# Risk Evaluation for Coating Thickness Conformity Assessment

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{®}456 device (Elcometer Limited, Edge Lane, 136 Manchester, UK). These probes for coating thickness gauge can measure ferrous and non-ferrous applications with automatic substrate detection. Measurements can be performed on smooth, rough, thin, and curved surfaces in accordance with national and international standards. Before starting the measurement of the dry coating film thickness, the device was calibrated using a set of calibration foils (Elcometer 990 Calibration Foils, Manchester, UK). In our measurement case, the device was calibrated to a thickness of 75.1 μm using the combined calibration foils or ‘shims’ with an uncoated substrate (Zero Test Plate). The results of the measurement and the pertaining standard uncertainty measures are provided in Table 1.

_{L}= 70 µm and T

_{L}= 80 µm. In order to analyze the impact of the quality of the measurement system on producer and consumer risk, an analysis was also carried out for the standard measurement uncertainties ${\mathrm{u}}_{\mathrm{m}}$ = 2 µm, for the straight part of the pipe, and for the standard measurement uncertainties ${\mathrm{u}}_{\mathrm{m}}$= 4 µm for the pipe junction.

_{0}(η). Considering that the coating thickness is always strictly greater than 0, for the value of the argument of prior distribution holds that η > 0. For modelling of the prior distribution, two quantities associated with the random variable Y are used: best estimate ӯ and standard deviation ${\mathrm{u}}_{0}.$ In accordance with [1], in this research, the gamma distribution Γ(η; α, λ) given by the following formula was chosen as the prior:

_{m}. The value of the measured quantity is denoted by ${\mathsf{\eta}}_{\mathrm{m}},$ and the associated standard measurement uncertainty is denoted by ${\mathrm{u}}_{\mathrm{m}}$ [20]. Those data are modelled via the likelihood function for the normal distribution denoted by $\mathrm{h}\left({\mathsf{\eta}}_{\mathrm{m}}|\mathsf{\eta}\right)$ which is given by the following formula:

_{L}, T

_{U}] is interval of permissible values of a item of interst, given by standard for given products. Labels T

_{L}and T

_{U}are lower and upper tolerance limit, respectively.

_{L}= 70 μm whose upper limit theoretically is placed in infinity, and the tolerance interval with the lower limit T

_{L}= 80 μm with the upper limit placed also at infinity. The conformance probability that the item of interest is within the tolerance interval can be calculated as:

_{L}, A

_{U}], where A

_{L}and A

_{U}are the lower and upper limits of the acceptance interval, respectively. The acceptance interval and the tolerance interval can be in a variety of relationships with one another [32]. In order to minimize the consumer’s risk, in this research the acceptance interval has been placed within the tolerance interval, Figure 1.

_{m}is within the acceptance interval, the global consumer’s risk R

_{C}can be calculated as follows:

_{m}is outside the acceptance interval, global producer’s risk R

_{P}can be calculated from the following formula:

## 3. Models and Results

#### 3.1. Straight, Outside Section of the Pipe

_{L}, W

_{U}]. The lower limit of the range for the coating thickness, W

_{L}, is obtained for r = −1, and the upper limit of the range for the coating thickness, W

_{U}, is obtained for r = 1. The upper limit of the coating thickness W

_{U}is also the lower limit of the acceptance interval, i.e., W

_{U}= A

_{L}is valid. The coating thickness ranges for models ${\mathrm{M}}_{1},{\text{}\mathrm{M}}_{2},{\text{}\mathrm{M}}_{3}$, and ${\mathrm{M}}_{4}$ are [66, 74] μm, [76, 84] μm, [62, 78] μm, and [72, 88] μm, respectively. In all of the models, the highest consumer risk and the lowest producer risk are achieved when the coating thickness is equal to W

_{L}. The lowest consumer risk and the greatest risk to the producer are achieved when the coating thickness is equal to W

_{U}, i.e., at the lower limit of the acceptance interval A

_{L}, Figure 2. The lower limits of the acceptance interval for models ${\mathrm{M}}_{1},{\text{}\mathrm{M}}_{2},{\text{}\mathrm{M}}_{3},$ and ${\mathrm{M}}_{4}$ are 74 μm, 84 μm, 78 μm, and 88 μm, respectively. For these values of the lower limits, the following consumer’s and producer’s risks were obtained. For model ${\mathrm{M}}_{1}$, when A

_{L}= 74 μm, the consumer’s risk holds R

_{C}= 0.018%, and the producer’s risk holds R

_{P}= 5.287%. For model ${\mathrm{M}}_{2},$ when the lower limit of the acceptance interval is A

_{L}= 84 μm, the consumer’s risk holds R

_{C}= 0.038%, and the producer’s risk holds R

_{P}= 9.145%. For model ${\mathrm{M}}_{3}$, for A

_{L}= 78 μm, the consumer’s risk holds R

_{C}= 0.032%, and the producer’s risk holds R

_{P}= 12.663%. For model ${\mathrm{M}}_{4}$ when A

_{L}= 78 μm, the consumer’s risk holds R

_{C}= 0.069%, and the producer’s risk holds R

_{P}= 19.025%.

_{L}, W

_{U}] = [W

_{L}, A

_{L}] is the lower limit of the T

_{L}tolerance interval. This value divides the interval [W

_{L}, A

_{L}] into two intervals, interval [W

_{L}, T

_{L}] and interval [T

_{L}, A

_{L}]. Interval [W

_{L}, T

_{L}] is not the interval of permitted values for the thickness of the epoxy coating, but due to the measured uncertainty, it may happen that the consumer is supplied with such a product. One of the goals of this paper is to reduce the possibility of the delivery of pipes in which the thickness of the epoxy coating is within the interval [W

_{L}, A

_{L}]. Interval [T

_{L}, A

_{L}] is the guard band interval. Outside this interval, in the area from A

_{L}to infinity, the consumer’s risk values fall, while the values for producer’s risk rise.

_{L}= 70 μm are the same and equivalent to 0.9251. This means that there is 92.51% of conforming coatings and 7.49% of non-conforming coatings. The percentage of falsely rejected products is obtained by subtracting the producer’s risk value from the value for conformance probability. The percent of falsely accepted coatings is obtained by subtracting the consumer’s risk from the percentage of non-conforming coatings, Table 2.

_{L}= 80 μm is significantly lower compared to models ${\mathrm{M}}_{1}$ and ${\mathrm{M}}_{3}$ and is equivalent to 0.7659. In these models, 76.59% of the measurements are in accordance with the standards and 23.41% of the measurements do not comply with the standards. It is notable that there are significantly more unfavorable results for the number of accepted measurements in the ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{4}$ models compared to models ${\mathrm{M}}_{1}$ and ${\mathrm{M}}_{3}$. The explanation for such differences stems from the difference between the values for the best estimation ӯ and the lower limit of the interval tolerance. For models ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{4}$, this difference is 12 μm, while for the ${\mathrm{M}}_{1}$ and ${\mathrm{M}}_{3\text{}}$, this difference is 22 μm.

_{L}, T

_{L}], it is necessary to increase the thickness of the epoxy coating. This increases the conformance probability and the number of accepted products that meet the standards and reduces the risk to consumers and the risk to producers.

_{L}= $\mathsf{\u04ef}-2{u}_{0}.$ In this case, for models ${\mathrm{M}}_{1}$ and ${\mathrm{M}}_{3},$ the best estimation ӯ = 102 μm, and for models ${\mathrm{M}}_{2}$ and ${\mathrm{M}}_{4}$ it holds that the best estimation is ӯ = 112 μm. Now, the new models were defined as: ${{\mathrm{M}}_{1}}^{\prime}\left(70,\text{}102,\text{}16,\text{}2\right),{{\mathrm{M}}_{2}}^{\prime}\left(80,\text{}112,\text{}16,\text{}2\right),{{\text{}\mathrm{M}}_{3}}^{\prime}\left(70,\text{}102,\text{}16,\text{}4\right)$, and ${{\mathrm{M}}_{4}}^{\prime}\left(80,\text{}112,\text{}16,\text{}4\right).$

_{C}= 0.005%. At the lower limit of the acceptance interval, the highest producer’s risk was determined, and it amounts to R

_{P}= 1.7%. If we want to ensure that the values for the thickness of the epoxy coating are outside the interval [W

_{L}, T

_{L}], then the consumer’s risk in model ${{\mathrm{M}}_{\mathrm{i}}}^{\prime},\text{}\mathrm{i}=1,2,3,4$ for r = −1, denoted by R

_{−1}, must be less than the consumer’s risk in model ${\mathrm{M}}_{\mathrm{i}},\text{}\mathrm{i}=1,2,3,4$ for r = 0. The risk obtained for r = 0, i.e., the risk that is obtained when T

_{L}= A

_{L}, is called the shared risk. This risk is denoted by R

_{0}. In model ${\mathrm{M}}_{1}$, shared risk, i.e., the consumer’s risk at the lower limit of the tolerance interval is equivalent to R

_{0}= 0.75%, and in the ${{\mathrm{M}}_{1}}^{\prime}$ model, the risk R

_{−1}= 0.76%. This means that it may happen that the thickness of the coating is within the interval of non-allowed values, and for this model, the value ӯ needs to be increased. It is enough to put that ӯ = 103 μm. Then, R

_{−1}= 0.63% and the coating thicknesses are within the allowed values. The ӯ = 103 μm risks at the lower limit of the acceptance interval are now R

_{C}= 0.004% and R

_{P}= 1.47%. In the ${{\mathrm{M}}_{2}}^{\prime}$ model, the minimum consumer’s risk, at the lower limit of the acceptance interval is R

_{C}= 0.005%, and the highest risk to the producer is R

_{P}= 1.72%. Since R

_{0}= 1.6% and R

_{−1}= 0.79%, the thicknesses of the coating are within the allowed values. In the case of model ${{\mathrm{M}}_{3}}^{\prime}$, the lowest consumer’s risk is R

_{C}= 0.008%, and the highest producer’s risk is R

_{P}= 5%. Here, R

_{−1}= 1.07% and R

_{0}= 1.31%, so the coating thicknesses are within the allowed values. For the ${{\mathrm{M}}_{4}}^{\prime}$ model, the lowest consumer’s risk is R

_{C}= 0.008% and R

_{P}= 5.02%. The coating thicknesses are within the allowed values because R

_{−1}= 1.12% and R

_{0}= 2.96%. The values ${\mathrm{u}}_{0},$ ${\mathrm{u}}_{\mathrm{m}}$, and T

_{L}remained unchanged.

#### 3.2. Pipe Junction

_{C}= 0.007%, and the highest producer risk is R

_{P}= 1.87%. For model ${\mathrm{M}}_{6}$, these risks are R

_{C}= 0.01% and R

_{P}= 2.55%. The risks for the ${\mathrm{M}}_{7}$ model are R

_{C}= 0.014% and R

_{P}= 4.84%. For the ${\mathrm{M}}_{8}$ model, they are R

_{C}= 0.019% and R

_{P}= 6.17%, Figure 4.

_{C}= 0.001%, and the highest producer’s risk is R

_{P}= 0.31%. For this model, R

_{−1}= 0.20% which is less than R

_{0}= 0.33% and thus ensures that the thickness of the epoxy coating is within the allowed values. For the ${{\mathrm{M}}_{6}}^{\prime}$ model, the consumer’s risk of R

_{C}= 0.001% and the producer’s risk of R

_{P}= 0.34% were obtained. Since R

_{−1}= 0.23% and R

_{0}= 0.46%, the thickness of the coating are within the allowed values. For the ${{\mathrm{M}}_{7}}^{\prime}$ model, it holds that R

_{C}= 0.002% and R

_{P}= 1.1%. The results obtained show that R

_{−1}= 0.29% and R

_{0}= 0.59%; these coating thicknesses are within the allowed values. The risks for the ${{\mathrm{M}}_{8}}^{\prime}$ model are R

_{C}= 0.002% and R

_{P}= 1.15%. The coating thicknesses are within the allowed values because R

_{−1}= 0.33% and R

_{0}= 0.86%.

#### 3.3. Common Risk

## 4. Discussion

_{0}and the standard measurement uncertainty ${\mathrm{u}}_{\mathrm{m}}$ which characterize the quality of the coating application process and the quality of the measurement system. The models for the evaluation of the conformity with the condition of the lesser standard measurement uncertainty of the measurement system ${\mathrm{u}}_{\mathrm{m}}$ have been simulated. The models for the straight and junction sections of the pipe have been tested. The analysis was carried out for the recommended epoxy coating thickness, in accordance with the EN 877 standard, in the amount of T

_{L}= 70 μm and T

_{L}= 80 μm.

_{L}, the same layer thickness ӯ, and the same standard deviation ${\mathrm{u}}_{0},$ but with different standard measurement uncertainties ${\mathrm{u}}_{\mathrm{m}}$, then models with a smaller value of the standard measurement uncertainty ${\mathrm{u}}_{\mathrm{m}}$ have lower risks, as shown in Figure 2 and Figure 4. This speaks to the importance of measured uncertainty and its impact on risk assessment.

## 5. Conclusions

- The lesser value of the measurement uncertainty ${\mathrm{u}}_{\mathrm{m}}$ leads to a reduced probability that the rejected product is conforming. This benefits the producer. A larger value of standard uncertainty leads to a lager probability that the accepted product is non-conforming.
- The lesser value of the standard measurement uncertainty ${\mathrm{u}}_{\mathrm{m}}$ benefits the consumers in that it protects them from non-conforming products, but it also benefits the producers in the sense that it reduces the likelihood of a wrongful rejection of the conforming product when global risk is calculated. The greater value of the standard measurement uncertainty um harms producers in the sense of the wrongful rejection of the conformity of the product and it also damages the consumer in the sense of the use of non-conforming products when calculating global risk.
- Measurement uncertainty plays a significant role in the conformity assessment process, especially with decisions based on the results of those measurements that are close to the tolerance limit and may be inaccurate and may lead to unintended consequences.
- In order for the likelihood of wrongful decisions being made to be contained within acceptable parameters, a consensus between producers and consumers is paramount. When estimating risk, it is essential to take stock of the measurement uncertainty of the measurement results.
- The rule of divided risk can be proposed as a consensus between producers and consumers. When such a rule is deployed, both the producer and the consumer accept or reject a sample as conforming or non-conforming. As the very name suggests, divided risk says that by using this decision rule, both the producer and the consumer share the consequences for the wrongfully made decisions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Acceptance and tolerance interval [32].

**Figure 2.**(

**a**) Consumer’s risk and (

**b**) producer’s risk, for models ${\mathrm{M}}_{1},{\text{}\mathrm{M}}_{2},{\text{}\mathrm{M}}_{3}\text{and}{\text{}\mathrm{M}}_{4}$.

**Figure 3.**The relationship between the applied thickness of the coating and the consumer’s and producer’s risk. (

**a**) Change in consumer risk with increasing coating thickness, and (

**b**) change in consumer risk with increasing coating thickness, for models ${\mathrm{M}}_{1},{\text{}\mathrm{M}}_{2},{\text{}\mathrm{M}}_{3}\text{and}{\text{}\mathrm{M}}_{4}$.

**Figure 4.**(

**a**) Consumer’s risk and (

**b**) producer’s risk, models ${\mathrm{M}}_{5},{\text{}\mathrm{M}}_{6},{\text{}\mathrm{M}}_{7}\text{and}{\text{}\mathrm{M}}_{8}$.

**Figure 5.**Relationship between the applied thickness of the coating and the consumer’s and producer’s risk. (

**a**) Change in consumer risk with increasing coating thickness, and (

**b**) change in consumer risk with increasing coating thickness, for models ${\mathrm{M}}_{5},{\text{}\mathrm{M}}_{6},{\text{}\mathrm{M}}_{7}\text{and}{\text{}\mathrm{M}}_{8}$.

Arithmetic Mean ӯ, μm | Standard Deviation ${\mathsf{u}}_{0},$ μm | Standard Measurement Uncertainty ${\mathsf{u}}_{\mathbf{m}},$ μm | |
---|---|---|---|

Straight part | 92 | 16 | 4 |

Pipe junction | 170 | 69 | 8 |

Model | p_{c} | Accepted/% | Falsely Rejected/% | Rejected/% | Falsely Accepted/% |
---|---|---|---|---|---|

M_{1} | 0.9251 | 87.224 | 5.287 | 7.472 | 0.018 |

M_{1′} | 0.9860 | 96.900 | 1.7 | 1.395 | 0.005 |

M_{2} | 0.7659 | 67.175 | 9.145 | 23.372 | 0.038 |

M_{2′} | 0.9852 | 96.800 | 1.72 | 1.475 | 0.005 |

M_{3} | 0.9251 | 79.847 | 12.663 | 7.458 | 0.032 |

M_{3′} | 0.9860 | 93.600 | 5 | 1.392 | 0.008 |

M_{4} | 0.7659 | 57.565 | 19.025 | 23.341 | 0.069 |

M_{4′} | 0.9852 | 93.500 | 5.02 | 1.472 | 0.008 |

Model | p_{c} | Accepted/% | Falsely Rejected/% | Rejected/% | Falsely Accepted/% |
---|---|---|---|---|---|

M_{5} | 0.9593 | 94.06 | 1.87 | 4.063 | 0.007 |

M_{5′} | 0.9956 | 99.25 | 0.31 | 0.439 | 0.001 |

M_{6} | 0.9325 | 90.7 | 2.55 | 6.74 | 0.01 |

M_{6′} | 0.9949 | 99.15 | 0.34 | 0.509 | 0.001 |

M_{7} | 0.9593 | 91.09 | 4.84 | 4.056 | 0.014 |

M_{7′} | 0.9956 | 98.46 | 1.1 | 0.438 | 0.002 |

M_{8} | 0.9325 | 87.08 | 6.17 | 6.731 | 0.019 |

M_{8′} | 0.9949 | 98.34 | 1.15 | 0.508 | 0.002 |

Model | M_{1} | M_{1′} | M_{2} | M_{2′} | M_{3} | M_{3′} | M_{4} | M_{4′} |
---|---|---|---|---|---|---|---|---|

M_{5} | 82.04 | 91.14 | 63.18 | 91.05 | 75.10 | 88.04 | 54.15 | 87.95 |

M_{5′} | 86.57 | 96.17 | 66.67 | 96.07 | 79.25 | 92.90 | 57.13 | 92.80 |

M_{6} | 79.11 | 87.89 | 60.93 | 87.80 | 72.42 | 84.90 | 52.21 | 84.81 |

M_{6′} | 86.48 | 96.08 | 66.60 | 95.98 | 79.17 | 92.80 | 57.08 | 92.71 |

M_{7} | 79.45 | 88.27 | 61.19 | 81.18 | 72.73 | 85.26 | 52.44 | 85.17 |

M_{7′} | 85.88 | 95.41 | 66.14 | 95.30 | 78.62 | 92.16 | 56.68 | 92.06 |

M_{8} | 75.95 | 84.38 | 58.50 | 84.29 | 69.53 | 81.51 | 50.13 | 81.42 |

M_{8′} | 85.78 | 95.29 | 66.06 | 95.19 | 78.52 | 92.05 | 56.61 | 91.95 |

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## Share and Cite

**MDPI and ACS Style**

Božić, D.; Samardžija, M.; Kurtela, M.; Keran, Z.; Runje, B. Risk Evaluation for Coating Thickness Conformity Assessment. *Materials* **2023**, *16*, 758.
https://doi.org/10.3390/ma16020758

**AMA Style**

Božić D, Samardžija M, Kurtela M, Keran Z, Runje B. Risk Evaluation for Coating Thickness Conformity Assessment. *Materials*. 2023; 16(2):758.
https://doi.org/10.3390/ma16020758

**Chicago/Turabian Style**

Božić, Dubravka, Marina Samardžija, Marin Kurtela, Zdenka Keran, and Biserka Runje. 2023. "Risk Evaluation for Coating Thickness Conformity Assessment" *Materials* 16, no. 2: 758.
https://doi.org/10.3390/ma16020758