Experimentally Validated Analytical Solutions to Homogeneous Problems of Electrical Impedance Tomography (EIT) on Rectangular Cement-Based Materials
Abstract
:1. Introduction
2. Theoretical Solutions
2.1. Shunt Model for the Boundary Conditions of Cement-Based Samples
2.2. Derivation of Analytical Solutions
2.2.1. Case 1: Current Flows in and out of the Same Boundary
2.2.2. Case 2: Current Flows in from one Boundary and out from the Opposite Boundary
3. Experiments
3.1. Materials and Sample Preparation
3.2. EIT Measurements
4. Results and Discussion
4.1. Cutoff of Infinite Series Solutions
4.2. Experimental Validation of Current Injection Models
4.2.1. Dirac Delta Function
Case 1: Current Flows in and out of the Same Boundary
Case 2: Current Flows in from one Boundary and out from the Opposite Boundary
4.2.2. Heaviside Step Function
Case 1: Current Flows in and out of the Same Boundary Surface
Case 2: Current Flows in from one Boundary and out from the Opposite Boundary
4.2.3. Gaussian Function
Case 1: Current Flows in and out of the Same Boundary Surface
Case 2: Current Flows in from one Boundary and out from the Opposite Boundary
5. Conclusions
- The Shunt model can generally describe the boundary conditions of cement-based materials, which have low electrical conductivity. This was shown by both theoretical and experimental approaches. However, there are a few data having a significant discrepancy between some the calculated and measured voltages. Therefore, it is necessary to consider uncertainties such as the inhomogeneity of the sample and the humidity of the surface electrode.
- The analytical solutions were derived in the form of infinite series. A cutoff up to the 50th was effective in comparing the theoretical to the measured voltages.
- For cement-based elements, the convergence between theoretical and experimental voltages is higher when the current flows from one boundary to the opposite boundary of a sample than in and out of the same boundary.
- In comparisons between theoretical and experimental voltages, the Gaussian function shows the lowest RAE, which suggests that it describes current injection more accurately than the Dirac delta and Heaviside step functions. Especially, when σs corresponding to the necessary constant of the Gaussian function is approximately half of the electrode width, the Gaussian function provides the lowest RAE.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Cement (Type I) | Silica Sand | FSSAs | Silica Fume | Silica Power | Water | Super-Plasticizer |
---|---|---|---|---|---|---|
1.0 | 0.50 | 0.50 | 0.15 | 0.25 | 0.20 | 0.042 |
Diameter (μm) | Density (g/cm3) | Electrical Conductivity (106 Sm−1) |
---|---|---|
<390 | 3.56 | 11.2 |
Experimental Configuration | Measured | Calculated | Residual (Mea.-Cal.) | |Res.|/Cal. (%) | ||
---|---|---|---|---|---|---|
Set #1 | V1 | 0.384 | 0.950 | −0.566 | 59.579 | |
V2 | 0.964 | 1.058 | −0.094 | 8.885 | ||
V3 | 0.964 | 0.765 | 0.199 | 26.013 | ||
Set #2 | V1 | 1.269 | 2.008 | −0.739 | 36.803 | |
V2 | 2.433 | 2.373 | 0.059 | 2.486 | ||
V3 | 2.561 | 1.823 | 0.738 | 40.483 | ||
Set #3 | V1 | 2.011 | 2.773 | −0.762 | 27.479 | |
V2 | 3.738 | 3.431 | 0.307 | 8.948 | ||
V3 | 3.127 | 2.773 | 0.354 | 12.766 | ||
Set #4 | V1 | 0.872 | 1.058 | −0.186 | 17.580 | |
V2 | 1.367 | 1.315 | 0.052 | 3.954 | ||
V3 | 1.708 | 1.058 | 0.650 | 61.437 |
Experimental Configuration | Measured | Calculated | Residual (Mea.−Cal.) | |Res.|/Cal. (%) | |
---|---|---|---|---|---|
Set #1 | V1 | 17.923 | 17.896 | 0.027 | 0.151 |
V2 | 8.716 | 8.882 | −0.166 | 1.869 | |
Set #2 | V1 | 16.638 | 16.826 | −0.188 | 1.117 |
V2 | 7.981 | 7.921 | 0.060 | 0.757 | |
Set #3 | V1 | 15.412 | 15.497 | −0.085 | 0.548 |
V2 | 7.090 | 6.852 | 0.238 | 3.473 | |
Set #4 | V1 | 14.015 | 14.428 | −0.413 | 2.862 |
V2 | 6.064 | 6.079 | −0.015 | 0.247 | |
Set #5 | V1 | 6.936 | 7.401 | −0.464 | 6.269 |
V2 | 13.440 | 13.298 | 0.143 | 1.075 | |
Set #6 | V1 | 4.392 | 5.558 | −1.167 | 20.997 |
V2 | 12.895 | 12.337 | 0.558 | 4.523 | |
Set #7 | V1 | 1.386 | 3.160 | −1.774 | 56.139 |
V2 | 12.166 | 11.268 | 0.898 | 7.969 | |
Set #8 | V1 | 1.332 | 1.130 | 0.202 | 17.876 |
V2 | 11.050 | 10.495 | 0.555 | 5.288 |
Experimental Configuration | Measured | Calculated | Residuals (Mea.−Cal.) | |Res.|/Cal. (%) | |
---|---|---|---|---|---|
Set #1 | V1 | 0.384 | 0.949 | −0.565 | 59.536 |
V2 | 0.964 | 1.058 | −0.094 | 8.885 | |
V3 | 0.964 | 0.766 | 0.198 | 25.849 | |
Set #2 | V1 | 1.269 | 2.007 | −0.738 | 36.771 |
V2 | 2.433 | 2.373 | 0.059 | 2.486 | |
V3 | 2.561 | 1.824 | 0.737 | 40.406 | |
Set #3 | V1 | 2.011 | 2.773 | −0.762 | 27.479 |
V2 | 3.738 | 3.431 | 0.307 | 8.948 | |
V3 | 3.127 | 2.773 | 0.354 | 12.766 | |
Set #4 | V1 | 0.872 | 1.058 | −0.186 | 17.580 |
V2 | 1.367 | 1.315 | 0.052 | 3.954 | |
V3 | 1.708 | 1.058 | 0.650 | 61.437 |
Experimental Configuration | Measured | Calculated | Residual (Mea.−Cal.) | |Res.|/Cal. (%) | |
---|---|---|---|---|---|
Set #1 | V1 | 17.923 | 17.881 | 0.041 | 0.229 |
V2 | 8.716 | 8.816 | −0.101 | 1.146 | |
Set #2 | V1 | 16.638 | 16.828 | −0.189 | 1.123 |
V2 | 7.981 | 7.871 | 0.110 | 1.398 | |
Set #3 | V1 | 15.412 | 15.518 | −0.106 | 0.683 |
V2 | 7.090 | 6.817 | 0.273 | 4.005 | |
Set #4 | V1 | 14.015 | 14.464 | −0.449 | 3.104 |
V2 | 6.064 | 6.053 | 0.010 | 0.165 | |
Set #5 | V1 | 6.936 | 7.324 | −0.387 | 5.284 |
V2 | 13.440 | 13.320 | 0.120 | 0.901 | |
Set #6 | V1 | 4.392 | 5.506 | −1.115 | 20.251 |
V2 | 12.895 | 12.375 | 0.520 | 4.202 | |
Set #7 | V1 | 1.386 | 3.143 | −1.757 | 55.902 |
V2 | 12.166 | 11.321 | 0.845 | 7.464 | |
Set #8 | V1 | 1.332 | 1.143 | 0.189 | 16.535 |
V2 | 11.050 | 10.558 | 0.492 | 4.660 |
Experimental Configuration | Measured | Calculated | Residual (Mea.−Cal.) | |Res.|/Cal. (%) | |
---|---|---|---|---|---|
Set #1 | V1 | 0.384 | 0.948 | −0.564 | 59.494 |
V2 | 0.964 | 1.058 | −0.094 | 8.885 | |
V3 | 0.964 | 0.768 | 0.196 | 25.521 | |
Set #2 | V1 | 1.269 | 2.006 | −0.737 | 36.740 |
V2 | 2.433 | 2.373 | 0.059 | 2.486 | |
V3 | 2.561 | 1.826 | 0.735 | 40.252 | |
Set #3 | V1 | 2.011 | 2.774 | −0.763 | 27.505 |
V2 | 3.738 | 3.431 | 0.307 | 8.948 | |
V3 | 3.127 | 2.774 | 0.353 | 12.725 | |
Set #4 | V1 | 0.872 | 1.058 | −0.186 | 17.580 |
V2 | 1.367 | 1.315 | 0.052 | 3.954 | |
V3 | 1.708 | 1.058 | 0.650 | 61.437 |
Experimental Configuration | Measured | Calculated | Residual (Mea.−Cal.) | |Res.|/Cal. (%) | |
---|---|---|---|---|---|
Set #1 | V1 | 17.923 | 17.857 | 0.066 | 0.370 |
V2 | 8.716 | 8.709 | 0.006 | 0.069 | |
Set #2 | V1 | 16.638 | 16.826 | −0.188 | 1.117 |
V2 | 7.981 | 7.786 | 0.195 | 2.504 | |
Set #3 | V1 | 15.412 | 15.545 | −0.133 | 0.856 |
V2 | 7.090 | 6.755 | 0.334 | 4.944 | |
Set #4 | V1 | 14.015 | 14.514 | −0.499 | 3.438 |
V2 | 6.064 | 6.007 | 0.056 | 0.932 | |
Set #5 | V1 | 6.936 | 7.203 | −0.267 | 3.707 |
V2 | 13.440 | 13.356 | 0.085 | 0.636 | |
Set #6 | V1 | 4.392 | 5.424 | −1.033 | 19.045 |
V2 | 12.895 | 12.432 | 0.463 | 3.724 | |
Set #7 | V1 | 1.386 | 3.113 | −1.727 | 55.477 |
V2 | 12.166 | 11.402 | 0.764 | 6.701 | |
Set #8 | V1 | 1.332 | 1.159 | 0.173 | 14.927 |
V2 | 11.050 | 10.653 | 0.396 | 3.717 |
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Yoon, S.; Jeon, D.; Oh, J.-E.; Kim, M.-K.; Kim, D.-J. Experimentally Validated Analytical Solutions to Homogeneous Problems of Electrical Impedance Tomography (EIT) on Rectangular Cement-Based Materials. Appl. Sci. 2023, 13, 335. https://doi.org/10.3390/app13010335
Yoon S, Jeon D, Oh J-E, Kim M-K, Kim D-J. Experimentally Validated Analytical Solutions to Homogeneous Problems of Electrical Impedance Tomography (EIT) on Rectangular Cement-Based Materials. Applied Sciences. 2023; 13(1):335. https://doi.org/10.3390/app13010335
Chicago/Turabian StyleYoon, Seyoon, Dongho Jeon, Jae-Eun Oh, Min-Kyoung Kim, and Dong-Joo Kim. 2023. "Experimentally Validated Analytical Solutions to Homogeneous Problems of Electrical Impedance Tomography (EIT) on Rectangular Cement-Based Materials" Applied Sciences 13, no. 1: 335. https://doi.org/10.3390/app13010335
APA StyleYoon, S., Jeon, D., Oh, J.-E., Kim, M.-K., & Kim, D.-J. (2023). Experimentally Validated Analytical Solutions to Homogeneous Problems of Electrical Impedance Tomography (EIT) on Rectangular Cement-Based Materials. Applied Sciences, 13(1), 335. https://doi.org/10.3390/app13010335