Analysis of fouling characteristic in enhanced tubes using multiple heat and mass transfer analogies

    Analysis of fouling characteristic in enhanced tubes using multiple heat and mass transfer analogies Zepeng Wang, Guanqiu Li, Jingliang Xu, Jinjia Wei, Jun Zeng, Decang Lou, Wei Li PII: DOI: Reference:

S1004-9541(15)00258-X doi: 10.1016/j.cjche.2015.07.011 CJCHE 337

To appear in: Received date: Revised date: Accepted date:

8 June 2014 31 January 2015 3 June 2015

Please cite this article as: Zepeng Wang, Guanqiu Li, Jingliang Xu, Jinjia Wei, Jun Zeng, Decang Lou, Wei Li, Analysis of fouling characteristic in enhanced tubes using multiple heat and mass transfer analogies, (2015), doi: 10.1016/j.cjche.2015.07.011

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ACCEPTED MANUSCRIPT Energy, Resources and Environmental Technology Analysis of fouling characteristic in enhanced tubes using multiple

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heat and mass transfer analogies Zepeng Wang(王泽鹏) 1, Guanqiu Li(李冠球) 2，Jingliang Xu(徐进良) 3， Jinjia Wei(魏进家) 4，Jun Zeng(曾军) 5,

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Decang Lou(娄德仓)5, Wei Li(李蔚) 2,* 1

College of Electromechanical Engineering, Qingdao University of Science and Technology, Qingdao 266061, China College of Energy Engineering, Zhejiang University, Hangzhou 310027, China 3 School of Energy, Power, and Mechanical Engineering, North China Electric Power University, Beijing 102206, China 4 State Key Laboratory of Multiphase Flow in School Energy and Power Engineering of Xi'an Jiaotong University, Xi’an 710049, China 5 China Gas Turbine Establishment, 6 Xin-Jun Road, Xidu district, Chengdu, China

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Article history: Received 8 June 2014 Received in revised form 31 January 2015

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Accepted 3 June 2015

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*Corresponding author. E-mail address: [email protected]

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Abstract This paper provides a comprehensive analysis on cooling tower fouling data taken from seven 15.54 mm I.D. helically ribbed, copper tubes and a plain tube at Re=16000. There are two key processes during fouling formation: fouling deposition and fouling removal, which can be determined by mass transfer and fluid friction respectively. The mass transfer coefficient can be calculated through three analogies: Prandtl analogy, Von-Karman analogy, and Chilton-Colburn analog. Based on our analyses, Von-Karman analogy is the optimized analogy, which can well predict the formation of cooling tower fouling. Series of semi-theoretical fouling correlations as a function of the product of area indexes and efficiency indexes were developed, which can be applicable to different internally ribbed geometries. The correlations can be directly used to assess the fouling potential of enhanced tubes in actual cooling tower water situations. Keywords Fouling, Enhanced tube, Turbulent flow, Mass transfer coefficient, Cooling tower water

1. Introduction Fouling is the solid substance deposited on heat transfer surfaces contacting with unclean fluid, which can significantly reduce the heat transfer efficiency and therefore has been attracted many research interests. There are three issues to be addressed in fouling research

[1]

. The first one is the theoretical and experimental analysis of fouling 1

ACCEPTED MANUSCRIPT formation, providing a common and precise predicting model for heat exchanger design. The second is fouling monitor techniques. The last one is the countermeasure of fouling. Many researches had been focused on the theoretical and experimental analysis on

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fouling mechanisms, which are crucial to control of fouling formation. Regard to modeling the fouling data, Kern and Seaton

[2]

published an analysis

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which is considered as a landmark in the beginning of modern scientific study of fouling.

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They assumed two key processes that control the fouling resistance, namely fouling deposition and fouling removal. In principle, they brought forward a model which can [3]

. Several

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embrace fouling involving all forms of growth and removal processes

significant investigations addressed the foulant deposition and removal rates in enhanced [4]

made a detailed research on three internal repeated rib tubes.

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tubes. Kim and Webb

Heat-mass transfer analogy was utilized to calculate the mass transfer coefficient (Km) in the diffusion region. Within the test range, the fouling resistance increased as geometric

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parameter e/D decreased and p/e increased. Li and Webb [5] developed the first long-term

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fouling model for practical cooling tower water flowing inside the enhanced tubes by using the Chilton-Colburn analogy to calculate mass transfer coefficient. Recently, Jun [6]

applied a 2D fouling model to predict the fouling performance of plate heat

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and Puri

exchangers, which coupled a 2D dynamic model with material balance equations. The model showed a strong relation between fouling and the operating conditions, i.e. flow rate and deposit formation. Their model can determine the sensor locations that can provide necessary information for fouling formation to monitor not only fouling during thermal process but also cleaning condition. Quan et al.[7] conducted an experimental study to investigate the fouling process of calcium carbonate on enhanced heat transfer surface in forced convective heat transfer. The fouling behaviors were examined under different factors including fluid velocity, hardness, alkalinity, solution temperature, and wall temperature. Webb [8] reported heat transfer and friction characteristics of three tubes having a conical, three-dimensional roughness on the inner tube surface with water flow in the tube. The Nusselt number (Nu) of 3-D TC3 truncated cone tube was 3.74 times higher than a plain tube. Accelerated particular fouling data were also provided for this tube and compared to the results of helical-ribbed tubes studied in Li

[9]

. The results

showed that the 3-D tube provided the highest heat transfer coefficient but also had the

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ACCEPTED MANUSCRIPT highest asymptotic fouling resistance. Webb and Li

[10]

described the experimental results of long-term fouling tests for

cooling tower water flowing inside enhanced tubes. Fouling data were measured for the

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seven helical rib roughened tubes for more than a 2500 h operating period. There exist combined precipitation and particulate fouling (PPF) in cooling tower systems. The test

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geometries include seven enhanced copper tubes and a plain copper tube. In this study,

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the main focus is to analyze the experimental results in ref. [10] to quantitatively define the effect of rib height, rib axial pitch, and helix angle on the tube fouling performance.

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No analysis has been done to establish the fouling model that can fully reflect the realistic flow conditions. From the previous studies, fouling deposition rate determined

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by mass transfer coefficient is the key parameter to estimate the fouling resistance. In this study, new models for calculating the deposition rate are developed. Comprehensive analyses about three widely used analogies, i.e. Prandtl analogy, Von-Karman analogy,

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and Chilton-Colburn analog, are discussed to develop the mass transfer coefficient Km.

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Semi-theoretical fouling models based on three analogies have been established to correlate experimental data on fouling. It is revealed that Von-Karman analogy is the

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most appropriate for providing the most accurate fouling model to represent the experimental results.

2. Fouling model

There are four different patterns for fouling formation: the linear increasing pattern, linear decreasing pattern, power pattern, and asymptotic pattern

[12]

. In this work, the

fouling pattern in cooling tower systems is considered to follow asymptotic pattern:

Rf  Rf * 1  e Bt  , Rf * 

K m PCb  , B s  s kf  f 

The asymptotic fouling resistance ratio Rf*/Rfp* is utilized to obtain the fouling characteristics in the enhanced tubes. For rough surfaces, a fraction of the pressure drop may be due to the profile drag on roughness elements. Webb et al.

[13]

reported j and f

factors for the clean tube, which can be used for characterize the heat transfer and flow performance in the helical-ribbed tubes. They showed that f ∝Re-0.283, and the results indicated that the pressure drop was dominated by wall shear stress, and that the pressure 3

ACCEPTED MANUSCRIPT drag component was negligible. In this study, the profile drag can be ignored, comparing with the wall shear stress, which can be calculated with τs=△PAc/Aw=0.5fρu2. We define σ=(P/Pp)(ξ/ξp), then the following equation can be obtained:

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Rf*/Rfp*=σ (Km/Kmp)/(τs/τsp)

(1)

σ is a fouling process index, which is determined from the fouling tests. The asymptotic

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fouling resistance ratio Rf*/Rfp* can be obtained after calculating Km and τs. 2.1. Calculation of Km [14-15]

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Three analogies are used to calculate the mass transfer coefficient, however, the Reynolds analogy is not included since it assumed that Pr≈1 and Sc≈1. It is not suitable

Prandtl

[16]

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for the experimental data because of the wide range in the operating conditions. assumed the turbulent boundary can be divided into laminar region and

turbulent region, which can amend the Reynolds analogy. The momentum and heat

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transfer are attributed to molecule transfer, and the thickness of the laminar region is thin.

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Thus, the temperature and velocity are distributed linearly. Prandtl analogy divides the boundary into laminar region and turbulent region and ignores the influence of the buffer

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region. This brings the deviation. The equations of Prandtl analogy are listed below: St=j×Pr-2/3= (f/2/)/[1+(f/2)0.5×5(Pr-1)]

(2a)

St'= Km /u0= (f/2/)/[1+(f/2)0.5×5(Sc-1)]

(2b)

Comparing to the Reynolds analogy and the Prandtl analogy, Von Karman analogy [17]

includes buffer region into the model. The usage of the three-region model makes the

turbulent heat and mass transfer theory more close to the realistic flow conditions. Figure 1 shows the temperature distribution and velocity profile of the turbulent flow inside a tube. The deviation of the Von-Karman analogy is primarily caused by neglecting turbulent transfer in laminar region and heat transfer in turbulent region. There are large deviations when Pr number is extra large or small. The equations of Von-Karman analogy are listed below: St=j×Pr-2/3= (f/2/)/{1+(f/2)0.5×[5(Pr -1)+5ln((5Pr +1)/6)]}

(3a)

St'= Km /u0= (f/2/)/{1+(f/2)0.5×[5(Sc -1)+5ln((5Sc +1)/6)]}

(3b)

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Fig. 1. Three regions in Von-Karman analogy

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Chilton and Colburn [18] modified the Reynolds analogy based on their experimental data. Their modification make the new analogy not restrict by Pr≈1 and Sc≈1, and it is

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extended to systems having fluid friction. Chilton-Colburn analogy presents: (Km/u0) Sc2/3=(h /ρcp u0) Pr2/3

(4)

The correlations of mass transfer coefficient Km are obtained through eliminating f

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For Prandtl analogy,

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factor in above equations:

Km /u0=4/[b12m12+(2a1b1-2b12)m+(b12-2a1b1)]

(5)

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where m1=[1+4/(b12St)]0.5，a1=5(Sc-1)，b1=5(Pr-1). For Von-Karman analogy,

Km/u0=4/[b22m22+(2a2b2-2b22)m+(b22-2a2b2)]

2

(6)

0.5

where m2=[1+4/(b2 St)] , a2=5(Sc-1)+5ln[(5Sc+1)/6], b2=5(Pr-1) +5ln[(5Pr +1)/6]. For Chilton-Colburn analogy,

Km/u0= j×Pr2/3×Sc-2/3

(7)

2.2. Correlation of f and j factors Webb et al. [13] developed the following heat transfer and friction factor equations for the seven tubes tested in their study: f=0.108Re-0.283ns0.221 (e/Di)0.785α0.78

(8)

j=0.00933Re-0.181ns0.285 (e/Di)0.323α0.505

(9)

Average deviation of friction and heat transfer correlations is 4.9% and 3.8%, respectively. For smooth tube, the Seider-Tate correlations fp=0.079Re-0.25 and jp=0.027Re-0.2 are used to calculate f and j factors for smooth tubes[5]. 5

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2.3. Analysis of fouling model Three analogies are used to calculate Km. Table 1 provides the range of the Sc and Pr

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numbers, which assures the deviation being acceptable. Substituting Eqs. (5), (6) and (7) and τs=PAc/Aw=0.5fu2 into Eq. (1) at u0=constant, the formula of Prandtl analogy,

(10)

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Rf*/Rfp*=σi(ji'/jpi')/(f/fp)

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Von-Karman analogy, Chilton-Colburn analogy are obtained as follows:

where ji'/jpi'=Kmi/Kmpi. For Prandtl analogy, i=1; for Von-Karman analogy, i=2; for

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Chilton-Colburn analogy, i=3. Note that the ji'/jpi' ratio indicates the relative foulant deposition rate and the f/fp ratio indicates the relative foulant removal rate. Therefore, the

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assumption we obtained shows the fouling characteristic can be determined through two processes, which are the heat transfer and fluid friction. The j and f factor is the key parameter in the fouling formation. Define ηi=(ji'/jpi') /(f/fp), which is determined by tube

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efficiency index η=(j/jp) /(f/fp). η is normally provided by tube manufacturer. Table 1 The available range of the Sc and Pr numbers [16-18] 1. Prandtl

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Analogy

Available range

2. Von-Karman

analogy

analogy

0.7< Pr <20

0.46< Pr <324

3. Chilton-Colburn analogy

0.6< Sc <2500, 0.6< Pr <100

3. Analysis of fouling data Table 2 gives the fouling data and tube geometry parameters of a plain tube and seven helical-rib tubes in Webb and Li

[10]

. Figure 2 shows tube geometry detail of

helical-rib tubes. Tubes 2, 3 and 5 are with p/e <5; The rest tubes are with p/e >5 . The fouling resistance Rf*/Rfp* is based on the nominal inside area (Ai=πDiL). Figure 3(a) shows Rf*/Rfp* vs. ηi as presented in Table 2. The comparisons among the three analogies are presented. Since the analysis is based on the heat-mass transfer analogies, it is automatically assumed that the fouling deposit is formed by a large amount of individual particles. The effect of cohesion between the particles and the tube surface is not included. This is valid for particulate fouling, which is the accumulation of suspended solid on the heat transfer surface. However, it is not valid for PPF in cooling tower systems. The

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ACCEPTED MANUSCRIPT cooling tower by its designed evaporative cooling multiplies the non-evaporative impurities of water. An important property of non-evaporative impurity of particular interest is its hardness. As hardness is originated from dissolved inverse solubility salts

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(mostly CaCO3 and MgCO3, their solubility decreasing with increasing temperature); these salts can precipitate onto the heated wall of the condenser, resulted in precipitation

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fouling. Cooling tower water contains usually rather high concentration of a large variety

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of salts, each exhibiting different crystalline formations. Consequently, crystalline clusters build up in irregular patterns, forming cavities between them which permit

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deposition of suspended particles, further decreasing the crystalline cohesion. When cooling tower waters are cooled by atmospheric air, the tiny particles in the atmospheric

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air get into the cooling waters. Rust and dust particles are commonly contained in cooling tower waters. Some of the particles can act as catalysts and undergo complex reactions.

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They imbed into the loose crystalline structure common to cooling tower water.

(a)

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(b)

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Fig. 2. Description of helically ribbed tubes (a) Photos of helically ribbed tubes, (b) Cross section drawing of rib

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Table 2 Experimental data and geometric parameters of helical-rib tubes [10]

(Di=15.54 mm, γ=41 deg, tt=0.024 mm, Rfp*=4.410-6 m2K/W, Pr=5.2, Sc=0.7, Re=16000, ppm=1300) Tube

Rf*/ Rfp

β

η

η1

η2

η3

ns

e (mm)

Α (deg)

p/e

1

1

1

1

1

1

1

n/a

n/a

n/a

n/a

7.44

1.66

1.18

1.752

2.25

1.18

45

0.33

45

2.81

3.25

1.56

1.05

1.511

1.899

1.05

30

0.4

45

3.5

4

1.94

1.24

0.95

1.152

1.313

0.95

10

0.43

45

9.88

5

5.65

1.76

1.04

1.53

1.938

1.04

40

0.47

35

3.31

6

2.69

1.52

1.01

1.361

1.641

1.01

25

0.49

35

5.02

7

2.47

1.52

1.05

1.326

1.526

1.05

25

0.53

25

7.05

8

2.03

1.4

0.98

1.222

1.373

0.98

18

0.55

25

9.77

2 3

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(a)

(b)

(a) Rf*/Rfp*vs. ηi; (b) Rf*/Rfp*vs. βηi

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Fig. 3.

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Even though the adherence forces of PPF of cooling tower water is smaller than that

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of pure precipitation fouling due to the existence of rust and dust particles, the effect of adherence forces between particles and heat transfer surface is an important factor for the

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formation of PPF deposit. Contrary to modeling of particulate fouling data, the effect of

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adherence forces must be considered in the process of modeling PPF data. The particulate fouling deposits primarily on the surface between the ribs and on rib tip surface. The

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nominal internal surface area based on plain tube (A=πDiL), the reference surface in above equations, has been use widely in particulate fouling data analysis of enhanced tubes. With respect to precipitation fouling or PPF, we should use wetted internal surface area Aw rather than nominal internal surface area A for the analysis. The adherence forces are sufficient to describe the formation of fouling. The PPF deposits on the total internal surface including the surfaces between ribs, rib tip surfaces, and side surfaces of ribs, due to deposit cohesion. In order to analyze the difference between Aw and A, we introduced an area index, β = (Aw /A)/(Ac /Acp) into Eq. (10): Rf*/Rfp*=σiβηi

(11)

3.1. Correlations By correlating Rf*/Rfp* with βηi, we obtained the following correlations which are shown in Fig. 3(b): For Prandtl Analogy, Rf* / Rfp*=1.536βη1-0.478 (p/e>5)

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(12a)

ACCEPTED MANUSCRIPT Rf* / Rfp*=7.511βη1-14.507 (p/e<5)

(12b)

For Von-Karman Analogy, Rf* / Rfp*=1.099βη2-0.0297 (p/e>5)

(13a)

Rf* /

(13b)

Rfp =5.416βη2-12.803 (p/e<5)

For Chilton-Colburn analogy,

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Rf* / Rfp*=2.836βη3-1.702 (p/e>5)

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*

(14b)

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Rf* / Rfp*=13.015βη3-18.098 (p/e<5)

(14a)

The deviation of Eqs. (12), (13) and (14) between the experimental data and

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calculated value is 5.4 %, 4.0% and 9.3%, respectively, but in contrast the deviation of corresponding equations without β between the experimental data and calculated value is

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8.9 %, 7.2% and 12.3%, respectively. However, as the change of Pr and Sc numbers (for instance Pr>100), the deviation will become much larger. For those points, the three layers model of Von-Karman analogy can express the actual turbulent situation inside the

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tubes, but the deviation of Prandtl analogy and Chilton-Colburn analogy are more than

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15%, which are not satisfactory. There are three sources for the deviation in Eqs. (12), (13) and (14). The first deviation is from the three analogy equations, which is relatively small.

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For instance, considering Von-Karman analogy, the deviation is produced due to the moderate value of Pr and Sc number (Pr=5.2, Sc=0.7). The second is the deviation from Eqs. (8) and (9), which is 4.9% and 3.8%, respectively. The last and main error is produced during the calculation on plain tube by Seider-Tate correlations.

3.2. Discussion

There are two ranges for each line corresponding to an analogy. In the enhanced tubes with intermediate helix angles used in this study, the flow inside the tubes has much greater momentum in the axial direction than the angular momentum caused by the ribs, which would result in the flow separation at the ribs

[3]

. We primarily considered the

effect of the reattachment of the fluid to the surface between the ribs. Reattachment does not occur when the pitch of roughness is reduced to less than approximately five rib heights (p/e<5) for tubes 2, 3, and 5, and the main flow is forced to "glide over" the ribs and a secondary flow is created between the ribs. However, when p/e is large enough (p/e >5), flow reattachment would occur downstream of a rib and the secondary flow

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between two ribs caused a lower wall shear stress as shown in Fig. 4.

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Fig. 4. Boundary layer separation at different p/e

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The relations between ηi and j factor is shown in Fig. 5. It can be easily found that η2> η1> η3, and j factor can reflect the magnitude of heat transfer coefficient. Therefore, ηi

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could reflect the heat transfer characteristic in certain degree. It could be considered that the heat transfer characteristic of Von-Karman Analogy is largest. From the view of deviation, the deviation of Von-Karman analogy is the least. Making the Von-Karman

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analogy as the standard, comparison of Prandtl analogy to Von-Karman analogy is

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instructive. The laminar region and turbulent region are considered in both analogies, but the influence of the buffer region is also considered in Von-Karman analogy. The heat

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transfer coefficient in the turbulent region is much larger than the other two regions, therefore j1' and j1p' (Prandtl analogy) is larger than j2' and j2p' (Von-Karman analogy). However, the internal ribs of the enhanced tubes could disrupt the laminar flow, thereby increasing the heat transfer coefficient inside the enhanced tubes. The disruption of laminar region in enhanced tubes could result the increasing rate of the smooth tube (j1p'/ j2p'), comparing with the rate of the enhanced tubes (j1'/ j2'). Therefore, η1 is smaller than η2 and the heat transfer characteristic of Von-Karman analogy is larger than Prandtl analogy. For Chilton-Colburn analogy, one region model is the reason that the heat transfer characteristic of Chilton-Colburn analogy is the least.

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ηi vs. j/jp

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Fig. 5.

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As shown in Eq. (14), η3 equals to the tube efficiency index η, which is an advantage of Chilton-Colburn analogy. Comparing βηi to βη, a relationship can be obtained and it can help us to understand more about the fouling deposition mechanism, which is shown

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in Fig. 6. The results indicate that a linear dependence can be found between βηi and βη,

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which can be used for further prediction.

Fig. 6.

βηi vs. βη

Besides calculated from the correlations above, the value of ηi can be obtained from simulation, which can simplify the calculation process. As shown in Fig. 5a, a power-law curve fit of (ji'/jp') vs. (j/jp) generated the following correlation: j1'/jp'=0.961×(j/jp)1.556

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(15)

ACCEPTED MANUSCRIPT j2'/jp'=0.917×(j/jp)1.889

(16)

Eqs. (15) and (16) has an average deviation of 1.43% and 2.98%, and j3'/jp' equals to j/jp. The deviation of Eqs. (15) and (16) is brought by the plain tube, and the deviation of

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enhanced tube is less than 0.05%. Therefore, we have

(17a)

η2=(j2'/jp')/(f/fp)=0.0902Re0.0689ns0.317 (e/Di)-0.175α0.173

(17b)

η3=η= 0.253 Re0.052ns0.064 (e/Di)-0.462α-0.275

(17c)

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η1=( j1'/jp')/(f/fp)= 0.135Re0.0626ns0.222 (e/Di)-0.282α0.00578

ηi can be calculated from Eq. (17). Combining Eqs. (8) and (9), Eq. (17) can be applied to

(a)

Fig. 7.

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different internal ribbed geometries within the dimensional range of those tested tubes.

(b)

(a) ji'/jp' vs. j/jp; (b) Approximate relation between heat transfer and fluid flow process

As shown above, a new model is established based on the model of Kern and Seaton [2]

. The assumption is that the fouling factor Rf*/Rfp* can be calculated through two

parameters (j and f factors), which represent heat transfer process and fluid flow process. Through these two process and the analogies in turbulent boundary, a new method can be found to analyze the fouling characteristics in enhanced tubes. And even there are some relationships between the heat and flow process in Fig. 7(a). As shown in Table 2, the available range of the Sc and Pr numbers for Prandtl analogy is a bit small comparing with the other two analogies. From the above analysis, analogies can be a suitable method to predict the mass transfer coefficient to further understand the fouling deposition process. Von-Karman analogy having higher accuracy and wider applicable range is the most suitable analogy for simulating the flow and mass transfer inside the tube, 13

ACCEPTED MANUSCRIPT comparing with the other two analogies. Although Fig. 7(b) is only an approximate relation between heat transfer and fluid flow, the new model can be testified in another aspect. More research progresses

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including the experimental and theoretical investigation about fouling formation are

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needed to further understand the true mechanisms and improve the fouling model.

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4. Conclusions

A semi-empirical fouling model considering two main parameters (j and f factor) has

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been established to predict the cooling tower fouling resistance in the helical-rib tubes. The asymptotic fouling resistance ratio of the helical-rib tubes were found to be higher than the plain tubes. The mass transfer coefficient Km is calculated by Prandtl analogy,

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Von-Karman analogy, and Chilton-Colburn analogy. The relation between ηi and the fouling resistance of the enhanced tubes has been discussed. Also, the results showed that

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ηi can be simulated through tube efficiency index η.

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Referring to smooth tubes, three groups of linear correlations Rf*/ Rfp* vs. βηi are developed to predict the cooling tower fouling inside the helical-rib tubes. The predicted

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model is validated by the practical experimental results at Re=16,000. The predictions have good accurancy and can be segmented when the key parameter p/e (pitch/rib height) is equal to 5. The deviation of the Prandtl analogy ， Von-Karman analogy, and Chilton-Colburn analogy, and is

5.4%, 4.0%, and 9.3% , respectively. The revised

parameter β is also used the comparison about ηi vs. η. The correlations developed in the study can be used to predict the fouling resistance in helical-rib tubes in practical cooling tower systems.

Nomenclature Ac cross-sectional area, m2 Ai nominal internal surface area based on plain tube, m2 Aw inside wetted surface area, m2 B

time constant, 1/s

cp specific heat at constant pressure, J/kg Cb bulk particle concentration, kg/m3

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ACCEPTED MANUSCRIPT dp diameter of the particle, m DAB Brownian diffusivity (=(KBT)/(3πμdp)), m2/s

f

fanning friction factor (=Di/2ρU2)

h

heat transfer coefficient based on Ai, W/m2 K

j

j -factor (=StPr2/3)

kf

thermal conductivity of deposit, W/m K

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KB Boltzmann constant (=1.38E-23), J/K Km particle transfer coefficient

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ns number of starts Nu Nusselt number (=hDi/k) axial element pitch, m

P

sticking probability

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p

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△P tube-side pressure drop, N/m2 Pr Prandtl number (=μcp/k) heat transfer rate, W

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q

Re Reynold number (=ρUDi/μ) Rf fouling factor, m2 K/W

Rf* asymptotic fouling factor, m2 K/W Sc Schmidt number(=ν/DAB) St

Stanton number (=h/ρUCp)

T

time, s

tt

fin tip thickness, m

U

fluid velocity, m/s

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internal rib height (average value), m

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e

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Di internal tube diameter, or diameter to root of fins, m

u* friction velocity, m/s Greek letters α

helix angle, degrees

β

area index, (Aw/Awp)/(Ac/Acp), dimensionless

γ

included angle between sides of ribs, degrees

μ

dynamic viscosity, Ns/m2

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ν

kinematic viscosity, m2/s

ξ

deposit bond strength

ρ

density of fluid, kg/m3

ρf

density of deposit, kg/m3

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Fouling process index, dimensionless

s

shear stress, N/m2

υ

deposition or removal rate, kg/m2s

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σ

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ρp density of particle, kg/m3

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Subscripts and superscripts deposition

p

plain surface

r

removal

'

revised function of Chilton-Colburn and efficiency index η

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Dedication

This paper is dedicated to the memory of the late Doctor Guanqiu Li (1988 – 2015).

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Dr. Guanqiu Li obtained his Ph.D. with Wei Li in the Department of Energy Engineering, Zhejiang University, in China in June, 2012 and passed away on December 25, 2014 when he was serving as a post doctor in the Department of Mechanical and Materials Engineering, Masdar Institute of Science and Technology, in Abu Dhabi in UAE at age of 27.

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ACCEPTED MANUSCRIPT low-velocity cooling tower water”, Int. J. Heat Mass Tran., 45(8),1685-1691(2002). [6] Jun, S., Puri, V.M., “A 2D dynamic model for fouling performance of plate heat exchangers”, J. Food Eng., 75(3), 364-374(2006). forced

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Graphical abstract

Fouling is the solid substance deposited on heat transfer surfaces contacting with unclean fluid,

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which can significantly reduce the heat transfer efficiency. It has been known as the major unsolved problem in heat exchangers. This paper provides a comprehensive analysis by using Prandtl analogy,

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Von-Karman analogy, and Chilton-Colburn analogy on cooling tower fouling data taken from seven 15.54 mm I.D. helically ribbed, copper tubes and a plain tube currently used in the industry. Series of

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semi-theoretical fouling correlations were developed, which can be applicable to different internally ribbed geometries. The correlations can be directly used to assess the fouling potential of enhanced

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tubes in actual cooling tower water situations.

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