Hao Huang
email: hao.huang@etu.univ-nantes.fr

Mounira Harzallah

Fabrice Guillet

Ziwei Xu

25th International Conference on Knowledge-Based and Intelligent Information & Engineering Systems Core-Concept-Seeded LDA for Ontology Learning

Keywords: Ontology Learning, Core Ontology, LDA, Term Clustering, Seed Knowledge, Prior Knowledge, Semantic Coherence, Word2vec

Ontologies are powerful semantic models applied for various purposes such as improving system interoperability, information retrieval, question answering, etc. However, building domain ontologies remains a challenging task for humans, especially when the concepts and properties are large or evolving, and also when they are built from large-scale textual data. Machine learning allows to automate the building of ontologies from texts. In particular, clustering techniques have a promising ability on the concept formation task by identifying the cluster of semantically closed terms as a concept. However, current works encounter issues in learning relevant domain-specific clusters or in identifying the relevant concept labels for each cluster. To solve these issues, we propose both to use core concepts from a domain ontology as prior knowledge, and to adapt term clustering with seed knowledge-based LDA models in order to take these core concepts into account. First, each topic is associated with a set of seed terms of a single core concept, then the learning is guided by these seeds to gather in the same topic the terms that refer to its core concept. We evaluate our proposal on two textual corpora and compare it to the baselines (LDA, K-means, and SMBM). The results show that our approach performs significantly better than other methods on the class-balanced dataset and works well on the class-imbalanced dataset with a proper number of topics for each core concept.

Introduction

With the increasing availability of textual resources on the web, ontology learning from texts becomes a challenging issue. The traditional process for building an ontology consists of five main tasks: terms extraction, concepts formation, taxonomy extraction, ad-hoc relationships extraction, and axioms extraction [START_REF] Cimiano | Ontologies[END_REF]. Many machine learning approaches have been proposed for achieving these tasks, generally classified into two main categories: pattern-based approaches and distributional approaches [START_REF] Asim | A survey of ontology learning techniques and applications[END_REF]. For the first four tasks, pattern-based approaches show quite high precision whereas their recall is low because of the large variability in natural language for expressing a meaning [START_REF] Ortega-Mendoza | Using lexical patterns for extracting hyponyms from the web[END_REF].

Distributional approaches, either supervised or unsupervised, are based on co-occurrence context distribution to detect semantic relationships between term pairs [START_REF] Sahlgren | The distributional hypothesis[END_REF]. Supervised methods learn a model for predicting the class of terms or semantic relationships between term pairs. Unsupervised approaches are either measure-based approaches or clustering-based approaches. The former computes a score of the semantic closeness or the relatedness of two terms based on symmetric or inclusion measures and the latter gathers semantically closed terms on the same cluster. Generally, the supervised approaches outperform non-supervised approaches [START_REF] Shwartz | Hypernyms under siege: Linguistically-motivated artillery for hypernymy detection[END_REF]. However, unlike unsupervised approaches, it requires additional effort to build a sufficiently large labeled training data set.

We are interested in term clustering methods for concept formation, and more specifically in clustering terms according to the core concepts (CCs) of a domain ontology. CCs are the minimal concepts that allow defining the other concepts of their domain [START_REF] Burita | K-gate ontology driven knowledge based system for decision support[END_REF]. In our works, we claim an ontology building approach guided by a core ontology (i.e. a model composed of CCs and core relationships between them). To be specific, according to the previous ontology building process, after terms extraction (step 1), we recommend first classify each term in a class associated with a CC (called in the remainder CC-class), then steps 2 and 3 are performed inside each class to form sub-concepts of CCs and hypernym relations between them. A CC-class is a class of terms that refer to this CC, i.e. its synonyms, hyponyms, or semantically closed terms. This approach allows to bound the look for synonym or hypernym relationships and then to reduce the computing time.

We focus on Latent Dirichlet Allocation (LDA) [START_REF] Blei | Latent dirichlet allocation[END_REF] for term clustering since it can deal with a huge number of documents and tackle the issue of text sparsity. LDA-based approaches provide a probability distribution of terms for each topic. From each topic, a cluster of the most relevant terms may be extracted according to the probability distribution (often the top-k most probable terms), then a cluster is labeled either by a domain expert or as terms with high probability. However, two key issues remain to be addressed. First, the semantic coherence of a cluster is not ensured, i.e the terms of a cluster do not refer necessarily to the same concept, thus the cluster is not entirely meaningful [START_REF] Chang | Reading tea leaves: How humans interpret topic models[END_REF][START_REF] Mimno | Optimizing semantic coherence in topic models[END_REF]. Second, these approaches do not deal with the relevance of clusters for the targeted knowledge domain. Indeed, a cluster could be out of the target, i.e. a cluster of terms referring to an irrelevant concept for the ontology domain. To solve these issues, we propose the Core-concept seeded LDA that aims to perform automatically the task of classification of terms on CC-classes by resorting to clustering. The approach is an adaptation of seed based LDA models by choosing as seed terms those referring to CCs and labeling a priori each topic by a CC.

In this paper, we consider four seed based LDA models (Z-labels LDA [START_REF] Andrzejewski | Latent dirichlet allocation with topic-in-set knowledge[END_REF], Seeded LDA M1, Seeded LDA M2, Seeded LDA [START_REF] Jagarlamudi | Incorporating lexical priors into topic models[END_REF]). In these models, a latent topic is about a subject that may be a large notion, and even if a topic is carried with high probability, it does not refer necessarily to a unique concept. For example, in [START_REF] Jagarlamudi | Incorporating lexical priors into topic models[END_REF], the topic of "earn" has as seed terms "company" and "quarter" that don't refer to the same concept. Further, the seed based LDA models are dedicated to document clustering. To the best of our knowledge, none of them has been used for term clustering for ontology learning. The paper is organized as follows. Section 2 presents term clustering approaches for ontology learning with a focus on LDA based approaches. Section 3 provides background knowledge of LDA and seeded based LDA models, with a focus on the principles of each. In section 4, we present our approach for adapting these models to term clustering over CCs. In section 5, we present several experiments conducted on two corpora. We compare the approach to unsupervised and semi-supervised baselines, we analyse the effectiveness of the proposed seed sets kind and we investigate the performance of the approach on imbalanced corpus before the conclusion.

Related Work

The main idea of distributional approaches is grouping terms by using the distribution of contexts where they appear, based on Harris' distributional hypothesis: terms that occur in similar contexts tend to have similar meanings [START_REF] Harris | Mathematical structures of language[END_REF]. Pereira et al. [START_REF] Pereira | Distributional clustering of english words[END_REF] originally proposed the distributional clustering of terms by using Kullback-Leibler (KL) distance to measure the term similarity. Caraballo [START_REF] Caraballo | Automatic construction of a hypernym-labeled noun hierarchy from text[END_REF] adapted a bottom-up clustering method for clustering noun terms and build up the hypernym relations among clusters. To improve the cluster coherence, Cimiano [START_REF] Cimiano | Learning concept hierarchies from text with a guided agglomerative clustering algorithm[END_REF] recommended a guided agglomerative clustering algorithm for inducing concept hierarchies from the text corpus. Often researchers used as contexts the words occurring with target terms in a fixed-size window [START_REF] Biemann | Automatic acquisition of paradigmatic relations using iterated co-occurrences[END_REF][START_REF] Mahn | Tuning co-occurrences of higher orders for generating ontology extension candidates[END_REF]. Recently, word embedding techniques are introduced to exploit large contextual information. Several works [START_REF] Albukhitan | Arabic ontology learning using deep learning[END_REF][START_REF] Ayadi | Ontology population with deep learning-based nlp: a case study on the biomolecular network ontology[END_REF][START_REF] Casteleiro | Deep learning meets ontologies: experiments to anchor the cardiovascular disease ontology in the biomedical literature[END_REF] explored the possibility of using word2vec for concept formation. To do so, first, seed terms are predefined and used for representing the domain concepts. Then the domain corpus is used for learning term representation by word2vec. Finally, candidate terms of each concept are selected by measuring the similarity between terms and seed terms of concepts. Those having high similarity with a seed term are chosen to enrich the concept. Some works cast term clustering as a graph partition problem. The idea of graph-based representation of terms has its origin in [START_REF] Widdows | A graph model for unsupervised lexical acquisition[END_REF]. A graph is defined as G = (V, E, W), where V is a set of vertices representing terms, E is a set of edges between terms, and W represents the weights of edges. Based on this graph, Matsuo et al. [START_REF] Matsuo | Graph-based word clustering using a web search engine[END_REF] introduced a graph clustering algorithm (Newman clustering) for word clustering. To define the weight of an edge, different measures (point-wise mutual information, Jaccard coefficient, chi-square) are used based on the frequency of each word and that of its word pair. An edge between two words is eliminated if its weight is lower than a threshold. Lee and Luo [START_REF] Lee | Word clustering for parallelism in classical chinese poems[END_REF] used the cosine similarity and word2vec for word representation to compute edge weights. Then, they proposed a term clustering method based on Louvain community detection algorithm [START_REF] Blondel | Fast unfolding of communities in large networks[END_REF], where each term community acts as a concept. Like the previous works, the author of [START_REF] Qiu | Automatic concept extraction based on semantic graphs from big data in smart city[END_REF] applied a community detection algorithm on terms graph for concept formation. The edge weights are computed in three ways: term co-occurrence based, word2vec based, and LDA based (i.e. the joint probability distribution of two words is calculated based on LDA theory). Thaiprayoon et al. [START_REF] Thaiprayoon | Graph and centroid-based word clustering[END_REF] introduced a hierarchical agglomerative clustering to find similar words according to the criterion of distance range on a graph where each edge is weighted by the inverse of word pair frequency.

A recent study [START_REF] Nematzadeh | Evaluating vector-space models of word representation, or, the unreasonable effectiveness of counting words near other words[END_REF] shows that word embeddings are unable to predict some of the characteristics of human similarity judgment. Compared with word embeddings, LDA can capture relevant word associations. With a special focus on the relation between concepts and terms, Rani et al. [START_REF] Rani | Semi-automatic terminology ontology learning based on topic modeling[END_REF] explored an LSI&SVD based method and a Mr.LDA (a map reduce LDA) based method for concept formation. Experimental results suggest the Mr.LDA based method is better than the LSI&SVD based method on several criteria. However, these methods need extra human effort to label each topic as a meaningful concept. In addition, in this work, they didn't consider the impact of irrelevant terms (terms of other domains) on topic building. To solve this problem, Xu et al. [START_REF] Xu | Modular ontology learning with topic modelling over core ontology[END_REF] proposed a twice-trained LDA where an LDA model is trained on a corpus the first time to identify irrelevant terms. Terms with a low probability on each topic are removed from the corpus. Then the LDA model is trained a second time with the cleaned corpus to build up topics. However, the author didn't exam the loss of domain-relevant terms. The approach in [START_REF] Colace | Terminological ontology learning and population using latent dirichlet allocation[END_REF] also used different thresholds to pick up aggregate terms to represent the concepts formed from topics. For labeling topics, they used LDA to compute the conditional probability between term pairs and determine the aggregate roots as terms whose occurrence is not implied by the occurrence of other terms of the corpus. Finally, the aggregate root of each topic is used to label it. However, most approaches based on the LDA model didn't consider the semantic coherence of term clusters. Moreover, extracted concepts (represented by term clusters) may be out of desire. Our approach, Core-Concept-Seeded LDA, aims at improving the cluster semantic coherence and providing human expected concepts.

Background

LDA is a probabilistic approach modeling topics for document clustering. A topic is assumed to be a distribution over words (φ), and a document is assumed to be a distribution over topics (θ). Documents are clustered based on the similarity of their topic distribution. The generative process is vital to understand the philosophy behind LDA. Blei [START_REF] Blei | Latent dirichlet allocation[END_REF] described it as following: 1. For the k th topic in a set of K topics, draw a word distribution φ k ∼ Dirichlet(β). where α and β are respectively the prior parameter of Dirichlet distributions θ d and φ k . θ is a D × K matrix, φ is a K ×V matrix, V is the number of unique words in the vocabularies of the corpus. The idea behind LDA is exploiting the co-occurrence information of words to extract the latent topic structure by maximizing the probability of the corpus generated from the model. Usually, Variational Inference [START_REF] Blei | Latent dirichlet allocation[END_REF] or Gibbs Sampling [START_REF] Griffiths | Finding scientific topics[END_REF] is used for model training. The latter samples topic of words from a Markov chain which finally converges to a stable distribution.

However, LDA will pay more attention to these statistically prominent topics with frequent terms. The non-frequent terms referring to different real-world topics are more likely to be mixed in a semantically ambiguous topic. To solve this problem, seed based LDA models have been proposed. The general idea behind them is to use seed information as prior knowledge to guide LDA and deliver topics more relevant to the user's interests. The seed information is composed of S seed sets, each of which contains seed words related to a topic. Four models are considered in our work: Z-labels, Seeded LDA M1, Seeded LDA M2, and Seeded LDA.

In LDA model, a word is assigned to any topic with a certain probability. In Contrast, Z-labels model constrains the topics of seed words to be assigned to, while other words are free to sample any topic. The parameter π specifies the probability of a seed word generated from the constrained topic. The idea of Z-labels is that using partial supervision information over some words, the topic sampling of other words through the Markov chain is impacted. Seeded LDA is derived from Z-labels by using seed information differently. It is composed of two models: Seeded LDA M1 and Seeded LDA M2. In the first model, each topic k is represented as a mixture of a "regular topic" distribution φ r k and a "seed topic" distribution φ s k . The former can generate any words (including seed words), while the latter can only generate seed words. A parameter π specifies the probability of a seed word generated from a seed topic. The original (LDA) conditional distribution for topic sampling is q i,k = p (z i = k| Z -i , α, β). In Z-labels, the seed word w has the probability of q i,k to be generated by its constrained topic, and the probability of (1 -π)q i,k by other topics. While in Seeded LDA M1, w has the probability of πφ s k + (1 -π) φ r k to be produced by its constrained topics, and the probability of (1 -π) φ r k by under other topics. Looking back to the basic LDA, a symmetric β is used as the prior distribution of the word distribution for each topic, without bias on generating words. However, the idea of both Z-labels and Seeded LDA M1 is roughly like using an asymmetric β where seed words have more chance to be generated, thus each topic has its own preference on words. Seeded LDA M2 uses a symmetric prior parameter for word distribution, but considers an asymmetric prior parameter for topic distribution. In Seeded LDA M2, a document will sample a group to determine its topic preference. A group (also called seed topic), has its own group-topic distribution which is used as the prior for the document-topic distribution. During the group sampling process of a document, the number of times each group occurred in this document (ie. sum of the frequencies of all seed words in each group) is used to calculate a probabilistic distribution where a group id s is sampled for this document. Then, the group-topic distribution ψ s (an asymmetric prior distribution) is used as the prior of the topic distribution θ d . By this mechanism, a document will be more likely to choose topics relevant to its seed words.

Our Approach: Core-Concept-Seeded LDA

The Core-Concept-Seeded LDA is an adaptation of seed based LDA models for term partition over CCs. Seed based LDA models are originally designed for document clustering, and the authors suggest using the seed words with a strong ability to distinguish document categories. As a consequence, a better document-topic distribution is obtained for document representation. But in our approach, the topic-word distribution (φ) is the key for term clustering. We consider terms instead of words as the vocabulary of LDA models. Then, we propose a different CC-seed sets setting compared with their model. Specifically, a CC-seed set of a topic is designed to include hyponyms or synonyms of a CC. In addition, a hard constraint that no overlap of any two CC-seed sets is added, while this overlap is allowed in the original models (i.e. one seed word can be shared by two or more topics). Finally, unlike seed based LDA models, our approach provides non-overlap clusters.

Steps of the Core-Concept-Seeded LDA

The Core-Concept-Seeded LDA is composed of three main steps: 1) Material Preparing, 2) Model Training, 3) Cluster Formation (Fig 1). In step 1, we prepare the term dataset, the CC-seed sets, and the document-term matrix. Firstly, the corpus is pre-processed with text segmentation, part of speech tagging, dependency parsing, and noun phrase (NP) lemmatisation. NP lemmas are extracted and filtered to constitute the terms of the vocabulary. For filtering, NP lemmas with corpus frequency less than 3 or included in the "stop words" list are eliminated. Then, the CC-seed sets are extracted by either a domain expert or automatically (see the next section). Finally, the D × V document-term matrix is constructed where each entry represents the frequency of a term in a document. In step 2, the CC-seed sets and the document-term matrix are the inputs of seed based LDA models. We choose the number of topics the same as the number of CC-seed sets: K = S (one CC-seed set guides one topic). During the training process, a topic is labeled a priori by a CC and guided by its CC-seed set. Then a K × V topic-term probability matrix φ is generated. Each topic gathers terms by assigning each of them a probability value. A term w has a topic probability vector φ :,w representing its closeness to all topics, but implying topic overlap. In step 3, to deal with topic overlap, a cluster is associated with each topic. Then a term w is assigned to a cluster associated to a topic k if k = argmax 1≤t≤K φ t,w . Finally, a cluster associated with topic k is labeled by its CC k . Thus, K labeled clusters are formed, including the CC that labels it and its seed set.

CC-seed Sets

The CC-seed sets, as one of the inputs of seed based LDA models, are keystones in our approach and directly impact the quality of topics. The authors of seeded LDA [START_REF] Jagarlamudi | Incorporating lexical priors into topic models[END_REF] use CC-seed sets that have a strong ability for discriminating categories of documents. Therefore, they suggest the information gain based method that requires labeling each document by a CC. Unlike the idea of the previous method, we suggest using the frequent synonyms and hyponyms of a CC as seed terms of a topic labeled by this CC. We assume that a term occurs frequently with its hypernym. If we use as seed terms the synonyms and hyponyms of a CC, its labeled topic can cover more other terms referring to it.

Indeed, in the Gibbs Sampling process, the conditional probability of term w in document d under topic k is calculated by

n (d) -i,k +α K u (n (d) -i,u +α) × n (w) -i,k +β W w (n (w ) -i,k +β)
, where α and β are the hyper-parameters of the model, n (d) -i,k is the number of terms (except w) in document d assigned to topic k, and n (w) -i,k is the number of terms (except w) in the whole corpus assigned to topic k. The idea of seed based LDA models is to force seed words of a topic k to be sampled for it and utilize the topic information to impact the topic sampling of other terms. If w refers to CC k , then n (d) -i,k would be bigger than n (d) -i,u , u k. Based on our assumption, the seed terms of topic k will largely contribute to n (d) -i,k , therefore lead w to sample k as its topic. Then, the topic information of w will be added to guide the topic sampling of its hyponyms. We use as seed terms synonyms and hyponyms that covers the widest semantic space of their CC, so that all terms referring to it will get the impact of its hypernyms and synonyms. In this paper, to form CC-seed terms, we query hyponyms and synonyms of CCs from DBpedia, then a domain expert is involved to validate them.

Otherwise, the hybrid score based method proposed in [START_REF] Qiu | A hybrid-based method for chinese domain lightweight ontology construction[END_REF] may be adapted to extract CC-seed sets. Basically, the hybrid score measures the importance of a term to a target sub-domain. Like the method used by the authors of Seeded LDA, it requires the prior labeling of each document by a CC. However, it considers not only the discriminating ability but also the domain relevance and consensus of a term.

Experiments

We performed 2 kinds of experiments on 2 corpora. The first kind aims to evaluate our approach and compare it to some baselines for term clustering. We have conducted experiments on both relevant term dataset (i.e. dataset including only terms relevant to the ontology domain) and whole term dataset (i.e. dataset including all extracted terms, as explained in section 4.1). Experiments on the latter aim to analyse the impact of irrelevant terms on the performance of our approach. The second kind of experiments conducted also on the whole term dataset aims to compare the impact of seed set extraction methods, to analyse the effect of seed set size, and to investigate the adaptation of our approach for imbalanced dataset.

Experiment Settings

Corpora. To our knowledge, there is no benchmark for term clustering as a task of ontology building. Therefore, we considered two domain corpora and built manually a gold standard of each. The first corpus (CS) is about the computer science domain, which is a part of the dataset Web of Science offered by [START_REF] Kowsari | Hdltex: Hierarchical deep learning for text classification[END_REF], including 5747 documents. Each of them is an academic paper labeled by a sub-domain (used as a CC) of computer science, consisting of several keywords and the abstract. The second corpus (Music) is about the music domain. It contains 10000 unlabeled documents, which are randomly sampled from the original corpus provided by [START_REF] Camacho-Collados | Semeval-2018 task 9: Hypernym discovery[END_REF].

Gold Standard Building. All extracted terms (see section 4.1) are labeled manually to form a domain gold standard (GS), i.e. a term that refers to a CC is labeled by this CC, otherwise labeled by "Others". The labeling task was done by two annotating groups of computer science students. The Cohen's Kappa coefficient [START_REF] Cohen | A coefficient of agreement for nominal scales[END_REF] is used to measure interannotator agreement. We obtained the coefficient value around 0.885 for CS corpus, and around 0.813 for Music corpus. For those terms with contradiction in annotation, we discussed and decided their final labels. 10 CCs for CS corpus (subdomains of the original corpus) and 5 CCs for Music corpus (general concepts of the music domain from DBpedia) are used for labelling terms. The final gold standards for two corpora are depicted in Table 1. Baselines. We compare our proposal to 2 kinds of baselines: 1) unsupervised clustering baselines and 2) semisupervised clustering baselines. For the first one, we consider LDA and K-means with word2vec for term representation. For the second one, we consider the similarity measure based method (SMBM) [START_REF] Albukhitan | Arabic ontology learning using deep learning[END_REF] with word2vec for term representation and the cosine similarity between a term and each CC to add it to the cluster of its closest CC.

Evaluation Metrics. We use micro precision (P) and micro recall (R) [START_REF] Sokolova | A systematic analysis of performance measures for classification tasks[END_REF] as metrics to evaluate the performance of our approach and compare it with baselines. Let's C and GS be 2 partitions of sets of terms, C = [C 1 , C 2 , ..., C K ] are the K formed clusters and GS = [GS 1 , GS 2 , ..., GS S ] are the S CC-classes of the gold standard (class "Others" is excluded) associated respectively to CC 1 , CC 2 , ..., CC S . We don't take into account the False Positives caused by terms from the class "Others" since in this paper we don't deal with non-domain terms. Therefore in our experiments, P equals to R defined as:

P = R = k |C k ∩GS l k | s |GS s |
, where C k has the label of GS l k (i.e. CC l k ). For a semi-supervised method, the label CC l k (1 ≤ l k ≤ S ) of a cluster is assigned a priori (called "prior labels"), while it is not the case for unsupervised methods. However, to evaluate the performance of the latter, we have to label clusters a posteriori. For that, we use the majority labeling method as in [START_REF] Xu | Modular ontology learning with topic modelling over core ontology[END_REF]. The label of a cluster C k is CC l k where l k = argmax s |C k ∩ GS s | that means CC l k is the label CC s of the class GS s that has the largest overlap with C k . Metrics using the majority labeling measure the potential largest number of true positives (TP) that unsupervised clustering-based methods can achieve.

Parameters Setting. For all LDA models, we follow the parameter setting of α and β from [START_REF] Hoffman | Online learning for latent dirichlet allocation[END_REF] where α = 1/K, β = 0.01, and keep other parameters as their default setting. While for word2vec, we use the skip-gram model as it can achieve better performance [START_REF] Casteleiro | Deep learning meets ontologies: experiments to anchor the cardiovascular disease ontology in the biomedical literature[END_REF]. We also follow its parameter setting where vector dimension is 300, window size is 10, sub-sampling threshold is 1e-5, minimum count is 5, and learning rate is 0.025.

Experiment Results

We performed two kinds of experiments. In the first one, we compared our approach (the adaptation of the 4 models denoted CC-Seed, CC-SeedM1, CC-SeedM2, CC-Z-labels) with baselines (LDA, K-means, SMBM) over two corpora and their two datasets and with CC-seed sets. In the second one, only CS corpus and Seeded LDA model are used. Music corpus is not considered because its documents are unlabeled whereas labels are needed for CC-seed sets extracted by information gain and hybrid score based methods. For each metric, the experiments are repeated 20 times for each model.

Comparison with Baselines. Table 2 shows that CC-seed and CC-seedM1 outperform all the methods either on relevant or whole dataset of CS corpus. For both corpora, CC-seeded LDA models don't benefit a lot from the whole dataset where the CC-SeedM2 and CC-Z-labels even get worse results, which suggests that CC-seeded LDA models don't take advantages of irrelevant terms as context information. CC-seeded LDA models perform worse on Music corpus than on CS corpus. In addition, for Music corpus unsupervised approaches outperform semi-supervised approaches and CC-Seeded LDA models achieve worse than SMBM. To better understand the behavior of these methods, we analyse the composition of clusters learned from the whole dataset. Fig 2 shows the composition of clusters learned by CC-Seed, SMBM, LDA and K-means. For CS corpus, CC-Seed achieved the highest value of TP (1458), and each cluster includes a high number of terms referring to its CC. It learned clusters with good semantic coherence. However, other methods produced clusters with low semantic coherence, for instance, the clusters "C6" and "C8" formed by SMBM (Fig2(b)), the clusters "C6" and "C9" formed by LDA (Fig2(c)), the cluster "C4" learnt by K-means (Fig2(d)). In addition, the semantic separateness of these clusters is very bad: their composition is a mix of similar proportions of terms referring to several CCs. For Music corpus, the partitions of all the methods are bad. Semantic coherence and separateness of clusters are poor: the majority of clusters are dominated by the terms of the Musician class. For example, with majority voting the clusters produced by LDA are all labeled as "M" (Fig 2(g)); for Kmeans 3 clusters labeled as "M" and no cluster labeled as "A" or "I" (Fig 2(h)). The similar problem occurs for CS corpus: no cluster labelled as "CP", "AD", or "OS" for LDA (Fig 2(c)), no cluster labelled as "CP" or "AD" for Kmeans (Fig 2(d)). This situation leads to the omission of concepts when using these unsupervised approaches for domain ontology building. In addition, the fact that the performance of unsupervised methods is better than that of CC-seeded models (see above), does not mean that the quality of their partition is better than that of CC-seeded models. SMBM somehow produces a little better partition than others ("C0" and "C4" in Fig 2(f)). Efficiency of Seed Set Kinds. We have conducted experiments on the CS whole dataset with CC-Seed to check the pertinence of CC-seed sets kinds and to compare then our approach to the original seeded LDA. It is important to remind that the latter uses information gain score based method for seed set extraction. IG and Hybrid are the CC-seed sets extracted respectively by information gain score based method and hybrid score based method. As mentioned before, in our approach, a CC-seed set includes hyponyms and synonyms of a CC. Each CC-seed sets kind comprises 100 terms with 10 terms/set. The seed terms of IG and Hybrid are more or less semantically close to their CC, some of them are hyponyms of the CC. IG has 27 common terms with the GS, and Hybrid has 24. The results show that CC-seed sets are the best kind for term clustering over CCs (Fig 3(a)). IG and Hybrid can be also used, the performance with Hybrid is better. This result proves that our approach outperforms the original seeded LDA. Impact of the Size of CC-seed sets. We also experiment on all seed based LDA models with various sizes of the seed set L (L ∈ [1, 2, ..., 10]). Each value of L will go through 20 experiments and the average performance is used. Each time, we randomly choose L × S seed terms from CC-seed sets as seed terms (L terms for each CC), while CCs are always used as seed terms. The final result (Fig 3(b)) shows that the performance is boosting with the growth of L. However, a small increment can be expected when L is higher than 6. Among all these models, CC-Seed and CC-SeedM1 can make the most use of CC-seed sets even just with one seed term (the CC) for each topic.

Unbalanced Corpus and Core-concept CC-seeded LDA. Fig 2 shows that none of those methods provide relevant partition on the Music corpus. The analysis of this corpus and more specifically its gold standard (Table 1) allows highlighting a big variance between the size of its CC-classes that can cause the bad partition. Indeed, all LDA models produce the roughly equal size of clusters (see the online supplementary materials1 for the results of other seed based LDAs). However, the size of the CC-class of "M" (Musician) is six times the size of the CC-class of "I" (Instrument). Assigning to clusters associated with "M" or "I" the same number of terms will produce a big false positive rate.

But why the cluster of a dominant CC (i.e. CC of a CC-class with a large number of terms w.r.t the others CC-classes) can not be big and the cluster of non-dominant one can not be small? The reason is related to the LDA approach. LDA generates topics by distributing the relevant terms to a topic with the highest probability. However, the total probability mass (i.e. 1) is shared by all terms in a topic. If a topic refers to a dominant CC, each term under this topic can get merely a small probability. Therefore, some terms are easily absorbed by other topics associated with non-dominant CCs (these terms may have a bigger probability under these topics). We conducted some experiments on the Music corpus to verify our guess. We assumed that allocating more topics for a CC will increase the shared probability mass and will improve the partition quality. One way is to increase the number of topics associated with each CC in the same way (e.g. each of the five CCs gets 2 topics then K = 10). Another solution is to use the number of topics proportional to the number of terms of each CC-class).

We checked these two strategies with some experiments: exp1, exp2, exp3, and exp4 for the first one, and experiments exp5, exp6, and exp7 for the second one. Table 3 shows that the first strategy is not helpful and the performance goes lower as the K grows. However, the results of exp5, exp6, and exp7 verify our guess and support our second strategy. The CC-seed achieved better performance than the SMBM (0.4089 vs 0.3670 on relevant dataset, and 0.3888 vs 0.3670 on whole dataset). But the total number of topics K should be reasonably small since the performance is not ideal with a large K.

Conclusion

In this paper, we propose the Core-concept seeded LDA, a new semi-supervised approach for concept formation, and more specifically for the formation of clusters where each one includes terms referring to a single CC of an ontology domain. This approach is an adaptation of seeded LDA models. We tackled two problems of existing clustering-based methods, i.e., the difficulty of labeling clusters and the low semantic coherence of clusters. The experimental results indicate that our proposal works better than baselines on a corpus with balanced classes or a corpus with imbalanced classes and a suitable number of topics for each CC. Among all these seed based LDA models, we recommend using the seeded LDA for its higher performance and stability. Concerning the seed sets, we suggest using the synonyms and hyponyms of the CCs and we have shown that CC-seeded LDA outperforms the original seeded LDA. For future works, irrelevant term elimination would be an important direction because it impacts directly the behavior of LDA models and contributes to decreasing the number of False Positives. In addition, applying our approach to an imbalanced corpus without knowing a priori the size of CC-classes is a challenging task. For that, one can use some optimization algorithms to determine the number of topics for each CC.

2 .

 2 For the d th document in a corpus of D documents, draw a topic distribution θ d ∼ Dirichlet(α). 3. For each word w n of the N words in the d th document: (a) Select a topic z n ∼ Multinomial(θ d ). (b) Choose a word w n ∼ Multinomial(φ z n ).

Fig. 1 .

 1 Fig. 1. Core-Concept-Seeded LDA steps for term clustering over core concepts.

  (a) CC-Seed on CS Whole (b) SMBM on CS Whole (c) LDA on CS Whole (d) K-means on CS Whole (e) CC-Seed on Musics Whole (f) SMBM on Music Whole (g) LDA on Music Whole (h) K-means on Music Whole Fig. 2. Cluster composition of CC-Seed, SMBM, LDA and K-means. X-axis represents the clusters. Y-axis represents the number of terms (terms of the class "Others" are not counted) in each cluster. The CC label of each cluster (prior labels for semi-supervised methods, majority labels for unsupervised methods) is mentioned above its bar. The number of TP is mentioned above each sub-graph.

  (a) Efficiency of seed sets kinds (b) Impact of CC-seed set size Fig. 3. Experiments on seed sets. P or R is used as criteria.

Table 1 .

 1 The number of terms labeled by each CC class or by "Others" in the 2 corpora.

	CS corpus

Table 2 .

 2 The performance (P or R) of each method on different datasets. Semi-supervised methods are in red, unsupervised methods are in blue.

	Corpus Dataset	CC-Seed CC-SeedM1 CC-SeedM2 CC-Z-labels SMBM LDA	K-means
	CS	CS Relevant CS Whole	0.5804 0.6323	0.5605 0.6159	0.5503 0.4947	0.5160 0.4010	0.5071 0.3789 0.5345 0.5071 0.4566 0.5214
	Music	Music Relevant 0.3294 Music Whole 0.3326	0.3171 0.3151	0.2749 0.2389	0.2940 0.2381	0.3670 0.4548 0.4701 0.3670 0.4516 0.4817

Table 3 .

 3 Performance (P or R) of CC-seeded LDA on Music corpus with different combinations of topic number for CCs. ↑ means the performance is better than the original group, and ↓ means the worse result.

	EXP	M(1297)	A(484)	G(395)	I(211)	P(485)	Relevant	Whole
	original	1	1	1	1	1	0.3294 -	0.3326 -
	exp1	2	2	2	2	2	0.3041 ↓	0.3242 ↓
	exp2	3	3	3	3	3	0.2942 ↓	0.3089 ↓
	exp3	4	4	4	4	4	0.2997 ↓	0.3011 ↓
	exp4	5	5	5	5	5	0.2947 ↓	0.2920 ↓
	exp5	6	2	1	1	2	0.4089 ↑	0.3888 ↑
	exp6	13	5	4	2	5	0.3842 ↑	0.3761 ↑
	exp7	26	10	8	4	10	0.3767 ↑	0.3631 ↑

Details of the datasets and the cluster information of this paper https://github.com/jason-huanghao/Core-Concept-Seeded-LDA.


Douglas O Pino-Herrera
email: d.pinoherrera@brgm.fr

Yannick Fayolle

Eric D Van Hullebusch

David Huguenot

Giovanni Esposito

Yoan Pechaud
email: yoan.pechaud@u-pem.fr

Surface volatilization modeling of (semi-)volatile hydrophobic organic compounds: the role of reference compounds

Keywords: Volatilization, Mass transfer modeling, Mass transfer kinetics, Henry's law constant, HOCs, Reference compounds

Volatilization of hazardous hydrophobic organic compounds is often observed in many water, wastewater and soil treatment (bio)processes. Several models have been developed to quantify and predict gas-liquid pollutant transfer, being the proportionality coefficient model (PCM) one of the most commonly used, particularly in wastewater treatment. The PCM is based on the use of oxygen as a reference compound, which has a low resistance to the transfer in the gas phase. However, this resistance might be important for (semi-)volatile organic compoundsor (semi-)VOCs, which may render the use of the PCM model inaccurate. This study proposes an experimental methodology and a modeling approach for the use of the two-reference compound model (2RCM) that considers both the liquid-side and the gas-side resistances, by using water and oxygen as references. Results showed that the 2RCM predicts more accurately the overall mass transfer coefficients than the PCM for a VOC and two semi-VOCs tested in this study. In addition, the 2RCM was found to be a more robust method to estimate mass transfer coefficient of any compound and its use can be extrapolated to all substances. Finally, the relevance and limitations of both models was established.

Introduction 1.

Over the last decades, the occurrence of hazardous hydrophobic organic compounds (HOCs) as pollutants in the aquatic environments and soils has become a major environmental concern. Among the pollutants concerned, many are volatile and semi-volatile and can thus be transferred to the atmosphere due to mass transfer processes. Physicochemical and biological processes are often used to remove these pollutants during wastewater treatment, water purification treatments and soil remediation. In general, these processes need mixing to improve the homogeneity and the reactor performance and/or the introduction of a gas phase by a diffuser (aerobic biological treatment, ozonation, electro-Fenton, etc.) [START_REF] Fayolle | Oxygen transfer prediction in aeration tanks using CFD[END_REF][START_REF] Hübner | Evaluation of the persistence of transformation products from ozonation of trace organic compounds -A critical review[END_REF][START_REF] Monteil | A review on efficiency and cost effectiveness of electro-and bio-electro-Fenton processes: Application to the treatment of pharmaceutical pollutants in water[END_REF][START_REF] Mozo | Dynamic modeling of biodegradation and volatilization of hazardous aromatic substances in aerobic bioreactor[END_REF][START_REF] Pino-Herrera | Removal mechanisms in aerobic slurry bioreactors for remediation of soils and sediments polluted with hydrophobic organic compounds: An overview[END_REF]. In systems open to the atmosphere, the mechanical power input promotes the surface aeration of the reactor, but it favors simultaneously the transfer of the most volatile molecules to the gas phase. In the same way, bubble dispersion through the liquid phase favors the transfer of the desired gas and at the same time the stripping of some volatile and semi-volatile compounds [START_REF] Baeten | When and why do gradients of the gas phase composition and pressure affect liquid-gas transfer?[END_REF]. However, despite the environmental and public health issues, the volatilization process has been, in general, severely underestimated in wastewater treatment process [START_REF] Yang | Underestimated public health risks caused by overestimated VOC removal in wastewater treatment processes[END_REF] and even not considered in many research papers [START_REF] Pino-Herrera | Removal mechanisms in aerobic slurry bioreactors for remediation of soils and sediments polluted with hydrophobic organic compounds: An overview[END_REF].

The most susceptible compounds to transfer are usually called volatile organic compounds (VOCs) and other compounds exhibiting the same behavior but in lesser extent (e.g. polycyclic aromatic compounds or PAHs) are frequently referred as "semi-volatile compounds". In general, the Henry's law constant determines the degree of volatilization of any compound [START_REF] Luo | A review on the occurrence of micropollutants in the aquatic environment and their fate and removal during wastewater treatment[END_REF]. However, the limit between the "volatile" and the "semi-volatile" categories is not clearly defined and there is no consensus in the literature regarding this issue.

Furthermore, it is difficult to generalize because the extent of the volatilization does not only depend on the molecules properties but also on the local hydrodynamic conditions [START_REF] Pomiès | Modelling of micropollutant removal in biological wastewater treatments: A review[END_REF].

To predict the gas-liquid mass transfer rate of volatile and semi-volatile compounds, many authors have used oxygen as a reference molecule [START_REF] Mihelcic | Estimation of VOC Emissions from Wastewater Facilities by Volatilization and Stripping[END_REF][START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF][START_REF] Padalkar | Removal mechanisms of volatile organic compounds (VOCs) from effluent of common effluent treatment plant (CETP)[END_REF]. They relate the mass transfer coefficient of both the target compound and oxygen using a proportionality factor that only depends on the ratio of the diffusion coefficients of the two molecules. This approach assumes that the mass transfer is mainly controlled by the liquid-phase resistance, which tallies with the low solubility property of oxygen. Nevertheless, this is only valid for very volatile compounds. Conversely, for less volatile compounds for which the gas phase resistance cannot be neglected more complex models should be used [START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF].

In this sense, two models have been proposed to predict the mass transfer coefficient of (semi-)volatile molecules. The most complex is the two-reference compound model (2RCM).

To estimate the mass transfer coefficient of the molecule considered, the mass transfer coefficient of two reference molecules need to be known; one whose transfer is controlled by the gas phase and one whose transfer is controlled by the liquid phase [START_REF] Smith | Volatilization rates of intermediate and low volatility chemicals from water[END_REF]. The second model, called the proportionality coefficient model (PCM), uses the sum of resistances of one reference molecule (in general oxygen). Unlike the 2RCM, the PCM is based only on the measurement of the liquid-phase resistance. The gas phase resistance is then estimated. Only very few studies have been performed to validate these models and no comparison between them have been done [START_REF] Mihelcic | Estimation of VOC Emissions from Wastewater Facilities by Volatilization and Stripping[END_REF][START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF][START_REF] Smith | Volatilization rates of intermediate and low volatility chemicals from water[END_REF][START_REF] Hsieh | Estimating Emissions of 20 VOCs. II: Diffused Aeration[END_REF]. In addition, most models require values of Henry's law constant (𝐻 𝐶 ) of the targeted compound. For many compounds, and particularly for HOCs, the range of values can be very wide, comprising several orders of magnitudes in some cases [START_REF] Sander | Compilation of Henry's law constants (version 4.0) for water as solvent[END_REF] which can lead to important errors in the estimation of the volatilization rates [START_REF] Mackay | Determination of air-water Henry's law constants for hydrophobic pollutants[END_REF].

Therefore, there is a need to further investigate and to assess methodologies and models allowing predicting mass transfer of volatile and semi-volatile molecules. In this context, the objective of the present work was to study and model the gas-liquid surface mass transfer process of HOCs using both the PCM and the 2RCM. Three HOCs with different 𝐻 𝑐 values were selected as model molecules for the volatile and semi-volatile groups one VOC (toluene) and two PAHs (naphthalene and phenanthrene). The Henry's law constants of these compounds were experimentally estimated. Additionally, oxygen and water were used as the reference compounds. Experimental and modeling results were used to elaborate a comparative analysis of both the PCM and the 2RCM.

Mass transfer modeling 2.

The two-film theory

According to the two-film theory, when a compound is transferred between two phases, it passes through two thin films that are formed on each side of the interface between these phases (Figure 1). The gradient of concentration in each layer decreases in the direction of the mass transfer and the relation between the concentrations at the interface is given by the Henry's law (Eq. ( 1)).

Moreover, Henry's law defines also the equilibrium concentration of each phase through Eq.

(2) and Eq. ( 3) [START_REF] Charpentier | Mass-Transfer Rates in Gas-Liquid Absorbers and Reactors[END_REF][START_REF] Cockx | Global and local mass transfer coefficients in waste water treatment process by computational fluid dynamics[END_REF][START_REF] Kashid | Gas-liquid and liquid-liquid mass transfer in microstructured reactors[END_REF].

𝐶 𝐺,𝑖 = 𝐻 𝑐 𝐶 𝐿,𝑖 (1)
𝐶 𝐺 * = 𝐻 𝑐 𝐶 𝐿 (2)
𝐶 𝐿 * = 𝐶 𝐺 𝐻 𝑐 (3)
Where 𝐶 𝐺,𝑖 , 𝐶 𝐺 and 𝐶 𝐺 * are respectively the gas interface concentration, the gas phase bulk concentration and the equilibrium concentration in the gas phase (kg.m -3 ); 𝐶 𝐿,𝑖 , 𝐶 𝐿 and 𝐶 𝐿 * are respectively the liquid interface concentration, liquid phase bulk concentration and the equilibrium concentration in the liquid phase (kg.m -3 ) and 𝐻 𝑐 is the dimensionless Henry's law constant (-).

Three possible cases are depicted in Figure 1: (i) the transfer of a liquid substance, such as water, to the gas phase at a temperature below its boiling point, known as evaporation; (ii) the transfer of a substance present in the gas phase, such as oxygen in air, into the liquid phase, called absorption; and (iii) the transfer of a substance dissolved in the liquid phase, in this case an HOCs, into the gas phase, usually called volatilization.

The mass transfer rate 𝑁 ̇ for any compound moving from the liquid phase to the gas phase is given by Eq. ( 4) (in the liquid film) and Eq. ( 5) (in the gas film) [START_REF] Cockx | Global and local mass transfer coefficients in waste water treatment process by computational fluid dynamics[END_REF][START_REF] Kashid | Gas-liquid and liquid-liquid mass transfer in microstructured reactors[END_REF].

𝑁 ̇= 𝑘 𝐿 𝐴(𝐶 𝐿 -𝐶 𝐿,𝑖 ) (4)
𝑁 ̇= 𝑘 𝐺 𝐴(𝐶 𝐺,𝑖 -𝐶 𝐺 )

Where 𝑘 𝐿 and 𝑘 𝐺 are the individual mass transfer coefficients respectively in the liquid film and in the gas film (m.s -1 ) and 𝐴 is the interfacial area (m 2 ).

If no accumulation in the layers is assumed, Eq. ( 4) and. Eq. ( 5) can be equalized. Then, invoking Henry's law (Eq. (1-3)), the transfer rate by unit of volume of a compound from the liquid phase to the gas phase can be calculated using either Eq. ( 6) or Eq. ( 7), depending on the phase in which the relation is applied [START_REF] Charpentier | Mass-Transfer Rates in Gas-Liquid Absorbers and Reactors[END_REF].

𝑑𝐶 𝐿 𝑑𝑡 = 𝐾 𝐿 𝑎(𝐶 𝐿 * -𝐶 𝐿 ) (6)
𝑑𝐶 𝐿 𝑑𝑡 = 𝐾 𝐺 𝑎(𝐶 𝐺 -𝐶 𝐺 * )

Where 𝐾 𝐿 𝑎 and 𝐾 𝐺 𝑎 are the overall volumetric mass transfer coefficient defined respectively for the liquid phase and the gas phase (s -1 ).

The 𝐾 𝐿 𝑎 in Eq. ( 6) is equal to the inverse of the sum of the reciprocals of the mass transfer coefficient of both the liquid layer and the gas layer (corresponding to a sum of resistances), as shown in Eq. ( 8). This coefficient is related to the one defined for the gas phase (in Eq. ( 7))

by the Henry's law constant of the compound (Eq. ( 9)) [START_REF] Kashid | Gas-liquid and liquid-liquid mass transfer in microstructured reactors[END_REF].

𝐾 𝐿 𝑎 = 1 1 𝑘 𝐿 𝑎 + 1 𝐻 𝐶 𝑘 𝐺 𝑎 ; 1 𝑅 𝑇 = 1 𝑅 𝐿 + 1 𝑅 𝐺 (8)
𝐾 𝐺 𝑎 = 𝐻 𝐶 𝐾 𝐿 𝑎

Where 𝑅 𝑇 , 𝑅 𝐿 and 𝑅 𝐺 are respectively the total resistance to transfer, the liquid-side resistance to transfer and the gas-side resistance to transfer (s.m -1 ).

For some compounds, the resistance in one of the phases can be considered negligible compared to the other phase. This is often the case for gases with low solubility (when the liquid phase is water), such as oxygen, that at standard conditions encounter very low resistance in the gas phase (𝑅 𝐿 ≫ 𝑅 𝐺 ). Thus, 𝐾 𝐿 𝑎 may be assimilated to the individual volumetric transfer coefficient of the liquid phase (𝑘 𝐿 𝑎) (Eq. ( 10)) and it is common to call these kind of processes "liquid-phase controlled mass transfer" [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview[END_REF].

𝐾 𝐿 𝑎 ≅ 𝑘 𝐿 𝑎 [START_REF] Mihelcic | Estimation of VOC Emissions from Wastewater Facilities by Volatilization and Stripping[END_REF] Conversely, for the substances with a higher affinity for the liquid phase, the mass transfer can be considered "gas-phase controlled", meaning that the resistance in the liquid film can be considered negligible (𝑅 𝐿 ≪ 𝑅 𝐺 ). In this case, 𝐾 𝐿 𝑎 may be approximated to the individual volumetric transfer coefficient of the gas phase (𝑘 𝐺 𝑎) multiplied by the Henry's law constant (Eq. ( 11)).

𝐾 𝐿 𝑎 ≅ 𝐻 𝐶 𝑘 𝐺 𝑎 (11)

Mass transfer coefficient and diffusivity

Most models indicate that the mass transfer coefficient in each layer is proportional to the diffusivity raised to some power, as mentioned by Munz and Roberts [START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF] (Eq. ( 12) and Eq. ( 13)) .

𝑘 𝐿 ∝ (𝐷 𝐿 ) 𝑛 (12)
𝑘 𝐺 ∝ (𝐷 𝐺 ) 𝑚

Where 𝐷 𝐿 and 𝐷 𝐺 are the diffusivities respectively in the liquid and gas phase (m 2 .s -1 ) and 𝑛 and 𝑚 are respectively liquid and gas diffusivity exponents (-).

This means that, if the individual mass transfer coefficient in each layer of a reference compound and the exponent of the diffusivity term are known, it may be possible to estimate this parameter for any desired compound using Eq. ( 14) and Eq. [START_REF] Hsieh | Estimating Emissions of 20 VOCs. II: Diffused Aeration[END_REF]. Depending on the mass transfer theory used to relate the diffusivityand the mass transfer coefficient, as well as its underlying assumptions, m and n can take several values. Table 1 shows the expression of the relation in Eq. ( 14) for the main mass transfer theories existing in the literature. Analogously, the same relations can be applied for the gas film (Eq. ( 15)). Surface renewal theory 𝑘 𝐿 = √𝐷 𝐿 𝑟 𝑐 0.5 [START_REF] Danckwerts | Significance of Liquid-Film Coefficients in Gas Absorption[END_REF] Where 𝛿 𝐿 is the thickness of the liquid film (m), 𝑡 𝑐 is the contact time (s) and 𝑟 𝑐 is the surface renewal rate (s -1 ).

𝑘
In most studies, one of the theories and its corresponding value for 𝑚 and/or 𝑛 are chosen.

However, some authors have estimated these values from experimental data by combining Eq. ( 8) and the ratio of diffusivities (Eq. ( 14) and/or Eq. ( 15)), which results in Eq. ( 16).

However, no consensus exists among the authors on the ranges, and even much less, on the specific values that these exponents might take [START_REF] Mihelcic | Estimation of VOC Emissions from Wastewater Facilities by Volatilization and Stripping[END_REF][START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF][START_REF] Arogo | Mass Transfer Coefficient of Ammonia in Liquid Swine Manure and Aqueous Solutions[END_REF][START_REF] Soltanali | Modeling of air stripping from volatile organic compounds in biological treatment processes[END_REF].

Additionally, a number of empirical correlations exist that allow the calculation of the overall mass transfer coefficient for any substance in specific configurations [START_REF] Heymes | An experimental investigation of evaporation rates for different volatile organic compounds[END_REF][START_REF] Poós | Mass transfer coefficient for water evaporation by theoretical and empirical correlations[END_REF]. However, it is important to observe the conditions in which the mentioned correlations are applicable.

Materials and methods 3.

Experimental part

Reactor and operating conditions

The experiments were carried out in a standard 4.2-l glass reactor (working volume) with a thermal jacket controlled at 20 °C and four baffles. The dimensions of the reactor are specified by Pino-Herrera et al. [START_REF] Pino-Herrera | Gas-liquid oxygen transfer in aerated and agitated slurry systems with high solid volume fractions[END_REF]. Two reactor set-up configurations were used (Figure 2): A) for Henry's law constant determination of HOCs, the gas phase was injected from the bottom of the reactor through a porous glass sparger connected to a three-port L-shaped valve, providing the choice between an air flow and a nitrogen flow as needed (Figure 2A); and B) for surface oxygen, water and HOCs mass transfer, the gas phase was introduced to the reactor using a plastic tube passing through holes in the lid, directing the air flow to the wall of the reactor. In this way, when the gas phase enters the reactor, preferential pathways for a direct exit and perturbations on the liquid surface were avoided (Figure 2B).

Figure 2. Experimental set-up: (A) Bubble mass transfer and (B) Surface mass transfer

Mechanical agitation was supplied by a motor with digital controlled stirring speed coupled with a single marine propeller (𝑑 𝑖 = 𝐷/3, where 𝐷 is the reactor diameter). The reactor was operated varying the corresponding operational parameters per test in the ranges given in Table 2. The power input was calculated using the power number (Eq. ( 17)). The power number 𝑁 𝑃 is constant and equal to 0.35 for a marine propeller at turbulent conditions (𝑅𝑒 > 10 4 ), which is the case for all conditions tested in this study [START_REF] Hall | Rules of Thumb for Chemical Engineers[END_REF].

𝑃 𝑁 𝑉 = 𝑁 𝑃 𝜌𝑁 3 𝑑 𝑖 5 𝑉 𝐿 (17)
Where 𝑃 𝑁 /𝑉 is the mechanical power input (W.m -3 ), 𝑁 𝑃 is the power number (-), 𝜌 is the water density (kg.m -3 ), 𝑁 is the stirring speed (s -1 ), 𝑑 𝑖 is the impeller diameter (m) and 𝑉 𝐿 is the liquid volume (m 3 ).

HOC solutions and monitoring

HOC solutions

The HOCs used in this research work (toluene, naphthalene and phenanthrene) were obtained from Sigma-Aldrich chemicals (≥ 98% purity). Solvents (methanol and acetonitrile, HPLC grade) and phosphoric acid were obtained from VWR chemicals. Diffusivity of the compounds used in this study in air and water are shown in Table S1.

A solution of the three HOCs in water was prepared. For phenanthrene and naphthalene, a concentrated solution in methanol was previously prepared and 2 ml of this solution were added to 5 l of tap water, containing 3 ml of toluene. The amount of methanol in solution (< 0.04%) was low enough not to modify the mass transfer and the surface tension of the liquid in the system [START_REF] Munz | Air-Water Phase Equilibria of Volatile Organic Solutes[END_REF]. The solution was magnetically stirred until no droplets of toluene were observed and then filtered to remove any possible PAH crystals remaining in suspension before it was added to the reactor (4.2 L).

HOC concentration monitoring during mass transfer experiments

For each HOC mass transfer experiments, samples of the liquid phase were taken before and after starting either the bubble or the surface airflow at appropriate times and analyzed for HOC concentrations. Samples were measured using an HPLC (Hitachi LaChrom Elite® L-2400) coupled with UV/VIS detector (set to 254 nm) and a fluorescence detector (Excitation wavelength set to 250 nm and Emission wavelength set to 350 nm). The separation was performed using a RP C-18 end capped column (Purospher®, Merck) (5 mm, 25 cm × 4.6 mm) placed in an oven at 40 °C. The mobile phase was a mixture of water (at pH adjusted to 2.5 using phosphoric acid) and acetonitrile (25:75 v/v) with a flow rate of 0.8 ml.min -1 in isocratic mode. The injection volume was 20 µl.

HOC Henry's law constant determination in bubbly reactor

The theoretical development for the determination of Henry's law constant is developed in Appendix A. For the Henry's law constant determination of the selected HOCs, the reactor set-up configuration shown in Figure 2A was used. An HOC solution was prepared as explained in section 3.1.2 and introduced in the reactor (4.2 l) and, after introducing the bubbly flow, the depletion of the HOC concentrations in the liquid phase was measured over time. Experiments were performed adjusting the operating parameters within the ranges given in Table 2.

Surface mass transfer in gas and liquid films

Oxygen mass transfer coefficient

The oxygen transfer coefficients were obtained using the configuration shown in Figure 2B and were measured using the dynamic method (gassing out), described by García-Ochoa and Gomez [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview[END_REF]. The curve of oxygen absorption was recorded using an inoLab® Oxi 7310 DO sensor connected to a Cellox 325 probe (WTW). From the oxygen absorption curves, the surface oxygen mass transfer coefficients (𝑘 𝐿 𝑎 𝑂 2 𝑆 ) in Eq. ( 18) were estimated, taking into account the response time of the electrode. The influence of power input on this parameter was measured within the range shown in Table 2.

𝑑𝐶 𝐿,𝑂 2 𝑑𝑡 = 𝑘 𝐿 𝑎 𝑂 2 𝑆 (𝐶 𝐿,𝑂 2 * -𝐶 𝐿,𝑂 2 ) (18)
Where subscript 𝑂 2 refers to oxygen, superscript 𝑆 refers to surface.

Water mass transfer coefficient

The water transfer coefficients were obtained using the configuration shown in Figure 2B. An airflow was continuously introduced to the upper part of the experimental system and steady state conditions in the gas phase was reached. Then, the gas phase relative humidity and temperature was measured using a KIMO ® AMI 310 multifunction meter at the inlet and the outlet of the reactor. The airflow was introduced in the gas phase from the top of the reactor, avoiding disturbances in the liquid surface and the saturation of the gas phase. The surface water transfer coefficient was subsequently obtained by performing a mass balance for the humidity in the gas phase (Eq. ( 19)), knowing the psychrometric conditions of the air at the inlet and the outlet of the reactor.

𝐶 𝐺,𝑊 𝑖𝑛 𝑄 𝐺 + 𝑘 𝐺 𝑎 𝑊 𝑆 (𝐶 𝐺,𝑊 * -𝐶 𝐺,𝑊 ) = 𝐶 𝐺,𝑊 𝑜𝑢𝑡 𝑄 𝐺 [START_REF] Cockx | Global and local mass transfer coefficients in waste water treatment process by computational fluid dynamics[END_REF] Where 𝐶 𝐺,𝑊 is the vapor concentration in air, the superscripts 𝑖𝑛 and 𝑜𝑢𝑡 refer respectively to the inlet and outlet airflow, the subscript 𝑊 refers to water and 𝐶 𝐺,𝑊 * is the saturated vapor concentration calculated at the air temperature using the equilibrium vapor pressure correlation developed by Lowe [START_REF] Lowe | An approximating polynomial for the computation of saturation vapor pressure[END_REF]. Since 𝐶 𝐺,𝑊 varies throughout the reactor headspace, the logarithmic mean vapor concentration between the inlet and outlet airflow was used in order to estimate the driving force for water evaporation along the water surface.

Several airflow rates (𝑄 𝐺 ) were tested for all agitation conditions to check that this parameter did not affect the transfer coefficient and that the saturated vapor concentration was not reached in the outlet airflow. An average airflow rate on the surface of 2.9×10 -4 m 3 .s -1 was fixed. The influence of power input on this parameter was measured within the range shown in Table 2.

Surface HOCs mass transfer coefficient

The overall HOC surface mass transfer coefficients were obtained using the reactor set-up configuration in Figure 2B. An HOC solution was prepared as explained in section 3.1.2 and introduced in the reactor. An airflow was introduced from the top of the reactor through the lid avoiding disturbances in the liquid surface to remove any accumulation of HOCs in the gas phase. In this case, HOC gas phase concentration was considered negligible, since the gas phase was continuously renewed (concentration in the inlet gas 𝐶 𝐺 = 0). This hypothesis was further confirmed by demonstrating the nominal gas phase HOC concentration (maximal HOC mass flux divided by 𝑄 𝐺 ) was negligible in comparison with the 𝐶 𝐺,𝐻𝑂𝐶 * .

A batch volatilization experiment where the HOC concentration was measured as a function of time led to a first order equation from which the HOC surface mass transfer coefficient was easily calculated (Eq. ( 6)). The influence of power input on this parameter was measured within the range shown in Table 2.

HOC mass transfer modeling

In this study, two models to predict the individual and the overall gas-liquid mass transfer coefficients of the HOCs were tested and compared: the 2RCM and the PCM.

Two-reference-compound model (2RCM)

In liquid-controlled mass transfer processes, oxygen is often used as reference compound because its gas-side resistance can be considered negligible and Eq. ( 10) can be applied.

Conversely, in aqueous solutions, water presents a virtually non-existent transfer resistance in the liquid phase, which allows the use of Eq. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF]. Then, by introducing oxygen and water as reference compounds in Eq. ( 16), the HOC mass transfer coefficient could be estimated using Eq. ( 20).

The exponents of each diffusivity ratio (𝑚 and 𝑛) were chosen according to the hydrodynamic conditions expected at each side of the interface as explained in section 3.2.3. The HOC mass transfer coefficients estimated using the experimental oxygen and water mass transfer coefficients and the Henry's law constant for each HOC were compared to those obtained experimentally and the deviations were calculated for each compound tested.

Proportionality coefficient model (PCM)

Hsieh et al. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF] transformed Eq. ( 16) by using only oxygen as the reference compound and defining the proportionality coefficient (𝛹) as shown in Eq. ( 21). This relation was also tested by fixing both exponents 𝑛 and 𝑚 as described in section 3.

Figure 3 shows a scheme of the parameter estimation followed in this study. First, from the experimental data and an assumed 𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄ , the HOC overall mass transfer coefficients for each power input tested were calculated. Then, the sum of the squared errors between the calculated coefficients and the experimental ones was minimized by modifying the initial assumed parameters. Once the minimal error was reached, the estimated and calculated parameters for each model were analyzed and compared. Both models were tested and compared based on their accuracy and their robustness to predict HOC overall mass transfer coefficients.

Selection of the diffusivity exponents (𝒏 and 𝒎)

In order to predict HOC mass transfer coefficients using Eq. ( 20) and Eq. ( 21), it is necessary to estimate or to fix the values for the exponents m and n. Several approaches for this estimation have been used in the literature, but, in general, they imply some assumptions that are difficult to test experimentally. For example, Hsieh et al. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF], Soltanali and Shams

Hagani [START_REF] Soltanali | Modeling of air stripping from volatile organic compounds in biological treatment processes[END_REF] and Munz and Roberts [START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF], assumed that both exponents are equal, due to the

Minimal error

No Yes uncertainty in their calculation and the lack of information regarding the parameters influencing them. Additionally, the latter authors have found that the diffusivity exponents are virtually independent of the mixing intensity. Moreover, some authors, such as Chrysikopoulos et al. [START_REF] Chrysikopoulos | Modeling the emission and dispersion of volatile organics from surface aeration wastewater treatment facilities[END_REF], Smith et al. [START_REF] Smith | Volatilization rates of intermediate and low volatility chemicals from water[END_REF] and Libra [START_REF] Libra | Volatilization of organic compounds in an aerated stirred tank reactor[END_REF], have considered that the diffusivity exponents depend on the type of compound. Since there is no consensus among the authors, the values for these exponents were fixed according to the hydrodynamic conditions at each side of the interface in this study. For the liquid side, the exponent 𝑛 was fixed to 0.5, which correspond to Higbie's penetration theory that is typically applied for turbulent regimes [START_REF] Higbie | Penetration theory leads to use of the contact time in the calculation of the mass transfer coefficients in the two film theory[END_REF]. In the case of the gas side, the mixing regime was not completely defined. Thus, the exponent 𝑚 was fixed to a value of 1, corresponding to the two-film theory [START_REF] Danckwerts | Significance of Liquid-Film Coefficients in Gas Absorption[END_REF].

Results and discussion 4.

HOC Henry's law constant

To calculate the Henry's law constant by the method described in section 3.1.3, the compound saturation concentration when the bubbles leave the liquid phase in the reactor need to be reached. Given that it was not possible to measure the gas phase concentration for the experiments performed in this study, indirect methods to ensure the bubble saturation were used. A linear relation was observed between the slope for bubble volatilization experiments tested (𝑆𝑝 𝐵 ) and the airflow rate at different mechanical power inputs (Figure S1). Likewise, there is no significant effect of the mechanical power input on this parameter. These corroborations allow to draw the conclusions that, for the three molecules tested in this research work, saturation was reached [START_REF] Mackay | Determination of air-water Henry's law constants for hydrophobic pollutants[END_REF] and that the bubble residence time was negligible in comparison to the variation of 𝐶 𝐿 in time (Eq. (A.2)).

The results for the 𝐻 𝐶 calculation are reported in Table 3. The calculated values for the Henry's law constant are within the range of those found in the literature, which further confirms the hypothesis of the bubble saturation. Additionally, it is interesting to observe that the Henry's law constants for HOCs presented in Table 3 show a wide range of values. This might be due to the different experimental set-ups and conditions in which they have been measured. Therefore, as 𝐻 𝑐 is a key parameter, it should be obtained experimentally whenever possible.

Surface mass transfer

Oxygen and water

Figure 4 shows the results of the test performed to measure the influence of 𝑃 𝑁 /𝑉 on the surface mass transfer coefficients of oxygen (𝑘 𝐿 𝑎 𝑂2 𝑆 ) and water (𝑘 𝐺 𝑎 𝑊 𝑆 ). These coefficients are positively correlated to 𝑃 𝑁 𝑉 ⁄ . In addition the trends in both cases follow a power-type curve fit, which is in accordance with the literature for many compounds [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF][START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF][START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview[END_REF].

However, for oxygen transfer, the power coefficient is five times higher than the one for water transfer. The 𝑃 𝑁 𝑉 ⁄ increases can lead to two effects at the surface:(i) since the agitation is being directly applied to the liquid phase, it induces a faster surface renewal (or a decreased film thickness) at the liquid side of the surface; (ii) higher agitation produces an increase of the gas-liquid interfacial area due to surface deformation. This effect was visible to the naked eye in the reactor. Since oxygen is transferred from the gas phase to the liquid phase, both phenomena affect its transfer. However, the continuous liquid phase is constituted by water, which means that the water transfer occurs at gas side of the gas-liquid interface. Hence, the first effect does not have any consequence and only the gas-liquid interfacial area

P N /V (W/m 3 )
B modification is influencing this water transfer. These phenomena can explain the significant difference in the influence of the power input in the transfer of these two substances.

Additionally, the exponent of the power relation for the oxygen transfer is in agreement with the results found by Hsieh et al. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF] for similar operating systems.

HOC

In the same way as for water and oxygen transfer, the surface volatilization coefficient of HOCs depends on the power input. The relation between 𝑃 𝑁 𝑉 ⁄ and the mass transfer coefficient (𝐾 𝐿 𝑎 𝐻𝑂𝐶

𝑠

) can be assimilated to a power model in the range of 𝑃 𝑁 𝑉 ⁄ tested in this study (Figure 5).

𝐾 𝐿 𝑎 𝐻𝑂𝐶 𝑆 = 𝐵 ( 𝑃 𝑁 𝑉 ) 𝑝 ( 22
)
Where 𝐵 and 𝑝 are empirical constant values. Moreover, a linear positive dependence of 𝑝 and the natural logarithm of 𝐻 𝐶 of the HOCs can be established (Figure 6). This can be explained by the effect of 𝑃 𝑁 𝑉 ⁄ on the mass transfer coefficient of the liquid side of the interface. In fact, 𝑃 𝑁 𝑉 ⁄ has a direct impact on the liquid side and increases at a higher extent the overall mass transfer coefficient of the more volatile compounds. Thus, the volatilization of semi-volatile compounds is less affected by 𝑃 𝑁 𝑉 ⁄ , because the gas-side transfer resistance is comparable or higher than the liquid-side transfer

P N /V (W.m -3 )
resistance (𝑅 𝐺 ≈ 𝑅 𝐿 ). The correlation proposed in Figure 6 can be used to estimate the extent of the impact of the power input on the volatilization rate for HOCs.

Figure 6. Correlation between 𝑝 and ln(H C ).

It is interesting to notice that the 𝑝 value is higher for toluene than for oxygen, which is probably due to changes in the surface properties of the toluene solution. However, further research is needed on the impact of the modification of surface properties on the film resistances and the mass transfer.

Modeling of surface HOC mass transfer coefficient

The two previously presented models (PCM and 2RCM) were tested employing the experimental data obtained for the surface transfer of HOCs, water and oxygen.

The PCM required the use of oxygen as the only reference compound makes possible to neglect the resistance in the gas phase for the reference compound (due to the high 𝐻 𝐶 value for oxygen). For this model, the ratio of individual volumetric mass transfer coefficients for the oxygen (𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄ ) is estimated. Since each individual mass transfer coefficient is influenced differently by the mixing in the reactor, their ratio changes with a variation of 𝑃 𝑁 𝑉 ⁄ . Therefore, the estimation was made assuming one

𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄ for each considered 𝑃 𝑁 𝑉 ⁄ .
The 2RCM is a more general approach than the PCM for the volatilization modeling of semivolatile substances [START_REF] Soltanali | Modeling of air stripping from volatile organic compounds in biological treatment processes[END_REF]. To test the 2RCM, oxygen and water were used as reference compounds for the liquid-side and gas-side mass transfer resistance, respectively. No parameters need to be estimated using this model, since both individual mass transfer coefficient were calculated experimentally, and the diffusivity exponents were fixed.

[PLAGECELL]

[PLAGECELL]

[PLAGECELL] γ = 0.11*ln(Hc) + 0.97 R² = 0.99 0

0.2 0.4 0.6 0.8 1 -8 -6 -4 -2 0 p (-) ln(Hc) (-)
Figure 7 shows the correlation between the experimental and the calculated values for the overall mass transfer coefficient of the three HOCs tested in this study for the PCM and the 2RCM. The PCM fits rather well to the most volatile compound (toluene), but much higher error values are obtained for the least volatile ones (naphthalene and phenanthrene).

Moreover, the plot between the calculated and the experimental overall mass transfer coefficients presents a good correlation (R 2 = 0.96). On the other hand, the 2RCM model fits much better for naphthalene and phenanthrene, but the difference between the experimental and estimated 𝐾 𝐿 𝑎 𝐻𝑂𝐶 𝑆 increase for toluene. Additionally, the goodness of fit is slightly better than that of the PCM (R 2 = 0.978) and most values are within the 30% error. This means that the 2RCM seems to predict better the overall mass transfer coefficient for the three substances tested. In addition this result shows that depending on the 𝑃 𝑁 𝑉 ⁄ value and thus on the hydrodynamic conditions the error can vary. In terms of reference compounds, most authors have used the PCM model due to the convenience of utilizing oxygen as the only reference compound. Thus, very few studies have used other substances as reference compounds, much less for estimating the gas-side mass transfer coefficient. Monteith et al. [START_REF] Monteith | Effect of Bubble-Induced Surface Turbulence on Gas-Liquid Mass Transfer in Diffused Aeration Systems[END_REF] suggested the use of ammonia for this purpose due to its low Henry's law constant, but the ionization of this compound in water and the influence of pH on the mass transfer may complicate the experiments and the analytical processing of the data. In fact, high ion concentrations and changes in surface tension can modify the mass transfer coefficient of any compound [START_REF] Matter-Müller | Transfer of volatile substances from water to the atmosphere[END_REF]. In the present paper, we chose to use water as a reference compound for the surface mass transfer of semi-volatile compounds. This choice was based on the following advantages (Smith et al. [START_REF] Smith | Volatilization rates of intermediate and low volatility chemicals from water[END_REF]): i) its transfer has virtually no

R² = 0.969 1.E-06 1.E-05 1.E-04 1.E-03 1.E-06 1.E-05 1.E-04 1.E-03 Predicted K L a S HOC (s -1 ) Experimental K L a S HOC (s -1 ) +30% -30% A R² = 0.9783 1.E-06 1.E-05 1.E-04 1.E-03 1.E-06 1.E-05 1.E-04 1.E-03 Predicted K L a S HOC (s -1 )
Experimental K L a S HOC (s -1 ) +30% -30% B resistance in liquid-phase film; ii) the measurement of water concentration in the gas phase is relatively easy to perform; and iii) it is an economic and fast method to obtain information for the gas-side mass transfer.

Besides, using Eq. ( 15) and taking water as a reference compound, it is possible to obtain the gas-side oxygen mass transfer coefficient (𝑘 𝐺 𝑎 𝑂 ⁄ values for the PCM were then compared to the calculated ones using the 2RCM model (Table 4). It is important to highlight that the values obtained by both methods are in the same order of magnitude of those found by Munz and Roberts [START_REF] Munz | Gas-and liquid-phase mass transfer resistances of organic compounds during mechanical surface aeration[END_REF], but much lower than those found by Hsieh et al. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF]. This can be explained by the different geometry and the hydrodynamic properties of the reactor in both phases.

Moreover, the calculated ratio decreases as the mixing intensity augments. Considering that the 𝑃 𝑁 𝑉 ⁄ was directly applied to the liquid phase and that the gas phase was fed with the same 𝑄 𝐺 throughout all the experiments, the obtained relation seems quite logical. In fact, when 𝑃 𝑁 𝑉 ⁄ increases, the renewal rate at the liquid interface augments, increasing the individual oxygen mass transfer coefficient in the liquid phase (𝑘 𝐿 ), while the flow properties in the gasside of the interface remains almost unchanged. This can be translated in less important 𝑘 𝐺 variations in comparison with 𝑘 𝐿 variations, particularly for oxygen since its transfer is liquidphase controlled. Consequently, 𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄ decreases when 𝑃 𝑁 𝑉 ⁄ increases.

However, for the ratio estimated using the PCM, the ratio increase with 𝑃 𝑁 𝑉 ⁄ . This trend contradicts the results calculated using water as reference compound and it is a direct consequence of using semi-volatile compounds for fitting the PCM using oxygen. Indeed, small changes in the gas-phase film can highly influence the transfer of semi-volatile compounds and this cannot be taken into accounted by the PCM with oxygen as the sole reference compound, distorting the relation between power input and 𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄ . In addition, with the purpose of testing the hypothesis made previously on the exponent values 𝑛 = 0.5 and 𝑚 = 1 and since the 2RCM does not requires the fitting of any parameter, an estimation of both 𝑛 and 𝑚 was done using this model with a minimization of the sum of the squared errors between the calculated and the experimental mass transfer coefficients. As a result, the best fitting for 𝑛 and 𝑚 was found for values of 0.53 and 1.01, respectively (R 2 = 0.98). This confirms the hypothesis behind the choice of these parameter values.

(Semi-)Volatile nature of HOCs

Many authors consider that the "volatile" characteristic of a dissolved compound in terms of mass transfer is mainly given by its Henry's law constant. Although there is no definitive consensus on the matter, in general, compounds with 𝐻 𝐶 ≥ 0.19 are considered volatile and thus their mass transfer is considered to be liquid-phase controlled [START_REF] Roberts | Modeling Volatile Organic Solute Removal by Surface and Bubble Aeration[END_REF]. Besides, some authors consider that the liquid-side resistance is only completely negligible for compounds with 𝐻 𝐶 ≥ 0.55 [START_REF] Soltanali | Modeling of air stripping from volatile organic compounds in biological treatment processes[END_REF]. According to these definitions, toluene is placed around the limit of the "volatile" category (𝐻 𝐶 = 0.178). Low-molecular-weight PAHs (i.e. naphthalene and phenanthrene) are generally considered "semi-volatile" compounds [START_REF] Smith | Volatilization rates of intermediate and low volatility chemicals from water[END_REF][START_REF] Lucattini | A review of semi-volatile organic compounds (SVOCs) in the indoor environment: occurrence in consumer products, indoor air and dust[END_REF], presenting a much more important gas-phase resistance [START_REF] Valsaraj | Sediment-to-air mass transfer of semi-volatile contaminants due to sediment resusmension in water[END_REF]. Nevertheless, some authors also consider naphthalene as a volatile compound [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF][START_REF] Martí | Water-Air Volatilization Factors to Determine Volatile Organic Compound (VOC) Reference Levels in Water[END_REF]. This proves that the limit between the two categories is not well-defined, mainly because the rate of volatilization of a dissolved compound does not only depends on 𝐻 𝐶 , but also on the hydrodynamic properties of the phases where the transfer occurs as highlighted by the present study.

Using the results of the 2RCM, the relative liquid resistance (𝑅 𝐿 𝑅 𝑇 ⁄ ) for each molecule as a function of 𝑃 𝑁 𝑉 ⁄ was calculated. This parameter represents the proportion of the total mass transfer resistance that is due to the liquid-phase transfer resistance. The calculation is based on Eq. ( 8) and the results are depicted in Figure 8. For toluene, more than 90% of the resistance correspond to the liquid side, which admits classifying it as a "volatile" compound, as expected. On the other hand, phenanthrene (less than 10% of liquid-side resistance) and naphthalene (between 20% and 40% liquid-side resistance) can be confirmed as "semivolatile" substances at the conditions tested in this study. This means that, since three substances tested are comprised within a wide 𝑅 𝐿 𝑅 𝑇 ⁄ range, the conclusions of this research work may be applicable to most volatile and semi-volatile substances.

Relevance and limitations of the models

Traditionally, in VOCs surface volatilization and stripping modeling, the gas-side resistance is considered negligible [START_REF] Mozo | Dynamic modeling of biodegradation and volatilization of hazardous aromatic substances in aerobic bioreactor[END_REF][START_REF] Baeten | When and why do gradients of the gas phase composition and pressure affect liquid-gas transfer?[END_REF]. However, this assumption is not always correct because of the variability of the "volatile" condition of the compounds according to the process characteristics and its operating conditions as highlighted by the present study. Therefore, for semi-volatile compounds, using models that consider the gas-side resistance is a better option.

Moreover, in many aerobic processes, oxygen represents a useful reference compound.

Indeed, numerous oxygen mass transfer measurements methods and scale-up correlations based on oxygen exist in the literature [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview[END_REF]. Thus, the PCM, usually based on this molecule, is one of the most frequently used in the literature. Although the use of only one reference compound is an easy and practical way to calculate the mass transfer coefficient of other compounds, this method has several limitations. For example, only the parameters affecting exclusively the liquid phase are considered (since the gas-side resistance is negligible for this compound). In addition, several parameters should be either assumed or estimated for the specific system, i.e. the ratio of individual mass transfer coefficient and the exponents of the liquid and gas diffusivity ratio. Hsieh et al. [START_REF] Hsieh | Estimating emissions of 20 VOCs. I: Surface aeration[END_REF] explain that the mass transfer coefficient ratio is generally assumed to be ranged between 50 and 300 (with an average of 150), which is a wide range. Besides, this parameter varies according to the hydrodynamic conditions as shown in section 4.3. In the same way, the values of 𝑚 and 𝑛 can significantly vary. All this may lead to wrong estimations of the mass transfer coefficient.

P N /V (W.m -3 )
One may reduce the uncertainties or the estimations errors by developing an approach to calculate the gas-side resistance using the 2RCM. By directly estimating the individual mass transfer coefficients with the help of two reference compounds, this model leads to a more robust way to obtain the gas-liquid mass transfer coefficient of any compound, volatile or not.

This research work has proved that, for surface volatilization, the use of water as a reference compound produces better results than the traditional PCM for semi-VOCs. The use of the 2RCM avoids the introduction of errors associated to assumptions made for the ratio of individual mass transfer coefficients. Knowing both oxygen and water transfer behaviors in the system allows accounting for liquid-film and gas-film changes, respectively. Moreover, by means of simple equipment, basic experiments on water transfer can be performed in almost any system to obtain the necessary information regarding the gas-side resistance. This implies that, if the reference compounds' gas-liquid transfer is characterized, the transfer of any volatile or semi-volatile compound can be obtained by extrapolation.

In this sense, the approach proposed in this paper can be used in wastewater treatment plants, slurry reactors, soil washing processes or any process in which HOC volatilization is nonnegligible to quantify the gas-liquid mass transfer of compounds susceptible to volatilize.

Nonetheless, the major limitation of this practice is that air saturation can be reached easily and rapidly, especially in systems with low or no gas-phase circulation. The same limitation applies for aerated systems using bubble diffusers within the liquid phase in which, depending on the bubble size and the physicochemical properties of the compound, few centimeters may be sufficient to reach the mass transfer equilibrium. For these cases, a similar approach, but using other substances allowing higher equilibrium concentrations in the gas phase can be used.

It is important to highlight that the extrapolation of the results of this paper for larger scales should be considered thoroughly regarding the specific conditions of the mass transfer. For instance, in large tanks open to the atmosphere, gas-side mass transfer at the surface will be largely controlled by the wind action, which will not be necessarily homogeneous along the surface of the reactor. Moreover, theoretical expressions proposed in Table 1, in which the models studied in this article are based allow an accurate representation of the mass transfer phenomena in conditions similar to the reactor condition. However, in the case of fully turbulent gas-films, modifications to the theoretical expressions should be included (e.g.

eddies dynamics in the interfacial layer), which would produce complex correlations between the diffusivity and the k G a, hindering the application of the 2RCM [START_REF] Brutsaert | A theory for local evaporation (or heat transfer) from rough and smooth surfaces at ground level[END_REF][START_REF] Prata | A critical review on liquid-gas mass transfer models for estimating gaseous emissions from passive liquid surfaces in wastewater treatment plants[END_REF].

Conclusions

5.

On one hand, this study demonstrated the impact of the mechanical power input ( 𝑃 𝑁 𝑉 ⁄ ) on the surface volatilization. A power-type correlation was found between water, oxygen and HOC surface overall mass transfer coefficients and 𝑃 𝑁 𝑉 ⁄ . A correlation between the Henry's law constant and the exponent of the mentioned power-type correlation for the HOCs was established. These correlations can be used to estimate the extent of the impact of 𝑃 𝑁 𝑉 ⁄ on the volatilization rate for HOCs.

On the other hand, the role of the reference compounds on the modeling of the mass transfer coefficient on volatile and semi-volatile compounds has been clarified. The 2RCM can predict the gas-liquid mass transfer coefficient with a reasonable error using only the hypothesis for fixing the diffusivity 𝑛 and 𝑚. In addition, the use of the 2RCM avoids the error introduced by the estimation of the ratio of individual mass transfer coefficients if the PCM is used for semi-VOCs. This means that the former is preferable for the cases where a reference compound with gas-side controlled mass transfer can be used, due to its higher robustness and its extrapolatable characteristics regarding hydrodynamic changes in both gas-side and liquidside interfaces.

Figure 1 .

 1 Figure 1. Two-film theory schema depicting three possible gas-liquid mass transfer processes: (i) evaporation, (ii) absorption and (iii) volatilization

Figure 3 .

 3 Figure 3. Modeling approach using the proportionality coefficient modem (PCM) and the two-referencecompound model (2RCM)

Figure 4 .

 4 Figure 4. Influence of the power input on the surface mass transfer coefficient for (A) Oxygen and (B) Water.

Figure 5 .

 5 Figure 5. Influence of power input on the surface mass transfer coefficient for ( ) Toluene, (•) Naphthalene and (○) Phenanthrene

Figure 7 .

 7 Figure 7. HOC mass transfer coefficients correlation for: (A) the PCM; and (B) the 2RCM; for ( ) Toluene, (•) Naphthalene and (○) Phenanthrene in logarithmic scale.

Figure 8 .

 8 Figure 8. Relative liquid resistance for ( ) Toluene, (•) Naphthalene and (○) Phenanthrene

Funding 6 .

 6 This work was supported by the European Commission through the Erasmus Mundus Joint Doctorate Program "Environmental Technologies for Contaminated Solids, Soils and Sediments" ETeCoS 3 [grant agreement FPA no. 2010-0009] and the Île-de-France region through the DIM R2DS program [no. 2014-13].

  Where 𝑘 𝐿 𝑎 𝑟𝑒𝑓 and 𝑘 𝐺 𝑎 𝑟𝑒𝑓 are respectively the 𝑘 𝐿 𝑎 and 𝑘 𝐺 𝑎 values of the reference compounds and 𝐷 𝐿,𝑟𝑒𝑓 and 𝐷 𝐺,𝑟𝑒𝑓 are respectively the 𝐷 𝐿 and 𝐷 𝐺 values of the reference compound.

	𝐿 𝑎 𝑘 𝐿 𝑎 𝑟𝑒𝑓	= (	𝐷 𝐿 𝐷 𝐿,𝑟𝑒𝑓	)	𝑛	(14)
	𝑘 𝐺 𝑎 𝑘 𝐺 𝑎 𝑟𝑒𝑓	= (	𝐷 𝐺 𝐷 𝐺,𝑟𝑒𝑓	)	𝑚	(15)

Table 1 .

 1 Relation between the mass transfer coefficient and the diffusivity

	Theory	Expression	Exponent 𝑛 value	Reference
	Two-film theory	𝑘 𝐿 =	𝐷 𝐿 𝛿 𝐿	1	[22]
	Penetration theory	𝑘 𝐿 = 2√ 𝐷 𝐿 𝑡 𝑐	0.5	[23]

Table 2 .

 2 Operational parameters used in this study.

	Reactor configuration Operational parameter	Units	Range
	A	B
		×	𝑄 𝐺 (×10 4 ) a	nm 3 .s -1	2.9
	×		𝑄 𝐺 (×10 4 ) b	nm 3 .s -1	0.375 -2.18
	×	×	𝑃 𝑁 /𝑉	W.m -3	17.65 -94.52
	×	×	T	°C	20

a surface airflow, b bubble airflow

  2.3. However, in this case, the ratio of individual oxygen mass transfer coefficients (𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2

									⁄	𝑆	) was not known. Thus,
	this parameter was estimated using the experimental 𝛹 obtained for each HOC tested in this
	study at different power input conditions.
	𝛹 =	𝐾 𝐿 𝑎 𝐻𝑂𝐶 𝑆 𝐾 𝐿 𝑎 𝑂 2 𝑆	=	(	1 𝐷 𝐿,𝐻𝑂𝐶 𝐷 𝐿,𝑂 2	)	𝑛 +	1 𝐻 𝐶 (𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 1 𝑆 ⁄	) (	𝐷 𝐺,𝑂 2 𝐷 𝐺,𝐻𝑂𝐶	)	𝑚

Table 3 .

 3 Estimated Henry's law constants and comparison with literature values.

	Compound	𝐻 𝐶 estimated (-)	Range of experimental 𝐻 𝐶 reference values*

*According to the 𝐻 𝐶 compilation made by Sander

[START_REF] Sander | Compilation of Henry's law constants (version 4.0) for water as solvent[END_REF]

Table 4 .

 4 𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2

	⁄	𝑆	ratio calculated using water as reference compound and estimated for the PCM and
	calculated
			𝑃 𝑁 𝑉 (W.m -3 )	𝑘 𝐺 𝑎 𝑂 2 𝑆 𝑘 𝐿 𝑎 𝑂 2 𝑆 ⁄	(-)
				Estimated by the PCM	Calculated

Appendix A. Theoretical considerations for Henry's law constant determination

The mass transfer rate by bubble volatilization is given by Eq. (A.1.) In this case, the transfer area is the interface between the dispersed bubbles in the liquid phase and the liquid phase itself (𝐴 𝐵 ).

𝑉 𝐺

𝑑𝐶 𝐺 𝑑𝑡 = 𝐾 𝐿 𝐴 𝐵 (𝐶 𝐿 * ,𝐵 -𝐶 𝐿 )

Where 𝑉 𝐺 the total gas phase volume in the bubbles and the superscript is 𝐵 refers to bubbles.

Under certain conditions, it is possible to assume that the variation of 𝐶 𝐿 is negligible in relation to the variation of 𝐶 𝐺 inside a single bubble rising through the reactor. Therefore, 𝐶 𝐿 can be considered constant for the integration of Eq. (A.1). Then, using Eq. ( 3) and integrating in time from the moment in which a bubble enters into the reactor (𝐶 𝐺 = 0) to the moment it exits (𝐶 𝐺 = 𝐶 𝐺 𝑜𝑢𝑡 ) (i.e. the gas residence time) and for all the bubbles dispersed in the gas phase, Eq. (A.2) is obtained.

Where 𝑡 𝑟 is the residence time of the gas in the reactor and is defined as

is rearranged by using the relations established in Eq. (A.3) derived from Henry's law (which defines the saturation degree), it is possible to obtain the concentration of the desired compound in the bubble when it reaches the surface of the liquid as a function of the mass transfer coefficient, the Henry's law constant and the operational parameters of the reactor 𝑉 𝐿 and 𝑄 𝐺 (Eq. (A.4)).

According to Matter-Müller et al. [START_REF] Matter-Müller | Transfer of volatile substances from water to the atmosphere[END_REF], a mass balance in the liquid phase of a reactor in which only bubble volatilization occurs would lead to Eq. (A.5). Invoking Henry's law (Eq.

( By combining and rearranging Eq. (A.5) and Eq. (A.6), it is possible to obtain Eq. (A.7).

Then, if the concentration of the volatile compound in the liquid phase is recorded in time, a first order curve may be plotted, from which it can be possible to calculate the saturation degree and, hence, the diffused mass transfer coefficient. A linear correlation may be found between the natural logarithm of the normalized concentration of the volatilized product and time, and the slope of this correlation, 𝑆𝑝 𝐵 , can be defined by Eq. (A.8).

Hsieh et al. [START_REF] Matter-Müller | Transfer of volatile substances from water to the atmosphere[END_REF] consider three cases regarding the saturation degree in the bubble: i) when 𝑆 𝑑 ≤ 0.1, the slope, 𝑆𝑝 𝐵 is approximately equal to the bubble mass transfer coefficient; ii) when 𝑆 𝑑 ≥ 0.99, the bubbles exit the liquid phase near saturation and the slope will approximate -𝑄 𝐺 𝐻 𝐶 𝑉 𝐿 and iii) when 0.1 ≥ 𝑆 𝑑 ≥ 0.99, the mass transfer coefficient can be calculated using the expressions in Eq. (A.4) and Eq. (A.8). When bubble saturation (case ii) is reached in a range of air flows, a linear correlation between 𝑆𝑝 𝐵 and 𝑄 𝐺 can be obtained and, using the slope of this correlation, it is possible to calculate accurately the Henry's law constant of a particular compound using Eq. (A.8) and assuming 𝑆 𝑑 = 1 [START_REF] Mackay | Determination of air-water Henry's law constants for hydrophobic pollutants[END_REF].

Gas-liquid mass transfer

Resistance to transfer:

Vola�liza�on modeling