13} (t) = k_{\text{ImpBuild}} \cdot (A_{13} (t) + A_{23} (t)) $$

(7)

a _Build(t) is a parameter function describing the specific abrasion and corrosion rate per stock unit per year. k _ImpBuild(t) is the fraction of copper imported into buildings.

Envelope of buildings

Balance equation:

$$ \dot{M}_{\text{Outer}} (t) = I_{\text{Outer}} (t) - O_{\text{Outer}} (t) $$

(8)

Copper stock:

$$ M_{\text{Outer}} (t) = P_{\text{Outer}} (t) $$

(9)

Renewing:

$$ O_{\text{Outer}} (t) = \int\limits_{0}^{t} {k_{\text{Outer}} (t,t^{\prime } ) \cdot I_{\text{Outer}} (t,t^{\prime } ){\text{d}}t^{\prime } } $$

(10)

P _outer(t) is the copper stock in the OuterBuild. k _outer(t,t′) is the transfer function or lifetime distribution of copper installed in the OuterBuild at a time t′.

Interior of buildings

For M _Inner(t), I _Inner(t) and O _Inner(t) similar equations to (8)–(10) apply.

Recycling:

$$ A_{62} (t) = k_{\text{Rec}} (t) \cdot (A_{62} (t) + A_{68} (t)) $$

(11)

k _Rec(t) is the fraction of recycling in the total output of dismantling/demolition.

Appendix A2: Calibration of the parameter functions

Stock for buildings P _Inner, P _Outer

Time series for the past of these stocks were compiled in Wittmer (2006) on the basis of the data sources mentioned above.

These time series show growth behaviour which is typical for all kinds of stocks. Logistic or sigmoidal curves are simple and turned out to describe growth adequately Fisher and Pry (1970). Logistic growth curves, and in particular linear-logistic and double logistic growth curves, were therefore fitted to these time series.

Logistic growth is exponential at the beginning, linear in the middle of the growth phase and decreasing towards the end until saturation is reached. Linear-logistic growth shows a linear pattern at the beginning of the growth period, followed by logistic growth. The derivation is continuous at the transition point.

This growth behaviour can be described mathematically as follows:

$$ P_{1} (t) = \left\{ {\begin{array}{ll} {a_{1} (t - t_{0} )} & {t_{0} \le t \le t_{1} } \\ {p_{\text{init}} + {\frac{{p_{\text{sat}} - p_{\text{init}} }}{{1 + e^{{ - \alpha (t - t_{\text{turn}} )}} }}}} & {t_{1} \le t} \\ \end{array} } \right. $$

(12)

t ₀ is the initial point of the growth, a ₁ the slope of the linear growth, p _init the initial value of the logistic part in the distant past t → −∞, p _sat the saturation value in the far future t → −∞, α is proportional to the maximum growth rate and t _turn is the turning point of the growth curve respectively.

The linear-logistic curve was chosen for buildings and mobiles because it proved to be more appropriate to the data than a simple logistic curve. An important parameter of the growth curves is the growth rate α, since typical values of this parameter are known for industrial or economics processes. The results of the non-linear fit for α are α _InnerBuild = 0.069 and α_OuterBuild = 0.057, respectively. These values are lower or comparable to typically observed growth rates for industrial products up to 0.15, see Bader et al. (2006). The results are presented in Fig. 6.

Fig. 6

Fitted growth curves and data sets for stocks of buildings

Lifetime distribution: k _Inner, k _Outer

The lifetime distribution can be described by a transfer function k(t _Op, k _Ip) in the two variables t _Op and t _Ip describing the amount of input at time t _Ip which leaves the balance volume at time t _Op.

The following two-parameter Gauss function was used as the lifetime distribution:

$$ k(t,t^{\prime } ) = {\frac{1}{{N_{0} }}}e^{{ - {\frac{{(t - t^{\prime } - \tau (t^{\prime } ))^{2} }}{{2 \cdot (\sigma (t^{\prime } ))^{2} }}}}} $$

(13)

N ₀: Normalisation factor; τ(t′), σ(t′): average lifetime and widths of the lifetime distributions of copper inputs at input time t′. Note that τ and σ are the values of the maximum and width of the lifetime distribution rather than the average and standard deviation, respectively. This is because the lifetime distribution is a truncated normal distribution. For the usual cases where σ ≤ τ/2, however, the difference between these quantities is small, namely, 2.9% for σ = τ/2 and 0.1% for σ = τ/3, etc. We therefore refer to τ and σ as the average and standard deviations of the lifetime distribution.

This lifetime distribution was discussed in detail in Baccini and Bader (1996) and applied in many case studies (Zeltner et al. 1999; Müller et al. 2004; Binder et al. 2001; Bader et al. 2006).

Estimates for τ and σ were obtained as follows. Since no time series data are available, we assume τ and σ to be a constant function of time in the sense of a first approximation.

Buildings: k _Inner, k _Outer: τ and σ are obtained from the experience of experts and tradesmen.

The estimated parameters τ and σ are listed in Table 4.

Table 4 Values and uncertainties for parameter functions that are a constant function of time

Abrasion/corrosion coefficients: a _Build

The abrasion/corrosion was assumed to be proportional to the surfaces exposed. For the infrastructures and buildings, the relevant surfaces of cables and sheets were related to the corresponding masses. The specific abrasion and corrosion rates were found in the literature [von Arx (1998): abrasion rate: 0.0036/year, Faller (2001): corrosion rate: 1.8 g/(m² year), He et al. (2001): corrosion rate for Sweden 1.3 g/(m² year)].

Appendix A3: Adaptation of the model to specific scenarios

Scenario: Copper ban in the envelope of buildings

The adaptation of the model is very simple: Eq. 9 for the copper stock of the building envelope has to be changed as follows:

Copper stock:

$$ M_{\text{Outer}} (t) = P_{\text{Outer}} (t)\;\;\;\;{\text{for}}\;t \le t_{\text{Ban}}^{{({\text{Outer}})}} $$

(9a)

$$ I_{\text{Outer}} (t) = 0\;\;\;\; {\text{for}}\;t > t_{\text{Ban}}^{{({\text{Outer}})}} $$

t ^(Outer)_Ban is the time at which the copper ban begins for the envelope of the buildings.

Scenario: Copper ban + copper dismantling in the envelope of buildings

In addition to the modification equation (9a) above, Eq. 10 must also be adapted.

Renewal and dismantling:

$$ O_{\text{Outer}} (t) = \int\limits_{0}^{t} {k_{\text{Outer}} (t,t^{\prime } )} I_{\text{Outer}} (t,t^{\prime } ){\text{d}}t^{\prime } \;\;\;\;{\text{for}}\;t \le t_{\text{Ban}}^{{({\text{Outer}})}} $$

(10a)

$$ O_{\text{Outer}} (t) = O_{\text{Outer}}^{{({\text{Renew}})}} (t) + O_{\text{Outer}}^{{({\text{Dism}})}} (t)\;\;\;\;{\text{for}}\;t > t_{\text{Ban}}^{{({\text{Outer}})}} $$

where

$$ O_{\text{Outer}}^{{({\text{Renew}})}} (t) = \int\limits_{{t_{\text{Ou}} (t)}}^{t} {k_{\text{Outer}} (t,t^{\prime } )} I_{\text{Outer}} (t^{\prime } ){\text{d}}t^{\prime } $$

(10b)

$$ O_{\text{Outer}}^{{({\text{Dism}})}} (t) = P_{\text{Outer}}^{{({\text{Dism}})}} (t) \cdot M_{\text{Outer}} (t) $$

(10c)

O ^(Renew)_Outer (t) and O ^(Dism)_Outer (t) are the copper outputs of OuterBuild due to renewing and dismantling, respectively. P ^(Dism)_Outer (t) is the specific dismantling rate per stock unit per year. t _Ou(t) is the age (relative to the initial time 0) of the oldest copper in OuterBuild that has not yet been removed due to renewing and dismantling. For t _Ou(t), the following ordinary differential equation applies:

$$ {\frac{{{\text{d}}t_{\text{Ou}} }}{{{\text{d}}t}}} = {\frac{{O_{\text{Outer}}^{{({\text{Dism}})}} (t)}}{{I_{\text{Rest}} (t,t_{\text{Ou}} (t))}}} $$

(10d)

where

$$ I_{\text{Rest}} (t,t^{\prime } ) = I_{\text{Outer}} (t^{\prime } )\left[ {1 - \int\limits_{{t^{\prime } }}^{t} {k_{\text{Outer}} (t^{\prime \prime } ,t^{\prime } ){\text{d}}t^{\prime \prime } } } \right] $$

is the rest of the copper output into OuterBuild at a time t′ and not renewed until a time t.

Equations (10a) are a generalisation of the relevant equations in Müller et al. (2004) and Bader et al. (2006). For reasons of simplicity in Eq. 10a, it was assumed that the copper ban and forced dismantling have the same onset time.

However, this is not a restricting assumption. Note that Eq. 10c is a possible but reasonable approach to the dismantling flow O ^(Dism)_Outer , representing a dismantling process that is proportional to the stock.