Thermal Design of Condensers WALTER GLOYERI American Locomotive Company, Alco Products Division, Dunkirk, N . Y . T h e method of rating condensers presented in this paper is based on the well known principles of heat transfer by convection and conduction. The design procedure is applicable to all kinds of surface condensers, but application in this paper is limited to condensers of shell and tube design. Heat release curves are presented for the condensing vapor and increasing amount of condensate as condensation progresses. The analysis shows that this heat release versus temperature relationship gives curves of parabolic shape, resulting in an improved mean temperature difference (MTD) for the vapor and a reduced mean temperature difference for the condensate. A n analysis is made of the effect of a variable temperature difference upon the heat
transfer coefficient of condensation on horizontal tubes. I t is found that the coefficient is increased by a small amount. Heat transfer of convection of vapors and the effect of condensation is analyzed. A simple relationship is presented to approximate a correction factor applicable to this coefficient of heat transfer. The problems encountered in this case of nonconstant heat flux are discussed and a method for combining the heat transfer coefficients is presented. The problem of pressure drop through condenser shell is treated, including derivations for the average amount of vapor flowing through a condenser and the relative cross-sectional flow area for condensate and vapor.
T
From the equilibrium flash curve the heat release can be obtained by taking total heats for vapor and condensate a t the inlet temperature, various intermediate temperatures, and the outlet temperature. The difference in total heats at the various points will give the heat release curve. The heat release curve for hydrocarbon vapors is normally a nearly straight line with a slight curvature toward the temperature coordinate. Therefore, it is normally conservative to assume a straight line heat release versus temperature relationship for hydrocarbon vapors containing no steam. The heat release curve for vapors containing steam deviates, however, appreciably from a straight line. At the dew point of the steam the heat release curve has a sharp break. The slope of the curve is decreased a t this point because of the high value for latent heat of steam. A typical case is illustrated in Figure 2. To obtain a close approximation of the effective mean temperature difference (MTD), the heat release curve is subdivided into several zones. Mean temperature differences are determined for every zone as well as over-all heat transfer rates, resulting in a surface required for every zone. The sum of these individual surfaces is the total surface required for the condenser. Since it is impractical to list several mean temperature differences on the specification sheet, a so-called weighted mean temperature difference is determined as shown in Figure 2 and entered on the Condenser specification sheet. After a heat release curve is established, the heat duty has to be broken up into vapor copling, liquid cooling, and condensing duty. The method of doing this is shown in the folloming examples:
H E original purpose of this paper was the development of a rating method for condensers in commercial practice. This method has been in use in its simplest form since 1937 and has since been expanded and improved in many ways. It is based on the well known principles of convection and conduction heat transfer. Another method for the thermal design of condensers has been proposed by Colburn and Drew (6) and Colburn and Hougen (8). This method is based on the theory of diffusion. Unfortunately this excellent theory is not developed far enough to be readily applicable to routine calculations. A third method by Atkins ( 1 ) is based on the principles of convection and conduction and is similar t o the method presented a t this time by the author. Fields of application, with range of operating conditions, where the method has been successfully employed, are listed in Table I. The subjects encountered in rating condensers are discussed in the following sequence: Heat release between inlet and outlet tem eratures Effective mean temperature difference tetween condensing vapor and cooling medium Rate of heat transfer Pressure drop through condenser shells HEAT RELEASE
Thermal data desired by the condenser rating engineer are shown in Figure 1. Heat release, amount of vapor, and the molecular weights of vapor and condensate are shown for a particular case, and all are plotted against temperature. Further necessary information given is operating pressures and allowable pressure drop. Seldom, however, is such complete information available. If the composition of vapors entering a condenser is given, an equilibrium flash curve for vapors which are immiscible or miscible in the liquid state can be calculated by the method as presented by Huntington (14). For hydrocarbon vapors containing steam, the method for immiscible liquids is used to determine dew point and equilibrium flash curve of the steam; the equilibrium flash curve for the miscible hydrocarbon vapors is determined separately. For heavier petroleum fractions, it is customary to determine the equilibrium flash curve from American Society for Testing Materials (A.S.T.M.) or true boiling point distillation curves. Various empirical methods are in use-Le., the ones by Piroomov and Beiswenger (do), Packie (19),and Griswold and Klecka (18). Present address, American Division, Chicago, Ill. 1
Looomotive Company, Alco Products
TOTALCONDENSATION.Vapors ( A , pounds per hour) are condensed between 2'1 and T,; properties are evaluated a t average temperature.
TABLEI. FIELDS OF APPLICATION AND RANGEO F OPERATING CONDITIONS
Service Depropanizer overhead condensers Rerun tower overhead condensers (with or without steam) Atmospheric tower overhead condensers Partial condensers (oil cooled) Vacuum tower overhead condensers Gas compressor inter- and aftercoolers Air compressor inter- a n d aftercoolers Organic vapor oondensers
1361
Operating Pressure, Lb./Sq. Inch
300
Temp. Range, a. F. 300-100
30-100 0-50 200 to 35 mm.
400-100 400-100 350-100 400-100
0-50
300-100 300-100 400-100
abs. 0-400 0-300
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 42, No. 7
EFFECTIVE MEAN TEMPERATURE DIFFEKENCE BETWEEN CONDENSING VAPOR AKD COOLtY(: MEDIUB.1
D E B U T A N I Z E R OVERHEAD C O N D E N S E R OPERATING PRESS. 155 LB./SQ. 1N.G. ALLOW.PRESS.DROP SLB./SQ.IN.
As condensation progres,c.s in a condenser, the vapor quantity decreases and correspondingly the condensate quantity increases. $ssuming straight line condensation, heat release curves and effective mean temperature differences for vapor cooling and liquid cooling duties can be determined by the follol~ing analysis:
t?O ri ;*I60 I
2a 150
E 140 0
z
W I30
I-
Figure 3 shows that the vapor quantity incrcases with increasing temperature, M7 = a bt. If the condenser out.let temperature, ti, is made the origin the coordinate system by introducing new variable I = t - tl, then TV = bT. Generally spcaking. heat release of vapor with changing temperature is d& = W c d T . Applied to the case here and integrated.
+
120
0 I Z 3 4 5 0 U T Y MI L L IO N B.T. U./ HR *
6
-
7
10,000 LB./HR.VAPOR
ef
Figure 1. Condenser Thermal Data
one gets Q =
lT lT
ch7'd'l'
Wcdl' =
=
ch7'22,
which is a parabola. To ronform with Figures 1 and 2 where temperature is plotted a3 the ordinate, the inverse function T = constant X is used, which of course is also a parabola. Figure 4 shows that the liquid quantity decreases with increasing temperature, 1V = a - bt. If one makes the condenser inlet temperature, t 2 , t t e origin of the coordinate system by introducing a new variable 7' = t - t z , one has W' = - bT. Heat release of the liquid with changing temperature is d& = W c d T . Applied and integrated one gets
HEAT RELEASE, MILLION B.T.U./HR.
Figure 2. Heat Release Curve for Hydrocarbon Vapors Containing Steam-Weighted Mean Temperature Differences
Vapor coo~iiig:fi 2 Liquid cooling:
- ~ ~ ) c b
B
2 (Tl - T 2 ) c ~
Condensing: A L
PARTIAL COSDEXS.ITIOS.Fraction X of vapor 21 is condensed between T l and T,. Gas cooling: A(1 Vapor cooling: Liquid cooling:
9x 2
X)(?',- T 2 ) a
=
JTo
TPcdY'
=
-
JTQ
ch'l'd?'
+ c b ( P / 2 ) , nhich is a parabola. The inverse function which is dejired for this relationship is T = constant X Q1/2. The inverse functions of both heat release curves plotted between tl and t~ and superimposed in one diagram are shown in Figure 5 , with sample calculations for the effective mean temperature differences in Table 11.
Owing to the characteristics of a parabola, = 52, Figure 5 will reveal that the condensing vapor reIcases 75% of its sensible heat from condenser inlet temperature to the arithmetic mean between inlet and outlet temperature, and the remaining 2570 of its heat between this intermediate point and the condenser outlet temperature. This applies, of course, only to that part of the vapors which actually condenses. The liquid condensate, however, relpases only 25% of its heat between inlet and the intermediate temperature and 7576 of its heat over the remaining temperature range. The resulting effect is an improved effective mean temperature difference for the vapor5, which have a rather low rate of heat transfer. I t re-
(T1- ?',)cT.
AX -- ( T I 2
?'?)cL
Condensing: A X L PARTIAL CONDENSATION. Fraction X of vapor A is condensed in presence of steam ( 8 ,pounds per hour) between 1'2 and dely point of steam T D S .
Steam cooling: S( TI- T D S ) C S Gas cooling: A ( l - X ) ( T , - T D S ) C G
.1X ITaporcooling: - (TI - 'i'us)cr. 2
An
Liquid cooling: -- (TI - T D S ) C L 2 Condensing: A X L
=
Figure 3.
Heat Release Curve for Changing Vapor Q-cia n ti t y
I N D U S T R I A L A N D E N G I N E E R Il? G C H E M I S T R Y
July 1950
f
'I 3 0 i
1363
TABLE11. EFFECTIVE MEANTEIIPERATURE DIFFERENCES FOR VAPORA N D LIQUID COOLING DUTY Water temp.,
t2 b \
O
F.
Straight M T D 200 + 120' 100 100
TEMP. f
48
100 20
77
Eff. RITD-Vapor 750 250 1000
-
Figure 4.
Heat Release Curve for Changing Liquid Quantity
i
900
%
W
a 160
9.75 7.35 i17.10 = 58.5
.-t
120' 100
20
34
Eff. MTD-Liquid 250
750 ~1000
i77
=
t 34 =
3.25
22.05
i25.30
= 39.5
Figure 6 shows, in diagrammatic form, resistances for vapor cooling duty, condensing duty, and liquid cooling duty. The vapor duty has to overcome the resistance of the vapor film, condensate layer, dirt layer, metal tube wall, and cooling medium film. The average heat density or flux through the vapor film is Qs t surface A . Parallel to this, the latent heat of condensation is given off a t the surface of the condensate layer, and, therefore, has to overcome the resistance of this layer, the dirt layer, metal tube wall, and water film. The average heat density has increased t o (Q. Q c ) + surface A , a t the interface 1-1 between vapor film and condensate film. Parallel to this again, the sensible heat of liquid cooling is given off within the condensate layer and has, therefore, to overcome a partial resistance of this layer, dirt layer, metal tube wall, and water film. The average heat density increased to (Q. QC QL)t surface A within the condensate film. Considering this variable heat density, the temperature drop through the gas film can then be determined by application of the above stated equation and subsequent multiplication by the heat flux ratio:
+
4
a w n
E 120
8-
0
i77 = i34 =
160 100 60
200 -+ 160° 100 100 100 60
26 SO 76 HEAT RELEASE %
100
Figure 5. Heat Release for Vapor and Liquid Cooling Duty
duces, however, the effective mean temperature difference for the liquid condensate. This is rather harmless since its rate of heat transfer is high. Calculations for the effective mean temperature difference, as carried out in Table 11, are based on 1000 B.t.u. per hour for convenience. HEAT TRANSFER
The author's treatment of this subject is in disagreement with the widely accepted theory of diffusion, and represents a different viewpoint of condensation of mixed vapors. To keep calculations within reasonable limits, simplifying assumptions have been made and resulting formulas are empirical to a large extent. Much room is left for further investigation to clarify existing conditions and to bring theory and practice into closer agreement. Consider a vapor mixture consisting of more or less easily condensable components being condensed in a shell and tube condenser. As condensation progresses through the condenser, the vapor will change in composition, owing to a preferential condensation of the more easily condensable components. The less easily condensable components will accumulate in the remaining vapor and by their partial pressure effect will reduce the condensing temperature. This phenomenon occurs especially in the thin vapor film surrounding the tubes. The concentration gradient and the corresponding temperature drop through the vapor film are dependent upon the thickness of thg film, the amount of less easily condensable component present, and its physical properties. For the general case of constant heat flux as occursi.e,, in liquid to liquid exchangers, gas to liquid exchangers, wherein no condensation occurs, or in condensers where condensation takes place at constant temperature-the temperature drop across any film under consideration can be determined from the equation HMTD = h,AL. Solving this formula one obtains Atz = HMTD. For the special case of condensation, where hz condensation takes place within a temperature range, the heat flux is not constant.
+ +
CONDENSATION OF VAPORS. The rate of heat transfer from a pure vapor to a condensing surface is controlled by two resistances. First, a resistance must be overcome in bringing about condensation of the vapor molecules, and, secondly, the resistance to the conduction of heat through the film of condensate on the condensing surface has to be overcome. Colburn and
QV QL
I
I_
1-
w
11%
Ill.
Figure. 6. Temperature Diagram, Main Vapor Stream to Water
.
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INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
Vol. 42, No. 'I
variable temperature difference upon the average heat transfer coefficient of condensing vapors is discussed below: In determining the resistame of a condensate layer on a coliderising surface, film condensation is assumed, and a film of uniform t,hickness is made the basis for the derivations. Actual condenser operation will, hot\-ever, not result in a uniform film thickness. Water-cooled condensers normally have a large temperature difference between the incoming vapor and outgoing water, and a small temperature difference a t the vapor outlet. Since heat flow is a function of local temperature difference, the change in temperature difference along the condenser surface n-ill result in a thick condensat,e film at the eondenser inlet and a thin film a t the outlet'. This tends to increase the average condensing coefficient for a horizontally installed tube which can be proved as follows: TLe film coefficient for a condensate layer is inversely proportioniil to its thickness, f : 1
h = 9 t
If the film has a different thickness a t two points, an average 1't2) can be calculated. heat transfer coefficient h = 4 f p is const'ant,for any given apparatus and operaThe sum of tl tion. It, will be seen that the sum of t,he inverse values is a miniriiuni when t l equals t 2 . Any difference between tl arid tf will yield a greater value for the heat transfer coefficient.
+
The above theory was originated by Wulfinghoff (93). Calculations for a correction factor, to be applied to the mean film coefficient, were carried through for terminal temperature difference ratios from 25 to 1 dotm to unity. Detail calculations follow: rlccording to Kusselt, the heat transfer coefficient of condensation is h = constant ( I / At0.25) for a certain vapor and temperat,ure. The thickness of the condensate film can be determined from thermal conductivity s = kh- =heat, transfer Coefficient
INLET
S UR FACE
OUTLET
Figure 7. Mean Temperature Difference Curves
Hougen ( 7 ) showed the first iesisiance to be negligible compared to the second. A laige amount oi information has been collected by such research workers as Nusselt (18),Kirkbride (16), Colburn ( 4 ) , McAdams ( 1 7 ) , etc., in the last 30 years concerning the condensate layer resistance, and is now in readily usable form for engineering calculations. McAdams' (1'7) data has been used by the writer for vertica] as well as horizontal tubes since 1939. It may be pointed out here that the heat transfer coefficients determined from this data are mean values of h with respect to height of condensing surface Comparing average x i t h local coefficients within the viscous flow region for Reynolds numbers less than 1600, it is found that, a t any distance, R , from the point where condensation starts, the coefficient is three quarters of the mean coefficient for the length, R. This is pointed out by Walker, Lewis, McAdams, and Gilliland ( b d ) . The local condensing coefficient can be calculated in this manner for aftercondensers, where liquid condensate enters condenser along 'iT.ith vapors. It may be pointed out further that the condensing coefficient as calculated from the above is based on a constant temperature difference between vapor and condensing surface. The effect of a
By substitution S = constant X At0.Z5 A series of temperature difference curves will be found in Figure 7 . The surface coordinate is divided into 10 equal parts. A t the center of each part the At is read for each curve and the fourth root taken (see Table 111). Hereafter the sum of all A P Z s is found. For a constant At = 1, this sum is equal t o 10. Since the sum of all thicknesses has to be constant, all At0.26are proportioned to give Ato.2j = 10 for each temperature differenrc curve. The heat, transfrr coefficient, is proportional to thr. w-
2 3 5 7 1 0 RATIO OF TERMINAL At's
20
30
Figure 8. Correction Factor for Heat Transfer Coefficient of Condensation for Horizontal Tubes us. Ratio of Terminal t's between Vapor and Tube Wall
ciprocal of AtD 26 as stated above. For a constant At = 1, the sum of 1 / A F z5 is equal to 10. To obtain the correction factor, F , for all other curves 21/A.to z6 is then divided by 10. The resulting curve is shown in Figure 8. The increase as read from curve is rather small and can be omitted in most plactical rases. The derivation and results are included, however, t o shorn its magnitude and basis for the following paragraph.
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1950
CALCULATIONS FOR EFFECT OF VARIABLE TABLE111. DETAIL At
UPON
Surface Coordinate
CONDENSING COEFFICIENT At Atin
1/ At”25
At0.E
= 1, Atout
=
1
9.5 8.5 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.5
1365
COOLINGOF LIQUID CONDENSATE. As already mentioned above, the liquid cooling duty has to overcome only part of the condensate layer resistance. To arrive a t an analytical expression for this fractional layer thickness, a consideration of this problem is given in detail. When the condensate layer is in viscous flow-Le., when the Reynolds number ig less than 1600-the velocity distribution in the falling condensate film is given by the usual equation for flow of liquid layers
U, = (Bx p = -‘1/
Surface Coordinate
At
9.5 8.5 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.5
1.20 1.78 2.54 3.46 4.57 5.90 7.50 9.30 11.35 13.72
Surface Coordinate
At
9.5 8.5 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.6
1.30 2.14 3.25 4.70 6.41 8.65 11.30 14.50 18.30 22.68
15,
Proportioned 1/ Ato.2j
Atout
1,139 1.262 1.364 1,462 1.559 1.655 1.746 1.836 1.925 214.994
0.697 0.760 0.840 0.910 0.975 1.040 1.104 1.164 1.224 1.282 9.996
-
Ae.26
At0.26 Atout
1.0676 1.2095 1.3425 1.473 1.592 1.715 1.833 1.950 2.070 2.183 Z16, 4356
1.435 1.316 1,190 1.100 1.026 0.961 0.905 0.860 0.817 0.780 10.390
= 1
0.649 0.735 0.816 0.895 0.970 1.043 1.115 1.188 1.260 1.330 _____ 9.999
1.540 1,360 1.225 1,118 1.030 0.960 0,897 0.844 0.794 0.751 ~ 10.519
For condensation on vertical tubes, Drew supplied a curve for this correction factor which is published without proof by Keyes and Deem (15). Inspection of this curve reveals a correction factor which varies from larger to smaller than unity depending on relative location of the larger and smaller terminal temperature differences. For simultaneous condensation of hydrocarbon vapors and steam, the hydrocarbon vapors only should be considered in determinine the condensing coefficient. The steam will condense in drops on the oil film surface due to the high surface tension of water, as compared to the surface tension of oil. Neglecting the resistance of these water drops themselves, the latent heat of the steam, therefore, has also to overcome the resistance of the oil condensate layer. This theory is confirmed through visual observations by Baker and Tsao ( 2 ) and Hazelton and Baker L
(13).
For simultaneous condensation of several vapors, all vapors and their weighted thermal properties have to be considered.
as stated by Colburn, Millar, and Westwater (9).
lB U,dx will
give the amount of condensate passing a fixed point of unit width. By substitution and integration
which is the area under the curve. The question now is, what value must X have in relation to B to divide the condensate layer into two streams of equal mass? Omitting the constant, the equation for half the area reads:
Proportioned1/
519 F = 10 -L-10 = 1.0519
-
(see Figure 9)
= 1
1.046
Atin = 25,
-=I
25
At0.26
Atm
At0.26
10
):
Figure 9. Condensate Film Velocity Diagram
Solving this equation results in z = 0.65B. This means that from the center of mass, the sensible heat has to overcome only 0.65 of resistance of full condensate layer thickness. The heat transfer coefficient for cooling a falling condensate layer is, therefore, the condensing coefficient divided by 0.65. I n drip-type condensers, as well as split flow-type condensers, liquid condensate cooling is achieved by cooling the falling condensate. In single-pass shell condensers, however, part of cooling is done while the condensate flows along the bottom of the shell after it has dripped off the tubes. For this case, the liquid cooling duty is arbitrarily divided evenly into drip cooling duty and bottom flow cooling duty. For cooling the liquid which runs along the bottom of the condenser shell, Colburn’s ( 5 )correlation is used. COOLINGOF VAPORS. The transmission of the sensible heat of vapors is governed by the usual laws of convection and conduction heat transfer, for which Colburn’s ( 5 ) correlations are being used. Since Colburn’s original cross-flow correlation covers flow across staggered tube banks only, a second curve for inline tube banks has been drawn, located 6% below Colburn’s curve. Grimison’s (11 ) correlations are the basis for this lower curve. Condensation of vapors as they flow through a condenser affect the vapor cooling as can be explained by the following analogy:
_all
The rate of convection heat transfer to steam flowing through a pipe, when the pipe is heated from the outside, is determined by the thickness of the steam film formed on the surface. The steam film will disappear completely should the tube wall be cooled below saturation temperature of the steam. Since there is no resistance against condensation, every steam molecule will condense when it hits the metal surface. The heat transfer coefficient will then be determined by thickness of condensate layer on the surface. The same principle will apply in a similar manner to condensation of mixed vapors, as has been learned by visual observations, generally called fogging within the vapor film. Assume a mixture of hydrocarbon vapors condenses between temperatures T I and 2’2, as shown in Figure 10. The average vapor temperature will be T M and the corresponding colder condensate film
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
1366
surface temperature T,. A fraction, (Tx- To)/(5"1- Tz), of the vapor is, therefore, below its dew point. It will condense and reduce the vapor film thickness by the above ratio. If only a fraction, X , of the vapor is condensed between T1and Tz, the vapor film thickness will be reduced b y X [ ( T x - T , ) / ( T l - T z ) ] . The remaining fraction of the film thickness K = (1 - X)[(TM- T,)/ (@ - T z ) )represents the remaining fractional resistance to heat flow. Since heat is transmitted through this film mainly by conduction, an approximate corrected vapor film coefficient can be written as hcor. = l~,,/li, wherein h, is the vapor film coefficient for zero rontlrnsation. Application of this equation to limiting cases for rheching purposes gives the following:
or A =
Condenqation a t constant temperature: T I = 112, or - T , = 0 ; K = 1 - X [ ( T n r - T < ) / O=] 0; hcor = h,/O = a , or a vapor film does not evist under t!iese conditions. Xocondensation: S = 0; K = 1 - 0 [ ( T ~ r- T , ) / ( T I- T2)I = 1; h,,,. = h,, or the vapor film is unaffected.
Qv
(r1
+ r? +
r3
MTD
">
Vol. 42, No. 1
+
based on derivations discussed under effective mean temperature difference b e h e e n condensing vapor and cooling medium.
Ti
PRESSURE DROP THROUGH COKDENSER SHELLS
In commercial practice, heat transfer and pressure drop calculations are usually based on average quantities. I t is obvious that the arithmetic mean between vapor inlet and outlet quantities is the proper average vapor quantity only if a constant At prevails throughout the condenser, For the normal case of a variable At, the determination of the average vapor quantity is derived later. The basic differential equation for transfer of heat reads as follows:
W
a
3
I-
