CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Development of a Dynamic Fouling Model for a 

Heat Exchanger 

Khayyam Zahid, Rajnikant Patel, Iqbal Mujtaba* 

School of Engineering and Informatics, University of Bradford, Bradford, BD7 1DP, UK 

 I.M.Mujtaba@bradford.ac.uk 

Fouling in heat exchangers (HE) is a major problem in industry and accurate prediction of the onset or degree 

of fouling would be of a huge benefit to the operators. Modelling of the fouling phenomenon however, remains 

a challenging field of study. Cleaning of heat exchangers, coulpled with the down time, is a financial burden 

and for industrialized nations and costs can reach to almost 0.25 % of the country’s Gross National Product 

(Pritchard, 1988).This work presents the development of a dynamic fouling model based on experimental data 

collected using a laboratory concentric tube heat exchanger handling a saline system. Heat transfer 

coefficients were obtained from first principles as well as from either the Sieder-Tate or Petukhov-Kirillov 

correlations modified by Gnielinski depending on the flow regime. The outlet temperatures were calculated 

using the Effectiveness-NTU method. The dynamic fouling factor was based on the Kern and Seaton fouling 

model and validation was completed by comparing the experimental outlet temperatures with those predicted 

by the model. The model predicts the outlet temperatures with an average discrepancy of 1.6 °C and 0.4 °C 

for the cold and hot streams respectively.  

1. Introduction 

Fouling is a general term used to describe the deposition of foreign matter on a heat transfer surface. The 

gradual deposition process remains a great concern when considering heat exchanger installation as it can 

affect both heat transfer and fluid flow. Costs associated with heat exchanger fouling include increased energy 

consumption, downtime, cleaning, and can also exacerbate problems associated with membranes (Shirazi, 

2010). Heat exchanger maintenance costs arise from fouling deposit removal through chemical or mechanical 

means; as well as the replacement of corroded or plugged equipment. Other costs include increased capital 

cost due to oversized or redundant equipment and energy wastage due to an increased heat transfer 

coefficient (Georgiadis et al., 1998). Within the dairy industry, it is estimated that fouling removal via cleaning 

can account for around 15 % and 80 % of extra production time and costs (Mérian et al., 2012). The simulated 

temperature profiles can also be used to identify potential weak spots concerning fouling (Jun et al., 2004). An 

example of non-optimal control implementation involves maintaining a wall temperature that is too high or 

product flow rate that is too low which can ultimately increase the severity of fouling (Fryer, 1989). 

Various models have been proposed in an attempt to model and predict fouling behavior (Müller-Steinhagen, 

2011) and more recently Arsenyeva (2012). In order to improve model reliability, multiple experiments and 

simulations have been conducted; nevertheless fouling problems within industry are still not accurately 

predicted leading to non-optimal fouling mitigation. Recent fouling model developments involve using CFD to 

predict fouling and are in agreement with previously published experimental results and theoretical models 

(Haghshenasfard, 2015 and very recently (Pääkkönen, 2016). 

The aim of this work is to develop a model for fouling in a heat exchanger which can effectively predict the 

heat transfer behaviour when a saline fluid is introduced to the system. Discrete rather than dynamic fouling 

factors are often added to calculate overall heat transfer co-efficient in heat exchangers. This will not be the 

case in this study as a dynamic fouling factor will be used in order to account for losses in heat transfer 

efficiency over time. To prove the validity of this model, an experiment has been conducted to verify the model 

accuracy. 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652190 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Zahid K., Patel R., Mujtaba I. M., 2016, Development of a dynamic fouling model for a heat exchanger, Chemical 
Engineering Transactions, 52, 1135-1140  DOI:10.3303/CET1652190   

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2. Double pipe heat transfer coefficient 

The fouling model of a double pipe heat exchanger requires the calculation of four heat transfer coefficients 

namely that of the inner tube, annulus space, tube wall, and fouling surface. Heated saline water will be 

carried through the inner tube which will be cooled using water supplied through the annulus. For this reason 

the annulus space heat transfer coefficient will be modelled as a constant as there is no fouling within the shell 

and therefore the diameter will remain unchanged. The wall heat transfer coefficient will also be modelled as a 

constant. The following assumptions are made to reduce system complexity: No condensation; no leakage; no 

heat loss to environment; and constant tube diameter. The Nusselt number for the inner tube (𝑵𝒖𝒉𝒐𝒕) and 

annular space (𝑵𝒖𝒄𝒐𝒍𝒅) is a function of the Prandtl number (𝑷𝒓) and Reynolds number. For laminar flow, the 

Sieder and Tate correlation is used (Sieder et al., 1936). 

𝑁𝑢𝑖 = 1.86 (
𝑅𝑒𝑖 𝑃𝑟𝑖 𝐷𝑒,𝑖

𝐿
)

1
3

(
𝜇𝑖

𝜇𝑤𝑎𝑡𝑒𝑟
) (1) 

Where L is the tube length and Pr is within 0.48 and 16,700. The value of (
𝑅𝑒𝑡𝑃𝑟𝑡𝑑𝑖 

𝐿
)

1

3
 must also be less than 2. 

For turbulent flow, the following correlation developed by Petukhov-Kirillov modified by Gnielinski can be used 

(Gnielinski, 1975). 

𝑁𝑢𝑖 =
(

𝑓𝑖
8

) (𝑅𝑒𝑖 − 1000)𝑃𝑟𝑖 (1 +
𝐷𝑒,𝑖

𝐿
)

2
3

1 + 12.7 (
𝑓𝑖
8

)
0.5

(𝑃𝑟
𝑖
2/3

− 1)  

(
𝜇𝑖

𝜇𝑤𝑎𝑡𝑒𝑟
)

0.14

 (2) 

Where 𝑓𝑖  is the friction factor calculated using 

𝑓𝑖 = (0.782 𝑙𝑛(𝑅𝑒𝑖 ) − 1.51)
−2 (3) 

The overall clean heat transfer coefficient of the system (𝑈𝑐 ) is a function of the inner tube heat transfer 

coefficient (ℎℎ𝑜𝑡) annulus space transfer coefficient (ℎ𝑐𝑜𝑙𝑑 ) wall thermal conductivity (𝑘𝑤) and wall thickness, 

(𝑇𝐻𝐾𝑤 ) during the first pass of the system. 

1

𝑈𝑐 𝐴
=

1

ℎℎ𝑜𝑡𝐴𝑓,ℎ𝑜𝑡
+

1

ℎ𝑐𝑜𝑙𝑑 𝐴𝑓.𝑐𝑜𝑙𝑑
+

𝑇𝐻𝐾𝑤
𝑘𝑤 𝐴𝑓𝑤

   (4) 

The dynamic fouling factor can be thought of as an additional heat resistance layer and given as 

𝑈𝑓 =
𝑈𝑐

1 + 𝑈𝑐 𝑅𝑓
 (5) 

Where 𝑅𝑓 will be obtained experimentally in this work. 

3. Dynamic fouling factor 

In order to accurately model this specific system, the fouling factor 𝑅𝑓 must be obtained through experimental 

means. Examples of fouling factor generic models can be found but are not an all-encompassing solution to 

obtain a direct relationship between fouling and time as they do not take into account all parameters such as 

temperature, flow rate etc. for a specific system. For this reason, it is best to obtain a fouling factor through 

experimental methods.  

3.1 Experimental procedure 

Prior to system start up, a saline solution was made up using 3 grams of regular table salt per 100 grams of 

water giving a salt concentration of 30,000 ppm (typical seawater salinity). Initial checks were made to ensure 

all tubing was connected and the equipment was safe to use. The system was set to run in counter-current 

operation and the hot fluid temperature was set to 40 °C (313.16 K, typical of last stage temperature of 

Multistage Flash Desalination process) whilst the cold fluid temperature was set to 20 °C (293.16 K, typical of 

winter seawater temperature). Both hot and cold fluid flow rates were set to 1 L/min. Figure 1 shows the 

experimental set-up. The saline fluid was heated in a water bath before being pumped through the double pipe 

heat exchanger via a flow control valve. A refrigerated circulating bath was used to control the inlet cold water 

temperature and this was also pumped through the HE via a flow control valve. As the system is closed loop, 

the system temperature was allowed to stabilise before data logging began. The inlet and outlet temperatures 

for both streams were monitored and data was sampled every minute for a total duration of 12 h. 

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3.2 Experimental results 

Figures 2 and 3 show a cooling/heating gradient of around 4 °C was observed throughout the duration of the 

experiment. The 0.5 °C variance in cooling fluid temperature can be accounted for as the system was turned 

off overnight and restarted in the morning in order to minimize the risk of damage to the system by leaving the 

system unattended for an extended period of time. Data logging began when the system reached steady state 

which took around 90 min. The next step in calculating the fouling factor was to observe how the cooling or 

heating changed over time. 

  

Figure 1: Fouling experiment apparatus Figure 2: Cold stream outlet temperature (experiment 

and prediction) 

  

Figure 3: Hot stream outlet temperature 

(experiment and prediction) 

Figure 4: Temperature change between inlet and outlet 

against time 

Figure 4 shows the temperature change over time where the temperature change was calculated using Eq(6). 

In this time period, the cold stream temperature change decreases over time indicating a lower heat transfer 

rate. This trend is also observed for the hot stream where the temperature change decreases. The decrease 

in heat transfer can be explained due to the increase in salt deposit over time which creates a larger heat 

resistance layer. The calculation of temperature change at time i is given as: 

𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖 = (𝑇𝑜𝑢𝑡,𝑖 − 𝑇in,𝑖 ) (6) 

Although Figure 4 establishes that heat transfer decreases over time, the actual rate of heat transfer decay is 

not established i.e. effect of fouling on temperature change. Figure 5 shows the decay rate of temperature 

change over time. In order to establish a degeneration rate regarding heat transfer, an average temperature 

change is calculated by taking the mean of the first 120 values of temperature change and comparing them to 

the temperature change at the current time. The calculation assumes minimal fouling occurs in the first 120 

minutes and therefore baseline temperature change decay is close to 0. Figure 5 shows the average 

temperature change of the hot stream outlet has increased by around 0.1 °C after 800 min and the average 

temperature of the cold stream outlet has decreased by around 0.3 °C. The calculation at time i is given as: 

𝑇𝑑𝑒𝑐𝑎𝑦,𝑖 = 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖 −
∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖

𝑡=120
𝑡=0

120
 (7) 

The term 
∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖

𝑡=120
𝑡=0

120
 can be thought of as the average temperature change when little to no fouling has 

occurred. 

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Figure 6 shows the change in efficiency of the heat exchanger over time as a result of fouling. In order to 

establish the efficiency of the heat exchanger, the following equation is used: 

𝐻𝐸𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 (%) =
𝑇𝑑𝑒𝑐𝑎𝑦,ℎ𝑜𝑡 +

∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,ℎ𝑜𝑡
𝑡=120
𝑡=0

120
∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,ℎ𝑜𝑡

𝑡=120
𝑡=0

120

 (8) 

 

  

Figure 5: Temperature change degeneration between 

inlet and outlet against time 

Figure 6: Heat exchanger efficiency against time 

The term 
𝑇𝑑𝑒𝑐𝑎𝑦,𝑖+

∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖
𝑡=120
𝑡=0

120

∑ 𝑇𝑐ℎ𝑎𝑛𝑔𝑒,𝑖
𝑡=120
𝑡=0

120

 can be thought of as the heat transfer efficiency of the stream i as the addition of 

the decay term considers the effect of fouling. Figure 6 shows that the heat transfer efficiency starts at 100% 

and decays to around 96 % after 800 minutes.  

The clean heat transfer coefficient (𝑈𝑐𝑙𝑒𝑎𝑛 𝐴) and heat exchanger efficiency (𝐻𝐸𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 ) are known. In order 

to calculate the heat transfer coefficient decay, the following equations are used: 

𝑈𝑓 𝐴 = (𝑈𝑐𝑙𝑒𝑎𝑛 𝐴)(𝐻𝐸𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 ) (9) 

(𝑈𝐴)𝑑𝑒𝑐𝑎𝑦 = (𝑈𝑓 𝐴) − (𝑈𝑐 𝐴) (10) 

Figure 7 shows the relationship between 𝑈𝐴𝑑𝑒𝑐𝑎𝑦 and time. This relationship can be used as the fouling factor 

(𝑅𝑓). It can be seen that the heat transfer rate has decreased by 0.6 
𝑊

𝑚2𝐾
 over 700 min. The overall equation 

for 𝑈𝑓 𝐴 can be written as: 

𝑈𝑓 𝐴 = 𝑈𝑐 𝐴 + (0.00000003𝑡
2 − 0.0009𝑡) (11) 

Although the overall heat transfer equation has been established, it is not in the form of the Kern and Seaton 

asymptotic model (Kern and Seaton, 1959): 

𝑅𝑓 = 𝑅𝑓
∗ (1 − 𝑒

−
𝑡
𝑡𝑐 ) (12) 

Where 𝑅𝑓 is the fouling resistance, 𝑅𝑓
∗ the asymptotic fouling resistance, 𝑡 the current time and 𝑡𝑐 the 

residence time. The thickness of the fouling deposit is derived as: 

𝑇𝐻𝐾𝑓 =  
−𝑘𝑓 𝐴𝑓,𝑓 (𝑈𝐴)𝑑𝑒𝑐𝑎𝑦

((𝑈𝐴)𝑑𝑒𝑐𝑎𝑦 + 𝑈𝑐 𝐴) 𝑈𝐶 𝐴
 (13) 

Figure 8 shows that the thickness of the fouling deposit increases from 0 to around 0.0002 mm over 700 min 

which is expected as the flow rates and temperatures were very low. Since 𝐴𝑓,𝑓 is unknown, 𝐴𝑓𝑤 is used in its 

place as this is a conservative value assuming the fouling deposit area will not become greater than the heat 

transfer area of the wall. The Kern and Seaton equation can be written as: 

𝑇𝐻𝐾𝑓

𝑘𝑓 𝐴𝑓,𝑓
= 𝑅𝑓

∗ (1 − 𝑒
−

𝑡
𝑡𝑐 ) (14) 

 

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Figure 7: Dynamic fouling factor using heat transfer 

efficiency over time 

Figure 8: Fouling thickness against time 

Figure 9 shows the asymptotic fouling resistance over time which increases from 0 to 0.003. The gradient of 

the asymptotic fouling resistance is now known as well as all other parameters. The dynamic fouling factor can 

thus be written as:  

𝑅𝑓 = ((0.000004t) (1 − 𝑒
−

𝑡
𝑡𝑐 )) (15) 

  

Figure 9: Asymptotic fouling resistance over time Figure 10: Heat transfer coefficient over time 

In order to validate this model, it can be compared to the model obtained from Figure 7. Eq(11) shows the 

fouled heat transfer coefficient obtained by comparing the clean heat transfer coefficient to the efficiency of the 

heat exchanger over time. Eq(5) is combined with Eq(15) and is a representation of the overall Kern and 

Seaton model. 

𝑈𝑓 𝐴 =
𝑈𝑐 𝐴

1 + 𝑈𝑐 𝐴 ((0.000004t) (1 − 𝑒
−

𝑡
𝑡𝑐 ))

 
(16) 

Figure 10 shows that the Kern and Seaton model that has been developed is comparable to the model 

obtained based on the heat exchanger efficiency. This shows the Kern and Seaton model can be used for this 

work as it matches the fouling model obtained from the experiment to a high degree of accuracy. The 

derivation of the dynamic heat transfer coefficient has been completed and the model can now fully operate. 

4. Results and Discussion 

Figures 2 and 3 show the predicted and experimental outlet temperatures for the cold stream and hot stream 

respectively showing some degree of discrepancy between the predicted and experimental data. Outlet 

stream temperatures were calculated using the E-NTU method. The hot stream temperature is over-predicted 

but the error is smaller as the temperature was maintained in the hot water bath better than the cold bath. This 

is seen in the experiment at around 400 min where the cold water temperature increases by 0.5 °C. On the 

other hand, the cold stream temperatures are under-predicted by around 1.5 °C throughout the experiment. 

Other factors that may have affected the results include the fact that the system was not well insulated which 

1139



lead to heat loss to the surroundings and that the flowrate of the hot and cold flows may not have always been 

exactly 1 L/min. The experiment itself should have been continued if time restrictions did not apply in order to 

remove significant data artefacts.  

5. Conclusions 

Modelling of heat exchanger fouling remains a difficult field of study and fouling remains a burden in industrial 

operations. Although several mathematical models have been proposed to explain the fouling behaviour of 

various fluids, an all-encompassing model has not emerged due to the complexity of fouling mechanism and 

its dependency on various operating parameters. This research has focused on a particular fouling problem 

when a heated saline stream has a cooling requirement. Due to the nature of the experimental equipment, low 

flow rates were used in order to keep the system running at steady state temperature-wise and allow 

adequate fouling to occur. It was found that running the experiment at high flow rates led to unstable cold 

stream temperatures as the residence time in the cold bath was too low to facilitate adequate cooling of the 

coolant stream. The aim of this work was to develop a dynamic model which reasonably described the fouling 

behaviour of the saline stream and to validate the model using experimental data. The development of this 

fouling model was based on using traditional heat exchanger models and implementing a fouling factor based 

on the decrease in heat transfer efficiency over time, observed experimentally. The fouling behaviour could be 

explained by crystallisation and precipitation mechanisms. The experimental data was analysed and a 

dynamic fouling model for heat transfer was developed from the findings.  

The model has been validated by comparing the experimental temperature change against those predicted by 

the model. The model is able to accurately predict the temperature change with an average discrepancy of 1.6 

°C and 0.4 °C for the cold stream and hot stream.  

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