Local Heat Transfer Process for a GasâLiquid System in a Wall

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Local Heat Transfer Process for a Gas−Liquid System in a Wall Region of an Agitated Vessel Equipped with the System of CD6-RT Impellers Iwona Bielka, Magdalena Cudak, and Joanna Karcz* Department of Chemical Engineering, West Pomeranian University of Technology, al. Piastow 42, 71-065 Szczecin, Poland ABSTRACT: The results of the experimental studies of local heat process for a gas−liquid system in the region of the cylindrical wall of an agitated vessel equipped with the system of CD 6−RT impellers are presented. A lower CD 6 impeller (Smith turbine) and upper RT impeller (Rushton turbine) were located on the common shaft in a baffled agitated vessel of inner diameter equal to 0.3 m. Liquid height in the agitated vessel was equal to 0.6 m. Newtonian liquids of different physical properties were used as a continuous phase. Air was dispersed in the liquid. Local heat transfer coefficients were measured using both thermal and electrochemical methods. In total, 2280 experimental points were obtained. Distributions of the heat transfer coefficients were described by means of eqs 12−17 and 18−23 as a function of the Re and Pr numbers, dimensionless axial coordinate z/H, and modified Frg number, separately, for turbulent and transitional ranges of the fluid flow in the agitated vessel. Equations 12−17 and 18−23, concerning both coalescing and noncoalescing gas−liquid systems, have no equivalent in the open literature. require precise measuring sensors such as, for example, heat flux meters,23 microfoil sensors,17,18 or thermister probes.40 There are different variants of the measuring methods used to determine the local heat transfer coefficient, for example, the thermal method,12,23 where a local source of the heat transfer is applied (direct measuring method), or electrochemical method,3,42,50 based on an analogy between mass and heat transfer (indirect measuring method). Distribution of the heat transfer coefficient on the side of the agitated fluid is observed in a jacketed, baffled agitated vessel. This distribution depends, among others, on the number and type of impeller used (Figure 1). A single high-speed impeller generating radial fluid circulation in the vessel causes more differentiated profiles of the heat transfer coefficient (Figure 1b) than such an impeller which produces axial fluid circulation (Figure 1a). In both cases a and b, the highest values of the local heat transfer coefficient correspond to the level of the impeller. In tall vessels, more than one impeller is used in order to ensure thermal homogeneity along whole wall of the agitated vessel (Figure 1c and d). Literature data show that local heat transfer was studied for the Newtonian liquid of low viscosity agitated by single Pfaudler,3,50 HE 3,17,18 and CD 6 impellers,26 as well as by a single agitator eccentrically located in the vessel, namely, a propeller,31 HE 3 impeller,31 Rushton turbine,6 or A 315 impeller.6 Kanamori et al.22 compared distributions of heat transfer coefficients along the cylindrical wall of the agitated vessel equipped with single Rushton turbine, propeller, fourbladed pitched paddle, or MR210 impeller. Delaplace et al.8 investigated the heat transfer process for the Newtonian liquid of high viscosity agitated by means of the double helical ribbon supported by two vertical arms. Broniarz-Press and Rózȧ ńska4 determined distributions of the heat transfer coefficients for

1. INTRODUCTION Mixing is a unit operation which very often occurs in many industries, for example, the chemical, biochemical, food, cosmetics, or paints industries. Knowledge of hydrodynamics, mass, and heat transfer processes is necessary in order to correctly design an agitated vessel.44,47,59,61 The principles of the heat transfer process in agitated vessels and basic correlations which enable calculation of the heat transfer coefficient for different physical systems agitated in vessels of different geometrical configurations have been discussed in monographs39,44,58 and papers.43,25 Over the last 25 years the development of research in this field concerned mainly such problems as (a) mathematical modeling of the mean37,38 and local24 heat transfer coefficients; (b) optimization of the agitated vessel geometry for heat transfer process;5,25,29,33,36,45,56,59,60 (c) measurements of the mean heat transfer coefficient for viscous8,9,45,51,54,64 and non-Newtonian fluids,4,51,62,66 immiscible liquid mixtures,19 and nanosuspensions;49 (d) measurements of local heat transfer coefficient for liquid6,22,40 and gas−liquid systems;17,18,23,26,31,40 (e) testing of different types of the heating/cooling heat transfer surfaces in agitated vessels, i.e. jackets,29,32,35,41,45 helical coils,19,48,49,62 vertical tubular coils;13,34,54 (f) CFD modeling of heat transfer process in agitated vessels.1,9,21,52,67 Literature data show that the studies are focused on the heat transfer optimization and the effects of both rheological properties of the fluids agitated and geometry of the impeller−baffles−vessel system on heat transfer coefficients, as well as mathematical and numerical modeling of the heat transfer process. Developments in numerical modeling of this process follow from the possibility of applying such useful tools as commercial CFD codes. On the other hand, experimental methods used to determine distributions of the heat transfer coefficient on the heating/cooling surface of the jacket or coil © 2014 American Chemical Society

Received: Revised: Accepted: Published: 16539

July 28, 2014 August 28, 2014 September 29, 2014 September 29, 2014 dx.doi.org/10.1021/ie503003t | Ind. Eng. Chem. Res. 2014, 53, 16539−16549

Industrial & Engineering Chemistry Research

Article

patterns in a baffled vessel with two CD 6 impellers filled by a Newtonian liquid up to the height H = 1.4T (where the inner diameter of the vessel T = 0.48 m). Review of the literature results on the agitation of gas−liquid systems by means of single or dual CD 6 impellers shows that this impeller gives very good results compared to other types of impellers (for example, Rushton turbines). Taking into account wide possibilities of the application of such an impeller type in many processes, especially bioprocesses, full information on hydrodynamics,11 mass,15,46 and heat transfer are needed in order to control the agitation process. In this case, when heat transfer is the controlling mechanism, then knowledge of the heat transfer coefficient distributions on the wall of the jacketed agitated vessel is necessary in order to evaluate regions of different heat transfer intensity. Literature data28 show that correlations for the mean value of the heat transfer coefficient are presented only for gas−liquid systems agitated by means of the system CD 6 (lower impeller)−Rushton turbine (upper impeller) in a baffled agitated vessel within the turbulent and transitional ranges of the Newtonian fluid flow, respectively

Figure 1. Axial profiles of heat transfer coefficient α for single and doubles impellers: (a) axial flow, single impeller; (b) axial flow, double impellers on common shaft; (c) radial flow, single impeller;12 (d) radial flow, double impellers.

⎡ αT ⎤ Num = ⎢ ⎥ = 0.738Re 0.67 Pr 0.33Vi 0.14(1 − 13.63Frg 0.43) ⎣ λ ⎦m (1)

non-Newtonian fluid agitated using a single Rushton turbine or pitched blade turbines. The effects of the gas bubbles and solid particles on the distributions of heat transfer coefficients in gas−liquid,26 solid−liquid,27 and gas−solid−liquid systems30 were experimentally analyzed for the jacketed baffled agitated vessels equipped with single Rushton turbine30 or CD 6 impeller.26,27 When a gas−liquid system is agitated in a tall jacketed baffled vessel equipped with two high-speed impellers on the common shaft, the lower impeller should ensure sufficient dispersion of the gas phase whereas the upper impeller should maintain required fluid circulation. A radial flow Rushton turbine is often used as a lower impeller to disperse gas in a liquid phase. Karcz23 used a local heat source to measure of the heat transfer coefficient for gas−liquid systems in a jacketed baffled agitated vessel equipped with a double system of Rushton turbines. Distributions of this coefficient within the turbulent range of the Newtonian fluid flow were described by means of the equations. Literature data2,14,53,63,65 show that turbine disc impellers with curved blades can be used as a lower impeller in an agitated vessel because they enable good gas dispersion in a liquid phase and simultaneously consume less power than a Rushton turbine. A turbine disc impeller with six concave, semicircular blades, known as a CD 647 (or Smith) impeller is recommended for agitation of gas−liquid systems in many industrial processes and bioprocesses.20 For gas−liquid systems agitated using single CD 6 impellers, hydrodynamics and mass transfer,55,57,63,65,68 as well as heat transfer,26,27,32,33 have been experimentally studied. On the basis of experimental tests of dual impeller combination of CD 6 impellers in fermentation vessels, Junker et al.20 stated that the use of such impeller systems affects the increase of the mass transfer rate and the decrease of the power consumption in the gas−liquid system compared to the results obtained for the dual Rushton turbines on the common shaft. Recently, a method of numerical simulations has been used for the hydrodynamics and mass transfer modeling in gas−liquid systems agitated by means of the CD 6 impeller.10,16 Devi and Kumar,10 using methods of computational fluid dynamics (CFD), modeled the flow

⎡ αT ⎤ Num = ⎢ ⎥ = 2.497Re 0.58Pr 0.33Vi 0.14(1 − 2.25Frg 0.22) ⎣ λ ⎦m (2)

Equations 1 and 2 approximate the results within the following range of the dimensionless numbers 104 < Re < 9 × 104, 7 < Pr < 10, 0 ≤ Frg ≤ 1.27 × 10−5 and 70 < Re < 700, 1500 < Pr < 2000, 0 ≤ Frg ≤ 9.2 × 10−6, respectively. The modified Froude number Frg = wog2/gT describes the effect of the gas phase on the value of the mean heat transfer coefficient. Equations 1 and 2 describe experimental results with maximal relative errors of ±15% and ±12%, respectively. Exponents at Re and Frg numbers in eqs 1 and 2 depend on the range of the fluid flow in the agitated vessel. Both exponents are higher for the turbulent range of the fluid flow (eq 1) compared to those obtained for the transitional range of the fluid flow (eq 2). The value of the exponent at Re number equal to 0.67 in eq 1 agrees with the exponent value ascribed to others correlations proposed in literature44,58 within the turbulent range of the liquid flow. The value of the exponent at Re number in eq 2 is lower than 0.67 because turbulence has not been fully developed yet within the transitional range of the fluid flow. Exponent values at Frg numbers in eqs 1 and 2 differ by about two times. This shows that an effect of the gas phase on the mean value of the heat transfer coefficient is greater within the turbulent range of the fluid flow. However, the correlations to calculate distributions of heat transfer coefficient for such geometrical and physical systems are lacking in the literature. Therefore, the aim of the present work was to determine experimentally the distributions of the heat transfer coefficient for gas−liquid systems in the vicinity of the inner cylindrical wall of the agitated vessel equipped with system of the CD 6−RT impellers.

2. EXPERIMENTAL SECTION Experimental studies of local heat transfer coefficients α in a liquid phase and gas−liquid system were carried out in a tall, jacketed, baffled vessel, which was equipped with two highspeed impellers on a common shaft (Figure 2a). The 16540

dx.doi.org/10.1021/ie503003t | Ind. Eng. Chem. Res. 2014, 53, 16539−16549


Article

Figure 2. (a) Geometrical parameters of the agitated vessel used to measure heat transfer coefficients by means of the thermal method. (b) Arrangement of the measuring points in the electrochemical method.

0.75D was located at a distance e = 0.5D from the vessel bottom. Systems of air−Newtonian liquids differing in the viscosity of the continuous phase were agitated within the transitional and turbulent regime of the fluid flow. Distilled water, 30% aqueous solution of glucose, aqueous solution of electrolyte used in electrochemical method and technical glycerol were used as a liquid phase. Liquid viscosity was changed from the 10−3 Pa·s for the water to the 240 × 10−3 Pa·s for the glycerol. Coalescing and noncoalescing gas−liquid systems were tested. Gas bubbles in the air−glycerol system did not have the capability to coalesce, whereas the coalescence of the bubbles occurred in the systems with the distilled water and aqueous solutions of glucose or electrolyte as the continuous phases. The measurements were carried out for changing impeller speed n and superficial gas velocity wog (where wog = 4Vg/πT2, 0 < wog [m/s] < 6 × 10−3). The lowest impeller speeds used in the experiments enabled dispersion of gas bubbles under the lower impeller. Studies of the local heat transfer coefficients were carried out using thermal and electrochemical methods. Thermal measurements were conducted, according to idea described in refs 12 and 23 by means of the local heat source, which was built at the vessel wall at the height z (Figure 2a). Positions of the local heat source corresponded to the following seven values of the dimensionless axial coordinate z/H equal to 0.07, 0.16, 0.28, 0.42, 0.56, 0.67, or 0.82. The local heat flux meter (Figure 4) built into the agitated vessel wall (3) consisted of a measuring cylindrical element (1) of diameter 2R = 30 mm and

transparent cylindrical wall of the vessel was made of organic glass. The vessel of inner diameter T = 0.3 m was filled by a liquid up to the height H = 2T. Four (J = 4) planar baffles of width B = 0.1T were arranged symmetrically around the cylindrical wall of the vessel. Two impellers of diameter D = 0.33T were located on the shaft at the distance h1 = 0.167H and h2 = 0.67H, respectively, from the flat bottom of the vessel. Clearance between both impellers Δh was equal to h1 − h2 = T. A Rushton turbine (Figure 3a) and CD 6 impeller (Figure 3b) were used as upper and lower impellers, respectively. Both impellers had six blades (Z = 6) of length a = 0.25D and width b = 0.2D. Curvature radius R for the CD 6 impeller was equal to b/2. A gas sparger in the shape of the ring of diameter dd =

Figure 3. Geometrical parameters of the impellers used: (a) Rushton turbine; (b) CD 6 impeller (Smith turbine). 16541



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⎡ ⎛ l ⎞⎤ ⎛ l ⎞ Tw = T1⎢1 − ⎜ 2 ⎟⎥ + ⎜ 2 ⎟T2 ⎢⎣ ⎝ l1 ⎠⎥⎦ ⎝ l1 ⎠

(6)

Analysis of the effect of the size of the heat flux meter on the local heat transfer coefficient shows23 that the temperature profile within the thermal boundary layer is stabilized for the measuring element of diameter 30 mm, therefore the calculated value of α does not require correction. Electrochemical measurements of the local heat transfer coefficient α were performed using a computer-aided method where diffusion current Id, proportional to the local mass transfer coefficient kA, is measured and an analogy between mass and heat transfer processes is assumed, i.e. ⎛ Sc ⎞2/3 ⎛ Sc ⎞2/3 Id αanal = c pρ⎜ ⎟ kA = c pρ⎜ ⎟ ⎝ Pr ⎠ ⎝ Pr ⎠ CAzeFS

Figure 4. Scheme of the local heat flux meter. 1−measuring cylindrical element; 2−surrounding ring; 3−wall of the agitated vessel; 4−sleeve; 5−cover; 6−thermocouples; 7−gland; 8−collection plates; 9−distance ring; 10−O-ring; 11−silicon seal.

where cp and ρ are specific heat and density of the electrolyte, CA is the concentration of component A in the electrolyte, ze is the number of electrons taking part in the reaction, S is the surface area of the cathode, F is Faraday’s constant, Sc and Pr are the Schmidt and Prandtl numbers. The effect caused by developing of the concentration profile in the liquid laminar boundary layer should be considered when small electrodes are used as measuring points. In order to eliminate the entrance effect, the result of the measurement should be corrected by means of the calibration factor p discussed in ref 26 α = αanalp (8)

surrounding ring (2) of diameter 60 mm. Both elements (1) and (2) were made of stainless steel. The role of the surrounding ring (2) was to screen and limit thermal losses of the measuring element (1). Both elements (1) and (2) were fitted into the sleeve (4), made of polyamide. The cover (5) closed the sleeve (4) in such a way that a chamber was formed inside. The chamber was provided with steam of temperature 102 °C. Condensate continuously flowed out to the condenser pot. In the chamber were collection plates (8) and distance rings (9). Plates (8) were used to separate “false” condensate, achieved as a result of steam condensation on the cool walls of the housing, from the condensate obtained during the steam condensation on the back part of the measuring element. Additionally, plates (8) protected the wall of the inner block against a direct blow of the steam flow. A silicon seal was used to fix sleeve (4) with the vessel wall (3), and an O-ring (10) sealed the cover (5). Two thermocouples (6) Fe−CuNi collected in a narrow sleeve were located in the inner measuring block. The sleeve with thermocouples was sealed by means of the gland (7). The front thermocouple was situated nearer to the contact surface of the heat flux meter with the agitated liquid. The distance between both thermocouples was equal to l1 = 73 mm, whereas the distance between front thermocouple and the contact surface of the measuring element with agitated liquid was equal to l2 = 2 mm. Local heat transfer coefficient α was calculated from the following equation q qw α= w = ΔT Tw − Tm (3)

In the computer-aided electrochemical method, 64 nickel circular cathodes with a diameter 4 mm, built into the wall of the agitated vessel (Figure 2b) were used as measuring sensors. The cathodes were arranged in four columns on one-quarter of the vessel wall. The sensors in the third column were located at an angle position before the baffle equal to 22.5°, whereas those in the second column were at an angle position behind the baffle equal to 22.5°. Clearance between sensors in the fourth column was equal to 25 mm. Cathodic reduction of the potassium ferricyanide K3Fe(CN)6 was employed as a measuring system. In each measuring point (Figure 2b), the measuring digital voltage signal was sampled 200 times. Diffusion current Id representative for a given measuring point was calculated automatically based of the averaged value from the sampling. This experimental method was described in detail in refs 7, 26, and 27. For each of the 64 measuring points (four values of the angle coordinate and different values of dimensionless axial coordinate z/H) in the electrochemical method, experiments were carried out for 4 values of the impeller speed n and 4 values of the superficial gas velocity wog. A set of 1024 data points was obtained. For each of the 7 measuring points (different values of the z/H) in the thermal method, experiments were performed for 12 values of the impeller speed n, 5 values of the superficial gas velocity wog, and 3 types of liquid as the continuous phase. A set of the 1260 data points was obtained. Therefore, in total, 2284 experimental values of the heat transfer coefficient were obtained using both thermal and electrochemical methods.

where qw is the heat flux, ΔT is the driving difference of temperature between the outer wall of the measuring element Tw and agitated liquid Tm. For steady-state conditions of measurement, heat flux qw = q1 = q2, where q1 =

λ (T1 − T2) l1

(4)

q2 =

λ (T1 − Tw ) l2

(5)

(7)

3. RESULTS AND DISCUSSION For a gas−liquid system, local heat transfer coefficient α is a function of the position on the cylindrical wall of the vessel, i.e. axial coordinate z/H, physical properties of the fluid, impeller

Taking into account the constraint q1 = q2, wall temperature Tw was calculated from the equation 16542



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speed n, and superficial gas velocity wog. Therefore, this dependence can be expressed using the following equation ⎡ αT ⎤ Nu = ⎢ ⎥ = C(z /H , wog)Re APr BVi E ⎣ λ ⎦

(9) 58

Equation 9 with the assumed literature data values of the exponents B = 0.33 and E = 0.14 was applied to correlate our experimental values of the local coefficient α. The value of the exponent A was estimated statistically for each of the measurement series (at given levels of z/H and wog values) on the basis of the dependence Nu = C(z /H , wog)Re A Pr Vi 0.14 0.33

Figure 7. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for 30% aqueous solution of glucose (wog = 0) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using the thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82.

(10)

For the turbulent flow of the distilled water and 30% aqueous solution of glucose as continuous phases, A = 0.65 ± 0.09 was obtained independently on the level of the axial coordinate z/ H. The value A = 0.6758 was assumed in further analysis of the heat transfer coefficients within this region of the fluid flow. For the transitional flow of glycerol as the continuous phase, a lower value of the exponent of the Reynolds number equal to A = 0.58 ± 0.05 was obtained and used in further calculations. Examples of the experimental dependence Nu/Pr0.33Vi014 = f(Re) for liquid phase (wog = 0) and gas−liquid systems (wog = 5.1 × 10−3 m/s) are shown in Figures 5−10. All the points are

Figure 8. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for the 30% aqueous solution of glucose−air system (wog = 5.1 × 10−3 m/s) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82.

aqueous solution of glucose as the liquid phase, respectively, whereas Figures 9 and 10 illustrate the data for glycerol. Solid Figure 5. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for distilled water (wog = 0) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using the thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82.

obtained by means of the thermal method. Figures 5 and 6 and Figures 7 and 8 present the results for distilled water and a 30%

Figure 9. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for glycerol (wog = 0) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using the thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82.

lines in Figures 5−10 correspond to eq 10 with the values of exponent A = 0.67 and A = 0.58 for the turbulent and transitional ranges of the fluid flow, respectively. This equation well approximates experimental data for all the presented cases and the exponents of A do not depend on the axial coordinate z/H. These exponents are dependent only on the range of the Reynolds number. From analysis of the experimental data shown in Figures 5−10, it results that heat transfer coefficients depend on the

Figure 6. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for the distilled water−air system (wog = 5.1 × 10−3 m/s) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using the thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82. 16543



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responsible for the decreasing of the maximal value of the coefficient C in this region. The distributions C = f(z/H) in Figure 11 show that the most intensive heat transfer takes place in the vicinity of the vessel wall for single liquid phase (wog = 0). The presence of the gas phase in the agitated liquid system affects the diminishing heat transfer intensity, especially in the zones of both impellers used. Distributions of the heat transfer coefficient along the whole height of the cylindrical wall of the agitated vessel, i.e. within the range of the geometrical parameter z/H ∈ ⟨0; 1⟩ were approximated by means of the following equation ⎡ αT ⎤ Nu = ⎢ ⎥ = C1Re APr 0.33Vi 0.14f1 (z /H )f2 (Frg) ⎣ λ ⎦

Figure 10. Dependence Nu/(Pr0.33Vi0.14) = f(Re) for the glycerol−air system (wog = 5.1 × 10−3 m/s) and different positions of the dimensionless axial coordinate z/H. Experimental points obtained using the thermal method: z/H = (Δ) 0.07, (○) 0.17, (■) 0.42, (●) 0.67, (▲) 0.82.

(11)

where the value of exponent A is equal to 0.67 for the turbulent range of the fluid flow (104 < Re < 9 × 104), and A is equal to 0.58 for the transitional range of the flow (70 < Re < 700). Froude number for gas phase, defined as Frg = wog2/(gT), describes the effect of the superficial gas velocity wog on the value of the heat transfer coefficient. Functions f1 and f 2 have different forms depending on the ranges of the z/H and fluid flow. Functions C1 f1 f 2 in eq 11 are collected in Tables 1 and 2. Equations 12−17 describe the experimental results, obtained by means of both thermal and electrochemical methods, within the turbulent range of the fluid flow (104 < Re < 9 × 104; 6.4 < Pr < 21.5; 0 < Frg < 1.27 × 10−5). Equations 18−23 describe the data within the transitional range of the fluid flow (70 < Re < 560; 1766 < Pr < 2260; 0 < Frg < 9.21 × 10−5). As it results from the analysis of the equations given in Tables 1 and 2, function f1 depends only on the dimensionless axial coordinate z/H. However, function f 2 is more complicated because, with the exception of the impeller region, it takes into consideration the effects of superficial gas velocity wog and geometrical parameter z/H on the local heat transfer coefficient. Equations 13 and 16 corresponding to both lower (CD 6) and upper (RT) impeller zones and the turbulent range of the of the liquid flow (wog = 0; Frg = 0) show that local heat transfer coefficient is higher about of 22% for the upper Rushton turbine compared to the lower CD 6 impeller. This relation is analogous to the results obtained for the dual system of

position on the wall of the agitated vessel from which they are measured. Experimental points corresponding to the location of the lower and upper impellers (z/H = 0.17 and 0.67, respectively) are situated on the top of the diagrams. The points lying in the lowest part of the diagrams correspond to the value of the dimensionless axial coordinate z/H equal to 0.42, i.e. position between both impellers on the shaft. Examples of axial distributions of the coefficient C(z/H, wog) from eq 10 for the systems with different types of liquid phase are presented in Figure 11. Local values of this coefficient were calculated on the basis of experimental data obtained by means of the thermal method. Figures 11a and b show the data for the turbulent flow of the distilled water and 30% aqueous solution of glucose, whereas Figure 11c illustrates experimental data for the transitional range of the glycerol as liquid phase in the agitated vessel. All the distributions in Figure 11 have two maxima located on the height z/H corresponding to the positions of lower and upper impellers, respectively. Maximal values of the coefficient C depend on the type of the impeller. They are higher for the upper Rushton turbine which generates strictly radial circulation of the fluid at the impeller plane. The curvature of the blades of the CD 6 impeller and a large amount of nondispersed gas under the impeller weaken radial fluid circulation in the region of the lower impeller, and they are

Figure 11. Dependence C(z/H, wog) = f(z/H) from eq 10 for the air−liquid systems differing in the physical properties of the liquid phase: (a) distilled water; (b) 30% aqueous solution of glucose; (c) glycerol. Data obtained using the thermal method: wog = (●) 0, (□) 3.4 × 10−3; (■) 4.3 × 10−3; (Δ) 5.1 × 10−3; (▲) 6.0 × 10−3 m/s. 16544



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Table 1. Functions C1 f1 f 2 in Equation 11 for Gas−Liquid Systemsa no.

range of z/H

C 1 f1 f 2

1

(0; 0.15)

⎤ ⎡⎛ z ⎞ ⎤0.615 ⎡ ⎛ z ⎞1.18 C1f1 f2 = 3.178⎢⎜ ⎟ + 0.005⎥ exp⎢− 399.4⎜ ⎟ Frg 0.5⎥ ⎝ ⎠ ⎝ ⎠ ⎣ H ⎦ H ⎣ ⎦

(12)

2

⟨0.15; 0.184⟩

⎛ z ⎞0 C1f1 f2 = 1.01⎜ ⎟ (1 − 39.84Frg 0.5) ⎝H⎠

(13)

3

(0.184; 0.458⟩

−0.88⎡

⎤ ⎛z⎞ ⎛z⎞ ⎢5.255⎜ ⎟ − 1.746⎜ ⎟ + 1.927⎥ ⎝ ⎠ ⎝ ⎠ H H ⎣ ⎦

⎛z⎞ C1f1 f2 = 0.128⎜ ⎟ ⎝H⎠

2

⎡ ⎛ z ⎞−0.636 0.5⎤ exp⎢− 14.99⎜ ⎟ Frg ⎥ ⎝H⎠ ⎣ ⎦ 4

(0.458; 0.65)

(14)

−7.22 ⎡

⎛z⎞ C1f1 f2 = 0.1285⎜ ⎟ ⎝H⎠

⎤ ⎛z⎞ ⎛z⎞ ⎢15.895⎜ ⎟ − 15.456⎜ ⎟ + 3.7606⎥ ⎝ ⎠ ⎝ ⎠ H H ⎣ ⎦ 2

⎧⎡ ⎫ ⎤ ⎛ z ⎞2 ⎛z⎞ exp⎨⎢− 678.9⎜ ⎟ + 698.8⎜ ⎟ + 207.5⎥Frg 0.5⎬ ⎝ ⎠ ⎝ ⎠ H H ⎦ ⎩⎣ ⎭ 5

⟨0.65; 0.683⟩

6

(0.683; 1)

⎪

⎪

⎪

⎪

(15)

⎛ z ⎞0 C1f1 f2 = 1.236⎜ ⎟ (1 − 38.12Frg 0.5) ⎝H⎠ −0.176 ⎡

(16)

⎤ ⎛z⎞ ⎛z⎞ ⎢1.718⎜ ⎟ − 4.81⎜ ⎟ + 3.336⎥ ⎝ ⎠ ⎝ ⎠ H H ⎣ ⎦

⎛z⎞ C1f1 f2 = 1.354⎜ ⎟ ⎝H⎠

2

⎧⎡ ⎛ z ⎞ 2 ⎫ ⎤ ⎛z⎞ exp⎨⎢616⎜ ⎟ − 1095.2⎜ ⎟ + 420.7⎥Frg 0.5⎬ ⎝ ⎠ ⎝ ⎠ H H ⎦ ⎩⎣ ⎭ a

⎪

⎪

⎪

⎪

(17)

Turbulent range of the fluid flow (104 < Re < 9 × 104; 6.4 < Pr < 21.5; 0 < Frg < 1.27 × 10−5).

Table 2. Functions C1 f1 f 2 in Equation 11 for Gas−Liquid Systemsa no.

range of z/H

1

(0; 0.15)

⎤ ⎡⎛ z ⎞ ⎤0.755 ⎡ ⎛ z ⎞0.34 C1f1 f2 = 16.56⎢⎜ ⎟ + 0.0062⎥ exp⎢− 11.8⎜ ⎟ Frg 0.26⎥ ⎝ ⎠ ⎣⎝ H ⎠ ⎦ H ⎣ ⎦

C 1 f1 f 2

(18)

2

⟨0.15; 0.184⟩

⎛z⎞ C1f1 f2 = 4.08⎜ ⎟ (1 − 3.068Frg 0.21) ⎝H⎠

(19)

3

(0.184; 0.42⟩

⎤ ⎛ z ⎞−2.16⎡ ⎛ z ⎞2 ⎛z⎞ ⎢11.89⎜ ⎟ − 4.263⎜ ⎟ + 0.831⎥ C1f1 f2 = 0.2355⎜ ⎟ ⎝H⎠ ⎝H⎠ ⎝H⎠ ⎣ ⎦

0

⎧⎡ ⎫ ⎤ ⎛ z ⎞2 ⎛z⎞ exp⎨⎢− 2.83·102⎜ ⎟ + 1.8·102⎜ ⎟ − 29.19⎥Frg 0.25⎬ ⎝ ⎠ ⎝ ⎠ H H ⎣ ⎦ ⎭ ⎩ 4

(0.42; 0.65)

⎪

⎪

⎪

⎪

⎛z⎞ C1f1 f2 = 10.084⎜ ⎟ ⎝H⎠

2.12 ⎡

(20)

⎤ ⎛z⎞ ⎛z⎞ ⎢31.32⎜ ⎟ − 32.51⎜ ⎟ + 9.215⎥ ⎝ ⎠ ⎝ ⎠ H H ⎣ ⎦ 2

⎡ ⎤ ⎛ z ⎞1.68 exp⎢− 87.25⎜ ⎟ Frg 0.4 ⎥ ⎝ ⎠ H ⎣ ⎦

a

(21)

5

⟨0.65; 0.683⟩

⎛z⎞ C1f1 f2 = 5.33⎜ ⎟ (1 − 15.369Frg 0.33) ⎝H⎠

6.

(0.683; 1)

⎫ ⎤ ⎛ z ⎞ ⎧⎡ ⎛ z ⎞2 ⎛z⎞ C1f1 f2 = 0.932⎜ ⎟exp⎨⎢2.06 × 102⎜ ⎟ − 5.84⎜ ⎟ − 1.35 × 102⎥Frg 0.4 ⎬ ⎝ H ⎠ ⎩⎣ ⎝H⎠ ⎝H⎠ ⎦ ⎭

0

(22)

⎪

⎪

⎪

⎪

(23)

Transitional range of the fluid flow (70 < Re < 560; 1766 < Pr < 2260; 0 < Frg < 9.21 × 10−5).

Rushton turbines on the common shaft23 where the local heat transfer coefficient for impeller zones corresponding to the upper Rushton turbine was higher than that for the lower impeller, but by only about 5%. Comparison of the results obtained for both RT−RT and CD 6−RT impeller systems in the impeller zones and liquid phase shows that lower values of the local coefficient α are ascribed to the CD 6−RT impeller system by about 23% and 5%, respectively, for the lower and upper impellers. Analysis of the eqs 19 and 22 corresponding to

both lower (CD 6) and upper (RT) impellers in the impeller zones and transitional range of the liquid phase (wog = 0; Frg = 0) shows that local coefficient α is higher by about 30% for the upper Rushton turbine compared to the data for the lower CD 6 impeller. Comparison between experimentally obtained values of local heat transfer coefficient and those calculated from equations collected in Tables 1 and 2 show Figures 12 and 13, where the dependences Nu/ReAPr0.33Vi014 = C1 f1 f 2 = f(z/H) are presented 16545



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for different liquid phases and both cases, i.e. without (wog = 0, Figures 12a and 13a) and with gas phase (wog = 4.3 × 10−3 m/s, Figures 12b and 13b), in the physical system. In Figures 12 and 13, solid lines correspond to the eqs from Tables 1 and 2. Equations presented in Tables 1 and 2 approximate experimental data with good accuracy. Mean relative errors of both sets of equations (12−17 and 18−23), describing turbulent and transitional ranges of the fluid flow, are equal to ±10% and ±13%, respectively. Therefore, it may assume that local heat transfer coefficients for mechanically agitated gas− liquid systems in a jacketed vessel can be determined on the basis of the eqs 12−23. Moreover, it follows from the fact that experimental data and those calculated from proposed equations (expressed in the form of the C1 f1 f 2) compared in Figure 14 for turbulent and transitional ranges of the fluid flow are consistent. Figure 12. Comparison between experimental values of the function [Nu/(Re0.67Pr0.33Vi0.14)] = Cf1 f 2 = f(z/H) and those calculated from eqs 12−17: (a) wog = 0; (b) wog = 4.3 × 10−3 m/s. Liquid: distilled water (•); 30% aqueous solution of glucose (+); electrolyte (○). Gas: air; turbulent range of the fluid flow.

4. CONCLUSIONS The results of the experimental study of the local heat transfer process at the vicinity of the wall region of the jacketed baffled vessel equipped with a CD 6−RT impellers system were obtained for coalescing and noncoalescing air−Newtonian liquid systems within the turbulent and transitional regimes of the fluid flow. These findings obtained on the basis of the 2284 data can be summarized as follows: Experimentally it was proved that the distributions of the heat transfer coefficient are strongly differentiated on the vessel wall significantly depending on the position of the measuring point on the vessel wall (axial coordinate z/H), as well as Re, Pr, and Frg numbers. Local heat transfer coefficients have the highest values at the impeller zones; however, for the region of wall corresponding to the lower CD 6 impeller, these values are lower about of 23% and 30%, respectively, for turbulent and transitional regimes of the fluid flow, compared to the upper Rushton turbine. Moreover, local heat transfer coefficients diminish with the increase of the superficial gas velocity, especially, at the impeller zones. The distributions of the heat transfer coefficient were approximated mathematically by means of the dimensionless eqs 11−23 as functions of the z/H, Re, Pr, and Frg numbers. Equations 12−17 and 18−23 have no equivalent in the open literature. The results obtained have of importance for such

Figure 13. Comparison between experimental values of the function [Nu/(Re0.67Pr0.33Vi0.14)] = Cf1 f 2 = f(z/H) and those calculated from eqs 18−23 for the glycerol−air system: (a) wog = 0; (b) wog = 4.3 × 10−3 m/s; transitional range of the fluid flow; • − experimental points.

Figure 14. Comparison of the C1 f1 f 2 values obtained experimentally and calculated from eqs 12−17 for gas−liquid systems. (a) Turbulent range of the flow. Liquid phase: distilled water (○); 30% aqueous solution of glucose (□); electrolyte (Δ). (b) Transitional range of the flow. Liquid phase: glycerol (■). 16546



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mixing processes where heat transfer is the controlling mechanism, for example, for bioprocesses. Moreover, these results can be useful in order to verify the data obtained using numerical methods.

■

■

AUTHOR INFORMATION

Corresponding Author

*Telephone number: +48 91 449 43 35. Fax number: +48 91 449 46 42. E-mail: [email protected].

Nu = αT/λ Pr = cpη/λ Re = nD2ρ/η Sc = ν/DA Vi = η/ηw

REFERENCES

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Notes

The authors declare no competing financial interest.

■

NOMENCLATURE A, B, E = exponents in Nusselt equation a = length of the impeller blade, m B = width of baffle, m b = width of the impeller blade, m C = constant in Nusselt equation CAo = concentration of the component A in an electrolyte, kmol/m3 cp = specific heat, J/kg·K D = impeller diameter, m DA = diffusion coefficient, m2/s dd = sparger diameter, m e = clearance between gas sparger and vesell bottom, m F = Faraday’s constant g = acceleration due to gravity, m/s2 H = liquid height in the agitated vessel, m h1, h2 = distance of the impellers from the vessel bottom, m Id = diffusion current, A J = number of baffles kA = mass transfer coefficient, m/s l1 = distance between thermocouples, m l2 = distance between front thermocouple and contact surface of heat flux meter with liquid, m n = impeller speed, 1/s p = correction factor qw = heat flux, W/m2 R = curvature radius of the impeller blade, m R = radius of the measuring element, m S = surface area of the cathode, m2 T = inner diameter of the agitated vessel, m T = temperature, K Tm = mean temperature of the agitated liquid, K Tw = temperature of the vessel wall, K Vg = gas flow rate, m3/s wog = superficial gas velocity, m/s Z = number of impeller blades z = axial coordinate, m ze = number of electrons taking part in the reaction

Greek Letters

α = local heat transfer coefficient, W/m2·K η = dynamic viscosity of the liquid, Pa·s λ = conductivity, W/m·K ν = kinematic viscosity, m2/s ρ = liquid density, kg/m3

Subscripts

anal = denotes analogy m = averaged value w = denotes wall Dimensionless Numbers

Frg = wog2/gT 16547



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