**4. Fouling measurement and monitoring**

The fouling resistances can be measured experimentally or analytically. The main measurement methods include:

1.Direct weighing: the simplest method for assessing the extent of deposition on laboratory test surfaces is to weigh directly. The method requires an exact balance in order to be able to detect relatively small changes in the mass of deposits. It may be necessary to use thin walled tube to reduce the tare mass in order to increase the accuracy of the method.

*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process DOI: http://dx.doi.org/10.5772/intechopen.88936*


### **5. Prediction of fouling factor**

As noted above, fouling has the effect of forming on the heat transfer surface a substantially solid deposit of low thermal conductivity, through which heat is to be

where *α*, *β*, *γ* and *δ* are parameters determined by regression, *τ<sup>w</sup>* is the shear stress at the tube wall and *T*film is the temperature of the fluid film (average of local bulk and local wall fluid temperatures). The relationship in Eq. (1) indicates the possibility of identifying combinations of temperature and velocity below where fouling rates will be negligible. Ebert and Panchal [16] present this as the "threshold condition." The model in Eq. (1) suggests that the geometry of the heat exchanger which affects the surface and film temperatures, velocities and shear stresses can be effectively applied to maintain the conditions below "threshold conditions" in a

1.Falling fouling is the type of fouling where the fouling rate decreases with time, and the deposit thickness does not reach a constant value, although the fouling rate never drops below a certain minimum value. In general falling fouling is due to an increase of removal rate with time. Its progress can often be described by two numbers: the initial fouling rate and the fouling rate after

2.Asymptotic fouling rate is where rate decreases with time until it becomes negligible after a period of time when the deposition rate becomes equal to the deposit removal rate and the deposit thickness remains constant. This type of fouling generally occurs where the tube surface temperature remains constant while the temperature of the flowing fluid drops as a result of increased resistance of fouling material to heat transfer. Asymptotic fouling may also result from soft or poorly adherent suspended solid deposits upon heat transfer surfaces in areas of fast flow where they do not adhere strongly to the surface with the result that the thicker the deposit becomes, the more likely it is to wash off in patches and thus achieve some average asymptotic value over a period of time. The asymptotic fouling resistance increases with increasing particle concentration and decreasing fluid bulk temperature, flow velocity, and particle diameter. The asymptotic fouling model was first described by Kern and Seaton [17]. In this model, the competing fouling mechanisms result in asymptotic fouling resistance beyond which any additional increase in

3. Saw-tooth fouling occurs where part of the deposit is detached after a critical residence time or once a critical deposit thickness has been reached. The fouling layer then builds up and breaks off again. This periodic variation could be due to pressure pulses, scaling, trapping of air inside the surface deposits during shutdowns or other reasons. It often corresponds to the moments of

system shutdowns, startups or other transients during operation.

The fouling resistances can be measured experimentally or analytically. The

1.Direct weighing: the simplest method for assessing the extent of deposition on laboratory test surfaces is to weigh directly. The method requires an exact balance in order to be able to detect relatively small changes in the mass of deposits. It may be necessary to use thin walled tube to reduce the tare mass in

given heat exchanger.

a long period of time.

*Inverse Heat Conduction and Heat Exchangers*

fouling does not happen.

**4. Fouling measurement and monitoring**

order to increase the accuracy of the method.

main measurement methods include:

**50**

transferred by conduction. But as the thermal conductivity of the fouling layer and its thickness are not generally known, the only possible solution to the heat transfer problem is to introduce a fouling factor to take into account the additional resistance to heat transfer and possible calculation of the overall coefficient of heat transfer. A fouling coefficient is also sometimes specified, it is the reciprocal value of the fouling factor. When carrying out heat transfer calculations, the selection of fouling factors must be made with caution, especially when the fouling resistances completely dominate the thermal design.

The influence of inherent uncertainties in fouling factors is generally greater than that of uncertainties in other design parameters such as fluid properties, flow rates and temperatures [18]. An important fouling factor is sometimes adopted as a safety margin to cover uncertainties on the properties of fluids and even in the knowledge of the process, but the use of an excessively large fouling factor will result in an oversized heat exchanger with two or three times more area than is necessary. Although many tabulations based on the experiment are available and provide typical fouling factors such as the TEMA RGP-T-2.4 table [19], an acceptable assessment of the effects of fouling needs to be judged and evaluated for each particular application. Such tabulations can, however, serve as a guide in the absence of more specific information.

A number of semi empirical models have been proposed over the years for the prediction of the rate of fouling in heat exchangers or for estimating a fouling factor to be used in heat transfer calculations.

The first work on this subject began in the late 1950s with Kern and Seaton [17].

The modeling resulting from this work is based on the assumption that two processes act simultaneously. The first process is that of particle deposition characterized by a deposition flux that is constant if the concentration is also constant. The second process is that of the re-entrainment of particles characterized by a reentrain flow *ϕ<sup>r</sup>* dependent on the mass of particles (*mp*) deposited. The particle balance of the deposit is expressed according to the following equation:

$$\frac{dm\_p}{dt} = \phi\_d - \phi\_r \tag{2}$$

Or

*kp* is the transport coefficient.

*DOI: http://dx.doi.org/10.5772/intechopen.88936*

*C*<sup>1</sup> a dimensional constant.

Equation (2) thus becomes:

The solution of Eq. (5) is thus:

Assuming that τ= <sup>1</sup>

resistance:

in [m2

expressed in [m2

completely removed.

determine the value *Rf*\*

**53**

.

*Cb* the particle concentration in the fluid. *Cw* the particle concentration at the wall.

*dmp*

*τ<sup>w</sup>* the shear stress exerted by the fluid on the deposit.

*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process*

*mp* <sup>¼</sup> *Kp* <sup>∗</sup> ð Þ *Cb* � *Cw C*<sup>1</sup> ∗ *τ<sup>w</sup>*

*<sup>C</sup>*<sup>1</sup> <sup>∗</sup> *<sup>τ</sup><sup>w</sup>* and *<sup>m</sup>*<sup>∗</sup>

We can thus express the equation as follows:


All models and theory of fouling are based on this model.

*<sup>p</sup>* =

*mp* <sup>¼</sup> *<sup>m</sup>*<sup>∗</sup>

*Kp* ∗ ð Þ *Cb*�*Cw C*<sup>1</sup> ∗ *τw:*

Considering that the initial fouling flow is equal to the deposition flow and that the thermo physical properties of the deposit (conductivity and density) are constant, it is thus possible to express Eq. (7) in the form of a thermal fouling

*Rf t*ðÞ¼ *Rf* <sup>∗</sup> <sup>∗</sup> <sup>1</sup> � *exp* � *<sup>t</sup>*


An apparent weakness of the Kern and Seaton [17] model is that the re-entrain flow depends on the thickness of the deposition layer. As a result, it is only once a significant deposit thickness has accumulated that the role of the re-entrain term becomes significant [20]. In the case of high speed flow, the deposit would be

We also note that this model requires to go back on the values of *Rf*\* and *τ*. In general, there is no way to predict these values unless detailed experimental work has been done [21]. These values are thus established on a given installation and especially under given operating conditions. Thus, a modification, even minor, of the operating conditions (for example, water quality, flow modification) significantly modifies the parameters of the model and leads to a bad modeling of the fouling [22]. However, we note various works that make it possible to know the impact of the

flow velocity (*Um*) on the value of the asymptotic fouling resistance *Rf*\*

With *Rf*(*t*), the evolution of the fouling resistance as a function of time

*<sup>p</sup>* <sup>∗</sup> <sup>1</sup> � *exp* � *<sup>t</sup>*

*dt* <sup>¼</sup> *kp* <sup>∗</sup> ð Þ� *Cb* � *Cw <sup>C</sup>*<sup>1</sup> <sup>∗</sup> *<sup>τ</sup><sup>w</sup>* <sup>∗</sup> *mp* (5)

*τ*

*τ*

, the asymptotic value of the fouling resistance expressed

∗ ½ � 1 � exp ð Þ �*C*<sup>1</sup> ∗ *τ<sup>w</sup>* ∗ *t* (6)

h i � � (7)

h i � � (8)

, or even to

The deposition process is designed as the serialization of particle transport and adhesion mechanisms. The following assumptions are made:


The particle wall transport phase controls the deposition process while the shear stress controls the re-entrain phase of the particles. Thus, considering the proportionality of *ϕ*<sup>d</sup> as a function of the deposited mass of particles, we can write the following equations:

$$
\phi\_d = k\_p \ast (\mathbf{C}\_b - \mathbf{C}\_w) \tag{3}
$$

$$
\phi\_r = \mathbf{C}\_1 \ast \tau\_w \ast m\_p \tag{4}
$$

*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process DOI: http://dx.doi.org/10.5772/intechopen.88936*

Or

transferred by conduction. But as the thermal conductivity of the fouling layer and its thickness are not generally known, the only possible solution to the heat transfer problem is to introduce a fouling factor to take into account the additional resistance to heat transfer and possible calculation of the overall coefficient of heat transfer. A fouling coefficient is also sometimes specified, it is the reciprocal value of the fouling factor. When carrying out heat transfer calculations, the selection of fouling factors must be made with caution, especially when the fouling resistances

The influence of inherent uncertainties in fouling factors is generally greater than that of uncertainties in other design parameters such as fluid properties, flow rates and temperatures [18]. An important fouling factor is sometimes adopted as a safety margin to cover uncertainties on the properties of fluids and even in the knowledge of the process, but the use of an excessively large fouling factor will result in an oversized heat exchanger with two or three times more area than is necessary. Although many tabulations based on the experiment are available and provide typical fouling factors such as the TEMA RGP-T-2.4 table [19], an acceptable assessment of the effects of fouling needs to be judged and evaluated for each particular application. Such tabulations can, however, serve as a guide in the

A number of semi empirical models have been proposed over the years for the prediction of the rate of fouling in heat exchangers or for estimating a fouling factor

The first work on this subject began in the late 1950s with Kern and Seaton [17]. The modeling resulting from this work is based on the assumption that two processes act simultaneously. The first process is that of particle deposition characterized by a deposition flux that is constant if the concentration is also constant. The second process is that of the re-entrainment of particles characterized by a reentrain flow *ϕ<sup>r</sup>* dependent on the mass of particles (*mp*) deposited. The particle

The deposition process is designed as the serialization of particle transport and

• not taken into account of the phase of initiation of the deposit and the state of

• constancy of the properties and thermo-physical characteristics of the fluid and

The particle wall transport phase controls the deposition process while the shear stress controls the re-entrain phase of the particles. Thus, considering the proportionality of *ϕ*<sup>d</sup> as a function of the deposited mass of particles, we can write the

*dt* <sup>¼</sup> *<sup>ϕ</sup><sup>d</sup>* � *<sup>ϕ</sup><sup>r</sup>* (2)

*ϕ<sup>d</sup>* ¼ *kp* ∗ ð Þ *Cb* � *Cw* (3) *ϕ<sup>r</sup>* ¼ *C*<sup>1</sup> ∗ *τ<sup>w</sup>* ∗ *mp* (4)

balance of the deposit is expressed according to the following equation:

*dmp*

adhesion mechanisms. The following assumptions are made:

• consideration of a single type of fouling;

• homogeneity of the deposit;

surface;

the deposit.

following equations:

**52**

completely dominate the thermal design.

*Inverse Heat Conduction and Heat Exchangers*

absence of more specific information.

to be used in heat transfer calculations.

*kp* is the transport coefficient. *Cb* the particle concentration in the fluid. *Cw* the particle concentration at the wall. *C*<sup>1</sup> a dimensional constant. *τ<sup>w</sup>* the shear stress exerted by the fluid on the deposit. Equation (2) thus becomes:

$$\frac{dm\_p}{dt} = k\_p \ast (\mathbf{C}\_b - \mathbf{C}\_w) - \mathbf{C}\_1 \ast \tau\_w \ast m\_p \tag{5}$$

The solution of Eq. (5) is thus:

$$m\_p = \frac{K\_p \* (C\_b - C\_w)}{C\_1 \* \tau\_w} \* \left[1 - \exp\left(-C\_1 \* \tau\_w \* t\right)\right] \tag{6}$$

Assuming that τ= <sup>1</sup> *<sup>C</sup>*<sup>1</sup> <sup>∗</sup> *<sup>τ</sup><sup>w</sup>* and *<sup>m</sup>*<sup>∗</sup> *<sup>p</sup>* = *Kp* ∗ ð Þ *Cb*�*Cw C*<sup>1</sup> ∗ *τw:* We can thus express the equation as follows:

$$m\_p = m\_p^\* \ast \left[1 - \exp\left(-\frac{t}{\tau}\right)\right] \tag{7}$$

Considering that the initial fouling flow is equal to the deposition flow and that the thermo physical properties of the deposit (conductivity and density) are constant, it is thus possible to express Eq. (7) in the form of a thermal fouling resistance:

$$R\mathcal{f}(t) = R\mathcal{f} \stackrel{\*}{\*} \* \left[\mathbf{1} - \exp\left(-\frac{t}{\tau}\right)\right] \tag{8}$$

With *Rf*(*t*), the evolution of the fouling resistance as a function of time expressed in [m2 -K/W] *Rf*\* , the asymptotic value of the fouling resistance expressed in [m2 -K/W] (this value characterizes the situation where the deposition rate equals the breakout speed). *t*, the time expressed in [*s*] *τ*, the characteristic time expressed in [*s*] and whose value is generally attributed to the time required for the fouling resistance to reach its asymptotic value if the evolution of this kinetics was linear.

The Kern and Seaton [17] model therefore provides a mathematical description of the concept of simple fouling. This equation verifies the asymptotic behavior of the formation of a particulate deposit on the exchange surface of a heat exchanger. All models and theory of fouling are based on this model.

An apparent weakness of the Kern and Seaton [17] model is that the re-entrain flow depends on the thickness of the deposition layer. As a result, it is only once a significant deposit thickness has accumulated that the role of the re-entrain term becomes significant [20]. In the case of high speed flow, the deposit would be completely removed.

We also note that this model requires to go back on the values of *Rf*\* and *τ*. In general, there is no way to predict these values unless detailed experimental work has been done [21]. These values are thus established on a given installation and especially under given operating conditions. Thus, a modification, even minor, of the operating conditions (for example, water quality, flow modification) significantly modifies the parameters of the model and leads to a bad modeling of the fouling [22].

However, we note various works that make it possible to know the impact of the flow velocity (*Um*) on the value of the asymptotic fouling resistance *Rf*\* , or even to determine the value *Rf*\* .

Different authors thus propose a relationship of proportionality of type:

$$R\mathcal{f}^\* \sim Um\_i \tag{9}$$

With regard to the tube exchangers: for Kern and Seaton [17] the value of *i* is �1. For Watkinson [23], this constant takes the value of �1.2 to �2.

As far as plate heat exchangers are concerned, Muller-Steinhagen [24] has in its study demonstrated a relation of proportionality between the asymptotic resistance of fouling *Rf*\* and the inverse of the speed squared (i.e., an exponent *<sup>i</sup>* � 2 in Eq. (9)), without providing a general relationship.

In the same context, Grandgeorge [25] proposes an empirical relation resulting from several experiments on different industrial size plate heat exchangers. In this context, Grandgeorge [25] established that the use of the initial pressure drop in the heat exchanger (Δ*Po*) in place of the flow velocity makes it possible to correlate with the aid of a single relationship the value of the asymptotic resistance *Rf*\* . The relationship is as follows:

$$Rf^\* = \frac{1}{4 \ast \Delta Po} \tag{10}$$

3.The thermal losses were neglected.

*Simplified diagram of the phosphoric acid concentration unit.*

*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process*

*DOI: http://dx.doi.org/10.5772/intechopen.88936*

extremities of the heat exchanger.

extremities of the circulation pump.

of the fouling resistance of the phosphoric acid was studied.

*Rf t*ðÞ¼ <sup>1</sup>

Us � 1 Up

**7. Calculation method**

**Figure 3.**

**7.1 Calculation of** *Us*

**55**

course, via the expression:

4.The inlet and outlet temperatures of the two fluids are determined at the

The experimental data was collected out during 1 year. The method that we used to follow the fouling evolution consists in carrying out a heat balance at the boundaries of the heat exchanger by the intermediary of measurements of the inlet and outlet temperatures pump suction and discharge pressure measurements and acid density measurement (**Figure 4**). The latter was taken each 2 h during all the day. This method, albeit indirect, makes it possible to detect the necessary moment to shut down the installation for cleaning. In the current study, the temporal evolution

The calculation of the fouling resistance was done using the following relation:

<sup>¼</sup> <sup>1</sup>

The overall heat transfer coefficient at the dirty state was given in the time

*U t*ð Þ � <sup>1</sup>

*U t*ð Þ <sup>¼</sup> <sup>0</sup> (11)

5.Pump suction and discharge pressure measurements are taken at the

Based on these observations, this model has been revised and modified by various researchers with various descriptions of the term deposition and re-entrain: Only empirical parameters were added and derived solely from the experimental study [20, 25].

### **6. Experimental procedure**

The phosphoric acid concentration loop is allowed to concentrate—by evaporation—the phosphoric acid from 28 to 54% P2O5 in a forced-circulation evaporator closed loop, functioning under vacuum feeded by a barometric condenser. The system used for concentration composed of a stainless steel tubular heat exchanger, a centrifugal pump, a boiler or expansion chamber, a barometric condenser and a basket filter [26].

The inclusion of the dilute acid is done at the basket filter where it mixes with the circulating acid in order to protect the pump from abrasion and to limit the heat exchanger fouling, which reduces the stop frequency for washing. The circulation pump then aspirates the mixture formed and sends it to the inlet of the heat exchanger at a temperature in the order of 70°C. The heat exchanger allows heating the phosphoric acid at a temperature in the order of 80°C. The steam undergoes a condensation at a temperature of around 120°C at the level of the exchanger. The condensate will be sent to a storage tank before being returned to the utility center.

The overheated mixture of the acid outgoing the exchanger then passes into the boiler where an amount of water evaporates and the production of concentrated acid is done by overflow in inner tube of the boiler and the rest will be recycled. The condenser also ensures the re-entrain of incondensable outgoing of the boiler by the effect of water tube created by the waterfall. At the foot of the barometric guard, the seawater is gathered in a guard tank before being rejected to the sea (**Figure 3**).

Our experimental study is based on the following hypotheses.


*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process DOI: http://dx.doi.org/10.5772/intechopen.88936*

**Figure 3.**

Different authors thus propose a relationship of proportionality of type:

For Watkinson [23], this constant takes the value of �1.2 to �2.

Eq. (9)), without providing a general relationship.

*Inverse Heat Conduction and Heat Exchangers*

relationship is as follows:

**6. Experimental procedure**

study [20, 25].

basket filter [26].

constant.

**54**

With regard to the tube exchangers: for Kern and Seaton [17] the value of *i* is �1.

As far as plate heat exchangers are concerned, Muller-Steinhagen [24] has in its study demonstrated a relation of proportionality between the asymptotic resistance of fouling *Rf*\* and the inverse of the speed squared (i.e., an exponent *<sup>i</sup>* � 2 in

In the same context, Grandgeorge [25] proposes an empirical relation resulting from several experiments on different industrial size plate heat exchangers. In this context, Grandgeorge [25] established that the use of the initial pressure drop in the heat exchanger (Δ*Po*) in place of the flow velocity makes it possible to correlate with the aid of a single relationship the value of the asymptotic resistance *Rf*\*

*Rf* <sup>∗</sup> <sup>¼</sup> <sup>1</sup>

Based on these observations, this model has been revised and modified by various researchers with various descriptions of the term deposition and re-entrain: Only empirical parameters were added and derived solely from the experimental

The phosphoric acid concentration loop is allowed to concentrate—by evaporation—the phosphoric acid from 28 to 54% P2O5 in a forced-circulation evaporator closed loop, functioning under vacuum feeded by a barometric condenser. The system used for concentration composed of a stainless steel tubular heat exchanger, a centrifugal pump, a boiler or expansion chamber, a barometric condenser and a

The inclusion of the dilute acid is done at the basket filter where it mixes with the circulating acid in order to protect the pump from abrasion and to limit the heat exchanger fouling, which reduces the stop frequency for washing. The circulation pump then aspirates the mixture formed and sends it to the inlet of the heat exchanger at a temperature in the order of 70°C. The heat exchanger allows heating the phosphoric acid at a temperature in the order of 80°C. The steam undergoes a condensation at a temperature of around 120°C at the level of the exchanger. The condensate will be sent to a storage tank before being returned to the utility center. The overheated mixture of the acid outgoing the exchanger then passes into the boiler where an amount of water evaporates and the production of concentrated acid is done by overflow in inner tube of the boiler and the rest will be recycled. The condenser also ensures the re-entrain of incondensable outgoing of the boiler by the effect of water tube created by the waterfall. At the foot of the barometric guard, the seawater is gathered in a guard tank before being rejected to the sea (**Figure 3**).

Our experimental study is based on the following hypotheses.

1.The flow of two fluids (Phosphoric acid and steam) is at counter current.

2.Values of the thermo-physical properties of the fluids were considered

*Rf* <sup>∗</sup> � *Umi* (9)

<sup>4</sup> <sup>∗</sup> *<sup>Δ</sup>Po* (10)

. The

*Simplified diagram of the phosphoric acid concentration unit.*


#### **7. Calculation method**

The experimental data was collected out during 1 year. The method that we used to follow the fouling evolution consists in carrying out a heat balance at the boundaries of the heat exchanger by the intermediary of measurements of the inlet and outlet temperatures pump suction and discharge pressure measurements and acid density measurement (**Figure 4**). The latter was taken each 2 h during all the day.

This method, albeit indirect, makes it possible to detect the necessary moment to shut down the installation for cleaning. In the current study, the temporal evolution of the fouling resistance of the phosphoric acid was studied.

The calculation of the fouling resistance was done using the following relation:

$$R\!f(t) = \frac{1}{\mathbf{U}\_{\mathbf{s}}} - \frac{1}{\mathbf{U}\_{\mathbf{p}}} = \frac{1}{U(t)} - \frac{1}{U(t=\mathbf{0})} \tag{11}$$

## **7.1 Calculation of** *Us*

The overall heat transfer coefficient at the dirty state was given in the time course, via the expression:

**Figure 4.** *The measurement method at the boundaries of the heat exchanger.*

$$U\_s = U(t) = \frac{\dot{\mathbf{m}}\_{\rm ac, cir} \* \mathbf{C} p\_{ac} \* (T\_{out, ac} - T\_{in, ac})}{A \* F \* \Delta T\_{ml}} \tag{12}$$

treated phosphoric acid. As it appears clearly as the fouling resistance increases with the time until reaching a maximum value varied from 1.38 \* 10<sup>4</sup> to 1.61 \*

*Variation of the fouling resistance as a function of time for the stainless-steel-tubular heat exchanger.*

*Tubular Heat Exchanger Fouling in Phosphoric Acid Concentration Process*

*DOI: http://dx.doi.org/10.5772/intechopen.88936*

The series functioned for more than 5 days, a sufficient period so that the resistance asymptotic value is reached. The fluctuation observed on these curves are due to the variation of flow, which, acting on the shear stress to the wall, causes re-entrain deposit particles or their deposition according to the sent flow value.

From Eq. (11), we notice that the overall heat transfer coefficient is inversely

*Variation of the overall heat transfer coefficient as a function of time for the stainless-steel-tubular heat*

**8.2 Temporal evolution of the overall heat transfer coefficient**

10<sup>4</sup> m<sup>2</sup>

**Figure 6.**

*exchanger.*

**57**

**Figure 5.**

.K.W<sup>1</sup> .

proportional to the fouling resistance.

This relation is taken by the evaluation of energy on the heat exchanger by supposing the isolated system and the physical properties of the two fluids, as well as, the heat transfer coefficients stay constant along the exchanger.
