# Characterizing the Urban Mine—Simulation-Based Optimization of Sampling Approaches for Built-in Batteries in WEEE

^{1}

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## Abstract

**:**

## 1. Introduction

^{3}containers [33]. However, WEEE and built-in batteries are more challenging due to their high (product) heterogeneity, locally different collection groups, inconsistent nomenclature, unharmonized classifications, as well as various influencing factors on the composition, such as short innovation cycles and varying product lifetime.

- Identification of WEEE with and without battery compartment and determination of the proportion of remaining batteries.
- Statistical description and analysis of distribution patterns for WEEE mass, battery mass (BM), and battery mass share (BMS) of built-in batteries in WEEE.
- Recommendation for determining MSS in the case of small data sets and unknown or inconclusive distribution patterns for BMS of built-in batteries in WEEE.

## 2. Materials and Methods

#### 2.1. Sampling and Classification

#### 2.2. Statistical Analysis

#### 2.3. Data-Driven Simulation: Bootstrapping

_{1}, x

_{2}, ..., x

_{n}) of size n

_{orig}with replacement is drawn from the original dataset. For each bootstrap sample, the resample statistics are calculated [64], indicated by an asterisk (*). The distribution of a resample statistic is called bootstrap distribution S* = S(x

_{1}, x

_{2}, ..., x

_{n}), which approximates the characteristics (center, spread, shape) of the population distribution [64]. For example, the approximated population mean (µ*) is the arithmetic mean of the bootstrap means and can be calculated according to Equation (1).

_{orig}, the number of repetitions is set to B = 5000. Adequate bootstrap simulation results depend on the minimum number of data points in the original dataset [58]. Guided by [58], we chose 15 as the minimum number of data points for bootstrap simulations (n

_{orig}≥ 15), which reduces the number of combinations for simulation of batteries and WEEE. Nevertheless, the robustness of this method is always based on the number of original data and repetitions B. While B is a matter of computational capacity, the challenge is to collect a sufficient amount of harmonized raw data for the simulation to increase robustness.

#### 2.4. Determining the Minimum Sample Size (MSS)

#### 2.4.1. Parametric Approach (PA): Assumption of Data Distributions

_{α}

_{/2}is 1.96 [31]. The coefficient of variation (VC) is used to express the data uncertainty gained by, e.g., a pilot study. We accept a relative error of ±10% (e

_{rel}= 0.1) to describe the whole population sufficiently accurate [33].

_{α}

_{/2}and e

_{rel}constant in Equation (2), the number of samples depends solely on the VC of the samples. The PA is performed for both the original data (VC) and the bootstrap distribution (VC*) to illustrate the influence of bootstrapping on the variation of the data.

#### 2.4.2. Non-Parametric Approach (NPA): Data-Driven Simulation with Bootstrapping

_{i}= n

_{orig}). The generated probability distribution approximates the population characteristic and is used to calculate the bootstrap confidence interval (95% CI*) (Figure 1a). Only combinations of WEEE and batteries (UNUkey-BATT, UNUkey-BATTkey) with at least 15 data points in the original dataset were used for simulation to ensure reasonable results [58] (see Supplementary Materials).

_{i}= 5 and increasing n

_{i}to the original number of samples n

_{orig}. For each subsample of n

_{i}, 1000 subsamples (B = 1000) were drawn [60], and the arithmetic mean (bootstrap mean ${\mathrm{x}}_{\mathrm{i}}^{*}$) was calculated (see Figure 1b).

_{i}. Thus, the MSS is approximated at n

_{i}= MSS, if the coverage increases above 90% (see arrow in Figure 1c). In other words, a sample number n with 90% coverage means that in 90% of samples with this sample size, the resulting value is likely to be within the 95% CI* of the population and consequently representative.

## 3. Results and Discussion

#### 3.1. Share of Waste Electrical and Electronic Equipment (WEEE) with and without Battery Compartment

#### 3.2. WEEE Characteristics

#### 3.3. Battery Characteristics

#### 3.3.1. Mass and Mass Share of Built-in Batteries

#### 3.3.2. Product-Specific Battery Characteristics

#### 3.3.3. Distribution Pattern and Bootstrap Simulation

#### 3.4. Minimum Sample Size to Determine Battery Mass Shares in WEEE

_{,}which was simulated with the initial number of sample in the data set (compare Figure 6e,f).

#### 3.5. Sampling Recommendation

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

BATTkey | n | LiPrim | LiRecharge | NiCd | NiMH | Pb | Zn | Unspecified |
---|---|---|---|---|---|---|---|---|

195 | 135 | 44 | 130 | 13 | 257 | 16 | ||

Mass (BM) | $\overline{\mathrm{m}}$ [g] | 3.9 | 74 | 310 | 69 | 1300 | 38 | 47 |

SD [g] | 3.2 | 120 | 440 | 110 | 1400 | 66 | 100 | |

VC [-] | 0.8 | 1.6 | 1.4 | 1.7 | 1.1 | 1.7 | 2.2 | |

$\tilde{\mathrm{m}}$ [g] | 3.0 | 23 | 67 | 26 | 800 | 23 | 7.0 | |

MAD [g] | 0.074 | 8.1 | 71 | 21 | 430 | 17 | 7.4 | |

95% CI [g; g] | [1.9; 8.8] | [8.9; 430] | [14; 1500] | [10; 490] | [15; 4300] | [2.3; 150] | [2; 300] | |

SW/SWlog [-] | 0/0 | 0/0 | 0/0 | 0/0 | 0/0 | 0/0 | 0/0 | |

S/Slog [-] | 5.4/1.9 | 2.2/0.99 | 2.1/0.28 | 3.0/0.93 | 1.3/−1.1 | 8.4/−0.27 | 2.7/0.7 | |

K/Klog | 42/4.7 | 3.5/1.2 | 4.5/−1.5 | 8.4/0.76 | 0.59/0.046 | 92/1.6 | 6.3/−0.87 | |

Mass share (BMS) | $\overline{\mathrm{x}}$ [%] | 2.3 | 20 | 23 | 19 | 46 | 18 | 12 |

SD [%] | 4.3 | 9.0 | 16 | 12 | 19 | 15 | 14 | |

VC [-] | 1.90 | 0.46 | 0.66 | 0.61 | 0.42 | 0.76 | 1.1 | |

$\tilde{\mathrm{x}}$ [%] | 0.045 | 20 | 22 | 18 | 48 | 15 | 8.4 | |

MAD [%] | 0.031 | 7.2 | 14 | 11 | 12 | 11 | 12 | |

95% CI [%; %] | [0.02; 12] | [1.9; 43] | [0.5; 46] | [0.65; 45] | [8.8; 74] | [2.0; 53] | [0.096; 43] | |

SW/SWlog | 0/0 | 0/0 | 0/0 | 0/0 | 0.28/0 | 0/0 | 0/0.12 | |

S/Slog | 1.8/0.92 | 0.83/−4.0 | 1.6/−2.8 | 0.72/−2.4 | −0.58/−1.5 | 1.4/−0.69 | 1.1/−0.76 | |

K/Klog | 1.5/−0.87 | 2.9/21 | 4.9/8.7 | −0.083/8.2 | −0.4/1.1 | 2.2/0.27 | 0.43/−0.57 | |

Occurrence in | UNUkeys | 19 | 16 | 11 | 15 | 7 | 16 | 9 |

subKeys | 23 | 27 | 15 | 24 | 8 | 29 | 10 |

UNUkey | n | Battery Mass | |||||

$\overline{{m}}$[g] | SD [g] | VC [-] | $\tilde{\mathrm{m}}$[g] | MAD [g] | 95% CI | ||

0201 | 23 | 26 | 28 | 1.1 | 23 | 12 | [2; 98] |

0202 | 13 | 39 | 35 | 0.89 | 46 | 64 | [2.9; 100] |

0204 | 29 | 360 | 240 | 0.67 | 260 | 260 | [80; 820] |

0205 | 48 | 26 | 9.6 | 0.36 | 23 | 6.2 | [14; 47] |

0301 | 51 | 120 | 690 | 5.7 | 12 | 16 | [1.1; 220] |

0302 | 118 | 3.4 | 3.2 | 0.94 | 3 | 0 | [2.8; 6.5] |

0303 | 29 | 210 | 190 | 0.92 | 250 | 300 | [2.1; 480] |

0305 | 52 | 33 | 54 | 1.7 | 22 | 15 | [11; 84] |

0306 | 93 | 29 | 18 | 0.61 | 21 | 5.1 | [15; 77] |

0401 | 121 | 22 | 14 | 0.66 | 22 | 16 | [3; 48] |

0402 | 29 | 73 | 160 | 2.2 | 24 | 12 | [2.9; 440] |

0403 | 7 | 71 | 73 | 1 | 37 | 21 | [10; 200] |

0406 | 18 | 36 | 63 | 1.8 | 18 | 11 | [2.7; 200] |

0501 | 16 | 79 | 120 | 1.6 | 33 | 39 | [3; 380] |

0506 | 15 | 240 | 710 | 3 | 32 | 27 | [3; 1900] |

0601 | 32 | 470 | 560 | 1.2 | 340 | 460 | [12; 2000] |

0701 | 42 | 38 | 25 | 0.66 | 34 | 24 | [3; 90] |

0702 | 5 | 33 | 6.2 | 0.19 | 35 | 4.1 | [24; 38] |

0901 | 31 | 10 | 11 | 1 | 7.9 | 0 | [2.6; 46] |

all | 750 | 76 | 270 | 3.6 | 21 | 23 | [2.1; 546] |

Battery mass share | |||||||

UNUkey | n | $\overline{\mathrm{x}}$ [%] | SD [%] | VC [-] | $\tilde{\mathrm{x}}$ [%] | MAD [%] | 95% CI |

0201 | 23 | 12 | 12 | 0.94 | 8.7 | 6.5 | [1; 38] |

0202 | 13 | 4.5 | 6.1 | 1.4 | 2.3 | 3.1 | [0.15; 18] |

0204 | 29 | 26 | 14 | 0.54 | 20 | 9.4 | [11; 54] |

0205 | 48 | 19 | 7.8 | 0.41 | 18 | 8.4 | [7.9; 38] |

0301 | 51 | 11 | 14 | 1.3 | 5.2 | 5 | [0.88; 56] |

0302 | 118 | 0.045 | 0.082 | 1.8 | 0.031 | 0.009 | [0.019; 0.098] |

0303 | 29 | 8.7 | 7.6 | 0.87 | 13 | 6.3 | [0.061; 20] |

0305 | 52 | 16 | 10 | 0.66 | 11 | 7.2 | [3; 38] |

0306 | 93 | 25 | 9.1 | 0.36 | 23 | 5.4 | [14; 47] |

0401 | 121 | 19 | 9.7 | 0.52 | 17 | 9.6 | [5; 41] |

0402 | 29 | 17 | 13 | 0.78 | 13 | 7.7 | [2.1; 57] |

0403 | 7 | 19 | 24 | 1.3 | 3.5 | 3.1 | [1.5; 60] |

0406 | 18 | 9.9 | 8.5 | 0.85 | 6.9 | 7.8 | [0.27; 28] |

0501 | 16 | 34 | 22 | 0.64 | 38 | 19 | [2.7; 69] |

0506 | 15 | 25 | 16 | 0.64 | 23 | 18 | [3.1; 54] |

0601 | 32 | 26 | 19 | 0.74 | 22 | 16 | [0.14; 79] |

0701 | 42 | 15 | 14 | 0.92 | 11 | 9.9 | [0.23; 38] |

0702 | 5 | 19 | 9.4 | 0.49 | 25 | 1.8 | [8.3; 26] |

0901 | 31 | 11 | 5.6 | 0.51 | 12 | 0 | [0.6; 23] |

all | 750 | 16 | 14 | 0.89 | 13 | 14 | [0.02; 48] |

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**Figure 1.**Non-parametric approach (NPA) to approximate the minimum sample sizes (MSS) with data-driven simulations. (

**a**) Simulated bootstrap population, (

**b**) sub-sampling from datasets with increasing sample size and 1000 repetitions, (

**c**) coverage of sub-sample results in percent lying within the population 95% confidence interval (CI*).

**Figure 2.**The proportion of WEEE devices (UNUkeys) with and without battery compartment (

**a**) and the proportion of WEEE with a battery compartment in which batteries remained or were missing (

**b**). The total number of appliances (N) is shown in white at the bottom. Figure data available in Supplementary Materials Table S3.

**Figure 3.**Battery mass (

**a**), battery mass share (

**b**), and WEEE mass (

**c**) differentiated according to their chemical systems (BATTkey). The mass is illustrated with a logarithmic scale. The lower and upper hinges of the boxplots correspond to the first and third quartiles (the 25th and 75th percentiles). The median is drawn as a bold horizontal line; the mean is shown as a square. Upper/lower whisker is the largest/smallest observation less/greater than or equal to upper/lower hinge +/− 1.5 ∗ interquartile range (IQR). The black dots represent values that lie outside of this range.

**Figure 4.**Occurrence and count of battery types (BATTkeys) in UNUkeys. Total number of devices considered: 790. Figure data are given in the Supplementary Materials Table S7.

**Figure 5.**Mass (

**a**) and mass share (

**b**) of batteries remaining in WEEE classified as UNUkeys and distinguished by their chemical system (BATTkey). The number of observations (n) is displayed on top of the graph (a). The lower and upper hinges of the boxplots correspond to the first and third quartiles (the 25th and 75th percentiles). The median is drawn as a bold horizontal line; the mean is shown as a square. Upper/lower whisker is the largest/smallest observation less/greater than or equal to upper/lower hinge +/− 1.5 ∗ IQR. The data of this figure can be found in the Supplementary Materials Table S7, Table S8, Table S10, Table S12, and Table S14.

**Figure 6.**Histogram and density distribution for the battery mass share (BMS) of LiRecharge and NiMH for mobile phones (UNUkey 0306). Original data (

**a**,

**d**), log-transformed data (

**b**,

**e**), and bootstrap sample means with B = 5000 (

**c**,

**f**). The (log-)normal distribution curve for the sample mean and SD is shown in dark red. The 95% CI* is drawn as a semi-transparent area in the corresponding BATTkey color.

**Figure 7.**NPA: Simulation of MSS using bootstrap simulation. The proportion of sub-samples that lie within the simulated 95% CI* (coverage) is plotted against the number of samples used to draw the sub-samples. A coverage of 90% is considered acceptable to achieve representable results with the given sample size (dashed line). The graphs show the three possible combinations using small information technology (IT) (0301), desktop personal computers (PCs) (0302), mobile phones (0306) and small consumer electronics (0401) as examples: (

**a**) all WEEE with and without battery compartment, (

**b**) only WEEE with battery compartment without battery specification (BATT), (

**c**) WEEE with batteries and BATTkey specification.

**Table 1.**Classification of waste electrical and electronic equipment (WEEE) categories (UNUkey main structure, the detailed classification can be found in Supplementary Materials Tables S16 and S17) and battery systems (BATT keys).

UNUkey | Description | BATTkey | Description |
---|---|---|---|

0001 | Central Heating (CH, household installed) | LiPrim | Lithium-based batteries, primary |

0002 | Photovoltaic panels (PV) | LiRecharge | Lithium-based batteries, rechargeable |

010x | Large household appliances (LHA) | Zn | Zinc-based batteries |

020x | Small household appliances (SHA) | NiCd | Nickel-cadmium based batteries |

030x | IT and telecom equipment (ITCE) | NiMH | Nickel-metal hydride batteries |

040x | Consumer equipment (CE) | Pb | Lead-acid batteries |

050x | Lighting equipment (LE) | Other | Other batteries (e.g., silver-oxide) |

060x | Electrical and electronic tools (EET) | Unspecified | Not specified or identifiable |

070x | Toys, leisure, and sports equipment (TLS) | ||

080x | Medical devices (MD) | BATT | No distinction of the battery system. |

090x | Monitoring and control instruments (MCI) | ||

100x | Dispensers (D) |

**Table 2.**Comparison of the approaches to determine the MSS with the PA using the original PA(VC) and bootstrap coefficient of variation PA(VC*) as well as the NPA.

UNUkey | n | BATTkey | PA(VC) | PA(VC*) | NPA | |||
---|---|---|---|---|---|---|---|---|

VC | MSS | VC* | MSS | 95% CI* | MSS | |||

0301 | 51 | BATT | 1.32 | 670 | 0.18 | 12 | [7.2; 15] | 40 |

29 | Zn | 1.03 | 420 | 0.19 | 14 | [6.9; 14] | 20 | |

0302 | 118 | BATT | 1.84 | 1300 | 0.17 | 11 | [0.03; 0.06] | 70 |

116 | LiPrim | 0.63 | 150 | 0.10 | 4 | [0.03; 0.04] | 20 | |

0306 | 93 | BATT | 0.36 | 50 | 0.04 | 1 | [24; 27] | 70 |

24 | NiMH^{ND} | 0.40 | 60 | 0.08 | 2 | [26; 35] | 20 | |

69 | LiRecharge | 0.29 | 30 | 0.06 | 1 | [22; 25] | 60 | |

0401 | 121 | BATT | 0.52 | 100 | 0.05 | 1 | [17; 20] | 90 |

105 | Zn | 0.51 | 100 | 0.06 | 2 | [17; 21] | 80 |

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## Share and Cite

**MDPI and ACS Style**

Mählitz, P.M.; Korf, N.; Sperlich, K.; Münch, O.; Rösslein, M.; Rotter, V.S.
Characterizing the Urban Mine—Simulation-Based Optimization of Sampling Approaches for Built-in Batteries in WEEE. *Recycling* **2020**, *5*, 19.
https://doi.org/10.3390/recycling5030019

**AMA Style**

Mählitz PM, Korf N, Sperlich K, Münch O, Rösslein M, Rotter VS.
Characterizing the Urban Mine—Simulation-Based Optimization of Sampling Approaches for Built-in Batteries in WEEE. *Recycling*. 2020; 5(3):19.
https://doi.org/10.3390/recycling5030019

**Chicago/Turabian Style**

Mählitz, Paul Martin, Nathalie Korf, Kristine Sperlich, Olivier Münch, Matthias Rösslein, and Vera Susanne Rotter.
2020. "Characterizing the Urban Mine—Simulation-Based Optimization of Sampling Approaches for Built-in Batteries in WEEE" *Recycling* 5, no. 3: 19.
https://doi.org/10.3390/recycling5030019