4.
r
a
R T T W
TO U
V V
X X++ Y 2
P AH
heat flux through outer wall of rotating cylinder, (B.t.u.)/(hr.) (sq. ft.) radial distance from center of rotating cylinder, ft. net rate of homogeneous reaction, (moles of A reacted)/(hr.) (cu. ft.) radius of rotating cylinder, ft. temperature, O R. temperature on outer wall of rotating cylinder temperature in bulk gas surrounding rotating cylinder tangential vt:locity at surface of rotating cylinder, fi./hr. product of chemical reaction stoichiometric coefficient of species V radial distance from surface of rotating cylinder, ft., equal to r - R
xu )(; V
mole fraction of species A reduced radial position = r/R; 2’ when used as dummy variable in definite integral constant in Elquation 9, taken equal to 1.77 in this study enthalpy change of chemical reaction per mole of A reacted = uH, - HA, (B.t.u.)/(mole of A)
+ F*eF9,k ,
7
1
7)
FT.LrH equilibrium heat capacity FuCp (1 4- 6y0)’ divided by frozen heat capacity eddy diffusivity, sq. ft./hr. u - 1, increase in moles due to reaction kinematic viscosity of fluid mixture, sq. ft./hr.
equilibrium thermal conductivity
divided by frozen thermal conductivity
e
0 Y
l+--
f zdz’ -
E
J L dimensionless E radial position f ” /I,’) J, t ’ ~ I -
P PM
= mass density of gas mixture, lb. m/cu. ft. = molal density of mixture,. Ob. moles tOtal)/(CU. ft.) .
U
@ @m
ic.
T,,, -. T o = h/h* = h/h* = (T
evaluateld when m = To)/(TU To)
-
-
m
SUBSCRIPTS A = species A = refers to bulk gas surrounding rotating cylinder 0 V = species V
W
=
cylinder surface
SUPERSCRIPTS * = physical heat transfer References (1) Altman, D., Wise, H., A R S Journal 26, 256 (1956). (2) Brian,P. L. T., A.Z.CI1.E. J . 9,831 (1963). (3) Brian, P. L. T., Bodman, S. W., IND.ENG.CHEM.FUNDAMENTALS 3, 339 (1964). (4) Brian, P. L. T., Reid, R. C., A.Z.CI1.E. J . 8 , 322 (1962). (5) Brian, P. L. T., Reid, R. C., Bodman, S. W., Ibid., 11, 809 (1965). (6) Broadwell, J. E., J . Fluid Mech. 4, 113 (1958). (7) Brokaw, R. S., “Correlation of Turbulent Heat Transfer in a Tube for the Dissociation System N 2 0 4 % 2 NO2,” Natl. Advisory Comm. Aeronaut., NACA RM E57K19a (March 1958). (8) Brokaw, R. S., J . Chem. Phys. 35, 1569 (1961). (9) Cotter, J. E., Schmidt, G. L., S.B. thesis, Chemical Engineering Department, Massachusetts Institute of Technology, Cambridge, Mass., 1956. (10) Deissler, R. G., Natl. Advisory Comm. Aeronaut., NACA 1210 (1953). (11) Dropkin, D., Carmi, A., Tram. A S M E 79, 741 (1957). (12) Eisenberg, M., Tobias, C. W., Wilke, C. R., Chem. Eng. Progr. 51, Symp. Ser. No. 16, l(1955). (13) Fan, S. S. T., Mason, D. M., Rozsa, R. B., Chem. Eng. Sci. 18. 737 (1963). ( l i j ’ F a y , JT A,’Riddell, F. R., J . Aero. Sci. 25, 73 (1958). (15) Irving, J. P., Smith, J. M., A.Z.Ch.E. J . 7,91 (1961). (16) Kays, W. M., Bjorklund, I. S., Trans. A S M E 80,70 (1958). 117) Krieve, W. F.. Mason, D. M., A.Z.Ch.E. J . 7.277 (1961). . . (18) Lees, L., Jet Propulsion 26, 259 (1956). (19) Lin, C. S., Moulton, R. W., Putnam, G. L., Znd. Eng. C h m . 45. 636 (1953). ~. .~~ (20)’Metzdorf:H. J., J . Aero Sci. 25, 200 (1952). (21) Moore, L. I., Zbid., 19, 505 (1952). (22) Murphree, E. V., Znd. Eng. Chem. 24,726 (1932). (23) Richardson. J. L.. Bovnton, F. P.,. Ene, Mason, D. M., -. K. Y., ‘ Chem. Eng. Sci.’13,130 ((961): (24) Schotte. W.. Ind. Ene. Chem. 50. 683 11958). (25j Sherwood, T. K., Clem. Eng. Phgr. Symp. Ser. 55, 71 (1959). (26) Sherwood, T. K., Ryan, J. M., Chem. Eng. Sci. 11, 81 (1959). (27) Srivastava, B. N., Barua, A. K., J . Chem. Phys. 35, 329 (1961). (28) Theodorsen. T.. ReEcier, A.. Natl. Advisorv Comm. Aeronaut., ‘ NACA T R 7 9 3 (1944r . ’ (29) Thievon, W. J., Sterbutzel, G. A., Beal, J. L., “Influence of Gas Dissociation on Heat Transfer,” WADC T R 59-45 (June 1959). (30) Vieth, W. R., Porter, J. H., Sherwood, T. K., IND.ENG. CHEM.FUNDAMENTALS 2 , l (1963). RECEIVED for review May 26, 1966 ACCEPTEDOctober 19, 1966 Industrial and Engineering Chemistry Summer Symposium on Applied Kinetics Reaction Engineering, Washington, D. C., June 1966.
ENHANCEMENT OF FILM CONDENSATION HEAT TRANSFER RATES ON VERTICAL TUBES EIY VERTICAL WIRES DAVID G . THOMAS
Oak Ridge National Laboratory, Oak Ridge, Tenn.
WHEN considering ways for increasing film condensing heat transfer coefficients, one thinks primarily of promoting dropwise condensation. This subject has been studied extensively and values for the dropwise condensing coefficient are in the range 3000 to :7O,OOO B.t.u./(hr.)(sq. ft.)(’F.) (70) compared with values of 500 to 2000 B.t.u./hr.(sq. ft.)(OF.) commonly observed in film condensation (72). Some success
has recently been reported (4) on developing long-lived dropwise condensation promoters, and increases in over-all heat transfer coefficients of u p to 82% were reported by coating copper-nickel tubes with a noble metal ( 4 ) . Since Nusselt (75)derived theoretical relations for the rate of film condensation in 1916, the subject has been studied extensively (77)and it is clear that the primary problem in preVOL. 6 NO. 1
FEBRUARY 1967
97
Vertical wires, 0.030 and 0.062 inch in diameter, loosely attached to a '/Z-inch 0.d. vertical tube 421/2 inches
x
104 B.t.u./ long markedly increase the film condensation heat transfer coefficient. At a heat flux of 2 (hr.)(sq. ft.), four wires obstructing 7.670of the surface increased the condensing coefficient by a factor of 3.1 4 (a 2 14% increase), while 12 wires obstructing 23% of the surface increased the condensing coefficient by a factor of 4.53 (a 353% increase). A qualitative model based on the film and rivulet hydrodynamics predicted the observed increase in condensing coefficient with decreasing rate of condensation and increasing fractional surface coverage by wires. In addition the model and the data showed a broad maximum for a fractional surface coverage of about 18%.
dicting rates of condensation involves the hydrodynamics of flow in thin films ( 5 ) . Little effort has been devoted to developing ways of enhancing film condensation coefficients. A notable exception is the recent studies on fluted tubes which report (2,6) condensing coefficients4l/2 to 7 times greater than that obtained for a comparable smooth tube a t a given heat flux. Recent data ( 2 ) were obtained with a tube 33/8 inches in 0.d. and 12 inches high having 80 uniformly spaced flutes around the periphery. The results were influenced by the heat flux, temperature level, vapor velocity, tube orientation, tube length or diameter, turbulence in the film, and presence of noncondensables, although no data were reported ( 2 ) . In other studies an increase in heat transfer was achieved by wrapping l/a-inch diameter wire around the outside of a 1-inch diameter tube in a helical spiral (8). Although no estimate was made of the magnitude of the increase in the condensing coefficient, it was postulated (8) that the wire wrapped around the tube decreased the effective drainage height by channeling the bulk of the condensate in a helical path along the wire down the tube, thus decreasing the condensate film thickness over the remainder of the tube. Mathewson and Smith (73) showed that pulsation in the sonic range of frequencies increased the condensing coefficient by 10 to 60%. The increase depended upon the vapor flow rate, a critical pulse amplitude, and was almost independent of pulse frequency. Surface roughness effects on film condensation a t zero interfacial shear have been calculated (74) using a fluid film velocity profile based on the turbulent profile for the fully developed rough region proposed by Rota (79). The analysis predicted (74) an upper limit above which, a t a given Prandtl number, no increase in heat transfer can be obtained. Tests with roughness elements up to 0.020 inch high on a vertical tube 2 inches in diameter and 6 feet long were between the curve for the predicted upper limit for fully rough surface and the curve for smooth tubes (74). During the course of detached turbulence promoter studies we discovered that small diameter wires loosely attached to a vertical tube with the wires parallel to the tube axis caused a marked increase in the condensing heat transfer coefficient. This paper describes a systematic study of the effect of such wires on the rate of condensation.
and the mean coefficient for the entire tube is defined as
A momentum balance on a small element of fluid assuming laminar film flow relates the local rate of condensation per unit tube perimeter, r,, to the condensate film thickness:
(3) Combining Equations 1 through 3 and integrating from 0 to L and from 0 to r gives
(4) Stretching wires along the surface in the direction of flow changes the flow from substantially two-dimensional to threedimensional flow with a large component of velocity normal to the wires-that is, when wires appreciably larger in diameter than the condensate film thickness are wetted by the condensate, liquid is drawn into the cavity between the wire and the tube surface by capillary forces as illustrated in Figure l a
u. FILM GEOMETRY
Qualitative Model
I n dhiving theoretical relations for the rate of condensation of a saturated vapor on a vertical tube, Nusselt (75)assumed laminar flow in the condensate film and neglected interfacial shear between the vapor and the liquid. Details of Nusselt's analysis are readily available (7,75) ; a brief outline is given below. The local condensation heat transfer coefficient at a distance z below the top of the tube, assuming heat is transferred solely by conduction through a film of thickness &, is defined as 98
I h E C FUNDAMENTALS
b. SIMPLIFIED MODEL
Figure 1.
Condensate film geometry in presence
of wires
(the curvature and thicltness of the film between the wires are shown on a greatly exaggerated scale). The magnitude of the capillary force due to the curvature of the fluid interface is given by Laplace's formula (9) :
ratio of film thickness in the absence of wires to the film thickness with wires, 6,/6:
(5) with positive surface pressure in the medium whose surface is convex. The radius of curvature of the condensate in the rivulet next to the wire is much less than the radius of curvature in any other part of the. condensate film. Thus, from Equation 5, there is a strong pressure gradient, due to the decreased surface pressure within the rivulet, which draws condensate toward the wire. The flow in the bulk of the rivulet is substantially parallel to the wire and continuously carries the condensate down the wire under the action of gravity. The combined action of the pressure gradient and the carryoff of the condensate by the rivulet near the wire means that the condensate film is poter,tially thinnest just beside the rivulet; how ever, thinning of the condensate layer causes a convex surface in the film of fluid adjacent to the rivulet which feeds fluid to within the range of influence of the strong pressure gradient within the rivulet. I n the ensuing discussion it is assumed that the range of influence in the condensate film of the radius of curvature of the rivulet is proportional to the condensate film thickness, 6. This assumption implies that the relative increase in condensing coefficient caused by a wire is independent of the distance between wires. The general features of the effect of the wires on the condensation heat transfer coefficient were determined using a greatly simplified modd. The geometry of the model is shown in Figure 1,b. The radius of curvature of the condenser tube is assumed to be infinite compared to the thickness of the condensate film, ,md the condensate film between the wires is assumed to have a uniform thickness, 6, in the direction normal to the wire. Since the wires are only resting against the surface, substantially no heat will be transferred through the tube surface beneath the wire and the thickened fillet of condensate adhering to the wire. Then, from Equation 1, the ratio of the condensing heat transfer coefficient with wires to the coefficient in the absence of wires, h/h,, is the ratio of the mean film thickness in the absence of wires, 6,, to the film thickness with wires, 6, corrected for the amount of surface blocked by the wires and fillet of water. The flow down the length of the tube may be divided into two regions, fluid drawn it0 the vicinity of the wires by capillary forces flowing with mean velocity V A through a fillet of crosssectional area A I , and a thin film of fluid flowing down the tube with velocity V,. From the continuity relation the sum of these flows must equal the flow in the absence of the wires at the same heat flux: 21V VAAlp,
+ NVBBpf = V&pf
Allowing for the amount of surface blocked by wires and fillets of water and assuming that the value of R/d is 3/4, the ratio of condensing heat transfer coefficients given by Equations 1 and 9 is
Before Equation 10 can be compared with experimental data, it is necessary to relate the terms V A , AI, and 6 to system variables. When the value of h/ho is much larger than 1, the bulk of the condensate must flow down the tube in the vicinity of the wires. Since condensation is taking place over the entire tube surface not covered by wires and the adjacent fillet of water, there is a large component of velocity in the film (region B, Figure 1) normal to the wires. The thickness of the condensate film, 6, in the vicinity of the rivulet may be determined from a force balance on a small element of fluid in the region where the rivulet joins the condensate film. This gives A-
1
Now if, as outlined above, we assume that the range of influence of the radius of curvature in the rivulet is proportional to 6, Equation 11 can be integrated to give the average velocity:
and when the bulk of the condensate flows along the wires:
P
Combining Equations 12 and 13 and assuming that
R
= and,
For laminar flow down along the wires assume that
j = - =Tw- 2gc
ff3
-
ff3
PfVA' DHVA _ _
0.430 R 2 V ~ dv
Y
From a force balance, (6)
When Equation G is divided by TD,the term on the right is the mass flux per unit width, r (Equation 3). From a force balance, the mean velocity in a film of fluid in laminar flow down a vertical tube is given by (75)
rV = 0.107
p
gc
Rz d
(16 )
Combining Equations 15 and 1G and solving for V A gives
(7) Neglecting the film thickness, 6, the area, A I , is given by Area B is related to the film thickness and the distance between wires by
B = 6(irD/N
-d
- 2R)
(8)
Combining Equations 3, 6, 7, and 8 and rearranging gives the
Ai = 0.215R2 = 0.215c~z~dz
(18)
Combining Equations 10, 14, 17, and 18 gives an expression for h/h, with one constant, a, which must be determined empirically: VOL 6
NO. 1
FEBRUARY 1967
99
hjh, =
The first term in parentheses on the right of Equation 19 is always less than or equal to 1 and makes little contribution to the variability of h/h,. Differentiating Equation 19 and setting the derivative equal to zero to find the value of N d / r D for maximum h/h, gives iVd/rD = 0.18, substantially independent of the value of (gcoL/rv) and @ for the range of interest in this study. For fraction of surface covered by wires much less than 0.18, Equation 19 predicts that h/h, increases almost in direct proportion to N d / r D . For any given value of N d l r D , Equation 19 also predicts that h/ho increases as the mass flux per unit of tube perimeter, r, decreases. Equipment and Procedure
The effect of longitudinal wires parallel to the axis of a vertical tube on the rate of condensing heat transfer was determined by measuring the over-all rate of forced convection from condensing steam to water and the average condensing and forced convection heat transfer coefficients were calculated assuming the resistances to be additive (76). Forced convection heat transfer coefficients in the absence of turbulence promoters were within +12% of the Sieder-Tate relation (20).
Test Section. The test section consisted of two vertical concentric tubes, a glass pipe 1 inch in i.d. by 43 inches long with removable insulation to permit visual inspection of the mode of condensation and an aluminum condenser tube inch in 0.d. by 4 feet long, with a 0.028-inch thick wall. Although the majority of the tests were made with steam condensing on the outer surface of the aluminum tube, in a few tests the system was modified to permit condensation on the inside of the tube. The over-all cooled length of the condenser tube was 42’/2 inches. The upper end of the condenser tube was connected to a steam chest maintained at a pressure of 4 to 5 p.s.i.g. Wire Promoters. The wires used to enhance the rate of condensation were either stainless steel or aluminum, 0.031and 0.062-inch diameter. In the majority of the tests the wires were stretched along the tube parallel to the axis and spot-welded at their upper and lower ends; a single loop of 0.005-inch wire was tied around the bundle at about 1-foot intervals. In one series of tests, four wires were wrapped in a spiral around the tube with each wire having a 4-inch pitch. I n another series of tests, four wires were supported away from the surface with nominal gaps beneath the wires of 0.030 and 0.0625 inch; the supports for the wires were small stubs spaced about 8 inches apart.
Rohsenow, Webber, and Ling’s (78) and Dukler’s (3) correlation for film condensation on a vertical surface with appreciable vapor shear. The interfacial shear was calculated directly from measured pressure drop data and was in good agreement with Bergelin’s (7) correlation for the friction factor for gas flowing in a tube with a liquid layer on the wall, using Lehtinen’s ( 7 7) recommendation for calculating the average mass velocity of the vapor. Visual inspection of the tube surface indicated that dropwise condensation did not occur during the tests when steam was condensed on the outside of the tube. From the good agreement of the data for steam condensing on the inside of the tube with those for steam condensing on the outside, it was assumed that dropwise condensation did not occur to an appreciable extent on the inside of the tube. Values of the ratio hlh, are shown in Figure 2 as a function of heat flux for l/Anch diameter tubes with three, four, eight, and 12 0,030-inch diameter wires loosely stretched along the tubes. The condensing coefficient with wires increases markedly with decreasing heat flux-Le., with three wires from hjh, = 1.36 to 2.94 as the heat flux decreased from lo5 to 2 x 104 B.t.u./(hr.)(sq. ft.). The condensing coefficient increased to a somewhat smaller extent as the number of wires was increased-i.e., at heat fluxes of 2 X lo4, h/h, was 2.94 with three wires and 4.55 with 12 wires. At higher heat fluxes the data for eight wires were somewhat above data for 12 wires. The effect of the number of wires is shown in Figure 3, in which h/h, is plotted against ( N d l r D ) , the fraction of the surface covered by wires. Data are shown for three, four, eight, 10, and 12 0.030-inch diameter wires and four 0.062inch diameter wires stretched along a ‘/*-inch diameter tube with the heat flux as a parameter. For any given heat flux,
I
I
I
5
’ 4
2 2
3
Experimental Results
The over-all heat transfer coefficient as a function of velocity was measured for four different ranges of mean temperature differences, At, between the condensing steam and the water coolant; typical values of At were 7’ to 8’, 11’ to 13O, 47’ to 60°, and 104’ to 112’F.; the heat flux range was 2 X 104 to plots of the 105 B.t.u./(hr.) (sq. ft.). Wilson plots-Le., reciprocal of the heat transfer coefficient us. the reciprocal of the velocity to the 0.8 power-were prepared for constant heat fluxes from plots of heat transfer coefficient us. heat flux. Assuming that the heat transfer resistances are additive, the intercept on the Wilson plot is the resistance of the metal wall plus the condensate film (76). Since the resistance of the metal wall is known, the condensing heat transfer coefficient can be easily calculated. Condensing heat transfer coefficients for the aluminum tube in the absence of wires were in excellent agreement with 100
l&EC FUNDAMENTALS
2
1
0
2
4
(PIA) x
6
8
10
t2
14
HEAT FLUX (Etu/hr f t 2 1
Figure 2. Effect of heat flux on enhancement of condensing heat transfer coefficient b y vertical wires loosely attached to vertical tube length 42 !h inches -Tube Calculated from simplified model
the value of h/ho increases as the number of wires is increased until a maximum is apparently reached at a value of ( N d / n D ) = 0.18. The data for four 0.062-inch diameter wires are in substantial agreement with the data for eight 0.030-inch diameter wires, both combinations of wires having almost the same fractional surface coverage-that is, Nd/nD 0.16. The data for eight and 10 wires differed in one respect from those for three, four, and 12 wires. With three, four, and 12 wires the plots of U us. q / A were reproducible straight lines for all velocities on a log-log plot. The data for eight and 10 wires were also reproducible for heat fluxes greater than 3 X l o 4 B.t.u./ (hr.)(sq. ft.) and gave the data shown in Figure 3. However, for velocities greater than 10 feet per second and heat fluxes less than 2.5 X 10%B.t.u./(hr.)(sq. ft.), the heat transfer coefficient, Lr, was mulitivalued a t a given heat flux. The largest values of U gave values of h/ho in good agreement with the upper curve of Figure 3, but since the reason for the multivalued results was not clearly established, these points were not included on the figure. One explanation for the multivalued heat transfer coefficients is that some of the loosely attached wires bowed away from the tube in some tests, so that some wires were less effective at the lowest heat fluxes. The effect of displacing four wires a known distance from the surface is shown in Figure 4. At the largest heat flux [ q / A = 105 B.t.u./(hr.)(sq. ft.)] positioning the wires with a gap of one wire diameter between the wire and the surface caused an increase in h/h, from 1.56 to 1.93; increasing the gap resulted in a decrease in the value of h/h, until there was no effect of the wires for a gap of four wire diameters. At the lowest heat 104 B.t.u./(hr.) (sq. ft.)] a fluxes used in this study [ g / A = 2 gap of one wire diameter caused a reduction in the value of h/h, from 3.08 to 2.62 and the effect of the wires was no longer discernible for a gap of four wire diameters. These data are consistent with the explanation of the effect of bowing of the wires advanced above. I n another series of tests, four wires were wrapped around the tube in a spiral with a pitch of 35/8 inches-Le., the vertical distance between individual wires was -0.9 inch. Such a spiral arrangement increased the length of wires by 10% over the value for straight vertical wires. The ratio h/ho was 2.0, 2.2, and 2.3 for heat fluxes of 10, 5, and 2 X lo4 B.t.u./(hr.) (sq. ft.), respectively (see Figure 2). At the highest heat flux the value of h/h, was greater than the value for four vertical wires determined from Figure 3 even after correcting the fraction of surface covered for the extra length of wire on the tube. At the intermediate heat flux the value of h/h, for four vertical wires determined from Figure 3 and experimental values for the spiral agreed and at the lowest heat fluxes the value of h/h, for the spiral was substantially lower than the value for four vertical wires determined from Figure 3 (h/ho = 3.47 us. 2.3 actual). All of the results described above were obtained with condensation and wires on the owaide of the tube. A few tests were made with steam condensing on the inside of a '/Z-inch tube having an inside diameter of 0.444 inch. Data a t g / A = 5 x 104 for a single 0.035-inch diameter wire wound in a spiral with a */*-inch pitch and for four straight 0.035-inch diameter wires on the inside of a tube were in excellent agreement with the curve shown in Figure 3 (h/ho = 1.70 and 2.46 for Nd/irD = 0.04 and 0.10, respectively). However, the data for four '/ls-inch diameter vertical wires were below the curve for q / A = 5 X 101 shown in Figure 3 (h/h, = 2.24 for N d / n D = 0.18).
5 NUMBER
WIRE 0,AMETEI
WIRES
A 0
-
0
A 10 12
4
5 .c
V 7
3
2
I
0
01 FRACTION
02 03 OF SURFACE COVERED BY WIRES ( N d / n D )
0.4
Figure 3. Effect of number and diameter of wires loosely attached to vertical tube on condensing heat transfer coefficient
-Calculated from sirnplifled model h 3
I
*Yx10;
Btu/hr f t 2
0
r \ r
2
0
2
6
DISTANCE FROM SURFACE (gap/wire diameter)
Figure 4. Effect of distance of wires from surface on condensing heat transfer coefficient Wires. Tube.
0.030-inch diameter 0.50-inch diameter, 42
Vi inches long
All the data for vertical wires loosely attached to the surface are fitted for (Nd/nD) C 0.15 by the empirical expressions
h/h,
=
[l
+ 6.8 X 106(Nd/nD)(A/q)]
(20)
[l f 6.8 X 102(Nd/nD)(L/P)]
(21)
or h/h,
=
when the numerical constant in Equation 20 has the dimensions B.t.u./ (hr.)(sq. ft.), and in Equation 21 the dimensions (lb.)m/(hr.) (sq.ft.). The direct proportionality between (h/h,) 1 and N d / n D supports the assumption that the range of influence of the radius of curvature of the rivulet is independent of the distance between wires.
-
VOL. 6
NO. 1 F E B R U A R Y 1 9 6 7
101
Comparison of Model with Data
The constant a of Equation 19 was calculated from the data for three, four, eight, and 12 wires; the average value was 8 = 1.7 (h0.3) X This valueof 8 was then used with Equation 19 to calculate the curves shown as solid lines in Figures 2 and 3. The model predicts the dependence of the present experimental data for h/ho on heat flux, number of wires, and wire diameter surprisingly well except for the two points for (,$‘d/rD) > 0.15 and q/A = IO5 B.t.u./(hr.)(sq. ft.) shown fitted with a broken line in Figure 3. Despite the excellent agreement of the model with the data for two different wire diameters tested, it seems clear that one very large wire could not be as effective as many small wires; in addition, the shape of the wirz enters only through the hydraulic radius term in Equation 15 and in the calculation of the area, A , Equation 18. A more detailed model is required to determine more accurately the effect of wire dimensions and wire shape as well as the three-dimensionality of the flow.
D DH
=
f g,
= Fanning friction factor, Equation 15, dimensionless = conversion factor, (lb.m/lb.f)(ft./hr.z)
= tube diameter, ft.
hydraulic diameter, ft. gravitational acceleration, ft./hr.2
g~
=
h
= heat transfer coefficient with wires, B.t.u./(hr.) (sq. ft.)
h,
= heat transfer coefficient without wires, B.t.u./(hr.) (sq.
h,
= latent heat of vaporization, B.t.u./lb.,
(OF.)
ft.) ( O F . )
k L N
=
pl
=
thermal conductivity, B.t.u./(hr.) (sq. ft.) (OF./ft.)
= tube length, ft. = number of wires, dimensionless
pressure in medium l,,lb.l/sq. ft.
pz
= pressure in medium 2, 1bnf/sq.ft. q / A = heat flux, B.t.u./(hr.) (sq. ft.) R = radius of curvature, ft.
At = temperature gradient, OF. Q / A = heat flux, B.t.u./(hr.)(sq.ft.) U = heat transfer coefficient u = local velocity, ft./sec. V = velocity, ft./sec. dy = differential distance normal to tube, ft. dz = differential distance in direction of flow, ft.
Conclusions
GREEKLETTERS Vertical wires loosely attached to the surface of a vertical tube cause an appreciable increase in the condensing heat transfer coefficient. This may offer certain practical advantages in comparison with the fluted tube geometry as it is now conceived-that is, either a thick wall or a fluted inner surface. With wires stretched along the tube the highest heat transfer coefficient occurs over that portion of the tube where the metal wall is thinnest; in addition, the inner surface is cylindrical, permitting the insertion of twisted tapes or detached turbulence promoters. For the present data, when the wires covered less than 15% of the surface the fractional increase in condensing coefficient (h - h,)/ho was directly proportional to the fraction of surface covered and inversely proportional to the heat flux, A maximum in the value of h/ho occurred for fraction of surface covered by wires, Nd/rD, of about 0.18. A qualitative model indicated that the decrease in h/h, for Nd/nD > 0.18 was due to the reduction in tube surface available for heat transfer by the wires and the associated rivulet of water flowing along the wire. The condensing heat transfer coefficient was more than tripled by loosely attaching four 0.030-inch diameter wires to the surface of a l/Z-inch diameter tube when the heat flux was 2 x 104 B.t.u./(hr.)(sq. ft.); increasing the number of wires to 12 increased the value of h/ho to 4.5 a t the same heat flux. At high heat fluxes [>5 x 104 B.t.u./(hr.)(sq. ft.)] it was beneficial for the wires to be supported away from the surface; a t smaller heat fluxes it was necessary for the wires to be in contact with the surface. Acknowledgment
The author acknowledges the support and suggestions of Kurt A. Kraus and the assistance of P. H. Hayes in performing the experimental measurements. Nomenclature
A A1 a B
C d 102
= total surface area of tube, sq. ft. = rivulet cross section area, sq. ft. = constant in Equation 19 = 1.72 x 10-6, dimensionless = cross-sectional area of film between wires, sq. ft. = cross-sectional area of film without wires, sq. ft. = wire diameter, ft. l&EC FUNDAMENTALS
a1 a2
= constant in Equation 11, dimensionless
a3
= constant in Equation 15, dimensionless
r
=
constant, R/d, dimensionless
mass flow per unit perimeter, lbJ(hr.) film thickness with wires, ft. film thickness without wires, ft. p viscosity, lb.,/(ft.)(sec.) v kinematic viscosity, sq. ft./hr. n 3.1415 . . . , . . p = density, lb.,/cu. ft. u = surface tension, lb.t/ft. rW = wall shear stress, lb.f/sq. ft.
6 6,
=
(ft.)
= = = = =
SUBSCR~PTS
A B
= area A
C f m z
= = = =
=
area B area C fluid mean value a t height z
literature Cited
(1) Bergelin, 0. P., et al., Heat Transfer and Fluid Mechanics Institute, ASME, 1949, p. 19. (2) Carnavos, T. C., First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965, Paper S W D / 17. “Thin Film Distillation.” (3) Dukler, A. E., Chem. Eng. Progr. Symp. Ser. 56 (30), 1 (1960). (4) Erb, R. A., Thelen, Edmund, First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965; Paper SWD/81, “Dropwise Condensation.” (5) Fulford, G. D., Aduan. Chem. Eng. 5 , 151-236 (1964). (6) Gregorig, Romano, Z A M P 5 , 36-49 (1954). (7) Jakob, Max, “Heat Transfer,” pp. 658-95, Wiley, New York, 1949. (8) Kumm, E. L., Dept. Commerce, OSW Progress Rept. 103, PB 166241 (1964). (9) Landau, L. D., Lifshitz, E. M., “Fluid Mechanics,” p. 230, Pergamon Press, New York, 1959. (10) Le Fevre, D. J., Rose, J. W., Intern. J. Heat Mass Trarrfer 8, 1117 (1965). (11) Lehtinen, J. A., Sc. D. thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., June 1957. (12) McAdams, W. H., “Heat Transmission,” 3rd ed., pp. 330-3, McGraw-Hill, New York, 1954. (13) Mathewson, W. F., Smith, J. C., Cham. Eng. Progr. Symp. Ser. 59 (41), 173-9 (1963). (14) Medwell, J. O., Nicol, A. A., ASME-AIChE Heat Transfer Conference, Los Angeles, Calif., Aug. 8-11, 1965, Paper 65HT-43. (15) Nusselt, W., Z. VDZ60, 541, 569 (1916).
(16) Perry, J. H., ed., “Clhemical Engineers’ Handbook,” 3rd ed., up. 464-70, McGraw-Hill, New York, 1950. (17) Rohsenow, W. M., Ihkhatme, S. P.’, “Developments in Heat Transfer,” W. M. Rohsenow, ed., pp. 293-318, M I T Press, Cambridge, Mass., 1964. (18) Rohsenow, W. M., ‘Weber, J. H., Ling, A. T., Trans. ASME 78, 1637-44 (1956). (19) Rota, J., Zng. Arch. 18, 277 (1950).
(20) Sieder, E. N., Tate, C. E., Znd. Eng. Chem. 28, 1429 (1936). RECEIVED for review May 13, 1966 ACCEPTED September 19, 1966
Research sponsored by the Office of Saline Water, U. S. Department of the Interior, under Union Carbide Corp. contract with the U. S. Atomic Energy Commission,
EFFECT OF M E A N FLOW ON W A L L D A M PI NC3 OF FI N ITE-AM PLITU D E GAS PULSATIONS G. E. K L I N Z I N G ’ A N D A . 0 . CONVERSE2 Carnegie Institute of Technology, Pittsburgh, Pa.
To study the effect of mean flow on the wall damping of finite-amplitude gas pulsations, experiments were carried out in an air-filled 1.5-inch diameter smooth tube having an oscillating piston a t one end and the other end opeii to the atmosphere. The mean flow was introduced through a critical-flow orifice in the tube wall near the piston. The pressure amplitude at resonance decreased with increasing Mach number. This effect was most pronounced a t low Mach numbers, where flow reversal occurred. The experimental results were compared to the predicted values by two types of mathematical analyses: a linearized closed form solution of the wave equation including mean flow and damping, and a nonlinear numerical solution using the method of characteristics. Both techniques were able to predict the effect of mean flow, the latter being considerably more accurate. This comparison suggests that the nonlinear effects are important. In the two cases tlhe friction factor was chosen so that the theory agreed with experiment when no mean flow was present. No additional empirical parameters were used in the treatment of the flow effect. HE DESIGN of compressor piping systems and the stability of Trocket motors are two areas in which gas pulsations are important. I n these processes the gas pulsations are superimposed upon a mean flow. Whereas wall damping of acoustic waves has been thoroughly studied (9, 74,76) and wall damping of finite-amplitude waves without mean flow has had some attention (7, 3, 7), the effect of mean flow on wall damping of finite-amplitude waves has been neglected. Existing treatments of oscillatory flow (5, 8, 70, 72) are not applicable to the present case, where the flow may fluctuate from turbulent flow in cine direction to turbulent flow in the other several times per second (2). Flows with finite-amplitude oscillations and mean jlow have been analyzed to obtain numerical solutions (method of characteristics) (4,7 7, 73). However, the technique has not been used previously to study the effect of mean flow on wall damping. T h e present work was undertaken in order to determine experimentally the effect of mean flow on the wall damping of finite-amplitude waves and determine the validity of a linearized analysis, and hence the importance of the nonlinear effects. An understanding of this effect is important in the evaluation of the acoustic admittance of solid propellants from the results of the so-called T-motor experiments (75).
Experimental Equipment and Procedure
The equipment consistled of a long tube open at one end and closed at the other by a reciprocating piston. The tube was of a smooth-drawn variety with 1.5-inch diameter and could 1 2
Present address, University of Pittsburgh, Pittsburgh, Pa. Present address, Dartmouth College, Hanover, N. H.
easily be adjusted to various lengths. The sections of tubes were fitted together by tight sleeves, sealed with Pyseal. The piston was driven by a Scotch yoke mechanism; hence, the displacement contained no harmonics. The frequency could be varied between 0 and 15 C.P.S. A sonic orifice consisting of a n annular ring of holes in the tube wall a few inches from the piston face was used to introduce air for the mean flow experiments. A sonic orifice was used so that the standing wave tube would be acoustically isolated from the feed system. The pressure oscillations were measured a t a point close to the piston face by a Statham differential pressure transducer. I n the mean flow measurements at high frequency turbulence components near the entrance orifice caused the pressure signal to be distorted, the frequency being considerably above that of the piston and its near harmonics. A Kronhite band pass filter was used to eliminate this noise. The speed of the piston was controlled electronically and varied by a 10-turn potentiometer. I n all the runs, the frequency of the piston was varied until the pressure amplitude was at a maximum. I n this manner, the resonant pressure amplitude was found. The resonant amplitude is most sensitive to damping, because without damping the amplitude would be infinite. This is not true of the nonresonant frequencies where the amplitude is limited also by wave cancellation. Experimenfal Results
Figure 1 shows the effect of the mean flow on the pressure amplitude a t resonance. When the mean flow is small, there is flow reversal with each oscillation a t every position within the tube. As the mean flow is increased, this flow reversal ceases first a t the piston end where the velocity oscillations are smallest and finally a t the open end of the tube where the velocity oscillations are maximum. To the left of the X on the curves drawn through the data are the regions in which part, but not VOL. 6
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