ENGINEERING,
Heat Transfer Design Characteristics
AND PROCESS DEV'&QPAENT
.-
Water Suspensions of Solids JEROME J. SALAMONE'
AND
MORRIS NEWMAN
Department o f Chemical Engineering, New York Universify, New York 53, N . Y.
W
HEN thifi investigation was initiated in 1950, only fragmen-
tary data on heat transfer to suspensions were available (10,14). Data on heat transfer to fluidized systems (16) indirated a substantial increase in the film coefficient of heat transfer for the gas-solid system compared to that of the gas alone. I t was thought that perhaps the presence of a suspended solid in a liquid would likewise increase the film coefficient of heat transfer of a liquid flowing inside a pipe. Since this investigation was started two independent studies of heat transfer to suspensions have been reportcd. Bonilla and coworkers ( 7 ) investigated the heat transfer properties of chalk-water slurries a t different concentrations. In correlating the data, the thermal conductivity of the fluid medium was used, and the viscosity was calculated from a theoretical relation involving the viscosity of the fluid and the volume fraction of the solid. The other properties used in the correlations were weighted avrrages of the individual proprrties of the liquid and solid. On this basis, a plot of Nu/Pr"S verBUS Re with percentage solid as a parameter showed that the value of Nu/Prl'a decreased with increasing solid concentration and that the effect was more pronounced at lower Reynolds numbers. The reduction in the value of Nu/Pr"a was shown to be approximately a linear function of the concentration of solid in the suspension. Orr and Dalla Valle (16) investigated various suspensions of powdered solids in water and ethylene glycol. The viscosity of the suspension was calculated by the use of an empirical equation involving the volume fraction of solids and the settled bulk density. The calculated values of the viscosity were checked experimentally with a Saybolt-type viscometer. The thermal conductivity of the suspension was evaluated from a relationship suggested by Maxwell for the analogoup electrical situation. Experimentally determined conductivities substantiated this relation. The other properties used were all weighted averages of the individual properties of the liquid and solid. On this basis the data were correlated fairly well by the Dittus-Boelter equation as modified by Siedw and Tate (18).
Neither of these investigators considered the effect of the change in the viscosity with flow rate for a non-Newtonian suspension. Moreover, viscosities and conductivities determined for laminar flow were used to correlate heat transfer data for extremely turbulent conditions. For a simple or Newtonian fluid a plot of shearing stress versus the time rate of shearing strain yields a straight line through the origin. The slope of this line is the viscosity and is constant for fixed conditions of temperature and pressure. For a non-Newtonian fluid the apparent viscosity, or the ratio of stress t o strain a t any given shearing stress, is a function of the velocity through the conduit. Previous investigations of the flow of sus1
Present address, Newark College of Engineering, Newark 2, N. J.
February 1955
pension (8, 4 , 6,6,19) 80) have showii that suspensions are nonNewtonian in character. Manv suspensions have been found to be non-Newtonian fluids of the pReudoplastic type (8, 20), where the apparent viscosity decreases with increasing velocity. The suspensions studied in this investigation were found to brhave as pseudo plastics. The numerical value for the apparent viscosity of a non-h'ew tonian fluid may vary markedly, depending on whether a torsional or pipeline viscometer is used for the measurement. Although some corrrlation has been established between the two values ( 2 ) )in the presrnt study the viscosities employed in the corrrlations of the heat transfer data were measured with a pipeline viscometer (1, d, 6) of the same diameter as the heat transfer pipe. The viscometer and heat exchanger were in series, and thus the slurry rate was exactly the same in both sections of the apparatus. The temperature in the viscometer was substantially the same as the average temperature in the heat exchanger. Experimental system indudes heating section, slurry cooler, and pipeline viscometer
The heat exchange section was similar to t h a t used by Bonilla and coworkers (7). The slurry was prepared in a 55-gallon drum equipped with an agitator, A . Thorough mixing was further ensured by means of a by-pass line recycling slurry into the tank. An AllisChalmers open impeller centrifugal pump, Type SS-BL, 2 X 2 inches, circulated the slurry from the tank, through the by-pass and the system, and back into the tank. The system (Figure 1 ) consisted of a heating section where the heat transfer measurements were made, a slurry cooler, M , where t h e slurry was cooled to approximately the average temperature in the heating section, and a pressure drop section or pipeline viscometer, L. The heat transfer section consisted of a l/rinch International. pipe size (I.P.S.) brass pipe inside a Il/,-inch I.P.S. brass r p e . , T h e I'/r-inch pipe was surrounded by a 21/rinch I.P.S. rass pipe. Both annular spaces carried steam, with the outermost annulus serving as a guard heater. Standard brass tees at the ends of the 2l/.2- and ll/r-inch pipes provided the inlet and outlet for the steam in both annular spaces. A brass cap, with a hole slightly larger than the outer diameter of the ll/4-inch pipe, was screwed to each of the 2'/*-inch tees and brazed to the 11/4-inch pipe to seal the outer annulus. The inner annulus was sealed with a packing gland at each end. The suspension was heated in the '/2-inoh pipe by steam flowing in the inner annulus over a length of 8'/2 feet. A steam calorimeter, G, operatin at a vacuum of about 8 t o 10 inches of mercury was used to &eck the quality of the steam. Provision was made t o collect and weigh the condensate from the inner annulus. The inside '/$-inch pipe was 12 feet long over-all, thus providing a calming section of l a / , feet a t each end. Each end was connected to an enlarged chamber containing a thermometer well. The thermometers, T , used to measure the inlet and outlet slurry temperatures were graduated in 0.1" C. Brass flanges with rubber gaskets were installed between each end of the ljz-inch pipe and the enlarged chamber to reduce end effects due to heat conduction between the heating section and the rest of the apparatus. Thermocouple Installation. Six copper-constantan thermocouple junctions, No. 22 gage wire, were attached to the outer surface of the l/&nch pipe at the top and bottom near the ends
INDUSTRIAL AND ENGINEERING CHEMISTRY
283
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT through a rotary switch to a Leeds 8.z Northrup portable precision potent,iometer, No. 8662. -411 ice bath was used as a reference junction. The entire heating section was insulated with
85% magnesia and thin aluminum foil. The suspension cooler was a concentric-type lieat exchanger consisting of 1-inch I.P.S. brass pipe inside a 2-inch standard iron pipe. Cold water was circulated countercurrent t o the slurry through t'he :iunular 'space over a length of 8
feet.
Figure A.
8.
Agitator Cooling coil
C.
Slurry tank
D. E.
By-pass valve Pump Calming section Steam calorimeter
F. G.
1.
Apparatus
H.
1. J. K.
L. M. N. T.
Steam regulator Steam pressure g a g e Steam traps Condensate cooler Pressure dror, section Slurry cooler Traps Thermometers
and the center of the inner annulus, !,\The thermocouples were installed t ~ ashown in Figure 2. Circumferential grooves ( 1 2 ) were cut in the pipe and the pipe was notched a t one point in the groove. The notch was then filled with molten solder and the thermocouple was embedded in the solder flush with the outer surface of the pipe (3). The exme58 solder was filed so that the solder was flush with the pipe surface. The thermocouple wire, which was covered with an impervious layer of enamel, was wrapped around the circumferential groove. The circumferential grooves were filled with a litharge-glycerol cement and the entire pipe surface was polished with fine emery pitper. Seal for Thermocouple W i r e . A '/,-inch hole way S drilled and tapped in the tees a t each e n d of t h e i n n e r annulus. The wires for three of' t h e thermocouples were pulled o u t through the hole a t one end; the wires for the other three thermocouples were led t h r o u g h t h e hole a t the other e n d . A '/c-inch pipe-to-tubingS compression fitting SECTION S-S was slipped over Figure 2. Thermocouple installation the bundle of wires and screwed into A. Notch filled with solder the '/&-inch hole in B. Thermocouple junction the tea, a8 shown C. Thermocouple wires in Figure 3. A piece D. Litharge-Glycerol Cement of thin-walled glass t u b i n g waa t h e n fitted around the bundle of wires, and a small section of copper tubing inserted over the glass tube. The compression fitting was then sealed Finally a piece of heavy walled rubber tube was slipped over the bundle of wires and fitted snugly over that portion of the coppei tubing extending beyond the compression fitting. Two screw clamps were placed on the rubber tubing about six inches apart to seal the thermocouple wires inside the rubber tubing. The wires were connected t o a terminal board which functioned as a zone box (3) and from this point copper leads were connected 284
Viscometer. The pipeline viscometer consisted of :m insu1:tted l/*-inch 1.P.S. brass pipe with pressure taps spaced 1 7 l / ~inches apart. A cnlniing section 25 inchos long preceded the preswre drop stxction, and an enlarged section with a thermometer well was installed about 10 inches beyond the pressure drop section. The pressure drop w m measured with a carbon tetmchloride manometer. Traps were installed immediately after t,he pressure taps t o prevent suspension particles from getting into the manometer lines. The lines froin the traps to the manometer were made of transparent Tygoii tubing so that if any particles got beyond the traps they could easily be seen. The manoineter system was so arranged that tlie traps and Tygon liner could be flushed with water. These lines were flushed before each reading i n order to remove sediment from the traps. The return line to the slurry tank was equipped with a by-pass and a set of quick-opening valves so that the slurry could be diverted into a weighing tank for measurement of the flow rate. T h e slurry tank was equipped with a cooling coil in order to maintain isothermid conditions. The solids used t o prepare the slurries are descrihd in Table I. Acceptable heat transfer data for water are used to evaluate operating procedure
The apparatus was first tested with water arid the acw'pted lieat transfer data for water (13) , . were reDroduced. :LS shown iu Fiaure 4 -4total of 23 runs were ni:tite n-ith w:tt~~r--18of thein ini-
H A
Figure 3. A. 6. C.
D. E. F. G.
H.
Seal for thermocouple wires
Thermocouple wires from ,unctions Glass tube Copper tubing Tee Compression fitting Rubber tubing Screw clamp Thermocouple wires to potentiometer circuit
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 2
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT and outlet slurry t,cmpcratures, viscometer teniperaturc, m a n o m e t e r r e a d i n g , conSpecific Thermal Heat a t Conducdensate temperatme, and calAv. Partirle Size Density 60° C.a, tivity,a orimeter t e m p e r a t u r e a n d D?am., at 20° C.”, B.t.u./(Lb.) B.t.u./(Hr.) microns Mesh P.) (Ft.)(O F.) Po\l.drr G./Cc. Soul ce pressure were recorded. The 21 (Fract. A) -325 0.0932 220 Copper Charles Hardy, New York 8 92 inlet and outlet slurry tem4.5 (Fract. R ) -200 4-325 (electrolytic copper pan dor, +ZOO 56 (Fract. C) Types B & C) peratures and the manometer 2 0 0.208 3.0 10 ... Carbon United Carbon Inc., reading were observed several New York 2 32 0.194 0.20 1.5b ... Exner Sand and Gravel Carp., Silica times, and the averages were New York (silica flour) 0.209 0.40 2 ?la Thompson, Weinman and Co., 2.5a ... Chalk recorded to minimize thc effect Montelair, N. J (atomite) of small fluctuations. The 0 From Chemical Engineers’ Handbook (9). steam rate was determined by b Specified by supplier. weighing the condensate collected during a given time interval The suspension flow rate was measured after all other readings had been rccorded by diverting the flow from the tank into a weigh tank and observing the time required to collect 75 pounds of slurry. The density of the suspension was obtained by weighing 4 liters of the slurry. This density was used t o determine the weight fraction of solid in the slurry from previously prepared curves based on known concentrations. Where information on average particle size was not supplied by the manufacturer it r a s estimated from sedimentation rates, by assuming the particles to be spherical. In the rase of copper various size ranges wcre separated by using 200-, 270-, and 325-mesh screens. The sizes calIO4 2 3 4 5 10 2 3 4 5 IO6 culated for each fraction from sedimentation measurements were R4: 0 9 found to be consistent with the screen dimensions. Figure 4. Heat transfer correlation for water Representative experimental data are tabulated in Table 11. Complete tables of the experimental data arc on filc with the American Documentation TnPtitutc. tiaily and five after 90 runs with various suspensions had been made. The two s e b of data for water agree well. The pipeline Apparent bulk viscosity and effective conductivity of viscometer was also calibrated with water, and the data are suspensions are determined over range of flow rates plotted in Figure 5 . This type of plot was used to facilitate the drawing of a smooth line through the points according to the von In the majority of cases heat balances were obtained to within Iiarman equation (8) for turbulent flow in smooth pipes. After 5%, and the rate o€ heat transfer for the calculation of the film a series of runs with any one slurry, the apparatus was drained and flushed. The 1-inch plugs and thermometer wells inserted in the 1-inch crosses comprising the enlarged chamber a t each end of the heating section were removed and the heating section was cleaned by pulling a stiff brush through the entire section. Prom time to time the pipeline viscometer was removed and cleaned in the same manner. For each set of runs with a given slurry, water entered the mixing tank until the tank was about two thirds full and circulated through the system. The steam was then turned on and after the water was hot the solid was added to form the suspension. The valve settings on the pump outlet were adjusted npproximately to the desired rate, as roughly indicated by the prcssure drop reading on the manometer in the pipeline viscometer, The system was then allowed to come to steady state, as evidenced by the attainment of constant readings on the inlet and outlet slurry thermometers. This generally required about to a/( hour beR. JT x 10.’ tween runs and approximately l l / * hours for the first run. After-steady state was reached, the thermocouple readings, inlet Figure 5. Calibration of pipeline viscometer Table
I.
Sources and Physical Properties of Slurry Materials
(0
Table II. Run No.
1 19 5.5
66 76 89 105 110
Solid in Suspension None Copper A Copper B Copper C Silica Chalk None Silica
February 1955
.If anom eter Readma. Cm. CClr 31.2 108.4 93.7 85.5 92.0 87.8 108.8 91.8
Viscometcr Temp., O C. 57.7 60.5 64.0 65.4 65.5 66.5 64.0 65.5
Bulk Density Lb./Cu.’ Ft. 61.42 67.86 63.48 62.96 62.49 62,81 61.25 64.40
Solids Concn.,
Wt. % 0 10.0 4.30 2.90 3.40 4.00 0 9.10
Representative Data Slurry Rate Lb./Min. 69.37 140.97 129.98 127.11 125.83 127.33 142.6 129.53
CalorimCalorimeter Coneter Press., densate S1urry Temp., O C. Temp., Cm. Hg Temp., Inlet Outlet O F. Abs. e C. 43.79 65.89 198 59.2 99.4 50.65 68.35 35.5 175 92.0 54.75 70.85 204 89.0 59.3 55.00 58.5 70.98 199 87.0 55.18 71.12 200 60.5 85.0 55.90 71.35 57.0 199 89.0 54.70 68.25 201 60.7 83.0 55.05 71.14 202 61.7 86.0
INDUSTRIAL AND ENGINEERING CHEMISTRY
Condensate Rate I,b./Mln. 2.905 4.154 3.795 3.572 3.625 3.210 3.672 3.647
Av. Outer Pipe Surface Temp., 0
c_.
91.40 96.62 100.76 98.90 98.os 94.69 92.13 96.41
285
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT ordinate from Figure 4 was used to calculate an effective slurry thermal conductivity for use in the Dittus-Boelter equation (IS). Typical plots of this eff cctive thrrmal conductivity versus flow rate for each suspension are shown in Figure 9. Equation is developed for prediction of film coefficient of heat transfer
Dimensional Analysis. The film coefficient ef heat transfer to a suspension in turbulent flow inside a pipe should be a function of the concentration of the solid, pipe diameter, thermal conductivities of the fluid and of the suspended solid, heat capacity of the fluid and of the suspended solid, particle size and particle shape of the suspended solid, density of the fluid and of the suspended solid, and velocity and apparent viscosity of the suspension. If the effect of particle shape is neglected and it is assumed that the particles are spheriral, incorporating the density of the solid, the solid concentration and the density of the fluid into a bulk density of the suspension
FLOW R A T E
Figure 6.
h = dDj
- La/uiti
coefficient was based on the slurry temperature rise and the average weighted heat capacity. The temperature rise of the slurry varied from 13' to 30" C., depending on the rate. The film coefficient of heat transfer was calculated using the conventional equation (2)
From pressure drop measurements and the density of the suspension, a friction factor wa,s calculated for each run and the c o r m sponding Reynolds number \vas read from Figure 5. From the Reynolds number the apparent viscosity was calculated ( 1 , 2, 5 ) . To account for the small difference between the viscometer temperature and the average temperature in the heating section a correction was made in each case; the correction was assumed to be the same as that for water. These apparent viscosity values rate of flolr for each suspension (Figures to ere plotted 8). For a series of runs with a,ny ono suspcnsion the average temperature in the heating section varied only slightly. The Reynolds number in the heating section was calculated using the appropriate value of the viscosity read from the smooth curves of Figures 6 to 8. For this Reynolds number the corresponding
I
~
F
(2 1
CF,Cas Da, ka)
I
Equation 3 was derived by employing the usual dimensional analysis approach. Arbitrary sets of data were used to evaluate the ronstants in the equation.
Apparent viscosity of copper slurries
h = q / A A&,
ob, Pbt f i b ,
( __ D ~ ) o ~ 6 z
hD
&
(gs)o.'"6
050')';(
($;)0.36
=
(3)
The experimental data are plotted in Figure 10. Effective Thermal Conductivity. As shown in Figure 11, the effective thermal conductivity attains a fairly constant figure for most of the suspensions above a Reynolds number of 50,600. The volume fraction of solids 7%-ascalculated from t.he density of the suspension a,nd t'he weight fractt>ionand density of the solids. The solid surface area per cubic foot, of suspension was computed from the volume fraction of solids and the avenge particle size, by assuming that the particles x-ere spherical in ellape. If the average efiective thermal conductivity of each suspension above a Reynolds number of 50,000 is plotted against the solid surface area per cubic foot' of suspension, a fairly linear relationship results (Figure 11). Moreover, if each line is extrapolated
0 5.0% Cu 1454 0 2.80. C" (45.
I
0 9 c9,
LOO
OAd
V
?? -
>
1
0.4
I
I
I
I
m
I
0 4.0% C H A L K + 10.5% CHALK
0
.
5
0
~
7
7
?
7
7
=
7
7
080 0 3050
Figure 7. Apparent viscosity of copper and carbon slurries
286
70
90
110
FLOW R A T E
FLOW RATE - LB./MIN.
Figure 8.
Apparent viscosity of chalk and silica slurries
Figure
INDUSTRIAL AND ENGINEERING CHEMISTRY
9.
-
130
150
LB/MlN.
Effective thermal conductivity of slurries
Vol. 47, No. 2
7
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT to zero concentration of yolid material, a value close to the thermal conductivity of water is obtained. The solid of highest thermal conductivity has the greatest slope while that of lowest conductivity has the smallest slope. The slope increases with increasing thermal conductivity of the solid. Validity of correlating equation i s substantiated by work of independent investigators
The slurries used in this investigation produced the following variations in the groups in Equation 3: hD -
273-500
kF
__-
14,00O-140,000
Fb
cFpb 3 kF
4-12 7
5 kF
0.53-583
I’ D”
282-1 0,500
FF
0 09-0 22
Orr and Dalla Valle (16) used a heat exchange section 3 / ~inch in diameter (I.P.S.). Bonilla and coworkers ( 7 ) used a heat exchange section l l / ~inches in diameter (I.P.S.). Their data foi water suspensions fall in line with Equation 3 for those slurries for which pipeline viscosities could be estimated from the viscosity versus flow data of the present investigation. This would indicate the vahdity of Equation 3 over the range of diameters from a/& to 1’/2 inches. The data of Bonilla and coworkrrs ( 7 ) ,when plotted in the form of Nu/Pr’/a versus Re, fall below the values predicted by thc. Dittus-Bmlter equation (or modification thereof) a t the 1owc.r Reynolds numbers. This discrepancy is probably due to the neglect of the non-Newtonian characteristics of the suspension, since viscosity data presented in the present paper indicate that the suspensions are pseudoplastic. .4t very high Reynolds numbers the apparent viscosity reachw Some limiting value, and for this reason the data in this region can be correlated even though the slurries are really non-Newtonian. On the other hand, the plots of effective thermal conductivity versus flow rate indicate that the thermal Conductivity increases somewhat with a de-
104
1.5
2
5
105
1.s
2
Figure 10. Heat transfer correlation for slurries
February1955
crease in flow rate. Hence there are two factors that have opposite effects on the film coefficient of heat transfer. However, the effective thermal conductivity does not vary as rapidly as the apparent viscosity with flow rate. As a result, a fair correlation of the data is obtained in the range of moderately high Reynolds numbers because in this range the change in viscosity is small. The data of Orr and Dalla Valle (16) correlate fairly well over the entire turbulent region, The Sieder and Tate (18) modification of the Dittus-Boelter equation correlates their data within approximately 20%. The deviation is probably due to the nonNewtonian behavior of the suspensions. The correlation obtained is partly due to the two opposing effects discussed above on the film coefficient. Bonilla and coworkers ( 7 ) used chalk-water slurries and Figure 9 indicates that the effective thermal conductivity for this system is practically independent of the flow rate over the range studied This is probably the reason for the reduction in the value of Nu/ Pr1’3 as the Reynolds number decreases below approximatefy 50,000. In this region the viscosity begins to increase as the flow rate decreases, but for these suspensions there is no compensating increase in the effective thermal conductivity to offset this increased viscosity.
I 0
I
1~000
Figure 1 1 ,
I
I
I
20,000 30,000 4opoo Sa FT OF SUSPENDED PMTICLE SURFACE C U FT OF SUSPENSION
54000
Average effective thermal conductivity correlation for slurries
The average effective thermal conductivity plot shown in Figure 11 exhibits two interesting features. All of the suspensions studied had higher effective thermal conductivities than the suspension medium, except for a few runs with the largest copper particles. In all other cases, including the silica and chalk slurries, the effective conductivities were larger than that of water. It is postulated that in the case of the largest copper particles the attainment of steady-state conduction within the particle is sufficiently retarded to reduce the over-all rate of heat transfer. For the silica and chalk particles, which have much lower thermal conductivities than copper, the attainment of steady-state conduction is still rapid because of the small particle sizes. The first three groups in Equation 3 are the Nusselt, Reynolds, and Prandtl groups, respectively. The appearance of these groups is to be expected, since in most cases of heat transfer the relation takes the form of the Nusselt group as a power function of the Reynolds and Prandtl groups. The remaining three groups might be regarded as correction factors accounting for the presence of the solids. Of these remaining groups, the ratio, C,/Cp, is the most significant; its exponent is 0.35 compared with 0.05 for the other two exponents. This indicates that convective transport of heat by the suspended particles is an important mechanism in the over-all heat transmission. The low exponents on the D/Dl and k./kp groups indicate that under most conditions the effect of these
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
’
282
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT groups is small.
Only for suspensions of very small particles
(E)
0.06
will the value of of
depart very much fromunity. The value
onlyvaries from 0.9 for the silica particles t o 1.1 for
(2)”‘06
thc copper particles. Consequently, the effect of these groups on the film coefficient is still somewhat in doubt. Unfortunately, the systems studied were such that k, and D,varied in the same direction. The over-all precision of the experimental work is considered to be approximately 10%. The average deviation of the experimental data from the dimensional equation is within this range. Conclusions
Water suspensions of solid powdeis such as copper, raibon, chalk, and silica behave as pseudoplastic non-Newtonian fluids, so that their apparent viscosity decreases with increasing rate of flow. I n fully turbulent flow the apparent viscosity approaches some limiting value which is greater than the viscosity of the dispersion medium. Because of the non-Newtonian behavior of these suspensions the ordinary design equations for heat transfw to Newtonian fluids cannot be applied Equation 3 may be used as a design equation for heat transfer t o solid-liquid suspensions in turbulent flow inside pipes, provided that the apparent viscosity of the suspension a t the existing flow conditions is used. This apparent viscosity can easily bc determined from pipeline viscometer measurements. For suspensions similar t o those studied in this investigation the viscosi t y versus flow data of Figures 6 t o 8 can be used t o estimate viscosities for use in applying Equation 3. The validity of Equation 3 is further substantiated by data of two independent investiga&om, although viscosities of the suspensions used in these separate investigations had t o be estimated from viscosity data contained in the present work. Acknowledgment
The authors wish t o aclrnowlcdge that the topic for this research \vas suggested by Frank Maslan. They also wish t o thank Robert E . Treybal for his helpful criticism. Nomenclature
heat transfer surface, sq. ft. specific heat of fluid, B.T.G./(lb.)( O F.) specific heat of suspended solid, B.T.V. ’(lb.)( F.) pipe diameter, ft. = average diameter of suspended solid, ft = film coefficient of heat transfer, B.T.U./(hr.)(sq.
= A C , CI; = C. = D =
D. h
+-+ \/o ‘U.,(
D \ I.,
thermal conductivity of fluid 01 suspending medium, B.T.U./(hr.)(’ F.)(ft.) thermal conductivity of solid in suspension, B.T.U./ (hr.)(O F.)(ft.) effective thermal conductivity of juspension, B.T.U./
(hr.)( F.)(ft.)
i At
average effective thermal conductivity of suspension Re greater than 50,000, B.T U./(hr.)(” F.)(ft.) heat transfer rate, B.T.U./hr. = temperature, O F. = logarithmic mean temperature difference between average inside pipe surface ttmperature and inlet and outlet slurry temperature, F.
u
= linear velocity of fluid, ft./sec.
Vb
= h e a r velocity of suspension, based on bulk density of
Nu
Pr Re p plr i* pto lit
suspension, ft./sec. Nusselt number, h D / k , dimensionless = Prandtl number, Cp/lc, dimensionless = Reynolds number, D V p l p , dimensionlem = density of fluid, lb./cu. ft. = bulk density of suspension, lb./cu. ft. = viscosity of fluid, Ib./(ft.)(sec.) = viscosity of fluid a t wall temperature, lb./(ft.)(sec.) = apparent bulk viscosity of juspension, lb./(ft.)(sec.) =
References
(1) ;tlves, G. IC., Chem. #no., 56, 107 (1949). (2) Alves, G . E., Boucher, D. F., and Pigford, R. L., Chem. Eng. Progr., 48, 385(1952). (3) American Institute of Physics, “Temperature, its Measurernent and Control in Science and Industry,” pp. 180-204, 855-861,
Reinhold, New York, 1941. Babbit, H. E., and Caldwell. D. H., ISD.EM. CHEILI., 33, 249 (1941). ( 5 ) Binder, R. C., and Busher, J. E., Truns. -4m.SOC.Mech. E n ~ r s . , 13, A 101 (1946). (6) Bingham, E. C., “Fluidity and Plasticity,” 1lcGraw-Hill Book Co., New York, 1922. (7) Bonilla, C. F., Cervi, A., Jr., Colven, T. J., Jr., and TVarig, S. J., presented as part of the Symposium on Heat Transfer, 44th Annual Meeting, Am. Inst. Chem. Engrs., December (4)
1951. (8) Brown, G. G., and coworkers, “Unit Operations,” 143, 438442, John M7iley & Sons, K‘ew York. 1950. (9) Chemical Engineers’ Handbook (Perry, J. H., Editor), AlcGraw-
Hill Book Co., New York, 1950. W.,and coworkers, Cheni. Eng. Rept. S-111, Columbia University, 1945. (11) International Nickel Co., Xew York, “Properties of Sorne Metals and Alloys,” 1948. (12) Kern, R., and Othmer, D. F.. T7u7m. A m . Inst. Chenz. EngTs., 39, 517 (1943). (13’1 . . XIcAdams, W. H., “Heat Transmission,” 2nd ed.., Chan. VIT --, McGraw-Hill Book Co., New York, 1942. (14) MacLaren, D. D., and Stairs. R. G., master’s thesis in chemical engineering, Columbia University. 1948. (15) hlickley, H. S.,and Trilling, C . -4., IND.EKG.CHm., 41, 1135 (1949). (16) Orr, C., Jr., and Dalla Valle, J. >I., presented as part of the Symposium on Heat Transfer, Annual Meeting, Am. Inst. Chem. Engrs., Preprint 13, December 1953. (17) Rouse, H., “Elementary Fluid Mechanics,” John Wiley & Sons, New York. 1945. (IS) Sieder, E. S . , and Tate, G . E.. IXD.E r c . CHEM.,28, 1429 (1936). (19) Wilhelm, R. H., \17roughten, D. bI.,and Loeffel, W. E?., Ibid., 31, 622 (1939). ( 2 0 ) Winding, C. C., Bauman. G. P., and Kranich, \V, I,., C’henL. Eng. Progr., 43, 527, 613 (1947). (10) Hoopes, J.
..
RECEIVED for review March 30,1954.
ACCEPTEDJuly 27, 1964. Abstracted from t h e doctoral thesis submitted by Jerome J. Salamone in June 1954 in partial fulfillment of the requirements for the degree, Doctor of Engineering Soience, at New York University. Material supplementary to thie artlcle has been deposited a8 Document No. 4459 with the AD1 Auxihary Publications Proieet Photoduplication Service. Library of Congress, Wajhington 25. D. C. A COPY may he secured by citing the docunlcnt number nnd b y rrmitting 8 2 . 6 0 for photoprints or 51.75 f o r 36-mm. microfilm. Advance payment is required. 1Iake checks or money orders payable to Chief, Photoduplication dervirr.. Library of Congress.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 2
