Study on the Selection of Power Cable Rejuvenation Fluids Based on Relative Diffusion Coefficients

The present work presents and formulates a method for calculating the relative diffusion coefficient of rejuvenation fluid in injection cable rejuvenation technology, and the permeability characteristics of rejuvenation fluid in polyethylene insulation materials are calculated and compared. The main components of the rejuvenation fluid such as (Phenylmethyl-dimethoxysilane) PMDMS, (Diphenyldimethoxysilane) DPDMS and (Trimethylmethoxysilane) TMMS, and the catalysts such as (Tetra-isopropyl titanate) TIPT and (Tetra-isobutyl titanate) TIBT were analysed in depth. The article uses theoretical calculation method, combined with Helmroth and Limm-Hollifield models, to calculate the diffusion coefficients and relative diffusion coefficients of these compounds in polyethylene. Through comparative analysis of theoretical calculations, the optimal parameters for evaluating the permeability characteristics of rejuvenation fluids were determined. The research results show that TMMS has the best permeability characteristics in polyethylene, followed by PMDMS and DPDMS. In the catalyst, TIPT has better permeability characteristics than TIBT. And when selecting the composite formula for cable rejuvenation fluid, it is more convenient to use the relative diffusion coefficient as an index of diffusion characteristics. This research provides important theoretical basis and practical guidance for the formulation selection of cable rejuvenation fluids.


Introduction
Polyethylene (PE) insulated cables has been widely used in power transmission engineering due to their superior insulation and mechanical properties.However, these cables are subjected to various harsh conditions in complex operating environments, including moisture ingress, cracks, the formation of defects caused by water and electrical trees.These issues can seriously affect the insulation performance of cables and even lead to accidents.Direct replacement of aged cables is not only costly but also involves a cumbersome construction process that can cause serious environmental pollution.Therefore, the injected cable rejuvenation technology was proposed [1].This technology can effectively rejuvenate the insulation performance of cables.The mechanism is that after injecting organic rejuvenation fluid inside the cable, rejuvenation fluid penetrates into the insulation of the cable and polycondensation occurs in contact with water.
In injection cable rejuvenation technology, the selection of an appropriate rejuvenation fluid formula is a critical factor influencing the effectiveness of rejuvenation.Currently, commonly used rejuvenation fluid formulations include Phenylmethyldimethoxysilane (PMDMS) [2], and Diphenyldimethoxysilane (DPDMS) proposed by Hirazi A and Hosseini S H [3], McCary J L et al. proposed Tolylethylmethyldimethoxysilane (TEM) and Cyanoalkylmethyldimethoxysilane (CAM) [4], as well as Trimethylmethoxysilane (TMMS) that can be used to improve the diffusion performance of rejuvenation fluids [5].
The rejuvenation fluid needs to have appropriate permeability characteristics to ensure that it can reach the area to be rejuvenated at a predetermined diffusion rate [6].The permeability characteristics of rejuvenation fluid can be characterized by the diffusion coefficient parameter [7], and the diffusion coefficient can be obtained through experimental measurement or theoretical calculation methods.However, both methods have their advantages and disadvantages.The experimental measurement method obtains more accurate values, but its operation is complex, the experimental cost is high, and time-consuming [8].The accuracy of theoretical calculation methods is slightly insufficient, but they have high computational efficiency and strong flexibility.Therefore, it is more suitable for selecting rejuvenation fluids.
Currently, there are three main types of prediction models used to describe diffusion behavior in polymers: diffusion models based on blocking effects, hydrodynamics models, and free volume theory models [9].Among them, Helmroth proposed a semi-empirical relationship between diffusion coefficient and molecular weight based on free volume theory [10].Limm-Hollifield further modified and derived the model, and obtained the prediction equation for diffusion coefficient in polyolefins [11].
This article analyzes and compares two models, Helmroth and Limm-Hollifield, based on theoretical calculation methods.And proposes a calculation method for the relative diffusion coefficient of rejuvenation fluid based on cable rejuvenation.To evaluate the permeability characteristics of three rejuvenation fluids (PMDMS, TMMS, and DMDB) and two rejuvenation fluid catalysts (TIPT and TIBT) in polyethylene, it is necessary to use two models to calculate the values of diffusion coefficient D and relative diffusion coefficient Dr.These values serve as indices for ranking the permeability characteristics.And a comprehensive analysis of D and Dr is conducted, elucidating the advantages and disadvantages of the two models.

Diffusion Coefficient
The random motion that causes material molecules to transfer from high concentration areas to low concentration areas in a medium is called diffusion.And the diffusion coefficient is a physical quantity that describes the diffusion ability of material molecules in a medium.It is related to the properties and temperature of the migrating material.
The Helmroth model is based on the free volume theory to calculate the diffusion coefficient of small molecule substances towards the diffused substance.It takes into account the influence of the molecular weight of the migration material and the free volume of the polymer on the diffusion process.The diffusion coefficient and molecular weight of this model satisfy equation (1) [10]: where D is the diffusion coefficient of the diffusion component in the polymer(cm 2 /s), M is the molecular weight of the diffusate rejuvenation fluid, M * is the molecular weight reference value, and a and b are model parameters.For polyethylene usually M * is taken as 1 g/mol, a is taken as 1.2×10 -6 and b is taken as 0.37 [10].
The Limm-Hollifield model is based on Fick's law and free volume theory, and is calculated based on the migration constants obtained from actual migration experiments.Therefore, the results it gets have a high degree of matching with the experimental results of the diffusion coefficient.The equation is as follows (2) [11]: where β is the coefficient in front of the index, γ and λ are adjusted parameters of the model.For polyethylene usually β is take e -4.16 , γ 0.555 and λ take 1140.5 [11,12].

Relative Diffusion Coefficient
In the application of cable rejuvenation technology, in order to achieve a rapid comparison of the permeability characteristics of each component of the rejuvenation fluid.The concept of relative diffusion coefficient Dr is introduced by analogy to the definition of relative dielectric constant εr.

Basic Calculation Model.
The relative diffusion cefficient Dr is the ratio of D to D0 in the same diffusion system.The defining equation is shown in equation (3).
where D is the diffusion coefficient (cm 2 /s) of the component to be evaluated in the polymer, D0 is the coefficient (cm 2 /s) of the reference component, and Dr is the relative diffusion coefficient of the component to be evaluated.
The relative diffusion coefficient reflects the degree of obstruction of the evaluated component in the medium, and also more intuitively reflects the differences with the reference component under specific conditions.Compared with the reference component, the larger the relative diffusion coefficient, the smaller the hindrance to the molecular diffusion of the evaluated component.On the contrary, the smaller the relative diffusion coefficient, the greater the hindrance to the molecular diffusion of the evaluated component.
By substituting the calculation model in section 1.1 into the definition equation (3), the calculation equation for the relative diffusion coefficient of each model with the same permeating material and experimental conditions can be obtained.The simplified calculation equation are as follows: The Dr equation for the Helmroth model is shown in equation ( 4).

Simplified Calculation Model based on the Pmdms-Pe System
PMDMS was chosen as the reference component.This is because it is the most widely used and wellstudied rejuvenation fluids material.Polyethylene (PE) was chosen as the penetrated medium.This is because it is the most commonly used cable insulation.Under the condition of 323.15K, substitute the various parameters into equations (4 and 5).Further simplification, the calculation model of the relative diffusion coefficient Dr based on the PMDMS-PE system was obtained.As shown in table 1.
Table 1.Relative diffusion coefficient definition equation.

Calculation Result
According to the calculation formulas of the Helmroth model and Limm-Hollifield model, the diffusion coefficients of the aforementioned five substances are calculated.Concurrently, the relative diffusion coefficient is calculated using PMDMS as the reference component.The calculated results are shown in table 3.
Table 3. Calculation results of diffusion coefficient and relative diffusion coefficient.

Discussion
Some experimental values obtained through literature [2,5] research is shown in table 4.
Table 4. Calculation results of diffusion coefficient and relative diffusion coefficient.In order to conduct in-depth analysis of the diffusion characteristics of the rejuvenation fluid components, tables 3 and 4 show the calculated and experimental values of the diffusion coefficient model.According to the results, the permeability characteristics of different components can be observed.Two of the calculation results showed that TMMS in the main formula of the rejuvenation fluid had the best permeability characteristics in polyethylene, followed by PMDMS and DPDMS.Among the catalysts, TIPT has better permeability than TIBT.

Experimental value Diffusion coefficient D(cm
The diffusion characteristics of the rejuvenation fluid components plays a crucial role in the rejuvenation effect during cable rejuvenation.In the actual rejuvenation process, the main formula and other components of the rejuvenation fluid are injected into the cable under pressure due to differences in the diffusion characteristics of different components.Therefore, in the process of diffusion or permeation, different components will have different diffusion situations.For different rejuvenation tasks, sometimes it is desired to have similar diffusion characteristics to better synergistic rejuvenation.Conversely, sometimes it is desirable to have a large difference in diffusion characteristics to better utilize the effects of different components.Therefore, an evaluation index is needed to screen the components.For the evaluation indexes of permeability characteristics, the results of D and Dr can provide a comparison of permeability characteristics.However, when selecting an appropriate reference component, the calculation of the relative diffusion coefficient is simple, and the comparison with the reference component is more intuitive.In different models, using relative diffusion coefficients can reduce experimental errors caused by model parameters.Therefore, when selecting the composite formula for cable rejuvenation fluid, it is more convenient to use the relative diffusion coefficient as an index of diffusion characteristics.
The calculation methods have better computational efficiency and flexibility in the selection of rejuvenation fluids, which is more conducive to the application requirements of rapid selection of rejuvenation fluids.Compare the diffusion coefficient calculation results of the Limm-Hollyfield model and the Helmroth model.It was found that the Limm-Hollifield model showed significant advantages compared to experimental values, and the diffusion coefficient calculation results of the Helmroth model were one to two orders of magnitude smaller.However, when using relative diffusion coefficient calculation, the difference in calculation results between the two models is not significant.This is because calculating the diffusion coefficient eliminates most of the parameters, and the reduction of parameters correspondingly reduces the calculation error, making the calculation of the relative diffusion coefficient more concise and effective.This simplified calculation method can be better used to compare the diffusion characteristics of different rejuvenation fluid components, and can also provide a more intuitive understanding and evaluation of the permeation characteristics of rejuvenation fluid.

Conclusions
This paper addresses the permeability characteristics of three rejuvenation fluids (PMDMS, TMMS and DPDMS) and two rejuvenation catalysts (TIPT and TIBT) in polyethylene.Diffusion coefficients and relative diffusion coefficients were calculated using the Helmroth and Limm-Hollifield models.The main conclusions obtained are as follows: (1) Among the calculated results of the three rejuvenation fluids under the same conditions, TMMS had the best permeability characteristics, while PMDMS and DPDMS were relatively worse than the former.Among the two catalysts, TIPT had better permeability characteristics than TIBT.
(2) The Helmroth and Limm-Hollifield models differ significantly in the calculation of diffusion coefficients.The calculation results of Limm-Hollifield model are closer to experimental values, and the difference in relative diffusion coefficients is not significant.
(3) When an appropriate reference component is selected.The calculation of the relative diffusion coefficients is relatively simple.And the comparison with the reference component is more intuitive.Therefore, in the selection process of composite formulations for cable rejuvenation fluids, it is more convenient to choose the relative diffusion coefficient as the evaluation index of diffusion characteristics.
IntroductionThis research focuses on three commonly used rejuvenation fluid components, PMDMS, TMMS, and DPDMS, as well as two commonly used catalyst components, TIPT and TIBT.The molecular structure and relative molecular mass of the five components are present as figure1and table 2.

Figure1.
Figure1.Schematic diagram of the molecular structure of each component.