Heat Transfer in Aerated Tower Filled with Non-Newtonian Liquid Masabuml Nlshlkawa,' Hldeo Kato, and Kenji Hashimoto Department of Chemical Engineering, Kyoto University, Kyoto 606, Japan
Correlations of heat transfer coefficients from a jacket wall or an immersed cooling coil were obtained for the aerated tower including physical properties of the liquid, viscosity correction term, liquid superficial velocity, and gas superficial velocity. It was also determined that heat transfer coefficients for non-Newtonian liquids could be correlated by the same correlative equations for Newtonian liquids using the apparent viscosity calculated from the flow curve and the local average shear rate defined in this study. As the local average shear rate at the wall surface was much larger than that near the cooling coil, it was more efficient to use a cooling coil for thermal control of the aerated tower.
Many chemical or biological reactions of industrial importance are suited to the use of aerated towers, and thermal control of the liquid in an aerated tower sometimes has a significant effect on the result. Heat transfer between a liquid and solid surface in the aerated tower has been studied by Kblbel et al. (1960), Fair et al. (1962), Kast (1962), Kat0 (1962), Yoshitome et al. (1964), Konsetov (1966), and Nagata et a]. (1975). However, almost all workers dealt with only water and no reliable correlation for predicting heat transfer coefficient including variables in wide range has been available. In this paper, the correlative equation for heat transfer coefficients from a tower wall or an immersed cooling coil in the aerated tower is reported including variables such as physical properties of the liquid, gas velocity, and liquid velocity. It is also reported in this paper that heat transfer coefficients for non-Newtonian liquids can be correlated as Newtonian liquids using the apparent viscosity which is calculated using the flow curve of a liquid and the local average shear rate defined in this study. Experimental Section Apparatus. The equipment flow sheet is shown in Figure 1. The dimensions of the two aerated towers used in this study, 15 and 5.1 cm i.d. towers, are shown in Table I. The 15-cm i.d. tower is made of polyvinyl chloride resin and is provided with a double jacket and an cooling coil as shown in Figure 1. However, the 5.1-cm i.d. tower is provided only with a double jacket. The double jackets used in this study are made of brass. The cooling coil is made of '/3-in. copper pipe wound in a helix. Air and liquid are introduced to the bottom of the tower through a single vertical nozzle attached a t the center of the bottom. A constant liquid level is maintained in the tower by means of an overflow. Thermometers are used to measure liquid temperature in towers, 8 points in the 15-cm i.d. tower and 10 points in the 5.1-cm i.d. tower. Experimental Liquids. Tap water, millet jelly solutions, and glycerin solutions were used for Newtonian liquids and carboxy methyl cellulose (CMC) solutions in various concentrations for pseudo-plastic liquids. The physical properties are shown in Table 11. Estimation of Heat Transfer Coefficients. Heat transfer measurements were all performed under steady-state conditions. The amount of heat transferred from the jacket was obtained by measuring the amount of condensing steam from the inner jacket. QS
= D, r pf
(1)
where D,,r , and pf are rate of steam condensation, latent heat
of steam condensation, and density of condensate, respectively. The amount of heat used to cool the liquid was obtained by measuring the temperature rise of the cooling water and the liquid flowing through the aerated tower. Qc
=
Qw
+ 81
where Qw
QI
- t,,)CP, = wl(ti0 - tiJCpi
= WJtC,
(3) (4)
w , and W I in above equations mean mass flow rate of cooling water and liquid flowing through the tower, respectively. As the difference in Q, and Q c was less than a few percent, the mean value was adopted for calculation of heat transfer coefficients. The Nusselt equation (Nusselt, 1916) eq 5 , for film type condensation heat transfer was used io evaluate the heat transfer coefficient on the steam side jacket wall. (h,/k ) (cL2/p2g) = i.47( 4 r /p)-1/3
(5)
Therefore, the heat transfer coefficient h, on the jacket wall of the aerated liquid side was estimated by substituting the overall heat transfer coefficient U , and h, into the following equation. llh, = 1/U, - l l h ,
(6)
The heat transfer coefficient inside the cooling coil h, was calculated using the following equation by Jescke (1926).
Accordingly, the heat transfer coefficient h , of the cooling coil on the aerated liquid side was estimated as follows l l h , = 1/U,
- l/hw
where U , is the coil side overall heat transfer coefficient. In calculating h, and h,, the heat transfer resistances due to thermal conduction through the jacket or coil wall were neglected as they were negligibly small. Estimation of Apparent Viscosity. In Newtonian liquids, the relation between shear stress and shear rate is linear and the viscosity is evaluated independently of shear rate. 7
= fij,
(9)
However, the linear relation between shear stress and shear rate is not valid for pseudo-plastic liquids, and the flow curve is usually expressed by the power law model. 7
= Kq"
( m 5 1.0)
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
(10) 133
I
I
non-Newtonian
rL?rb P
@
0
0
flow curve
8 8
0water z d uoutlet t e r Osteam jacket Inlet
elzzon: ; : [ 3 mpressr 4 nater inlet manometer mnometer
8
ccoling toll
dram collector
a boiler oddrain dp miiector o thermometer SFQW
I
mist
Figure 2. a, Evaluation of apparent viscosity; b, evaluation of average shear rate.
Figure 1. Experimental apparatus.
Table I. Dimensions of Aerated Tower
10000
;e N
D H dn
Lj
Hj DC dc
LC
HC
Tower A
Tower B
15.0 cm
5.1 cm
E
3 Y
180.0 cm 3.0 cm 15.0 cm 32.0,107.0 cm 11.0 cm 1.0 cm 31.0 cm 100.0 cm
5033
U
2J k 2000
180.0 cm 1.6 cm 15.0 cm 97.0 cm
i m0'
water-air
io
50 Ug
-
260
160 Cmihrl
54
Figure 3. Heat transfer coefficient in aerated tower. Table 11. Properties of Materials Used (60 "C)
CP, By analogy with Newtonian liquids, the apparent viscosity for the power law liquid is defined as follows.
As in the aerated tower shear rate distribution has not been reported, the apparent viscosity cannot be calculated, although many aerated towers deal with non-Newtonian liquids in industry. The procedure proposed by the present authors to evaluate apparent viscosity and average shear rate for heat transfer in the aerated tower is as follows. (1) A heat transfer coefficient-viscosity correlation curve is drawn for a given aerated tower with a Newtonian liquid as shown in Figure 2a. (2) The heat transfer coefficient for a pseudo-plastic liquid is measured at a given superficial gas velocity. Then the corresponding apparent viscosity is found as shown by Figure 2a. (3) From the flow curve measured by a rheometer, the corresponding average shear rate +av to bLacan be estimated as shown in Figure 2b. In the case of power law liquids, the following relation exists. jiav = (Ka/K)l'(m-l)
(12)
Results a n d Discussion Effect of Wall Temperature Difference. As shown in Figure 3, hj is larger than h, a t the same superficial gas velocity, though both are proportional to the superficial gas velocity to the 0.25 power as reported by several authors (Fair et al., 1962; Hart, 1976; Kolbel et al., 1960; Nagata et al., 1975). This difference may be due to differences of wall temperature 134
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
P,
Materials Water Glycerin solution Millet jelly solution CMC solution Air
kg/m3 983 11801230 1200-
p
or pa, CP
0.469 24-95 7-700
1400
983 1.07
5-600
0.096
kcal/kg "C
k, kcal/m h O C
1.00 0.53-0.725
0.563 0.25-0.256
0.57-0.75
0.38-0.45
1.oo 0.25
0.563 0.022
at the jacket and cooling coil. In correlating turbulent heat transfer coefficients on the pipe wall or mixing vessel, the viscosity correction term, V~SO.'~, by Sieder and Tate (1936) has been applied. However, the exponent 0.14 is too large in the case of the aerated tower and 0.05 gives the best fit correlation as shown in Figure 4. Accordingly, heat transfer coefficients for aerated towers with water are correlated by the following equation. h = 1350U,1/4Viso.05
(13)
Though eq 13 was obtained for 25 to 45 "C water in this study, this equation is applicable for water of 10 to 90 "C within 20% error by considering changes of physical properties of liquid with temperature. Turbulence produced by bubbles rising through the liquid promotes the rate of heat transfer. This promoting effect of heat transfer is larger at the cooling coil than at the tower wall as bubbles are apt to rise through the tower along the center part. Accordingly, it may be reasonable to give different correlations for the jacket and cooling coil side using VisO.14.
c looool
1
0 locket side
0 Jacket 51de 0 coil side
water- air
ug
Cm/hrl
Figure 4. Effect of viscosity correction term on heat transfer coefficient in aerated tower.
-
05
1
2
5
10
P
'r?r
U,
However, one equation is presented in this study for the sake of convenience. Effect of Liquid Velocity. The effect of superficial liquid velocity on heat transfer coefficients is shown in Figure 5, and the following relations are obtained.
h
0:
Uf3
Ul1I4
(Ul I54 m/h)
(14)
(54 I U1 I 5 0 0 m/h)
(15)
Accordingly, the effect of liquid velocity cannot be neglected at high superficial liquid velocity. Effect of Liquid Viscosity. The change of heat transfer coefficients with change of liquid viscosity is shown in Figure 6, which gives the following equation. h
a p-l/3
(16)
As a result, heat transfer coefficients for an aerated tower filled with a Newtonian liquid can be correlated with the following equations considering the effects of physical properties of liquid. ( h / p C p )( p 2 / A p p g )lI3( C p ~ / k Vis ) ~ -O/ ~O5 = 0.054U,1/4 (U1 I 54 m/h)
(17)
( h/ p c p ) (p*/Appg ) 1/3(c p p / k )2/3 vis -0 "5 = 0.0541J,'/4(U1/54)1/4 (54 I Ul I500 m/h)
(18)
In the above equations ( h / p C p ) ( p 2 / A p p g ) 1 /is 3 the Stanton number considering buoyancy force of bubbles, and ( C p p / k ) is the Prandtl number. Accordingly, ( h / p C p ) ( p ' / A p p g ) 1 / 3 ( C p p / k ) 2 / ~is the j factor for heat transfer. In Figure 7, estimated values of heat transfer coefficients using eq 17 and 18 are compared with the observed values. I t is clear from this figure that the correlative equations obtained in this study give good correlation. These equations are also applicable for various tower diameters, as they represent experimental values reported by several authors using various diameters of aerated towers as shown in Figure 7. It is also observed in Figure 7 that jhVis-O O5 gives a constant value of 0.3 at a larger superficial gas velocity than 1000 m/h regardless of the physical properties of the liquid when the superficial liquid velocity is small.
Vis- O "5 ( U i I54 m/h, U , 2 1000 m/h)
( h/pC p ) (p 2 / A p p g l I 3 ( C p p / k ) 2/J
= 0.3
200
'0
Crnlhrl
0 "7
>
0:
100
M
Figure 6. Effect of liquid viscosity on heat transfer coefficient in -.---l aerated tower.
Figure 5. Effect of superficial liquid velocity on heat transfer coefficient in aerated tower.
h
20 CCPl
(19)
........... Faireta121
3
___-_ Kato" ~ = 5D c= 4m5 7 , 1 0 6 7 ~ ~ - - _ _ Nagata et aiio1D = 3 O c m,
,' ,
152 229 437
,' , I
I.
The same tendency was reported by Kat0 (1962) and Yoshitome et al. (1964). As a t high superficial liquid velocity, the superficial gas velocity at which slugging of large bubbles is initiated becomes larger with liquid velocity as reported by Kat0 (1957) and Inoue and Unno (1972), eq 18 is applicable for a larger superficial gas velocity. According to Kolbel et al. (1958), the following relations were obtained for the laminar range of low superficial gas velocity and the turbulent range of high superficial gas velocity. h h
0:
Ug0.161p-0.139(laminar range)
a U,o.113p-o.248 (turbulent
range)
(20) (21)
The tendency of a smaller exponent for U , a t higher superficial gas velocity is same as that obtained in this work; however, exponents for U , or p are rather smaller than the values by other workers. It was also reported that the critical Reynolds number from laminar to turbulent condition was almost inversely proportional to the liquid viscosity. This observation indicates that the critical Reynolds number can be decided only by the superficial gas velocity and supports the result obtained by the present authors. The correlation by Nagata et al. (1975) is a little smaller than that by the present authors as compared in Figure 7. This may be due to the small ratio, 1.33, of height to diameter of the aerated tower used by them, as conditions of bubble dispersion in the aerated tower are stabilized after bubble ascending of one tower diameter length from the sparger as reported by Akita (1972). Hart (1976) reported the following dimensionless equation including the data by Kolbel et al. (1958) in the correlation.
(h/CpU,p) ( C , F / ~ ) O = . ~0.12~4U,3p/pg)-".25
(22)
The values of the exponents for various properties given by Hart are almost the same as those in eq 17 in this work. Equation 22 is the first dimensionless correlation of heat transfer for an aerated tower. His equation, however, is only applicable in the limited range of the superficial gas velocity Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1. 1977
135
bo'
500 5001
slope 1 0
5000
u
2003 w
E
5 lKI0
n
u
W
500
U
Ug
Cm/hrl
10
U g CmIhrJ
Figure 8. a, Jacket side heat transfer coefficient in aerated tower with non-Newtonian liquid: b, coil side heat transfer coefficient in aerated tower with non-Newtonian liquid. I
0 Jacket side
0 coil side
0
05
1
2 u4
COIL
side
5
10
20
fcrnisecl
Figure 10. Average shear rate and superficial gas velocity. 1
I
[sec-'~
M
50 100 203 Ug C m i h r l
M
Figure 9. Example of apparent viscosity measured and flow curve of a CMC solution.
because the exponent for U , is not constant a t 0.25 at lower gas velocity than 15 m k or at a larger gas velocity than several hundred m/h as reported by several authors (Kato, 1962; Kolbel et al., 1958; Nagata et al., 1975; Yoshitome et al., 1964). It may not be rational to make a dimensionless equation in which the effect of the operating variable is not free from the effects of physical properties of the liquid. Although the terms of left-hand side of eq 17 and 18 are nondimensional, the right-hand side is not. However, the power per unit mass of liquid in the aerated tower is equal to the product of the superficial gas velocity and gravitational acceleration, and it is not necessary to introduce a nondimensional number instead of the superficial gas velocity to the right-hand side of the correlative equations. This can be supported by the fact that the various phenomena in the aerated tower can be properly correlated only by the superficial gas velocity. Accordingly, it is considered by the present authors that the dimensional correlation using the superficial gas velocity is reasonable in the present condition. Non-Newtonian Liquids. Examples of heat transfer coefficients observed for non-Newtonian liquids are shown in Figure 8. It is known from this figure that coil-side heat transfer coefficients indicate almost the same tendency as Newtonian liquids. However, jacket-side heat transfer coefficients decrease rapidly at superficial gas velocity lower than a hundred and several tens of m/h. As this effect is not observed for Newtonian liquids whose viscosity is widely varied, it should arise from the non-Newtonian nature of the liquid. In this study, this effect is explained by the difference of the local apparent viscosity of the liquid, that is the difference of the local average shear rate in the aerated tower. Examples of apparent viscosity at the jacket surface and cooling coil calculated according to the procedure stated above are shown in Figure 9 with the flow curve of a liquid. As can be seen from Figure 9, the apparent viscosity at the jacket surface gives a much larger value than that at the coil surface at small superficial gas velocity, though almost the same values are observed at a larger superficial gas velocity than about 150 136
Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 1, 1977
m/h. This may be due to the fact that turbulence generated by rising bubbles is so strong a t large superficial gas velocity that the shear rate difference in the cross-sectional area of the tower is small. However, turbulence generated at smaller superficial gas velocity is mainly dissipated in the center part of the tower and less turbulence energy is transported to the part near the tower wall where gas holdup is smaller than that observed in the center part. This tendency is promoted by the nature of the non-Newtonian liquid. In Figure 10 the local average shear rate observed in the 15 cm i.d. tower is plotted vs. the superficial gas velocity and the following relations are observed regardless of the difference of concentration of CMC solutions experimented in this study. Yav
= 5 0 . 0 ~ ~ (upL 4.0 cm/s)
(23)
yav,c= ~ O O U , ~ . ~ (ug I4.0 cm/s)
(24)
yav,j= 0 . 1 9 5 ~ , ~ , ~(upI 4.0 cm/s)
(25)
Though the effect of vessel diameter on the local average shear rate is not determined in this study, it may be reasonable to consider that eq 23 is applicable regardless of the vessel diameter, as the gas holdup and the average bubble diameter are almost the same value in aerated towers of various diameters at same superficial gas velocity. However, the jacket side and coil side local average shear rate may change with vessel diameter at low superficial gas velocity. As the local average shear rate a t a cooling coil is much larger than that at a wall surface, it is more favorable to use a cooling coil for a thermal control device in the aerated tower, especially the tower filled with a non-Newtonian liquid. The calculated heat transfer coefficients using local average shear rate obtained in this study and flow curves are shown in Figure 8 as solid lines and they represent measured values very well. Accordingly, it is determined that heat transfer coefficients for an aerated tower with a non-Newtonian liquid can be correlated by eq 17 and 18 using the apparent viscosity calculated from the flow curve and the local average shear rate found in this study. Conclusions Correlations of heat transfer coefficients from a jacket wall
or an immersed cooling coil were obtained for the aerated tower as follows
(h/pC p ) (p 2 / A p pg )
( C pp / k ) 2/3 Vis -0.05 = 0.054Ug1l4
when U l I 54 m/h and U , I1000 m/h
( h / p C p) (p 2 / A p w g ) ( C pp / k )2/3 Vis -O, O5 = 0.3 when U I I54 m/h and U , I1000 m/h, and
(h/pC p ) (p 2 / Ap pg ) ' I 3 ( C pp / k ) ' I 3 Vis -O. O5 = 0.054U,'/4( U154)'/~
when 54 m/h. Heat transfer coefficients for an aerated tower with anonNewtonian liquid can also be correlated by equations for Newtonian liquids by using the apparent viscosity calculated from the flow curve and the local average shear rate which is obtained as follows in this study. jlav = 5 0 . 0 ~ ~
when u y I4.0 cm/s ( U , 2 144 m/h), and = 100u,0.5; yavj= 0.195u2.0
when u gI4.0 cm/s.
r = latent heat of steam condensation, kcal/kg ti = temperature at inlet, "C t o = temperature at outlet, "C u = velocity, cm/s U = velocity, m/h uh = overall heat transfer coefficient, kcal/m2 h "c w = mass flowrate, kg/h y = shearrate,s-' qav = average shear rate, s-l r = rate of condensation, kg/h m I.( = viscosity at bulk temperature, P wa = apparent viscosity, P ww = viscosity a t wall temperature, P p = density, kg/m3 pf = density of condensate, kg/m3 Ap = density deference between gas and liquid, kg/m3 r = shear stress, dyn/cm2 jt, ( = (h/Cpp)(p2/Appg)1/3(Cpp/k)2/3) = j factor Vis ( = p w / p ) = viscosity correction term
Subscripts c = cooling coilside g = gasside j = jacketside 1 = liquidside s = steamside w = cooling water side Literature Cited
Nomenclature C p = specific heat, kcal/kg "C d, = inside diameter of coil tube, m d, = nozzle diameter, m D = inside diameter of aerated tower, m D, = diameter of coil helix, m D, = rate of steam condensation, m3/h g = gravitational acceleration, m/h2 h heat transfer coefficient, kcal/m2 h " C H = liquid height in aerated tower, m H , = lowest coil level, m H , = lowest jacket level, m k = thermal conductivity, kcal/m h "C K = fluid consistency index, dyn sm/cm2 L = overall height of heat transfer device, m m = exponent in power law rheological equation Q = amount of heat transferred, kcal/h
Akita, K., Ph.D. Thesis, Kyoto University, 1972. Fair, J. R.. Lambright, A. I.,Andersen, J. W., Id.Eng. Chem., Process Des. Dev., I,33 (1962). Hart, W. F., lnd. Eng. Chem., Process Des. Dev., 15, 109 (1976). Inoue, I., Unno, H., Kagaku Kogaku, 36, 65 (1972). Jescke, D., Zh. Ver. Deut. lng., 23, 69 (1926). Kast, W., lnt. J. Heat Mass Transfer, 5, 329 (1962). Kato, Y , Repr. Fac. Eng. Yamanashi Univ. Japan, 8, 31 (1957). Kato. Y., Kagaku Kogaku, 26, 1068 (1962). Kolbel, H., Borchers, E., Muller, K.. Chem. lng. Tech., 30, 729 (1958). Kolbel, H., Borchers, E., Martins, J., Chem. hg. Tech., 32, 84 (1960). Konsetov, V. V.. lnt. J. Heat Mass Transfer, 9, 1103 (1966). Nagata. S., Nishikawa. M.. Itaya, M., Ashiwake, K., KagakuKogaku, Ronbunshu, 1, 5 (1975). Nusselt, W., Zh. Ver. Deut. lng., 60, 541. 569 (1916). Sieder, R. A., Tate, G. E., ind. Eng. Chem., 28, 1429 (1936). Yoshitome, H., Makihara, M.. Tsuchiya, Y., Kagaku Kogaku, 28, 228 (1964).
Received f o r reuieu May 11, 1976 Accepted September 8,1976
Chebychev Polynomial Correlation Equations of Composition and Bubble/Dew Point Temperature of Binary Mixtures Containing Ethane with Propane, Butane, and Pentane Bruce R. Corn, James H. Weber, and Luh C. Tao' Department of Chemical Engineering, University of Nebraska, Lincoln, Nebraska 68588
Correlation equations based on Chebychev polynomials are presented for the bubble and dew temperatures of ethane-propane, ethane-butane, and ethane-pentane binaries. They cover the entire composition range and the pressure range of 100 to 600 psia.
For a narrow pressure interval, vapor pressures of pure substances may often be correlated by a simple equation such as the Antoine equation. To cover a wide pressure range up
to the vicinity of the critical regions, Gibson (1967) used the orthogonal Chebychev polynomial to obtain the vapor pressure equations of eight chemical species. Utilizing this concept Ind. Eng. Chem.,
Process Des. Dev., Vol. 16, No. 1, 1977
137
