^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Oil sand pumps are widely used in the mining industry for the delivery of mixtures of abrasive solids and liquids. Because they operate under highly adverse conditions, these pumps usually experience significant wear. Consequently, equipment owners are quite often forced to invest substantially in system maintenance to avoid unscheduled downtime. In this study, an approach combining relevance vector machines (RVMs) with a sum of two exponential functions was developed to predict the remaining useful life (RUL) of field pump impellers. To handle field vibration data, a novel feature extracting process was proposed to arrive at a feature varying with the development of damage in the pump impellers. A case study involving two field datasets demonstrated the effectiveness of the developed method. Compared with standalone exponential fitting, the proposed RVM-based model was much better able to predict the remaining useful life of pump impellers.

Slurry pumps are widely used to remove mixtures of abrasive solids and liquids in wet mineral processing operations. These pumps usually experience severe erosive and/or corrosive wear even under normal working conditions. Consequently, their performance becomes severely comprised over time and at a certain point, the pumps will begin to fail without warning. The prevention of such unscheduled downtime requires substantial investment to maintain the system near the initially intended maximum level of efficiency. In particular, it becomes necessary to implement a scheduled preventive maintenance program capable of predicting the trend of degradation and estimating the remaining useful life of the pumps, to ensure a safe, economical, and efficient operation of the pump systems in the field. The remaining useful life of an asset or system is defined as the length of time from the present time to the end of the asset's useful life [

A review of the literature on the degradation of slurry pumps shows that several studies have investigated the process of wear associated with the machines and the possibility of improving pump performance through the use of more durable materials [

However, compared with the studies on wear, few research articles have been devoted to the problem of monitoring the condition of slurry pumps, and even fewer have reported on the issue of fault diagnosis in slurry pumps [

The research reported in this paper was conducted in response to a particular requirement in oil mining whereby slurry pumps need enhanced monitoring because they are prone to sporadic catastrophic breakdowns. In the oil-mining sector, equipment owners need to be aware when their pumps require an overhaul or when the related pump components will shortly need to be replaced to avoid unplanned pump downtime. To reduce potential costs, it is of great practical importance to have available a method to monitor the condition of the pump that is capable of determining when it should be overhauled or replaced, or how long its useful life is expected to be. Experience has shown that slurry pumps wear mainly because of impeller failure that can be indicated in advance by a decrease in impeller diameter [

Recent years have witnessed the rapid development of RUL prediction methods for maintenance [

As the data were sampled from oil sand pumps under extremely complicated and adverse working conditions, which may include all kinds of disturbances, the use of data pre-processing, feature extracting, and model building in this study was much more challenging than in research based on laboratory datasets. To the best of our knowledge, little effort has been devoted to pump prognosis based on field datasets in the literature. In this study, relevance vector machines (RVMs) were combined with the sum of two exponential functions to arrive at a method capable of predicting the degree of wear of field impellers and their remaining useful life. RVM—a method first introduced by Tipping [

The remainder of the paper is organized as follows: after an introduction to the basic theory of RVM in Section 2, Section 3 presents a prediction of the deterioration trend and RUL in a field oil sand pump derived from vibration-based degradation signals. In Section 4, the results of this prognostic procedure are presented and the prognostic performance of the developed model applied to real data is analyzed. We conclude the paper in Section 5.

An RVM is a Bayesian sparse kernel model that introduces a prior distribution over the model weights that are governed by a set of hyper-parameters [

The RVM starts with the concept of linear regression models that are generally used to find the parameter vector _{0},_{1},_{2},…,_{N}^{N}^{th}_{i}_{n}_{n}_{1}),_{n}_{2}),…,_{n}_{N}_{n}^{2} In this way, the likelihood of the dataset can be written as:

In many applications, due to the singularity of the coefficient matrix in

In the RVM learning process, the parameter vector _{i}_{i}_{1},_{2},…,_{M}_{i}

By Bayes' rule, the posterior probability over all of the unknown parameters can be expressed as:

However, the solution of the posterior ^{2}|

According to Bayes' rule, the posterior distribution over weights can be expressed as:
_{0},_{1},…,_{N}

The probability distribution over the training targets can be obtained by integrating the weights to obtain the marginal likelihood for the hyper-parameters:
^{−2}^{−1}^{T}

Thus, the estimated value of the parameter weights ^{2} can be estimated by maximizing _{new}_{new}

Slurry pumps are used to deliver a mixture of bitumen, sand, and small pieces of rock from one site to another in wet mineral processing operations. Experience has shown that the components of slurry pumps undergo a great variety and degree of abrasiveness and erosion. Often, the pump wear results in sudden downtime. This leads to huge economic losses due to the interruption of the mineral processing operations. Hence, it is of critical significance to have a method that is capable of helping to decide when a pump should be taken out of service and overhauled. In this study, a prognostic method is developed to assess the pump's performance degradation and to predict the RUL of the pump. The schematic diagram of the developed method is depicted in

Field data were collected from the inlet and outlet of slurry pumps operating in an oil sand mine. Vibration signals using the same sampling frequency rate (51.2 kHz) were obtained from four accelerometers mounted at four different pump locations. These four accelerometers were named as casing 1, casing 2, casing 3, and casing 4, respectively, in

The fast Fourier transform (FFT) technique converts time-domain signals into frequency-domain signals and can thereby identify salient features in machines [

Three representative frequency spectrums of the oil sand pump are shown in

In the mineral processing field, some critical components of slurry pumps frequently fail earlier than their expected service time. For example, according to field observations, the vanes of impellers were usually the first component to wear out due to abrasion from the fluid-solid materials [

The details of the feature extraction process are given as follows. At the first step, the vibration data _{new}(

A Fourier transform-based sliding-window averaging technique was then used to obtain averaged FFT amplitude values

Then, the averaged FFT amplitude values

Finally, the sequential standard deviation values

When compared with the progression of pump damage demonstrated by the energy evolution based on averaged FFT amplitude as shown in

The RVM learning process was performed on the pair of vectors {_{j}_{1},_{2},…,_{j}_{j}^{2} in

An exponential function, a polynomial function and a sum of two exponential functions were the potential candidates to approximate the pump degradation curve. The reasons why we chose the sum of two exponential functions were given as follows. First, compared with the exponential function and the polynomial function with a low degree, the sum of exponential functions was more flexible to fit a complex degradation curve, which had been proven in reference [

Hence in this study, the sum of two exponential functions was used to fit the degradation evolution of the pump impeller on the basis of the vector constructed by the mean values of the sparse dataset, referred to as

The future evolution of degradation was predicted by extrapolating the fitted model along the inspection file number and the degradation trajectories were traced up to a pre-defined failure threshold; thus simultaneously the mean values of the remaining useful life (RUL) were obtained. The corresponding point _{j}_{F}_{j}

The correspondence variance vector,
_{l}_{j}_{u}_{j}

_{j}_{j}_{j}_{j}

The application of the prognostic procedure to the calculation of the estimated _{j}_{j}

In the case of T2G1C3, the vibration signals were sampled from the Suction Pipe. During the feature extraction phase, the sliding window width was selected as 5. The data contained in the first 100 files were taken to represent the steady state of the impeller. The feature extraction results are plotted in _{j}

Comparisons of the results for datasets

As shown in _{j}

Correspondingly, the comparisons of the results for datasets

The performance of the developed procedure for estimating the impeller RUL is further evaluated by using the weighted average accuracy of prediction. The weighted average accuracy of prediction may be calculated using the following formula [

The weights _{j}_{j}_{200} = 200/2700 = 0.0741, _{300} = 200/2700 = 0.1111 Note that late predictions were penalized more heavily than early predictions. Hence, the designed performance indicator corresponded with actual needs and was more trustworthy in practical applications. The results with the weighted accuracy of prediction from the above applications are summarized in

From the results listed in

This paper has presented a model combining relevance vector machines (RVMs) and a sum of two exponential functions that can be used for pump impeller prognosis and for the estimation of the pump's remaining useful life (RUL). The data used in the case study were all sampled from the field,

The proposed procedure was found to be capable of treating degradation signals for RUL estimation and yielded better performance than conventional standalone exponential fitting. However, owing to the extremely complicated running environment of the field pump, the weighted average accuracy of the prediction was not as high as expected. There is certainly room for improvement, and the authors propose to devote their future research efforts to the development of novel ensemble prognostic models that can further improve the predictive accuracy of this model.

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 122011) and a grant from City University of Hong Kong (Project No. 7008187). The authors would like to appreciate the comments from Dong Wang.

The authors declare no conflict of interest.

Schematic diagram of the developed method.

The measurement locations of the four accelerometers mounted on a slurry pump (the four accelerometers named as casing 1, 2, 3 and 4) [

(

The frequency spectra of the vibrations collected from the oil sand pump.

Feature extraction process.

(

RUL estimation process (

Inspection file number _{200} = 200.

Inspection file number _{300} = 300.

Inspection file number _{400} = 400.

Inspection file number _{500} = 500.

Inspection file number _{600} = 600.

Inspection file number _{700} = 700.

Comparison of prognosis performance using the RVM-based model and exponential fitting for dataset

(

Inspection file number _{200} = 200.

Inspection file number _{300} = 300.

Inspection file number _{400} = 400.

Inspection file number _{500} = 500.

Comparison of prognosis performance using the RVM-based model and exponential fitting for dataset

Goodness of fit statistics used for comparing three functions.

Sum of two exponential functions | 0.9149 | 0.9147 | 0.1348 |

One exponential function | 0.4874 | 0.4869 | 0.3305 |

Quadratic polynomial | 0.8688 | 0.8686 | 0.1673 |

Values of _{j}

_{j} |
_{A}_{j} |
_{j} |
_{j} |
---|---|---|---|

200 | 690 | 359.4 | 2,000 |

300 | 590 | 268.7 | 1,400 |

400 | 490 | 333.6 | 1,600 |

500 | 390 | 332.1 | 901 |

600 | 290 | 275.4 | 483 |

700 | 190 | 107.2 | 249 |

Values of _{j}

_{j} |
_{A}_{j} |
_{j} |
_{j} |
---|---|---|---|

200 | 368 | 33.8 | 181.6 |

300 | 268 | 24 | −2.9 |

400 | 168 | 79.3 | −32.2 |

500 | 68 | 23.1 | −50.5 |

The weighted average accuracy of the prediction for pump impeller.

| |||
---|---|---|---|

RVM-based model | Exponential fitting | RVM-based model | Exponential fitting |

70.51 % | 25.31 % | 28.85% | 7.05% |