ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Spined Tubes in xchangers HEAT TRANSFER CHARACTERISTICS M. HOBSON
AND
J. H. WEBER
Department of Chemiafry and Chemical Engineering, University of Nebrarkcr, lincoln, Neb.
XTESDEII surface? for heat transfer have long been available commercially in a wide variety of designs. Heat and momentum transfer data, however, are available for only a relatively few basic types, such as the longitudinal and transverse fin. The present study considers the performance characteristics for a unique type of extended surface in the form of a large number of spines shaved from the surface metal of the base tube. The spines produce a continuous helix along t'hc length of the tube. A typical surface is illustrated in Figure 1. The purpose of this investigation was to provide fundamental data for comparing the performance of thwe zurfaces to smooth or other extended surfaces. Three spined tubes, wi1,h significant variations in surface properties, were used as the inner elements in a total of eight concentric pipe exchangers. I n all instances, air was used as the test fluid in the annulus. The experimental data for both heat, transfer and pressure drop are presented in correlations that fire generally applicable t'o spined surfaces of this type in annuli. Heat and Momentum Transfer Data Are Available for Special Types of Extended Surfaces
Other factors being equal, an improvement in the rate of heat transfer by the use of extended surfaces is obtained from increased surface area normal to the flow of heat, from increased turbulence, or from a combination of both effects. I n general, the ext,cnded surface is not as effective for heat transfer as t,hearea of the parent tube because of the temperature gradient for conduction along the fin or spine. As a result, there may be a significant reduction in the driving force for convcction between the spine and the fluid as the distance from the base increases. The decrease in the effectiveness of the extended surface has been t'reated quantitatively by a number of investigators in terms of a fin efficiency. Nayan ( 5 ) offered a solution for fin efficiency by applying a geometrical electrical analogy to the flow of heat. Gardner (5) developed solutions t o the Fourier equation for a number of extended surfaces by relating the temperature variation to the geometry and conductivity of the fin. Heat transfer coefficients for the combined fin-tube aurface have been correlated for a limit'ed number of extended surfaces in annuli. De Lorenxo and Anderson ( 2 ) presented data for longitudinal fins in the form of the Colburn heat, transfer factor modified by the Sieder-Tate viscosity correction. The conventional Reynolds number for annuli m s used for flow charact,erixat'ion.
The same variables were applied to studies on both continuous and discontinuous longitudinal fins by Gunter and Shaw (4). Knudsen and Katx (6) modified the Dittus-Boelter equation with appropriate configuration factors to correlate data for helical transverse fins. I n all instances, the Fanning equation or a
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inodification involving the radial visrosity correction factor used to correlatr frictional losses.
wag
Three Spined Tubes with Significant Variations in Surface Properties Are Used in Study
The propert,ies of the spined SurEace are suhjeut to a large numbcr of varist,ions that have a direct bearing on heat and momcntum t'ransfer. The number of spines, the pitch of the spined helix. and the length of t'he Epines are variable for any iiomin:11 size of the tube. Furthermore, a change in any one of these variables may completely alter t'hc shape of the spines. IXffcrences in the shape of the spines lire emphasized by thc crosssectional vien-s ol t'he three t,est, surfaces shown in I'igurt: 2. The degrec of curvature and the slope of the spines re1:itive to the b a i c surface are also subject t,o cliarige. The nat,urc of these variations cwmplicates a generalized trcatmerit of the performanre d a h . Such standard flow criteritt as the equivalent diameter and the cross-sectional area available for flou must be arbitrarily defined. In evaluat'ing the Reynolds number for irregular conduits, the practice has been to include an equivalent diameter which is four timcs the ratio of the Tree cross-sect,ional area t o t'he iT-ett,edperimeter. This definition has little meaning in terms of the spine surface. A volumetric equivalent dianiet,er, defiried as four t8imesthe ratio of the frec voluine t o the wetted surface area, has been used in correlating flow through hanks of tubes. Since the production of spines from i t thicker walled base tube does not create any additional volume, this dcfinition may be readily applied to annuli of this type if the nominal dimensions of the tuhos and the area of tho spined
Figure 1.
Typical Spined Surfaces
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 11
.
surface are known. In correlating the experimental data, hoaever, a Reynolds number based on the volumetric equivalent diameter did not offer significant improvement over a Reynolds group which includes thc conventional equivalent diameter for smooth annuli De = Dio - Dt (2) The spine variations may be treated quantitatively from the st'andpoint of their influence on the relative roughness of the surface, and the arbitrary definition of Equation 2 is most convenient for this development. A similar approach has been used by Braun and Knudsen ( I ) in correlating frictional data for t,ransverse fins in annuli. Satisfactory flow characterization also requires definition of the free cross-sectional area of the annulus. The usual definition is not practical in the caw of thn spined surface. A section taken normal to flow will include only a fraction of the spines in a complete turn of the helix and, since t.he spines are inclined relative to t,he basic surface, will segment different parts of adjacent spines. I n view of these properties, thc free cross-sectional area has been based on a complete turn of the spines. The definition represents the minimum cross-sectional area in a plane which deviates from the normal as a function of the pitch. Values of this variable were determined experiineiibally for each of the twt, surfaces by planimetering phot,ographic enlargements of thc sectional views shown in Figure 2 . Under the test, conditions, fin efficiencies for all the experimental surfaces approach unity. The effective area has been assumed to be equal to the total area, of the spined surface. This assumption is, undoubtedly, within the accuracy of any direct determination of the fin efficiency. Gardner ( 3 ) has shown that for a number of extended surfaces the fin efficiency decreases with the variable wqh/kYb I
BNOINEERINO, DESIGN, AND PROCESS DEVELOPMENT The total heat, transfer surface consists of the surfaces of the base tube and the spines. The area of the tube may be determined readily. The area of a single spine was approximated by considering individually a number of small sections that approach a regular configuration. The total estended surface was computed from the area per spine and t,hc number of spines per unit length. Since successive approsiniations of this kind are subject t'o error, t'he values reported in Table I are averages of a number of independent estimates. The maximum deviation between estimates was approximately 10%. Table I summarizes the dimensions of the three spined surfaces. There are variations in the number and length of t,he spines. The pitch of the spined helix is 3/16 inch for all of the surfaces, howver.
Table I.
Dimensions o f Spined Surfaces
Nominal Size of Spines/ Pitch, Diam. Base Tube, Inoh Turn Inoh Tube, Inch 13 8/16 0.376 26 8/16 0.495 26 8/16 0.626 Based on one complete turn of spines.
8/8 '/2
a
Surface Cross Section Area, Sq. of Tubea, In./Ft. Sa. F t .
DS
E 2.04 2.67 1.96
71 164 160
0,00274 0.00597 0.00715
Dimensional data for each of the test annuli are given in Table 11. A total of eight annular combinations were considered in the studies of frictional losses; six in the heat transfer tests. With one exception, the test exchangers included 4-foot lengths of spine surfacc. All of the exchangers were extended approxiinately 3 feet on either side of the test section to eliminate entrance and exit effects. The validity of this design was substantiated by pressure drop measurements over various sections of the IO-foot exchanger.
where h = fin side coefficient, k = thermal conductivity of the metal, w = fin length, and ,y6 = half thickness of fin base. Because of t.he low ail, film coefficients and the high thermal conductivity of copper, values of' this variable are approximately 0.1, and the corresponding fin cficiencies are above 0.95. The experimental results of t,his study are generally applicable to situations where the fin efficiency is less than unity, if the appropriate fin efficiency is known. In many instances tho application of these surfaces may well be limited by the fin cfficiency, since spines of this type are inherently thin a t the base.
Table II.
Dimensions o f Annuli
Xominal Size, Inches
Length of Equiv. Diam. Exchanger, F t . (Dd,F t . >/a x 1 10 0.0458 1/4 x l l / a 4 0.0733 '/a x l l / a 4 0,0925 3/8 x 2 4 0,1202 3/8 x 2'/2 4 0.1537 '/2 x 11/2 4 0.0691 4 0.1070 1/2 x 2 4 0.1407 I/% x 2'/2 Based on one complete turn of spines
%L%Area of Free De Flowa, Sq. Ft. 0.954 0.00326 0.591 0,00766 0.468 0.01140 0,722 0.01733 0.565 0.02725 0.904 0.00699 0.584 0.01615 0.444 0.02607
Pressure differentials were determinedwith a verticalmanometer and with an inclined draft gage for differentials below 2 inches of water. All temperature measurements were made with copperconstantan thermocouples used in conjunction with a Leeds & Northrup Type K potentiometer. Three Fischer and Porter F'lowrators of varying capacity were used for air flow mrasurement. Flow rates for the heating medium, steam condensate, wwc obtained by weighing the total flow over appropriate time intervals. t\ll of the experimental data u ere obtained under steady-state conditions. The bulk of the data apply to fluid flow in the same direction as the slope of the spines. The problem of flow counter to the slope of the spines was considered for two exchangers and the results are reported separately. Frictional Data Are Correlated by Parameters Characterizing Roughness of Surface
Figure 2. A. 6. C.
November 1954
Sections of Test Surfaces
The pressure drop data were interpreted in terms of the Fanning equation
Nominal size, 3/8 inch Nominal size, 1/2 inch Nominal size, 1 /4 inch
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
(3)
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ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 06
surfaces as n-ell as the relative influence of tEiese propertie.. v i t h variations in the dimensions of tiic annulus. The effect of spine length in relation to the dimensions of the annulus niny be expressed hy an annular configuration factor
I
0.4
02
L
I
01
0
0 08
o a i L
0.06
lx
representing t'he fraction of the equivalent dianirter occupied by the spines. The linding v a I u ( ~ ~ of the variable are zero Cor the smooth tube an(! unit>- lor the case in iv1iic-h the spines contact tlict outcar ~ h c l l . Friction factors for each of the test mri'awP as a [unction of this unnular configuraiion facto], are shown in Figure 5 , The d a h for th(. nominal :/4- and ',/?-inc-hspiiid surfaces arc ill t h c s form
I-
2
0.04
z1 I-
o a LL
I
001 I
q
I
i
l
l
I
I
0008 I
0.006
1 -
I
where f' = friction factor for smooth tube and C = function of the Reynolds number.
1
SMOOTH PIPE I
1
It is possible to estimate friction factors 101 these t n o spined surfaces directly from the fiiction factors for the smooth tube and the annului confieuration factor. The small variations in thc Figure 3. Friction Factors Based on Superficial Mass Velocitv constant, C, with the Reynolds number are indicated in Figure 6. Despite differences in t,hr Phape and number of the ppines, friction factors for both the For convenience, the friction factors were initially expressed on and 1/2-inch surfaces are essentially the same when compared a t the basis of the superficial fluid velocity. When plot,ted as a functhe same Reynolds number and configuration factor. Table I tion of the corresponding Reynolds number, each of the test indicates that the rat,io of the spine diameter to the tube diameter annuli produced a separate friction fact,or relationship, as indiis similar for both surfaces. These results suggest the possible imcated in Figure 3. The curves are similar in nature to the conportance of the D,lDi ratio as a description of the relative roughventional friction factor plot's for cylindrical surfaces with varying ness of the t'ube. The friction factors for the nominal 3/%-inch:surroughness. For all t'hree test surfaces, the effect of the Reynolds face are higher t'han those for the other two surfaces a t comparable number becomes less as the equivalent diameter of the annulus decreases. Since the correlat,ingvariables are based on t,he supervalues of the annular configuration factor. Since the D,/Dt ratio is higher for t'he 3/s-inch surface, this result is in accord with the ficial velocity, Figure 3 permits a direct comparison between fricuse of the ratio as a measure of surface roughness. tion factors for the test exchangers and for smooth annuli a t the same mass flow rate. The data of Figure 3 may be adjusted t'o any desired crosssectional area or equivalent diameter by means of appropriate rat,ios I
2
4
6
810 Re (
20
40
6o
loo
Y
L
(5)
I E
P
0
The minimum sectional area for flon. offered by a complete turn of spines can be determined accurately by using photographic techniques. The use of the minimum area provides a more satisfact,ory approximation for the average fluid velocity than does the superficial basis. The area correction factors are given in Table I1 for each of -the annuli. The adjusted data are presented graphically in Figure 4. Since the velocity has been increased by redefinition of the free cross-sectional area, the net effect is a reduction in the friction factor and an increase in the Reynolds modulus. There are significant variat'ions in ilie relative positions of the curves for the annuli with the smaller equivalent diameters. The differences in the friction factor relationships of Figure 4 may be considered from tjhe standpoint, of the relative roughness of thc surface. Quantitatively, a description of the roughness must include the effect of variations in the properties of individual
2292
2 Z
0
t
0
[L:
LI
0008
0006
I
2
Figure
4.
I 4
6
~ 810
20 f i e (10~') Based on
Friction Factors Cross-Sectional Area
INDUSTRIAL AND ENGINEERING CHEMISTRY
SMOOTH PIPE -r---r
40
A
60 83
Minimum
Free
Vol. 46,No. 1 2
J
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 04
counter to the slope of the spines is shown in Figure 8. For both of the surfaces studied, an increase of about 11% is indicated for the counterflow
02 r
'
5 < LL
01 008 006
Correlations Permit Prediction of Heat Transfer Coefficients in Different Annuli
004
z
P
g
002
Heat transfer data were limited t o six of the eight exchangers. These tests are summarized in Table 111. Over-all heat transfer coefficients were measured experimentally. The film coefficients for the fluid inside the tube were calculated from the Dittus-Boelter equation and the annular coefficient was estimated using the resistance concept.
E 001 0008 0006
0
02
04
06
08
IO
0
04
02
06
08
IO
0
02
04
06
08
@A @e
Figure
5.
Variation of Friction Factor with Annular Configuration Factor
The influence of the surface roughness was studied using a series of plots similar to Figure 7 at various values of ( D l D t ) / D e while the RegnoldP group is c o n s t a n t . When expressed in the form
40
1= -1 A ,_ _ _ - -l_ h, U , hiAi
35
30
-
f
25
4
8
12 Re
16
20
24
(167
an average value of n = 2 1% as obtained over the range of the experimental data. Minor variations in the exponent resulted with changes in ( O s - D t ) / D . and the Reynolds number. Since only three surfaces were involved in the study, however, a more accurate definition of the exponent is not warranted. The effect of both configuration factors may be combined into a single empirical expression f = 0.254f'(Ds/D,)2C (Os- Dt)'De)
November 1954
(Nu)/(P~)~ =' ~ +(Re)
(10)
The heat transfer data in this formare much less sensitive 01 to the roughness pa008 rameters than the fluid friction data. FLOW W I T H SLOPE OF S P I N E S There are no signifiFLOW COUNTER T O SLOPE cant differences in I I I 1 1 1 I I t h e heat, transfer *O 40 f a c t o r s , (XU)/ Re, (Id3) (Pr)l'3, for the t n o Figure 8. Variation of Friction e x c h a n g e r s USi% the I/*-inch spined Factor with Direction of Flow surface. The data for the X 2 inch annulus are slightly higher than the mean values for the other exchangers, which may reflect the influence of the D,/Dt ratio. In the largest annulus tested, the performance of the l/n-inch surface was essentially similar to the other surfaces. The values of the heat transfer factor for t,he other exchangers involving the l/Z-inch surface are below the bulk of the data and appear to decrease in a regular manner with the configuration factor, ( D , D t ) / D e . A general interpret'ation of this variat'ion is not justified from the limited data available. The results for four of the exchangers, including data for a11 of the spined surfaces, are represented by the equation, 02
d(Da/Dt)" ( 7 )
With one exception, this relationship reproduces the experimental results of Figure 4 with a maximum deviation of about 7%. The results for the x 11/4 inch annulus show a maximum deviation of approviniately 20% from those predicted by the correlation. I t is concluded that friction factors for any spined surface in a variety of annuli may be predicted with a reasonable degree of precision from the dimensions of the exchanger. All of the data reported are for surfaces with 3/~b-inch pitch with the flow in the same direction as the inclination of th_e spines. The increase in friction factor for flow
Tube metal resistances were negligible under the test conditions. The total area of the spined surfaces, A,, was used as the basis for the coefficient, h,. The annular film coefficients are presented in Figure 9 in the form
Figure 6. Variation of Coefficient C of Equation 6 with Reynolds Number
zA, k A,
(8)
I
'
'
Nu
-
D, Figure 7. Variation of Friction Factor with Ratio of Spine Diameter to Tube Diameter
.
= 0.00168 (Re)l.Og(Pr)l'3
(11)
The use of the relationship is questionable under conditions wheie the value of (I), - D t ) / D e approaches unity. This does not appear to be a serious limitation, however, since exchangers Tyith high values of the ratio are not likely to be of practical importance because of the excessive pressure drops involved. A comparison of Equation 11 with the analogous relationship for smooth surfaces in annuli indicates the increased effect of the Reynolds number in the case of the spined surfaces. Values of the heat transfer factor for both types of surfaces have the same order of magnitude.
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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ENGINEERING, DESlGN, AND PROCESS DEVELQPMENT 200
Table 111.
Fluid Friction and Heat Transfer Data
FLCID FRICTION Area CoriecIloiiiinal tion for Area CorrecSize of Reynolds tion for FricExolinnger, Number tion Factor Reynolds Inches (Equation 3) (Equation 4) Xumber 114 x 1 1.461 0.590 3,500-37,000 1/4 x I'/n 1.204 0.791 3,400-30,000 L/r X 11/! 1,142 0,856 3,100-29,000 3/8 x 2 1.244 0.718 3,500-31,000 1 1 8 x 2'/! 1.160 0.808 3,100-29,000 0.490 3,500-40,000 1.684 X 1l/r 1.259 0.721 3,400-33,000 1/2 x 2 0.817 3,100-30.000 '/2 1 , 167 1/1 X 2
IO0 80
No. of Testa
2 -
18 24 14 21 8 16 11 10
3
Air Film Coefficient, (ha) B . t . u . / (Hr.)(Sq. 8 t . j
-
Reynolds Nuinher
Log Mean Temp. Difference (Atm), O C.
40
jx
k1xA-r T m r a x m
Nominal Size of Exchanger, Inohes
60
'dx
20
S o . of Tests
(OF.)
x
1/4 1/4 '/4
7s',
'/S 11~
..
1
X 11/r X 1,'s X 2 x 2'11
5,300133,000 6,000-27,000 4,500-24,400
X 2
8,400-19,000 5,000-19,000
x
' / a X 21/2
A.Ol29.4 3.1-20.4 3.5-13.0
s,zooliS,ooo
2.31ii.4 3.3-8.0 1.8-6.9
21.89125.91 27.85-31.59 20.15-25.43 18.9632.63 28.55-26.19 29.81-31.50
IO
0 6 13
2
Figure 9.
k = =
= = =
= =
=
Nu = AP =
n z
area for heat transfer, sq. ft. function of Reynolds number, Equation 6 diameter, ft. equivalent diamet'er, ft. diameter of spines, f t . diameter of base tubc, it. mass velocity, Ib./(hr.)(sq. it.) length, ft. h 1) Nusselt group, k pressure drop, Ib./sq. it.
Pr
= Prandtl group, g
Re
=
S
F DG Reynolds group, --
= cross-sectional area for floiv, sq. it.
h
=
over-all heat transfer coefficient, B.t.u./(hr.)(sq. ft.) / O D \ ( I.I
Ti cp
=
velocity, ft./sec.
R.t.u. /(lb.) (' F ) friction facto1 = conversion factor, 32 l i poundals 'pound force = individual heat transfer coefficient, l3.t.u. '(hr.)(sq. it.)
= heat capacitv a t constant pressure,
f
=
'g
h
( " F.)
= Colburn's heat transfer factor
,j
C. E. DRYDEN'
8 1 0
20
40
60
p M
():
Spined Tube Heat Transfer Correlation
thermal conductivity, B.t.u./(hr.)(sq. ft.) exponent of ( D , / D t )ratio, Equation 7 = pipe wall thickness, ft. = density, Ib./cu. ft. = viscosity, lb./(hr.) (it.) = =
( O
F./ft.)
= Seider-Tate viscosity correction factor
Subscripts a refers to annulus i refers to inside of inner tube io refers to inside of outer tuhc m refers to mean value o refers to outside of inner pip? literature Cited
M
U
6
Re (IO')
8 5 6
Nomenclature
A C D D, D, Dt G L
4
..
AND
(1) Braun, F. W., Jr., and Knudsen, J. C . , C h e n ~ .Eng. Progr., 48, 523-7 (1952). (2) De Lorenzo, B., and Anderson, E. D., Tiaras. L4nr. SOC. M e c h . E n g r ~ . 67, , 697-702 (1945).
( 3 ) Gardnor, K. A , , I b i d . , 67, 621-31 (1945). (4) Gunter, A. T.. a i d Shan-, W.A , , Ihid., 64, 795-802 (1942). (5) Hayan, C. F.. ISD. ENG.CHEM..40, 1 0 4 4 9 (1948). (6) Knudsen, J. G., and Iiatz, D. L.. Ciieni. Eng. Progr., 46, 490-500 (1950).
RECEIVED for review February 24,1953.
A c c n r w x Julie 14, 1034.
W. B. KAY
D e p a r f m e n f o f Chemical Engineering, The O h i o State University, Columbus, O h i a
S adsorption, the solid removes solute, also termed absorbatc,
from the liquid phase by physically or chemically adsorbing the solute on the available surface of the solid. I n desorption, the reverse procedure occurs and solute is removed from t,he surface of the Bolid. When the rates of adsorption and desorpt,iori are equal, equilibrium is established between the solid and the liquid phase and no net transfer of solute t'alccs place. Many commercial operations employ batch adsorption rtnd desorpt,ion techniques. I n such processes, the adsorbent is stirred in a tank of liquid for. a definite period of time until equilibrium is 1
Present address, Battelle Memorial Institute, Columbus, Ohio.
2294
attained. The liquid and solid phases are separated b y centriiugation or filtration. The petroleum and edible oil industries furnish numerous examples of batch adsorption from liquid phase. I n operating such processes, the rate of approach to equilibrium is an important design variable, and yet little quant,itative information is available. For Solution to Finite Bath Theory, Heat Transfer Equations Are Applied to Analogous Mass Transfer
Freundlich (6) discussed briefly the subject of kinetics of batch adsorption. Eagle and Scott ( 3 ) were the first .to apply diffu-
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 11
