(8
ied Humidification Calculations Systems Other than Air and Water
a
JOHN G. LEWIS AND ROBERT R. WHITE Departrnenf o f Chemical a n d M e t a l l u r g i c a l Engineering, University of M i c h i g a n , Ann A r b o r , M i c h .
T
Equation 5a may be employed to construct plots of enthalpy versus modified enthalpy n-ith temperature as a parameter, a$ shown in Figure 1. Provided sp,f,and b are constant, the plots are straight and parallel, and linear interpolation is possible. Furthermore, if two points are determined for any one teinperature, only one point need he determined for successive trmperatures and the lines drawn parallel to the first line.
HE
analysis of humidification problems for systems other than air and water usually involves laborious trialand-error calculations in order t o determine the course of the operation or the required height of a tower ( 1 , 2 ) . A modification of the calculation technique of Mickley ( 2 ) when applied to a single run on laboratory equipment will provide information permitting the calculation of individual heat and mass transfer coefficients for the gas film and the heat transfer coefficient for the liquid film. Trial and error is required, but the trials can be made quickly. A similar modification has been proposed by hfizushina and Kotoo ( 3 ) .
Application of the Method
Line PQ in Figure 2 is a plot of the enthalpy of air saturated with toluene versus the temperature at which saturation has been attained. RS is a plot of the modified enthalpv of air saturated with toluene versus the temperature a t v hich saturation has been attained.
Basis of Method
The equations describing mass and mass transfer in a humidifier previously derived are ( 1 , 2 )
If, in addition, it is assumed that the ratio of the area active in heat transfer and mass transfer are constant then
aH = f
(3)
ann Now define the “modified latent heat of vaporization” as
and the “modified enthalpy” as
+ HA‘
H‘ = sgTg
Introducing Equations 2, 3, 4, and 5 into the heat transfer relations
GlcidTz
-
= ~ L U H (TI
Ti) d Z
(6)
and GbsgdTg = - h g a H (To - Tz)dZ
Locus of Saturation
(7)
2
and combining gives
Enthalpy v s Modified
and
(9) and
1
MIJOIFIED AIF
(10)
NTHALPY, BTU PEI
Figure
486
i
LB. DRY A I R
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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487
be calculated by solving Equation 10 simultaneously with the saturation curve represented by the line RS. This s o h tion would be accomplished graphically by determining the intersection with RS of a straight line draum from point A’ with a slope of
When the interface conditions were known, Equation 8 would give the slope of a plot of the enthalpy of the gas versus the temperature of the gas. I n graphical terms, the slope of the curve describing ent,halpy as a function of temperature would be equal to the slope of the line X’Al drawn. Since the ratio
0 20
1
40
60 TEMPERATURE
Figure
80 ,OF
P
RS in Figure 2 was plotted from line PQ by the use of Equation 5a.- A 1M is a plot of gas enthalpy versus gas temperature. The slope of this line is given by Equation 8. A M is a plot of gas enthalpy versus liquid temperature. This line is known as the operating line, and the slope of this line is given by Equation 1. Lines A’&, B’B,, etc., are described by Equation 10, and are known as the tie lines. At point 2 the enthalpy of the inlet gas stream is plotted versus the temperature of the inlet gas stream. The air was assumed to be free of other materials at this point. At A the enthalpy of the inlet gas stream is plotted versus t h e temperature of the outlet liquid stream. At @ the enthalpy of the outlet gas stream is plotted versus the temperature of the outlet gas stream. At M the enthalpy of the outlet gas stream is plotted versus the temperature of the inlet liquid stream. At A’the modified enthalpy of the gas stream is plotted versus the temperature of the gas stream. At point A ’ the modified enthalpy of the gas stream is plotted versus the temperature of the liquid stream. Figure 3 is an enlargement, not to scale, of the lower right side of Figure 2. I f the ratio
hi
c
b ko
were known, the interface conditions (at which the gas is saturated with vapor of the liquid) on a modified enthalpy basis could
is unknown, a straight line is drawn from A ’ upward and to the left to intersect RS in point A,. Line A’A1 is drawn with an assumed slope. Equation 10 shows that the slope of A’A1 is a function of the ratio of the heat transfer coefficient in the liquid film t o the mass transfer coefficient in the gas B is film. Line Z’Al is drawn. drawn parallel to >’A1 for a short arbitrary distance. Equation 8 shows - _t h a t by this procedure the slope of A B is IO0 made t o approximate the slope of the plot of gas enthalpy versus gas temperature. The ordinate of point 3 is projected to the right to intersect A M in point B . Point 3‘ is located from B just as 2’ is located from 2. B’ is then located from E’ jupt as A’ is located from Line B‘B1 is drawn parallel to A’A,; B’B, is drawn; then line segment B is drawn for a short B was drawn. arbitrary distance just as The process described is repeated until the successive line segments extended from 3 reach or exceed the coordinates of point 2.-If the last line segment of the series of segments starting from A does not pass through %, then the process must be repeated with a different slope assumed for the lines A ’ A I , B‘B,, C‘C1, etc. If the last line segment passes through the 2, then the slope of the lines A’AI, B’BI, C’CI, etc. (the tie lines) has been correctly assumed. Once a correct slope has been found for the tie lines, the left side of Equation 9 may be integrated graphically. Corresponding values for H,, HB‘,and Hi’,are represented in Figure 2 by points A , A’, and A I respectively. All factors on the right side of Equation 9 are assumed to be constant with the exception of Z . Therefore, the right side of the equation can also be integrated and k&M can be found, since Z is assumed to be known. Equations 2 and 3 now permit evaluation of hoax. The slope of the tie lines and Equation 10 now permit solution for h ~ a ~ . If it is desired to extend the results of laboratory investigations to full scale equipment, a procedure similar to t h a t described for the determination of the rate coefficients may be employed. If the rate coefficients have been determined for the mass rates of flow of gas and liquid t o be used in full scale equipment, the required height of tower may be found as follows (see Figure 2 and Figure 3). Points A and and the abscissa of M will be fixed by the requirements of the operation to be conducted. The
A’.
c
A
INDUSTRIAL AND ENGINEERING CHEMISTRY
488
right side of Equation 10 is known, hence the slope of the tie - and - lines is fixed. Successive points A , B, C, etc., will be found as before on the plot of gas enthalpy versus gas temperature. The location of a satisfactory terminus for this plot is variable. If a maximum permissible temperature for the exit gas is given, then the abscissa of fi? is fixed. The given set of conditions must permit %, as finally reached by successive extensions of the plot
Vol. 45, No. 2
The humid heat varies by approximately 50% of the lo\\er value for the system and temperature interval chosen. The aritlnmetic average value was chosen for computation. The magnitude of this error becomes less as the inteival of temperature and humidity considered is decreased. The ratio of the heat transfer coefficient for the gas film t o the m w s transfer Coefficient for the gas film was assumed to be piopoi tional to the humid heat of the gas phase. The cffect of evaporation or condensation Upon the mass rate of liquid flow was neglected.
Nomenclature
IS
Equation ( 1 1 )
/
7
x
CL
= area of transfer surface per unit of gross
volume, sq. f t . / cu. f t . ; a x for heat transfer; u.52 for mass transfer h b = ratio defined by Equation 2 bs, c = heat capacity, B.t.u.j(lb.) ( “ F . ) ;ci for liquid * ti = prefix indicating differential f = ratio U I I , / U . , ~ G = niass velocity, lb. mass/(hr.) (sq. ft,. of gross cross section); Go for drg- gas; Gz for liquid € = Iabsolute humidity, lb. of vapor/lh. of dr>7 gas; H, a t T,; Hi at Ti \ Y h = surface coefficient of h e a t t r a n s f e r , E.t.u./(hr.) (sq. it.) ( O F . ) ; h, for gas phase; hl for liquid phase H = enthalpy of mixture of gas and vapor: -,,,./I Equation (3b) A A equal to B.t.u./lb. of dry gas; H, a t Tg;H , at, Ti HI = modified enthalpy of mixture of gas and vapor, defined by Equation 5 , B.t..u./ 112. of dry gas; H I , at T o ; H’, a t Ti k , = mass transfer coefficient for gas phase, . TEMPERATURE, “F. lb./(hr.) (sq. ft. of wetted surface) (Hi - Hg) Figure 3 = liuinid heat, B.t.u./(lh. dry gas) ( ” F.); s sB for hulk conditions of gas I , 1 = temperature, F . : Ti for interface tempera.ture; To cleof gas enthalpy versus gas temperature, to represent an acceptnotes reference teinpzrature; 2’, for gas; 2’1 for liquid able value of gas enthalpy. An acceptable value of the gas enZ = effective tower height, ft. thalpy is one which will satisfy the over-all enthalpy balance for = latent heat of vaporization of liquid at the reference temthe given conditions without exceeding the maximum permissible perature, To, U.t.u.i’lb. exit gas temperature. An additional condition to be met is that -_ A ’ = modified latent heat of vaporization of liquid a t the refer-4 M should not cross PQ, since t o do so would indicate the presence temperature, 7 ’ 0 : defined by Equation 4. H.t.u./lh. ence of fogging within the equipment. Once the terminal conditions of all streams have been determined, the left side o f Equation 9 can be integrated graphically. Then the right side of Literature Cited Equation 9 may be solved for the tower height, 2,vhich is the (1) Brow-n, G. G., and associates, “Unit Operations,” p. 548, New only unknon-n. C.
-
Assumptions The proposed method employs several assumptions which are not as valid for many systems as they are for air and water.
CORRECTION Owing to the fact that the proof of the above article by John G . Lewis and Robert R. White was not received before press run started, the following changes should be made on the illustrations: Figure 1, Equation (13) should read (5a). Figure 2, Equation (3b) should be ( l ) ,and Equation (11) should be (10). Figure 3, Equation (3b) should be ( l ) , and Equation (12) should be (8).
York, John Wiley & Sons, 1991. ( 2 ) Rlickley, H. S., Chem. Eng. Pi.ogr., 45, 739 (1949). ( 3 ) Mizushina, T., arid Kotoo. T., C’iiem. Eng. J a p a n . 13,75 (194‘3).
RECEIVED for review May 7, 19.5%.
ACCEPTEDOctsber 4 , 1952.
