Ind, Eng. Chem. Process Des. Dev. 1983,22,367-376
367
Experimental Studies of Heat Transfer between a Bundle of Horizontal Tubes and a Gas-Solid Fiuidized Bed of Small Particles Nanak S. Grewal Mechanlcai Englneering Department, University of North Dakota, Grand Forks, North Dakota 58202
Satkh C. Saxena' Deperhnent of Chemical Englneerlng, U n h # y of Illlnols at Chicago Circle, Chlcago, Illinois 60680
Experimental data are obtained for heat transfer coefficient between el~lcallyheated horlzontai tube bundles (12.7 and 28.6 mm diameter) and square fluidized beds of silica sand ( d p = 167 and 504 pm) and alumina ( d p = 259 pm) as a function of air fluidizing velocity. The staggered tube bundle has its tubes located at the vertices of equllaterai triangles with pitch varying between 1.75 to 9 tbms the tube diameter. Heat transfer data for a bundle of tubes are compared with a slngk tube under otherwise identical condltlons and are compared wlth the existing correlations and theoretical models in the literature for maximum heat transfer coefficient. A correlation for the maximum heat transfer coefficient between horlzontai tube bundles and a gas-solid fluidized bed of small particles (75 < Ar < 20 000) Is proposed which includes the influence of tube pitch in the bundle. The predicted values of the maximum heat transfer coefficient from the proposed correlation are generally within f20% of the experimental data available In the literature when the contribution due to radiation is also included.
Introduction Fluidized-bed combustion offers a great potential for the utilization of high-sulfur coal and low-rank coal in an environmentally acceptable way. In fluidized-bed coal combustion crushed coal is burnt in the presence of an inert material (dolomite or limestone for high-sulfur coal and silica sand, alumina, or its own ash for low-rank coal) while they are held in suspension by upward flowing air. The fluidized-bed combustion is an efficient method to generate steam because of high heat transfer rates to immersed boiler tubes. The sulfur dioxide formed during combustion of high-sulfur coal reacts with limestone or dolomite in the bed and forms a solid sulfate material which can be disposed of as a dry solid waste along with the coal ash. The amount of nitrogen oxide produced in industrial-scale atmospheric-fluidized-bed-combustionboilers generally has been observed to be significantly lower as compared to pulverized coal combustors without special combustion modifications (Newby et al., 1980). The large quantities of solid waste produced during the combustion of highsulfur coal is one of the disadvantages of the fluidized-bed combustion. The results of initial investigations (Newby et al., 1980; Benett et al., 1980) indicate that the solid waste can be processed for utilization purposes and/or to minimize the environmental impact of disposal. Fluidized-bed combustion technology has been under development for many years supported by several agencies. A number of pilot plant facilities such as ones at Pope, Evans, and Robbins; Exxon; General Electric; FosterWheeler; Babcock and Wilcox; Morgantown Energy Technology Research Center; Oak Ridge National Laboratory; Grand Forks Energy Technology Center; etc. (Grewal, 1979) are employing horizontal tube bundles in a fluidized-bed to generate steam. An efficient boiler heat exchanger design requires the knowledge of heat transfer coefficient between the tube bundle and the bed, and its dependence on fluidized bed operating conditions, tube dimensions, tube pitch, and thermal and physical properties of bed material and fluidizing gas. A large number of experimental investigations have been reported on the measurement of heat transfer rate between 0196-4305/83/1122-0367807.50/0
either a horizontal tube or tube bundle and gas-solid fluidized beds. Several reviews of these works have been reported in the literature (Botterill, 1975; Gelperin and Ainshtein, 1971; Grewal, 1979; Gutfinger and Abuef, 1974; Kunii and Levenspiel, 1969; Saxena et al., 1978; Zabrodsky et al., 1976, and Zabrodsky, 1966). To a first approximation, the total heat transfer coefficient, h,, between an immersed tube and a gas-solid fluidized bed can be regarded as equal to three additive components of particle convection, &, gas convection, &, and radiation, h, (Botterill, 1975)
h, = h,, + h,, + h, (1) The particle convection involves first the heat transfer by unsteady-state conduction from the heat transfer surface to the aggregates of solid particles directly adjacent to it and then the convection of the heated particles to the bulk of the bed. For particle sizes between 40 and 800 bm, which fall in Geldart's groups A and B (Geldart, 1973),the contribution by the pmticle convection is predominant for nonpressurized systems. The gas convection contributes to the process of heat transfer by convective mixing which augments the heat transfer in the gas gaps between the particles and the heat transfer surface and between neighboring particles. It is important for denser and larger particles and high operating pressures. The radiative component of the heat transfer coefficient becomes significant only for bed temperatures above 870 K (Botterill, 1975). In this paper, we shall limit our discussion to heat transfer in fluidized beds of small particles (d < 1 mm). In our previous pul!hcations (Grewal et al., 1979; Grewal and Saxena, 1980; 1981), we have reported the effect of size, shape,density, and specific heat of particles; tube size; bed depth; heat flux and distributor design on h, and h, between a single horizontal_ tube and a gas-solid fluidized bed of small particles (dp.< 1 mm). Existing correlations of h, and h, for a single horizontal tube have been examined and more reliable correlations for h, and h,,, have been proposed (Grewal and Saxena, 1980; 1981; Grewal, 1982). In this paper, we report data for 0 1983 American Chemical Society
368 Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
- -t--2MA42:6 355
BRONZE TUBE-
-
THERMOCOUPLE CONTACT LOCATIONS CALROD
G i A 'I'
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304 8
I
1
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NYLON SEAL
- Figure 1. Details of heat transfer tube assembly: (a) A = 44.5, B 355
l
1
= 40.6, C = 28.6, D = 12.7,E = 38.1, and F = 40.7; (b) A = 25, B = 21, C = 12.7, D = 9.4, E = 19, and F = 22.2. All dimensions are in millimeters.
average heat transfer coefficient for staggered horizons tube bundles immersed in a fluidized bed of silica sand (dp = 167 and 504 pm) and alumina = 259 pm) at room temperature as a-function of fluidizmg velocity. Existing correlations for h, between horizontal tube bundles and a fluidized-bed have been examined recently by Grewal (1981). In the present paper, the existing correlations and theoretical models for h, mu as listed in Table I have been examined on the basis of our present data. A new correlation for the maximum heat transfer coefficient, h,, , has been proposed which also includes the influence of relative tube pitch in tube bedles. The predictions from the proposed correlation for h, are compared with the existing data in the literature for small particles C1 mm). This correlation will be particularly useful in the design of low-rank coal fluidized-bed combustor where crushed coal is burnt in an inert bed of silica sand or alumina of particle size less than 1 mm (Goblirsch and Sondreal, 1977).
(4
(ap
Experimental Apparatus and Procedure Experiments are carried out in a square fluidized bed which is described in earlier publications (Grewal, 1979; Grewal and Saxena, 1977,1979,1980, and 1981). The flat head bubble cap distributor is used in all the experiments on tube bundles. The horizontal tubes in a bundle are arranged in a staggered array and are located at the vertices of equilateral triangles. The design of a heat transfer tube is shown in Figure 1. The tubes (DT = 12.7 and 28.6 mm) are heated by electric calrod heaters. Three ironconstantan thermocouples are bonded to the tube surface in milled grooves with technical quality copper cement; 12.7" diameter tubea are made of copper while 28.6-mm tubes are made of bronze. The ends of the tubes are provided with Teflon supports to reduce axial heat loss. The end heat loss is estimated to be less than 1% of the total power fed to the tube. In all runs, the thermocouple at the middle of heat transfer tube is kept at the top side of the tube. In all experiments on single tubes, the height of the tube center is kept 213 mm above the bubble cap distributor plate. On the other hand, for tube bundles the centers of the bottom row 12.7-mm and 28.6" tubea are kept 139 and 125 mm above the distributor, respectively. In all runs with tube bundles, only the middle tubes of each row are heated. A dc power supply with a voltage regulation of *O.l% is used to energize the heater. Voltmeters and ammeters with a precision of 1% are employed to determine the power fed to the heater. The relative tube pitch, PIDT, is varied from 1.75 to 9.0 as shown in Table 11. Five thermocouples are employed to measure the bed temperature and an arithmetic average value has been used in the calculation of heat transfer coefficient. The thermocouples are connected to a Leeds and Northrup Nu-
I
I
I
I
06
04
G, kg/m2s
Figure 2. Performance of 12.7-mm tube bundles in a fluidized bed of silica sand; d, = 167 pm. 320 r
Y
$
k
2901
1 260
230
1 c
i 04
05
06
G, k g / m 2 s
Figure 3. Performance of 12.7-mmtube bundles in a fluidized bed of silica sand; d, = 504 pm.
matron temperature recorder with 0.1 K resolution and 21 column digital printer. The static bed height in all experiments is about 35 cm. The steady state is assumed to be established when the bed temperature variation is less than 0.5 K/h. The temperatures at each of the other locations are recorded over a period of time and an arithmetic average value is used to calculate the total heat transfer coefficient, h,, for each of the heated tubes from the following relation h, =
8
(1)
Aw(Tw - Tb) The total heat transfer coefficient for the tube bundle, h,, is obtained by taking arithmetic average of the values of h,for all the heated tubes in the bundle. The maximum error in the measurement of heat transfer coefficient is estimated to be 8%. The precision of heat transfer measurement is found to be *2%. Silica sand (a = 167 and 504 pm) and alumina = 259 pm) are used as bed material. The average diameter is obtained from screen analysis and the following relation
(ap
The particle density of the solid particles, pa, is determined by the displacement of methanol in a graduated cylinder. The particle diameter and density are given in Table 11. Results and Discussion The heat transfer coefficients for smooth tube bundles and single tubes are shown plotted as a function of superficial mass fluidizing velocity in Figures 2 to 8. The value of the heat transfer coefficient for both single tube and tube bundles increases yith increase in the value of G. The rate of increase in h, is greater at low values of G than at large values of G. The heat transfer coefficient attains its maximum value at the higher end of the fluidizing velocity range investigated here. The initial increase is due to decrease in particle residence time at the tube surface which is due to particle mixing caused by rising bubbles in the fluidized bed. Further, the residence time
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 360
1
I
/I,
3501
Y
225
"E
0 SINGLETUBE
1c .
3
A
1
0
2 250
I
BED HEIGHT, CM 36
A
ir
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04
02
300
1
1
25
i
-
G, k g l m 2 s
Figure 4. Performance of 12.7" of alumina; 2, = 259 pm.
3.50 0 SINGLE TUBE
0.3
0.1
0.5
G. k g l m 2 s
Figure 6. Performance of 28.6-mmtube bundles in a fluidized bed of silica sand; d, = 167 pm. 2601
'
>i
I
N
E
5
,,
240 220
B
mn -_-
04
06
05 G kglm's
Figure 6. Performance of 28.6-mm tube bundles in a fluidized bed of silica sand; dp = 504 pm.
2 200
0 SINGLETUBE 1
02
1
0.4
0.6
G. kg/m*s
Figure 8. The effect of bed height on & for a bundle of tubes (28.6 mm) of 3 row8 and a bed of alumina; d, = 259 pm.
8
Y
0.2
tube bundles in a fluidized bed
1
04
I
I
06
G. kglm's
Figure 7. Performance of 28.6-mmtube bundles in a fluidized bed of alumina, dp = 259 pm.
of the particle decreases with increase in G due to increase in bubble induced particle circulation. However, at larger values of G the rate of increase in h, decreases due to the opposing effect caused by increase in surface area of the tube being engulfed by rising bubbles. It is seen from Figure 2 that for 167-pm silica sand particles a single curve can be used to effectively represent the heat transfer coefficient both for a single tube and tube bundles with a relative pitch of 4.5 to 9.0 with a maximum deviation of about f 2 % . The reproducibility of our measurements of heat transfer coefficient is also f2%. The values of h, for tube bundles with a relative pitch of 2.25 are smaller as compared to that for a single tube. This decrease for tube bundles is attributed to an increase in the particle residence time close to the heat transfer surface. The latter is due to the increase in the obstruction to solid particle movement when the tubes are located that closely. The maximum value of the heat transfer coefficient for the tube bundle with the relative pitch of 2.25 is about 5% smaller as compared to its value for the single tube.
It may be noted from Figure 3 that there is also no effect of P/DT on hw_fortube bundles and silica sand particles of larger size (d, = 504 pm) as long as the relative pitch is varied from 9 to 4.5. A further decrease in the rel_ative pitch to 2.25 results in the decrease in the value of h, mar by about 7%. The effect of relative pitch on h, for alumina (dp = 259 pm) is shown in Figure 4. Here again, a single curve can represent the data for a single tube and a tube bundle with a relative pitch of 4.5. However, when the relative pitch is decreased to 2.25, the value of 6, max reduces again for the tube bundle and it is about 6% smaller than the value for the single tube. The dependence of h, for tube bundles of a larger diameter tube (DT = 28.6 mm) for silica sand and alumina is shown in Figures 5 through 7. In all three figures, the decrease in h, for tube bundle with P/DT of 3.50 as compared to a single tube is very small and is of the order of 1% . The decrease in h,, for the tube bundle with the relative pitch of 1.75 as compared to the single tube is about 6 4 7 . 5 , and 8.5% for silica sand (d, = 167 pm), silica sand (dp = 504 pm) and alumina (dp = 259 pm), respectively. The valyes of h, for a tube bundle with P/DT = 1.75 for alumina (d, = 259 pm) as a function of G are shown plotted in Figure 8 for two beds of slumped heights of 25 and 36 cm. A single curve can represent the data for both the bed heights within a maximum deviation of f l %. Thus, at least in this range of bed heights, there is no detectable effect of bed height on heat transfer coefficient for tube bundles. The Proposed Correlation. Recently Grewal and Saxena (1981) have proposed the following correlation for the maximum heat transfer coefficient for a single tube immersed in a fluidized-bed of small particles (300 < Ar < 10000)
Nu,
max
= 0.9(ArD12.7/DT)0.21 (Cps/Cpf)46.6Ar4'(3)
The correlation of eq 3 is based on data for silica sand
(ap= 167-504 pm), dolorflite (dp = 293 pm), alumina_(d,
= 259 pm), glass beads (dp = 241-427 em), copper (d, = 136 pm), nickel (fl, = 136 pm), solder (dp= 136 pm), and silicon carbide (d, = 178, 362 pm). The tube diameter varied between 12.7 mm and 28.6 mm. This correlation includes the effect of specific heat of solid particles. The predicted values of the heat transfer coefficient fram eq 3 are generally in good agreement with the available data on small particles (300 < AF < 10000). Grewal(l982) modified the correlation of eq 3 to extend ita validity to particles smaller than 136 pm. The modified correlation for Nuarpmar is of the following form
(
Nuwmar= 0.9 Ar-;;)21($>"'
(4)
370
Id.Erg. Chem. Process Des. Dev., Vol. 22, No. 3. 1983
VI
VI
0
u
0
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h
. $
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 371 Table 11. Heat Transfer ExDeriments with Tube Bundles
no. of
material silica sand
P ~ kg/m3 ,
2670
d,, ctm 167
silica sand
2670
504
alumina
4015
259
re1 pitch, PJDT
rows in a vertical line
no. of tubes in a bundle
2.25 4.50 6.75 9.00 2.25 4.50 1.75 1.75 3.50 2.25 4.50 6.75 9.00 2.25 4.50 1.75 1.75 3.50 2.25 4.50 1.75 3.50
3 3 3 3 5 5 3 5 3 3 3 3 3 B 5 3 5 3 3 3 3 3
23 11 8 5 38 18 14 23 14 23 11 8 5 38 18 14 23 8 23 11 14 8
DT, 12.7 12.7 12.7 12.7 12.7 12.7 28.6 28.6 28.6 12.7 12.7 12.7 12.7 12.7 12.7 28.6 28.6 28.6 12.7 12.7 28.6 28.6
The predicted values of Nu,, from the correlation of eq 4 are generally within f20% of the existing data (dp C 800 Km, Tb < 870 K, 75 < Ar < 20000). Based on the correlation of eq 4 for a single tube, a , max correlation of the following form is proposed for % for horizontal tube bundles
Nu,,
= 0.9(Ar D12.7/DT)0*21 (C,/Cpf)o.2CF
(5)
where CF is a correction factor to account for the effect of the relative pitch of the tube bundles on the maximum heat transfer coefficient. From the observed qualitative variation of hw, for tube bundles, the correction factor, CF, of the following form is proposed CF = 1- a(P/DT)*
c
,s
8
1
I
1
(6)
The values of the constants a and b are obtained by the regression analysis of the present experimental data. The , mar for tube bundles is given by final correlation for % eq 7.
-
Nu,mflx O . ~ ( A ~ & ~ . , / D T(c,/cpf)0'2 ) ~ . ~ ~ [1 - O.~~(P/DT)-~'"] (7) for 1.75 IPIDT 5 9 ; 75 IAr I20000. All the necessary properties of the fluidizing gas are calculated at the average of the bed and tube surface , for the temperatures. The predicted values of,% tube bundles -are within *6% of the present experimental values of Nu,,; see Figure 9. Next we shall check the reliability of the proposed correlation and the existing correlations and theoretical models as listed in Table I in the light of the existing data for small particles. Comparison of Existing Data with Various Correlations and Theoretical Models. A detailed comparison between calculated and Table of hwmax from the proposed correlation of eq 7 and various other correlations and theoretical models (Aerojet Energy Conversion Company (AECC), 1980; Chekansky et al., 1970; Gelperin et al., 1968, 1969; Martin, 1982; Staub, 1979; Xavier and Davidson, 1981; Zabrodsky et al., 1981) and the experimental values of &- from the present data and the data of eleven other investigators (AECC, 1980; Bansal, 1978; Bartel and Genetti, 1973; Borodulya et al., 1980; Canada and
Figure 9. Comparison of calculated values of Nuvmlu from the u ,. For proposed correlation with the experimental values of N explanation of symbols, see Figure 10.
__---
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AECC DATA
V
t
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---_---I 102
GELPERIN
EIT d~ ' CORRELATION
A
PRESENT DATA
I
,
,
-d-i
= 259ym
ALUMINA,
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,
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4
6
8
lo3
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AI
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7
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0 10'
-
#
Figure 10. Comparison of calculated values of Nu, from the Gelperin et al. (1969) correlation with experimental values of N u , , . For explanation of symbols, see Figure 9.
McLaughlin, 1978; Chandran et al., 1980; Gelperin et al., 1969; Grewal and Hajicek, 1982; Xavier and Davidson, 1978) is shown in Figures 9 through 15 and Table III. T h e
372
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
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Ind. Eng. clrem.Proce~~s Des. Dev., Vol. 22, No. 3, 1983 373 6
5
4
4
3
2
-
NuWpmox(ExptlI
Figure 11. Comparison of calculated values of Nu,,
from the Chekansky et al. (1970) correlation with the experimental values of Nu, -. For explanation of symbols, see Figures 9 and 10. 5
4
3
2
-
NoO
Q, rl
rlN
99
I
4
2
00
3
E
2
NuWpmax (Exptl)
Figure 12. Comparison of calculated values of Nu,,
(1980)correlation with the experimental values of N u ,. explanation of symbols, see Figures 9 and 10.
from AECC For
correlations of Catipovic et al. (1980), Glicksman and Decker (1980), and Zabrodsky et al. (1981) are applicable for fluidized beds of large particles > 1mm). Therefore, the predictions from these three correlations are not compared with the data. Moreover, the correlation of Catipovic et al. (1980) does not predict characteristic maximum in the heat transfer coefficient. The predicted values of from the present correlation of eq 7 are compared with the present and AECC data in Figure 9. For a tube bundle with staggered array, P is taken equal to the minimum distance between the
(ap
-
374
I d . Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 5
5
/
/
/'
4
3
I
I
3
4
S
I
I
1
2
-3
4
5
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N~~~,,~(Exptl)
Nu
J
1
2
WPmax
(Exptl)
Mgue 13. Comparieon of calculated values of N%from Xavier and Davideon (1981) model with the experimentalvalues of Nu,,.
Fwre 16. Comparieon of calculated values of N%from Martin (1982) model with the experimental values of Nuapmsl.For explanation of symbols, see Figures 9 and 10.
For explanation of symbols, see Figures 9 and 10.
c-
1
AECC DATA
PRESENT DATA 4 c
-
NuWpm,,(Exptl)
Figure 14. Comparison of calculated values of Nu, (1979) model with the experimental values of Nu, nation of symbols, see Figures 9 and 10.
, from Staub
,-.
For expla-
centers of the adjacent tubes. On the other hand, for tube bundle with in-line arrangement, P is taken equal to the horizontal pitch, Y. The propoaed correlation predicts the present and AECC data very well. The maximum devia, does not exceed tion of the calculated values of % 15%. Further, it is seen from Figures 10 through 15 that none of the existing correlations and the theoretical models predict the present and AECC data as well as the present Correlation of eq 7. The correlations of Chekansky et aL (1970) and Gelperin et al. (1969) do not include the effect of specific heat of solid particles on the heat transfer Coefficient. This is one
-
of the reasons why these two correlations do not predict the AECC data for nickel and/or solder particles well; see Figures 10 and 11. The AECC correlation (1980) does not take into account the effect of the thermal conductivity of the fluidizing gas and the tube pitch on the heat transfer coefficient. It is due to the fact that the correlation is based on data for which bed temperature (Tb= 477 K) and tube pitch were not varied. The correlation generally overpredicts the present data and AECC data for nickel and solder because these data have been obtained for bed temperatures less than 477 K; see Figure 12. The predictions from the Xavier and Davidson model (1981) are compard with the present and AECC data in Figure 13. The values of U , Uopt,Db,and k, are calculated from the correlations proposed hy Wen and Yu (1966), Goroshko et al. (1958), Mori and Wen (1975), and Krupiczka (1967),respectively. The Xavier and Davidson (1981) model predicts the present data with a maximum error of 25%. However, it OverpredictsJhe AECC data on nickel, solder, and expanded alumina (dp = 163 bm) by as much as 50%. As seen from Figure 14, the Staub model (1979) underpredicts the present and AECC data by as much as 50%. It is mainly due to the bubbly nature of the fluidized bed under experimental conditions of the present and AECC data, whereas the Staub model (1979) has been developed for a fluidized bed in turbulent flow regime. The predicted values of % , from the Martin model (1982) are compared with the present and AECC data in Figure 15. The gas convective component, h,, is calculated from the correlation of Baskakov et al. (1973). All the necessary properties of the fluidizing gas are evaluated at the bedtemperature. The maximum error in predicted values of Nu, mlu does not exceed 28 % . All the correlations and the theoretical models listed in Table I are applicable for low bed temperature (Tb< 870 K)where contribution by radiation is not significant. A t high bed temperatures (Tb> 870 K), the radiative component of the heat transfer coefficient, h, is important and is estimated from the Baskakov et al. (1973) data. It
-
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 375
-
is seen from Table 111, that the root-mean-square deviation of the calculated values of hw using the proposed correlation from the experimental values of h,, does not exceed 20% except for the data of Bartel and Genetti (1973). Further, it is evident from Table I11 that the present correlation, in general, predicts the existing data better than any other existing correlations and theoretical models. The correlation of Gelperin et al. (1969) predicts imaginary values of hw, under the experimental conditions of Bansal (1978). Conclusions Based on the present experimental study and comparison of predictions from the correlations and theoretical models available in the literature with the existing data, it may be concluded that the proposed correlation of eq 7 predicts the existing data on small particles (75 < Ar < 20000) very well. For high-temperature applications, the radiative component is important and is estimated following Baskakov et al. (1973). The proposed correlation is applicable to Geldart’s groups A and B powders and the scale of the apparatus employed in the reported studies. The use of the proposed correlation a t different conditions will require some caution and may result in a greater uncertainty. Acknowledgment This research is partly supported by the Department of Energy under Contract No. Ex-77-c-01-1787 and Grant No. DEAB18-80FC10120 (Task 19-DEAT1882 FC10555) and partly by the National Science Foundation under Grant No. ENG77-08780A01. This material is partly based upon a thesis by N. S. Grewal for the doctoral degree a t the University of Illinois a t Chicago Circle. The authors are thankful to S. S. Kumbhat for his help in conducting some of the experimental runs. Nomenclature A, = surface area of a smooth tube, m2 Ar = Archimedes number, = d:pt(p, - pf)gp-2, dimensionless C, = specific heat of fluidizinggas at constant pressure, kJ/kg K C F = correction factor defined by eq 6 - specific heat of solid particles, kJ kg K d, = average particle diameter defined y eq 2, m dPi= arithmetic average diameter of the successive screens, m D b = diameter of a sphere having same volume as a bubble, m DT = outside diameter of a smooth heat transfer tube, m DI2., = heat transfer tube 12.7 mm in diameter, m D, = heat transfer tube 20 mm in diameter, m g = acceleration due to gravity, m/sz G = superficial mass fluidizing velocity, kg/m2 s h, = total average heat transfer coefficient for a tube, W/m2 K h, = particle convective component of heat transfer coefficient for a tube, W/m2 K h, mar = maximum particle convective component of heat transfer coefficient for a tube, W/m2 K h , = gas convective component of heat transfer coefficient for a tube, W/m2 K h, = maximum heat transfer coefficient for a tube, W/m2 K h, = radiative component of heat transfer coefficient for a - smooth tube, W/m2 K h, = average heat transfer coefficient for a smooth tube bundle, W/m2 K hwc= particle convective component of heat transfer coefficient for a tube bundle, W/m2 K
c,
r:
hwm = gas convective component of heat transfer coefficient for a tube bundle, W/m2 K hwe= heat transfer coefficient for a tube bundle without any
- solid material in the bed, W/m2 K h,, = maximum heat transfer coefficient for a tube bundle, W/m2 K k, = stagnant bed thermal conductivity, W/m K k f = thermal conductivity of air, W/m K kd = particulate phase thermal conductivity, W/m K M = number of transfer units, = 2kfNumar/psCps[gdp3(eOpt cd)/5(1 - !&(l - topt)]O.S,dimensionless Nu, = maxmum Nusselt number for brief contact time, = 4[(1 + (2u/d,) X In (1 + (d /2a)) - 11, dimensionless Nu, = maximum Nusseyt nuplber based on particle conductive component = h, ,dp/ kf, dimensionless Nu = Nusselt number based on particle diameter = dimensionless Nupcp, = maximum value of Nusselt number = (hw,dp)/kf, dimensionless P = center-to-center distance of adjacent tubes, m Pr = Prandtl number = pCPf/kf, dimensionless Q = electrical power supplied to heater, W Red = R_eynoldsnumber at minimum fluidizing velocity = (Umfp&,/p), dimensionless Re, = Reynolds number at optimum fluidizing velocity = (boptpJp/p), dimensionless Tb = average fluidized bed temperature, K T , = average surface temperature of the heat transfer tube, K U = superficial fluidizing velocity, m/s Ub = bubble velocity, m/s U d = minimum fluidizing velocity, m/s U,,, = fluidizing velocity at which maximum value of heat transfer coefficient occurs, m/s U, = particle superficial velocity, m/s w = weight fraction of particles in a specified size range, dimensionless Y = horizontal pitch of tube bundle, m ym = mixing length, m 2 = vertical pitch of tube bundle, m Greek Letters 0 = time fraction that the tube is in contact with bubbles, dimensionless y = accomodation coefficient, dimensionless A = mean free path of a gas molecule, m A = minimum distance between surfaces of adjacent tubes, m 6 = bubble fraction in the bed, dimensionless emf = void fraction at minimum fluidization conditions, dimensionless Copt = void fraction at the optimum fluidization conditions, dimensionless p = viscosity of fluidizing gas, N s/m2 pf = fluidizing gas density, kg/m3 pmf = particulate phase density, kg/m3 ps = density of solid particles, kg/m3 Literature Cited
1r$,)/kf,
,
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Received for review March 20, 1981 Revised manuscript received October 6, 1982 Accepted December 20, 1982
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Effect of Vapor Maldistribution on Tray Efficiency Tadlnad Mohan, Krowldl K. Rae,' and D. Prahlada Rao’ Department of Chemical Engineering, Indlan Institute of Technology Kanpw, Kanpur 2080 76, India
The effect of vapor maldistribution on the Murphree tray efficiency has been studied accounting for the variation of point efficiency that arises due to vapor maldistribution. I t is found that, even in the case of perfectly mixed and plug flow models of the liquid phase, vapor maidistribution leads to considerable reduction in tray efficiency. The maximum reduction is found to occur at a Peciet number of about 5. The variation of efficiency with the liquid flow-path length of a rectangular tray of a fixed area has been studied. Tray effeciency is found to increase with the Iiquld fbw-path length well up to the point where dumping is encountered, provided uniform liquid flow distribution is ensured.
Introduction Excessive liquid gradient on a tray leads to vapor maldistribution. It is known to have a detrimental effect on tray efficiency. However, only a few attempts were made to analyze this effect with a view of providing a rational basis for the design of large diameter trays. Holm (1961) was the first to make an attempt to study the effect of linear vapor-flow gradient on tray efficiency. Later, Furzer (1969) showed that the linear vapor-flow gradient has no effect on tray efficiency for the perfectly mixed and plug model of liquid phase. For the dispersed flow of liquid, he found a marginal reduction in tray efficiency which is maximum a t a Peclet number of about 10. Recently, Lockett and Dhulesia (1980) extended Furzer’s analysis for the plug flow of vapor between the trays and the liquid flow in the same direction as well as in opposite directions on successive trays. They found that tray efficiency is unaffected by the linear vapor-flow gradient in the case of the plug flow and perfectly mixed flow of liquid phase, and there is a very small effect for trays Department of Chemical Engineering, University of California, Davis, CA 95616. 0196-4305/83/1122-0376$01.50/0
with dispersed flow conditions. In contrast, Bolles (1963) reported that investigations of bubble-cap tray columns exhibiting poor performance because of high liquid gradient showed that the vapordistribution ratio was in excess of 0.5, and recommended a value of the ratio 0.5 as the design limit. To meet this condition either multipass trays or other remedial measures are recommended. Furzer (1969) and Lockett and Dhulesia (1980) analyses do not indicate that the vapor maldistribution leads to poor tray performance. However, they considered the point efficiency to be constant over the tray in spite of wide variation of vapor velocity along the liquid flow path. In the design of large diameter columns, the tray selection-whether single or multipass-should be based on a consideration of the variation of the tray efficiency with the liquid flow-path length. In the present work, equations have been obtained to relate tray efficiency to point efficiency taking into account its variation due to vapor maldistribution. A computer simulation study of the effect of various parameters on the tray efficiency is presented. The dependence of tray efficiency on the liquid flow-path length has been examined with a view of evolving a criterion for tray selection. 0 1983 American Chemical Society
