Abstract
This paper presents the use of fouling targets as a design objective in heat exchanger design. The type of fouling addressed in this work is the scale formation in the operation of coolers. A mathematical model validated against published experimental data is used to predict the buildup of scaling as a function of time. In this work, a fouling target is the fouling resistance value that corresponds to a maximum permitted cooler heat load reduction that is reached in a fixed operating time. Fouling targets are related to operating conditions through a family of curves that relate fouling resistance to fluid velocity, fluid bulk temperature, CaCO_{3} concentration, and operating time. The design of the exchanger for the targeted fluid velocity sets the basis for the control of fouling from the design stage. The approach is demonstrated on a case study.
Introduction
The dynamic characteristics of fouling create many adverse situations that affect the heat transfer performance of heat exchangers. Some of the most common are the reduction of the heat transfer capacity of the unit as a result of the increase in the thermal resistance to heat transfer and the increase of pressure drop as the freeflow area gets reduced with the deposition of unwanted material on the heat transfer surface. Due to the implications of fouling in terms of increased energy consumption, increased capital costs, and increased production losses, a large amount of research has been dedicated to three broad aspects of fouling: development of cleaning technologies, the accurate prediction of the rate of fouling growth, and the minimization of fouling from the design stage. Two of the most common types of fouling that have been the focus of predictions through the development of mathematical models and the development of mitigation strategies are chemical reaction, typically found in crude oil processing, and scale formation that is found in the operation of cooling systems that uses water as the cooling medium.
Chemical reaction and scale formation fouling are often modeled based on the concept of the competition between the rate of deposition and the rate of suppression (Wilson 2005) of the foulant on a heat transfer surface. These mechanisms have been shown to strongly depend on two main process parameters, wall temperature and fluid velocity. The main understandings about this behavior are as follows (Ebert and Panchal 1997):

Fouling rates increase with increased operating temperature.

Fouling rates decrease with increased fluid velocity.
The concept of threshold fouling was first introduced by Ebert and Panchal (1997), and it refers to the set of operating conditions where the fouling rate is near zero. The concept came to change the traditional use of fixed fouling resistance values in design. When worst case fouling values are used, the result is an overdesigned exchanger. Close analysis of the original fouling threshold model of Ebert and Panchal (1997) shows that it fails to accurately represent real plant data since the model is more sensitive to velocity than experimental data reveals. Polley et al. (2002a) modified the model by introducing some parameters and obtained better predictions. The model was further modified to improve the accuracy of the predictions (Polley et al. 2010).
As the modeling and prediction of fouling became more robust, Polley et al. (2002b) and (Yeap et al. 2004) set the basis for the incorporation of the threshold fouling model in the design of heat exchangers and for the right positioning and selection of configurations to minimize fouling in crude preheat trains. Another significant contribution is the simultaneous consideration of thermal and hydraulic effects of fouling in heat recovery networks presented by Polley et al. (2009, 2010).
Over the past four decades, a considerable amount of research has been carried out in the development of passive techniques for the mitigation of fouling in heat exchangers. Aubin (2015) presents a review of the most important technologies developed. The principle behind the use of turbulence promoters for fouling mitigation is the continual removal of the boundary layer these devices can achieve, which impedes the deposition of the foulant on the heat transfer surface by maintaining the fouling particles suspended in the fluid (Aquino et al. 2007).
In terms of the fouling that leads to scale formation, most research work is focused on the study of the morphology of the crystals and the rate of deposition (Karabelas, 2002). Many authors have proposed analytical models for the prediction of scaling. Most of these models are based on the mechanism proposed by Hasson (1968). The mechanism comprises two stages: mass transfer by diffusion though the boundary layer and the combination of the ionic species at the surface typically modeled as a chemical reaction. Most of the models developed so far have been introduced to demonstrate the accurate fit to experimental data; however, little has been done in terms of the use of the model for design purposes. For instance, Bohnet (1985) developed a model for CaSO_{4} as a foulant where the main assumption is that the deposition takes place as a result of particles being formed at the bulk of the fluid as opposed to the combination of ions on the surface. The model by Andritsos (1996) was developed based on the experimental observations that deposition rate depends strongly on the fluid velocity, concluding that the process is mass transfer controlled. Epstein (1994) proposed a fouling model and included a residence time correction factor to consider that if within a given length of time the Ca and CO_{3} ions do not react, they return to the stream. Pääkkönen (2015) published a model based on the assumption that not all the material that reaches the surface will deposit on it. He considered the need to account for a residence time within which the combination between ions takes place, otherwise no fouling is formed.
Loo and Bridgwater (1985) proposed a novel model to predict the deposition of CaCO_{3} in heat exchangers that incorporates the rate of removal based on the rupture due to thermal stress. Babuška et al. (2018) validated a model to predict scale breakoff against experimental data obtaining good results.
Studies of scaling in different heat exchanger geometries have been undertaken by Arsenyeva et al. (2013) who developed a mathematical model for scaling prediction on plate and frame heat exchangers. Pääkkönen et al. (2016) used a CFD numerical model to study scaling on a flat plate. The characterization of scaling using indices was described by MüllerSteinhagen and Branch (1988). The modeling of scaling on heat exchangers has also been carried out through solution of a set of differential equations as proposed by Delrot et al. (2011).
Research has also been conducted to understand the effect of velocity and temperature on fouling in heat exchangers. Liang et al. (2016) found that the fluid velocity reduces fouling. However, the effect depends on the flow regime. That is, under laminar flow, fouling increases with velocity and under turbulent flow, fouling decreases.
Lugo and Picón (2018) developed an approach to model scaling that accounts for the viscous and inertial effects that tend to mitigate fouling. The model was validated against several sets of experimental data published in the open literature and was used to predict the buildup of scaling with time, velocity, and concentration.
The aim of this work is to show a way in which scale fouling can be incorporated as a design target for coolers. The work is organized as follows: first, the model by Lugo and Picón (2018) is further analyzed to demonstrate the accuracy of its predictions by comparing it with experimental data from other authors; then, the model is used to establish fouling targets with time. The results are the set of operating conditions (fluid velocity, foulant concentration, and operating temperature) under which a fixed fouling resistance or fouling threshold is reached for a set operating time. This approach allows the designer to have a stricter control on the design and establish the appropriate cleaning times of the exchangers. The criteria for the fixing of the fouling target values are the operating time, the maximum reduction of the exchanger heat load, or the maximum permitted increase of pressure drop. The results are problem sensitive since the surface temperature has a major impact upon the rate of reaction term in the fouling rate equation. Therefore, the application of the methodology is demonstrated on a case study.
Prediction of Deposition by Scaling
The mechanism that more closely describes the process whereby deposits accumulate on the surface of a heat exchanger is the one proposed by Hasson (1968). According to this mechanism, mass transfer of the ionic species takes place by diffusion through the boundary layer to eventually reach the surface where crystals are formed. The removal of crystals once formed is very difficult and for practical purposes this term is considered negligible. However, fouling reduces as the exchanger operates at higher velocities as ions are moved away from the points of crystal formation. It has been observed that the incorporation of a factor (α) that includes the inertial and viscous effects gives a better approximation to experimentally measured data. A model that features this correction factor is represented in Eq. (1) (Lugo and Picón 2018). This model assumes that once formed, scale is hard to remove; this led to consider it negligible. The validity of the model was demonstrated by comparing its results with experimental data reported by the following authors: Young et al. (2013), Paakkonen et al. (2015), Andritsos et al. (1996), and Wu and Cremaschi (2013).
where the terms C_{1} and C_{2} are the concentrations of Ca^{2+} and CO_{3}^{2−} contained in water; β is the mass transfer coefficient (Venkatesan and Fogler, 2004); Ksp is the solubility constant of calcium carbonate. It can be computed as K_{sp} = S^{2}, where S is the solubility of calcium carbonate that can be obtained from an expression as a function of the temperature and pH of the solution as (Lugo and Picón 2018):
The coefficient of reaction k_{r} is obtained from the Arrhenius equation.
where k_{0} is the constant of reaction; E_{a} is the activation energy, T_{i} is the temperature of the surface, and R is the universal gas constant. The values of E_{a} and k_{o} are 113 kJ/mol and 2.06 × 10^{15} m^{4}/kg s (Agustin and Bohnet 1995).
The term α from Eq. 4 is a dimensionless correction factor that accounts for the inertial and viscous forces. These forces become important as the fluid circulates at higher velocities, and its effect is to reduce the rate of deposition of calcium carbonate on the surface of the exchanger. The correction factor is given by:
where Re is the Reynolds number, and the coefficient a and exponent b are obtained from experimental data as demonstrated by Lugo and Picón 2018. The values are a = 191 and b = − 1.67. The friction factor f is calculated from:
where K_{t} is the absolute roughness at a time t (see Eq. 11). For a hydraulically plain surface, K_{t} ≤ 19.25 D/(Re^{7/8}), and for a hydraulically rough surface, K_{t} ≥ 560 D/Re (Nikuradse, 1933).
The term β is the mass transfer coefficient that is a function of the diffusivity (D_{f}), the diameter (D), and the Sherwood number (Sh).
For Reynolds number greater than 400, the Linton–Sherwood correlation applies (Venkatesen 2004):
where Sc and Re are the Schmidt and Reynolds numbers given by:
where μ is the fluid viscosity, ρ is the density, and \( \overline{v} \) is the fluid mean velocity. The expression to calculate the fouling thermal resistance is:
where ρ_{f} and λ_{f} are the density and thermal conductivity of the salt; \( {\dot{m}}_{\mathrm{r}} \) is the mass flow removed from the heat transfer surface. Based on the work of Quan et al. (2008), Wu and Cremaschi (2013) proposed a way to determine the rate of removal; however, since the estimation of these rates using the proposed approach results in very low quantities, as mentioned above, in the present work the removal term is considered negligible.
Figure 1 shows a comparison of the prediction of fouling resistance as a function of time for a fluid velocity of 1 m/s, fluid temperature of 80 °C, and a salt bulk concentration (C_{b}) of 418 ppm, and data is shown in Table 1. For these same data, the models of Andritsos (1996), Cremaschi and Wu (2014), and Pääkkönen (2015) are compared with the new model in Fig. 1. One feature that stands out from Fig. 1 is that compared with the new model, the predicted values of thermal resistance of other models are much higher. The values obtained with the new scaling model are within reasonable ranges. Overall, the results demonstrate that the scaling model provides congruent results.
The fact that fouling is a function of the fluid temperature, the foulant concentration, the fluid velocity, and time makes it possible to link the growth of fouling with the length of a tube as the fluid temperature increases. In the next section, two case studies are analyzed to study the difference in performance when a variable fouling rate is used compared with the use of a fixed value of fouling resistance.
Case Study
A cooler is designed using the geometry given in Table 2. Cooling water (30,000 kg/h) at 17 °C is used to condense residual steam at 67 °C and 0.2 bar with a mass flow rate of 50,000 kg/h. Typical published fouling resistance for water in cooling systems varies between 0.9 × 10^{−4} and 3.5 × 10^{−4} m^{2} K/W (Bott, 1995). For design purposes, a fouling resistance of Rs = 1.7 m^{2} K/W is employed. The allowable pressure drop on the water side is 10.3 kPa.
For the sake of comparison, the cooler is also designed under no fouling resistance. The results are presented in Table 3.
With the use of a timedependent fouling model, valuable information can be obtained, for instance, starting from the design under clean conditions, a question that could be posed is what would be the operation time wherein a fixed fouling resistance value is reached? An analysis is performed for a foulant concentration of 300 ppm and showed in Fig. 2, where it is seen that a fouling resistance of 1.7 × 10^{−4} (m^{2}K/W) is reached when the exchanger has been in operation for approximately 2700 h.
As fouling develops, the pressure drop across the tube increases due to the change of surface roughness and the reduction of the free flow area represented by the reduction of tube diameter. The way absolute roughness changes with time can be represented by (Colebrook and White 1937):
where K_{t} is the new absolute roughness, K_{o} is the absolute roughness of a clean tube, t is the number of years of service, and c is the annual roughness growth index. The value of c is determined experimentally. For cases of fouling due to corrosion, Colebrook and White (1937) used values of c between 0.0006 and 0.001 m/year. However, in this work using scaling data reported in the literature, the value of c was found to be 0.0001 m/year. The fouling resistance at time t can be expressed as:
where X_{f,t} is the layer thickness at time t. For a given time (t), X_{f,t} can be calculated from:
In the same way, if we assume that the layer of scale develops uniformly on the surface of the tube, the reduction in diameter is calculated from:
where D_{t} is the new diameter of the tube and D_{o} is the diameter of a clean tube. The pressure drop across a tube is calculated from:
where f is the friction factor given by Eq. (5) evaluated for K_{t} and D_{t}, L_{t} is the tube length, N_{p} is the number of passes, and v is the fluid average velocity.
A hydraulic analysis can now be performed seeking to answer the following question: starting from the design under clean conditions, what is the operation time that must elapse for the pressure drop to increase to the allowable value of 10.3 kPa? Figure 3 shows that after approximately 23000 h operation, the pressure drop has reached the allowable value.
The development of fouling can be followed along the length of the tube. Figure 4 shows such analysis where the local fouling resistance increases with length.
As fouling builds up, the heat load of the exchanger drops (Fig. 5a). One way to maintain it is by the increase of the water flow rate as shown in Fig. 5 b.
A designer can now explore the effect of designing for different fluid velocities. Figure 6 shows a set of curves for a fixed fluid temperature where fouling factor is plotted vs fluid velocity. Each curve represents a specified operation time. Valuable information can be extracted from these curves. For example, is it possible to specify the maximum fouling resistance and determine the operating conditions to achieve it and the operation time? Say for a maximum fouling resistance of Rs = 0.00005 m^{2}K/W, Fig. 6 indicates that the minimum fluid velocity for fouling not to exceed this value after 9000 h operation is 0.65 m/s. For an extended operation time of 12,000 h, the minimum velocity is 0.21 m/s.
This case demonstrates that the wrong choice of the fouling factor at the design stage can lead to over expenditure of capital. The use of a rather conservative fouling factor as in this case study results in a surface area excess of 66%.
Influence of the Material of Construction on Scaling
The material of construction has a significant influence on the fouling resistance principally due to its thermal conductivity since higher conductivity materials are prone to exhibit higher metal wall temperature. Additionally, surface roughness is another factor that influences the rate at which fouling develops. The effect of material of construction on fouling is determined by calculating the surface temperature (T_{i}) and the coefficient of reaction (k_{r}) given the thermal conductivity of the material. Under clean conditions, the surface absolute roughness (K) determines the friction factor (Eq. 5) which in turn affects the term α (Eq. 4). The variation of the absolute roughness with time is calculated from Eq. 9. All these effects are brought together into Eq. 1 to calculate the mass deposition. The fouling resistance is calculated from Eq. 10. Table 4 presents the physical properties and roughness for different materials of construction. The plot of fouling resistance vs time for the different materials indicates that when the tubes are made of copper, higher fouling resistances are produced (Fig. 7).
To validate the model, theoretical predictions are compared against experimental data reported by Teng et al. (2017). Figure 8 shows the variation of the mass of CaCO_{3} deposited on the surface of tubes of different materials where the match between the fouling model and the experimental data is reasonably good. Analysis of the data indicates that copper exhibits larger CaCO_{3} deposition; its roughness is higher than carbon steel and slightly lower than stainless steel. Its thermal conductivity is much higher than the other materials (401 W/mK) which brings about higher inner wall temperature, increases the water temperature, and, consequently, increases the deposition as the solubility of the salt reduces. Carbon steel exhibits higher salt deposition compared with stainless steel but it also has a higher thermal conductivity and roughness.
Fouling Design Curves
Having demonstrated the validity and accuracy of the scaling model, it is now used to find the design and operating parameters that will allow the engineer to be in control of fouling and predict with certainty the right time for cleaning. The objective is to develop general design rules based on fouling design curves. This is, a plot that provides the designer with the information regarding fouling resistance, as a function of temperature, fluid velocity, foulant concentration, and time.
In terms of design, fouling design curves can be used in different ways. For instance, it may be of paramount importance to fix a maximum reduction of the heat load of the cooler to take cleaning action. This is, as soon as the heat load of the cooler reaches a certain value, it may be time for taking the unit out of operation for cleaning. Since the reduction of the heat duty is mainly attributed to the buildup of fouling, the determination of the fouling resistance responsible for the reduction of the heat load can readily be obtained. Such fouling resistance can be used as a reference or limiting value. Important to note, though, that in this scenario, time is treated as a dependent variable. Now, the problem can be treated differently, for instance, if we fix beforehand the time where we would expect cleaning to take place, we will not only calculate the fouling rate that causes the heat load to reduce to a certain level but we also are in the position to determine the operating conditions, namely fluid temperature and velocity, at which this fouling rate occurs. If the exchanger operates outside these parameters, the fouling resistance could exhibit lower or higher values. In quantitative terms, at higher temperatures and lower velocities, fouling increases; at lower temperatures and higher velocities, fouling decreases. In the next section, the fouling model is used to produce different types of fouling design curves.
Results and Discussion
As mentioned earlier, typical fouling resistances for water used as cooling fluid vary from 0.9 × 10–4 (m^{2}K/W) and 3.5 × 10–4 (m^{2}K/W) (Bott, 1995). In this section, a fouling resistance of 1.0 × 10^{−4} (m^{2}K/W) is used as a reference to demonstrate the use of the model. This value is assumed to be low enough to prevent a rapid accumulation of fouling. Figure 9 shows the reduction of the heat removed with the fouling resistance. The plot shows the reduction of the cooling water outlet temperature.
For design purposes, it would be useful to have the overall picture where the fouling resistance is represented as a function of time, fluid velocity, and fluid bulk temperature. Figure 10 fulfills that goal. Here, the fouling resistance at the exit of the tube is plotted vs fluid velocity. The group of curves represent a constant fluid bulk temperature and the whole system determined considering one year of operation for a salt concentration of 300 ppm. This information can be used as follows: in order to maintain a fouling resistance at a value of 1.0 × 10^{−4} m^{2}K/W when the maximum water bulk temperature is 45 °C, the minimum velocity at which water must circulate through the exchanger is approximately 1.62 m/s.
It is possible to create different types of fouling design curves which would give the minimum velocity for the desired maximum fluid outlet temperature that will produce a fixed fouling resistance at a given time of operation.
An alternative way of presenting the fouling design plot is by means of a plot that shows the fouling resistance vs operation time for different velocities and a fixed temperature. This type of plot is useful if the question is: how long would it take for the fouling factor to reach the value of 1.0 × 10–4 m^{2}K/W if the fluid velocity is 0.6 m/s? From Fig. 11, it is seen that once 4800 h of operation have been completed, the exchanger will exhibit a fouling factor of that magnitude.
As it has been mentioned, the rate of fouling deposition depends on temperature and velocity; however, the concentration of the foulant is also of vital importance in this process. Figure 12 presents the fouling resistance as a function of time for an operating velocity of 0.6 m/s, a temperature of 45 °C, and different concentrations of CaCO_{3}. If the foulant concentration is 250 ppm, a fouling resistance of 0.0001 m^{2}K/W is reached after 7300 h of operation. If the concentration were 300 ppm, the same fouling resistance will be reached in only 4800 h of operation.
Another variation of the fouling threshold design curves is the one shown in Fig. 13. Here, the fouling resistance is presented as a function of velocity and operating time for fixed foulant concentration and fluid temperature. The curves were generated for a concentration of 300 ppm and a temperature of 45 °C. From these curves, the maximum operation time for fouling not to exceed the fixed fouling resistance can be determined. On the other hand, it is possible to determine the velocity that will render control over fouling. For instance, Fig. 13 indicates that maintaining a velocity of 0.9 m/s, it is possible to have, after an operation time of 15,000 h, a fouling resistance of 0.0001 m^{2}K/W. If a lower fouling resistance is desired within this operating time, the exchanger will have to be designed at a higher water velocity.
If the outlet temperature of the water can be as high as 52 °C in the case under consideration, the following analysis can be derived. Figure 14 shows the variation of the thickness of the layer of fouling as a function of the fouling resistance. It can be observed that for a fouling resistance of R_{s} = 0.00065 m^{2}K/W the thickness is 0.001 m (1.0 mm). Figure 15 indicates that a fouling thickness of 1.0 mm brings about a temperature reduction of 14 °C. If the thickness increases to 0.002 m (2.0 mm), the temperature reduction due to the extra 1 mm thickness is only 7 °C. The lower temperature reduction is the result of the increase of the thermal resistance due to conduction through the layer of fouling. This causes the temperature at the surface of the layer to be lower which reduces fouling. Further information extracted from Fig. 15 is that for a temperature reduction of 4 °C from the original water outlet temperature, the fouling resistance must be 0.00025 m^{2}K/W.
It is important to note, that in real operation, coolers are subject to a continuous change of the operating conditions. This includes the concentration of CaCO_{3} and fluid inlet temperature. Some of the reasons that cause the salt concentration to change are the continuous evaporation of water if the cooling system includes an evaporative cooling tower, and the salt deposited on the surfaces of coolers and piping. In the case of the inlet water temperature, daily and seasonal atmospheric conditions reduce or increase the cooling tower capacity, bringing about changes in the temperature of the cooling water. This dynamic behavior does not invalidate the results here presented but opens the opportunities for further analysis.
Conclusions
This work presents the development of fouling design curves using a validated model for the prediction of scaling. The fouling design curves provide valuable design information as they establish a link between fouling resistance, concentration of CaCO_{3}, fluid velocity, fluid temperature, and time. The main conclusions of this work are:

Fouling design curves are general and provide straightforward design parameters for coolers.

The main operating design parameters that determine the fouling resistance in a cooler given the salt concentration are the fluid velocity and the operating maximum temperature.

Minimum velocity and maximum operating temperature can be determined for different design targets.

Some of these design targets are: desirable operating time before cleaning and maximum permitted reduction of heat load before cleaning.

The model is limited to steadystate conditions. Extension to include dynamic behavior can readily be undertaken.
Abbreviations
 a:

Coefficient of Eq. 4
 b:

Exponent of Eq. 4
 c:

Annual roughness growth index (m/year)
 C _{1} :

Concentration of Ca^{2+}
 C _{2} :

Concentration of CO_{3}^{2−}
 C _{b} :

Bulk concentration (ppm)
 D :

Diameter (m)
 Dc:

Shell diameter (m)
 D _{f} :

Diffusivity (m^{2}/s)
 D _{t} :

New tube diameter (m)
 d _{i} :

Tune inner diameter (m)
 d _{o} :

Tube outer diameter (m)
 E _{a} :

Activation energy (kJ/mol)
 f :

Friction factor
 K :

Absolute roughness (m)
 K _{o} :

Absolute roughness of a clean tube (m)
 K _{t} :

Absolute roughness at time t (m)
 K _{sp} :

Solubility constant (kg/m^{3})^{2}
 k _{r} :

Coefficient of reaction (m^{4}/kgs)
 k_{0} :

Constant of reaction (m^{4/*−/}/kgs).
 Lt:

Tube length (m).
 Ltc:

Distance between tubes (m)
 m :

Mass deposited (kg/s)
 \( \dot{m_{\mathrm{d}}} \) :

Rate of mass deposition (kg/m^{2} s)
 \( \dot{m_{\mathrm{r}}} \) :

Rate of mass removal (kg/m^{2} s)
 N _{p} :

Number of tube passes
 N _{t} :

Number of tubes
 Pt:

Tube pitch (m)
 pH:

Potential hydrogen
 Q :

Heat duty (kW)
 R :

Gas constant (J/mol∙K)
 Re:

Reynolds number
 R _{s} :

Fouling resistance due to scaling (m^{2} K/W)
 R _{s,t} :

Fouling resistance at time t (m^{2} K/W)
 S :

Solubility (kg/m^{3})
 Sc:

Schmidt number
 Sh:

Sherwood number
 T :

Temperature (°C)
 T _{i} :

Temperature of the surface (°C)
 t :

Time (s)
 U :

Overall heat transfer coefficient (W/m^{2} K)
 v :

Fluid velocity (m/s)
 \( \overline{v} \) :

Mean flow velocity (m/s)
 X _{f,t} :

Thickness of fouling layer (m)
 α :

Correction coefficient
 β :

Mass transfer coefficient (m/s)
 λ _{f} :

Thermal conductivity of CaCO_{3} (W/m K)
 μ :

Fluid viscosity (kg/m s)
 ρ :

Fluid density (kg/m^{3})
 ρ _{f} :

Density of CaCO_{3} (kg/m^{3})
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LugoGranados, H., Tamakloe, E.K. & PicónNúñez, M. Controlling Scaling in Heat Exchangers Through the Use of Fouling Design Curves. Process Integr Optim Sustain 4, 111–120 (2020). https://doi.org/10.1007/s41660020001098
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Keywords
 Scale fouling
 Fouling targets
 Fouling modeling
 Fouling design curves