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Condensation of refrigerants R-11 and R-113 in the annuli of horizontal double-tube condensers with an enhanced inner tube
Condensation of Refrigerants R-11 and R-113 in the Annuli of Horizontal Double-Tube Condensers with an Enhanced Inner Tube Hiroshi Honda Institute of Advanced Material Study, Kyushu University, Fukuoka, Japan
Shigeru Nozu, Yoichi Matsuoka, Tohru Aoyama Department of Mechanical Engineering, Okayama University, Okayama, Japan
Haruo Nakata Daikin Industries, Ltd., Osaka, Japan
IIExperiments were performed to study the flow and heat transfer characteristics during condensation of R-I 1 and R-113 in the annuli of horizontal doubletube test condensers. The condensers are made up of a 19.1 mm O.D. corrugated inner tube with wire fins soldered on the outer surface and three outer smooth tubes of 24.8, 27.2, and 29.9 mm I.D. The mass velocity of the test fluids ranged from 50 to 300 kg/(m 2 s), and the condensation temperature difference ranged from 0.7 to 20 K. The frictional pressure gradient was correlated fairly well by using the Lockhart-Martinelli parameters. The local heat transfer coefficient was 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes. An empirical equation for the local heat transfer coefficient was developed, in which the dimensionless parameters based on the surface tension controlled flow and the vapor shear controlled flow models were introduced for the low and high vapor velocity regimes, respectively.
Keywords: condensation, heat exchangers, heat transfer augmentation
INTRODUCTION Water-cooled horizontal double-tube condensers are commonly used in air-conditioning systems of relatively small size. Condensation may occur in the annulus or in the inner tube depending on the conditions. For systems using fluorocarbon refrigerants, coiled condensers with enhanced inner tubes, with vapor flowing through the annular passage, are commonly adopted because of their high performance and compact size. Several experimental studies have been reported of condensation in annuli with smooth and/or enhanced inner tubes [16]. Miropolsky et al [3] and Cavallini et al [5] showed that empirical equations for forced convection condensation inside smooth tubes are also applicable to vertical annuli consisting of smooth inner and outer tubes. Hashizume [4] compared the annulus-side and tube-side condensation data for horizontal double tubes with smooth and corrugated inner tubes. The heat transfer enhancement of the horizontal annulus using the corrugated inner tube was less than 25%. The average heat transfer performance of coiled double-tube condensers with various types of enhanced inner tubes have also been reported [6]. Although considerable heat transfer enhancement has been achieved, the range of experimental data is limited and the mechanism of heat transfer enhancement has not been given theoretical consideration. The present study was undertaken to obtain a better understanding of the flow characteristics and heat transfer during condensation of fluorocarbon refrigerants in horizontal annuli with an enhanced inner tube. The effect of the annular space was studied by using three test condensers with inner
tubes of the same diameters and outer tubes of different diameters.
EXPERIMENTAL APPARATUS AND PROCEDURE The experimental apparatus, shown schematically in Fig. 1, consists of forced circulation loops of test fluid and cooling water. The vapor generated in an electrically heated boiler (1) flows upward via a superheater (2) to a horizontal test section (3). The vapor is condensed almost completely in the test section. The condensate flows downward through a condensate receiver (4), then horizontally through a circulation pump (5), a strainer (6), and a flow control valve (7), and then returns to the boiler. The loop is thermally insulated with fiberglass. The cooling water is pumped from a cooling water tank (8) by a pump (9) to the test section via a rotameter (10). After exchanging heat with the test fluid, the cooling water returns to the tank. The cooling water temperature is regulated by the use of a chilling unit (14) and a heater (15). The test section is a horizontal double-tube condenser with the test fluid flowing through the annular passage. The cooling water flows countercurrently through the inner tube. Three test sections with inner tubes of the same diameters and outer tubes of different diameters are used to study the effect of the annular space. Details of the inner tube are shown in Fig. 2. The inner tube is a corrugated copper tube with copper wire fins soldered onto the outer surface (wire-finned tube). The outer surface area is 3.04 times that of an equivalent smooth tube. The outer tubes are smooth tubes with insider diameters D; of 24.8, 27.2,
Address correspondence to Professor Hiroshi Honda, Institute of Advanced Material Study, Kyushu University, Kasuga-shi, Pukuoka 816, Japan.
Experimental Thermal and Fluid Science 1989; 2:173-182
© 1989 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/89/53.50 173
174 H. Honda et al.
r
2000
(11)
~
(1a )
!
)
Test
................
fl~I--~
f (2)
(16)
{4) II
o
:
, 1o
.,--,( 5 i" l '1 ~ d ~
t-
U
Cooiint::J w c l l e r - - ~
. l 2 3 4 5 6 7 8 9
i
am
I
I
.
Boiler Superheater Test tube Condensate receiver Circulation pump Strainer Needle valve Cooling water tank Cooling water pump
.
. lO II 12 13 14 15 16 17 18
.
.
.l
Rotameter Pressure gage Sight glasses Vacuum pump Chiller Heater Pressure taps Cooling water pump A u x i l i a r y condenser
Figure 1. Schematic diagram of experimental apparatus. and 29.9 mm. The 2 m long test section is subdivided into eight 250 mm long subsections, which are called the 1st, 2nd . . . . . and 8th subsections from the vapor inlet. The test sections with Di = 24.8 and 27.2 mm are provided with three sight glasses having almost the same Di as the outer tube at the 2nd, 5th, and 8th subsections.
Diameter at fin t l p Diameter at fin root Crest pitch Fin height Wire fin diameter Fin pitch Tube thickness Corrugation pitch Internal ridge height
13o
~1
[Outer tube I Static mixer(Right hand helix) I Inner t u b e l
(12)
"0 0
~-
DO 20.6 Dr 19.1 Pc 1.8 hf O. 8 df O. 3 pf 0.48 0.95 Per 7.0 hcr O. 3
Figure 2. Details of inner tube.
mm mm mm mm mm mm mm mm mm
-i
U .........
, ...................................
T-~L J ' ~ , ~ T ~
~
I •
'"
C~in9
i "°''
........... ...................... 7 ........
C f .~stantan.. wire(l~StaticO.127)pressurehoieCOpper(61 )w re 1~0.37 i
Figure 3. Details of subsection. Details of a subsection are shown schematically in Fig. 3. Nine T-type thermocouples for measuring the cooling water temperature at the inlet and exit of each subsection are inserted in the inner tube. These thermocouples consist of a 0.3 mm diameter enamel-coated copper wire fixed at the tube axis and nine 0.127 mm diameter Teflon-coated constantan wires soldered to the copper wire at 250 mm intervals. A 130 mm long static mixer consisting of a couple of right- and left-hand helices made of polyvinyl chloride (PVC) is inserted in the inner tube at the middle of each subsection. The inner tube is electrically insulated from the outer tube to measure the average wall temperature of each subsection by resistance thermometry. The inner tube (about 0.08 milliohm) and a standard resistor (1 milliohm) are connected in series by a lead wire to a d.c. power supply, and a constant current of 40 A is passed through the circuit. Nine voltage taps (PVC-coated 0.2 rnm diameter copper wire) are soldered to the wire fins at the same longitudinal positions as the cooling water thermocoupies. These copper wires are taken out through small holes on the outer tube. Nine pressure taps with 1 mm diameter holes are drilled at the bottom of the outer tube at the same longitudinal positions as the voltage taps.The adjoining pressure taps are connected to inverse U-tube manometers, reading to 1 ram, to measure the difference in condensate level. The vapor pressure at the inlet of the test section P0 is measured by a precision Bourdon tube reading to 102 Pa gauge. The mixed mean temperature of the cooling water at the test section exit and the refrigerant temperatures at the test section inlet and in the strainer are measured by 1 mm diameter K-type sheathed thermocouples. The local refrigerant temperature at the midpoint of each subsection is also measured by the same type of thermocouple for the test section with outer tube of 29.9 mm I.D. - The outputs of all thermocouples and the voltage drops of all subsections and the standard resistor were read to 1 #V using a data acquisition system. To avoid the effect of parasitic voltage, readings of the voltage drops were repeated by reversing the d.c. current, and the average values of the two measurements were adopted as the experimental data. Experiments were conducted using R-11 and R-113 as test fluids. The test fluid was held near dry saturation at the test section inlet. The exit quality was less than 0.1. Table 1 shows the ranges of experimental conditions, where T~0 is the vapor temperature at the inlet and G is the mass velocity of the test fluid. The T~o value agreed within 0.1 K with the saturation temperature Ts corresponding to the vapor pressure at the inlet. The G value was calculated from the heat balance of the
Condensation of Refrigerants R-I 1 and R-113 Table 1. Experimental Conditions
Test Fluid Inner Diameter Mass Vapor Temperature Test of Outer Tube, Velocity, at Test Fluid Inlet. Fluid Dj (mm) G [kg/(m 2 s)] (K) R-11
24.8 27.2
R-I13
29.9
51-282 57-301 52-201 49-260
313-319 313-318 312-314 324-327
electrically heated boiler as
4 (Qb-Qt~ 7r(D i - D r)
(1)
where Qb is the heat input to the boiler, Q1 is the heat loss from the boiler and the piping up to the vapor inlet, and h0 and h~t are the enthalpies of the test fluid at the vapor inlet (assumed to be dry saturated) and at the strainer, respectively. The Q~ value was estimated by assuming one-dimensional conduction through the layer of fiberglass insulation. The Ql/Qb ratio was less than 2 %. The local heat flux q. and the local heat transfer coefficient ct. are defined on the basis of the nominal surface area as
q.= Q/~rDr AZ,
t~. = q . / ( T s - Tw)
(2)
>
175
where Q is the heat transfer rate of a subsection calculated from the flow rate and enthalpy change of the cooling water, Az = 250 mm, Ts is the saturation temperature corresponding to the local pressure, and T , is the wall temperature at the fin root. The Tw value was obtained from the measured local wall temperature, making a small correction (less than 0.15 K) for the radial conduction. The mixed mean temperature of the cooling water Tom at each subsection (except for the inlet) was estimated from the measured temperatures Tc and Tw as T~,, = Tc + a(Tw - Tc), where a ( = 0 . 0 5 ) was calculated from the measured values of T~m, To, and T~ at the exit. The local quality of the test fluid X was obtained from the heat balance. The uncertainty in the measured o~, value is estimated to be + 13 %. In the data reduction the condensate properties were evaluated at the reference temperature T, + 0.3(Ts - Tw). EXPERIMENTAL RESULT AND DISCUSSION
Flow Pattern Figure 4 compares the flow patterns observed for R-11 with Di = 24.8 mm at O = 72.0 and 282 kg/(m 2 s). In Figs. 4a and 4b for G = 72.0 kg/(m 2 s), the inner tube surface can be observed clearly. In Figs. 4c and 4d for G = 282 kg/(m 2 s), on the other hand, the inner tube surface is obscured by the wavy condensate film on the sight glass and the entrained droplets
Flow direction
(a) X = 0.87
(b) X = 0.17 G = 72.0 kg/m2s
(a) X = 0.81
(b) X = 0.II G = 282 kg/m2s
F ~ u ~ 4. Comparison of flow patterns, R-11, Di = ~ . 8 nun. (a) X = 0.87, O = 72.0 ~ / ( m 2 s); (b) X = 0.17, O = 72.0 kg/(m 2 s); (c) X = 0.81, G = 282 kg/(m 2 s); (d) X = 0.11, G = 282 kg/(m 2 s).
176
H. Honda et al.
(a) X = 0 . 8 7
(b) X = 0 . 1 7
Figure 5. Close-ups of condensate on inner tube surface, R-11, G = 72.0 kg/(m 2 s), Di = 24.8 ram. (a) X = 0.87; (b) X = 0.17. flowing through the annulus. The condensate film at the bottom of the outer tube becomes thicker as condensation proceeds. In Figs. 4b and 4 d for the 8th subsection, the lower half of the annulus is f'dled with condensate. Figure 5 shows close-ups of the inner tube surface corresponding to Figs. 4a and 4b. In Fig. 5a for X = 0.87, the condensate is retained between some of the wire fins and the tube surface. In Fig. 5b for X = 0.17, on the other hand, the space between the wire fins and the tube surface is completely filled with the condensate,
o. --
(a) Rll.
oo
Di =24.8
mm
_~
G = 78.2 kg/m's Po = 1.7 3 b a r
t -'~
0.2
1 150
Axial Distribution of Measured Quantities
-
- 5 0 - -
Or--
-
-
IIEI-
- ~j
T=
rw
-
~r-
Figure 6 compares the axial distributions of c~., q , , Ts, Tw, Tc, the wetness fraction (1 - X ) , and the static pressure drop (P0 - P ) for R-11 with Di = 24.8 m m a t G = 78.2 and 182 kg/ (m 2 s). The static pressure decreases sharply near the test section inlet and tends to a constant value near the exit. This is due to the decrease in the vapor velocity along z/D,. The static pressure drop for G = 182 kg/(m 2 s) is about five times as large as that for G = 78.2 kg/(m 2 s). The condensation temperature difference (Ts - Tw) increases monotonically with the distance z for both cases. In Fig. 6a for G = 78.2 kg/ (m 2 s), the c~n value decreases near the inlet, takes almost a constant value at the 2nd to 6th subsections, and then decreases sharply at the 7th and 8th subsections. In Fig. 6b for G = 182 kg/(m 2 s), the an value decreases monotonically with increasing z. These results for different G suggest that the effect of vapor velocity on the value of con is not significant at the low-G regime. A sharp decrease of t~, at the downstream subsections is due to the submergence of the inner tube surface in the stratified condensate at the lower half of the annulus.
O I
"
I,,-
T¢
"
d
50
z I Dr
(b) R l l .
Di=24.8
G =182 -
100
mm
kg/m2s
.
".o
a. i
o~
1 - 50
~5o .? 3o L-. (,J
Pressure D r o p The static pressure gradient is divided into frictional and momentum components as
-if-_ 7-,-~ 0--
C
50
z/O r
100
Figure 6. Axial distributions of measured quantities.
1
Condensation of Refrigerants R- 11 and R- 113 The momentum pressure gradient can be written as
.... '
(dP) M= G2~d ( p-~ X2+ ~(~--~ (1 - X ) 2 X ~ /
-(a) Rll,
(4)
• e •
where/~ is the void fraction. The value of/~ is estimated using the Fauske equation [7]
''
Di = 2 4 . 8 m m G kg/m2s 78.2 • 182 91.3 © 200 112 • 250
....
I
'
'
'
'
....
177
I
/ Equotion(lO) \ \
/
~'l~&
i
1
L
, ,~,,~
\P'/
....
7
An attempt is made to correlate the frictional pressure gradient (dP/dz)F in terms of the Lockhart-Martinelli parameters [8]
{ (dP/dz)F'~
5
-(b)
,n
1
i
"""'l
,I
I
I
, ....
',
', , . . . .
t
.
1.
•
1
Equotion(lO)
/ i /
o 2sl
-
6
5
i i ' ' ' ' .... D= = 2 9 . 9 m m
o
G kglm2s 52.5 • 1~3
,
75.8
,, 170
e •
94.0 123
• 201
/ / -
\ ~
t
I '
1
i ,
,
, I ....
.J _
" ~ Equation(10)
/
_
•/.~"
\ \• I~e
e~-~
(7)
/,o=2(Di-D~)
f = 0.34 Re -1/4
=,,I
7
where f is the friction factor and u is the velocity. The f value was obtained experimentally using a superheated R-11 vapor. The results are shown in Fig. 7 on the coordinates of f versus Re = u(Di - D~)/v, along with the correlation for a smooth annulus [9]
,
.... I
5
.
,
, I , , , ,l
,
, ' I .... I
=
.
,
, I,
, ,,[
, i .... I
.(d) R113. Di = 2 9 . 9 m m
"
G kglm2s 49.4 , ~70 88,I " 200
.
9,7
o
~->
/ /
Equation(lO) \
.28o
• \ ~ . -
(8)
The present data are about four times as large as the smooth annulus value. The data are correlated by the expressions f = 0 . 8 Re -0.2
.
(6)
(fpu2)l't'
",
G kglm2s 82.4• 203
.... i (c) R l l ,
(dP/dz)t and (dP/dz)~
"
DL = 2 7 . 2 m m
- o ,~,1
7 5
X,, = ( (dP/dz)l "~1/2 (dP/dz)~ /
(dP)
R~l e
and
The single-phase pressure gradients are obtained from
',
0.7
.... 5
}
,
10-2
,
, I ,,J,{ 5
, 10 -1
,
, I .... 5
[ 100
2
Xmt
for R e < 8 x 10 a
8. Frictional pressure gradient data correlated in terms of the Lockhart-Martinelli parameters. Figure
and f=0.084
for Re>_8x 104
(9)
Figure 8 shows the frictional pressure gradient data for the
0.2
'
'
'
I
''"1
'
'
Equation
~ . = 1 + 1.8XtO'9
(9)
0.1
0.0
Oi
Heat Transfer mm,
::t: t
'4--
t
7. V a r i a t i o n superheated v a p o r .
Figure 9 shows the local heat transfer data for Di = 29.9 mm plotted on a,lot,~ versus X, coordinates. The single-phase liquid heat transfer coefficient based on the total flow O~,L is obtained from C=,L= 0.023 Re °'s Pr °'4 XJ(Di-D,)
104
Figure
(10)
to a mean absolute deviation of 7.3%.
5
0.01
condensing flows plotted on Ov versus X, coordinates. The present data are slightly affected by G and Di except the region X, < 0.01 for Di = 24.8 and 27.2 mm, where ~, < 1. Both the R-11 and R-113 data are correlated by the expression
5 of friction
105 factor w i t h
Re Reynolds
5 number;
(11)
where ReL = G(Di - D,)/#l. Also shown in Fig. 9 are the correlations for horizontal annuli with smooth and corrugated inner tubes [4]. The present data are 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes, with the t~,/ot.~, ratio decreasing as G increases. An attempt is made to develop a correlation for the local heat
178 H. Honda et al. 500
,,,[
,
(o) R l l , 200
,
,
i
~ ,,,
I
~
O
o
0
O
100
e
%
O
t
e
o
®
:
o
52.5
•
143
75.8
o
170
94.0
© 20l
o
R
10
2
e
e
o
®
oo
e
®
~
iTil~ll,rlllll,lllr:llllf*ll,i
®
C o r r u g o t e d inner
S m o o t h inner t u b e
" Tube
2
5
10-~
2
5
100
' '''l
,
(b) R113 o
200
,
' I ,
,j
'
O
O e
o
G
O
~
kg/m2s
49.4
•
170
e
85.1
•
200
e
94.7
•
260
®
o
o
o
o o
5O
-... •
.
.
e
.
I
dg/dX = d 38/d~ 3= 0
%
Hoshizume
20
•
•
./.
S m o o t h inner t u b e I
,
10.2
2
,
,
I
5
....
~
I
,
10-I
2
,
,
d~/d~
= t a n E,
~ I
5
....
I
100
2
Xet
9. Variation of tx./c~.L with X . and G.
Figure
transfer coefficient. First we consider the low-G regime. Figure 10 shows the physical model of film condensation on the wire f'm and coordinates. The condensate formed on the fin surface is driven by the combined gravity and surface tension forces to the fin root. Then the condensate flows circumferentially down the tube and falls off the tube bottom. The space between the wire fin and the tube surface is filled with condensate. The mathematical formulation of the condensate flow (assumed to be laminar) is basically the same as that for a horizontal low finned tube [10]. The basic equation for the condensate fdm thickness 8 in the thin-film region is written as
G_.d~3
sin 0 -~+cos_
3
~
0
_ ( 8 3 s i n ~b)
=~
at ~ = 0
(14)
and
R2~'~"~.~----.~CC o r r u go ted 7nner 10
(13)
The dimensionless parameters Gd and Sa represent the effects of gravity and surface tension, respectively. Comparison of the Gd and Sa values for the present experimental conditions reveals that Od/S d < 0.08. T h U S , the first term on the lefthand side of Eq. (t2) can be neglected. The boundary conditions are
150
o
5
section
[(1 + 28) 2 + (dg/d:~) 2] 3/2 ?= [2 + 8 (1 + ~)8 + 4(dg/dX) 2 - (1 + 28)(d28/dX2)]
' ) ' '''l
o
o
O o
....
cross
surface. The r value is written as
'
=29.9mm
Di o
O
100
Fin
2
Xtl
500
cross section
surface
F i g u r e 10. Physical model of film condensation on the wire fin and coordinates.
10-2
5
Tube
~ ~ . ~ . -
i )11I
5
i
o
Hoshizume
2O
Is~ o n d e n s a t
123
O O
• •
I
o
O
d
, ~,,
kg/m2s
e
:
i G
e o
o
~ so ff
,
,
Di = 2 9 . 9 m m
02)
where X = xldy, y = y/dy, ~ = 6/d/, Gd = pah/sd~/Xtvt(T~ - T:), Sd = oh:sd//Xtvt(T, Tf), ¢ = r/d/, Tf istbe fm surface temperature, and r is the radius of curvature of the film
f = - ?b
at ,~ = , ~ b
(15)
The values of e, fo, and .% are related by a geometric condition. These values depend on the local conditions and are not known a priori. However, since the object of the present analysis is to obtain a rough estimate of the heat transfer rate, the numerical solutions of F,q. (12) (with Ga = 0) were obtained for e = x/12-x/6 and several values of-rb using a finite difference scheme described in Ref. 10. The average heat transfer coefficient txf and the average Nusselt number Nuf of the fin surface are defined based on the actual surface area as
2xt
,~b 1
or:d:
(16)
Figure 11 shows the measured and calculated values of Nu/for 29.9 mm plotted against Sa. The procedure for calculating or: from the experimental data is given in the Appendix. The data for R-11 and R-113 at large Sa (which
Di -
correspond to the upstream subsections at small G) agree well with the numerical results for ,% = x/6 and 5x/12, respectively. A similar tendency was observed for the tubes with other Di. The R-11 and R-113 data at large Sa are respectively correlated by the expressions NUl= 0.485 T M
for R- 11
Nuf= 0.565 °'25
for R-113
and (17)
The difference between the R- 11 and R- 113 results is related to the difference in the flooding angle ~y between the two fluids.
Condensation of Refrigerants R-11 and R-113
200
i
1 ,
0)Rll,
,
,
. . . .
I
G
I
Di=29.9mm
100 Numericol result
o
52.5
• e e
75.8 94.0 123
i
I
I
'
p/ Outer
i
'il
kglm2s • 143[ e 170-.q • 201
tube
> /,
0
10 7
'rrl8 I l
'
l(b)R113,
I
J
I
'
'
I
'
D~ 29.9
I
I I I
I
'
''
'
I
mm
100 Io
1
~110 ~'120 I
I
i
J
i
a
i
'
'
I
'
'
'
o
G kglm2s 49 4 e 170
~
85.1
:
e
-
•
94.7 150
-
Numerical result
179
65
® 2oo•
260
0
Condensate
f i l m on i n n e r "tube
Figure 12. Model of condensate flow on the tube surface. 10
entrainment, and assuming laminar condensate flow, the condensate fdm thickness ~ is determined by the vapor shear Fv
=I
7 106
I
!
I
5
I
i l l i
I
i
'
107
'
5
'
'''"
= f , pou2/2 as 108
$d
=\
Figure 11. Variation of Nuf with Sd and G. The expression of ~bffor a low finned tube [10] can be extended to the present case as
~kf=cos-'(ptg(p~-_df)Do
l)
(18)
Thus, the ~kfvalues for R-11 and R-113 are calculated as 1.31 and 1.51 rad, respectively. The ratio of these values is roughly in accord with the ratio of Nuy for the two fluids at large Sd. Next we consider the large-G regime, where the condensate flow is controlled by the vapor shear. In this regime the condensation process on the fm surface is supposed to be similar to the forced convection f'dm condensation on a cylinder normal to the vapor flow. The expression of the average Nusselt number for the latter case is written as [11]
Nuf= b(1 + A ) i/3 (u~df/~,t) i/2
(19)
where b = 0.9, A = ltlhfs(pop.v/plltl)l/2/hl(Ts - Tf). Equation (19) may be applied to the present case by treating b as a parameter accounting for the lower effective vapor velocity as compared to uo and the decrease of effective f'm surface area due to submergence in the condensate flow on the tube surface. The effect of condensate stratification at the lower part of the annulus will be considered separately. Figure 12 shows the model of condensate flow on the tube surface. Neglecting the effects of gravity and condensate
/
=
Re*
Z
(20)
wherefi is the interfaciai friction factor and Re* = GDr(1 X){ (Di/Dr) 2 - 1 }/#t. The actual condensate flow is affected by the circumferential gravity force F s = Ptg~ sin ~, and tends to the gravity-drained flow solution with increasing z. The Fg/F~ ratio is written as
Fg = ptg8 sin ~b
Fo f, pvu~/2 2
~
\
~
,] sin ~
(21)
Neglecting the variation of ~ , the dimensionless parameter determining/~ is derived as F = ( P / ~ I / 2 Re* i/6 (g~'t)'/3
(22)
\po/ uo Figure 13 shows the Nu:/Nu/c ratio plotted against F, where Nuf denotes the experimental data and Nufc is the calculated value from Eq. (19) with b = 1. As is evident from Eq. (22), a small F value corresponds to the upstream subsections at large G. The Nuy/Nuyc ratio for a given G first decreases, then takes a minimum value, and then increases again with increasing F. This indicates the transition from the vapor shear controlled regime to the combined gravity and surface tension controlled regime. Comparison of the results for different D, reveals that the data at F < 0.05 are higher for smaller Di. This may be attributed to the increase in effective vapor velocity with
180 H. Honda et al.
1.0
I(
, i .... o)RI'I,
0.5
I
'
Di = 24.8mrn
G e e
(23)
"-'~quation XX~e V
Z
i
1 12
•
146
]
l
2.0
i r.
kglm2s 182
78.2 91.3
o
v
e
1.0
.... ~
i
I
~
'
~
~
l
'
,
i
~
T ....
i
I
'
~
,
~
,
r ....
I
- -
Di = 2 9 . 9 t u r n
__
v_
u
200
e 250 • 282
2:.2 0.5
Z
Z @
0.1
O Ir
.
.
.
i
i
I
.
l i i i l
•
I ....
(b)R11,
Di = 27.2mm •
(23)
Equution
I i
I
I l
I ,
I I , i ,
o
e ~ e leon®,..__Z•,..i>i ®o 0.1 1.0
--
•
--L
I
I
i
I
i
i i i r
I
I
I
l
I
I i
'
'
'
I
' '''I
I
I
I
I
I
'
(c)R11,
0.5
G
Di =29.9 mm
Equation (23)
O
52.5
e
75.8 94.0 123
o
Z
0
=T
o
e
kg/m2s • 143 {i 170 © 201
I I.
--
O0
oo
oQ
Z
1.0
I
I i Jill
I
(d)R113, Di =29.9 mrn
0.5
\/
@~ •
Z
0.1
(23)
Equution
o
:
94.7
,
10 -2
i
~
I
S
,
,
O
O O
•
O
I
I
I
i
i
i
I
i
i w l_
kglm2s e 170 © 200 • 260
G 49.4 85.1
o
Z
5
10-2
O
i
10"1
I
i
l
5
F Figare 13. Variation of Nuf/Nuyc with F and G.
100
increasing relative fin height h'y = hf/(Di Dr). The data in the vapor shear controlled regime are correlated by the expression -
Nu/=0.12
1 +(5 X 10 -7)
I+(5X
10-I
5
100
(24)
where Nuys and Nufv denote the Nuf values given by Eqs. (17) and (23), respectively, and c is the parameter accounting for the effect of submergence of the inner tube in the stratified condensate at the downstream subsections. By analogy with the stratified two-phase flow in a horizontal tube [12], the major parameter determining the condensate level is considered to be Xtt. Figure 14 shows the Nuf/Nuyc ratio plotted against Xtt, where Nuy denotes the experimental data and Nufc the calculated value from Eq. (24) with c = 1. It is seen from Fig. 14 that the Nuf/Nufc ratio is approximately unity at X , ,~ 0.08 and deviates toward a lower value with increasing X , at Xtt '~ 0.08. It is also noted that the Nuf/Nuyc ratio at Xtt ~ 0.08 is somewhat higher for larger Dr. All data are correlated by the expression
105)h'~] Ii4
1 + IO0(2Dr/D#- l).X~t
(25)
to a mean absolute deviation of 16.2%. Practical Significance
$o •
,,I
5
Xtt
Nuy= I
24~8
~o
0.1 I
EquQtion(25) Rl13
Nuf= c(Nu}s + Nu~o)1/4
o
.•
•
J
equation is assumed to be
Z Z
29.9
Fig-re 14. Variation of Nuy/NUycwith Xtt.
G kg/m2s 8 2.4© 203 144 e 251 165 • 301
e
0.5
I i
I
299
=
0.1
e
O I
1.0
0~
--
m
(23)
The experimental as a whole data may be expressed by the combination of Eqs. (17) and (23). Following the case of a horizontal cylinder [11], the functional form of the empirical
Local pressure gradient and heat transfer data were obtained for condensation of R-11 and R-113 in horizontal annuli with a wire-finned inner tube. The dimensionless correlations for the local frictional pressure gradient and the local heat transfer coefficient, Eqs. (10) and (25), can be used to accurately predict the pressure drop and heat transfer characteristics of double-tube condensers of this type. The analytical model of film condensation in a horizontal annulus developed in this study can be extended to double-tube condensers with different types of enhanced inner tubes. CONCLUSIONS 1. The local frictional pressure gradient was correlated fairly well using the Lockhart-Martinelli parameters. The effect of the annular space was small except for the region near the upstream end. Overall, the data were correlated by Eq. (10) to a mean absolute deviation of 7.3%. 2. The local heat transfer coefficient was 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes. The heat transfer enhancement ratio decreased with increasing mass velocity. 3. The heat transfer processes on the fin surface at the low and high vapor velocities were controlled by the surface tension
Condensation of Refrigerants R-11 and R-113
from the Cavallini et al equation [5]), respectively. Then, q,r and q,/are obtained from Eqs. (A.3) and (A. 1), respectively. Finally, c~y is obtained from Eq. (A.2). According to the calculated results, the q,r/q, ratio was less than 0.11.
Ts Y
II VI I
2b
-~
A - A
T.,
i
\ Solder
Fi
i\.*X//\.
/,,
'
/-+
L,
A
1
~,s
r,,,
n /~
J-
~'¢ ~
i x
.I
J
Z
2a s = 0 . 6 mm Af = 0 . 0 ] mm @ = O, 87 t a d
2b s = 0 . 2 5 Ar = 0.02
mm
NOMENCLATURE Di Do Dr df F Fg Fo f G Gd
mm
Figure 15. Details of wire fin soldered on the inner tube. and the vapor shear forces, respectively. The data were correlated by Eq. (25) to a mean absolute deviation of 16.2%. APPENDIX. Determination of %, from the E x p e r i m e n t a l D a t a Figure 15 shows the details of the wire fins soldered on the inner tube. The base area of the soldered portion is approximated by an ellipse with semimajor axis as and semiminor axis bs. The thicknesses of the solder layer on the tube surface and at the fin base are A, and A/, respectively. The local heat flux qn can be divided into two parts, qn = qnf+ q,r
(A.1)
where q,¢ and q,, are the components of heat transfer rates on the fin surface and the tube surface, respectively, q,,f is related to (~y by
g h hf hfg Nuf P Pr q Re Re* Sd T u X Xtt x, y z
inner diameter of outer tube, m diameter at fin tip, m diameter at fin root, m wire fin diameter, m parameter, Eq. (22), dimensionless circumferential gravity force, N/m 2 axial vapor shear, N/m 2 friction factor, dimensionless refrigerant mass velocity, kg/(m 2 s) parameter
(A.2)
(A.3)
where T~, is the tube surface temperature and A f = wa~bs/ PfPc is the ratio of fin base area to tube surface area. Since the solder layer is very thin, it is sufficiently accurate to assume
T~r=Tw.
c~ /~ 6
Sub~eripts ^.4 Xl/4
+ tzrv!
Tf)],
local heat transfer coefficient, W / ( m 2 K) void fraction, dimensionless condensate film thickness, m fin efficiency, dimensionless )~ thermal conductivity, W/(m K) dynamic viscosity, kg/(s m) v kinematic viscosity, m2/s o density, kg/m 3 o surface tension, N/m ~o Lockhart-Martinelli parameter, Eq. (6), dimensionless i ~k angle measured from tube top, rad
The ar value is estimated by -4 Ot r - - ( 0 / r g
-
Greek Symbols
where ;/ = tanh ( l / ~ ) / ( i f ~ ) is the fin efficiency, // is the fin length per fin crest pitch, T,, is the interfacial temperature between the tube and the solder layer, and Rs is the thermal resistance of solder layer at the fin base. q,, is related to the heat transfer coefficient at the tube surface by q , , = ctr( T s - Tw,)(1 - A f )
[ = plghfgd3f/)~lVl(Ts
dimensionless gravitational acceleration, m/s 2 specific enthalpy of test fluid, J/kg fin height, m specific enthalpy of evaporation, J/kg Nusselt number ( = ctfdf/Xt), dimensionless static pressure, N/m 2 Prandtl number, dimensionless local heat flux, W / m E Reynolds number [ = u(Di - D,)/v], dimensionless parameter [=GDr(I - X ) { ( D i / D r ) 2 - 1}//~t], dimensionless parameter [ = ohfsdf/Xtvt( Ts - Ty)], dimensionless temperature, K velocity, m/s mass quality, dimensionless Lockhart-Martinelli parameter, Eq. (6), dimensionless coordinates measured along fin surface, m distance measured from test section inlet, m
T,-Tw qnf = P f P c (1 / 7rdflfotf~ 1 + Rs )
181
(A.4)
where a,~ and u,~ are the heat transfer coefficients in the gravity-controlled regime (calculated from the Nusselt equation [13]) and the vapor shear controlled regime (calculated
c F
coolant friction component
f
fin
fc
calculated value
182
H. Honda et al.
fs fv L 1 M n
surface tension controlled r e g i m e v a p o r shear controlled r e g i m e based o n total flow assumed liquid liquid momentum component based o n n o m i n a l surface area (surface area o f a smooth tube with 19,)
s
saturation
o w
vapor tube wall
0
test section inlet
6.
7.
8.
9.
REFERENCES 1. Borchmann, J., Zur Kondensation von R l l und Wasserdampf bei hoben Dampfgeschwindigkeiten, Kdltetechnik-Klimat., 19, 208213, 1967. 2. Sensyu, T., Murata, K., and Sega, H., Heat Transfer Coefficients for Condensation of R-22 and Performance of Condensers (in Japanese), Refrigeration, 49, 599-605, 1974. 3. Miropolsky, Z. L., Shneerova, R. I., and Treputnev, V. V., The Influence of Steam Flowing and Condensing in a Duct on Heat Transfer to Liquid Film, Proc. 6th Int. Heat Transf. Conf., Toronto, Vol. 2, pp. 431--436, 1978. 4. Hashizume, K., Refrigerant Two-Phase Flow Heat Transfer in Double Tube Condensers (in Japanese), Proc. 18th Nat. Heat Transfer Syrup. of Japan, Sendai, pp. 544-546, 1981. 5. Cavallini, A., Frizzerin, S., and Rossetto, L., Condensation of
10.
11.
12.
13.
Refrigerants Inside Annuli, Proc. 7th Int. Heat Transfer Conf., Munich, Vol. 5, pp. 45-51, 1982. Sakon, Y., Sakitani, K., and Uemura, S., Performance of Double Tube Condensers with SpiRted Fin Tubes (in Japanese), Proc. 17th Air-Conditioning and Refrigeration Joint Meeting, Tokyo, pp. 1-4, 1983. Fauske, H. K., Critical Two-Phase Steam Water Flows, Proc. Heat Transfer and Fluid Dynamics Inst., Vol. 79, Stanford University Press, 1961. Lockhart, R. W., and Martinelli, R. C. Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes, Chem. Eng. Prog., 45, 39--48, 1949. Kays, W. M., and Pekins, H. C., Forced Convection, Internal Flows in Ducts, in Handbook of Heat Transfer Fundamentals, W. M. Rohsenow, J. P. HartneU, and E. N, Ganic, Eds., pp. 7.1-7.180, McGraw-Hill, New York, 1985. Honda, H., and Nozu, S., A Prediction Method for Heat Transfer During Film Condensation on Horizontal Low Integral-Fin Tubes, J. Heat Transfer, 109, 218-225, 1987. Fujii, T., Uehara, H., and Kurata Ch., Laminar Filmwise Condensation of Flowing Vapor on a Horizontal Cylinder, Int. J. Heat Mass Transfer, 15, 235-246, 1972. Taitel, Y., and Dukler, A. E., A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow, AIChE J., 22, 47-55, 1976. Nusselt, W., Die Oberfliichenkondensation des Wasserdampfes, Z. Ver. Deut. Ing., 60, 541-546, 1916.
Received April 15, 1988; revised November 10, 1988.
Shigeru Nozu, Yoichi Matsuoka, Tohru Aoyama Department of Mechanical Engineering, Okayama University, Okayama, Japan
Haruo Nakata Daikin Industries, Ltd., Osaka, Japan
IIExperiments were performed to study the flow and heat transfer characteristics during condensation of R-I 1 and R-113 in the annuli of horizontal doubletube test condensers. The condensers are made up of a 19.1 mm O.D. corrugated inner tube with wire fins soldered on the outer surface and three outer smooth tubes of 24.8, 27.2, and 29.9 mm I.D. The mass velocity of the test fluids ranged from 50 to 300 kg/(m 2 s), and the condensation temperature difference ranged from 0.7 to 20 K. The frictional pressure gradient was correlated fairly well by using the Lockhart-Martinelli parameters. The local heat transfer coefficient was 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes. An empirical equation for the local heat transfer coefficient was developed, in which the dimensionless parameters based on the surface tension controlled flow and the vapor shear controlled flow models were introduced for the low and high vapor velocity regimes, respectively.
Keywords: condensation, heat exchangers, heat transfer augmentation
INTRODUCTION Water-cooled horizontal double-tube condensers are commonly used in air-conditioning systems of relatively small size. Condensation may occur in the annulus or in the inner tube depending on the conditions. For systems using fluorocarbon refrigerants, coiled condensers with enhanced inner tubes, with vapor flowing through the annular passage, are commonly adopted because of their high performance and compact size. Several experimental studies have been reported of condensation in annuli with smooth and/or enhanced inner tubes [16]. Miropolsky et al [3] and Cavallini et al [5] showed that empirical equations for forced convection condensation inside smooth tubes are also applicable to vertical annuli consisting of smooth inner and outer tubes. Hashizume [4] compared the annulus-side and tube-side condensation data for horizontal double tubes with smooth and corrugated inner tubes. The heat transfer enhancement of the horizontal annulus using the corrugated inner tube was less than 25%. The average heat transfer performance of coiled double-tube condensers with various types of enhanced inner tubes have also been reported [6]. Although considerable heat transfer enhancement has been achieved, the range of experimental data is limited and the mechanism of heat transfer enhancement has not been given theoretical consideration. The present study was undertaken to obtain a better understanding of the flow characteristics and heat transfer during condensation of fluorocarbon refrigerants in horizontal annuli with an enhanced inner tube. The effect of the annular space was studied by using three test condensers with inner
tubes of the same diameters and outer tubes of different diameters.
EXPERIMENTAL APPARATUS AND PROCEDURE The experimental apparatus, shown schematically in Fig. 1, consists of forced circulation loops of test fluid and cooling water. The vapor generated in an electrically heated boiler (1) flows upward via a superheater (2) to a horizontal test section (3). The vapor is condensed almost completely in the test section. The condensate flows downward through a condensate receiver (4), then horizontally through a circulation pump (5), a strainer (6), and a flow control valve (7), and then returns to the boiler. The loop is thermally insulated with fiberglass. The cooling water is pumped from a cooling water tank (8) by a pump (9) to the test section via a rotameter (10). After exchanging heat with the test fluid, the cooling water returns to the tank. The cooling water temperature is regulated by the use of a chilling unit (14) and a heater (15). The test section is a horizontal double-tube condenser with the test fluid flowing through the annular passage. The cooling water flows countercurrently through the inner tube. Three test sections with inner tubes of the same diameters and outer tubes of different diameters are used to study the effect of the annular space. Details of the inner tube are shown in Fig. 2. The inner tube is a corrugated copper tube with copper wire fins soldered onto the outer surface (wire-finned tube). The outer surface area is 3.04 times that of an equivalent smooth tube. The outer tubes are smooth tubes with insider diameters D; of 24.8, 27.2,
Address correspondence to Professor Hiroshi Honda, Institute of Advanced Material Study, Kyushu University, Kasuga-shi, Pukuoka 816, Japan.
Experimental Thermal and Fluid Science 1989; 2:173-182
© 1989 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010
0894-1777/89/53.50 173
174 H. Honda et al.
r
2000
(11)
~
(1a )
!
)
Test
................
fl~I--~
f (2)
(16)
{4) II
o
:
, 1o
.,--,( 5 i" l '1 ~ d ~
t-
U
Cooiint::J w c l l e r - - ~
. l 2 3 4 5 6 7 8 9
i
am
I
I
.
Boiler Superheater Test tube Condensate receiver Circulation pump Strainer Needle valve Cooling water tank Cooling water pump
.
. lO II 12 13 14 15 16 17 18
.
.
.l
Rotameter Pressure gage Sight glasses Vacuum pump Chiller Heater Pressure taps Cooling water pump A u x i l i a r y condenser
Figure 1. Schematic diagram of experimental apparatus. and 29.9 mm. The 2 m long test section is subdivided into eight 250 mm long subsections, which are called the 1st, 2nd . . . . . and 8th subsections from the vapor inlet. The test sections with Di = 24.8 and 27.2 mm are provided with three sight glasses having almost the same Di as the outer tube at the 2nd, 5th, and 8th subsections.
Diameter at fin t l p Diameter at fin root Crest pitch Fin height Wire fin diameter Fin pitch Tube thickness Corrugation pitch Internal ridge height
13o
~1
[Outer tube I Static mixer(Right hand helix) I Inner t u b e l
(12)
"0 0
~-
DO 20.6 Dr 19.1 Pc 1.8 hf O. 8 df O. 3 pf 0.48 0.95 Per 7.0 hcr O. 3
Figure 2. Details of inner tube.
mm mm mm mm mm mm mm mm mm
-i
U .........
, ...................................
T-~L J ' ~ , ~ T ~
~
I •
'"
C~in9
i "°''
........... ...................... 7 ........
C f .~stantan.. wire(l~StaticO.127)pressurehoieCOpper(61 )w re 1~0.37 i
Figure 3. Details of subsection. Details of a subsection are shown schematically in Fig. 3. Nine T-type thermocouples for measuring the cooling water temperature at the inlet and exit of each subsection are inserted in the inner tube. These thermocouples consist of a 0.3 mm diameter enamel-coated copper wire fixed at the tube axis and nine 0.127 mm diameter Teflon-coated constantan wires soldered to the copper wire at 250 mm intervals. A 130 mm long static mixer consisting of a couple of right- and left-hand helices made of polyvinyl chloride (PVC) is inserted in the inner tube at the middle of each subsection. The inner tube is electrically insulated from the outer tube to measure the average wall temperature of each subsection by resistance thermometry. The inner tube (about 0.08 milliohm) and a standard resistor (1 milliohm) are connected in series by a lead wire to a d.c. power supply, and a constant current of 40 A is passed through the circuit. Nine voltage taps (PVC-coated 0.2 rnm diameter copper wire) are soldered to the wire fins at the same longitudinal positions as the cooling water thermocoupies. These copper wires are taken out through small holes on the outer tube. Nine pressure taps with 1 mm diameter holes are drilled at the bottom of the outer tube at the same longitudinal positions as the voltage taps.The adjoining pressure taps are connected to inverse U-tube manometers, reading to 1 ram, to measure the difference in condensate level. The vapor pressure at the inlet of the test section P0 is measured by a precision Bourdon tube reading to 102 Pa gauge. The mixed mean temperature of the cooling water at the test section exit and the refrigerant temperatures at the test section inlet and in the strainer are measured by 1 mm diameter K-type sheathed thermocouples. The local refrigerant temperature at the midpoint of each subsection is also measured by the same type of thermocouple for the test section with outer tube of 29.9 mm I.D. - The outputs of all thermocouples and the voltage drops of all subsections and the standard resistor were read to 1 #V using a data acquisition system. To avoid the effect of parasitic voltage, readings of the voltage drops were repeated by reversing the d.c. current, and the average values of the two measurements were adopted as the experimental data. Experiments were conducted using R-11 and R-113 as test fluids. The test fluid was held near dry saturation at the test section inlet. The exit quality was less than 0.1. Table 1 shows the ranges of experimental conditions, where T~0 is the vapor temperature at the inlet and G is the mass velocity of the test fluid. The T~o value agreed within 0.1 K with the saturation temperature Ts corresponding to the vapor pressure at the inlet. The G value was calculated from the heat balance of the
Condensation of Refrigerants R-I 1 and R-113 Table 1. Experimental Conditions
Test Fluid Inner Diameter Mass Vapor Temperature Test of Outer Tube, Velocity, at Test Fluid Inlet. Fluid Dj (mm) G [kg/(m 2 s)] (K) R-11
24.8 27.2
R-I13
29.9
51-282 57-301 52-201 49-260
313-319 313-318 312-314 324-327
electrically heated boiler as
4 (Qb-Qt~ 7r(D i - D r)
(1)
where Qb is the heat input to the boiler, Q1 is the heat loss from the boiler and the piping up to the vapor inlet, and h0 and h~t are the enthalpies of the test fluid at the vapor inlet (assumed to be dry saturated) and at the strainer, respectively. The Q~ value was estimated by assuming one-dimensional conduction through the layer of fiberglass insulation. The Ql/Qb ratio was less than 2 %. The local heat flux q. and the local heat transfer coefficient ct. are defined on the basis of the nominal surface area as
q.= Q/~rDr AZ,
t~. = q . / ( T s - Tw)
(2)
>
175
where Q is the heat transfer rate of a subsection calculated from the flow rate and enthalpy change of the cooling water, Az = 250 mm, Ts is the saturation temperature corresponding to the local pressure, and T , is the wall temperature at the fin root. The Tw value was obtained from the measured local wall temperature, making a small correction (less than 0.15 K) for the radial conduction. The mixed mean temperature of the cooling water Tom at each subsection (except for the inlet) was estimated from the measured temperatures Tc and Tw as T~,, = Tc + a(Tw - Tc), where a ( = 0 . 0 5 ) was calculated from the measured values of T~m, To, and T~ at the exit. The local quality of the test fluid X was obtained from the heat balance. The uncertainty in the measured o~, value is estimated to be + 13 %. In the data reduction the condensate properties were evaluated at the reference temperature T, + 0.3(Ts - Tw). EXPERIMENTAL RESULT AND DISCUSSION
Flow Pattern Figure 4 compares the flow patterns observed for R-11 with Di = 24.8 mm at O = 72.0 and 282 kg/(m 2 s). In Figs. 4a and 4b for G = 72.0 kg/(m 2 s), the inner tube surface can be observed clearly. In Figs. 4c and 4d for G = 282 kg/(m 2 s), on the other hand, the inner tube surface is obscured by the wavy condensate film on the sight glass and the entrained droplets
Flow direction
(a) X = 0.87
(b) X = 0.17 G = 72.0 kg/m2s
(a) X = 0.81
(b) X = 0.II G = 282 kg/m2s
F ~ u ~ 4. Comparison of flow patterns, R-11, Di = ~ . 8 nun. (a) X = 0.87, O = 72.0 ~ / ( m 2 s); (b) X = 0.17, O = 72.0 kg/(m 2 s); (c) X = 0.81, G = 282 kg/(m 2 s); (d) X = 0.11, G = 282 kg/(m 2 s).
176
H. Honda et al.
(a) X = 0 . 8 7
(b) X = 0 . 1 7
Figure 5. Close-ups of condensate on inner tube surface, R-11, G = 72.0 kg/(m 2 s), Di = 24.8 ram. (a) X = 0.87; (b) X = 0.17. flowing through the annulus. The condensate film at the bottom of the outer tube becomes thicker as condensation proceeds. In Figs. 4b and 4 d for the 8th subsection, the lower half of the annulus is f'dled with condensate. Figure 5 shows close-ups of the inner tube surface corresponding to Figs. 4a and 4b. In Fig. 5a for X = 0.87, the condensate is retained between some of the wire fins and the tube surface. In Fig. 5b for X = 0.17, on the other hand, the space between the wire fins and the tube surface is completely filled with the condensate,
o. --
(a) Rll.
oo
Di =24.8
mm
_~
G = 78.2 kg/m's Po = 1.7 3 b a r
t -'~
0.2
1 150
Axial Distribution of Measured Quantities
-
- 5 0 - -
Or--
-
-
IIEI-
- ~j
T=
rw
-
~r-
Figure 6 compares the axial distributions of c~., q , , Ts, Tw, Tc, the wetness fraction (1 - X ) , and the static pressure drop (P0 - P ) for R-11 with Di = 24.8 m m a t G = 78.2 and 182 kg/ (m 2 s). The static pressure decreases sharply near the test section inlet and tends to a constant value near the exit. This is due to the decrease in the vapor velocity along z/D,. The static pressure drop for G = 182 kg/(m 2 s) is about five times as large as that for G = 78.2 kg/(m 2 s). The condensation temperature difference (Ts - Tw) increases monotonically with the distance z for both cases. In Fig. 6a for G = 78.2 kg/ (m 2 s), the c~n value decreases near the inlet, takes almost a constant value at the 2nd to 6th subsections, and then decreases sharply at the 7th and 8th subsections. In Fig. 6b for G = 182 kg/(m 2 s), the an value decreases monotonically with increasing z. These results for different G suggest that the effect of vapor velocity on the value of con is not significant at the low-G regime. A sharp decrease of t~, at the downstream subsections is due to the submergence of the inner tube surface in the stratified condensate at the lower half of the annulus.
O I
"
I,,-
T¢
"
d
50
z I Dr
(b) R l l .
Di=24.8
G =182 -
100
mm
kg/m2s
.
".o
a. i
o~
1 - 50
~5o .? 3o L-. (,J
Pressure D r o p The static pressure gradient is divided into frictional and momentum components as
-if-_ 7-,-~ 0--
C
50
z/O r
100
Figure 6. Axial distributions of measured quantities.
1
Condensation of Refrigerants R- 11 and R- 113 The momentum pressure gradient can be written as
.... '
(dP) M= G2~d ( p-~ X2+ ~(~--~ (1 - X ) 2 X ~ /
-(a) Rll,
(4)
• e •
where/~ is the void fraction. The value of/~ is estimated using the Fauske equation [7]
''
Di = 2 4 . 8 m m G kg/m2s 78.2 • 182 91.3 © 200 112 • 250
....
I
'
'
'
'
....
177
I
/ Equotion(lO) \ \
/
~'l~&
i
1
L
, ,~,,~
\P'/
....
7
An attempt is made to correlate the frictional pressure gradient (dP/dz)F in terms of the Lockhart-Martinelli parameters [8]
{ (dP/dz)F'~
5
-(b)
,n
1
i
"""'l
,I
I
I
, ....
',
', , . . . .
t
.
1.
•
1
Equotion(lO)
/ i /
o 2sl
-
6
5
i i ' ' ' ' .... D= = 2 9 . 9 m m
o
G kglm2s 52.5 • 1~3
,
75.8
,, 170
e •
94.0 123
• 201
/ / -
\ ~
t
I '
1
i ,
,
, I ....
.J _
" ~ Equation(10)
/
_
•/.~"
\ \• I~e
e~-~
(7)
/,o=2(Di-D~)
f = 0.34 Re -1/4
=,,I
7
where f is the friction factor and u is the velocity. The f value was obtained experimentally using a superheated R-11 vapor. The results are shown in Fig. 7 on the coordinates of f versus Re = u(Di - D~)/v, along with the correlation for a smooth annulus [9]
,
.... I
5
.
,
, I , , , ,l
,
, ' I .... I
=
.
,
, I,
, ,,[
, i .... I
.(d) R113. Di = 2 9 . 9 m m
"
G kglm2s 49.4 , ~70 88,I " 200
.
9,7
o
~->
/ /
Equation(lO) \
.28o
• \ ~ . -
(8)
The present data are about four times as large as the smooth annulus value. The data are correlated by the expressions f = 0 . 8 Re -0.2
.
(6)
(fpu2)l't'
",
G kglm2s 82.4• 203
.... i (c) R l l ,
(dP/dz)t and (dP/dz)~
"
DL = 2 7 . 2 m m
- o ,~,1
7 5
X,, = ( (dP/dz)l "~1/2 (dP/dz)~ /
(dP)
R~l e
and
The single-phase pressure gradients are obtained from
',
0.7
.... 5
}
,
10-2
,
, I ,,J,{ 5
, 10 -1
,
, I .... 5
[ 100
2
Xmt
for R e < 8 x 10 a
8. Frictional pressure gradient data correlated in terms of the Lockhart-Martinelli parameters. Figure
and f=0.084
for Re>_8x 104
(9)
Figure 8 shows the frictional pressure gradient data for the
0.2
'
'
'
I
''"1
'
'
Equation
~ . = 1 + 1.8XtO'9
(9)
0.1
0.0
Oi
Heat Transfer mm,
::t: t
'4--
t
7. V a r i a t i o n superheated v a p o r .
Figure 9 shows the local heat transfer data for Di = 29.9 mm plotted on a,lot,~ versus X, coordinates. The single-phase liquid heat transfer coefficient based on the total flow O~,L is obtained from C=,L= 0.023 Re °'s Pr °'4 XJ(Di-D,)
104
Figure
(10)
to a mean absolute deviation of 7.3%.
5
0.01
condensing flows plotted on Ov versus X, coordinates. The present data are slightly affected by G and Di except the region X, < 0.01 for Di = 24.8 and 27.2 mm, where ~, < 1. Both the R-11 and R-113 data are correlated by the expression
5 of friction
105 factor w i t h
Re Reynolds
5 number;
(11)
where ReL = G(Di - D,)/#l. Also shown in Fig. 9 are the correlations for horizontal annuli with smooth and corrugated inner tubes [4]. The present data are 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes, with the t~,/ot.~, ratio decreasing as G increases. An attempt is made to develop a correlation for the local heat
178 H. Honda et al. 500
,,,[
,
(o) R l l , 200
,
,
i
~ ,,,
I
~
O
o
0
O
100
e
%
O
t
e
o
®
:
o
52.5
•
143
75.8
o
170
94.0
© 20l
o
R
10
2
e
e
o
®
oo
e
®
~
iTil~ll,rlllll,lllr:llllf*ll,i
®
C o r r u g o t e d inner
S m o o t h inner t u b e
" Tube
2
5
10-~
2
5
100
' '''l
,
(b) R113 o
200
,
' I ,
,j
'
O
O e
o
G
O
~
kg/m2s
49.4
•
170
e
85.1
•
200
e
94.7
•
260
®
o
o
o
o o
5O
-... •
.
.
e
.
I
dg/dX = d 38/d~ 3= 0
%
Hoshizume
20
•
•
./.
S m o o t h inner t u b e I
,
10.2
2
,
,
I
5
....
~
I
,
10-I
2
,
,
d~/d~
= t a n E,
~ I
5
....
I
100
2
Xet
9. Variation of tx./c~.L with X . and G.
Figure
transfer coefficient. First we consider the low-G regime. Figure 10 shows the physical model of film condensation on the wire f'm and coordinates. The condensate formed on the fin surface is driven by the combined gravity and surface tension forces to the fin root. Then the condensate flows circumferentially down the tube and falls off the tube bottom. The space between the wire fin and the tube surface is filled with condensate. The mathematical formulation of the condensate flow (assumed to be laminar) is basically the same as that for a horizontal low finned tube [10]. The basic equation for the condensate fdm thickness 8 in the thin-film region is written as
G_.d~3
sin 0 -~+cos_
3
~
0
_ ( 8 3 s i n ~b)
=~
at ~ = 0
(14)
and
R2~'~"~.~----.~CC o r r u go ted 7nner 10
(13)
The dimensionless parameters Gd and Sa represent the effects of gravity and surface tension, respectively. Comparison of the Gd and Sa values for the present experimental conditions reveals that Od/S d < 0.08. T h U S , the first term on the lefthand side of Eq. (t2) can be neglected. The boundary conditions are
150
o
5
section
[(1 + 28) 2 + (dg/d:~) 2] 3/2 ?= [2 + 8 (1 + ~)8 + 4(dg/dX) 2 - (1 + 28)(d28/dX2)]
' ) ' '''l
o
o
O o
....
cross
surface. The r value is written as
'
=29.9mm
Di o
O
100
Fin
2
Xtl
500
cross section
surface
F i g u r e 10. Physical model of film condensation on the wire fin and coordinates.
10-2
5
Tube
~ ~ . ~ . -
i )11I
5
i
o
Hoshizume
2O
Is~ o n d e n s a t
123
O O
• •
I
o
O
d
, ~,,
kg/m2s
e
:
i G
e o
o
~ so ff
,
,
Di = 2 9 . 9 m m
02)
where X = xldy, y = y/dy, ~ = 6/d/, Gd = pah/sd~/Xtvt(T~ - T:), Sd = oh:sd//Xtvt(T, Tf), ¢ = r/d/, Tf istbe fm surface temperature, and r is the radius of curvature of the film
f = - ?b
at ,~ = , ~ b
(15)
The values of e, fo, and .% are related by a geometric condition. These values depend on the local conditions and are not known a priori. However, since the object of the present analysis is to obtain a rough estimate of the heat transfer rate, the numerical solutions of F,q. (12) (with Ga = 0) were obtained for e = x/12-x/6 and several values of-rb using a finite difference scheme described in Ref. 10. The average heat transfer coefficient txf and the average Nusselt number Nuf of the fin surface are defined based on the actual surface area as
2xt
,~b 1
or:d:
(16)
Figure 11 shows the measured and calculated values of Nu/for 29.9 mm plotted against Sa. The procedure for calculating or: from the experimental data is given in the Appendix. The data for R-11 and R-113 at large Sa (which
Di -
correspond to the upstream subsections at small G) agree well with the numerical results for ,% = x/6 and 5x/12, respectively. A similar tendency was observed for the tubes with other Di. The R-11 and R-113 data at large Sa are respectively correlated by the expressions NUl= 0.485 T M
for R- 11
Nuf= 0.565 °'25
for R-113
and (17)
The difference between the R- 11 and R- 113 results is related to the difference in the flooding angle ~y between the two fluids.
Condensation of Refrigerants R-11 and R-113
200
i
1 ,
0)Rll,
,
,
. . . .
I
G
I
Di=29.9mm
100 Numericol result
o
52.5
• e e
75.8 94.0 123
i
I
I
'
p/ Outer
i
'il
kglm2s • 143[ e 170-.q • 201
tube
> /,
0
10 7
'rrl8 I l
'
l(b)R113,
I
J
I
'
'
I
'
D~ 29.9
I
I I I
I
'
''
'
I
mm
100 Io
1
~110 ~'120 I
I
i
J
i
a
i
'
'
I
'
'
'
o
G kglm2s 49 4 e 170
~
85.1
:
e
-
•
94.7 150
-
Numerical result
179
65
® 2oo•
260
0
Condensate
f i l m on i n n e r "tube
Figure 12. Model of condensate flow on the tube surface. 10
entrainment, and assuming laminar condensate flow, the condensate fdm thickness ~ is determined by the vapor shear Fv
=I
7 106
I
!
I
5
I
i l l i
I
i
'
107
'
5
'
'''"
= f , pou2/2 as 108
$d
=\
Figure 11. Variation of Nuf with Sd and G. The expression of ~bffor a low finned tube [10] can be extended to the present case as
~kf=cos-'(ptg(p~-_df)Do
l)
(18)
Thus, the ~kfvalues for R-11 and R-113 are calculated as 1.31 and 1.51 rad, respectively. The ratio of these values is roughly in accord with the ratio of Nuy for the two fluids at large Sd. Next we consider the large-G regime, where the condensate flow is controlled by the vapor shear. In this regime the condensation process on the fm surface is supposed to be similar to the forced convection f'dm condensation on a cylinder normal to the vapor flow. The expression of the average Nusselt number for the latter case is written as [11]
Nuf= b(1 + A ) i/3 (u~df/~,t) i/2
(19)
where b = 0.9, A = ltlhfs(pop.v/plltl)l/2/hl(Ts - Tf). Equation (19) may be applied to the present case by treating b as a parameter accounting for the lower effective vapor velocity as compared to uo and the decrease of effective f'm surface area due to submergence in the condensate flow on the tube surface. The effect of condensate stratification at the lower part of the annulus will be considered separately. Figure 12 shows the model of condensate flow on the tube surface. Neglecting the effects of gravity and condensate
/
=
Re*
Z
(20)
wherefi is the interfaciai friction factor and Re* = GDr(1 X){ (Di/Dr) 2 - 1 }/#t. The actual condensate flow is affected by the circumferential gravity force F s = Ptg~ sin ~, and tends to the gravity-drained flow solution with increasing z. The Fg/F~ ratio is written as
Fg = ptg8 sin ~b
Fo f, pvu~/2 2
~
\
~
,] sin ~
(21)
Neglecting the variation of ~ , the dimensionless parameter determining/~ is derived as F = ( P / ~ I / 2 Re* i/6 (g~'t)'/3
(22)
\po/ uo Figure 13 shows the Nu:/Nu/c ratio plotted against F, where Nuf denotes the experimental data and Nufc is the calculated value from Eq. (19) with b = 1. As is evident from Eq. (22), a small F value corresponds to the upstream subsections at large G. The Nuy/Nuyc ratio for a given G first decreases, then takes a minimum value, and then increases again with increasing F. This indicates the transition from the vapor shear controlled regime to the combined gravity and surface tension controlled regime. Comparison of the results for different D, reveals that the data at F < 0.05 are higher for smaller Di. This may be attributed to the increase in effective vapor velocity with
180 H. Honda et al.
1.0
I(
, i .... o)RI'I,
0.5
I
'
Di = 24.8mrn
G e e
(23)
"-'~quation XX~e V
Z
i
1 12
•
146
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l
2.0
i r.
kglm2s 182
78.2 91.3
o
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1.0
.... ~
i
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~
'
~
~
l
'
,
i
~
T ....
i
I
'
~
,
~
,
r ....
I
- -
Di = 2 9 . 9 t u r n
__
v_
u
200
e 250 • 282
2:.2 0.5
Z
Z @
0.1
O Ir
.
.
.
i
i
I
.
l i i i l
•
I ....
(b)R11,
Di = 27.2mm •
(23)
Equution
I i
I
I l
I ,
I I , i ,
o
e ~ e leon®,..__Z•,..i>i ®o 0.1 1.0
--
•
--L
I
I
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I
i
i i i r
I
I
I
l
I
I i
'
'
'
I
' '''I
I
I
I
I
I
'
(c)R11,
0.5
G
Di =29.9 mm
Equation (23)
O
52.5
e
75.8 94.0 123
o
Z
0
=T
o
e
kg/m2s • 143 {i 170 © 201
I I.
--
O0
oo
oQ
Z
1.0
I
I i Jill
I
(d)R113, Di =29.9 mrn
0.5
\/
@~ •
Z
0.1
(23)
Equution
o
:
94.7
,
10 -2
i
~
I
S
,
,
O
O O
•
O
I
I
I
i
i
i
I
i
i w l_
kglm2s e 170 © 200 • 260
G 49.4 85.1
o
Z
5
10-2
O
i
10"1
I
i
l
5
F Figare 13. Variation of Nuf/Nuyc with F and G.
100
increasing relative fin height h'y = hf/(Di Dr). The data in the vapor shear controlled regime are correlated by the expression -
Nu/=0.12
1 +(5 X 10 -7)
I+(5X
10-I
5
100
(24)
where Nuys and Nufv denote the Nuf values given by Eqs. (17) and (23), respectively, and c is the parameter accounting for the effect of submergence of the inner tube in the stratified condensate at the downstream subsections. By analogy with the stratified two-phase flow in a horizontal tube [12], the major parameter determining the condensate level is considered to be Xtt. Figure 14 shows the Nuf/Nuyc ratio plotted against Xtt, where Nuy denotes the experimental data and Nufc the calculated value from Eq. (24) with c = 1. It is seen from Fig. 14 that the Nuf/Nufc ratio is approximately unity at X , ,~ 0.08 and deviates toward a lower value with increasing X , at Xtt '~ 0.08. It is also noted that the Nuf/Nuyc ratio at Xtt ~ 0.08 is somewhat higher for larger Dr. All data are correlated by the expression
105)h'~] Ii4
1 + IO0(2Dr/D#- l).X~t
(25)
to a mean absolute deviation of 16.2%. Practical Significance
$o •
,,I
5
Xtt
Nuy= I
24~8
~o
0.1 I
EquQtion(25) Rl13
Nuf= c(Nu}s + Nu~o)1/4
o
.•
•
J
equation is assumed to be
Z Z
29.9
Fig-re 14. Variation of Nuy/NUycwith Xtt.
G kg/m2s 8 2.4© 203 144 e 251 165 • 301
e
0.5
I i
I
299
=
0.1
e
O I
1.0
0~
--
m
(23)
The experimental as a whole data may be expressed by the combination of Eqs. (17) and (23). Following the case of a horizontal cylinder [11], the functional form of the empirical
Local pressure gradient and heat transfer data were obtained for condensation of R-11 and R-113 in horizontal annuli with a wire-finned inner tube. The dimensionless correlations for the local frictional pressure gradient and the local heat transfer coefficient, Eqs. (10) and (25), can be used to accurately predict the pressure drop and heat transfer characteristics of double-tube condensers of this type. The analytical model of film condensation in a horizontal annulus developed in this study can be extended to double-tube condensers with different types of enhanced inner tubes. CONCLUSIONS 1. The local frictional pressure gradient was correlated fairly well using the Lockhart-Martinelli parameters. The effect of the annular space was small except for the region near the upstream end. Overall, the data were correlated by Eq. (10) to a mean absolute deviation of 7.3%. 2. The local heat transfer coefficient was 2 to 13 times as large as those for horizontal annuli with smooth and corrugated inner tubes. The heat transfer enhancement ratio decreased with increasing mass velocity. 3. The heat transfer processes on the fin surface at the low and high vapor velocities were controlled by the surface tension
Condensation of Refrigerants R-11 and R-113
from the Cavallini et al equation [5]), respectively. Then, q,r and q,/are obtained from Eqs. (A.3) and (A. 1), respectively. Finally, c~y is obtained from Eq. (A.2). According to the calculated results, the q,r/q, ratio was less than 0.11.
Ts Y
II VI I
2b
-~
A - A
T.,
i
\ Solder
Fi
i\.*X//\.
/,,
'
/-+
L,
A
1
~,s
r,,,
n /~
J-
~'¢ ~
i x
.I
J
Z
2a s = 0 . 6 mm Af = 0 . 0 ] mm @ = O, 87 t a d
2b s = 0 . 2 5 Ar = 0.02
mm
NOMENCLATURE Di Do Dr df F Fg Fo f G Gd
mm
Figure 15. Details of wire fin soldered on the inner tube. and the vapor shear forces, respectively. The data were correlated by Eq. (25) to a mean absolute deviation of 16.2%. APPENDIX. Determination of %, from the E x p e r i m e n t a l D a t a Figure 15 shows the details of the wire fins soldered on the inner tube. The base area of the soldered portion is approximated by an ellipse with semimajor axis as and semiminor axis bs. The thicknesses of the solder layer on the tube surface and at the fin base are A, and A/, respectively. The local heat flux qn can be divided into two parts, qn = qnf+ q,r
(A.1)
where q,¢ and q,, are the components of heat transfer rates on the fin surface and the tube surface, respectively, q,,f is related to (~y by
g h hf hfg Nuf P Pr q Re Re* Sd T u X Xtt x, y z
inner diameter of outer tube, m diameter at fin tip, m diameter at fin root, m wire fin diameter, m parameter, Eq. (22), dimensionless circumferential gravity force, N/m 2 axial vapor shear, N/m 2 friction factor, dimensionless refrigerant mass velocity, kg/(m 2 s) parameter
(A.2)
(A.3)
where T~, is the tube surface temperature and A f = wa~bs/ PfPc is the ratio of fin base area to tube surface area. Since the solder layer is very thin, it is sufficiently accurate to assume
T~r=Tw.
c~ /~ 6
Sub~eripts ^.4 Xl/4
+ tzrv!
Tf)],
local heat transfer coefficient, W / ( m 2 K) void fraction, dimensionless condensate film thickness, m fin efficiency, dimensionless )~ thermal conductivity, W/(m K) dynamic viscosity, kg/(s m) v kinematic viscosity, m2/s o density, kg/m 3 o surface tension, N/m ~o Lockhart-Martinelli parameter, Eq. (6), dimensionless i ~k angle measured from tube top, rad
The ar value is estimated by -4 Ot r - - ( 0 / r g
-
Greek Symbols
where ;/ = tanh ( l / ~ ) / ( i f ~ ) is the fin efficiency, // is the fin length per fin crest pitch, T,, is the interfacial temperature between the tube and the solder layer, and Rs is the thermal resistance of solder layer at the fin base. q,, is related to the heat transfer coefficient at the tube surface by q , , = ctr( T s - Tw,)(1 - A f )
[ = plghfgd3f/)~lVl(Ts
dimensionless gravitational acceleration, m/s 2 specific enthalpy of test fluid, J/kg fin height, m specific enthalpy of evaporation, J/kg Nusselt number ( = ctfdf/Xt), dimensionless static pressure, N/m 2 Prandtl number, dimensionless local heat flux, W / m E Reynolds number [ = u(Di - D,)/v], dimensionless parameter [=GDr(I - X ) { ( D i / D r ) 2 - 1}//~t], dimensionless parameter [ = ohfsdf/Xtvt( Ts - Ty)], dimensionless temperature, K velocity, m/s mass quality, dimensionless Lockhart-Martinelli parameter, Eq. (6), dimensionless coordinates measured along fin surface, m distance measured from test section inlet, m
T,-Tw qnf = P f P c (1 / 7rdflfotf~ 1 + Rs )
181
(A.4)
where a,~ and u,~ are the heat transfer coefficients in the gravity-controlled regime (calculated from the Nusselt equation [13]) and the vapor shear controlled regime (calculated
c F
coolant friction component
f
fin
fc
calculated value
182
H. Honda et al.
fs fv L 1 M n
surface tension controlled r e g i m e v a p o r shear controlled r e g i m e based o n total flow assumed liquid liquid momentum component based o n n o m i n a l surface area (surface area o f a smooth tube with 19,)
s
saturation
o w
vapor tube wall
0
test section inlet
6.
7.
8.
9.
REFERENCES 1. Borchmann, J., Zur Kondensation von R l l und Wasserdampf bei hoben Dampfgeschwindigkeiten, Kdltetechnik-Klimat., 19, 208213, 1967. 2. Sensyu, T., Murata, K., and Sega, H., Heat Transfer Coefficients for Condensation of R-22 and Performance of Condensers (in Japanese), Refrigeration, 49, 599-605, 1974. 3. Miropolsky, Z. L., Shneerova, R. I., and Treputnev, V. V., The Influence of Steam Flowing and Condensing in a Duct on Heat Transfer to Liquid Film, Proc. 6th Int. Heat Transf. Conf., Toronto, Vol. 2, pp. 431--436, 1978. 4. Hashizume, K., Refrigerant Two-Phase Flow Heat Transfer in Double Tube Condensers (in Japanese), Proc. 18th Nat. Heat Transfer Syrup. of Japan, Sendai, pp. 544-546, 1981. 5. Cavallini, A., Frizzerin, S., and Rossetto, L., Condensation of
10.
11.
12.
13.
Refrigerants Inside Annuli, Proc. 7th Int. Heat Transfer Conf., Munich, Vol. 5, pp. 45-51, 1982. Sakon, Y., Sakitani, K., and Uemura, S., Performance of Double Tube Condensers with SpiRted Fin Tubes (in Japanese), Proc. 17th Air-Conditioning and Refrigeration Joint Meeting, Tokyo, pp. 1-4, 1983. Fauske, H. K., Critical Two-Phase Steam Water Flows, Proc. Heat Transfer and Fluid Dynamics Inst., Vol. 79, Stanford University Press, 1961. Lockhart, R. W., and Martinelli, R. C. Proposed Correlation of Data for Isothermal Two-Phase, Two-Component Flow in Pipes, Chem. Eng. Prog., 45, 39--48, 1949. Kays, W. M., and Pekins, H. C., Forced Convection, Internal Flows in Ducts, in Handbook of Heat Transfer Fundamentals, W. M. Rohsenow, J. P. HartneU, and E. N, Ganic, Eds., pp. 7.1-7.180, McGraw-Hill, New York, 1985. Honda, H., and Nozu, S., A Prediction Method for Heat Transfer During Film Condensation on Horizontal Low Integral-Fin Tubes, J. Heat Transfer, 109, 218-225, 1987. Fujii, T., Uehara, H., and Kurata Ch., Laminar Filmwise Condensation of Flowing Vapor on a Horizontal Cylinder, Int. J. Heat Mass Transfer, 15, 235-246, 1972. Taitel, Y., and Dukler, A. E., A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow, AIChE J., 22, 47-55, 1976. Nusselt, W., Die Oberfliichenkondensation des Wasserdampfes, Z. Ver. Deut. Ing., 60, 541-546, 1916.
Received April 15, 1988; revised November 10, 1988.