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Fault Detection of Multi-Rate Two Phase Reactor Condenser System with Recycle Using Multiple Probabilistic Principal Component Analysis^{ †}

^{*}

^{†}

## Abstract

**:**

^{2}and SPE statistics are generated for fault detection in TPRCR systems, and the MPPCA approach’s effectiveness for fault detection is satisfactory.

## 1. Introduction

^{2}and SPE statistics.

## 2. MPPCA Method

_{1}∈ R

^{K1×M1}, x

_{2}∈ R

^{K2×M2}, and x

_{3}∈ R

^{K3×M3}are three different rate measurements classes in which x

_{3}is the slowest and x

_{1}is the fastest measurement. ${\xd8}_{1}$ ∈ R

^{M1×D}, ${\xd8}_{2}$ ∈ R

^{M2×D}and ${\xd8}_{3}$ ∈ R

^{M3×D}are loading matrices with three different sampling rates. t ∈ R

^{D}is a latent variable which extracts a restricted link between data with varied sampling rates and helps develop one single model. The latent variable is assumed to have a Gaussian distribution with a zero mean and unit variance. ε

_{1}∈ R

^{M1}, ε

_{2}∈ R

^{M2}and ε

_{3}∈ R

^{M3}are used to model the corresponding isotropic Gaussian noises.

_{3}), the following sample variables have dimensions M1 + M2 (V

_{2}), and the last one contains only M1 (V

_{1}) variables. As a result, the entire observation set is expressed as a union of all three.

^{2}-distributed approximation: SPE~g.${\chi}_{h}^{2}$, in which g and h are the parameters of the χ

^{2}distribution, and they are given by [5].

## 3. Two-Phase Reactor–Condenser System with Recycle

_{A}and F

_{B}and temperatures T

_{A}and T

_{B}, respectively, in the vapour and liquid phases. Reactant A diffuses into the liquid phase at rate N

_{A}

_{1}, where an exothermic reaction occurs, which is given by Equation (10).

_{A}is the rate at which reactant A is consumed at temperature T

_{1}. The preexponential factor and activation energy are denoted by k

_{10}and E

_{a}, respectively. ${M}_{1}^{l}$ is the liquid molar holdup in the reactor, and $\rho $ is the liquid density. x

_{A}

_{1}and x

_{B}

_{1}are A and B mole fractions in the liquid phase. For the sake of simplicity, heat capacity, density, and molar heat of vaporisation are considered to be constant and equal for all species. The liquid and vapour phases are suitable combinations. The liquid stream from the reactor is withdrawn at a constant flow rate F

_{1l}, while the vapour stream enters the condenser at a flow rate F

_{1v}. The vapour in the condenser is cooled to T

_{2}to improve product purity by eliminating reactant A from the liquid.

_{2l}, while the product vapour phase departs the condenser at a flow rate of F

_{2v}and a composition of y

_{A}

_{2}.

## 4. Fault Detection Using MPPCA for the TPRCR System

_{1}). Medium-rate measurements include molar holdups available every fifteen seconds (x

_{2}), and slow-rate measurements include mole fractions available every sixty seconds (x

_{3}).

^{2}statistics. Table 3 shows the false alarm rates for normal data and the missing detection rates for faults, where Fault 0 represents normal test data and that the monitoring results are false alarm rates. The false alarm rate is the fraction of normal data that is interpreted as problem data. Similarly, the missing detection rate is the fraction of the defect data that are treated as normal data. Table 3 shows the monitoring results of all faults using T

^{2}and different SPE statistics for the MPPCA model.

_{A}), which suggests that this fault affects all three SPE statistics.

## 5. Conclusions

^{2}and three different SPE statistics for each measurement class. Six different types of faults are used to check the effectiveness of the developed MPPCA model, and, from the monitoring results, we can clearly say that the MPPCA model can detect faults with a high detection rate.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameter | Description | Value | Unit |
---|---|---|---|

a | Interfacial mass transfer area/unit liquid holdup | 1000 | m^{2}/m^{3} |

C_{p} | Molar heat capacity | 80 | J/mol K |

E_{a} | Activation energy | 110 | kJ/mol |

K | Proportional gain of pressure controller | −8 | mol/s atm |

K_{10} | Preexponential factor | 2.88 × 10^{11} | m^{3}/mol s |

k_{a} | Overall mass transfer coefficient for A | 0.2 | mol/m^{2} s |

k_{c} | Overall mass transfer coefficient for C | 0.8 | mol/m^{2} s |

M_{1}^{l} | Liquid molar holdup in reactor | 14.52 | kmol |

M_{2}^{l} | Liquid molar holdup in condenser | 15 | kmol |

M_{1}^{v} | Vapour molar holdup in reactor | 3.75 | kmol |

M_{2}^{v} | Vapour molar holdup in condenser | 3.90 | Kmol |

P_{1} | Pressure in reactor | 50 | atm |

P_{2} | Pressure in condenser | 48.69 | atm |

P* | Set point for reactor pressure | 50 | atm |

T_{A} | Temperature of feed A | 315 | K |

T_{B} | Temperature of feed B | 300 | K |

T_{1} | Temperature in reactor | 330 | K |

T_{2} | Temperature in condenser | 304.16 | K |

V_{1T} | Volume of reactor | 3 | m^{3} |

V_{2T} | Volume of condenser | 3 | m^{3} |

$\rho $ | Liquid molar density | 15,000 | mol/m^{3} |

$\Delta {H}_{r}$ | Heat of reaction | −50 | kJ/mol |

$\Delta {H}^{v}$ | Heat of vaporization | 10 | kJ/mol |

Fault No. | Fault Type | Fault Introduced (s) |
---|---|---|

1 | Step jump in flow rate of A (F_{A}) | 2400 |

2 | Step jump in flow rate of B (F_{B}) | 2400 |

3 | Step jump in temperature of A (T_{A}) | 2400 |

4 | Step jump in temperature of B (T_{B}) | 2400 |

5 | Ramp jump in flow rate of A (0.0004 × t) | 2400 |

6 | Ramp jump in temperature of A (0.003 × t) | 2400 |

Fault No | T^{2} | SPE_{1} | SPE_{2} | SPE_{3} |
---|---|---|---|---|

0 | 0.021 | 0.001 | 0.004 | 0.0001 |

1 | 0.035 | 0.023 | 0.09 | 0.008 |

2 | 0.067 | 0.065 | 0.011 | 0.034 |

3 | 0.001 | 0.001 | 0.001 | 0.001 |

4 | 0.854 | 0.673 | 0.765 | 0.231 |

5 | 0.313 | 0.452 | 0.023 | 0.045 |

6 | 0.201 | 0.121 | 0.111 | 0.201 |

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**MDPI and ACS Style**

Gandhi, D.; Srinivasarao, M.
Fault Detection of Multi-Rate Two Phase Reactor Condenser System with Recycle Using Multiple Probabilistic Principal Component Analysis. *Eng. Proc.* **2023**, *37*, 91.
https://doi.org/10.3390/ECP2023-14669

**AMA Style**

Gandhi D, Srinivasarao M.
Fault Detection of Multi-Rate Two Phase Reactor Condenser System with Recycle Using Multiple Probabilistic Principal Component Analysis. *Engineering Proceedings*. 2023; 37(1):91.
https://doi.org/10.3390/ECP2023-14669

**Chicago/Turabian Style**

Gandhi, Dhrumil, and Meka Srinivasarao.
2023. "Fault Detection of Multi-Rate Two Phase Reactor Condenser System with Recycle Using Multiple Probabilistic Principal Component Analysis" *Engineering Proceedings* 37, no. 1: 91.
https://doi.org/10.3390/ECP2023-14669