T H E J O U R N A L OF I-VDCSTRIAL A N D ENGINEERING CHEMISTRY
May, 1921
427
THE SYMPOSIUM ON DRYING ~
~~
~
~~
Papers1 presented before the Division of Industrial and Engineering Chemistry a t the 6 l s t Meeting of the American Chemical Society, Rochester, N. Y., April 26 t o 29, 1921
The Rate of Drying of Solid Materials By W. I(.Lewis DEPARTMENT OF CHEMICAL ENGINEERING, MASSACHUSETTS INSTITUTE OB TECHNOLOGY, CAMBRIDGE, MASS.
I n t h e design of drying equipment and processes t h e engineer must provide 1-An adequate supply of heat for the evaporation of moisture. 2-Sufficient air t o sweep away the moisture under the conditions in question. 3-Such control of temperature and humidity as will protect the product against injury. 4-Sufficient time for the escape of the moisture from the material being dried. T h e factors governing t h e first three of these are well understood and are covered in present designing practice; regarding t h e fourth, there is little in t h e literature, a n d mistakes in design are not infrequent. It is t h e purpose of this article t o present t h e conditions governing t h e r a t e of drying of solids. FUNDAMENTAL FACTORS AFFECTING
DRYING
RATE
Because i t has such a n important relation t o drying r a t e i t will first be necessary t o call attention t o one factor, t h e character of which is already well understood. This factor is t h e moisture retained on solids under ordinary conditions of temperature and humidity. Most solids hold a certain amount of moisture,
10
40 30 40
SO
60
X,
Br Cmf Re/crfive Humidify
FIG. 1
80 90 /W
/O
10 30 40 50 60 70 80 90 ID0 &r Cenf Re/at/va Humidify
FIG.2
even when in contact with unsaturated air. This moisture is probably adsorbed on t h e surface of t h e solid; a t any rate, t h e amount retained under equilibrium conditions is a definite function of temperature and humidity. Thus, a t ordinary temperatures cott o n in contact with air of 50 per cent humidity retains 6 per cent of moisture. Cotton holding less moisture t h a n this will pick u p moisture from air of 50 per cent humidity; cotton damper t h a n this will lose moisture i n such air. It is obvious, therefore, t h a t air of 50 per cent humidity cannot dry cotton below 6 per cent moisture, because this moisture content of cotton represents a t r u e equilibrium with t h e air. 1
Received April 5 , 1921.
T h e moisture content of a material which corresponds t o equilibrium with t h e air with which t h e material is in contact will be spoken of as “equilibrium moisture.” T h e total moisture content less this equilibrium moist u r e represents t h e moisture which can be evaporated by drying in t h e air in question. This difference will be called “free moisture.” Moisture will be reported as pounds per 100 lbs. of dry material (or, in some cases, per pound of dry material). While t h e equilibrium moisture content of a material varies with both temperature and humidity, i t changes b u t slightly with t h e temperature of air, t h e relative humidity of which is held constant. It is therefore more convenient t o plot t h e equilibrium moisture cont e n t against t h e relative humidity rather t h a n against t h e absolute humidity. If t h e temperature in question does not vary widely i t is allowable t o draw a single such curve; where t h e temperature variation is large, a series of curves, one for each specific temperature, should be drawn, a n d results interpolated between them. T h e equilibrium water content of wood, leather, soap, and textiles is shown in Figs. 1 t o 4. Such
/O
20 30 40 Z !.
Per Cenf &/af/vr
@ 7V & S /OO
Humid, ty
FIG. 3
/O
20 30 40 50 60 70 L@ 90 85
Per CenP Re/af/veHumidify FIG.4
curves must be determined experimentally in each individual case. T h e r,ate of drying of any material is obviously determined b y t h e temperature and humidity of t h e air with which i t is in contact, b y t h e velocity of t h a t air past its surface, and b y t h e heat supply t o which i t is exposed. These controlling factors, characteristic of t h e external surroundings of t h e material being dried rather t h a n of t h e material itself, will be referred t o as t h e “drying conditions’’ of t h e problem in hand. Fig. 5 represents a typical drying curve for a solid material, t h e moisture of which is evaporating under constant drying conditions. This particular materiaL has a n equilibrium moisture content of 9.5 per cent.
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T H E J O U R N A L OF I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
It will be noted t h a t the rate of drying is a t first rapid, and then decreases, t h e moisture content falling off in a characteristic “fade-away” curve. This curve becomes asymptotic t o t h e equilibrium moisture, but theoretically never really reaches this limiting value. Even in the case of very thin films t h e curve has this
proximation by assuming t h e line AB straight. Call E F the average concentration of moisture in t h e sheet y. Call t h e surface concentration of moisture ys. Call t h e total weight of moisture in t h e sheet for L each unit of surface w. Obviously, w = -y. The 2 rate of diffusion of moisture from the interior of t h e sheet will be proportional t o t h e difference in concentration ( y - y s ) , and inversely proportional t o t h e distance t o be traveled, L/4; t h e proportionality cons t a n t we shall call A. T h e rate of surface evaporation will be equal t o a coefficient, R, times t h e surface concentration. These two must be equal t o each other and equal t o t h e rate of loss of moisture b y t h e sheet, i. e., dw d8
_ - =
3 4 5 6 T / m e in M t n u t c s Fro 5-RATE OF TWINE DRYING
shape, so i t seems reasonable t o assume t h a t t h e rate of surface evaporation falls off as t h e moisture content of t h e surface decreases. The simplest assumption is t h a t t h e evaporative rate is proportional t o t h e free moisture content of t h e surface. Probably t h e actual surface exposure of moisture itself is proportional t o t h e moisture content of t h e surface, and it is obvious t h a t t h e rate of evaporation would be proportional t o t h e water surface actually exposed. The drying of a solid of necessity involves two independent processes, first, t h e evaporation of t h e moisture from t h e surface of t h e solid, and, second, t h e diffusion of t h e moisture from t h e interior of t h e solid out t o t h e surface. The evaporation of a n extremely small amount of moisture from t h e surface will leave t h e surface practically dry unless and until fresh moisture diffuses from t h e interior of t h e solid out t o t h e surface t o restore its moisture content. Since t h e surface can be conceived as extremely thin, no appreciable evaporation can take place without sufficient diffusion t o compensate quantitatively for surface evaporation. T h e two must, therefore, be equal?
Vol. 13, No. 5
Rys =
4A(y
-
whence
dw --
8ARw
+
=
L d% L(4A RL)’ This may be looked upon as t h e basic differential equation governing t h e drying of solid materials in sheet form. Since t h e water content appears as dw/w,this expression is independent of t h e units in which water is measured. One may therefore call W t h e free water content of any desired quantity of t h e material and write d W / W in place of dw/w. The quantity R is obviously a function of t h e drying conditions. Furthermore. when t h e water content W becomes very high (100 t o 200 per cent on t h e d r y material, depending on t h e substance) t h e drying r a t e no longer increases with increasing moisture, b u t remains constant, i. e., t h e surface is water-saturated. This condition is not often met, and this discussion assumes t h e water content less t h a n this critical amount. This equation was derived on t h e assumption t h a t t h e equilibrium moisture was negligible. Where this is not true t h e moisture content t o be used in t h e equation is t h e free moisture; or t h e total moisture W less t h e equilibrium moisture E, i. e., -d(W - E)/(W - E)d6 = 8AR/L(4A RL).
+
I
D E R I V A T I O N OF D R Y I N G F O R M U L A S
For purposes of derivation, assume a sheet material, t h e thickness of which is L. Assume t h a t t h e equilibrium moisture of t h e material in question is negligibly small. Let Fig. 6 represent a cross-section of t h e sheet, the line C M representing t h e surface and D N t h e center line of t h e sheet. From D C as a base, plot t h e concentration of moisture in parts by weight per unit volume vertically upward. Call t h e initial concentration equal t o CM, so t h a t t h e area under line N M represents t h e initial moisture content of t h e sheet. When surface evaporation starts t h e moisture content of t h e surface will drop t o some point such as B. Diffusion will immediately start and at t h e time in question t h e moisture content will have fallen t o some such condition as A B . This line AB will not be straight, b u t its equation will be determined by t h e integral of t h e diffusion equation, i P y / ~= --sy/se.
The exact integration of this equation is, however, SO involved t h a t we have chosen t o integrate i t by ap-
C
w i -
FIG.7
FIG.6
Assuming constant drying conditions, and a given material t h a t does not shrink greatly during drying, t h e right-hand side of this equation is a constant, which may be called t h e drying coefficient K. Integration constant, or, calling t h e gives log (W - E) = - KO initial content a t time zero W,, KO. log (W. - E)/(W - E) Figs. 8 t o 11 represent experimental d a t a on t h e drying of twine passing around steam-heated drums.
+
May, 1921
T H E J O U R N A L O F I N D U S T R I A L A N D E N G I N E E R I' N G C H E M I S T R Y '
429
07
06
30
/. 0
Time in M i n u t e s FIG.8
50 40
30 20
/O
8 6 5 4 3
5
2.0
FIG. 10
FIG. 9
I n Fig. 8 t h e free moisture is plotted against t h e time t h e twine is in t h e dryer. From this curve t h e slope was read off graphically at each point, a n d also plotted against t h e time. Finally, t h e slope divided by t h e free moisture was plotted, giving, with exception of t h e last point, substantially constant values. The graphical determination of slopes is always inaccurate, and i t is far more satisfactory t o use t h e integrated expres-
2
T/me /n Minu fes
7im e in Minu f e s
sion and plot t h e logarithm of the free moisture against t h e time, as is done in Figs. 9 and 10. I t is more convenient t o use semi-logarithmic paper, as in Fig. 11. S P E C I A L CASES-SURFACE
E V A P O R A T I O N THE LI%lITING FACTOR
+
T h e drying coefficient, K = 8 A R / L ( 4 A RL), varies with rate of diffusion and of surface evaporation, a n d with thickness. Two special cases are of importance. First, if diffusion is very rapid in comparison with surface evaporation, R L may be neglected in comparison with 4.4, and K = 2R/L. Second, if diffusion is very slow compared with surface evaporation, 4A is negligible compared with R L . and K = 8A/L2. For sheet materials with rapid diffusion we have no d a t a available except for such as shrink greatly in drying. Results are given for one of these, heelboard, made of a pulp of ground leather and paper, t h e thickness of a given sheet of which is found experimentally t o increase linearly, with t h e moisture content, i. e., L = Lo (1 a W ) , where Lo is t h e thickness of t h e dry sheet, and a is a constant coefficient. Therefore, t h e drying coefficient, K = 2R/(1 a W ) L o . Sub-
+
+
100
ZcW
300
400
rime in Mirlufes
Fro. 12
Jra?
/00
ZGQ
3W
4W
Tlmc in Minutes
Fro 13
500
T H E J O U R N A L OF I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
430
13, NO. 5
1701.
I I I I I I I I J
4 6 8 /O /2 /4 G /.8 20 Wafer vapor pQrfio/ ressure difference,
2
IO
mL
FIG.14
stitution of this value into t h e differential equation and integration gives aW 2.3 loglo (W - E) = KO const.
+
FIG.16
FIG.15
+
If t h e left-hand side of this equation be plotted against t h e time a straight line should result. Figs. 12 a n d 13 show t h e d a t a of two drying runs on this material. The logarithmic plots are also included t o show t h e magnit u d e of t h e correction for variation in thickness of sheet: As long as t h e free moisture content of a material being dried is reasonably high, in t h e absence of direct exposure t o a heating element t h e material will remain at t h e wet bulb temperature of t h e drying air. When t h e free moisture becomes very low, t h e material is heated u p t o t h e temperature of t h e air. T h e transition is gradual, and unless t h e drying is carried, say, close t o equilibrium i t is a safe approximation t o assume t h e stock at wet bulb temperature throughout t h e drying operation. The rate of surface evaporation is determined by t h e rate of diffusion of water vapor through t h e stationary film of air surrounding t h e sheet. This diffusion is proportional t o t h e difference between t h e partial pressure of t h e water on t h e surface of t h e sheet (wet bulb temperature) and t h a t in t h e drying air. This difference can be read off directly from t h e usual psychrometric tables or charts, and will be called p . T h e r a t e of diffusion will increase with increasing air velocity, due t o decreasing thickness of t h e air film. At constant velocity, therefore, K = -2R = * L L
where b is a proportionality constant, and K is t h e slope of t h e logarithmic plotscof Figs. 9 t o 11, a n d of t h e corrected logarithmic plots of Figs. 1 2 a n d 13. Fig. 1 4 shows K for five separate runs on heelboard at cons t a n t sheet thickness and air velocity, b u t at variable ~p (due t o wide variation>finboth temperature and humidity of t h e drying air), plotted against A$. K is proportional t o Ap within t h e experimental error. Fig. 15 shows K plotted against 1/L for two series of runs in which air velocity and Ap were kept constant. Table I shows t h e constancy of b = KL/A p a t a fixed
air velocity for runs in which L and A p each vary fivefold. Fig. 16 indicates t h e variation of b = K L / A ~ with air velocity. 1 TABLE I-SHOWINGCONSTANCY OA K -
A@
AT
CONSTANT AIR VELOCITY:
(Velocity 180.7)
K
Run No. 1 2 3 4 5 9 13 17
1
,= 1
9
-r
3.9 3.9 3.9 3.9 8.0 12.1 15.1 19.7
0.000795 0.000870 0.000859 0.000723 0.000790 0.000838 0.000874 0.000940
DIFFUSION T H E L I M I T I N G F A C T O R
Where t h e ratio of t h e diffusion of moisture inside t h e material itself t o t h e r a t e of surface evaporation is low, i. e . , where K = 8A/LZ, t h e concentration of free moisture in t h e outside surface layer is negligible, air velocity has no marked effect except as i t increases heat transfer, and t h e temperature of t h e stock is much nearer t h a t of t h e drying air. A case i n point is t h e drying of closely twisted cord traveling over rotating, steam-heated drums. T h e process may be looked upon as a diffusion of heat into t h e cord as much as of moist u r e out. Experimental results are given in Fig. 17 t o demonstrate t h e proportionality of drying r a t e t o t h e inverse square of t h e diameter (the thickness of t h e 0
0.5
Yonah'on of r a f e o f drying
--
w / f h dibmefer Qfcord
0
/
T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
hilay, 1921
sheet of cords traveling over t h e drum) rather t h a n t o t h e inverse thickness as in t h e preceding case.
n log (W, where
= log
e
+ log K,
K = 4A(Wo - E)"/L2,
SKIN EFFECT
I n the drying of a thick layer of a material in which internal diffusion is very slow ( e . g., soap, glue, jellies, or
- W)
431
a n d log (W, - W) plotted against log 8. The curves are straight within t h e experimental error, a n d for-this case t h e value of n is seen t o be $bout 1.93. G E N E R A L S I G N I F I C A N C E OF THE F O R M U L A S
FIG. 18-ADIABATICAIR D R Y I N G O F SOAP
wood), the diffusion gradient is not quickly set u p into t h e center of t h e mass so t h e basic differential equation given above must be modified. Let G D (Fig. 7) be a cross section of t h e sheet. Plot water ccncentrations u p from CD, G E being t h e initial concentration, y o , a n d B t h e surface concentration, y s . ys will be substantially equilibrium concentration with t h e drying air. Diffusion will start, t h e gradient being t h e line AB, t h e point A moving back as drying proceeds. Call t h e thickness, AE, through which diffusion is actually t a k ing place, 1. This is a variable in t h e equation
- aw/ae
=
A(Y,
While these equations have been derived for sheet materials, they apply satisfactorily t o lumpy, granular solids, except t h a t t h e correction for lump or grain size must be modified. Furthermore, these equations assume constant drying conditions throughout the drying operation, a state of affairs never met in equipment of the heat design. We have integrated and tested out these equations for t h e most important cases arising in industrial practice, but i t would require too long t o present them here. Usually, however, in calculating t h e drying time i t is sufficiently exact t o employ for A p its average value during t h e drying period, using the arithmetic mean of the initial and final values if these differ by less t h a n two- or three-fold, b u t using t h e logarithmic mean of t h e terminal values if differing more t h a n this. For E use its value a t t h e end of t h e drying period or operation, because a t this point W - E will be small a n d must be accurately determined. The use of this value of E will introduce no serious error in t h e earlier stages of drying, where W is large.
%i*
-
/
O
+
/
2
3
4
5
6
7
- ys)/i.
Thislayer AE is t h e "skin effect" of t h e drying operation. By comparison of areas in this diagram one sees t h a t
L(% -Y)/(Yo - Y J . - w = L(yo - y ) / 2 ,
1 =
Furthermore, wo
Z(r0 -- y s ) / 2 ; whence - (w, (W,
- E)2
=
- w)dw = 2A(w0 4AB/L2.
and w o
- E
=
- E)dO/L2,or (W, -W)a/b
Since t h e water content appears as a ratio, i t may be t a k e n for any desired amount of materihl, e . g., if V7 = water per 100 lbs. of dry material, (Wo - W)'/(W,
- E)2
=
4AB/L2.
According t o this equation, t h e drying time is proportional t o t h e square of t h e loss in moisture since t h e s t a r t of drying over t h e initial free water, a n d t o the square of t h e sheet thickness. Because of the simplifying approximations used in its derivation it is better t o interpret it as indicating t h a t t h e drying time is a power function of these quantities, the power being nearly 2 . An analogy is found in t h e case of liquid friction, which is usually assumed proportional t o t h e square of t h e liquid velocity, b u t is actually a power function with a n exponent of about 1.8. When the point A has receded t o the center of t h e sheet, F, t h e character of t h e drying curve will change and transform itself into t h e case originally considered. Illustration is found in t h e drying of bar soap, d a t a for which are plotted in Fig. 18. Because t h e exact exponent is unknown, t h e equation has been thrown into t h e form
Time in Minutes FIG. 19-ADIABATICAIR DRYING
With regard t o the two basic drying coefficients, A and R, the variation of R with temperature, humidity, and air velocity has already been shown. A varies for each specific solvent and material, b u t always increases rapidly with temperature. This is made use of in drying materials which shrink and harden upon evaporation of t h e solvent, b u t which must not be allowed t o crack, as will happen if t h e surface dries a n d contracts around a still swollen and incompressible
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T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
rime /n #inufes FIG.2Cb-ADIABATIC AIR DRYING
interior. Such substances, e. g., wood, varnish, films, artificial leather coatings, etc., are dried a t high humidity so t h a t even t h e outer surface is not too dry because of the high equilibrium moisture, and a t high temperature, so diffusion will be rapid. There is a certain concentration gradient which may be maintained through t h e surface layer without straining it t o rupture. The humidity is adjusted t o get this gradient, a n d the temperature raised t o get t h e most rapid diffusion possible with this limited concentration difference. As drying proceeds t h e surface layer gets thicker so t h a t a greater total concentration difference is allowable without increasing t h e concentration gradient, i . e . , the humidity can with safety be progressively reduced. The values of the drying coefficients should where possible be determined from the measured performance of full-scale equipment. The result of plant tests can even be used t o determine the equilibrium moisture.
Vol. 13, No. 5
T h u s Fig. 19 shows t h e rate of loss of water of a porous, spongy, lumpy material, exposed a t a point in a commercial dryer where the drying conditions are substantially constant. By reading the slopes off this curve a n d plotting against t h e total water, t h e intercept of t h e line obtained gives t h e equilibrium moisture, E = 8.5, a t which evaporation ceases. One can now draw t h e logarithmic drying curve for this material (Fig. 20) from which the time required t o reduce t h e moisture content t o a n y required point can be determined. 70 The slope of this last line is t h e drying coefficient K. From 60 runs under other drying conditions t h e variations of K determine A and R. Fig. 2 1 shows t h e application of these general equations t o t h e drying of a n organic solvent from a fibrous material. 30
It is believed these facts demonstrate t h a t t h e drying of a 20 solid material represents a balance between a process of diffusion of moisture through t h e substance a n d of evaporation from its sur0 10 20 30 4 50 60 7ime in Minutes face; and t h a t these processes can be FIG.2 l - D R Y I N G OF ORGANIC SOLVENT FROM F I s R o u s MATERIAL AT CONSTANT DRYING quantitatively repre- CoNDITroNs sented by t h e differR ) , which ential equation, --dw/d8 = 8ARw/L(4A can, after modifications dependent on the material being dried, be integrated into simple a n d usable formulas which answer the question as t o drying rate.
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The Theory of Atmospheric Evaporation-With Compartment Dryers
Special Reference to
By W. H.Carrier CARRIERENGINEERING CORPORATION, 39
I n this paper an a t t e m p t is made t o state as concisely as possible the fundamental theory involved in air dryers with particular reference t o compartment drying, although the greater part of the theory developed applies equally well t o t h e tunnel type, the continuous type, and the spray type of dryers. We have endeavored t o make t h e theory general t o apply t o the evaporation of any volatile liquid in a n y kind of atmosphere. I n this respect, we believe t h e theory is somewhat new. Moisture exists in material in two distinct forms-as free moisture, and as -hygroscopic or absorbed moisture.
CORTLANDT
Sr., NEW Y O R K , N. Y.
Evaporation is t h e term usually applied t o the converting of a liquid into a vapor in a n atmosphere whose pressure is above t h a t of the vapor pressure of t h e evaporating liquid, i. e . , causing vaporization below t h e boiling point. The heat of vaporization is usually taken entirely from t h e air itself, and this will be chiefly t h e basis of t h e theory considered. The theory will b e considered (1) with reference t o the evaporation of free moisture, a n d (2) with reference t o hygroscopic moisture. The rate of evaporation depends upon: 1-The vapor tension of the moisture in the material corresponding to its temperature.
