External Coefficient of Heat Transfer by Convection in Mixed Vessels

Article pubs.acs.org/IECR

External Coefficient of Heat Transfer by Convection in Mixed Vessels Using Vertical Tube Baffles Vitor da Silva Rosa,*,† Thiago Cesar de Souza Pinto,‡ Aldo Ramos Santos,‡ Carlos Alberto Amaral Moino,§ Karina Tamiaõ ,‡ Luiz Renato Bastos Lia,‡ Elias Basile Tambourgi,⊥ Marcílio Dias Lopes,§ Juliana Tófano de Campos Leite Toneli,† and Deovaldo de Mores Júnior‡ †

Department of Energy, UFABC, Santo André, SP, Brazil Department of Chemical Engineering and §Department of Mechanical Engineering, Santa Cecília University, Santos, SP, Brazil ⊥ Department of Chemical Engineering, UNICAMP, Campinas, SP, Brazil ‡

ABSTRACT: Many correlations in the literature have been proposed to predict the external heat transfer coefficient, but most of them are suggested for batch operations. This paper aims to experimentally investigate a steady state condition (usually found in industrial processes) of mixed vessels equipped with a four 45° pitched-blade turbine and vertical tubular baffles to determine the external heat transfer coefficient. An empirical correlation based on the work of Sieder−Tate [Ind. Eng. Chem. 1936, 1429− 1435] for a steady state condition using standard configuration of a mixing system is being proposed. A sucrose solution of 20% and 32% concentrations (w/w) and a mixing apparatus with a 50 L cylindrical flat bottom vessel are employed. Water was used as heating fluid running inside the vertical tubular baffles with an inlet temperature range from 25 to 45 °C and an impeller speed from 30 to 330 rpm. A new fit correlation was proposed based on the Sieder−Tate modeling, yielding good agreement with experimental data in steady states with deviations of less than 15% between observed and predicted external heat coefficients.

1. INTRODUCTION Agitated vessels are often used for reactors, suspension of solids, homogenization of miscible liquids including heat and mass transfer operations and are widely used in chemical, petrochemical, biochemical, and food industries.1 The area in question concerning heat exchange in the tanks mentioned above is usually based on the overall coefficient of heat transfer as a function of dimensionless groups, Reynolds, Prandtl, and Nusselt. The classical equations to project this area are shown in eqs 1 and 2. A s = Q /U × LMTD

(1)

1/Ud = 1/αio + 1/αo + Rd

(2)

consistent results in batch given the rapid changes in liquid temperature. Many researches as Kumpinsky,6 Pratt,7 Cummings,8 Kraussold,9 Oldshue,10 and Bourne11 found the exponents of eq 3, but they recommend using empirical correlations only when process similarity exists. When the ratio of surface/ volume of the tank diminishes, thus undermining the heat transfer process through jackets, a larger heat transfer surface is necessary. A usual solution for that is to adopt the helical coil. As an alternative, the vertical tube baffles can be used to increase the surface area of heat exchange and also prevent circular motion of the agitated fluid, generating some axial mixing. The baffles also have the advantage comparable to the helical coil namely the ease of cleaning and maintenance. Mohan12 also highlights that the baffles can increase the heat transfer coefficient by as much as 37% in the turbulent region. Tatterson13 reports that the standard baffles promote a better distribution of turbulence in the tank.

The hio coefficient relative to fluid running inside a pipe is found in the literature for industrial processes to be in the turbulent region.2 The pioneering work of Sieder−Tate,3 established a semiempirical correlation to determine the external coefficient of heat transfer for the turbulent region (ho), as shown by eq 3. Many other correlations have been proposed in the literature, but most of them are suggested for batch operations. Nu = KRe aPr bVi c

2. ENERGY BALANCE

(3)

The time to achieve the steady state in the mixed tank can be predicted by the first law of thermodynamics, as depicted by eq 4.

As an industrial process usually requires a continuous operation, this paper aims to determine the parameters of eq 1, for a mixing system with a four 45° pitched-blade turbine using standard vertical tubular baffles in a cylindrical flat bottom vessel operating in a steady state. According to Uhl and Gray,4 the batch process requires much simpler equipment than the steady state process. However, Chilton et al.5 states that it is difficult to obtain © 2013 American Chemical Society

Received: Revised: Accepted: Published: 2434

July 11, 2012 November 13, 2012 January 22, 2013 January 22, 2013 dx.doi.org/10.1021/ie301841q | Ind. Eng. Chem. Res. 2013, 52, 2434−2438

Industrial & Engineering Chemistry Research

Article

Figure 2. The unit is composed of an axial mechanical impeller with a four 45° pitched-blade turbine and four vertical banks of

∑ Q vc − ∑ Wvc + ∑ mh (h + v 2 /2 + gz) −

∑ mc(h + v 2/2 + gz)

= dEvc/dθ

(4)

Assuming the input and output mass flow rates are the same in the system Figure 1, the fluctuation of kinetic and potential

Figure 2. Sketch of experimental unit. (1) Tank of cold fluid. (2) Temperature indicator of cold fluid. (3) Temperature indicator of endapproach of cold fluid. (4) Impeller. (5) Baffles. (6) Feed tank. (7) Temperature indicator of end-approach of hot fluid. (8) Temperature indicator of hot fluid. (9) Speed meter. (10) Motor supported by bearings and a frequency inverter. (11) Preheater. (12) Thermal controller.

copper, used as baffles. The tank is made of transparent acrylic and presents a volume of 50 L. The vertical tubular baffles present an internal diameter (di) of 11.25 mm and external diameter (de) of 12.6 mm. The geometric relation of the system is shown in Figure 3, and their values are displayed in Table 1.

Figure 1. Mass and energy balance in tank.

energy is negligible compared to the total heat flow rate inside the tank. Thus, eq 2 assumes the form depicted in eq 5. Q − W + mh (h1 − hT → T 2) − mc(h2 − ht → t 2) = dEvc/dθ (5)

A proportionality constant (K*) is given and the output temperature of the hot fluid is a function of the output temperature of cold fluid (eq 6).

T = K *t + y

(6)

Replacing eq 5 for eq 6 and integrating the final form of eq 7 enables us to observe the transition point for the steady state inside the tank. t = (1/K1){(K 2e[(K1/ K3)θ]) − K4}

(7)

Where the constants K1, K2, K3, and K4 are depicted by eqs 8−11. K1 = αA − mcCpc − K *mh Cph

Figure 3. Geometrical parameters of the system: (a) devices arrangement (b) baffles location.

(8)

K 2 = (αA − mcCpc − K *mh Cph )t1 + mcCpct1 + mh Cph T1 − αAt00 − W − ymh Cph

K3 = MCpc K4 = mcCpct1 + mh Cph T1 − αAt00 − W − ymh Cph

Table 1. Geometrical Parameters of Feed Tank

(9) (10) (11)

3. MATERIALS AND METHODS 3.1. Materials. The sketch of the experimental unit used to determine the external film coefficient (ho) is presented in 2435

parameters

dimension (m)

Dt Da H E W J

0.40 0.13 0.40 0.13 0.0112 0.04

dx.doi.org/10.1021/ie301841q | Ind. Eng. Chem. Res. 2013, 52, 2434−2438


Article

The apparatus presents an electrical motor of 1450 W, a proprotional−integral derivative (PID) thermostat bath control, and 2 boilers with 5000 W each. A sucrose solution of 20% or 32% concentration by mass (w/w) was employed as cold fluid, and distilled water was used as hot fluid. The temperature taps are located in the input and output of the vertical tubes and in the input and output of the tank. Table 2 presents the fluid properties.

Table 4. Results for Dimensionless Parameters for Sucrose Solution at 32% Sucrose Solution of 32%

Table 2. Physical Properties of Fluids at 25 °C14 parameter

water

sucrose 20%

sucrose 32%

μ (cP) ρ (kg/m3) Cp (J/(kg K)) k (W/(m °C))

1.00 996.64 4184.88 0.61

1.710 1090.00 3920.00 0.47

5.206 1090.00 3638.00 0.43

speed (rpm)

Reynolds

Prandtl

Nusselt

Vi (μ/μw)

30 60 90 120 150 180 210 240 270 300 330

6053.02 12625.31 19397.98 26545.25 34466.73 42534.52 50930.74 59739.78 69115.65 78818.02 88983.58

13.14 12.60 12.30 11.99 11.54 11.22 10.93 10.65 10.36 10,09 9.83

465.98 605.79 650.27 812.93 755.81 856.84 879.89 831.49 878.55 818.95 754.73

1.4918 1.4362 1.3965 1.3607 1.3364 1.2995 1.2712 1.2461 1.219 1.1948 1.1712

3.2. Methods. The experimental conditions of this study were the impeller speed rotation (30−330 rpm) and the cold fluid input temperature range (25−45 °C). The fixed parameters were the temperature and flow rate of hot fluid (60 °C and 1.4 L/min), respectively. Equation 5 was used to determine the lapsed experimental time to obtain the steady state operation which was reached after 90 min for the total mass of the feed tank. The output temperature was monitored every 2 min for all tested conditions. There were 22 runs, 11 for each sucrose solution.

4. RESULTS AND DISCUSSION The results were divided into two stages. The first stage concerns the experimental determination of heat transfer coefficients, and the second one refers to the parameters for the proposed parameters in eq 6. Tables 3 and 4 depict the Table 3. Results for Dimensionless Parameters for Sucrose Solution at 20%

Figure 4. Effect of Reynolds number.

Sucrose Solution of 20% speed (rpm)

Reynolds

Prandtl

Nusselt

Vi (μ/μw)

30 60 90 120 150 180 210 240 270 300 330

5823.82 12281.62 18919.04 26105.63 33358.50 42170.30 50859.49 60153.41 70034.02 80530.96 91978.23

13.59 12.89 12.55 12.13 11.78 11.26 10.89 10.53 10.17 9.83 9.47

451.78 538.08 535.39 595.63 681.51 655.08 681.25 743.02 770.77 864.62 1043.44

1.4618 1.4219 1.3984 1.3692 1.3406 1.3097 1.2824 1.2542 1.2294 1.2024 1.1760

dimensionless Numbers according to Nusselt, Reynolds, Prandtl, and the viscosity relations as a function of the impeller speed. The bulk viscosity (μ) was determined by the mean temperature of the fluid after the steady state was achieved, and the wall viscosity (μw) was obtained using the mean temperature at the external surface of the tube baffles. In order to evaluate the effect of each dimensionless number of eq 5, the exponents “a”, “b”, and “c” as well as the constant k were determined based on a log−log plot and depicted by Figures 4−7. The same procedure was also adopted by Chilton et al.5 The magnitude of each exponent is in agreement with

Figure 5. Effect of Prandtl number.

the findings in literature in that they were determined for a steady-state condition. After the effect of each parameter was determined, the final proposed model for steady state operation is depicted by eq 12. Nu = 17.8848.Re 0.2686Pr 0.2855Vi 0.3645

(12)

Equation 12 is held to the following conditions: (a) Reynolds number range 6000−90 000; (b) an axial impeller with a four 2436



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Figure 6. Effect of the viscosities ratio.

Figure 8. Comparison between observed and predicted Nusselt.

Figure 7. Linear regression for “K” determination.

Figure 9. Comparison with different models for Nusselt versus Reynolds.

45° pitched-blades turbine; (c) steady state and continuous process; (d) four banks of vertical tube baffles; (e) water as heater fluid with no changing phase; and (f) heating the sucrose solution of 20% and 32%. Figure 8 presents the error between the observed external Nusselt number and the predicted values determined by eq 10, yielding a maximum deviation of less than 15%. A comparison with different models (those of Dunlap,15 Havas,16 and Dostál17) was applied for a given condition, and the results are shown in Figure 9, showing a discrepancy among them. An inference on this result can be made for the required similarity process of each correlation. The exponents of the Sieder−Tate (1927) correlation for several authors are illustrated in Table 5. Most of them are based on the first suggested values 50 years ago, as 2/3 for Re, 1/3 for Pr, and 0.14 for Vi. The input exponents of the proposed model are an attempt to describe a correlation for steady state operation, using vertical tubular baffles as a heat transfer device.

Table 5. Comparison of the Parameters of Equation 6 Proposed by Different Correlations model

Re (a)

Pr (b)

Vi (c)

K

Pratt7 Dunlap/ Rushton15 DeMaerteleire18 Ishibashi et al19 Havas16 Dostál17 this work

0.5000 0.6500

0.8000 0.3333

0.1400

34.0000 0.0900

batch batch

0.6280

0.3333

0.2000

1.7780

batch

0.3333 0.6670 0.6670 0.2686

0.3333 0.4000 0.3333 0.2855

0.1400 0.1600 0.1400 0.3645

2.0000 0.1300 0.5700 17.8848

operation

batch batch batch continuous

geometrical parameters such as type and dimension of impeller, vessel shape, internal parts of the mixing system, and also physical properties of fluids. The empirical correlation proposed in the foregoing work has presented a satisfactory prediction of the external heat transfer coefficient in a steady state operation, yielding a deviation of less than 15% between the observed and predicted heat transfer coefficients. The value of constant k is dependent on geometric parameters and type of impeller. For the proposed model, with a four 45° pitchedblades turbine and a steady state condition, a magnitude for the

5. CONCLUSIONS Many empirical correlations have been developed to predict the external heat transfer coefficient, and most of them have been designed for batch operation, regardless of any specific 2437



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constant k of 17.4 was found, corroborating with findings in literature for k range of (0.5−34). Further investigations with different kinds of fluids and larger vessels are justified.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Mailing address: 277, Oswaldo Cruz street, Santos, SP, Brazil. Phone: + 55 13 32027119.

μb = dynamic viscosity of agitated liquid at mean temperature, kg/(m s) μw = dynamic viscosity of agitated liquid at wall temperature of tube baffle, kg/(m s) ρ = density of agitated liquid, kg/m3

REFERENCES

(1) Karcz, J.; Szoplik, J. An Effect of the Exccentric Position of the Propeller Agitator on the Mixing Time. Chem. Pap. 2004, 58 (1), 9− 14. (2) Paul, E. L.; Atiemo-Ubeng, V. A.; Kresta, S. M. Handbook of Mixing, 1st ed.; Wiley-Interscience: New York, 2004. (3) Sieder, E. N.; Tate, G. E. Heat Transfer and Pressure Drop of Liquids in Tubes. Ind. Eng. Chem. 1936, 1429−1435. (4) Uhl, V. W; Gray, J. B. Mixing Theory and Practice, Academic Press: New York, 1966; Vol. I, Chapter V. (5) Chilton, T. H.; Drew, T. B.; Jebens, R. H. Heat Transfer Coefficients in Agitated Vessels. Ind. Eng. Chem. 1944, 36, 510. (6) Kumpinsky, E. Heat Transfer Coefficients in Agitated Vessels. Latent heat models. Ind. Chem. Res 1996, 35, e938−e942. (7) Pratt, N. H. The Heat Transfer in a Reaction Tank Cooled by Means of a Coil. Trans. Inst. Chem. Eng., London 1947, 25, 163. (8) Cummings, G. H.; West, A. S. Heat transfer data for kettles with jackets and coils. Ind. Eng. Chem. 1950, 42, 2303−2313. (9) Kraussold, H. Der Wärmeübergang in Rührgefässen (German). Chem. Ing. Tech 1951, 23, 177. (10) Oldshue, J. Y.; Gretton, A. T. Helical Coil Heat Transfer in Mixin vessels. Chem. Eng, Progr. 1954, 50, 615−621. (11) Bourne, J. R.; Dossenbach, O.; Post, T. Local and Average Mass and Heat Transfer due to Turbine Impellers. Fifth European Conference on Mixing, Wurzburg, West Germany, June 10−12; Stanbury, J., Ed.; 1985; pp 199−207. (12) Mohan, P.; Emery, A. N.; Al-Hassan, T. Review Heat Transfer to Newtonian Fluids in Mechanically Agitated Vessels. Exp. Therm. Fluid Sci. 1992, 5, 861−883. (13) Tatterson, G. B. Fluid Mixing and Gas Dispersion in Agitated Tanks, 2nd ed.; McGraw Hill: North Carolina, 1991; (14) Poling, B, E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 2nd ed.; McGraw-Hill: New York, 2000. (15) Dunlap, I.; Rushton, J, H. Heat Transfer Coefficients in Liquid Mixing using Vertical-Tube Baffles. Chem. Eng. Prog. Symp. Ser. 1953, No. 19, No. e137. (16) Havas, G.; Deak, A.; Sawinsky, J. Heat Transfer Coefficients in an Agitated Vessel using Vertical Tube Baffles. Chem. Eng. J. 1982, 23, 161−165. (17) Dostál, M.; Petera, k.; Rieger, F. Measurement of Heat Transfer Coefficients in an Agitated Vessel with Tube Baffles. Acta Polytechnica 2010, 50 (2), 46−57. (18) De Maerteleire, E. Heat Transfer in Turbine Agitated GasLiquid Dispersions. International Symposium On Mixing, Mons, Belgium, Feb 21−24; Eur. Federation of Chemical Engineering, 1978; pp XC7−XC35. (19) Ishibashi, K.; Litchtman, R. S.; Mahoney, L. H. Heat Transfer in Agitated Vessels with Special Types of Impellers. J. Chem. Eng. Jpn. 1979, 12 (3), 230−235.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS ́ University, and Unit Operations UFABC, Santa Cecilia Laboratory are fully acknowledged for supporting this research.

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NOMENCLATURE As = heat transfer area, m2 A = external surface area of the tank, m2 a = exponent in eq 3 b = exponent in eq 3 c = exponent in eq 3 Cpc = specific heat of could fluid, J/(kg °C) Cph = specific heat of hot fluid, J/(kg °C) Da = impeller diameter, m Dt = tank diameter, m E = distance of impeller from the bottom of the tank, m H = height of liquid at rest in the tank, m αi0 = heat transfer coefficient at the internal surface in relation of external surface, W/(m2 °C) αo = heat transfer coefficient at the external surface, W/(m2 °C) J = bank of tubes diameter, m k = thermal conductivity of fluid, W/(m °C) LMTD = logarithmic mean temperature difference, °C N = impeller rotation speed, 1/s Q = heat transfer rate in control volume, W Nb = number of bank tubes Nu = Nusselt number Pr = Prandtl number Re = Reynolds number T1 = temperature of water entering the tube baffle, °C t1 = temperature of sucrose solution entering the tank, °C T = temperature of water out of the tube baffle, °C t = temperature of sucrose solution out of the tank, °C ts = temperature of the outer surface of the tank, °C t00 = outside ambient temperature, °C U = overall heat transfer coefficient, W/(m2 °C) Vi = viscosity ratio me = mass flow of liquid in the tube baffle, kg/s ms = mass flow of liquid in the tank, kg/s W = length of blade impeller, m K = constant in eq 3 K* = constant in eq 6 y = constant in eq 6 W = workflow in control volume, W h = specific enthalpy, kJ/kg v = velocity of fluid, m/s g = acceleration of gravity, m/s2 z = elevation, m Evc = global energy in control volume, kJ M = mass of fluid in the tank, kg 2438


External Coefficient of Heat Transfer by Convection in Mixed Vessels

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