Conclusions
literature Cited
When an aerosol moves within a duct. the change in particle concentration caused by settling and by turbulent diffusion to the duct wall can be related mathematically to particle size, distance a n d velocity of movement. physical properties of the material dispersed and of the suspending medium. and the conditions of flow. ‘Ihe results of experimental work. in Lvhich a dibutyl phthalate aerosol was moved through a duct a t several velocities. correlate well with a semiempirical mathematical expression. However. a direct use of the data as shown in Figure 7 is probably to be preferred.
(1) Frbdlander, S. K.. Johnstone. H. F.. Ind. En?. Chem. 49, 1151 (195i).
( 2 ) Mitchell? R. I.. Pilcher. J. M.. Ibid.. 51, 1039 (1959). (3) R a m . W.E., “Principles of Inertial Impaction.” Tech. Rept. 1, U. S. Public Health Service Grant S-19. (4) Kosin, E.. Kammler. E.. Intelman, i V , . V I I I %. 76, 433 (1932).
RECEIVED for review June 25. 1962 RESUBMITTED July 9, 1964 . ~ C C E P T E D July 9, 1964 47th Annual A.1.Ch.E. meeting. May 1962. Baltimore, Md. Based on work sponsored by Fort Detrick, Frederick. Md,. under Contract DA-18-064-404-CML-122.
RATE OF SCALE FORMATION IN TUBULAR HEAT EXCHANGERS Mathematical Analysis of Factors Ir2Jueni.ing Rate of Decline of Over-all Heat Transfer Coeficients B
.
J
.
R E I T Z E R,’ Rpsearch DPpartment.
Iron Co.; Oak Brook: I l l .
Chicago Bridge
Relationships showing the effects of inverse solubility, crystallization velocity, scale properties, and heat transfer conditions on the rate of decline of the over-all heat transfer coefficient are presented for tube scaling conditions. Equations and data are shown for the important case of constant heat transfer as well as for constant operating conditions. For the latter case, with an additional restriction, the familiar equation
+
1 / U 2 = 1 / U O 2 /30 is obtained. It is shown that a simple relationship exists between /3 and the pertinent The equations presented should b e of help in interpreting and projecting commercial and pilot factors. plant heat transfer data. occurring on heat transfer surfaces such as in a n a crystalline deposition caused by precipitation from solution of a material which has a n inverse solubility relationship ( 1 ) . This t)-pe of scale build-up which. of course. causes a decreasing over-all coefficient of heat transfer. C7. also occurs \vith substances having a normal solubility curve in the case of cooling-for example. in a cooling crystallizer. Prcvention of scalt in most instances is impossible or uneconomical. It is necessary for both design and operation not only to understand this phenomenon but also to have means of predicting and follo\ving the course of the rate of deterioration of heat transfer ability. Combining the equation for the rate of crystallization with that for heat flux, with crrtain limiting assumptions, provides equations for following the change of 1.bvith time for the case of constant heat transfer as \vel1 as for constant operating conditions. T h e case where the solution in an evaporator. for example. is unsaturated in the vesscl but supersaiurated a t the tube wall is also treated. Btcaucr of the simp1ic:ations and assumptions used, the equations presented cannot be expected to hold for all cases of tube waling that may be encountered. T h e work does. however, fit into the pattern of past experience and shows quantitatively what can be expected from a change in the process variables and or scale properties. CALE
S evaporator is cornmonly
1
Deceased
Consider the mechanism for the formation of scale caused b y a substance which has an inverted solubility curve from a solution saturated with respect to the scaling component in a forced circulation evaporator. Figure 1 shows a hypothetical solubility curve for such a substance. ,4t the hot tubular surface the solubility will be the lowest. T h e solution in the laminar boundary layer will thus become supersaturated by the amount, s = c 1 - c s . It is thus a t the surface that nucleation will occur. Because of the temperature gradient in the tube wall itself, the first particles to form will crystallize in and adhere to the intergranular interstices and general imperfections or roughness a t and near the surface, forming a crystalline
I
tl Temperature
Figure 1.
’S
Typical inverted solubility curve VOL. 3
NO. 4
OCTOBER
1964
345
thin film of scale. Subsequent scaling proceeds on this smooth dense surface. The scale, because of its relatively low conductivity, becomes a barrier to hezt transfer. The temperature of the inner scale surface decreases, thus decreasing the supersaturation, and the rate of decline of U thereby decreases.
Constant Operating Conditions
Equation 9 can be rearranged and integrated as follows:
to yield
Rate of Scale Formation
I
The following arguments consider the scale thickness to be small compared to the tube diameter. The change of scale thickness can then be related to the mass build-up of scale by
dm
- _-
Ap
de
Substituting Equation 10 into Equation 6 gives the change in U with time
dx
-
de
Crystallization velocity depends on the rates of diffusion to and reaction a t the solid surface (5). The rate of growth can generally be followed ( 3 ) by the relationship
dm d6
C2
s = b(ts
- ti)
(3)
Combining Equations 1 , 2 , and 3 and solving for the rate of increase of scale thickness with time yield
dx kb - = -(ts de P
-
tl)"
1
c2
- tl
=
(5)
While the case of constant operating conditions is of considerable interest, a more important and common commercial problem is that of constant total heat transfer. This is caused by the fact that a constant evaporation or production rate is required. For this case
(6)
(g) At
Combining Equation 13 with Equation 6 results in
- ti
(14)
(7)
Combining Equations 6 and 7 and rearranging, ts
+ 0s originally presented
by McCabe and Robinson ( 6 ) . Badger and Othmer ( 7 ) presented a considerable amount of experimental data to back this relationship and showed its practical applications. The effects of the process variables and scale properties on the change of I' with time now become apparent. .4 substance possessing a steep inverted solubility curve, a relatively low density, and low thermal conductivity will tend to scale faster. Miterial that has a rapid rate of crystal growth will naturally also scale faster. Use of a high At will decrease L more rapidly. When reaction a t the surface controls, k will be relatively independent of velocity, and a high velocity u i t h increased hi should reduce the rate of scale formation. Fluid velocity should have little effect in the case of mass transfer controlling, since the effect of velocity will be about the same on both hi and k . The effect of velocity has been discussed (2, 4 ) .
Equating the total heat transfer rate to the rate through the inner liquid film leads to ts
1
- = LTU2
Constant Heat Flux
in which the h's and k t are corrected for the ratio of inside to outside diameter. Equation 5 can be reduced to
% + LO ' kt
+-2bkAt pkth,
(4)
T h e equation defining IJ is
1 1 - -
cb2
Equation 11 is of the form
1 , Equation 11 reduces to
=
1 - __ 1 -
= kAs"
At the limit, where reaction a t the crystal surface is controlling, n represents the order of the reaction. This case will hold for many slow scaling materials, especially in view of the high tubular velocities usually encountered. Where mass transfer is controlling, n = 1 . Between these limits. Equation 2 can be considered empirical. Bransom (3)demonstrates its applicability with data in this range. The pseudocrystallization velocity constant, k. may be dependent on the crystal surface temperature. However, over the usual range of temperature considered it can be assumed to be constant, or an average value over the range can be used. Over the relatively small range of temperature solubility covered for this type of work, a linear relationship c = a - bt may be assumed, leading to
U
With the restriction that n
Substituting Equation 14 into Equation 9 with subsequent cancellation of the term involving x yields
ktAt/ht kt
= --
(15)
- + x
LT0
Substituting Equation 8 into Equation 4 gives
346
I&EC PROCESS DESIGN A N D DEVELOPMEN1
This shows that the thickness of scale increases linearly with time. Integrating Equation 15 between 0 to 0 and 0 to x , and substituting in Equation 6 gives
I
l
k
U
Uu
pkt
(1 6)
weeks or even months, following which a steady rather slow decrease in U is observed. This may be explained by the following. T h e liquor being concentrated is unsaturated with respect to the scaling component but slightly supersaturated a t the hot tube surface. Figure 3 shows this condition graphically. T h e supersaturation in this case is well within the metastable region where nucleation is rare. Eventually, however, nucleation may occur and U will begin to decrease. Because the supersaturation i s low, the change of C‘ with time must also be slow. When incipient nucleation occurs-that is, the initial film of scale is formed-there is, as indicated, from Equations 1 and 2
70 ,Lc
60
3 50 40
dx - k dB P
But since s
= c1
-
cs
Jn
we have
dx
-
do
=
k
-
(cl
-
CJn
P
Combining Equations 7, 8, and 18 with c 1 ’ = a - 611, in which cl’ is the hypothetical saturated concentration in equilibrium a t t l , it follows t h a t :
9 , Hours
dx
Figure 2. Linear relationships for typical constant heat flux d a t a
d0
D
L
With n # I , Equation 19 will not give a simple closed solution upon integration. For constant operating conditions and n = 1, integration between the same limits as before yields the transcendental equation
I
I
I
~
1,
Figure 3.
1; Temperature
_ PX _ k(ci - GI))
1
=e
~(20)
1s
Scaling from an unsaturated solution
For the case of constant heat flux, combining Equations 14 and 19 again eliminates terms involving x , resulting upon integration in bUoAto
I n following heat transfer performance, it is convenient to observe the change in At required for constant production. Substituting Equation 16 into Equation 14 gives
Therrfore. both 1 / U and At are linearly dependent on time, with the variables afTecting the slopes of these changes as indicated in Equations 16 and 17. In addition to similar effects discussed previously, it is seen that these slopes are increased by a factor of the heat flux raised to the nth and (n f 1 ) th po\ver .for 1/ U and At, respectively. ‘Typical evaporator data are presented in Figure 2 for a constant product concentration and production rate. T h e fit of the data to the straight line would be even closer if minor fluctuations in heat flux were taken into account. Scaling from Unsaturated Solutions
It is often found in evaporator operations and heat exchange in general that scaling does not occur for a long period, perhaps
P
and by following the same procedures as before we obtain
and
The slopes of these curves are the same as those of Equations (c1 - ~ 1 ’ ) .
16 and 17 except for the term General Remarks
Theories concerning crystallization and the equations of heat transfer have been related to explain the mechanism of scale build-up in tubular heat exchangers. Most of the illustrations are for the case of heat transfer associated with evaporator operation but can readily be applied to heat exchange in general. An implicit assumption of the work is that the VOL. 3
NO. 4
OCTOBER 1964
347
temperature i: \ \ h i c h \elution \vith L: lvould be saturated t l = temperature of liquid in tubes t, = temperature at inner surface of scale At = total temperature droll from heating to cooling medium c' = over-all coefficient of heat transfer C, = initial IT x = scale thickness
t l ' = hypothetical
temperature change of the liquid through the heat exchanger is small compared with the total temperature drop from the heating or cooling medium to the liquid. This is often the case. T h e relatively simple relationships should prove of value in interpreting experimental data. By proper interpretation the work can be extended to cases of stepwise changes in At and to multiple effect evaporation problems. Nomenclature
GREEKLETTERS p = constant
A
p
= density of solid scale 0 = time All in consistent units
= area
constant b slope of solubility curve c, concentration of liquid cl' h>.pothetical concentration of liquid saturated with sealing component a t t l cs = concentration of saturated liquid a t inner surface of scale h , = film coefficient, inside tubes h, = film coefficient, outside tubes corrected for inside area h , = equivalent coefficient of tube wall corrected for inside area k = combined diffusion-reaction mass transfer coefficient k , = thermal conductivity rn = mass of scale component n = constant exponent s = supersaturation a
= = = =
Literature Cited
(1) Badger, \V, L..Othmer. D. F , , Trnns. A m . Inst. C'heni. Eri,qrs, 16, Part 11. 164 (1924). ( 2 ) Ranchero. .J. T.. Gordon, K. F.; A d r a n . Chern. Ser.. S o . 27, 105 (1 960).
(,., 3 i Rraksorn. S H.. Bri/ Chem
E n p . 38. 838 11960'1 .i. H. 'Perry; ed., 3rd ed.; p. 313, .McGraw-Hill. New York. 1950. (5) Hixson. A. LV.,Knox. K. I... In(/. En-. Cheni. 43, 2144 (1951). (6) McCabe, \V. I.., Robinson. C:. S.. Ibid.. 16, 478 (1924). ~~
~~~~
~
~
(4) Chemical Engineers' Handbook:
liEcEivm
for re\.ie\i July 3. 1963 ACCEPTEDJuly 9 , 1964
PRESSURE AND ADDITIVE EFFECTS ON FLOW OF BULK SOLIDS P F R
. U.
. .
A
,
A
.
U L S A R A , Polytrchnic Iw/i:itte of Brooklyn. B i o o k l ~ n ..\'. 1 ~ ~ ~ a ~RoJyn . Horboi. L. I...V. 1'. Z E N Z , Sqzutps I i ~ / r r t ~ o t i oIric.. . Gnrden City. L . I . . .V. 1.. E C K E R T , : l ~ s o c i ' n t r d . ~ u c i p o n i c sInr.. B
In contrast to the conventional addition o f fine anticaking agents to render "sticky" powders free-flowing, the addition o f coarse inert particles, in a quantity sufficient to make these the bulk of the mixture, will also result in a free-flowing mass at ambient conditions. Even without the use of any sort of additives, sticky powders can b e made to flow b y properly applying pressure differentials or reducing total ambient pressure. The effect of pressure differentials on free-flowing solids i s in accord with the conventional orifice equation.
u
NUMEROVS
po\vder-processing or handling operations it is
I-not 'only desirable but necessary that the powder be free-
flo\ving and exhibit no caking or agglomeration. Highly successful conventional practice calls for the addition of small amounts of flo\v-conditioning agents. These conditioners. usually in amounts of less than 1 weight yG,act principally as selective adsorbents for gases or vapors otherwise adsorbed on the surfaces of the particles of the sticky poLvder. T h e conditioners are usually of submicron particle diameter and thus exhibit an erormous surface area for adsorption. Certain conditioners also promote free flow by dispersing themselves throughout the interstices and particle surfaces of the poLvder. giving a lubricative effect; the effectiveness of some is also attributed to residual electrical charge, which counteracts agglomeration of a po\vder as the particles of the dispersed conditioner repel each other within the bulk mixture. Conventional submicron conditioners have won \vide acceptance because they are effective a t concentrations so low as to have a negligible effect on the functional properties of the po\vder to Lvhich they are added. Many of them are acceptable as additives even to food products and pharma348
I&EC PROCESS DESIGN A N D DEVELOPMENT
ceuticals. It Ivould. hoivever. in relatively rare instances be desirable to remove all foreign matter from the final product. and in some i:Istances submicron. highly adsorptive solids might be harmful to the product. In such cases it is possible to carry the sticky potvder through various processing phases in admixture with a n inert, free-flou.ing material of coarse particle size. Gravity Flow of Sticky Powders
Coarse Additives, 'To explore the ranges of admixture concentrations \vithin Lvhich coarse plus fines \ \ o d d still exhibit the free-floLv characteristics of the coarse additive. various mixtures of sand and po\vders Lvere made up by stirring lveighed batches in a large pail and pouring them into a cylindrical bin. T h e bin consijred of a 5.625-inch I.D. Lucite tube, 28 inchej high, closed at the bottom Lvith a flat plate. in the center of which a 1-inch hole \vas rapped and fitted \vith bushings, permitting threaded long- and short-tube orifice openings of 0.493- and 0.838-inch I.D. to be inserted. The cylinder \vas a small flat-bottomed bin \vith variable discharge porr characteristics. .2 standard 20- to SO-mesh Otto\va sand \vas used as the free-flo\\-ing coarse additive and fly ash and Xficro-Ckl rrprc-
.
