Generalized Reynolds Number and Viscosity Definitions for Non-Newtonian Fluid Flow in Ducts of Non-Uniform Cross-Section

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1 2 3 4 5 6 7 8 9 Generalized Reynolds number and viscosity definitions 10 11 for non-Newtonian fluid flow in ducts of non-uniform 12 13 cross-section 14 15 16 D. Cresp´ı-Llorensb, P. Vicenteb, A. Viedmaa 17 a 18 Dep. Ing. Térmica y de Fluidos. Universidad Politécnica de Cartagena. Dr. 19 Fleming, s/n (30202). Cartagena (Spain). 20 bDep. Ing. Mecánica y Energ´ıa. Universidad Miguel Hernández. Av. Universidad, s/n 21 (03202). Elche (Spain). [email protected] 22 23 24 25 26 Abstract 27 28 29 This work presents and experimental study of the generalization method 30 31 of the Reynolds number and the viscosity of pseudoplastic fluid flow in ducts 32 of non-uniform cross-section. This method will permit to reduce 1 degree of 33 34 freedom of hydrodynamical and thermal problems in those ducts. A review 35 36 of the state of the art has been undertaken and the generalization equation 37 38 proposed for ducts of uniform cross section has been used as a starting point. 39 40 The results obtained with this equation have not been found satisfactory and 41 42 a new one has been proposed. 43 44 Specifically, the procedure has been developed for two models of scraped 45 46 surface heat exchanger with reciprocating scrapers. For both models, the 47 48 scraper consists of a concentric rod inserted in each tube of the heat ex- 49 50 changer, mounting an array of plugs that fit the inner tube wall. The two 51 52 models studied differ in the design of the plug. 53 54 The procedure to perform the generalization method out of experimental 55 data is accurately detailed in the present document. 56 57 58 59 Preprint submitted to Journal name February 6, 2015 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 Keywords: non Newtonian, viscosity, generalized Reynolds number, 10 11 non-uniform cross-section, Power Law 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 2 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 1 Nomenclature 10 11 n 12 2 m flow consistency index (rheological property), [Pa.s ] 13 14 15 3 D inner diameter of the heat exchanger pipe, [m] 16 17 4 D inner diameter of the viscometer pipe, [m] 18 v 19 20 5 d diameter of the insert device shaft, [m] 21 22 23 6 Dh hydraulic diameter Dh = D d, [m] 24 − 25 26 7 Lp pipe length between pressure ports of test section, [m] 27 28 8 29 Lv viscometer pipe length between pressure ports, [m] 30 31 9 N number of measures for each experiment 32 33 34 10 P pitch of the insert devices, [m] 35 36 37 11 p pressure, [Pa] 38 39 40 12 pL pressure drop by length unit, [Pa/m] 41 42 3 13 Q flow rate, [m /s] 43 44 2 45 14 S main cross-section, [m ] 46 47 48 15 ub bulk velocity, [m/s] 49 50 51 16 Dimensionless numbers 52 53 54 17 n flow behaviour index (rheological property) 55 56 18 Re Reynolds number, Re = ρubDh/µ 57 58 59 3 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 19 2 f Fanning friction factor, f = ∆pDh/2Lρub 10 11 12 20 ξ pressure drop constant dependent on the duct geometry 13 14 15 21 a to e correlation constants 16 17 22 Greek Symbols 18 19 −1 20 23 α exponent of Reb in experimental correlations, [s ] 21 22 −1 23 24 γ shear rate, [s ] 24 25 25 µ fluid viscosity (exact definition indicated by the subindex), [Pa.s] 26 27 28 26 φ function of n 29 30 3 31 27 Ψ unknown function, [kg/m ] 32 33 28 3 34 ρ fluid density, [kg/m ] 35 36 29 τ shear stress, [Pa] 37 38 39 30 Subscripts 40 41 42 31 b Reynolds number or viscosity defined by Eq. 2 43 44 32 g Reynolds number or viscosity defined by Eqs. 18 and 19 45 46 47 33 MR defined by Metzner and Reed (1955) (Eqs. 4and 5) 48 49 50 34 DL defined using the equation from Delplace and Leuliet (1995) (Eqs. 51 52 35 6 and 7) 53 54 36 ξ = an generalization based on pressure drop in annulus, where ξ is ob- 55 56 37 tained from Kozicki et al. (1966)) 57 58 59 4 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 38 ξ = exp ξ in Eqs. 6 and 7 is obtained by experimental correlation 10 11 12 39 v belonging to the viscometer 13 14 15 40 w at the inside pipe wall 16 17 18 41 1. Introduction 19 20 21 42 Many fluids in the food and chemical or petrochemical industries are 22 23 43 non-Newtonian. In such applications the determination of parameters such 24 25 44 as the friction factor and the Nusselt number is necessary for the calculation 26 27 45 of pressure losses and heat transfer rates or temperature distributions in heat 28 29 46 exchangers. This can be achieved experimentally or theoretically by solving 30 47 31 the appropriate transport equations for typical common geometries (circular 32 48 ducts, flat ducts, etc.). An important characteristic of these fluids is that 33 34 49 they have large apparent viscosities; therefore, laminar flow conditions occur 35 36 50 more often than with Newtonian fluids. 37 38 51 Pseudoplastic fluids are the most common non-Newtonian fluids in the 39 40 52 process industry Chhabra and Richardson (2008); Cancela et al. (2005). For 41 42 53 this fluids, in a certain range of shear stress, the viscosity decreases as shear 43 44 54 stress increases. To describe this behaviour, various mathematical models 45 46 55 can be used. Among them, the Power Law model is widely used because of 47 48 56 its simplicity. The model can be used to explain the viscosity of a specific 49 50 57 fluid in a limited range of shear rates. The Power Law model (Eq. 1) has two 51 58 52 parameters: the flow behaviour index n and the flow consistency index m. 53 59 Thus, the hydrodynamic and thermal problems have one additional degree 54 55 56 57 58 59 5 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 60 of freedom, which increases their complexity. 10 11 n 12 τ = mγ (1) 13 14 61 For example, let us consider the study of pressure drop in fully developed 15 16 62 flow in pipes for forced convection. The list of significant variables can be 17 18 63 pL = Ψ(D,ub, ρ, m, n). Through the Pi Theorem the problem simplifies to 19 20 64 three non-dimensional numbers f = Ψ(Re, n). Consequently, the relation 21 22 65 between Re and the friction factor will be different for fluids with different 23 24 66 n. With the previous list of variables, the Reynolds number for power law 25 67 fluids would be, 26 2−n n 27 ρub D ρubD Reb = = (2) 28 m µb 29 n−1 30 68 , where viscosity would be defined by µb = m(ub/D) . Other viscosity 31 32 69 definitions, with the same dimensional equations, are possible and will be 33 34 70 more useful for the study of pressure drop in heat exchangers. 35 71 Metzner and Reed (1955) where the first to use the so called generalization 36 37 72 method. They analytically obtained the relation between the friction factor 38 39 73 f and the Reynolds number Reb for the fully developed laminar flow in a 40 41 74 pipe. Then, they defined a new Reynolds number ReMR, being the one 42 43 75 which multiplied by the friction factor gave the same result that the one 44 45 76 given by a Newtonian fluid. 46 47 48 49 f ReMR =16 (3) 77 × 50 ρu2−nDn ρu D 51 b b ReMR = n−1 n = (4) 52 m 8 ((3n + 1)/(4n)) µMR 53 78 , being the generalized viscosity for the flow in pipes 54 − 55 n 1 n ub n−1 3n +1 56 µMR = m 8 (5) 57 Dh 4n 58 59 6 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 79 Kozicki et al. (1966) obtained a relation between friction factor and Reynolds 10 11 80 number for various simple geometries (circular pipes, parallel plates, concen- 12 13 81 tric annuli and rectangular, isosceles triangular and elliptical ducts) as a 14 15 82 function of two parameters. Afterwards, Delplace and Leuliet (1995) re- 16 17 83 duced those parameters to one. Therefore, the definition of Metzner and 18 19 84 Reed (1955) can be applied to geometries with uniform cross-section as a 20 21 85 function of a single geometric constant. 22 2−n n ρub Dh 23 ReDL = n (6) 24 n−1 24n+ξ m ξ (24+ξ)n 25 86 × 26 n−1 n ub n−1 24n + ξ 27 µDL = m ξ (7) Dh (24 + ξ)n 28 87 29 f ReDL =2ξ (8) 30 × 31 88 For duct geometries of uniform cross-section different from the ones stud- 32 33 89 ied by Kozicki et al. (1966), similar relations can be obtained either exper- 34 35 90 imentally or numerically. This simplification leads to significant reduction 36 37 91 in the study cases of a particular problem. This has been called a general- 38 39 92 ization method because it allows to express the pressure drop behaviour of 40 41 93 Newtonian and non-Newtonian fluids with a single curve. Consequently, the 42 43 94 Reynolds number and viscosity defined by this method are known as the gen- 44 45 95 eralized Reynolds number and the generalized viscosity (Kaka¸cet al., 1987; 46 47 96 Chhabra and Richardson, 2008). Besides, the generalized viscosity can be 48 49 97 used to generalize other dimensionless numbers such as the Prandtl number 50 51 98 in non isothermal flows (Hartnett and Kostic, 1985; Delplace and Leuliet, 52 99 53 1995). 54 100 The described method has been used by many authors until recent days 55 56 101 (Gratao et al., 2006, 2007; Giri and Majumder, 2014). But, as mentioned 57 58 59 7 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 102 before, it can only be applied to ducts with uniform cross-section, where the 10 11 103 shear-stress at the wall is uniform along the duct. 12 13 104 Enhanced heat exchangers EHE (Hong and Bergles, 1976; Marner and 14 15 105 Bergles, 1985) are widely used in the process industry in order to enhance 16 17 106 heat transfer and they work often with non-Newtonian fluids. Webb (2005) 18 19 107 classified enhancement techniques into active, if they require external power, 20 21 108 and passive, if they do not. Active techniques as scraped surface heat ex- 22 109 23 changers SSHE are specially designed to avoid fouling and enhance heat 24 110 transfer. This last kind of enhanced heat exchanger is specially useful for the 25 26 111 work with non-Newtonian fluids because of their high viscosity (Nazmeev, 27 28 112 1979). In most EHE designs, specially in SSHE, the cross-section varies along 29 30 113 their length or else the cross-section is uniform but complex and has not pre- 31 32 114 viously been studied. Therefore, the generalization method must be based 33 34 115 on experimental or numerical results and it is not straightforward. 35 36 116 To overcome this inconvenience, most authors have considered their geom- 37 38 117 etry to be very similar to one of the simple uniform cross-section geometries 39 40 118 studied by Kozicki et al. (1966) or Metzner and Reed (1955). This is the case 41 42 119 of corrugated pipes or pipes with wire coil or twisted tape inserts. Manglik 43 44 120 et al. (1988); Oliver and Shoji (1992); Patil (2000); Mart´ınez et al. (2014) 45 121 46 took this option for their studies of passive EHE performance with non- 47 122 Newtonian fluids and Igumentsev and Nazmeev (1978) did so for his study 48 49 123 of SSHE. However, there are complex geometries where this assumption is 50 51 124 not valid at all. For those cases, Delplace and Leuliet (1995) proposed the 52 53 125 use of experimental methods to obtain the value of ξ. Based on the previous 54 55 126 research of Rene et al. (1991), they proposed to use ξ = 56.6 for a plate heat 56 57 58 59 8 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 127 exchanger type. Afterwards some other researchers have broaden Rene et al. 10 11 128 (1991) and Delplace and Leuliet (1995) studies with numerical simulations in 12 13 129 the same plate heat exchangers model (Fernandes et al., 2007, 2008) varying 14 15 130 some design parameters. 16 17 131 Our extensive literature search has not yielded further researches about 18 19 132 the generalization method of viscosity in complex geometries with non-uniform 20 21 133 cross section. In view of this situation the present study was undertaken. 22 134 23 The present paper presents a simplified generalization method for the 24 135 Reynolds number and fluid viscosity, based on the studies of Metzner and 25 26 136 Reed (1955) and Delplace and Leuliet (1995), which can be applied to ducts 27 28 137 of non-uniform cross-section. In order to prove its validity, pressure drop 29 30 138 has been measured experimentally in two different pipe axial reciprocating 31 32 139 scraped surface heat exchangers AR-SSHE. These geometries are shown in 33 34 140 Fig. 1. 35 36 37 141 2. Experimental Set-up 38 39 40 142 The experimental setup shown in Fig. 2 has been used to measure pressure 41 42 143 drop for different flow regimes in axial reciprocating scraped surface heat 43 44 144 exchangers (Fig. 1(a) and Fig. 1(b)). The experimental facility consists of 45 46 145 two independent circuits. The primary circuit, which contains the test fluid, 47 48 146 is divided in two sub-loops. The test section is placed in the main one, 49 50 147 including a gear pump (2) driven by a frequency controller (3). The test 51 52 148 fluid in the supply tank (1) is continuously cooled in the second sub-loop 53 54 149 through a plate heat exchanger (13) with a coolant flow rate settled by a 55 150 56 three-way valve (15). The coolant liquid of the secondary circuit is stored in 57 58 59 9 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 (a) EG1 geometry of an scraped surface heat exchanger. 31 32 33 34 35 36 37 38 39 40 41 42 (b) EG2 geometry of an scraped surface heat exchanger. 43 44 45 Figure 1: Analysed geometries. 46 47 48 49 50 51 52 53 54 55 56 57 58 59 10 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Figure 2: Experimental set-up. (1) Test fluid tank, (2, 12) gear pumps, (3) frequency 42 43 converter, (4) immersion resistance, (5) Coriolis flowmeter, (6) RTD temperature sensor, 44 (7) pressure transmitter, (8) stainless steel tube with an insert scraper and with inlet 45 46 and outlet immersion RTDs, (9) pressure ports, (10) smooth stainless steel pipe used as 47 viscometer, (11) hydraulic piston (14, 17) centrifugal pumps, (15) three-way valve with a 48 49 PID controller, (16) coolant liquid tank, (18) cooling machine. 50 51 52 53 54 55 56 57 58 59 11 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 151 a 1000 l tank (16) from where it flows to a cooling machine. The thermal 10 11 152 inertia of this tank, with a capacity of 1000 l, together with the operation of 12 13 153 the PID-controlled three way valve provides stability to the temperature of 14 15 154 the test fluid in the supply tank, which can be accurately fixed to a desired 16 17 155 value. The test section was placed in the main circuit and consisted of a 18 19 156 thin-walled, 4 m long, 316L stainless steel tube with an insert scraper. The 20 21 157 inner and outer diameters of the tube were 18 mm and 20 mm, respectively. 22 158 23 Two oversize, low-velocity gear pumps (one on each circuit) were used for 24 159 circulating the working fluid, in order to minimize fluid degradation during 25 26 160 the tests. Mass flow rate and fluid density was measured by a Coriolis flow 27 28 161 meter, which performs properly when working with non-Newtonian fluids 29 ◦ 30 162 (Fyrippi et al., 2004). Four pressure taps separated by 90 were coupled to 31 32 163 each end of the pressure test section of 1.85 m length. A long test section has 33 34 164 been used to improve measurement precision. Pressure drop ∆pE1 and ∆pE2 35 36 165 was measured by means of two highly accurate pressure transmitters LD-301 37 38 166 configured for different ranges. Pressure measurement ports were separated a 39 167 distance Lp = 20 P , and consisted of four pressure holes peripherally spaced 40 × 41 ◦ 168 by 90 . Test section was preceded by a development region of Le = 6 P 42 × 43 44 169 length, in order to establish periodic flow conditions. 45 170 46 The rheological properties of the non-Newtonian test fluid n and m is 47 171 measured by an in-line viscometer, parallel to the testing tube. In that way, 48 49 172 measurements of the rheological properties could be done at the beginning 50 51 173 and at the end of each set of experiments, minimizing the thixotropy effect. 52 53 174 Further details are given in next section. 54 55 175 Further details of the working apparatus and the calibration procedure 56 57 58 59 12 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 176 are given in Solano et al. (2011) and Garc´ıaet al. (2005). 10 11 12 177 2.1. Test fluid characteristics 13 14 178 The test fluid was 1% wt aqueous solutions of carboxymethyl cellulose 15 16 179 (CMC), supplied by SigmaAldrich Co. CMC with different chain length 17 18 180 have been used: medium viscosity (ref. C4888, 250 kDa), high viscosity (ref. 19 20 181 C5013, 700 kDa) and ultra high viscosity (ref. 21904). The solutions were 21 22 182 prepared by dissolving the polymer powder in distilled water and then raising 23 24 183 the pH values of the solution to increase viscosity. This fluid shows a non- 25 26 184 Newtonian pseudoplastic behaviour well described by the Power Law model 27 28 185 of Eq. 1 for a big range of shear rates (Abdelrahim and Ramaswamy, 1995; 29 30 186 Ghannam and Esmail, 1996; Abu-Jdayil, 2003; Yang and Zhu, 2007), al- 31 −1 187 though it presents a Newtonian plateau for shear rates under 0.1 s (Bench- 32 33 188 abane and Bekkour, 2008). 34 35 189 All CMC thermophysical properties but the rheological parameters and 36 37 190 fluid density were assumed to be the same as pure water (Chhabra and 38 39 191 Richardson, 2008; Cancela et al., 2005). 40 41 192 Rheological fluid properties are strongly influenced by the type of CMC 42 43 193 powder employed, the preparation method and fluid degradation due to shear 44 45 194 stress and thermal treatment. The combination of those factors allows to 46 47 195 obtain fluids with different pseudoplastic behaviour, ranging from n = 0.45 48 49 196 to n = 1. 50 51 197 The values of n and m for the test fluid were obtained by using the in-line 52 198 53 smooth pipe as a viscometer. In the smooth pipe, flow rate Q and pressure 54 199 drop ∆p are measured 20 times for four different flow rates. Bulk velocity 55 56 200 ub and shear stress at the wall τw are obtained out of flow rate and pressure 57 58 59 13 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 101 10 Adjustment 11 Exp. data 12 13 100 14 f 15 16

17 Fanning, − 10 1 n = 0.65 18 m = 0.802 19 20 21 10−2 22 0 1 2 3 4 10 10 10 10 10 23 24 ReMR 25 26 27 Figure 3: Rheological properties measurement during one of the test. 28 29 30 201 31 drop respectively ∆pD 32 τ = v (9) 33 w 4L 34 35 202 , as the velocity profile of the fully developed isothermal flow of power law 36 37 203 fluids in pipes is well known, τw can also be derived from the constitutive 38 39 204 Eq. 1, 40 41 n 42 8ub 3n +1 43 τw = m (10) Dv 4n 44 45 205 , whose logarithm yields 46 47 8ub 3n +1 48 ln(τw)= n ln(ub)+ ln(m)+ n ln (11) 49 × × Dv 4n 50 51 206 , out of which the rheological properties m and n can be obtained by ad- 52 53 207 justing the experimental data with a least squared method. An example of 54 55 208 a rheological measurement result is shown in Fig. 3. 56 57 58 59 14 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 209 Because of fluid degradation, rheological properties must be obtained fre- 10 11 210 quently. Experiments are planned in sets of 15 to 25 and rheological proper- 12 13 211 ties are obtained before and after each set. A maximum of 3% deviation be- 14 15 212 tween rheological properties measurements has been obtained. Degradation 16 17 213 between measurements has been supposed to be linear with experimenting 18 19 214 time, so that m and n can be obtained for each experiment. 20 21 22 215 2.2. Accuracy of the experimental data 23 24 216 The experimental uncertainty was calculated by following the ”Guide to 25 26 217 the expression of uncertainty in measurement”, published by the ISO (1995). 27 28 218 On one hand, the Coriolis flowmeter has a repeatability of 0.025 % of the 29 3 30 219 flow rate measure, while its precision when measuring density is 0.2 kg/m . 31 220 32 On the other hand, pressure sensor has a repeatability of 0.075 of its range. 33 221 Uncertainties of the heat exchanger and viscometer dimensions have been as- 34 35 222 signed according to the measuring tool employed. The uncertainty associated 36 37 223 to rheological properties are obtained out of the least squares adjustment. 38 39 224 The maximum uncertainty of n and m are 0.01% and 0.4% respectively. 40 41 225 A summary of the uncertainties of dimensions and sensor measurements 42 43 226 is shown in Table 1. The resulting error for Reb and f are of 1.2% and 2% 44 45 227 respectively. 46 47 48 228 3. Results 49 50 51 229 Friction factor measurements in EG1 and EG2 geometries are plotted in 52 53 230 Fig 4 versus Reb defined by Eq. 2. As it can be appreciated the friction factor 54 55 231 f is a function of Reb and the flow behaviour index n. 56 57 58 59 15 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 (a) Dimensions. 18 19 Variable Value Uncertainty Units Uncert. (%) 20 √ 21 D 18 0.05/ 3 mm 0.2 22 d 5 0.05/√3 mm 0.6 23 24 D 13 0.04 mm 0.3 25 h 26 S 234.8 0.7 mm2 0.4 27 28 Dv 16 0.05/√3 mm 0.2 29 2 30 Sv 201.1 0.8 mm 0.4 31 32 Lv 1885 0.5/√3 mm 0.02 33 34 Lp 1850 0.5/√3 mm 0.02 35 36 (b) Sensors measurements. N is the number of measurements. 37 38 Variable Value N Uncertainty Units Max. Uncert. (%) 39 40 ∆pE1 10 405 20 10 0.07 0.09 mbar 0.7 41 − − − 42 ∆pE 400 2500 10 0.6 mbar 0.1 2 − 43 − 44 Q 30 2000 10 20 - kg/h 7.9 10 3 45 − − × 3 46 ρ 1000 1 0.1 kg/m 0.01 47 48 Table 1: Uncertainties in dimensions and sensor measurements. 49 50 51 52 53 54 55 56 57 58 59 16 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 103 14 n = 0.45 15 n = 0.66 2 n = 0.97 16 10 n = 1.00 17

18 f 19 101 20

21 Fanning, 22 100 23 24 25 10−1 26 10−1 100 101 102 103 27 Reb 28 29 30 (a) EG1

31 3 32 10 n = 0.50 33 n = 0.64 34 2 n = 1 35 10 36 37 f 1 38 10 39 40 Fanning, 41 100 42 43 44 10−1 − 0 2 3 45 10 1 10 101 10 10 46 Reb 47 48 (b) EG2 49 50 51 Figure 4: Reb versus Fanning friction factor for the geometries under study. Only most 52 representative results are shown. 53 54 55 56 57 58 59 17 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 232 3.1. Generalization based on annulus geometry 10 11 233 Geometries of EG1 and EG2 scraped surface heat exchangers are similar 12 13 234 to an annular passage. Therefore, a generalization method based on the 14 15 235 annulus geometry may be a good approach for these cases. For this value of 16 17 236 the radius ratio (d/D =5/18), the value of ξ = 11.69 can be obtained from 18 19 237 Kozicki et al. (1966). 20 21 238 In Fig. 5, f Reb versus n has been plotted for the experiments and for 22 × 23 239 the solution in annulus given by Eq. 6 with ξ = 11.69. As it can be observed, 24 25 240 annulus results underpredict experimental ones in 34% on average for EG1 26 27 241 and in 27% on average for EG2. Pressure drop results are shown in Fig. 6 28 242 29 and Fig. 7, where the generalized Reynolds number has been defined for 30 243 the mentioned value of ξ. As it can be observed, the results show different 31 32 244 curves for each fluid with different value of n and measurements do differ 33 34 245 from the theoretical solution in annulus. Therefore it can be concluded that 35 36 246 the generalization method is not valid in these cases. 37 38 247 However, some useful information can be obtained from Figures 6 and 39 40 248 7. For ReDL,ξ=an < 100 the flow is laminar and above this range the tran- 41 42 249 sitional flow starts. Besides, it can be observed that the distance between 43 44 250 experimental results and the line representing the annulus solution varies 45 −1 46 251 with ReDL,ξ=an, meaning that f Reb . 47 6∝ 48 49 252 3.2. Experimental value of ξ 50 51 253 In this subsection, the solution suggested by Delplace and Leuliet (1995) 52 53 254 has been used for the generalization method. They proposed to use Eq. 8, 54 55 255 what has been modified to include an exponent for the Reynolds number, α, 56 57 58 59 18 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 50 14 Exp. Data 15 Annulus 16 40 17 18 30 19 b 20 fxRe 21 20 22 23 10 24 25 0 26 0.4 0.5 0.6 0.7 0.8 0.9 1 27 n 28 29 30 (a) EG1 31 32 50 33 Exp. Data Annulus 34 40 35 36 30

37 b 38

39 fxRe 20 40 41 42 10 43 44 0 45 0.4 0.5 0.6 0.7 0.8 0.9 1 46 n 47 48 49 (b) EG2 50 51 Figure 5: Comparison of f Reb between experimental results and theoretical results for × 52 annulus. 53 54 55 56 57 58 59 19 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 3 14 10 15 n = 0.45 n = 0.66 16 2 n = 0.97 17 10 n = 1.00 18 Annulus 19 f 20 101 21 22 Fanning, 23 100 24 25 26 10−1 27 10−1 100 101 102 103

28 ReDL,ξ=an 29 30 31 (a) Whole Reynolds range 32 1 10 33 n = 0.45 34 n = 0.66 35 n = 0.97 36 n = 1.00 Annulus

37 f

38 0 39 10 40 Fanning, 41 42 43 44 10−1 45 1 2 3 46 10 10 10 47 ReDL,ξ=an 48 49 (b) Narrower Reynolds range 50 51 Figure 6: EG1. Re = versus Fanning friction factor. 52 DL,ξ an 53 54 55 56 57 58 59 20 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 3 14 10 15 n = 0.50 n = 0.64 16 2 n = 1 17 10 Annulus 18 19 f 20 101 21 22 Fanning, 23 100 24 25 26 10−1 27 10−1 100 101 102 103

28 ReDL,ξ=an 29 30 31 (a) Whole Reynolds range 32 1 10 33 n = 0.50 34 n = 0.64 35 n = 1 36 Annulus

37 f

38 0 39 10

40 Fanning, 41 42 43 44 10−1 45 1 2 3 46 10 10 10 47 ReDL,ξ=an 48 49 (b) Narrower Reynolds range 50 51 Figure 7: EG2. Re = versus Fanning friction factor. 52 DL,ξ an 53 54 55 56 57 58 59 21 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 256 as it has been explained in previous section, 10 11 n α n 24n + ξ 12 f Reb =2ξ (12) 13 × (24 + ξ)n 14 15 257 The experimental data has been correlated to obtain the value of ξ in 16 17 258 Eq. 12. For this, only experiments with Reynolds numbers under 40 (highly 18 19 259 laminar region) have been considered. The reason for doing this is that, 20 21 260 although laminar region ends at Reynolds number about 100, the exponent 22 23 261 of the Reynolds number in Eq. 12 decreases with the Reynolds number along 24 25 262 the laminar region, what becomes significant for Reynolds numbers above 26 27 263 40. The goal is not to obtain an experimental correlation for the data in the 28 264 laminar region but to obtain a proper definition of the generalized Reynolds 29 30 265 number, valid for the whole laminar region. 31 32 266 The experimental values of ξ for EG1 and EG2, and the corresponding 33 1 34 267 uncertainties for a confidence level of 95 % are shown in Table 2 . The exper- 35 36 268 imental data and Eq. 12 with the calculated values of ξ for both geometries 37 38 269 are plotted in Fig. 8. Besides, friction factor versus the generalized Reynolds 39 40 270 number (with this ξ) is plotted in Fig. 9. 41 42 43 Table 2: Experimental correlation for ξ in Eq. 12 (Delplace and Leuliet, 1995). 44 45 α ξ Error 46 47 EG1 0.974 19.38 17.0% 48 49 EG2 0.951 16.67 13.3% 50 51 52 53 54 1The procedure to obtain α is explained in section 3.4. 55 56 57 58 59 22 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 50 14 Exp. Data 15 ξ = 19.38 40 16 17

18 b 30 19 − b 20 fxRe 20 21 22 23 10 24 25 0 26 0.4 0.5 0.6 0.7 0.8 0.9 1 27 n 28 29 30 (a) EG1 31 32 50 Exp. Data 33 ξ = 16.67 34 40 35 36

37 b 30 − b 38

39 fxRe 20 40 41 42 10 43 44 0 45 0.4 0.5 0.6 0.7 0.8 0.9 1 46 n 47 48 (b) EG2 49 50 51 Figure 8: Comparison between experimental results and Eq. 12 with the experimental 52 values of ξ (see Table 2) 53 54 55 56 57 58 59 23 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 103 14 n = 0.45 15 n = 0.66 2 n = 0.97 16 10 n = 1.00 17 18 f 19 101 20

21 Fanning, 22 100 23 24 25 10−1 26 10−1 100 101 102 103 27 ReDL,ξ=exp 28 29 30 (a) EG1

31 3 32 10 n = 0.50 33 n = 0.64 34 2 n = 1 35 10 36 f 37 1 38 10 39 40 Fanning, 41 100 42 43 44 10−1 −1 0 1 2 3 45 10 10 10 10 10 46 ReDL,ξ=exp 47 48 (b) EG2 49 50 51 Figure 9: Generalized Reynolds number with Eq. 12 and the experimental values of ξ (see 52 Table 2). 53 54 55 56 57 58 59 24 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 271 α Results in Fig. 8 show an under prediction of the product f Reb for 10 × 11 272 n 0.45 and n 1 for both geometries. Furthermore, it can be observed in 12 ≈ ≈ 13 273 Fig. 9 that the experimental results for different n do not collapse to the same 14 15 274 curve. This effect is higher in EG1 geometry, where the flow is significantly 16 17 275 different from the annulus geometry. Results of this generalization method 18 19 276 are still not satisfactory. 20 21 22 277 3.3. Proposed experimental correlations 23 α 24 278 At this point, an experimental correlation for f Reb must be obtained 25 × 26 279 in order to apply the generalization method properly. With this objective, 27 28 280 different expressions will be tested: 29 30 281 31 1. Expression with two parameters, 32 − 33 f Reα = acn 1 (13) 34 × b 35 36 282 2. Expression with three parameters, 37 38 39 f Reα = acn−1 nd (14) 40 × b 41 42 283 3. Expression with four parameters, 43 44 c n2 + d n + e n 45 f Reα = a (15) 46 × b (c + d + e)n2 47 48 284 , where a, c, d and e are correlation constants (the letter b has been omitted 49 50 285 to avoid confusion). As in previous section, the exponent of the Reynolds 51 52 286 number α has been included due to the peculiar nature of the AR-SSHE, 53 54 287 where the flow does not exactly behave as in a uniform cross section geometry. 55 56 57 58 59 25 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Table 3: Correlation results. 17 18 19 (a) EG1. 20 21 Ec. 13 Ec. 14 Ec. 15 22 23 α 0.974 0.974 0.974 24 25 a 39.742 41.403 41.729 26 27 c 15.536 262.27 212.8 28 d -2.1177 -319.16 29 30 e 158.93 31 32 Error (%) 15.9 11.4 9.6 33 34 (b) EG2. 35 36 Ec. 13 Ec. 14 Ec. 15 37 38 α 0.951 0.951 0.951 39 40 a 33.786 34.070 34.078 41 42 c 12.574 80.555 75.852 43 44 d -1.4419 -95.224 45 46 e 50.102 47 48 Error (%) 12.7 9.0 9.0 49 50 51 52 53 54 55 56 57 58 59 26 60 61 62 63 64 65 1 2 3 4 5 6 7 8 1 9 288 The results of the different approaches can be seen in Table 3 . The three 10 11 289 correlations proposed perform better than the one proposed by Delplace and 12 13 290 Leuliet (1995). The lower error corresponds to Eq. 15 followed by Eq. 14, 14 15 291 both presenting good agreement with experimental data. Both correlations 16 17 292 are plotted in Fig. 10 versus experimental results. 18 19 293 To our understanding, Eq. 14 offers a good approach to experimental 20 21 294 results with just three parameters, two of which will appear in the generalized 22 295 23 viscosity definition. 24 296 In order to define a generalized Reynolds number and viscosity, according 25 26 297 to Delplace and Leuliet (1995) φ(1)=1 in Eq. 16, so 27 28 Reb 29 Reg = (16) 30 φ(n) 31 32 298 , consequently 33 n−1 d 34 φ(n)= c n (17) 299 35 2−n n 36 ρub D Reg = (18) 37 n−1 d 300 mc n 38 n−1 39 n−1 d ub µg = mc n (19) 40 Dh 41 42 301 Pressure drop results are shown in Fig. 11 and Fig. 12 with the generalized 43 44 302 Reynolds number defined by Eq. 18. The figure shows as the experimental 45 46 303 data for fluids with different pseudoplastic behaviour (different n) can be rep- 47 304 resented with a single curve in the laminar flow region, while some differences 48 49 305 arise in transition flow region. 50 51 306 The proposed generalization method has more parameters than the equa- 52 53 307 tion from Delplace and Leuliet (1995), but correlates better with the ex- 54 55 56 1The procedure to obtain α is explained in section 3.4. 57 58 59 27 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 50 15 Exp. Data Eq. 15 16 40 Eq. 14 17 18 30

19 α b 20

21 fxRe 20 22 23 24 10 25 26 0 27 0.4 0.5 0.6 0.7 0.8 0.9 1 28 n 29 30 31 (a) EG1 32 50 33 Exp. Data 34 Eq. 15 35 40 Eq. 14 36 37 30 38 α b 39 40 fxRe 20 41 42 10 43 44 45 0 46 0.4 0.5 0.6 0.7 0.8 0.9 1 47 n 48 49 (b) EG2 50 51 52 Figure 10: Comparison between experimental results and experimental correlations. 53 54 55 56 57 58 59 28 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 103 16 n = 0.45 17 n = 0.66 2 n = 0.97 18 10 n = 1.00 19

20 f 21 101 22

23 Fanning, 24 100 25 26 27 10−1 28 10−1 100 101 102 103 29 Reg 30

31 1 32 10 n = 0.45 33 n = 0.66 34 n = 0.97 35 n = 1.00 36 f

37 0 38 10 39 Fanning, 40 41 42 43 10−1 44 1 2 3 45 10 10 10 46 Reg 47 48 49 Figure 11: EG1. Generalized Reynolds number Reg (Eq. 18) versus Fanning friction 50 factor. 51 52 53 54 55 56 57 58 59 29 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 103 16 n = 0.50 17 n = 0.64 n = 1 18 102 19

20 f 21 101 22

23 Fanning, 24 100 25 26 27 10−1 28 10−1 100 101 102 103 29 Reg 30

31 1 32 10 n = 0.50 33 n = 0.64 34 n = 1 35 36 f

37 0 38 10 39 Fanning, 40 41 42 43 10−1 44 1 2 3 45 10 10 10 46 Reg 47 48 49 Figure 12: EG2. Generalized Reynolds number Reg (Eq. 18) versus Fanning friction 50 factor. 51 52 53 54 55 56 57 58 59 30 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 308 perimental data, being still very simple (see Table 2 and Table 3). This 10 11 309 generalization method allows to reduce complexity in hydrodynamical prob- 12 13 310 lems, where the dependence of n is included in the viscosity definition. The 14 15 311 method can be followed in similar heat exchangers devices in order to obtain 16 17 312 a valid expression for the generalized viscosity and the generalized Reynolds 18 19 313 number. 20 21 314 The expression obtained for the generalized Reynolds number and vis- 22 315 23 cosity (Eq. 18 and Eq. 19) will be valid for this design of heat exchanger, 24 316 working with any non Newtonian fluid whose behaviour can be modelled 25 26 317 with the Power Law model. Obviously, care must be taken that the values 27 28 318 of m and n, obtained for the working fluid, are valid in the working range of 29 30 319 shear stress. 31 32 33 320 3.4. Additional comments on the experiments and the correlations obtained 34 35 321 A total of 161 experiments for the EG1 geometry and 101 for EG2 have 36 37 322 been carried out. Those experiments belong to laminar, transition and tur- 38 39 323 bulent regions. For all the correlations in this work, only experiments under 40 41 324 Reg < 40 have been used to ensure they belong to the laminar region. All 42 43 325 the experiments with Reg < 40 are represented in Figures 5, 8 and 10. As, 44 45 326 at first, the definition of Reg is unknown, the first selection has been done by 46 47 327 using ReDL,ξ=an < 40 and corrected with Reg at the end if necessary. The 48 49 328 number of experiments which satisfy the previous condition are 61 and 47 50 51 329 for EG1 and EG2 geometries respectively. In spite of the restriction imposed 52 330 53 (Reg < 40), in Figures 11 and 12 it can be appreciated that the behaviour 54 331 of the different fluids in the AR-SSHE is represented by a single curve in the 55 56 332 whole laminar region (Reg < 100). This means that the generalized Reynolds 57 58 59 31 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 333 number and viscosity definitions are valid in that range. 10 11 334 In order to perform correlations of equations 12, 13, 14 and 15, the expo- 12 13 335 nent of the Reynolds number α has been obtained first. For that, a correlation 14 15 336 for the friction factor has been obtained in f = Ψ(Reb, n) as indicated by 16 17 337 each equation. Afterwards, that value of the exponent α has been used to 18 α 338 correlate f Re as a function of n according to each equation. This proce- 19 × b 20 21 339 dure allows to minimize the correlating error due to high scaling differences 22 340 23 in the Fanning friction factor. 24 25 26 341 4. Conclusions 27 28 29 342 In this work, a generalization method in ducts of non uniform cross- 30 31 343 section has been presented and experimentally evaluated in two commercial 32 33 344 scraped surface heat exchangers. 34 35 345 Pressure drop of a pseudoplastic non-Newtonian fluid has been ex- 36 • 37 346 perimentally determined in two scraped surface heat exchangers (EG1 38 39 347 and EG2) in static conditions. Experiments have been carried out in 40 41 348 a wide range of Reynolds numbers Reg = [0.3, 600] and with Newto- 42 43 349 nian and non-Newtonian fluids with different degree of pseudoplasticity 44 45 350 n = [0.45, 1]. 46 47 351 The performance of a generalization method based on the annulus ge- 48 • 49 50 352 ometry has been tested and found inadequate. Theoretical results for 51 α 353 f Re in annulus underestimates the experimental data on average in 52 × b 53 354 34% and in 27% in geometries EG1 and EG2 respectively (in laminar 54 55 355 region Re < 40). Furthermore, the representation of the friction factor 56 g 57 58 59 32 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 356 versus the generalized Reynolds number based on the annulus geome- 10 11 357 try is still dependent on the flow behaviour index f = Ψ(n, ReDL,ξ=an) 12 13 358 in the laminar region. Therefore, this generalization is invalid. 14 15 359 As suggested by Delplace and Leuliet (1995), an experimental value of 16 • 17 18 360 ξ in Eq. 8 has been obtained using the experimental data in laminar 19 361 20 region (ReDL,an < 40). This equation correlates with an error of 17% 21 362 and 13% in geometries EG1 and EG2 respectively. Furthermore, rep- 22 23 363 resentations of f versus ReDL,ξ=exp still shows some dependence on n. 24 25 364 This solution is very simple, as it only depends on 1 parameter, but 26 27 365 the results can be improved. 28 29 30 366 A more precise and still simple correlation for the generalization method • 31 α 367 has been proposed. The proposed correlation estimates f Re with 32 × b 33 34 368 an error of 11% and 9% in geometries EG1 and EG2 respectively and 35 369 36 the representation of f versus the generalized Reynolds number with 37 370 this method Re shows no appreciable dependence on n. 38 g 39 40 371 The generalized expressions of the Reynolds number and the viscos- 41 • 42 372 ity obtained in this work are valid for their use in this specific heat 43 44 373 exchanger working with any non Newtonian Power Law fluid. 45 46 374 The generalized method proposed can be applied to similar heat ex- 47 • 48 49 375 changer designs with complex non-uniform cross sections. 50 51 52 376 5. Acknowledgements 53 54 377 The first author thanks Mr. Mart´ınez and Dr. Solano for their invaluable 55 56 378 contribution to the project and their advise. 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Cellulose . 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 39 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 474 List of Figures 10 11 12 475 1 Analysedgeometries...... 9 13 14 476 2 Experimentalset-up...... 10 15 16 477 3 Rheological properties measurement during one of the test. . . 13 17 18 478 4 Reb versus Fanning friction factor for the geometries under 19 479 20 study...... 16 21 480 5 Comparison of f Reb between experimental results and the- 22 × 23 481 oreticalresultsforannulus...... 18 24 25 482 6 EG1. ReDL,ξ=an versus Fanningfriction factor...... 19 26 27 483 7 EG2. ReDL,ξ=an versus Fanningfriction factor...... 20 28 29 484 8 Comparison between experimental results and Eq. 12 with the 30 31 485 experimental values of ξ (seeTable2)...... 22 32 33 486 9 Generalized Reynolds number with Eq. 12 and the experimen- 34 35 487 tal values of ξ (seeTable2)...... 23 36 37 488 10 Comparison between experimental results and experimental 38 39 489 correlations...... 27 40 490 41 11 EG1. Generalized Reynolds number Reg (Eq. 18) versus Fan- 42 491 ningfrictionfactor...... 28 43 44 492 12 EG2. Generalized Reynolds number Re (Eq. 18) versus Fan- 45 g 46 493 ningfrictionfactor...... 29 47 48 49 50 51 52 53 54 55 56 57 58 59 40 60 61 62 63 64 65