I
J. A. SCANLAN Mechanical Engineering Department, University of Texas, Austin, Tex.
Effects of Normal Surface Vibration on laminar Forced Convective Heat Transfer Normal vibration of a heating surface can have a pronounced beneficial effect on heat transfer. And the local heat transfer coefficient can be predicted from an equation developed during the work reported here
MANY
techniques have been proposed or used to increase heat transfer between fluids and hot surfaces. I n most of these, disturbances are created in the relatively stagnant film of fluid on the heating surface, and bulk mixing of the stream is increased. I n some techniques, baffles, roughened surfaces, and other mechanical means are used to disturb the flow. Recently, the possibility of achieving this disturbance by vibrating the fluid or the surface has attracted attention. Investigations revealed significant increases in heat transfer rates for the conditions involved (3, 5, 9). The present investigation concerns the effects of vibrating a heating surface in contact with a liquid in a direction normal to the surface.
Experimental The system used gravity flow from an adjustable constant-head tank to ensure freedom from spurious vibrations, and the water was deaerated under vacuum (Figure 1). Water temperature was held close to room temperature to minimize heat loss or gain. The 0.2inch square annulus around the heater was partially filled with freshly mixed Dow-Corning RTV Silastic stock, which self-vulcanized at room temperature into a soft, white, rubbery substance, giving a waterproof seal flexible enough to permit considerable displacement of the heater in any direction. The Silastic was flat and flush with its surroundings to less than 0.001 inch. A maximum displacement of the heater of 0.002 inch from its flush midposition gave, in effect, a smooth rubber ramp of at least 100 to 1 angle of approach. Tests with this displacement at very low frequencies showed that the heat transfer coefficient was unchanged by the ramp. Fine scratches on the driving rod were observed through a microscope, calibrated to 0.00001 inch, as a measure of the amplitude of vibration. Excitation was by a calibrated 20 to 20,000-C.P.S.variable frequency oscillator. Although h seemed to be a weak func-
Significant Literature Background Effect of vibration on heat transfer by free convection from horizontal cylinder immersed in water Vibration of electrically heated wires in still air Application of pulsations to water in double-pipe heat exchanger No improvement in low N R range ~ Mechanical vibration of entire heat exchanger Effects of fluid pulsations on heat transfer from still surface to air by vibrating vertical air column at 125 to 2400 c.p.8.; improvements in over-all coeacients up to 54% and local coefficients to 200%
tion of the temperature difference between the surface and the fluid, the ratio of h, to h, was practically independent of such temperature difference. Therefore, the heating current was set so that a midpoint surface temperature of 125' F. was obtained (no vibration) for each flow rate and the same current (heat flux) was maintained for all tests at that flow rate.
Martinelli, Boelter (6) Lemlich (9) West and Taylor (9) Morris, Webb (6, 8) Andreas (1)
Harrison, others (8)
Predictions
With the
assumption that
hu = $(D, V , M > k , P, C , f , A )
(1)
the usual methods of dimensional analysis lead to the conclusion that a possible relationship among the above variables is DVp cp D2pf A
Thermocouples soldered
I x 0.002" Nlchrome V
Figure 1. The heater held water temperature close to room temperature R . Side View
.025" Copper busbor Vulcanized fiber block Thermocouple terminals Heoter terminalt No. 8-32 thd. for driving rod
Bottom Vlew VOL. 50,
NO. 10
OCTOBER 1958
1565
3.0
3.0
Figure 2. Effect of frequency and amplitude, N R e =
3 60 2.5
2.5
2.0
2.0
h"
hV -
h0
h0
I.5
I. 5
I .o
I.o Frequency, I
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The tendency for curves o f greater amplitude to peak at lower frequency values is in satirfactory agreement with predictions
1
1
3.0
5 h0
& A
0
Figure 3.
1.5
k
2.0
&
1.0
I.5
300 U I
2
400 I A
K
I
4
A = 0.002" x\
a
200 4I
100 I
h0
Figur 4. Effect of frequency and amplitude, NRe
0.001" 500 uI 6A( i
= 1460
500
400
600
cpn
Effect of frequency and amplitude, NRo=
1
!h
:0.004"
300
720
2.5
NRe = 1460
200
Frequency,
2.5
2.0
100
0
cps
a
I .o
' *
1
1
1
NRe= 2170
\\ \
A = 0.004"
1
1
1
1
1
100
200
300
400
500
Frequency,
Figure 5.
1
61
cpi
Effect of frequency and amplitude, Nno =
2170
Frequency, c p s
Table I.
Selected Data and Results
& J fp
Run No. 1 37 38 39 40 40a
50 51 52 73
1 566
PI!
A,
Lb., cyC.O.001 Sec.
1/121.8 1/60.1 1/60.9 1/61.7 1/60.7 1/60.5 1/58.9 1/64.6 1/63.3 2/59.4
Sec. In.
0 0 20 40 70 75 20 40 70 0
0 0 1 1 1 1 2 2 2 0
T8,
TW2
Mv.
O F .
1.413 1.414 1.425 1.414 1.310 1.380 1.396 1.344 1.397 1.317
95.5 95.5 96.0 95.5 91.0 94.1 94.8 92.5 94.8 91.3
INDUSTRIAL A N D ENGINEERING
Mv.
Mv.
1.466 2.127 1.448 2.139 1.459 2.119 1.464 2.343 1.349 1.803 1,420 1.824 1.434 2.139 1.384 1.930 1.440 1.747 1.340 2.084
CHEMISTRY
O F .
125.9 126.4 125.6 134.8 112.3 113.2 126.5 117.7 109.9 124.1
Hr.-
ss.
Lb., 1.11, Htr., __ Cu. Lb., Ft. Amp. Ft. Hr.- Ft. O F . 30.0 34.0 33.0 40.0 34.5 34.5 34.5 34.5 34.5 38.0
62.03 62.04 62.09 62.05 62.05 62.07 62.05
320 410 1.730 424 1.739 461 1.829 675 1.766 776 1.752 456 1.798 551 1.752 1008 530
Np,
Dhf UI
594 1180 1963 2175 585 1142 2050
NA, A De
Col. 14 ( ~ ) " ; ' , 7 z
0.0055 0.0055 0.0055 0.0055 0.0110 0.0110 0.0110
8wA
0.052 0.166 0.397 0.474 0.051 0.157 0.427
X
hu/ho
Col. 15 - 1
3370
1.492 1.492 1.492 1.492 2.22 2.22 2.22
0 0 0.0776 0.04 0.248 0.13 0.592 0.65 0.707 0.89 0.1133 0.11 0.349 0.35 0.949 1.45 0
HEAT TRANSFER This grouping is chosen in preference to several other possibilities on account of the presence of the Nusselt, Reynolds, and Prandtl numbers. For convenience, D2pf/,u is called the frequency number, N F , and A / D is called the amplitude number, NA. I n this investigation, the ratio of h, to h,, as suggested by Lemlich (3), was used :
In the limiting condition of no vibration this reduces to an identity. Change in h,/ho due to vibration can be expressed as ( h v / h o ) - 1 = ~ N FN ,A )
(41
This approach has the inherent advantages of emphasizing the phenomenon under investigation and minimizing any consistent spurious effects due to peculiarities of the apparatus, radiation, convection, instrumentation, or method of calculation. I t is obvious that Equation 4 will fit the boundary condition of no vibration. What will be the effects of increasing the vibration frequency and amplitude indefinitely? Will h,/h, increase without limit, or will some phenomenon extraneous to the assumption in Equation 1 enter the situation and establish an upper limit to the validity of Equation 4? If the acceleration of the heating surface becomes sufficiently great in the direction away from the liquid, cavitation will occur. The resulting blanketing effect would be expected to counteract the increase in coefficient caused by mixing. Below a certain combination of amplitude and frequency this blanketing will not occur. Above this point, the blanketing will increase with increased amplitude and frequency as that portion of a cycle during which the critical acceleration is exceeded becomes larger. Only during that half-cycle when the surface is pulling away from the liquid will the surface acceleration be directly effective in creating such a blanket. The dimensions of the flow system will affect the magnitude of the critical frequency at which the blanketing will begin. The physical properties of the fluid will enter into the situation, as will the contours, rigidity, and other properties of the containing walls. T o simplify the analysis, a n approximation to the actual system was used. The critical acceleration was predicted by assuming that at cavitation the instantaneous local pressure a t the surface would be equal to the vapor pressure of the liquid:
hap = Pa where
+ pgH/gc - worL/gc
(5)
pa represents pressure due to
atmosphere, pgH/gc is the static pressure due to the weight of H feet of liquid, dnd paerL/gc is the pressure due to an acceleration of aor applied to the end of a column of liquid L feet long (this dssumes that the liquid is incompressible, passage walls are rigid, and the liquid entirely fills a uniform passage for a distance L, at which point there is a surface free to move). With simple harmonic motion the critical frequency is
Various factors affect the critical frequency by an amount determinable best by experiment, although the dependency of critical frequency upon amplitude may remain evident. One is fluid friction, which tends to make the vibrating column sluggish in following the vibratory motion of the surfacei.e., the apparent value of p in the denominator (only) of Equation 6 will be increased, lowering the critical frequency. Any dissolved gas in the liquid will encourage the breaking away of the liquid from thC surface, lowering the critical frequency. Surface roughness or any other type of physical discontinuity also will encourage rupture of the liquid-to-surface interface. Surface tension will discourage the formation ol voids. It may be inferred that Equation 4 may have an upper limit of applicability as evaluated by Equation 6. Above this limit, the benefits due to vibration may decrease as the blanketing portion of each oscillatory cycle becomes larger. The experimental apparatus was designed to check the validity of the general form of Equation 4, evaluate its constants, and to check the existence of an upper limit for Equation 4 in terms of Equation 6. Calculations
The film coefficient, h, was determined from the relationship
The specific heat, c, properly should be the local value at every point; however. its value changed by an insignificant amount over the temperature range experienced by the water. Therefore, the value of c at the initial water temperature was arbitrarily used. The final water temperature, T,, was the bulk mixed temperature. The water temperature, T,, should be the local value a t some arbitrary distance away from the heating surface for each and every point on the surface, but in view of the small temperature change experienced by the water in these experiments, initial water temperature was used. Any particularities in the values of h
resulting from these choices essential11 disappear from h,/h,. During preliminary tests, values of h were calculated using the average of five measured surface temperatures as well as only the mid-point temperature. As the midpoint temperature was close to the average temperature, values of h were close. When the ratios of h,/h, were calculated, even these minor differences disappeared. For this reason only the midpoint surface temperature was used in calculating h. In determining the theoretical value of frequency for the experimental system, the density of water was evaluated at initial conditions. The static head, H, varied from 0.8 to 1.4 feet, so a mean value of 1.1 feet was used. The length, L, of the vibrating column of water to the free surface in the constant head tank was 4.75 feet. Extrapolation from observed surface temperatures gave an estimated trailing edge temperature of 153" F. Because cavitation would begin at the hottest point, the corresponding vapor pressure of 4.0 p s.i.a. was used. For a total amplitude of 0.001 inch of the heater, the axial motion of the column of water was 0.010 inch, beoause of the area ratios involved. With these values, critical frequencies of 102, 72, and 51 C.P.S.were predicted for heating surface total amplitudes of 0.001,0.002, and 0.004 inch, respectively. I t was estimated that the maximum experimental error in determining the water flow rate was 1%. In measuring the temperature rise of the water it was 6.25%. Errors in specific heat and area were negligible and tended to cancel out in h,/h,. 'The maximum error in temperature difference between the surface and the water was 0.70y0. These gave an estimated maximum possible total error of 7.95y0 in determining either h, or h,; hence the error in h,/ho could be as high as 16% under the worst possible combination of circumstances. The probable average experimental error was estimated to be about 6%. Results
Figures 2 to 5 show h,/h, as a function of frequency. Each figure is for a particular flow rate, with multiple curves showing the effect of amplitude of vibration. The peaks occur in the range of 50 to 80 C.P.S. with a tendency for the curves of greater amplitude to peak at the lower frequency values. This tendency and the values of critical frequency are considered to be in satisfactory agreement with predictions, considering the simplified assumptions previously discussed. The deviation from typicalness of the curves above the critical frequency and for large ampliVOL. 50, NO. 10
OCTOBER 1958
1567
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l
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acceleration, feet/sec2 British thermal units = specific heat at constant pressure. Bl1b.i-O . ... F. = efrective diameter of rectangular cross-section, feet, (2D1D& (Dl Dz) = frequency, cycles pcr second. C.P.S. = acceleration due to gravity, feet. sec.2 = dimensional constant, Ib.;,-ft./’1b.+ec2. = height of a column of liquid, feet = heat transfer coefficient, B/hr.-sq. ft.-” 1‘. = thermal conductivity, B-ft./hr.sq. ft.-O F. = length of column of liquid, reet = mass flow rate, 1b.Jhr. = amplitude number, dimensioiiless, AID = frequency, dimensionless, D2p / / p = Reynolds number, dimensionless. OVP/P = pressure, lb.j/sq. ft. = surface area of heater, sq. feet = temperature, O F. = velocity, feet per second = mu, viscosity, Ib.Jsec.-fi. or 1b.,/hr.-ft. = rho, density, 1b.Jcu. foot = phi, a function in general = psi, a function in general = =
1
I
A
+
Figure 6. Comparison of experiments shows good agreement with values obtained from Equation 9
I I 1 0.04 .061)0.I
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.2
.4
.6
tude of vibration, especially at the lower Reyndds numbers, was not predicted, as this range of frequency is above the upper limits of applicability of Equation 4 as evaluated by Equation 6. Experimental runs were made well above the predicted critical frequency, to determine the sharpness of this cutoff point; results in that range are shown in Figures 2 through 5 for the sake of completeness. One possible explanation for the large values of h,/h, for 0.004-inch amplitude and for the secondary peak at a frequency of about 400 C.P.S. is that of induced reversed local flow due to cavity resonance. These effects, caused by a resonance local velocity superposed upon the mean bulk velocity, would be expected to decrease at the higher flow rates (as they do) in contrast to the effects below the critical frequency, which are the subject of this paper. (An investigation of these high frequency effects is now under way.) To separate the effects of the two variables, A and f,it is assumed that ~ N FN ,A ) = ~ ( N FrndNa) )
(8)
Only points below and near the peaks in Figures 2 to 5 were used, as Equation 4 is not applicable a t frequencies greater than critical. Plots for cases of constant amplitude showed that the best single slope for all was reasonably good for each alone, and permitted evaluation of the effect of frequency. Upon using smoothed values from these curves cross-plotted us. A to obtain az(NA), Equation 4 was evaluated as
l
.e
l
I
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2
I
4
from Equation 9. Values of predicted in this manner:
/zu
may be
where h, may be obtained by any accepted method (4, 7) which is applicable to water with Reynolds numbers below 2200. Equation 10 is valid within the range of 0 < .VA < 0.022 and 0 < N F < 2370 for this experimental system; upper limits of NA and N F for other systems may be predicted by use of Equation 6 with properties evaluated at the initial temperature. Conclusions
Normal vibration of a heating surface can have a pronounced beneficial effect upon heat transfer. The local heat transfer coefficient can be predicted from Equation 10 within the described limitations. The effect of vibration on heat transfer depends on frequency and amplitude. Within the range investigated, there is a n upper limit of frequency for which the above equation is applicable. Above this point, heat transfer usually decreases sharply. The frequency limit can be predicted reasonably well by Equation 6 with the fluid properties evaluated a t initial conditions, except for the vapor pressure, which corresponds to the hottest surface temperature. Heat transfer coefficients can be almost tripled under the conditions investigated. Further investigation :is needed with extended operational limits. Nomenclature
I n Figure 6, observed values of (h,/h,) 1 are plotted against values obtained
1568
A
INDUSTRIAL AND ENGINEERING CHEMISTRY
= amplitude of vibration (total),
feet
SUBSCRIPTS a = atmospheric cr = critical J = force rn = midpoint of heating surface. o r mass o = without vibration s = heating surface u = with vibration w = water vap = vapor 1 = initial water, or first 2 = final (mixed) water, or second literature Cited
(1) Andreas, Arno, German Patent 717,766 (Feb. 5, 1942). (2) Harrison, W. B., Boteler, W. C., Jackson, T. W., Lowi, A., Jr., Thomas, F. A,, Jr., “Heat Transfer to Vibrating Air Columns,” NACA Rept. N-49857 (December 1955). (3) Lemlich, Robert, IND.ENG. CHEM. 47, 1175-80 (1955). (4) McAdams, W. H., “Heat Transmission,” 2nd ed., McGraw-Hill, New York. 1942. (5) Martinelli, R. C., Boelter, L. M. I
