ings rapidly. Therefore, t h e r e may be some snbstantial use of lead phthalate in tbis d i r e c t i o n in the future, nia.king it p o s s i b l e t,o use clear, transparent finishes w i t h a greater life expectants than heretofore. Esperitnents are being continuedin the directionof ernpl!J~iIig&tier the metallic phtlralatrs or pignimt,s eo-precipitated with them in ot,tier industries, sue11 a s p a p e r , c e r a m i c s , and fabrics, hut they have not yet hcen develoyd sufficiently f o r t h i s purpose to varrant immediate applic a t i o n . On tlie other hand, whim used in r u I) b e r e o n i p o u n d i ii g , lead pli t ha1 a t e k n d s t,o loI\-er tlie 'i( i s c osi t y of the compoiind during m i l l i n g and apparently increases tile speed of vnlcarlieation. During the past year the w r i t , e r I t a s f o u n d that plitlralic acid as \rellas tire metallic plrt~halatesh a w a &st,inet, effect i n discouraging t li e growth of mildew >Then u s e d i n paints that are exp o s e d in damp. WITH ExmnioR VARNISHAND Ikrossn warm c I i m a t.e s, FOR 6 MONTHS such as that of (Adou,) pmei coated w i t b a clear. untreated rarrrirh. (Abmei panel coated with varnish Florida. Mercury wmtainin$, S m r cent ired phthainte: this aiid copperphthalamount ~d not detract greatly from the clarity of the varnish but did function BS B ates h a v e heen l i g h t shield and add greatly to tile dursbility. f o u n d useful in Compositions for discouraging the attack of teredo and Limnoria upon submerged wood or barnacles upon the steel hulls of ships. These experiments are being carried out in the subtropical maters of Bineayne Bay, and it may be some time before qiiantitative data can he reported. R E C ~ Y EMarch ~ 20, 11437. Presented before the Division
oi Paint %lid VarniSh Chemistry s t the 9:3id Xeetina of the Amcriean Chernicsgl POeiCLY. Chapel ITill. N. C . . A ~ n 12 l to 15. 1937.
641
Integration of the Drying Equation at Constant Temperature A. E. MARKUAM 3330 East John Street. Seattle, Wash.
T
HE differential drying equation developed in chemical engineering textbooks ( I , 5) for variable conditions using an inert gas, where surface evaporation is the controlling factor, is: -dW 1V
*f") ~
(K~ Il)da
(1)
The integration of this equation becomes possible if the quant,ities bf(u), L, and (Hs - If) are constants or are expressible as functions of W or 6'. For drying a t constant temperabure, ( H , - H ) or A€f is a function of IV, but the relation has not hitherto been expressed in form suitable for fornial integration. Hence the integrated form for this case is:
The left side of this equation has always been evaluated graphically. Such mettiods at best consume considerable time. The object of this paper is t,o show relations between Wand A H xhich make the formal integration possible. The humidity chart (Figure 1) sliovs the nature of this problem when
FILUXE1.
1 I ~ i ~ 1 n Cmnr ir~
the drying operation takes place a t a constant temperature sucli as t. At any point in the dryer, the air has a humidity E-for example, I . At this point the wet-bulb humidity is represented by D, and A H is represented by the segment I N .
INDUSTRIAL AND ENGINEERING CHEMISTRY
642
At a point farther along the dryer, when W has decreased, H will have increased-for example, to K . At this point AH is represented by K M . Thus AH is a definite function of H . For any specific drying problem, H is a function of W . For countercurrent flow this relation is shown by the material balance : r ( H - HI) = W - WI (3) The derivation of a theoretically exact equation between I$, and W leads to complicated results. However, the relation can be expressed very satisfactorily by a n empirical equation. Such a n expression must be both accurate and of a form which will be integrable when substituted in the differential equation. The expression A - BH A H = ___ (4) 1 CH
+
satisfies both these requirements. The quantities A , E,and C are functions of the temperature of the drying operation. Their values between 80" and 140" F. are shown graphically in the curves of Figure 2 and are as follows: A 0.0071 0,0086 0.0100 0.0115 0.0129 0.0143 0.0162
Temp., F. 80 90 100 110 120 130 140
B
C
0.320 0.280 0.235 0.198 0.161 0.130 0.108
18.0 20.2 19.6 16.0 12 5 9.1 9.9
They may be calculated from the humidity chart at any temperature by choosing three values of H , finding the corresponding values of A H , and then substituting each pair of values in turn into Equation 4. This procedure gives three equations containing A , B, and C, and their values are easily found. If the values of H are chosen wisely the problem is considerably simplified. For instance if H is
a Col
01 80
I
90
1
I
I
100 110 120 TEMPERATURE, *F.
FIGURE 2. VALUES OF A , B, BETWEEN
80"
AND
lo
I
UO
I40
AND
140" F.
C
chosen equal to zero (point G in Figure I), then AH = A , which is the value G J , or it is the wet-bulb humidity for dry air. After A is determined, B is found by choosing F for the next point: H = H , and A H = 0 A = BH or B = A / H = GJ/GF
C is then easily found by choosing one intermediate humidity, such as K , finding the AH corresponding to this point or K M , and introducing these values in Equation 4. The agreement a t 110" F. between the values of A H , as read from Figure 1, and those calculated by means of Equation 4, using A = 0.0115, B = 0.198, and C = 16.0, is as f OlOTVS: H
AH (from Chart) 0.0580 0.0000 0.0486 0.0013
0.0401 0.0325 0.0259 0.0203
0.0025 0.0037 0.0047 0.0057
AH
(Equation 4)
H
AH
&tu
(from Chart) (Equation 4) 0,0068 0.0069 0.0162 0.0080 0.0080 0.0106 0.0063 0.0092 0.0094 0.0103 0.0104 0.0028 0.0115 0.0116 0.0000
VOL. 29, NO. 6
The agreement is much the same as that found for other temperatures. The errors are within those encountered in reading a humidity chart of the ordinary size. The integration of Equation 2 is now possible. From Equation 3: r H = rHi
- WI + W
Multiplying both numerator and denominator of Equation 4 by r: Ar r Ar AH = r
AH =
- BrH
+ CrH
- B(rH1 - WI + W ) + C(rH1 - WI + W )
(5)
Then Equation 2 becomes :
This is a form easily integrated, the result being:
Equation 7, although long and involved, is simpler to use than graphical integration and leads to the same results. Thus the example given by Walker, Lewis, and McAdams (4) is solved by this means. The temperature is 120" F. The values of the terms appearing in Equation 7 are: A = 0.0129, B = 0.161, C = 12.5, r = 43.7, H I = 0.0053, Wo = 2.33, W , = 0.111. Substitution of these values gives a value of 433 for the integral, compared to 432 obtained by graphical integration. The first problem presented by Hougen (3) can be integrated by Equation 7. In this case the temperature of drying is 140' F. and the constants are: A = 0.0162, B = 0.108, C = 9.9, H I = 0.0158, W o = 1.484, W1 = 0.049, r = 104. The value of the integral as found from Equation 7 is 286. Hougen obtains 300 by a graphical integration. In this graphical integration the change of equilibrium moisture with humidity was considered, so that one would expect a certain difference on this account. As Hougen points out, since A H does not vary greatly, a n average constant value can be assumed for integration. However, such an average is likely to be uncertain, especially as the changes in AH become greater.
Nomenclature humidity of air, lb. waterjlb. dry air d = differential operator W = free water concentration in stock r = Ib. air/Ib. bone dry stock A H = Ha - H H, = wet bulb humidity of air A , B, C = constants at any particular temperature Subscripts: 0 = conditions at end of dryer where stock enters 1 = conditions at end of dryer where stock leaves H
=
Literature Cited (1) Badger and McCabe, "Elements of Chemical Engineering," 2nd ed., p. 310, New York, McGraw-Hill Book Co., 1936. ( 2 ) Hougen, 0. A , , IND.Exa. CHEM.,26, 333 (1934). (3) WalkeT, Lewis, and McAdams, "Principles of Chemical Engineering," 2nd ed., p. 542, New York, McGraw-Hill Book Co., 1927. (4)Ibid., p. 543. RECEIVED January 8, 1937.
