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Velocity and temperature profiles, heat transfer coefficients and residence time distribution of a temperature dependent Herschel–Bulkley fluid in a tubular heat exchanger
Journal of Food Engineering 76 (2006) 632–638 www.elsevier.com/locate/jfoodeng
Velocity and temperature profiles, heat transfer coefficients and residence time distribution of a temperature dependent Herschel– Bulkley fluid in a tubular heat exchanger C. Ditchfield a, C.C. Tadini a
a,*
, R.K. Singh b, R.T. Toledo
b
Department of Chemical Engineering, Sa˜o Paulo University, Escola Polite´cnica, P.O. Box 61548, 05424-970 Sa˜o Paulo-SP, Brazil b Department of Food Science and Technology, UGA, 30602-7610 Athens, GA, USA Received 21 October 2004; accepted 5 June 2005 Available online 9 August 2005
Abstract Banana puree was studied as a temperature dependent Herschel–Bulkley fluid. Experiments were conducted in a tubular heat exchanger at three flow rates (2.5 · 105, 3.7 · 105 and 4.7 · 105 m3 s1), at three steam temperatures (110.0, 121.1 and 132.2 C) and two length/diameter ratios (250 and 500). The minimum, maximum and average residence times were calculated from on-line electrical conductivity measurements. Inlet and outlet temperatures were recorded. A model was proposed to obtain heat transfer coefficients and velocity and temperature profiles in the heat exchanger. The continuity, energy and momentum equations were integrated using a finite difference method. The heat transfer coefficients varied from 654.8 to 1070.4 W m2 K1 for average velocities of 0.24–0.40 m s1. There is good agreement between experimental parameters and those predicted by the model. 2005 Elsevier Ltd. All rights reserved. Keywords: Banana puree; Heat transfer; Residence time distribution; Velocity profile; Temperature profile
1. Introduction Banana puree has a complex rheological behavior, modeled as a Herschel–Bulkley fluid, with consistency index (K), flow behavior index (n) and yield stress (r0) varying significantly with temperature (Ditchfield, Tadini, Singh, & Toledo, 2004; Guerrero, Alzamora, & Gerschenson, 1997). Equipment design depends greatly upon reliable equations for explaining heat transfer, pressure loss and energy requirements. There are some models that have been developed for the flow of non-Newtonian fluids, whereas some models have considered laminar flow of Bingham plastic fluids (Forrest & Wilkinson, 1973; Guariguata, Barreiro, & *
Corresponding author. Tel.: +55 11 3091 2258; fax: +55 11 3091 2255. E-mail address: [email protected] (C.C. Tadini). 0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.06.015
Guariguata, 1979). Other models consider laminar and turbulent flow of power law fluids with constant wall temperature and constant heat flux (Luna, Me´ndez, & Mar, 2003; Son & Singh, 2002). Herschel–Bulkley flow was modeled by some authors (Chilton & Stainsby, 1998; Nakayama, Niwa, & Hamada, 1980; Nouar, Desaubry, & Zenaidi, 1998; Sayed-Ahmed, 2000). Nakayama et al. (1980) designed a pipe system to study pressure loss in isothermal pipe transportation of minced fish paste. Minced fish paste is a Herschel–Bulkley fluid with a large yield stress value. Chilton and Stainsby (1998) presented equations for laminar and turbulent flow of Herschel–Bulkley fluids. A numerical model was discussed and compared to experimental measurements, agreeing closely in the laminar flow region, though in turbulent flow predicted pressure loss is generally 15% lower than experimental values. Nouar et al. (1998) proposed an investigation
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
633
Nomenclature C Cp d D E F h K k L M N n P DP Q R T T0 t t
concentration (kg m3) specific heat (J kg1 K1) distance from tube wall (m) tube diameter (m) exit age distribution (dimensionless) cumulative exit age (dimensionless) heat transfer coefficient (W m2 K1) consistency index (Pa sn) thermal conductivity (W m1 K1) tube length (m) K(1/n) (Pa s) 1/n (dimensionless) flow behavior index (dimensionless) pressure (Pa) pressure drop (Pa) flow rate (m3 s1) tube radius (m) temperature (K) temperature (C) time (s) average residence time (s)
of thermal convection within a thermodependent Herschel–Bulkley fluid flowing in an annular duct with rotating inner cylinder. The authors carried out experiments with a polymer (Carbopol 940) with a 0.2% mass concentration and heat transfer was influenced by the rotation speed of the inner cylinder. Sayed-Ahmed (2000) investigated numerically laminar heat transfer to a Herschel–Bulkley fluid in the entrance region of a square duct, considering fully developed velocity profile. Higher flow behavior indices (n) resulted in higher velocity and lower Nusselt number. The objectives of this study were: (1) to determine the information on residence time distributions, velocity and temperature profiles, and heat transfer coefficients; and (2) to investigate how these variables are affected by the rheological changes observed in a Herschel– Bulkley fluid (banana puree) whose parameters (r0, K and n) vary significantly with temperature, in the heating section of a double pipe tubular heat exchanger. This will provide important knowledge for equipment design in processing of temperature dependent Herschel–Bulkley fluids.
2. Materials and methods 2.1. Banana puree Banana puree was obtained from a commercial source as acidified aseptic banana puree (22.1 ± 0.9
u v V
axial velocity (m s1) velocity (m s1) volume (m3)
Greek characters c_ flow shear rate (s1) q density (kg m3) r0 yield stress (Pa) 0Þ h dimensionless temperature, h ¼ ðTðTwT T 0 Þ Subscripts 0 initial value avg average value c calculated f final value max maximum min minimum p processing s steam w wall
Brix) in 225 kg drums. The puree had been acidified with a combination of citric and ascorbic acid to an average pH of 4.49 ± 0.03, heat treated for enzyme inactivation and aseptically packaged in plastic bags within the drums. 2.2. Experimental measurements Experiments were conducted in a tubular heat exchanger with two heating sections, each 3.05 m long with an internal diameter of 1.2 · 102 m. Three flow rates (2.5 · 105, 3.7 · 105 and 4.7 · 105 m3 s1), three steam temperatures (110.0, 121.1 and 132.2 C) and two length/diameter ratios (250 and 500) were studied. For the length/diameter ratio of 250, only one heating section was considered. A total of 36 experiments were carried out, since each condition was repeated once and approximately 4 ton of banana puree were processed. Temperature was measured with thermocouples before the heating section (T1), after the first heating section (T2), at the end of the heat exchanger (T3) and steam temperature (T4) and a data logging system was used to record the temperatures every 10 s (Fig. 1). A saturated sodium chloride solution (10 mL) was injected at the inlet of the heat exchanger as a tracer for measuring residence times. A conductivity transmitter (Compak, Signet Industrial, El Monte, CA) was placed at the outlet of the heat exchanger. Two inline mixers were used after the injection port and at the outlet of the heat exchanger to ensure
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C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638 Water out Conductivity probe Puree
Steam in
T3 Mixer Feed
T2
T4
Cooling unit
Mixer Pump
T1
Heating unit
Steam out
Water in Puree
Salt injection
Fig. 1. A schematic diagram of the experimental setup used for residence time distributions and temperatures in a tubular heat exchanger.
uniform distribution of the salt, as well as a correct measurement of the average temperature at the outlet of the heat exchanger.
The boundary conditions and initial conditions are as follows: oT ou ¼ 0; ¼0 T r¼R ¼ T w ; ur¼R ¼ 0; or r¼0 or r¼0
2.3. Velocity and temperature profiles The system considered was banana puree processed in a tubular heat exchanger under laminar flow conditions. The physical and thermal properties of banana puree (density, thermal conductivity and specific heat) were considered constant and were obtained from Charm (1962). Banana puree was modeled as a Herschel–Bulkley fluid. The temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) was determined for temperatures between 30 C and 120 C using a dynamic stress controlled rheometer (model SR 5000, Rheometric Scientific, Piscataway, NJ) (Ditchfield et al., 2004). The problem can be described by the following equations with the assumptions of steady laminar flow, incompressible fluid, constant wall temperature, negligible viscous dissipation, small axial conduction relative to radial conduction and insignificant free convection. The continuity, momentum and energy equations, considering the assumptions presented, can be written as follows:
0 6 r 6 R; 0 6 x 6 L
Momentum equation: ou dP 1 o qu þ ¼ ðrðr0 þ K cn ÞÞ ox dx r or Energy equation: oT 1 o oT rk q u Cp ¼ ox r or or
P ðt¼0Þ ¼ P 0;
The coupled continuity, momentum and energy equations were solved explicitly using the Du Fort Frankel finite difference method. Fig. 2 shows a fluxogram of the routine used to perform the calculations of the temperature and velocity profiles. The temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) was modeled as presented in Table 1 (Ditchfield et al., 2004). The velocity was calculated considering a Herschel– Bulkley fluid (Nakayama et al., 1980): For r > r0: ( N þ1 DP N þ1 ) DP R r0 2L r r0 2L 2L u¼ ð4Þ DP M N þ1 For r 6 r0: 2L u¼ DP M
DP 2L
R r0 N þ1
N þ1 ð5Þ
2.4. Residence time distribution
Continuity equation: oðq u rÞ ¼0 ox
T ðt¼0Þ ¼ T 0 ;
oT ¼0 or ðt¼0Þ
ð1Þ
ð2Þ
ð3Þ
The average residence time was calculated from measured salt concentrations using the following formula that is valid for discrete times obtained by a step injection technique (Chakrabandhu, 2000): Pt i¼0 t i C i Dt i t ffi P ð6Þ t i¼0 C i Dt i The variance was calculated as follows: Pt Pt 2 2 ðti tÞ C i Dti i¼0 t i C i Dt i r2 ffi i¼0Pt ¼ P ðtÞ2 t C Dt C Dt i i i i i¼0 i¼0
ð7Þ
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
635
1. Initial Data
2. Calculate r(j) and Rm(j) 3. Boundary conditions v and T 4. Calculate τ0(i,j), N(1,j) and M(1,j)
5. Calculate r0(1)
6. Calculate v(1,j) NO
7. 2
15. Calculate vavg(i)
9. Calculate τ0(i,j)
16. Calculate T avg(i)
10. Calculate N(i,j)
11. Calculate M(i,j)
12. Calculate r0(i)
13. Calculate v(i,j) YES 14. 2
j=j+1
i=i+1
NO Stop
Fig. 2. Fluxogram representing the routine used to obtain the velocity and temperature profiles of banana puree in a tubular heat exchanger.
The ratio between the maximum and average velocities (vmax/vavg) was calculated by vmax tavg ¼ ð8Þ vavg tmin
2.5. Heat transfer coefficients The energy balance applied to a small section of heating pipe provides (Son & Singh, 2002) dq ¼ hAðT w T Þdt ¼ C p qV dT
Table 1 Models used for temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) of aseptic banana puree Temperature range (C)
Model
30–120 30–50 50–60 60–120 30–50 50–60 60–120
r0 = 99.81 0.53T 0 K = 10.18 0.18T 0 K = 21.12 + 0.45T 0 K = exp(7.01 0.088T 0 ) n = 1/(3.29 0.032T 0 ) n = 1.60 0.02T 0 n = 0.149 + 0.009T 0
ð9Þ
Integrating between the limits T = T0 when t = 0 and T = T when t = t: Z t Z T dT hA ¼ dt ð10Þ q¼ C p qV 0 T 0 ðT w T Þ 2
Considering that, t ¼ Lu, A = pDL and V ¼ pD4 L: Z L ðT T w Þ 4hpDL ¼ dL ln ðT 0 T w Þ C p qpD2 Lu 0 ln
ðT T w Þ 4hL ¼ ðT 0 T w Þ C p qDu
ð11Þ
ð12Þ
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and thus, h¼
C p qDu ðT T w Þ ln 4L ðT 0 T w Þ
To prevent this problem, an inline static mixer was placed before the thermocouple and the temperature measurement rose, and therefore the average temperatures measured by the thermocouple and that measured in the puree collected were quite similar. The processing temperature reached was higher for the slowest flow rate and highest steam temperature employed and decreased with increasing flow rate and decreasing steam temperature, as expected. Another static inline mixer was placed after the injection port to ensure uniform distribution of the sodium chloride solution throughout the puree. The residence time distribution results indicate that the higher the flow rate, the lower the average residence time and
ð13Þ
The average heat transfer coefficient can be calculated considering that density, thermal conductivity and specific heat capacity are constant.
3. Results and discussion The experiments conducted yielded initial (T0) and processing (Tp) temperatures, flow rate measurements used to calculate the average velocity and conductivity measurements that were used to obtain the average, minimum and maximum residence times, residence time deviation and maximum velocity to average velocity ratio (vmax/vavg) for banana puree at each experimental condition (Table 2). Initially the thermocouple to measure the average processing temperature was placed at the heat exchanger outlet and at the center of the tube without the inline mixer. Despite the fact that the steam temperature was high (132.2 C) and the flow rate was the lowest possible (2.5 · 105 m3 s1) the outlet temperature reached a maximum of 50 C. It was perceived that the wall temperature was much higher and tests indicated that the thermocouple was measuring correctly. The puree was collected right after the outlet and the average temperature was measured. It was determined that the average temperature was much higher (approximately 90 C) than that measured by the thermocouple inserted in the tube.
1.00
E (t), F (t)
0.80 0.60 0.40 0.20 0.00 0
10
20
30
40
50
60
Time (s) E (t) Q1 E (t) Q2 E (t) Q3
F (t) Q1 F (t) Q2 F (t) Q3
Fig. 3. Exit age distribution (E(t)) and cumulative exit age distribution (F(t)) for steam temperature 132.2 C and different flow rates (Q1 = 4.7 · 105 m3 s 1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s1).
Table 2 Length/diameter (L/D) ratio, flow rate (Q), average velocity (vavg), average residence time (t), deviation, minimum residence time (tmin), maximum residence time (tmax), maximum to average velocity ratio (vmax/vavg), steam temperature (Ts), initial temperature (T0), experimental processing temperature (Tp), calculated processing temperature (Tc), difference between experimental and calculated processing temperature and average heat transfer coefficient (h) for each experimental condition Experiment
L/D
Q (m3 s1)
t vavg (m s1) (s)
Deviation (s)
tmin (s)
tmax (s)
vmax/vavg
Ts (C)
T0 (C)
Tp (C)
Tc (C)
Difference (%)
1 2 3 4 5 6 7 8 9
500
4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105
0.40 0.32 0.21 0.40 0.32 0.21 0.40 0.32 0.21
19.55 22.13 35.44 17.63 22.65 38.38 18.79 21.23 36.97
4.48 4.67 7.45 4.53 5.28 8.14 4.10 4.80 9.46
15.07 17.47 27.99 13.10 17.37 30.24 14.70 16.44 27.51
24.02 26.80 42.89 22.17 27.93 46.53 22.89 26.03 46.43
1.30 1.27 1.27 1.35 1.31 1.27 1.28 1.29 1.34
132.0 132.4 132.4 120.9 120.9 121.0 109.4 109.6 109.8
30.6 32.9 29.4 30.7 33.7 32.7 35.7 35.9 34.8
95.3 100.5 116.8 85.2 92.2 108.2 78.8 83.7 94.6
95.5 100.9 115.8 85.7 92.1 106.8 78.7 83.6 94.7
0.2 0.4 0.8 0.5 0.2 1.3 0.1 0.1 0.1
842.2 827.8 800.7 836.1 779.4 747.5 696.1 689.2 654.8
10 11 12 13 14 15 16 17 18
250
4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105
0.40 0.32 0.21 0.40 0.32 0.21 0.40 0.32 0.21
8.73 12.18 17.69 10.22 12.40 18.20 8.59 11.72 19.24
2.56 3.52 4.42 2.86 3.20 4.90 2.24 2.91 4.67
6.17 8.66 13.27 7.36 9.20 13.30 6.35 8.81 14.57
11.29 15.70 22.11 13.08 15.60 23.10 10.84 14.63 23.90
1.41 1.41 1.33 1.39 1.35 1.37 1.35 1.33 1.32
132.1 132.2 132.2 120.7 120.9 121.1 109.3 109.6 109.8
33.8 34.2 31.1 32.9 32.7 32.1 33.2 32.9 32.5
83.2 86.2 94.2 71.2 73.7 85.2 64.0 67.8 78.3
82.8 86.5 93.5 71.5 73.8 84.8 63.7 68.3 77.8
0.4 0.3 0.8 0.5 0.1 0.5 0.5 0.7 0.6
1070.4 957.1 826.4 886.3 807.1 743.5 820.4 803.4 735.5
h (W m2 K1)
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
also the lower the deviation, which is to be expected (Fig. 3). The relationship between maximum velocity and average velocity (vmax/vavg) indicates that the fluid presents a non-Newtonian behavior since the results are <2.0, thus the velocity profile is flatter than what would be expected for a Newtonian fluid (Table 2). The temperature profiles calculated indicate that there is a significant temperature difference between the banana puree that is near the wall of the heat exchanger and that at the center of the tube, as was confirmed by the measurements done at the heat exchanger outlet using thermocouples (Fig. 4). The 0Þ dimensionless temperature h h ¼ ðTðTwT is used to plot T 0 Þ
θ
the results in Fig. 4(a)–(c).The average outlet tempera1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
0.200
0.400
0.600
0.800
1.000
d/D Q1
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
Q2
Q3
tures calculated agree with the average experimental temperatures within 2% (Table 2). The temperature profile of banana puree at the heat exchanger outlet is flatter for a lower flow rate and the temperature reached is higher, as is expected (Fig. 4(a)). The lower the flow rate the smaller the difference between the wall and the center of the tube. An increase in steam temperature causes a more uniform temperature distribution (Fig. 4(b)). The first section of the heat exchanger was responsible for most of the heating. For the highest steam temperature studied, there was almost no change in banana puree temperature in the second section of the heat exchanger. However, there was still a significant temperature difference between the wall and the center of the tube (Fig. 4(c)). The velocity profiles obtained for banana puree are shown in Fig. 5. The velocity near the wall tends to be higher than that at the center of the tube because of the large temperature difference between the wall and the center. The lower the flow rate, the more uniform is the temperature profile, therefore the velocity profile is also more uniform and there is less difference between the conditions at the wall and at the center of the tube. For the highest flow rate, an interesting fact can be observed on Fig. 5(a). When the temperature reaches the region between 50 and 60 C there is a thickening 1.0 0.8
V/Vmax
θ
a
0.6 0.4 0.2
0.200
0.400
0.600
0.800
1.000
0.0 0.000
d/D b
T1
T2
0.400
0.600
0.800
1.000
d/D Q1
Q2
Q3
1.0 0.8
V/Vmax
θ
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
0.200
T3
a
0.6 0.4 0.2
0.200
0.400
0.600
0.800
1.000
0.0
d/D c
637
L0
L0.5
0.000
L1
Fig. 4. Temperature profiles of banana puree (a) at the heat exchanger outlet for three different flow rates (Q1 = 4.7 · 105 m3 s1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s 1) at a steam temperature of 121.1 C, (b) at the heat exchanger outlet for three different steam temperatures (T1 = 132.2 C, T2 = 121.1 C and T3 = 110.0 C) at a flow rate of Q = 3.7 · 105 m3 s1, (c) at the heat exchanger inlet (L0), after the first heating section (L0.5) and at the outlet (L1) for a steam temperature of 132.2 C and a flow rate of Q = 3.7 · 105 m3 s1.
0.200
0.400
0.600
0.800
1.000
d/D b
T1
T2
T3
Fig. 5. Velocity profile of banana puree at the heat exchanger outlet (a) for three different flow rates (Q1 = 4.7 · 105 m3 s1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s1) at a steam temperature of 121.1 C, (b) for three different steam temperatures (T1 = 132.2 C, T2 = 121.1 C and T3 = 110.0 C) at a flow rate of Q = 3.7 · 105 m3 s1.
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C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
of banana puree probably due to starch gelatinization, causing a disturbance in the velocity profile. The temperature is in fact within the region 50 and 60 C for the highest flow rate at the point were the velocity profile shows the disturbance. An increase in steam temperature reduces the variations between the profiles for the different flow rates, since temperature distribution is more uniform (Fig. 5(b)). The use of inline mixers to improve temperature distribution should be considered by the industry, improving heat transfer. The average heat transfer coefficients were calculated from the average values of initial and processing temperatures that were measured. The results obtained are higher for higher flow rates and higher steam temperatures, as expected. For lower L/D ratio, average heat transfer coefficient was higher, and the temperature change in the first heating section was always greater than that for the second section (Table 2).
4. Conclusions The equations for modeling banana puree rheological behavior can be used to predict fluid behavior with temperature under different processing conditions. This information is a valuable tool for designing equipment for banana puree processing. Models can be obtained for heat transfer and pressure loss prediction and applied to different situations. The use of inline mixers is recommended for the industry to even out the temperature distribution, otherwise the puree at the center of the tube will not reach the required processing temperature and the puree near the tube walls will be overprocessed.
Acknowledgments The authors acknowledge the State of Sa˜o Paulo Research Foundation (FAPESP) and the Brazilian Committee for Postgraduate Courses in Higher Education
(CAPES) for the scholarships granted to author Ditchfield. Special thanks to Mr. Carl Ruiz, Mr. David Peck and Dr. Nepal Singh for providing laboratory assistance during the experimental work of this project.
References Chakrabandhu, K. (2000). Aseptic processing of particulate foods: computational models and experimental studies. Ph.D. Thesis, Purdue University, West Lafayette, IN, USA. Charm, S. E. (1962). Calculation of center-line temperatures in tubular heat exchangers for pseudoplastic fluids in streamline flow. Industrial Engineering Chemistry Fundamentals, 1(2), 79–82. Chilton, R. A., & Stainsby, R. (1998). Pressure loss equations for laminar and turbulent non-Newtonian pipe flow. Journal of Hydraulic Engineering, 124(5), 522–529. Ditchfield, C., Tadini, C. C., Singh, R. K., & Toledo, R. T. (2004). Rheological behavior of banana puree at high temperatures. International Journal of Food Properties, 7(3), 571–584. Forrest, G., & Wilkinson, W. L. (1973). Laminar heat transfer to temperature-dependent Bingham fluids in tubes. International Journal of Heat and Mass Transfer, 16, 2377–2391. Guariguata, C., Barreiro, J. A., & Guariguata, G. (1979). Analysis of continuous sterilization processes for Bingham plastic fluids in laminar flow. Journal of Food Science, 44, 905–910. Guerrero, S., Alzamora, S. M., & Gerschenson, L. N. (1997). Flow behavior of processed banana puree. Food Science and Technology International, 3, 103–111. Luna, N., Me´ndez, F., & Mar, E. (2003). Transient analysis of the conjugated heat transfer in circular ducts with a power-law fluid. Journal of Non-Newtonian Fluid Mechanics, 111, 69–85. Nakayama, T., Niwa, E., & Hamada, I. (1980). Pipe transportation of minced fish paste. Journal of Food Science, 45, 844–847. Nouar, C., Desaubry, C., & Zenaidi, H. (1998). Numerical and experimental investigation of thermal convection for a thermodependent Herschel–Bulkley fluid in an annular duct with rotating inner cylinder. European Journal of Mechanics—B/Fluids, 17(6), 875–900. Sayed-Ahmed, M. E. (2000). Laminar heat transfer for thermally developing flow of a Herschel–Bulkley fluid in a square duct. International Communications in Heat Transfer, 27(7), 1024– 1031. Son, S. M., & Singh, R. K. (2002). Turbulence modeling and verification for aseptically processed soybean milk under turbulent flow conditions. Journal of Food Engineering, 52, 177–184.
Velocity and temperature profiles, heat transfer coefficients and residence time distribution of a temperature dependent Herschel– Bulkley fluid in a tubular heat exchanger C. Ditchfield a, C.C. Tadini a
a,*
, R.K. Singh b, R.T. Toledo
b
Department of Chemical Engineering, Sa˜o Paulo University, Escola Polite´cnica, P.O. Box 61548, 05424-970 Sa˜o Paulo-SP, Brazil b Department of Food Science and Technology, UGA, 30602-7610 Athens, GA, USA Received 21 October 2004; accepted 5 June 2005 Available online 9 August 2005
Abstract Banana puree was studied as a temperature dependent Herschel–Bulkley fluid. Experiments were conducted in a tubular heat exchanger at three flow rates (2.5 · 105, 3.7 · 105 and 4.7 · 105 m3 s1), at three steam temperatures (110.0, 121.1 and 132.2 C) and two length/diameter ratios (250 and 500). The minimum, maximum and average residence times were calculated from on-line electrical conductivity measurements. Inlet and outlet temperatures were recorded. A model was proposed to obtain heat transfer coefficients and velocity and temperature profiles in the heat exchanger. The continuity, energy and momentum equations were integrated using a finite difference method. The heat transfer coefficients varied from 654.8 to 1070.4 W m2 K1 for average velocities of 0.24–0.40 m s1. There is good agreement between experimental parameters and those predicted by the model. 2005 Elsevier Ltd. All rights reserved. Keywords: Banana puree; Heat transfer; Residence time distribution; Velocity profile; Temperature profile
1. Introduction Banana puree has a complex rheological behavior, modeled as a Herschel–Bulkley fluid, with consistency index (K), flow behavior index (n) and yield stress (r0) varying significantly with temperature (Ditchfield, Tadini, Singh, & Toledo, 2004; Guerrero, Alzamora, & Gerschenson, 1997). Equipment design depends greatly upon reliable equations for explaining heat transfer, pressure loss and energy requirements. There are some models that have been developed for the flow of non-Newtonian fluids, whereas some models have considered laminar flow of Bingham plastic fluids (Forrest & Wilkinson, 1973; Guariguata, Barreiro, & *
Corresponding author. Tel.: +55 11 3091 2258; fax: +55 11 3091 2255. E-mail address: [email protected] (C.C. Tadini). 0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.06.015
Guariguata, 1979). Other models consider laminar and turbulent flow of power law fluids with constant wall temperature and constant heat flux (Luna, Me´ndez, & Mar, 2003; Son & Singh, 2002). Herschel–Bulkley flow was modeled by some authors (Chilton & Stainsby, 1998; Nakayama, Niwa, & Hamada, 1980; Nouar, Desaubry, & Zenaidi, 1998; Sayed-Ahmed, 2000). Nakayama et al. (1980) designed a pipe system to study pressure loss in isothermal pipe transportation of minced fish paste. Minced fish paste is a Herschel–Bulkley fluid with a large yield stress value. Chilton and Stainsby (1998) presented equations for laminar and turbulent flow of Herschel–Bulkley fluids. A numerical model was discussed and compared to experimental measurements, agreeing closely in the laminar flow region, though in turbulent flow predicted pressure loss is generally 15% lower than experimental values. Nouar et al. (1998) proposed an investigation
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
633
Nomenclature C Cp d D E F h K k L M N n P DP Q R T T0 t t
concentration (kg m3) specific heat (J kg1 K1) distance from tube wall (m) tube diameter (m) exit age distribution (dimensionless) cumulative exit age (dimensionless) heat transfer coefficient (W m2 K1) consistency index (Pa sn) thermal conductivity (W m1 K1) tube length (m) K(1/n) (Pa s) 1/n (dimensionless) flow behavior index (dimensionless) pressure (Pa) pressure drop (Pa) flow rate (m3 s1) tube radius (m) temperature (K) temperature (C) time (s) average residence time (s)
of thermal convection within a thermodependent Herschel–Bulkley fluid flowing in an annular duct with rotating inner cylinder. The authors carried out experiments with a polymer (Carbopol 940) with a 0.2% mass concentration and heat transfer was influenced by the rotation speed of the inner cylinder. Sayed-Ahmed (2000) investigated numerically laminar heat transfer to a Herschel–Bulkley fluid in the entrance region of a square duct, considering fully developed velocity profile. Higher flow behavior indices (n) resulted in higher velocity and lower Nusselt number. The objectives of this study were: (1) to determine the information on residence time distributions, velocity and temperature profiles, and heat transfer coefficients; and (2) to investigate how these variables are affected by the rheological changes observed in a Herschel– Bulkley fluid (banana puree) whose parameters (r0, K and n) vary significantly with temperature, in the heating section of a double pipe tubular heat exchanger. This will provide important knowledge for equipment design in processing of temperature dependent Herschel–Bulkley fluids.
2. Materials and methods 2.1. Banana puree Banana puree was obtained from a commercial source as acidified aseptic banana puree (22.1 ± 0.9
u v V
axial velocity (m s1) velocity (m s1) volume (m3)
Greek characters c_ flow shear rate (s1) q density (kg m3) r0 yield stress (Pa) 0Þ h dimensionless temperature, h ¼ ðTðTwT T 0 Þ Subscripts 0 initial value avg average value c calculated f final value max maximum min minimum p processing s steam w wall
Brix) in 225 kg drums. The puree had been acidified with a combination of citric and ascorbic acid to an average pH of 4.49 ± 0.03, heat treated for enzyme inactivation and aseptically packaged in plastic bags within the drums. 2.2. Experimental measurements Experiments were conducted in a tubular heat exchanger with two heating sections, each 3.05 m long with an internal diameter of 1.2 · 102 m. Three flow rates (2.5 · 105, 3.7 · 105 and 4.7 · 105 m3 s1), three steam temperatures (110.0, 121.1 and 132.2 C) and two length/diameter ratios (250 and 500) were studied. For the length/diameter ratio of 250, only one heating section was considered. A total of 36 experiments were carried out, since each condition was repeated once and approximately 4 ton of banana puree were processed. Temperature was measured with thermocouples before the heating section (T1), after the first heating section (T2), at the end of the heat exchanger (T3) and steam temperature (T4) and a data logging system was used to record the temperatures every 10 s (Fig. 1). A saturated sodium chloride solution (10 mL) was injected at the inlet of the heat exchanger as a tracer for measuring residence times. A conductivity transmitter (Compak, Signet Industrial, El Monte, CA) was placed at the outlet of the heat exchanger. Two inline mixers were used after the injection port and at the outlet of the heat exchanger to ensure
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C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638 Water out Conductivity probe Puree
Steam in
T3 Mixer Feed
T2
T4
Cooling unit
Mixer Pump
T1
Heating unit
Steam out
Water in Puree
Salt injection
Fig. 1. A schematic diagram of the experimental setup used for residence time distributions and temperatures in a tubular heat exchanger.
uniform distribution of the salt, as well as a correct measurement of the average temperature at the outlet of the heat exchanger.
The boundary conditions and initial conditions are as follows: oT ou ¼ 0; ¼0 T r¼R ¼ T w ; ur¼R ¼ 0; or r¼0 or r¼0
2.3. Velocity and temperature profiles The system considered was banana puree processed in a tubular heat exchanger under laminar flow conditions. The physical and thermal properties of banana puree (density, thermal conductivity and specific heat) were considered constant and were obtained from Charm (1962). Banana puree was modeled as a Herschel–Bulkley fluid. The temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) was determined for temperatures between 30 C and 120 C using a dynamic stress controlled rheometer (model SR 5000, Rheometric Scientific, Piscataway, NJ) (Ditchfield et al., 2004). The problem can be described by the following equations with the assumptions of steady laminar flow, incompressible fluid, constant wall temperature, negligible viscous dissipation, small axial conduction relative to radial conduction and insignificant free convection. The continuity, momentum and energy equations, considering the assumptions presented, can be written as follows:
0 6 r 6 R; 0 6 x 6 L
Momentum equation: ou dP 1 o qu þ ¼ ðrðr0 þ K cn ÞÞ ox dx r or Energy equation: oT 1 o oT rk q u Cp ¼ ox r or or
P ðt¼0Þ ¼ P 0;
The coupled continuity, momentum and energy equations were solved explicitly using the Du Fort Frankel finite difference method. Fig. 2 shows a fluxogram of the routine used to perform the calculations of the temperature and velocity profiles. The temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) was modeled as presented in Table 1 (Ditchfield et al., 2004). The velocity was calculated considering a Herschel– Bulkley fluid (Nakayama et al., 1980): For r > r0: ( N þ1 DP N þ1 ) DP R r0 2L r r0 2L 2L u¼ ð4Þ DP M N þ1 For r 6 r0: 2L u¼ DP M
DP 2L
R r0 N þ1
N þ1 ð5Þ
2.4. Residence time distribution
Continuity equation: oðq u rÞ ¼0 ox
T ðt¼0Þ ¼ T 0 ;
oT ¼0 or ðt¼0Þ
ð1Þ
ð2Þ
ð3Þ
The average residence time was calculated from measured salt concentrations using the following formula that is valid for discrete times obtained by a step injection technique (Chakrabandhu, 2000): Pt i¼0 t i C i Dt i t ffi P ð6Þ t i¼0 C i Dt i The variance was calculated as follows: Pt Pt 2 2 ðti tÞ C i Dti i¼0 t i C i Dt i r2 ffi i¼0Pt ¼ P ðtÞ2 t C Dt C Dt i i i i i¼0 i¼0
ð7Þ
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
635
1. Initial Data
2. Calculate r(j) and Rm(j) 3. Boundary conditions v and T 4. Calculate τ0(i,j), N(1,j) and M(1,j)
5. Calculate r0(1)
6. Calculate v(1,j) NO
7. 2
15. Calculate vavg(i)
9. Calculate τ0(i,j)
16. Calculate T avg(i)
10. Calculate N(i,j)
11. Calculate M(i,j)
12. Calculate r0(i)
13. Calculate v(i,j) YES 14. 2
j=j+1
i=i+1
NO Stop
Fig. 2. Fluxogram representing the routine used to obtain the velocity and temperature profiles of banana puree in a tubular heat exchanger.
The ratio between the maximum and average velocities (vmax/vavg) was calculated by vmax tavg ¼ ð8Þ vavg tmin
2.5. Heat transfer coefficients The energy balance applied to a small section of heating pipe provides (Son & Singh, 2002) dq ¼ hAðT w T Þdt ¼ C p qV dT
Table 1 Models used for temperature dependency of the yield stress (r0), consistency index (K) and flow behavior index (n) of aseptic banana puree Temperature range (C)
Model
30–120 30–50 50–60 60–120 30–50 50–60 60–120
r0 = 99.81 0.53T 0 K = 10.18 0.18T 0 K = 21.12 + 0.45T 0 K = exp(7.01 0.088T 0 ) n = 1/(3.29 0.032T 0 ) n = 1.60 0.02T 0 n = 0.149 + 0.009T 0
ð9Þ
Integrating between the limits T = T0 when t = 0 and T = T when t = t: Z t Z T dT hA ¼ dt ð10Þ q¼ C p qV 0 T 0 ðT w T Þ 2
Considering that, t ¼ Lu, A = pDL and V ¼ pD4 L: Z L ðT T w Þ 4hpDL ¼ dL ln ðT 0 T w Þ C p qpD2 Lu 0 ln
ðT T w Þ 4hL ¼ ðT 0 T w Þ C p qDu
ð11Þ
ð12Þ
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and thus, h¼
C p qDu ðT T w Þ ln 4L ðT 0 T w Þ
To prevent this problem, an inline static mixer was placed before the thermocouple and the temperature measurement rose, and therefore the average temperatures measured by the thermocouple and that measured in the puree collected were quite similar. The processing temperature reached was higher for the slowest flow rate and highest steam temperature employed and decreased with increasing flow rate and decreasing steam temperature, as expected. Another static inline mixer was placed after the injection port to ensure uniform distribution of the sodium chloride solution throughout the puree. The residence time distribution results indicate that the higher the flow rate, the lower the average residence time and
ð13Þ
The average heat transfer coefficient can be calculated considering that density, thermal conductivity and specific heat capacity are constant.
3. Results and discussion The experiments conducted yielded initial (T0) and processing (Tp) temperatures, flow rate measurements used to calculate the average velocity and conductivity measurements that were used to obtain the average, minimum and maximum residence times, residence time deviation and maximum velocity to average velocity ratio (vmax/vavg) for banana puree at each experimental condition (Table 2). Initially the thermocouple to measure the average processing temperature was placed at the heat exchanger outlet and at the center of the tube without the inline mixer. Despite the fact that the steam temperature was high (132.2 C) and the flow rate was the lowest possible (2.5 · 105 m3 s1) the outlet temperature reached a maximum of 50 C. It was perceived that the wall temperature was much higher and tests indicated that the thermocouple was measuring correctly. The puree was collected right after the outlet and the average temperature was measured. It was determined that the average temperature was much higher (approximately 90 C) than that measured by the thermocouple inserted in the tube.
1.00
E (t), F (t)
0.80 0.60 0.40 0.20 0.00 0
10
20
30
40
50
60
Time (s) E (t) Q1 E (t) Q2 E (t) Q3
F (t) Q1 F (t) Q2 F (t) Q3
Fig. 3. Exit age distribution (E(t)) and cumulative exit age distribution (F(t)) for steam temperature 132.2 C and different flow rates (Q1 = 4.7 · 105 m3 s 1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s1).
Table 2 Length/diameter (L/D) ratio, flow rate (Q), average velocity (vavg), average residence time (t), deviation, minimum residence time (tmin), maximum residence time (tmax), maximum to average velocity ratio (vmax/vavg), steam temperature (Ts), initial temperature (T0), experimental processing temperature (Tp), calculated processing temperature (Tc), difference between experimental and calculated processing temperature and average heat transfer coefficient (h) for each experimental condition Experiment
L/D
Q (m3 s1)
t vavg (m s1) (s)
Deviation (s)
tmin (s)
tmax (s)
vmax/vavg
Ts (C)
T0 (C)
Tp (C)
Tc (C)
Difference (%)
1 2 3 4 5 6 7 8 9
500
4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105
0.40 0.32 0.21 0.40 0.32 0.21 0.40 0.32 0.21
19.55 22.13 35.44 17.63 22.65 38.38 18.79 21.23 36.97
4.48 4.67 7.45 4.53 5.28 8.14 4.10 4.80 9.46
15.07 17.47 27.99 13.10 17.37 30.24 14.70 16.44 27.51
24.02 26.80 42.89 22.17 27.93 46.53 22.89 26.03 46.43
1.30 1.27 1.27 1.35 1.31 1.27 1.28 1.29 1.34
132.0 132.4 132.4 120.9 120.9 121.0 109.4 109.6 109.8
30.6 32.9 29.4 30.7 33.7 32.7 35.7 35.9 34.8
95.3 100.5 116.8 85.2 92.2 108.2 78.8 83.7 94.6
95.5 100.9 115.8 85.7 92.1 106.8 78.7 83.6 94.7
0.2 0.4 0.8 0.5 0.2 1.3 0.1 0.1 0.1
842.2 827.8 800.7 836.1 779.4 747.5 696.1 689.2 654.8
10 11 12 13 14 15 16 17 18
250
4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105 4.7 · 105 3.7 · 105 2.5 · 105
0.40 0.32 0.21 0.40 0.32 0.21 0.40 0.32 0.21
8.73 12.18 17.69 10.22 12.40 18.20 8.59 11.72 19.24
2.56 3.52 4.42 2.86 3.20 4.90 2.24 2.91 4.67
6.17 8.66 13.27 7.36 9.20 13.30 6.35 8.81 14.57
11.29 15.70 22.11 13.08 15.60 23.10 10.84 14.63 23.90
1.41 1.41 1.33 1.39 1.35 1.37 1.35 1.33 1.32
132.1 132.2 132.2 120.7 120.9 121.1 109.3 109.6 109.8
33.8 34.2 31.1 32.9 32.7 32.1 33.2 32.9 32.5
83.2 86.2 94.2 71.2 73.7 85.2 64.0 67.8 78.3
82.8 86.5 93.5 71.5 73.8 84.8 63.7 68.3 77.8
0.4 0.3 0.8 0.5 0.1 0.5 0.5 0.7 0.6
1070.4 957.1 826.4 886.3 807.1 743.5 820.4 803.4 735.5
h (W m2 K1)
C. Ditchfield et al. / Journal of Food Engineering 76 (2006) 632–638
also the lower the deviation, which is to be expected (Fig. 3). The relationship between maximum velocity and average velocity (vmax/vavg) indicates that the fluid presents a non-Newtonian behavior since the results are <2.0, thus the velocity profile is flatter than what would be expected for a Newtonian fluid (Table 2). The temperature profiles calculated indicate that there is a significant temperature difference between the banana puree that is near the wall of the heat exchanger and that at the center of the tube, as was confirmed by the measurements done at the heat exchanger outlet using thermocouples (Fig. 4). The 0Þ dimensionless temperature h h ¼ ðTðTwT is used to plot T 0 Þ
θ
the results in Fig. 4(a)–(c).The average outlet tempera1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
0.200
0.400
0.600
0.800
1.000
d/D Q1
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
Q2
Q3
tures calculated agree with the average experimental temperatures within 2% (Table 2). The temperature profile of banana puree at the heat exchanger outlet is flatter for a lower flow rate and the temperature reached is higher, as is expected (Fig. 4(a)). The lower the flow rate the smaller the difference between the wall and the center of the tube. An increase in steam temperature causes a more uniform temperature distribution (Fig. 4(b)). The first section of the heat exchanger was responsible for most of the heating. For the highest steam temperature studied, there was almost no change in banana puree temperature in the second section of the heat exchanger. However, there was still a significant temperature difference between the wall and the center of the tube (Fig. 4(c)). The velocity profiles obtained for banana puree are shown in Fig. 5. The velocity near the wall tends to be higher than that at the center of the tube because of the large temperature difference between the wall and the center. The lower the flow rate, the more uniform is the temperature profile, therefore the velocity profile is also more uniform and there is less difference between the conditions at the wall and at the center of the tube. For the highest flow rate, an interesting fact can be observed on Fig. 5(a). When the temperature reaches the region between 50 and 60 C there is a thickening 1.0 0.8
V/Vmax
θ
a
0.6 0.4 0.2
0.200
0.400
0.600
0.800
1.000
0.0 0.000
d/D b
T1
T2
0.400
0.600
0.800
1.000
d/D Q1
Q2
Q3
1.0 0.8
V/Vmax
θ
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.000
0.200
T3
a
0.6 0.4 0.2
0.200
0.400
0.600
0.800
1.000
0.0
d/D c
637
L0
L0.5
0.000
L1
Fig. 4. Temperature profiles of banana puree (a) at the heat exchanger outlet for three different flow rates (Q1 = 4.7 · 105 m3 s1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s 1) at a steam temperature of 121.1 C, (b) at the heat exchanger outlet for three different steam temperatures (T1 = 132.2 C, T2 = 121.1 C and T3 = 110.0 C) at a flow rate of Q = 3.7 · 105 m3 s1, (c) at the heat exchanger inlet (L0), after the first heating section (L0.5) and at the outlet (L1) for a steam temperature of 132.2 C and a flow rate of Q = 3.7 · 105 m3 s1.
0.200
0.400
0.600
0.800
1.000
d/D b
T1
T2
T3
Fig. 5. Velocity profile of banana puree at the heat exchanger outlet (a) for three different flow rates (Q1 = 4.7 · 105 m3 s1, Q2 = 3.7 · 105 m3 s1 and Q3 = 2.5 · 105 m3 s1) at a steam temperature of 121.1 C, (b) for three different steam temperatures (T1 = 132.2 C, T2 = 121.1 C and T3 = 110.0 C) at a flow rate of Q = 3.7 · 105 m3 s1.
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of banana puree probably due to starch gelatinization, causing a disturbance in the velocity profile. The temperature is in fact within the region 50 and 60 C for the highest flow rate at the point were the velocity profile shows the disturbance. An increase in steam temperature reduces the variations between the profiles for the different flow rates, since temperature distribution is more uniform (Fig. 5(b)). The use of inline mixers to improve temperature distribution should be considered by the industry, improving heat transfer. The average heat transfer coefficients were calculated from the average values of initial and processing temperatures that were measured. The results obtained are higher for higher flow rates and higher steam temperatures, as expected. For lower L/D ratio, average heat transfer coefficient was higher, and the temperature change in the first heating section was always greater than that for the second section (Table 2).
4. Conclusions The equations for modeling banana puree rheological behavior can be used to predict fluid behavior with temperature under different processing conditions. This information is a valuable tool for designing equipment for banana puree processing. Models can be obtained for heat transfer and pressure loss prediction and applied to different situations. The use of inline mixers is recommended for the industry to even out the temperature distribution, otherwise the puree at the center of the tube will not reach the required processing temperature and the puree near the tube walls will be overprocessed.
Acknowledgments The authors acknowledge the State of Sa˜o Paulo Research Foundation (FAPESP) and the Brazilian Committee for Postgraduate Courses in Higher Education
(CAPES) for the scholarships granted to author Ditchfield. Special thanks to Mr. Carl Ruiz, Mr. David Peck and Dr. Nepal Singh for providing laboratory assistance during the experimental work of this project.
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