Robinson, R. L., Jacoby, R. H., Hydrocarbon Process., 44(4),141 (1965). Rowlinson, J. W., “Liquids and Liquid Mixtures”, 2d ed,Chapter 9,Butterworth, London, 1969. Saddington, A. W., Krase, N. W., J. Am. Chem. SOC., 56, 353 (1934). Sage. 0. H., Lacey, W. N., Schaafsman, J. G.,Ind. Eng. Chem., 26, 214
(1934).
Takenouchi, S.,Kennedy, G. C., Am. J. Sci., 262, 1055 (1964). Tanner, C. C., Masson, f., Pruc. Roy. Suc.,Ser. A, 126, 268 (1930). Towned, D. T. A., Bhatt, L. A.. Proc. Roy. SOC., Ser. A, 134,502 (1932). Vera, J. H., Prausnitz, J. M., Chem. Eng. J., 3, l(1972). Watson, G.W., Stevens, A. B., Evans, R. B., Ill, Hodges,D., Jr., lnd. Eng. Chem.,
46,362 (1954).
Sage, B. H., Lacey, W. N., Ind. Eng. Chem., 31, 1497 (1939). Sanchez, M., Lentz, H., High Temp-high press., 5, 689 (1973). Sass, A., Dodge, B. F., Bretton, R. H., J. Chem. Eng. Data, 12, 168 (1967). Selleck, F. T., Carmichel. L. T.. Sage, 8. H., Ind. Eng. Chem., 44, 2219
Welsch, H., Dissertation, Institute for Physical Chemistry at Karl-,
Gemany,
1973. Wiebe.
R.,Gaddy, V. L., J. Am. Chem. SOC., 60, 2300 (1938).
,.nc*,
(IYJLJ.
Received for review February 11,1976 Accepted June 7,1976
Solbrig, C. W., Ellington, R. T., Chem. Eng. Prog. Symp. Ser., 59, No. 44,127
(1963).
Heat Transfer to Pseudoplastic Fluids in an Agitated Vessel S. Suryanarayanan, B. A. Mujawar, and M. Raja Rao* Depadment of Chemical Engineering, Indian Institute of Technology, Bombay 400 076, India
The vessel inside and coil outside film heat transfer coefficients of water and dilute aqueous polymer solutions of sodium carboxymethyl cellulose (SCMC) and sodium alginate (SA) have been studied in a turbine-agitated vessel for standard and nonstandard vessel configurations with agitator diameter, depth of agitation, helix diameter, and coiled tube outside diameter as parameters. The jacket- and coil-side heat transfer results are correlated.
Introduction In recent years, there has been an increasing interest in the field of mechanically aided heat transfer to non-Newtonian fluids which are often encountered in processing industries such as petroleum, plastics, paints, paper, nuclear fuels, cosmetics, cellulose, and foods. There are many situations where such fluids have to be heated or cooled in an agitated vessel and non-Newtonian characteristics can be of considerable significance. Voluminous literature has been built up on heat transfer to Newtonian and nowNewtonian fluids in an agitated vessel which is conforming to standard vessel configuration. Many times, in industry, one has to make use of the available agitated vessel with its accessories, which are not conforming to standard vessel configuration (SVC). Under these circumstances it becomes very difficult (but essential) to know beforehand the heat transfer performance of such nonstandard vessel configuration (non-SVC) to put it into operation with a fair degree of confidence. The present work was, therefore, undertaken to study systematically the heat transfer characteristics of the agitated vessel for which agitator diameter, depth of agitation, helix diameter, and coiled tube outside diameter deviate from the values pertaining to the standard vessel configuration using water and different pseudoplastic test liquids of industrial importance. Previous Work Carreau et al. (1966) studied jacket side heat transfer for SVC with pseudoplastic liquids under unsteady-state conditions. These authors defined a generalized Reynolds number which is applicable only for straight-tube flow. Further, these authors were unable to define a generalized Prandtl number for agitated fluids and hence tried to use differential viscosity in the Prandtl number with little success. Hagedorn and Solamone (1967) used four different impellers to get a design correlation for the prediction of the batch heat transfer coefficient for pseudoplastic fluids in agitated vessels. The 564
Ind. Eng. Chern., Process Des. Dev., Vol. 15, No. 4, 1976
definitions of generalized agitated Reynolds and Prandtl numbers used by these authors are not appropriate. Pandian and Rao (1970) investigated the coil side and jacket side film heat transfer coefficients for pseudoplastic liquids in standard vessel configuration using effective viscosity in generalizing agitated Reynolds number. Since the Prandtl number generalized in this manner did not correlate their heat transfer results, differential viscosity was used in the Prandtl number. Polard and Kantyka (1969) studied the anchor agitator operating in jacketed vessels of flat and hemispherical bottoms for pseudoplastic fluids. These authors studied the effect of helix diameter and coiled tube outside diameter on heat transfer and proposed the following correlation
The exponent of dolDt and Dc/Dt were taken from Jha and Rao (1967), who studied heat transfer to Newtonian fluids. The significant study in the field of coil-side heat transfer was made by Skelland and Dimmick (1969), who carried out experiments in a propeller-agitated vessel with a submerged coil to account for the effect of the agitator diameter and coiled tube outside diameter; they brought out the following correlation >
Sandal1 and Pate1 (1970) investigated the vessel inside film heat transfer coefficients for pseudoplastic liquids (0.348 i n i 0.887) in a baffled, jacketed vessel, equipped with turbine and anchor impellers for standard vessel configuration only.
---S--%-
HOT WATER COOLING WATER DRAIN LINE
Figure 1. Schematic flow diagram of the experimental setup: 1,Hot water tank; 2, immersion heaters; 3, hot water pump; 4, needle valve; 5, rotameter; 6, cold water tank; 7, cold water pump; 8, needle valve; 9, rotameter; 10, thermometer; 11,baffle; 12, jacket; 13, agitated vessel; 14, thermometer; 15, helical coil; 16, turbine agitator; 17-19, thermometers; 20, agitator shaft; 21, reduction gear; 22, sliding pulley; 23, V-belt; 24, fixed step pulley; 25, induction motor; 26, three-way cock.
Using effective viscosity technique, Heinlein and Sandal1 (1972) reported some interesting work on heat transfer to pseudoplastic types of power law liquids and Bingham plastic slurries in an anchor-agitated vessel. Yagi and Yoshida (1975) have recently used an apparent viscosity technique proposed by Metzner and Otto (1957) in their gas absorption studies. The literature survey, thus, reveals that only two nonstandard vessel configuration parameters were studied at a time in order to correlate either coil-side or jacket-side heat transfer coefficients. Therefore, it was felt necessary to investigate the effect of four important parameters which deviate from SVC and thus get useful correlations for both jacket-side and coil-side heat transfer.
Theoretical Background For time-independent non-Newtonian fluids obeying the power-law model (3) the expression for an "apparent viscosity" is given by
Calderbank and Moo-Young (1961) obtained the following expression for an apparent viscosity
where B = 11 f 1090 for all impellers except anchors, and n 5 1 (Le., pseudoplastic fluids). This equation (6) was used successfully by Sandall and Pate1 (1970) in their heat-transfer studies. If one uses the flow model suggested by Metzner and Reed (1955) W
=K'(g)n'
(7)
then, over a particular range of shear rate n' = n 3n+1 n gcK' = K(-> Substituting eq 8 and 9 in eq 6, one gets gcK' (10) (BN)I-"' This expression (10) can be used to calculate the Reynolds number, Prandtl number, and viscosity ratio. I t should be noted that this apparent viscosity has been evaluated from power input measurements and hence it is more appropriate to the fluid in the region of the impeller rather than that near the heat transfer surface. However, this procedure has been shown to be satisfactory for the correlation of heat transfer data and as yet there is no convenient alternative. viscosity =
(4) Thus, to obtain a viscosity for use in the Reynolds number, Prandtl number, and viscosity ratio, an appropriate average shear rate for an agitated liquid has to be defined. There are different methods of predicting such an average shear rate in agitated vessels, developed from measurements of power requirements for stirring of Newtonian and non-Newtonian liquids. The reliable methods were proposed by Foresti and Liu (1959), Metzner and Otto (1957), and Calderbank and Moo-Young (1961). According to Metzner and Otto (1957) (5) which was used by Skelland and Dimmick (1969) for their heat transfer studies.
Experimental Section The experimental setup shown in Figure 1 consisted of a flat-bottomed, cylindrical, jacketed vessel equipped with a helical coil and a four flat-bladed turbine agitator. To prevent vortex formation, four equally spaced baffles of 30 mm width were installed vertically a t the wall of the vessel (i.d. = 356 mm). Nine copper-constantan thermocouples of 30 B.W.G. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
585
and
Table I. Summary of the Experimental Conditions (1) Test liquids: (a) 1%and 2% SCMC; (b) 1%and 2.5% S A . ; (c)
Nuj =
water (2) Speed range: 180-670 rpm (3) Agitator diameter (Da):7.8, 11.8,” 15.2 cm (4) Depth of agitation (Ha):9.0, 11.8,“ 15.2, 17.8,21.4cm ( 5 ) Helix diameter ( D J :16.9, 20.1, 24.0,a 27.8 cm (6) Coiled tube outside diameter (do)= 1.59,” 1.91,2.22 cm The standard vessel configuration used in the present work has the following geometrical relationships: (i) Da/Dt = 0.33, (ii) HalDt = 0.33; (iii) D,lDt = 0.675; (iv) dolDt = 0.0447; (v) H J D t = 1.0; (vi) wb/Dt = 0.0833. embedded in the tube wall at diametrically opposite points along the coil circumference were used to measure the metal wall temperature of the coiled tube. Likewise, nine thermocouples, embedded in the vessel wall in radial and axial directions, were used to measure the vessel wall temperature. The various experimental conditions have been summarized in Table I. The test liquids used were dilute aqueous polymer solutions of 1.0 and 2.0% (wt/wt) sodium carboxymethyl cellulose (SCMC) and 1.0 and 2.5% (wtlwt)sodium alginate (SA), with 1.0% by volume of saturated solution of sodium benzoate in water, added as a stabilizer. The rheological properties of these test liquids were determined by capillary tube viscometer at varios temperatures. These test liquids were found to obey the flow model given by eq 7 and the experimental values of the rheological constants n’, K’ are listed in Table 11. A measuring glass cylinder was filled with the test liquid. A steel ball of 3-mm diameter was allowed to sink into the test liquid. The steel ball did not oscillate in any of the test liquids used and thus viscoelastic properties were not observed with our test liquids. The other properties such as thermal conductivity, density, and specific heat were found to be the same as those of water by Boggs and Sibbit (19651, Ghosh (1969), and Metzner and Reed (1955), and hence the properties of water were used. For a constant speed of agitation in the range of 180-670 rpm, a batch of the agitated test solution was heated by hot water flowing through the jacket and cooled simultaneously by cold water passing through the submerged coil under such conditions to maintain a constant vessel liquid temperature of 45 & 1“C. The vessel liquid temperature was measured at different positions and average temperature was thus noted. After the attainment of steady state, the cold and hot water flow rate, inlet and outlet temperatures, vessel liquid temperature, average thermocouple readings, and speed of agitator were recorded. The maximum variation in temperature recorded by nine thermocouples a t the coil-side and jacketside was found to be 2 O C and hence arithmetic average temperature was used for wall temperature: Results a n d Discussion Defining the Nusselt number as Nu,=- h d t k
(for coil-side)
h.D
k
(for vessel inside)
(12)
the experimental heat transfer results were analyzed. The viscosity given by eq 10 with B = 11was used to calculate the Reynolds and Prandtl numbers in the present work. While performing the experiments, only one parameter was allowed to deviate from its SVC and thus the effect of various parameters on heat transfer characteristics was determined using the following equation
Thus determination of constants C, a, b, d, e, f, and g will give us the required correlations for jacket-side and coil-side heat transfer. Such an attempt is outlined below. Generalized Agitator Reynolds Number (Re‘). The effect of generalized Reynolds number on coil-side and jacketside heat transfer for all the pseudoplastic solutions for standard and nonstandard configurations is shown in Figures 2 and 3, respectively. For a constant temperature of the agitated liquid used, the Nuj and Nu, were found to vary respectively with the mean powers of 0.63 and 0.66 of the generalized Reynolds number, for both standard and nonstandard configurations. Generalized Agitator Prandtl Number (Pr’), The effect of the generalized Prandtl number on jacket-side and coil-side heat transfer for standard and one typical nonstandard configuration is shown in Figures 4a and b and the slope of the straight lines was observed to be 0.33, which is the exponent of (Pr’). Thus for both cases of heat transfer, the NusseltPrandtl number relationship observed in the case of pseudoplastic test solutions is similar to the one reported for Newtonian fluids. This could be due to use of the correctly generalized Prandtl number. Agitator diameter (Da).The jacket-side and coil-side heat transfer results expressed as N u ~ / ( R ~ ’ ) O . ~ ~ ( Pand ~ ’ )Nu,/ O.~~ (Re’)0.66(Pr’)0.33 were shown as a function of (Da/Dt)ratio in Figures 5a and b, respectively. The film coefficients of heat transfer showed an increasing trend with increase in agitator diameter, due to the higher order of turbulence, resulting from the increased Reynolds number with increase in agitator diameter. The exponents for the diameter ratio were found to be 0.14 and 0.17, respectively, for the jacket-side and coil-side heat transfer. It would be interesting to compare these values with a value of 0.18 reported by Nooruddin and Rao (1966) for coil heat transfer, and 0.10 reported by Oldshue and Gretton (1954) and Skelland and Dimmick (1969). For jacket-side heat transfer, Strek (1963) has reported an exponent of 0.13 for the diameter ratio. Agitator Depth (Ha).The graphical treatment of the results for the depth of agitation at both the heat transfer surfaces presented in Figures 6a and b revealed that the film heat transfer coefficients increased with increase in depth of agitation up to a ( H a / D t )ratio of l/2, due to the good mixing conditions and elimination of vortices realized, as the agitator is
Table 11. Rheological Properties of the Pseudoplastic Solutions at Different Temperatures Solution: Temp, “ C 30
40 50 60
65 K’ = kgen’/m2. 566
1.0% SCMC
n’
K )x 103
n’
0.81 0.84 0.89 0.91 0.93
11.6 7.52 4.38 3.76 2.37
0.43 0.50
2.096SCMC K‘ x 103
0.55
0.57 0.62
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
221 123 78 56 39
1.0%SA
n‘
K I x 103
n’
0.83 0.87 0.89 0.90 0.95
2.49 1.48 1.00 0.90 0.83
0.44 0.46 0.48 0.496 0.489
2.5% SA K’ x 103 100
75 60 49 45
/
/ I
I
." 102
lo3
106
lo4
Rd
Figure 3. Variation of Nusselt number with agitated Reynolds number for nonstandard vessel configuration (H,/Dt =
(Legend as in Figure
2.)
IO
-
'92 Pr
10
Figure 4. Effect of Prandtl number on coil-side and jacket-side heat transfer for standard and nonstandard vessel configuration (H,/D, = I$): -, coil side; - - -,jacket side; 0,1%SCMC; 0 ,2% SCMC; A , 2% SA; V, 2.5% SA.
taken higher up in the tank from the bottom. The values of exponents for (H,/D,) for jacket-side and coil-side heat transfer were found to be 0.09 and 0.13, respectively, comparable with the value of 0.12 of Strek (1963) for jacket-side heat
transfer to Newtonian liquids and 0.14 reported by Jha and Rao (1967) for coil side heat transfer to Newtonian liquids. Helix Diameter ( D J .It was found that the jacket-side heat transfer coefficients varied with a power (-0.21) of the ( D J D J ratio, while the coil-side coefficients varied with a power (-0.29) of the diameter ratio as is evident from Figures 7a and b. In the case of coil-side heat transfer, the value of the exponent of the (D,/DJ ratio was found to be in agreement with -0.27 reported by Jha and Rao (1966) for Newtonian fluids and -0.25 by Pratt (1947) for coil-side heat transfer. With a smaller coil diameter, the turbulence prevalent at both the heat transfer surfaces would increase in magnitude, resulting in the increased film coefficients of heat transfer. Coiled Tube Outside Diameter ( d o ) .The effect of coil tube outside diameter has been established for both the heat transfer surfaces as shown in Figures 8a and b. The slopes of the straight lines in these figures revealed that the jacket-side heat transfer coefficients vary with the (-0.35) power of the diameter ratio (do/DJ while the coil-side heat transfer coefficients vary with the (-0.45) power of the dimensionless ratio (do/Dt).The value of the exponent of ( d J D , ) obtained in the present investigation is comparable to the value of -0.5 reported by Skelland and Dimmick (1969) for coil-side heat transfer to agitated pseudoplastic fluids, and a value -0.48 of Jha and Rao (1967) for coil-side heat transfer. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
567
=u y
0.3
1.0
0.3
0.1
(DC/D~)
(b)
-3.0 0.1
1.0
0.3 ( Da/D1)
Figure 7. Effect of helix diameter on (a)jacket-side (b) coil-sideheat transfer for different test solutions. (Legend as in Figure 4.)
(b)
Figure 5. Effect of agitator diameter on (a)jacket-side (b) coil-side heat transfer for different test solutions. (Legend as in Figure 4.)
m
c?
0 -
2
.
0
7
1
0.4 0.1
10
03 (do/DI) (b)
Figure 8. Effect of coiled tube outside diameter on (a)jacket-side (b) coil-side heat transfer for different test solutions.
0.2 01
1.0
0.3 ( Ha/Dt
(b)
Figure 6. Effect of depth of agitation on (a)jacket-side (b) coil-side heat transfer for different test solutions.
number (Re’) for all pseudoplastic test liquids for standard and various nonstandard configurations in Figure 9 for jacket-side and coil-side heat transfer, resulting in the following correlations: jacket side:
The coil-side film heat transfer coefficient was observed to increase with decreasing coil tube outside diameter, due to the fact that, in a submerged coil of a smaller tube diameter, the dampening action of the coiled tube on turbulence intensity would much less pronounced than with a coil of larger tube diameter. Proposed Correlations. After determining the exponents of various dimensionless groups, the experimental results calculated as
D, (6) (Haz) 0.14
Nuj = 0.22(Re’)0.63(Pr’)0.33
0.09
(std dev = 7.8%) (14) coil side:
(z)(g)
Nu, = 0.21(Re’)0.66(Pr’)0.33
D,
0.17
Ha
0.13
(std dev = 7.1%) (15) and
D,
NUc’(pr’)0.33
0.17
Ha
0.13
D,
-0.29
d o -0.45
(5) iDt) (6) (6)
have been presented a function of generalized Reynolds 568
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
Equations 14 and 15 could be used to predict the film heat transfer coefficients of agitated pseudoplastic liquids, respectively, for jacket- and coil-side heat transfer for a generalized Reynolds number range from 200 to 21 700, a generalized Prandtl number range from 49 to 1220, n’ range from 0.47
lo2
lo3
lo5
104
106
R:!
Figure 9. Final correlation for jacket-side and coil-side heat transfer for standard and non-standard vessel configuration. (Legend as in Figure 2.)
to 1.0, and K’ range from 1.24 X m2.
to 100 X
kgpsn’/
Conclusions Thus it is possible to predict the heat transfer performance of any agitated vessel conforming to standard and nonstandard configurations with Newtonian and pseudoplastic solutions inside the vessel, by our reliable proposed correlations represented by eq 14 and 15, respectively, for jacket-side and coil-side heat transfer. Nomenclature a, b = constants in eq 13 B = empirical constant in eq 10 = 11,as used in the present work C, d, e, f, g = constants in eq 13 C, = specific heat D = inside diameter of the tube D, = agitator diameter D, = mean diameter of helical coil Dt = inside diameter of the vessel (tank) do = coiled tube outside diameter H a = depth of agitation from the bottom of tank HL = depth of liquid in agitated vessel hj = vessel inside (jacket) film heat transfer coefficient hoc = coiled tube outside film heat transfer coefficient K = flow consistency index defined in eq 3 K‘ =defined by eq 7 k = thermal conductivity of the agitated liquid k , = empirical constant used in eq 5 N = agitator speed n = flow behavior index defined by eq 3 n‘ = defined b y e q 7 V = average velocity of the fluid w b = baffle width Greek Letters 1 = viscosity PA = apparent viscosity
9 = average shear rate 7w
= wall shear stress
Dimensionless Numbers Nu, = coil-side Nusselt number = h,,DJk Nuj = jacket-side Nusselt number = h P J k Re = agitator Reynolds number = Da2Np/K Re’ = generalized agitator Reynolds number = Da2N P/pA = Da2Np (BN)l-n‘/gcK’ Re” = Da2N p ( k , N ) l - n / K Pr = Prandtl number = C,u/k Pr’ = generalized agitatoFPrandt1 number = Cp,,tA/k = C d C K ’ / k(BN)I-n’ Pr” = C&/k(k,N)l-n Literature Cited Boggs, J. H., Sibbit. W. L., Ind. Eng. Chem., 47,289 (1965). Calderbank, P. H., Moo-Young, M. B., Trans. Inst. Chem. Eng., 39, 337 (1961). Carreau, P., Charest, G., Corneille, J. L., Can. J. Chem. Eng., 44,3 (1966). Foresti, R., Liu, T., Ind. Eng. Chem., 51,860 (1959). Ghosh, M. K.. Ph.D. Thesis, Indian Institute of Technology, Bombay, 1969. Hagedorn, D., Salamone J. J., Ind. Eng. Chem., Process Des. Dev., 6, 469 (1967). Heinlein, H. W., Sandall, 0.C., Ind. Eng. Chem., Process Des. Dev., 11, 490 (1972). Jha, R . K.. Rao, M.R.. Int. J. Heat Mass Transfer, IO, 395 (1967). Metzner, A. B., Otto, R. E., A.I.Ch.E. J., 3, 3 (1957). Metzner, A. B.. Reed, J. C., A.I.Ch.E. J., 1, 434(1955). Nooruddin. Rao, M. R., Ind. J. Tech., 4, 131 (1966). Oldshue. T. Y.. Gretton, A. T., Chem. Eng. frog., 50,615 (1954). Pandian, J. R., Rao, M.R., Ind. Chem. Eng. 7,29 (1970). Polard, J., Kantyka, T. A., Trans. Inst. Chem. Eng., 47,T21 (1969). Pratt, N. H.. Trans. Inst. Chem. Eng., 25, 163 (1947). Sandall. 0.C.. Patel, K. G., Ind. Eng. Chem., Process Des. Dev., 9, 139 (1970). Skelland, A. H. P., Dimmick, G. R., Ind. Eng. Chem., Process Des. Dev., 0,267 (1969). Strek, F.. Int. Cbem. Eng., 3, 533 (1963). Yagi, H.. Yoshida, F., Ind. Eng. Chem. Process Des. Dev., 14,408 (1975).
Received f o r review January 31,1975 Accepted June 24, 1976
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
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