# Improving heat exchanger fouling detection for phosphoric acid concentration units: A hybrid inverse approach integrating genetic algorithms and the Levenberg-Marquardt technique

Ridha Zitouni, Ali Fguiri, Amine Aymen Assadi, M Rahman, A Amrane, Juan

Pedro Solano, Mohamed Razak Jeday

► To cite this version:

Ridha Zitouni, Ali Fguiri, Amine Aymen Assadi, M Rahman, A Amrane, et al.. Improving heat exchanger fouling detection for phosphoric acid concentration units: A hybrid inverse approach integrating genetic algorithms and the Levenberg-Marquardt technique. Case Studies in Thermal Engineering, 2025, 73, pp.106572. 10.1016/j.csite.2025.106572. hal-05245160

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Distributed under a Creative Commons Attribution 4.0 International License
Improving Heat Exchanger Fouling Detection for Phosphoric Acid
Concentration Units: A Hybrid Inverse Approach Integrating Genetic
Algorithms and the Levenberg-Marquardt Technique

Ridha Zitouni<fnref n="1" />, Ali Fguiri<fnref n="1" />, Amine Aymen Assadi<fnref n="2" />*, Mohammod Hafizur Rahman<fnref n="2" />, Abdetif Amrane<fnref n="3" />, Juan Pedro Solano<fnref n="4" />, Mohamed-Razak Jeday<fnref n="1" />

Correspondence : E-mail: AAAssadi@imamu.edu.sa, Tel: 00966(0)563260210,

**Abstract:**

This study aims to estimate the asymptotic fouling resistance and the time constant using an industrial database and the mathematical fouling model developed by Kern and Seaton. The results obtained with the Kern and Seaton model show an asymptotic evolution of the fouling resistance as a function of time, consistent with that observed in the experimental data. Subsequently, the influence of asymptotic resistance (Rf*) and characteristic time (τ) on this model was examined. This inverse problem is formulated as the minimization of the sum of the quadratic differences between the measured thermal resistance and that calculated by the Kern and Seaton model. Different optimization methods, such as the genetic algorithm (GA) and the hybridization of the genetic algorithm with the Levenberg-Marquardt (LM) method, are employed. The GA method converges to the global minimum more slowly compared to the deterministic method. On the other hand, the hybridization between GA and LM shows a rapid convergence toward the global minimum with high precision, reaching a maximum relative error as low as 0.15%, 0.10%, and 0.12% for the three studied heat exchangers. The asymptotic fouling resistance was estimated to be 8.60×10⁻⁶ m<fnref n="2" />.K/W for the stainless-steel tubular heat exchanger (HEx1), 5.42×10⁻⁶ m<fnref n="2" />.K/W for the graphite polyblock exchanger “Supplier A” (HEx2), and 1.54×10⁻⁵ m<fnref n="2" />.K/W for “Supplier B” (HEx3), confirming the method’s reliability. This approach combines the genetic algorithm’s global search capabilities with the

<footnote n="1">¹ University of Gabes, National School of Engineering of Gabes, Laboratory of Process Research, Energy, Environment & Electrical Systems, Street of Omar Ibn El Khattab, Zrig Gabes 6029, Tunisia.</footnote>

<footnote n="2">² College of Engineering, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia.</footnote>

<footnote n="3">³Univ Rennes, ENSCR, CNRS, ISCR-UMR 6226, F-35000 Rennes, France</footnote>

<footnote n="4">⁴ Dep. Ing. Termica y de Fluidos, Universidad Politécnica de Cartagena, Campus Muralla del Mar, 30202 Cartagena, Spain.</footnote>
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Levenberg-Marquardt method's local refinement, demonstrating both robustness and efficiency in parameter estimation.

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**Keywords:** Fouling resistance; Kern and Seaton model; Genetic algorithm; Levenberg-Marquardt method; Heat exchanger optimization

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# 1. Introduction

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Fouling is generally defined as a deposit of materials or substances on a heat exchange surface [1]. The fundamental process of fouling is classified by Epstein into five steps: initiation, transport of the particles from the fluid to the wall, adhesion of the particles, removal of the deposited particles, and aging of the deposit. There are many forms of fouling, such as fouling of particles by iron oxides and hydroxides transported by fluids in industrial environments, corrosion, shelling, and biofouling caused by the presence of algae and fungi. This deposit leads to an increase in overall heat transfer resistance, which reduces the heat exchanger's performance[2,3]. In addition, this deposit, which is formed on one or both sides of the exchange surface, also affects the flow section of the fluids leading to a greater loss of pressure in the apparatus. As a result, more experimental works are carried out to understand the consequences of fouling on heat exchangers[4-7]. This problem has many adverse effects. It reduces the heat efficiency of the heat exchanger, which can lead to a decrease in performance or an increase in energy consumption to achieve the same efficiency. In addition, fouling increases the operating cost due to the need to use more energy to compensate for the loss of thermal efficiency. Therefore, several research studies have been conducted to find ways to limit the fouling of heat exchangers, as Maximilian Waack et al. [8] have done. They carried out experiments to measure the amount of deposits formed under different conditions of film flow and with the addition of surfactants. The results showed that the addition of surfactants can reduce the amount of scale formed. However, this effect is limited to specific film flow conditions. S. Hatte et al. [9] worked on a study that provides a systematic experimental analysis to elucidate the dynamic fouling of calcium carbonate and calcium sulfate on smooth, superhydrophobic, and liquid-impregnated surfaces on a generalized basis. To do this, they manufactured copper tubes whose internal wettability was modified and subjected them to forced convection at different rates and concentrations of salt.
Based on the results of the study, superhydrophobic surfaces were identified as reducing deposition, while liquid-impregnated surfaces were instead associated with increased deposition. These observations suggest that surface treatment may be a viable solution to reduce the dynamic fouling of heat transfer surfaces. The two previous studies indicate that although reducing fouling is possible, it is difficult to avoid it completely. As a result, it is often necessary to shut down the process and perform cleaning to remove accumulated deposits on heat transfer surfaces. The effective cleaning method includes chemical cleaning, which involves the use of chemicals to dissolve deposits, such as acids, alkalis or solvents; mechanical cleaning, which involves the use of brushes, scrapers, high-pressure water jets or other mechanical tools to remove deposits; and physical methods that include the electric field, ultrasonic field, and magnetic field[10, 11]. Regular cleaning is required to maintain the heat exchanger's efficiency, which can lead to delays and additional costs for maintenance operations [12]. This is why the need to minimize maintenance costs leads us to consider that it is essential to quickly detect fouling in the device.

Since fouling on the surface of the heat exchanger is a time-consuming process, it is difficult to observe its formation in real time and propose effective measures to prevent it. Consequently, many researchers have turned to predicting the resistance to fouling on the surface of the heat exchanger to better understand the formation of fouling. Accordingly, the neural network, the grey prediction model, and the partial least squares algorithm are widely accepted models for predicting fouling[13-15]. Wang et al. [16] proposed an innovative approach to predict the fouling of heat exchanger surfaces using an alternating magnetic field. They used the IGWO algorithm to select the most relevant characteristics related to fouling and then apply support vector regression (SVR) to build a prediction model. The work from Schlüter et al. [17] focused on the use of experimental data to develop models of local fouling resistances in heat exchangers. The authors collected experimental data on fouling in different types of heat exchangers and used this data set to develop predictive models. Fguiri et al.[18]focused on the accurate estimation of the resistance to fouling in a heat exchanger used for the treatment of phosphoric acid by steam. The authors used an inverse method approach to estimate fouling resistances from experimental data. In this study, Fguiri et al. used the LM method in the inverse method to estimate model parameters. In another work, Fguiri et al [19] used the
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inverse method for the determination of the heat transfer coefficient, producing
satisfactory results.

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However, a major challenge is the need to provide the model with an initial estimate
very close to the final result. This has led to the exploration of new methods for
combining the inverse method with other approaches. Among these approaches is
the use of optimization methods inspired by nature and metaheuristics, which are
among the most commonly used techniques for solving complex engineering
problems[20-23]. These stochastic techniques are distinguished by their ability to
search for the global optimum without requiring a precise starting point or knowledge
of the gradient of a function [24]. These methods provide an effective exploratory
approach to solving optimization problems, randomly exploring the solution space to
find the best possible outcomes. Their performance was evaluated in the work of
Küdela et al. [25], who used them for the inverse identification of the effective thermal
capacity of phase-change materials, and in the work of Bouallègue et al. [26], which
explored optimization techniques inspired by nature, emphasizing their application in
the optimization of parameters of fuzzy control systems. Among the different
approaches available to solve optimization problems, the GA stands out as most
commonly used[27-29]. Their effectiveness has been demonstrated in finding optimal
solutions for a variety of problems, including predicting and studying fooling
parameters [30], [31]. Thus, the use of the GA has important advantages, particularly
in terms of exploration and exploitation of the research space. These advantages are
particularly useful for initial estimation in deterministic methods such as the LM [32],
[33]. This approach saves a lot of time during the estimation process. Based on its
excellent scalability, GA is combined with the LM algorithm, forming a hybrid GA-LM
algorithm. Thanks to the combination of the LM algorithm, GA search conditions can
be effectively reduced, and the stability of the GA has also been improved. This
hybrid method has applications in a variety of research studies, such as electrical[34],
thermal [35], and optics [36]. The work of Liang Zhang et al.[37] presents an
innovative early fault diagnosis method for lithium-ion batteries by combining an
equivalent circuit model (ECM) with a TCN-BiLSTM neural network. Using simulated
fault data, this approach reliably detects early internal short-circuit faults in energy
storage batteries.
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The work of Xiong Xu[38] addresses the detection of anomalous sounds generated by degraded hydro turbine operation using a lightweight information enhancement fusion network (IEFNet) that combines sound feature extraction with load data to improve real-time monitoring accuracy. The method enables reliable anomaly detection every 10 seconds, strengthening turbine operational reliability.

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In this paper, a set of industrial data taken from a phosphoric acid concentration process is analyzed to derive the thermal resistance due to fouling. A genetic algorithm (GA) and a novel hybrid GA-LM method are subsequently used to analyze the experimental results in the search for a reliable predictive model of the fouling resistance.

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## 2. Material and Methods

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### 2.1. Description of the industrial unit

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The unit of phosphoric acid concentration within the phosphoric acid plant of the Tunisian Chemical Group (GCT) is equipped with heat exchangers whose thermal performance is analyzed in this work. This process is commonly used to concentrate acid from 28% P₂O₅ to 54% P₂O₅ using a closed-circuit forced-circulation evaporator, which typically operates in vacuum tanks. The unit under study is equipped with an operating diagram clearly illustrated in Fig.1. At the heart of this facility is a central heat exchanger. In addition to the heat exchanger, the unit has other components such as a basket filter, a barometric condenser, a boiler, and a circulation pump[39].
<figure><img src="image_2.png" /><figcaption>Fig 1. Diagram of phosphoric acid concentration unit</figcaption></figure>

The flux of dilute phosphoric acid (1), which contains impurities, is fed continuously into the basket filter. It is carefully mixed with circulating phosphoric acid (2). The filtration objective is to retain any crusts and debris from phosphogypsum that may be present in the fluid. This approach avoids any risk of damage to the circulation pump and obstruction of the heat exchanger tubes. In addition, the circulation pump takes the mixture produced and sends it to a heat exchanger. The heat exchanger is responsible for raising the acid temperature from 70°C to about 80 °C. Condensation at the steam inlet (3) also occurs during the heat process, particularly at a temperature of 120 °C. The condensate outlet (4) is diverted to a storage tank before being sent back to the service station. The mixture of phosphoric acid leaving the heat exchanger (5) then evaporates at the boiling temperature in the boiler (6) to produce the concentrated acid (7).

It should be noted that the operational data collected include the phosphoric acid inlet and outlet temperatures, the pressure and temperature of the steam used for heating at the heat exchanger inlet, and phosphoric acid suction and discharge pressures at the circulation pump. These measurements are taken every two hours during each
process cycle. The measurement points are marked in Fig.2. Three temperatures (Acid: T<sub>in,ac</sub>, T<sub>out,ac</sub>, and steam: T<sub>st</sub>) are measured using PT-100 type A resistance temperature detectors, with an uncertainty of 0.3 °C. In addition, a DMA35 density meter was also used with an uncertainty of 0.05% for the phosphoric acid. In addition, the steam pressure (Acid: p<sub>s</sub>, p<sub>d</sub>) was measured with a membrane manometer with an uncertainty of 1.6%. Finally, the flowrate v<sub>ac,cir</sub> of the phosphoric acid is measured with a 2% uncertainty. Table 1 summarizes the operating limits of the various variables influencing heat exchanger efficiency.

<figure><img src="image_3.png" /><figcaption>**Fig 2.** Heat exchanger integration diagram with signal processing</figcaption></figure>

<table><caption>**Table 1.** Measurement ranges</caption><thead><tr><th rowspan="2">Variable</th><th colspan="2">Range</th></tr><tr><th>Min</th><th>Max</th></tr></thead><tbody><tr><td>T<sub>in,ac</sub> (°C)</td><td>38</td><td>87</td></tr><tr><td>T<sub>out,ac</sub> (°C)</td><td>38.6</td><td>88.5</td></tr><tr><td>T<sub>st</sub>(°C)</td><td>118</td><td>138</td></tr><tr><td>p<sub>s</sub>(bar)</td><td>0.4</td><td>1.6</td></tr><tr><td>p<sub>d</sub>(bar)</td><td>2.4</td><td>4.7</td></tr></tbody></table>
It should be noted that the phosphoric acid concentration units of the Tunisian Chemical Group plants in Gabes use three types of heat exchangers: a stainless-steel tubular heat exchanger (HEx1) [40]and two graphite poly-block heat exchangers (HEx2, HEx3) [41].

The calculation of the fouling resistance will be performed according to the procedure detailed in the flowchart in Fig 3. This flowchart defines the successive steps of the calculation, from the collection of input data until the result is obtained. The final result, expressed in (m².K/W), represents the resistance to fouling of the exchangers.

<figure><img src="image_4.png" /><figcaption>Fig 3. Calculation of the experimental fouling resistance</figcaption></figure>

## 2.2. Fouling model

The Kern and Seaton [42] method is used to calculate the fouling resistance, under the assumption of two processes working similarly: the first is particle deposition characterized by a $\Phi_d$ deposition flux. The second process is that of the re-entry of the characteristic particle by a re-entry flux $\Phi_r$. The particle balance of the deposit is expressed according to the following equation:
$$ \frac{dm_p}{dt} = \Phi_d - \Phi_r \qquad (1) $$

198 $\Phi_d$ was time-independent and $\Phi_r$ directly dependent on the actual value of $m_p$, i.e.:

$$ \Phi_d = k_p \times (C_b - C_w) \qquad (2) $$

$$ \Phi_r = C_1 \times \tau_w \times m_p \qquad (3) $$

201 with $k_p$ and $C_1$ are two constants, $C_b$ and $C_w$ the particle concentration in the feed load and at the wall, $\tau_w$ the shear stress at the wall. So, the equation becomes:

$$ \frac{dm_p}{dt} = k_p \times (C_b - C_w) - C_1 \times \tau_w \times m_p \qquad (4) $$

204 The solution of this equation:

$$ m_p = \frac{k_p \times (C_b - C_w)}{C_1 \times \tau_w} \times [1 - e^{(-C_1 \times \tau_w \times t)}] \qquad (5) $$

$$ m_p = m_p^* \times [1 - e^{(-\frac{t}{\tau})}] \qquad (6) $$

207 Considering that the initial fouling flux is equal to the deposition flux and that the thermophysical properties of the deposit are constant, the equation can therefore be formulated as a thermal fouling resistance:

$$ R_f(t) = R_f^* \times [1 - e^{(-\frac{t}{\tau})}] \qquad (7) $$

210 where $R_f(t)$ is the temporal evolution of fouling resistance (m².K/W), $R_f^*$ is the asymptotic value of fouling resistance (m².K/W) and $\tau$ is the characteristic time (h).

213 Focusing on the characteristic time $\tau$, a key parameter used by manufacturers in their graphical approach, expressed in hours, is highlighted [43]. This characteristic time is generally attributed to the period required for the fouling resistance to reach its asymptotic value, assuming that the kinetic evolution follows a linear trajectory.

217 According to previous works [44], it is recommended to experimentally opt for tests with a duration equivalent to three times the characteristic value (3 × $\tau$). This approach leads to basing the experimental method on the duration of the tests or, in an industrial context, on the duration of a running cycle, in order to arrive at a precise determination of these two parameters. In this study, the determination of these
specific parameters will be addressed using an inverse estimation method. To do this, we use techniques such as the GA and the hybridization between the GA and the LM method, which allow a more robust and reliable estimation.

## 2.3. Inverse problem formulation

This study's resistance to heat deposition, expressed using the Kern and Seaton model (Eq. 7), is essentially based on two main factors. The estimation of these two factors results from the inverse resolution of the problem. In general, the inverse problem is formulated in the form of an optimization, where the objective function consists of the sum of the squares of the differences between the measured and estimated values. Suppose that $P = (R_f^*, \tau)$ the estimated fouling parameter vector and $P = (P)$ is the sum of the quadratic difference between the experimental measurements and the mathematical modelling of the fouling thermal resistance:

$$f(P) = \sum_{i=1}^{m} (R_{f,exp} - R_{fi}(P))^2 \quad (8)$$

$R_{f,exp}$ and $R_{fi}$ represent the measured and estimated fouling resistance. Thus, the determination of these parameters is carried out using the inverse method based on stochastic techniques which are the GA and a hybridization between the GA and LM.

## 2.4. Method of estimation

This article examines the methods of estimation used to derive key parameters from the Kern and Seaton model. We will specifically address approaches for improvement, such as the genetic (GA) and the hybrid GA with LM (GA-LM).

### 2.4.1. Genetic algorithm method

The GA method is the most well-known evolutionary optimization metaheuristic widely used in research such as [30], [45]. These algorithms have the distinction of being inspired by the evolution of species in their natural environment [46].

A GA will reproduce this evolutionary model with the objective of finding solutions to a given problem. We then use terms borrowed from biologists and geneticists to better represent each of the concepts discussed [47]. In our case, a population will be a set of individuals. These individuals will be an answer to a given problem, whether or not it is a valid solution to the problem. For the gene and the generation, the first will be
251 part of an answer to the problem so that an individual and the second is an iteration
252 of our algorithm.

253 The steps of our GA are presented in the flowchart in Fig. 4. To start an estimation
254 procedure using the (GA) in MATLAB, the essential elements of the GA must be first
255 configured. We start by defining the population size to 100 individuals. This value
256 indicates how many individuals will be present in each generation of the population.
257 Then, we specify the tolerance of the function, with a value of the order of 1.10<sup>-6</sup>. This
258 tolerance is a convergence criterion, meaning that the algorithm continues to iterate
259 until the difference between the values of the objective function of successive
260 generations is less than this value, reaching a permissible error threshold ε<sub>min</sub> < 0.01

<figure><img src="image_5.png" /><figcaption>Fig 4. Genetic algorithm flowchart</figcaption></figure>

263 **2.4.2. Hybridization method**

264 The term hybrid refers to the implementation of the deterministic method in each
265 generation of the GA. In this work, we have chosen the LM algorithm as a
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deterministic method [48]. In each generation, the best individual of the population obtained from GA is given as an initial estimate to the LM algorithm. In this way, the advantages of GA and LM algorithms are combined.

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The advantages of the hybrid GA-LM for the inverse estimation of unknown parameters are the efficiency in improving the performance and the convergence speed and the possibility of reaching the overall optimal value [34]. GA-LM hybridization intelligently combines these two approaches. It starts by using GAs to generate an initial population of potential solutions. Then each solution is refined locally using the LM. This means that GA is responsible for the global exploration of solution space, while LM rapidly refines solutions in regions where convergence is possible [35].

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The term hybrid refers to the implementation of the deterministic method in each generation of the GA. In this work, we chose the LM algorithm as a deterministic method [48]. In each generation, the best individual of the population obtained from GA is given as an initial estimate to the LM algorithm. In this way, the advantages of GA and LM algorithms are combined.

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The advantages of the hybrid GA-LM for the inverse estimation of unknown parameters are the efficiency to improve the performance on the convergence speed and the possibility of reaching the overall optimal value [34]. GA-LM hybridization intelligently combines these two approaches. It starts by using GAs to generate an initial population of potential solutions. Then each solution is refined locally using the LM. This means that GAs is responsible for the global exploration of solution space, while LM rapidly refines solutions in regions where convergence is possible [35].

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The general overview of the procedure adopted in this work for the implementation of the hybrid GA-LM is presented in the form of a flowchart in Fig. 5.

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<figure><img src="image_6.png" /><figcaption>Fig 5. GA-LM hybridization flowchart</figcaption></figure>

## 3. Results and discussion

### 3.1. Experimental data

After analyzing the experimental data in Fig. 6 and 7, we found that the vapor pressure gradually increases over time while the flow rate decreases. What is particularly interesting is that during certain increases in pressure, the flow remains constant; conversely, when the flow decreases, the pressure remains stable. This suggests the formation of deposits on the surface. For example, for the HEx1 exchanger, after 8 hours of operation, the pressure remains stable at 1.8 bar, but the flow rate decreases from 30 to 24 m³/h. Similarly, for HEx2, after 16 hours of operation, the pressure remains constant while the flow rate continues to decrease over time.
<figure><img src="image_7.png" /><figcaption>Fig 6. Evolution of Flow Rate Over Time for HEx1, HEx2, and HEx3</figcaption></figure>

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<figure><img src="image_8.png" /><figcaption>Fig 7. Steam Pressure Variation in Heat Exchangers HEx1, HEx2, and HEx3</figcaption></figure>

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Analysis of the data from the experiment on three heat exchangers reveals the formation of fouling. This observation is confirmed when we plot the fouling resistance and heat flux curves in Fig. 8. As far as heat flow is concerned, a reduction in heat transfer is observed over time, and this reduction is significant for all three heat exchangers. For example, for HEx1, the heat flux decreases from $2 \times 10^4$ W to 8776 W at the end of the test. As for fouling resistance, we observe that there is an asymptotic tendency in the resistance to fouling. Each exchanger shows that the resistance to fouling converges (Fig. 8) to a stable value, defined as the asymptotic resistance to fouling. These limit values differ according to the exchanger under consideration. Indeed, for the tubular, graphite heat exchangers «Supplier A» and «Supplier B», the asymptotic fouling resistance values are $R_f^*=1.61 \ 10^{-4} \ m^2 \cdot K/W$, $R_f^*=1.79 \ 10^{-4} \ m^2 \cdot K/W$ and $R_f^*=2.71 \ 10^{-4} \ m^2 \cdot K/W$ respectively for 102, 116 and 120 hours of operation.

a)

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>c)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig 8.** Evolution of Fouling Resistance and Heat Flux: a) HEx1, (b) HEx2, c) HEx2</figcaption></figure>
## 3.2. Estimation process

The objective of the simulation is to study the feasibility of the inverse problem-solving procedure to estimate the resistance to fouling of the heat exchanger. Initially, the Kern and Seaton model is considered for the resistance to fouling of an industrial heat exchanger. The estimation of parameters by the GA and GA-LM methods is executed in MATLAB software, with the aim of reducing the quadratic sum of the differences between the mathematical model of the resistance to fouling and the measurements associated with the same moments.

To obtain reliable results from the analysis, we analyzed the Root Mean Square Error (RMSE). This error gives a synthetic measure of the overall error in a single value, which is expressed as follows [49]:

$$RMSE = \sqrt{\frac{\sum_{i=1}^{M} (R_{f,exp} - R_{f_i^{theorique}})^2}{M}} \quad (9)$$

where $R_{f_i^{theorique}}$ is the Kern and Seaton model of the fouling resistance determined.

The control parameters for the GA are chosen to be the same. Indeed, the probability of crossing with pc= 0.9 and the probability of mutation p_m=0.005. When solving an estimation or optimization problem, the selection of control parameters for stochastic methods is crucial. These parameters, such as population size and maximum number of iterations, must be carefully defined before launching the algorithm. An effective strategy is to test different combinations of these parameters to assess their impact on RMSE. This helps identify the most efficient configuration for the specific application in question. Table 2 presents such an optimization experiment and concludes with the choice of appropriate values.

<table><caption>**Table 2.** Comparison of simulation results with different numbers of iterations and population size</caption><thead><tr><th>Iteration</th><th>Population size</th><th>GA</th></tr></thead><tbody><tr><td rowspan="3">50</td><td>30</td><td>2.257·10<sup>-7</sup></td></tr><tr><td>50</td><td>3.80278·10<sup>-8</sup></td></tr><tr><td>100</td><td>1.56437·10<sup>-8</sup></td></tr></tbody></table>
<table><tr><td rowspan="3">100</td><td>30</td><td>1.41535·10<sup>-7</sup></td></tr><tr><td>50</td><td>2.89453·10<sup>-8</sup></td></tr><tr><td>100</td><td>1.46987·10<sup>-8</sup></td></tr><tr><td rowspan="3">250</td><td>30</td><td>8.35935·10<sup>-8</sup></td></tr><tr><td>50</td><td>2.543·10<sup>-8</sup></td></tr><tr><td>100</td><td>1.40245·10<sup>-8</sup></td></tr></table>

The genetic algorithm is a stochastic method that, although it does not produce identical results with each run, is extremely effective for overall optimization. A statistical analysis was conducted on this method, revealing that the execution of twenty iterations of this algorithm with three heat exchangers makes it possible to determine the optimal, average, and worst-performing cases, as well as the standard deviations resulting from these executions.

<table><caption>**Table 3.** Optimization results from 20 runs of problem</caption><thead><tr><th>Heat exchanger</th><th>Best case</th><th>Middle case</th><th>Bad case</th><th>STD</th></tr></thead><tbody><tr><td>stainless-steel tubular<br>(HEx1)</td><td>8.6036·10<sup>-6</sup></td><td>8.9279·10<sup>-6</sup></td><td>9.8134·10<sup>-6</sup></td><td>3.4855·10<sup>-7</sup></td></tr><tr><td>graphite poly block A<br>(HEx2)</td><td>5.4201·10<sup>-6</sup></td><td>6.1708·10<sup>-6</sup></td><td>8.2669·10<sup>-6</sup></td><td>7.4844·10<sup>-7</sup></td></tr><tr><td>graphite poly block B<br>(HEx3)</td><td>1.5417·10<sup>-5</sup></td><td>1.6566·10<sup>-5</sup></td><td>1.8830·10<sup>-5</sup></td><td>1.0618e-10<sup>-6</sup></td></tr></tbody></table>

The statistical analysis of heat exchangers through the use of a GA reveals very promising results. The standard deviations observed between the 20 executions for the three types of exchangers are indeed acceptable, ranging between 3.4855·10⁻⁷ and 1.0618·10⁻⁶ m².K/W indicating the reliability and accuracy of the algorithm. The algorithm provided results close to the best-case level for the three exchangers such as 8.6036·10⁻⁶ m².K/W for the stainless-steel tubular heat exchanger (HEx1), 5.4201·10⁻⁶ m².K/W for the graphite poly block heat exchanger «Supplier A» (HEx2) and 1.5417·10⁻⁵ m².K/W for the graphite poly block heat exchanger «Supplier B» (HEx3).
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## 3.3. Estimation result

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The convergence curve is crucial in analyzing numerical methods, especially for parameter estimation. It provides insight into the progression of the estimation error across iterations.

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In fact, Fig. 6, 7, and 8 illustrate the convergence histories related to the GA and its hybridization with the LM method for the test problem involving the three heat exchangers. These three figures show that the two methods offer excellent exploration of the search space, helping to avoid getting blocked in the local minimum. In addition, they make good use of the best individuals obtained to reach the global minimum. Over the generations, Figs. 9, 10, and 11 show that the two methods converge towards approximately the same values of the fitness function, which are very close to zero, thus obtaining the best solution for the asymptotic value of fouling resistance $R_f^*$ and the characteristic time $\tau$.

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<figure><img src="image_12.png" /><figcaption>**Fig 9.** Performance comparison of algorithms GA and GA-LM for the stainless-steel tubular heat exchanger (HEx1)</figcaption></figure>

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<figure><img src="image_13.png" /><figcaption>**Fig 10.** Performance comparison of algorithms GA and GA-LM for the graphite poly block heat exchanger «Supplier A» (HEx2)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig 11.** Performance comparison of algorithms GA and GA-LM for the graphite poly block heat exchanger «Supplier B» (HEx3)</figcaption></figure>
## 3.4. Comparison of methods

After the visualization of the convergence curves, the table below compares the performance of the GA and GA-LM methods, applied to the HEx1 stainless steel heat exchanger, based on the fitness value $f(P)$ obtained over 20 runs for each generation setting (30, 50, and 100). This table summarizes the mean values and standard deviations of the results obtained for each method. These statistics allows us to assess both the accuracy and the robustness of the two approaches.

<table><caption>**Table 4.** Fitness Value Comparison of GA vs. GA-LM on HEx1 (20 Runs, Generations: 30, 50, 100)</caption><thead><tr><th rowspan="2">Number of generations</th><th colspan="2">GA</th><th colspan="2">GA-LM</th></tr><tr><th>Mean</th><th>STD dev</th><th>Mean</th><th>STD dev</th></tr></thead><tbody><tr><td>30</td><td>2.1501 10<sup>-8</sup></td><td>1.4291 10<sup>-8</sup></td><td>4.0506 10<sup>-9</sup></td><td>4.8465 10<sup>-10</sup></td></tr><tr><td>50</td><td>2.6066 10<sup>-8</sup></td><td>2.3573 10<sup>-8</sup></td><td>3.8323 10<sup>-9</sup></td><td>1.2526 10<sup>-12</sup></td></tr><tr><td>100</td><td>1.789 10<sup>-8</sup></td><td>1.912 10<sup>-8</sup></td><td>3.8319 10<sup>-9</sup></td><td>2.5576 10<sup>-18</sup></td></tr></tbody></table>

Analysis of the results clearly shows that the GA-LM method outperforms the GA method in terms of robustness and convergence. For all three generation sizes, GA-LM has a lower and more stable mean, with a very small standard deviation, sometimes even almost zero at 100 generations. This indicates better reproducibility of results and a greater ability to reach an optimal solution on a regular basis.

This initial analysis will then be extended by a comparison of the two methods in terms of execution time and RMSE error value, in order to complete the overall assessment of their performance.

For the RMSE values, Figs. 12, 13, and 14 show that the two methods retrieve roughly the same values for the three heat exchangers. These values range from $8.5843 \times 10^{-6}$ for the stainless-steel tubular exchanger to $5.4165 \times 10^{-6}$ for the graphite polyblock exchanger from supplier A and to $1.5399 \times 10^{-5}$ for the graphite polyblock exchanger from supplier B. However, a notable difference appears in the execution time, where the GA-LM method shows faster performance than the GA method, as shown in Figs. 10, 11, and 12. For the stainless-steel tubular exchanger, the best execution time is obtained with the GA-LM method, at approximately 3.54 seconds,
which is lower than the GA method's time of about 5.32 seconds. For the graphite polyblock exchanger from supplier A, the hybridization method also shows the best performance with a time of 3.81 seconds, compared to 5.71 seconds for the GA method. Finally, for Supplier B, as with previous estimates, the GA-LM method again achieves the best run time, at approximately 3.6 seconds.

<figure><img src="image_15.png" /><figcaption>**Fig 12.** Comparison of performance for the stainless-steel tubular heat exchanger (HEx1) with execution time and RMSE</figcaption></figure>

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<figure><img src="image_16.png" /><figcaption>**Fig 13.** Comparison of performance for the graphite poly block heat exchanger «Supplier A» (HEx2) with execution time and RMSE</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig 14.** Comparison of performance for the graphite poly block heat exchanger «Supplier B» (HEx3) with execution time and RMSE</figcaption></figure>
As shown in Table 1, our hybrid GA-LM method significantly boosts computational efficiency for heat exchanger fouling parameter estimation, maintaining accuracy comparable to existing techniques. This offers a distinct practical advantage for industrial applications where rapid and reliable results are essential. In contrast, Wang et al.[50] prioritize interpretability and robustness in AI diagnosis models using GNNs, a crucial focus for safety-critical nuclear power plant systems that goes beyond mere predictive accuracy. Meanwhile, King and Yune [51] demonstrate accurate thermal resistance forecasting via DFNN-GA, and Aljaidi et al.[52] enhance PEMFC parameter estimation through their GTO-HBA hybrid. These diverse studies collectively underscore the versatility of algorithm hybridization across varied engineering disciplines. The overarching goal across these research efforts remains the robust and accurate determination of complex system parameters, optimizing aspects from efficiency to model transparency.

<table><caption>**Table 5.** Comparative Precision of Hybrid Optimization Approaches</caption><thead><tr><td>Study</td><td>Hybrid Method(s)</td><td>Problem Domain</td><td>Key Precision/Accuracy Metrics</td></tr></thead><tbody><tr><td>Our Work</td><td>GA-LM</td><td>Fouling Estimation (Heat Exchangers)</td><td>RMSE for fouling resistance: ~8.58×10<sup>-6</sup> to 1.54×10<sup>-5</sup>m<sup>2</sup>·K/W. Comparable RMSE to GA, faster execution.</td></tr><tr><td>Wang et al. [50]</td><td>GNN (with interpretability)</td><td>FaultDiagnosis (Nuclear Power Plant)</td><td>Qualitative & Quantitative interpretable diagnosis. Addresses "weak interpretability."</td></tr><tr><td>King & Yune[51]</td><td>DFNN-GA</td><td>Thermal Resistance Forecasting (Borehole Heat Exchanger)</td><td>Robust correlation and accurate forecasting. Outperforms other methods in accuracy.</td></tr><tr><td>Aljaidi et al. [52]</td><td>GTO-HBA</td><td>PEMFC Parameter Estimation</td><td>Enhanced PEMFC parameter estimation.</td></tr></tbody></table>
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**3.5. Experimental validation**

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This part aims to apply the GA-LM method to simultaneously estimate the asymptotic fouling resistance and the characteristic time using real experimental measurements of three heat exchangers used in the phosphoric acid concentration unit. To determine whether the estimate is accurate, a comparison between the calculated fouling resistance and the experimental measurements is made. Fig. 15 shows a good agreement for the three heat exchangers HEx1, HEx2, and HEx3 with a maximum relative error of 0.15%, 0.10%, and 0.12%, respectively.

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The hybridization method between the GA and the Levenberg-Marquardt is a fairly robust and stable estimation procedure whose main advantage is a good convergence rate. Essentially, it guarantees convergence to the global minimum with fewer generations than a stochastic method like GA alone.

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Next, we compare this method in comparison with the method of Fguiri et al [18], which estimates the fouling resistance using the Levenberg-Marquardt method combined with the graphical method. The GA-LM method produces results very similar to this recent estimate, except that our method offers industrial companies the advantage of reduced computation time, because the GA makes an excellent initial estimate- a value close to the overall optimum- for the LM method. So instead of using the graphical method as a first estimate for the LM method, we use the GA, which minimizes the estimation time.

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Ultimately, it can be concluded that this study offers industry professionals the opportunity to correctly determine the decision moment and, in a reduced time, to stop the process and clean the exchanger. This recommendation is based on the inverse estimation method.

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<figure><img src="image_18.png" /><figcaption>**Fig 15.** Comparison between measured and computed fouling resistance for the three heat exchangers HEx1, HEx2, and HEx3</figcaption></figure>

**4. Conclusions**

In this study, various optimization strategies were applied to estimate two key parameters: the asymptotic fouling resistance and the time constant ($\tau$), using data derived from an industrial environment. The framework of this analysis relied on the well-established Kern and Seaton fouling model, which was deemed appropriate based on the observed asymptotic behavior of thermal resistance over time.

The experimental evaluation confirmed the model's validity by highlighting how fouling resistance evolves with time, justifying the use of this specific mathematical representation. By minimizing the squared deviations between experimentally measured resistance values and those predicted by the model, the estimation process was guided through optimization techniques.

The use of GA and its hybridization with the LM method yielded successful estimations across three different heat exchanger types. The comparative analysis revealed that the GA is capable of reaching global minima, even amidst multiple local minima, and is especially robust for functions lacking smoothness or differentiability. Despite its slower convergence rate due to its population-based evolutionary operations (selection, crossover, and mutation), the GA offers a
broad set of ranked optimal solutions. Conversely, the hybrid GA-LM approach combines the exploratory power of GA with the rapid convergence of LM, leading to high-precision results more efficiently. These combined methodologies demonstrate clear advantages over conventional graphical methods typically employed by industrial operators, particularly in the phosphoric acid concentration units in Tunisia.

**Ethical Approval:** Not applicable (This study does not involve any experiments with human participants or animals)

**Funding statement:** This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP-RP25).

**Consent to Participate:** Not applicable

**Consent to Publish:** Not applicable

**Competing Interests:** The authors declare that they have no competing interests.

**Availability of data and materials:** Not applicable (The data presented in this study are available in the manuscript)

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