October, 1930
I N D US T RI A L A N D ENGINE E R I NG CHEMISTRY
salt is superior in some respects to sodium nitrate for use in meat preservation. The American Nitrogen Products Company was nearing financial stability with an expanding market before it; but the plant had reached a point of growth where further improvisation was impracticable. Further progress called for reconstruction of the factory; new buildings ‘were almost essential for continued operation and a removal of the plant to Tacoma or Seattle was eminently desirable. A. department
1103
for the manufacture of the alkali used in neutralization of the nitrous acid was needed and in general a modernization of all the equipment. Ordinarily with input of normal power there was little disturbance of continuous furnace operation, and the La Grande furnaces were a remarkably practicable electrical apparatus for the specific purpose for which they were intended, but important improvements were possible in new design and reconstruction.
The Condensation of Vapors’” C. C. Monrad and W. L. Badger DBPARTMENT OF CHEMICAL ENGINEERING, UNIVERSITY OF MICHIGAN, ANNARBOR, MICH.
Several important theoretical contributions have CRITICAL survey of the heat flowing. This means been made to the subject of vapor condensation that the literature on heat that any appreciable increase have not been generally recognized. The contributions transfer from condensin the amount of heat transof Nusselt in particular show the quantitative effects ferred m u s t b e m a d e b y ing vapors to solid surfaces of physical properties of the vapor and liquid, impurity, changing conditions on the shows that this subject is exsuperheat and velocity of the vapor, and the size and vapor side. tremely complicated theoretishape of the surface on the transmission of heat from cally, and that experimental Nomenclature condensing vapors. This theory is briefly reviewed. results to date seem to be Application of the Nusselt theory to data on vapor rather inconsistent and inThe writers have thought condensation on the outside of horizontal tubes shows complete. it advisable throughout this that the theory is valid for this case. A large amount of work paper to use the symbols Application of the theory to data on vertical tubes has been done on the 1,ransfer f o u n d m o s t g e n e r a l l y in shows that o n long tubes, or at high temperature of heat from steam to water American practice and a t the differences, the theory does not hold, probably owing to through a metal wall, but very same time not to use the same turbulence and drop formation. An equation for defew investigators ham: meassymbol for more than one termining the start of turbulence in a film of condenured the temperature of the p r o p e r t y . I n general the sate is derived, and it is shown that turbulence acmetal. Therefore, in most properties of the liquid are counts for most of the observed deviations from the cases only over-all coefficients used without subscript and Nusselt theory. of heat transfer have been obthose of the vapor symbolThe effects of superheat, vapor velocity, and impurities t a i n e d , which could be ized with the subscript u. are pointed out. Several corrections to the Nusselt changed by varying condiFor the most part all physitheory are made, especially on the effect of vapor vetions on either side of the cists and other scientists use locity. metal boundary. A rough the c. g. s. system for the units calculation of the meta’ surinvolved. but no such agreeface temperature could be made if the coefficient of heat trans- ment is found in technical practice. The European or rechfer for the water side were known, but such calculations are nical metric system was for a long time based on the meter, never accurate and may lead to erroneous conclusions. kilogram, large calorie, second, etc., and conformed in general The subject of heat transfer from condensing vapors to to the c. g. s. system except that the units of force were desolid surfaces has not received so much attention as that fined in kilograms, corresponding to grams instead of dynea from a solid surface to a non-boiling liquid because of the in the c. g. s. system. This had a marked effect on the units experimental and theoretical difficulties. Mori?over, until of various other properties such as viscosity. More recently very recently it was not necessary to know very much about a mixed set of units has been used in which the hour replaced this subject, because the resistance to heat flow of the con- the second for many of the properties. This has led to densing vapor has been much less than the other resistances considerable confusion. involved and often could be neglected. The application of The English system of units is in even worse condition. dimensional analysis to the problem is extrernely difficult Very few workers use the same units, and in general all use because of the large number of variables and because there a mixed system. Thus a very familiar example is the comis a change of state from vapor to liquid. mon Reynolds’ criterion, which is supposedly a dimensionVery recently, however, liquor velocities in evaporators less group. and condensers have been increased to such a point that the I n the English system this is usually vapor resistance is becoming of greater importance. I n foot pound second-centimeters fact, in the experiments by Badger, Monrad, and Diamond Dup = inch x 2 second cubic foot X gram ( 4 ) on the evaporation of caustic soda with diphenyl vapor, the major resistance to heat flow was on the vapor side, and the liquor resistance could be almost neglected in calculating If the correct units were used, the numerical value of the criterion would be independent of the particular system 1 Received June 10, 1930. Presented before the meeting of the Ameriused. It so happens that in the technical system Reynolds’ can Institute of Chemical Engineers, Detroit, Mich., June 4 i o 6, 1930. * Abstracted from a dissertation submitted in partial fulfilment of the criterion must be Dupjtg in order to be dimensionless. A large part of this paper deals with the theoretical work requirements for the degree of doctor of philosophy in the University of Michigan. of Nusselt, who used the older technical units throughout.
A
INDUSTRIAL A N D ENGINEERIAVG CHEMISTRY
1104
I n order to use Nusselt’s derivations in the most simple manner, it is necessary to adopt a technical English system that is comparable with the system used by him, and, most important of all, a system in which the fundamental units are consistent. Throughout this discussion the old technical metric system is used for fundamental calculations. The final results of heat-transfer coefficients, etc., are calculated on the hour basis by multiplying by the factor 3600. To convert to the English system, it is merely necessary to divide the value thus obtained by 4.88. VALUE
SYMBOL
a
A
Lp?k’ -
b
Caw% -
B
C
D
-
LPZ 4 kz(lt. - 1,)
..
8
3P
Width Constant Diameter of tube
E
CJP. 3 pk
f
Friction factor
g
Acceleration of gravity
G II
Weight of condensate Height of wall
k
Thermal conductivity
1
Tube length
L
Latent heat
m
Hydraulic radius
P
Pressure
i?
Heat Bowing
I
Radius
S,SI,SI
UNITS (TECHNICAL METRIC)
Friction force
I, T
Temperature
U
Velocity, x direction
U
Coefficient of heatytransfer
0
Velocity, y direction
W
Relative vapor velocity
YO
Condensate thickness
s
Viscosity
e 9
Temperature difference Angle
c
Density
m. m.
m. = 9.81 sec 2
kg. m. Cal.
m.’C. sec. m. Cal.
kg. m.
kg. m. 2 Cal. m.2 h r .
m.
kpm,2
c.,
A
m. see.
cd1. ni.2 ’ C iec m sec. m -
-
see. m. kg_ sec. _
m.:
c.
radians kg. m.3
Experimental and Theoretical Results to Date
Probably Joule (27) wasthe first one to understand clearly the fact that the major resistance t o heat flow from a fluid to a solid surface is a film of nearly stagnant fluid a t the interface. Both he and Lord Kelvin realized that the thickness of this film was indeterminate, since in general heat flows both by conduction and convection through the film. Kelvin suggested that, instead of ‘‘thermal conductivity,” the term “heat-transfer coefficient” be used for such cases. This coefficient was defined as the amount of heat passing through unit surface in unit time with unit temperature difference. I n 1873 Reynolds (48) studied the effect of air on the
Vol. 22, No. 11
over-all coefficients of heat transfer from steam to water. He concluded that there is no limit to the rate at which heat will flow from pure condensing steam to a metal surface except the power of the surface to carry away the heat. He found that a small amount of air in the steam has a very marked effect in decreasing the flow of heat. From the time of Reynolds until about 1910 no significant experiments were made. A large number of investigators worked on the over-all coefficients of heat transfer from steam to water in condensers and evaporators, but these were largely tests on specific apparatus. Since 1910 rather a large number of experiments have been made measuring the actual metal temperatures when steam was condensed on tubes. I n this way the heattransfer coefficients of the condensing vapor could be determined. Webster ( 5 4 , Clement and Garland (8), McAdams and Frost (%), Morris and Whitman (S7), and Othmer (40) have investigated the condensation of steam on the outside of a horizontal tube. McAdams and Frost also reported experiments on benzene and carbon tetrachloride. Jakob, Erk, and Eck (24) have studied steam cgndensation inside a horizontal tube. Jordan (,E),Callendar and Nicolson (e), Philipp ( 4 d ) , and Jakob and Erk (22) have studied steam condensation on vertical tubes. Badger, Monrad, and Diamond (4) have reported results obtained with diphenyl vapor condensing at high temperature differences on a long vertical tube. Othmer (/to), Merkel (35), Josse (84, Arzoomanian and Alpert ( I ) , Kerr (2.9) and Robinson (49), Chambers and Eskew (‘7) and very recently Colburn and Hougen (9) have studied the effect of air on the condensation of steam. The effects of vapor velocity have been studied experimentally by several of the above investigators, but no very definite conclusions were reached. The effect of superheat of the vapor has been studied by Jakob and Erk (22) and shown to have a very small effect on the transfer of heat. I n 1916 Nusselt (38) in an excellent mathematical paper, studied the effects of vapor and liquid properties, shape of the surface, and the purity and superheat of the vapor on the condensation of any vapor on a solid surface. This paper has not received the attention it merits, possibly because it has not been abstracted in this country. Nole-The
authors have found that a large amount of foreign literature
on such subjects as heat transfer, Euid motion, diffusion, etc., has not been
abstracted in English. Considerable advances in these fields have been made in Germany during the past twenty years that have never been noticed by the abstracting journals. It is especially to be noted that only very recently Chemical Abslracls has been covering these fields to any extent and thus should not be relied upon solely. Some additional information may be obtained in the Engineering I n d e x and the Industrial Arts I n d e x , but these subjects are covered most completely in the Technische Zeilschrijtenschaw.
This theory will be reviewed in some detail as it is of fundamental importance. Application of this theory to experimental work of the authors and others will be shown. I n 1925 Stender (52) applied Nusselt’s theory to superheated steam and calculated several formulas for this case. Merkel ( S / t ) , Grober (15) and ten Bosch ( 5 ) give excellent summaries of Nusselt’s work on the condensation of vapors. I n 1920 Parr (41) calculated steam-film coefficients on somewhat different assumptions than Nusselt but using the same general idea. His assumptions did not hold for conditions actually found in practice as well as those of Nusselt; and the treatment was not nearly so thorough. A brief comparison of the two theories is given by Jakob (21).
INDUSTRIAL AND ENGINEERING CHEMISTRY
October, 1930
Nusselt's T h e o r y of C o n d e n s a t i o n of Vapors
CASES CONSIDERED AND FUNDAMENTAL AssumTIoxsNusselt considered the following five fundamental cases: (1) Vapor condensing on a smooth, plane surface making the angle @ with the horizontal, assuming t h a t the vapor is pure and saturated and relatively stationary with respect t o t h e condensate. ( 2 ) Vapor condensing on the outside of a horizontal tube under the above conditions. (3) Vapor condensing on surface as in (1) but with appreciable vapor velocity. (4) Superheated vapor condensing on any surface. ( 5 ) Vapor condensing on any surface, vapor impure.
Nusselt first pointed out that surface tension of the condensate can have no effect on the thickness of film formed, and this thickness mag' be calcu'ated from hydrodynamical considerations. I n order to simplify the mathematical treatment, Nussrlt made sevcral assumptions, some of which he was later able to prove. These assnmptions should be clesrly borne in mind throughout the discussion and are essentially aa follows : (1) The film of condensate is so thin that the temperature gradient through it is a straight line. ( 2 ) The heat is all carried t o the metal surface by pure conduction in the direction perpendicular t o the surface. (3) Physical properties of the condensate may be taken a t the mean film temperature. (4) The surface is relatively smooth and clean. (5) The film of condensate always moves in Viscous motion. (6) The curvature of the film may be neglected. (7) The temperature of the solid surface is constant.
CASE 1. VAPOR CONDENSINQ ON INCLINED 'FUBE OR W A L d o t w i d e r a wall making the angle 4 with the horizontal and kept at a uniform temperature t,. Saturated pure vapor a t a temperature 1, is condensing on this surface. Let x be the direction parallel to the wall and y the direction perpendicular t o it-i. e., through the condensate film. At any point in the condensed film a shear stress, S, is set up owing to the viscosity of the condensate and the variation of velocity of the condensate with distance from the wall.
s
u=-
dY
p
sin + 22 (2Y Yo
- r2)
The mean condensate velocity, urn,a t any point down the wall x is given by the following equation:
If k is the thermal conductivity of the condensate, then the heat passing through unit surface in unit time at the point z is k
= Yo - (t,
Q
- t,)
If L is the latent heat and G the weight of vapor condensed, then
represents the amount of vapor condensed on unit surface in unit time, For simplicity consider a vertical wall of width €3 and height H. I n a unit time the amount of condensate flowing past a point x down the wall is
At dx lower
more condensate flows downward. This differential amount of condensate must be due to the vapor condensed in t,he strip B dx,or dG =
k(t,
- t,)B
dx
LY 0
or
Lp2 Y~~
x =
Therefore
du
4 kz(t,
+
- t,)
Since at x = 0, no liquid runs down from above, C = 0, and
d2U
dS = z - dy
and
If
= z -
Therefore
1105
dY2
is the density of the condensate, the differential shear stress set up as above must be counterbalanced by the differential weight of the condensate tending to pull the condensate downward, or p
dS
+
p
Thus Q at the point x =
sin + dy = 0
If the heat-transfer coefficient, U,, is defined by the equation
and
This integrates a t once to
QI
u = - - p sin +y2
2a
+ CIY +
c 2
Since a t y = 0, u = 0; therefore CZ = 0, and if the vapor is stationary there is no force tending t.0 shear the condensate at the surface and e
du
-dY
then
=
4
= U"(L - t,)
4 zx(t, L p 2k 3 t,) at the point x
To obtain a mean coefficient it is necessary to integrate
Q=over the length H and the width B. H
Therefore BQH = B JQ,dx =
0, a t y = yo
Thus
du
and
P sin +YO c1 -
0
=
B
4Lp2k3Ha(tu tw)3
42
G =0 i=
I
where A = Lp2ka/e and is a function of the condensate
INDUSTRIAL AND ENGINEERING CHEMISTRY
1106
properties alone. It may be plotted versus temperature for any fluid. If the wall is a t the angle 4 with the horizontal
Nusselt showed that from the equation of continuity there is a component of condensate velocity toward the wall, and therefore some heat is carried by convection to a point nearer the wall. This velocity may be calculated and the heat carried in this manner is found to be negligible. The error caused by neglecting this effect is doubtless no greater than that due to the assumption of a straight-line temperature gradient through the film. Figure 1 shows Nusselt's t h e o r y applied to s a t u r a t e d steam condensing on a vertical wall 1 meter 02 high. The vapor temperature is 100" C., and the wall 90" C. The condensate thickness a t 04 2 m/r any point is shown by the curve starting at 2 x = 0. Atfourpoints on the wall the velocity B x 06 components of the condensate in the x and y direction are given by the curves labeled u 08 and v, the scales of which are indicated at the right. Four stream lines are shown labeled I O as $ = constant, and 05 I 15 2 Y ~ d ~ ~ ~ t these ~ ~ ~ the path indicate Figure 1-Saturated Steam Condensine of the condensate as it on Wall 1 Meter High (Nusselt) moves down the tube. It is apparent that condensation is very rapid at the top of the wall and gradually becomes less. Nusselt pointed out that, since there is a temperature gradient in the film, some sensible heat is given up by the condensate and the above calculation is somewhat in error. The error is never more than 1 to 3 per cent even at very high temperature differences. He has developed a more exact formula which is of very little practical significance. CASE 2. VAPORCONDENSINQ ON HORIZONTAL TUBEFor horizontal tubes the same assumptions are made as above, but it is necessary to consider the force of gravity acting in the direction tangential to the tube surface. Nusselt found that the average coefficient over the whole tube may be given as
2 becomes 0.545, and becomes still less for tubes lower in the bank. Nusselt applied Equation 2 to the experiments of English and Donkin ( I S ) and found fair agreement in the case of smooth tubes, although these experiments were made on very crude apparatus and could not be expected to give very accurate results. The agreement between theory and experiment was of the order of magnitude of *20 per cent.
1:
c
Vol. 22, No. 10
Figure 2-Effect
of Vapor Velocity (Ten Bosch)
CASE3. EFFECT OF VAPORVELocIm-If vapor is moving with a velocity w with respect to the outside of the condensate film on a vertical wall or tube, the constant C1 can no longer be determined from the condition that there is no shearing force a t the surface, since the vapor exerts a force S per unit area on the condensate surface. For the case of steam flowing through a tube or pipe Nusselt was able to calculate the constant C1 by considering the pressure drop through the tube. The experiments of Eberle (21) showed that
43
15 I 0
I
I
I
I
40
20
60
I
M1
1
KIO
I
120
1
140
I
160
I lffl
1
2Oa
Temperature, *C
where D is the diameter of the tube. Nolr-Colburn and Hougen (9)have mentioned this equation and calculated the constant to be 0.61 instead of 0.725. This was app'arently due to the omission of the fourth root of 2.
(0.8024
.yi
= 0.61, but 0.8024
; I
Figure 3
-
q
F
for Water (Old Technical Metric TJnits)
Since the friction forces on the tube circumference must equal the pressure forces on the tube cross section,
= 0.72,)
SHD =
The upper half of the tube condenses 59.4 per cent of the vapor if no condensate flows from tubes above. If tubes are in banks and the condensate from one drips on the one below, the coefficient is decreased. For the case of a tube directly below a single tube the constant in Equation
n D 2 dP
4 dl
INDUSTRIAL A N D ENGINEERING CHEMISTRY
October, 1930
1107
or
If the steam velocity is in the direction of the condensate drainage the positive sign is used, and the negative if the flows are countercurrent. Proceeding in general as with stationary vapor, Nusselt was able to show that
y-;i:
where
a =
b = function of
LP2 4 kz(t, - t,)
Ten Bosch (5) has pointed out that if the above relationship is true, then
u r n .
' 4H (tV - t.1 dx
= function of
Ew2
-
(t.
t,)
;1~ -
A = -Lp2ka
where
Note-In the original article Nusselt had -y*o/h' and y4dh instead of -yo/h* and y40/4. This was probably due to a typographical error. Obviously if h becomes zero (no superheat), the equations should degenerate into the one previously calculated for saturated steam.
This complicated power series is not satisfactory to use in practice. Recently Stender (53) and Merkel (34) have shown that a very good approximation may be made, since the amount of heat flowing is proportional to the condensate thickness and to the cube root of the amount of condensate if t , and t, are the same. Therefore
where the values with the subscript s refer to the condensation of superheated vapor and those without subscript to the saturated vapor. /3 has been shown by Merkel to be a u, (t, - t,) alone . and may be plotted as in function of u (t. - t,) Figure 4. U,must be calculated as the usual heat-transfer coefficient from a superheated vapor or gas to a wall, and depends on the velocity, temperature, etc.
2 141
I
,
,
,
I
and
This function is shown in Figure 2 for concurrent flow of vapor and condensate. The value of i/;l is shown for water in Figure 3 in metric units. ( L may be taken a t mean film temperature for small temperature differences.') Nusselt discussed several very interesting cases for the flow of vapor countercurrent to condensate drainage and showed that sometimes all or part of the condensate is carried to the top of the wall. It is necessary to use very high vapor velocities in order to do this, and such cases would seldom be found in commercial practice. Nusselt did not attempt to apply the theory described above to any other vapor than steam and only then in the case of steam condensing inside of a tube since the constant CIwas unknown in all other cases. This is discussed later, and a method shown by means of which the effect of velocity of any vapor in any cross section may be calculated. CASE 4. EFFECT017 SUPERHEAT-There are two subcases to the case of heat transfer from superheated vapor. When the wall temperature is above that of saturated vapor a t the given pressure, no vapor will condense and the heat transfer will be the usual one from a gas or vapor to a solid surface. When the wall temperature is less than that of saturated vapor, condensation will take place, but the phenomenon is not the same as in the case of saturated vapor. There is a temperature difference between the vapor and the surface of the condensate corresponding to the degrees of superheat. Heat is carried to this interface by the usual gas-solid method. Thus if U , is the heat-transfer coefficient from vapor to interface,
or
u Figure 4-Effect
(t,-tw)
of Superheat (Merkel a n d Stender)
For an example, consider the condensation of saturated steam on a vertical wall 0.5 meter high; vapor a t 100" C.; temperature difference to wall, 5' C. The mean film temperature = 97.5" C., and therefore from Figure 3 0.943
X 3.03 X 3600 = SI90 Cal./m*/hr./" C. u=lt/m
Let the vapor be superheated to 200" C. Then 1. - t, = 100' C., and t, - t , = 5" C. Assume that U. = 50. Then
and from Figure 4 the ratio of the amount of heat flowing to that with saturated steam is 1.032. Thus, there is only a 3 per cent increase in the amount of heat, which is nearly negligible. If the steam velocity is increased, the value of U , is increased and the effect of superheat is greater. CASE 5. EFFECTO F NON-CONDENSABLE Gams-The effect of non-condensable gases on the heat-transfer coefficient is very important technically. However, exact calculation of the effect is very difficult and Nusselt merely indicated a general method. Colburn and Hougen have recently attacked this problem theoretically and experimentally with some definite results. Applications of Nusselt's Theory
Since hyo series,
