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Enhanced film condensation of steam on spirally fluted tubes
DESALINATION ELSEVIER
Desalination 101 (1995) 295-301
Enhanced film condensation of steam on spirally timed tubes N.H. Aly, S.D. Bedrose Reactors Department, Heat Transfer Laboratory and Desalination Laboratory, Atomic Energy Authority, Cairo, Egypt
Received 5 April 1993; accepted 20 December 1994
Abstract Film condensation of stagnant water vapor on both vertical and horizontal spir~ly fluted robes was studied theoretically. A sine flute was taken as a typical example. In the analysis two regions of the flute, namely the valley and the crest, are treated separately. The condensation process takes place on the crest of the flute and runs down to the valley by surface tension aud gravity. In the valley region no heat transfer occurs, and the condensate moves down stream by gravity to be drained. The equation of motion and energy balance are used to conclude relations between condensate flow rate and film thickness for both regions. By equating the condensation rate in the crest region to the change of flow rate in the valley region, the film thickness and 'he local heat transfer coefficient are calculated using numerical integra~on. The average heat transfer coefficient and Nusselt cumbers are calculated and compared to that of smooth robes. The results show that the enhancemem due to fluting may reach live times in the case of horizontal tubes, while for vertical tubes it is much .lower. The the~retical results are compared to the available experimental results of film condensation on ll~rizontal finned tubes and twisted vertical tubes, which are very similar to spirally fluted tubes, and the:; ~how good agreement. Keywords: Film condensation; Spirally fluted tubes
1. Introduction The cost of tubing in heat exchangers represents a substantial proportion of the capital cost. In the case of desalination plants, the cost of tubes may reach 15-20% of the total cost of the plant. A considerable reduction of the capital cost can be achieved by using enhancement heat transtbr surfaces. The required heat transfer surfaces may be reduced by 50% [1]. These enhancement surfaces SSD100lI-9164(95)00033-X
may be finned tubes or profded tubes such as roped, twisted or fluted ones. Important applications of fluted robes are the long-robe vertical desalination plants, MSF preheaters and naval condensers. These enhanced surfaces may break the condensate f'dm and a reasonable part of the surface, having a very thin f'dm thickJmss, which will result in higher condensation rates. Gregorig [2] described the phenomenon of
296
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
increased condensing side heat transfer coefficients of fluted surface by the effect of surface tension. Thereafter, some researches were conducted to confirm the importance of surface tension in film condensation on finned tubes [3-5]. Several attempts were made to solve the twodimensional flow problem of film condensation on fll~ted and finned horizontal tubes. Solutions for horizontal helically fluted tubes showed a large effect of helix angle [6]. Honda et al. [7,8] developed theoretical models to predict the condensation rate on horizontal low-finned tubes assuming a rectangular fin cross-section. Webb et al. [9,10] developed a model for prediction of the condensation coefficient on horizontal in~!egral fin tubes, neglecting the condensation rate in the condensate flooded region. They assumed a continuous profile shape of the fins. Recentlv Adamak and Webb [11] developed a model for prediction of film condensation on horizontal integral fin tubes, either the fins having continuous profile shapes or having a rectangular cross-section. The film condensation in this mod~l was divided into three distinct zones: the unflooded, the flooded and the drop-off. Also, the film surface was divided into segments, and the total condensation rate was calculated as the sum of condensation rates on these segments. The surface tension and gravity forces were taken into consideration in their model. Many experimems were conducted pertaining to condenser applications. However, few experiments were conducted on spirally fluted tubes. Often these experiments measured the overall heat transfer coefficients. Marto et al. [1] showed that the twisted horizontal tubes improved the overall heat transfer coefficient to reach about three times that of smooth tubes. Brodov [12] correlated the data for vertical twisted tubes and showed maximum enhancement of 40% for film condensation in this c~se. Yau et al. [13] investigated the effect of fin spacing on the condensation of steam on horizontal integral fin tubes. Winniaracchi et al. [ 14,15] conducted a set of experiments on condensation of steam on horizontal spacing, thickness
and height. Their results showed best performance of the finned tubes when the value of fin spacing was nearly equal to the fin ti~ickness (1-2 mm) and the maximum enhancement ralio wa3 about 5. In the present work a simple model was .'l~,,eloped based on Nusselt assumptions to predict heat transfer coefficients on both horizontal and vertical fluted tubes. The results of the proposed model were compared to both the previous models and the experiments results where available.
2. The proposed model The film condensation on both horizontal and vertical spirally fluted tubes was studied. A sine wave flute was guggest.~a as a typical example. The tube is helical!) grooved with helix angle 0. The flute dept~, is a and its pitch is p. The flute geometry and used coordinates are shown in Fi~ --i. The film of condensate is divided into two regions, namely the crest and valley. In the cre~t region the film thickness is minimum and he~t transfer takes place. The condensate is drained from the crest to valley region by surface tension and gravity forces. In the valley region no heat transfer occurs, and the condensate moves down stream by gravity in the direction of the helix. Thus, the condensate velocity is considered in the direction of the flute axis [6]. This motion happens whether the tube is placed horizontally or vertically. The system is described by a set of equations. These equations are solved to determine the film thickness distribution and hence the heat tra,sfer coefficient. The film thickness is calculated in the crest and valley regions and matched in a suitable point where 6b=O. 1 5max [4,6]. 2.1. The valley region The equation of motio~ in the valley region 02// + Oq2U P g / ~ . . . . =
0y 2
0z 2
/~
(1)
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
297
Fig. 1. Flute geometry and coordinates system.
where/3 may be approximated as: cos 0 for vertical tubes; cos 0 sin • for horizontal tubes. The boundary conditions of Eqn. (1) are: u,:--0 at Y=Yb; u = 0 at z = 0 ; (#u/#y)=O at y=O; (au/az)=0 at z=& To determine the velocity distribution, the film thickness should be known. Using the geometrica2 relations of the flute and approximating the tilm surface as a circular arc of radius R [7,8,11], the following relations are derived:
a (X) = amax(X) + R - ~
- y2
(2)
Yb = Yscos-I
(4) /a
~x
+
6---~=-- - 1 + l
where Ys is taken on natural coordinates. Eqs. (1)=(4) are solved using finite difference techniques to determine the velocity distribution in the valley region. The mass flow rate in the x direction can be expressed as
- a / 2 t_1-cos~r y Yb ~i
~a(x) = 2P l I p u dy dz
(3)
=P/2 I II+ [--~sin-~]2 dy
(5)
where x = e/cos 0 for vertical tubes and x ='I'r/cos 0 for horizontal tubes.
2.2 The crest regioa In this region the heat transfer takes place and the velocity of condensate has two components - u,v in x,y directions. Thus, the equations of
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
298 motion are:
Nu=hP
,.~
2a2u-'°g#;/~ Oz az 2a2v-IIaP+'°g~' ] ~ .
(6)
k 4rn Re = - ~ Ix w e
where P=f(o,tS) and may be expressed as [4,6]
a2a j
r
A simple computer program has been developed to solve the model equations and to calculate the required variables.
-o'__
aP _ a
ay2
3. Results and discussion where 3,=sin0 sin0 for vertical tubes; and 3,=cos0 cos 0 cos ~I, and 0 = t a n - l pl2a for horizontal tubes. The boundary conditions of Eq. (6) are: u - - v = 0 at z--0; and (aulaz)-- (avlaz) at z=a. The rate of condensation along a distance dx is given by Ys
dth(.-¢) = k A t I
1
dydx
(7)
Eqs. (6) and (7) are solved to give the film thickness distribution as a function of geometric and heat transfer parameters. By equating the rate ot condensation in the crest region to the increase of condensate flow rate in the valley region, the following equation is obtained: th(x+dx) - n~(x) = dth(x)
(8)
Thus, using the relations between flow rates and film thickness in Eqs. (5) and (7) to solve Eq. (8), the film thickness distribution A(y) is calculated. The heat transfer coefficients, Nu and Re numbers are calculated from the film thickness and velocity distributions using numerical integration, where x Ys
h=
k IIb"gdydx
The proposed model has been used to calculate the film thickness distributions and hence the heat transfer coefficient in film condensation on fluted tubes. By comparing average heat transfer coefficients on fluted tubes to that on smooth tubes, the enhancement ratio is predicted. The helix angle, the flute depth and pitch are varied to predict their effect. In Fig. 2 the two positions of the fluted tubes under investigation are shown. In Fig. 3 the depth of the flute is varied from .75-2 mm. The heat transfer coefficient decreases as the heat flux increases while the rate of this decrease is affected by variation of heat flux. However, it is noticed that thi~ rate of decrease is very small for flute depths between 1.5-2 mm. These results are very near to the experimental results on finned tubes [13,14]. The helix angle of horizontal fluted tubes largely affects the enhancement ratio. This effect is shown in Fig. 4 for different values of depth (.52 mm). The results show an increase of enhancement ratio as the helix angle increases. The best performance is obtained for a = 1.5-2 mm to give a maximum enhancement ratio of about 5. This result is very near to that of integral fin tubes
[14,15]. The variation of flute pitch may slightly increase the enhancemem ratio as shown in Fig. 5. This increase is due to the increase of the thin film area if the ratio between pitch and depth increases. Thus, more condensate is produced for the same length of the tube. However, this increase does not
N,H. Ai~, S.D. Bedrose /Desalination 101 (1~;95) 295-301
Rim
Rim
thinning
thickening
299
I Vertical t u b e
Horizontal tube Fig. 2. Horizontal and vertical spirally fluted tubes. ~0.00
,~000
..--"
.......................
°°
.~3.00 ~ a = 2 m m
J~"
--a - 1.5 m m ........ o = | ='rim ---o = .7,5 m m
10,00.
~.oo
°'°°o.=~'"'~:;~'""6:~'""~:~'"";;~"'"~:~"'";:~o
~'~o.~
Q, MW/m 2 Fig. 3. Variation of condensation heat transfer coefficient of horizontal tube with heat flux. -"
o
mm 1.5 mm 1ram .5 mm
2
_ _ _
o
==
........
o o
=
. . . . . . ~ . b ~ . . . . . . ~ . b o . . . . . . 8o.bb .....
"~:oo
Helix angle O Fig. 4. Effect of helix angle on enhancement ratio of horizontal abe.
6.00 I ~.0O ~
-"
5O.00
0 " =ram - ~.5"
¢1 0
•
4
, "~="%
40.00
,j 4,00 I
~
°i
~
o
O
("4 E 30.00
.3.0O
2.0O
:~ 1.00 ~
~ ¢.~llete¢ mod~ (~COW4nnklr~hCl~ ~tul(HI86).o-.5 mm ~ a ~ w~m'~mcm:n~ eto~tg86).o- t mm 44~441~ a c h c n , ~t41(Ig86).0-- I.~rnm ~w4~4cnclti ¢ta~19861,0,,, 2 mm
0.0O ! . . . .
,i~ . . . . . . . ~ i ~ . . . . . . i.~0 Pitch, mm Fig. 5. Variation of enhancement ratio with flute pitch on horizontal fluted tubes. o,0o
o ~o ......
.2o.oo J~ 10.00
~
fo
m
!
V~Q~r~you el ¢1l(1~05) ...... ~mc~hi( t 9a6)
°'00o.o~...... ~ : ~ ) " ' " ' ~ : ~ ...... ~:6o...... ;3o ...... ~,~6o Pitch, m m
Fig. 6. Effect of flute pitch on condensation heat transfer coefficient of horizontal finned and fluted tubes.
300
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
!.
o 4.00
o - 2 .
.
.
.
o
.
•
""-o...
.¢ ;. ~.-.o...~
71 Z 0.20
2.00
"~
~
"
% ,%N
1.~ _ __
o.oo
~
1 .....
,.oo
0.10 ...... ..n,. ococo rr"rv'~
hoeizontol tube raft,col tube
"~'6.b'ci.... ¥6.~6 .... ~6.~;~ . . . . "8"6.b'6.... ~~o'.oo Helix angle ®
Fig. 7. Ettect of helix angle on enhancement ratio of condensation on tluted tube. exceed 10% for different values of depth. For a further increase of pitch, the enhancement ratio will decrease since the surface of the tube approaches a smooth tube surface. In Fig. 5 a sample of experimental results [ 14,15] are presented and show that the proposed model agrees with the results for integral fin tubes within +20%. The comparison between the results of the Adamek model [11] and the experimental results for integral fin horizontal tubes [13,14] is shown in Fig. 6. The proposed model agrees with these results within +20% as shown in the same figure. Also, the maximum enhancement is obtained for p = 1.5-2 mm. The comparison between the vertical and horizontal cases of spirally fluted tubes proves the advantage of a horizontal position. The enhancement ratio against a helix angle was plotted for various ~,alues of depth in the two cases as shown in Fig. 7. The maximum values of enhancement ratio is about 3 using vertical tubes while it exceeds 5 in using horizontal tubes. The helix angle increase has an opposite effect in the two tubes due to its definition as the angle between the flute and tube axes. To coml,are the results of the proposed model to experimental data for film condensation on vertical profiled tubes, N u and R e numbers are calculated for different values of a / p . The relation
O¢OgoHd moet,tl.,.L'2.. I e,ar'~se~t . - ~ e l . u , D=.15 c . o o o l c o mo~e,.a,'a=,." Beoao~ * t a l ( t g S T ' a ! o - . ! Broaov eto~( ~IJoT~.o/p..; ~ 8tO4ov etol( l g a - ) , o / p - . 2
Fie Fig. 8. Variation of Nu with Re for vertical fluted tubes. between N u and R e is shown in Fig. 8. The proposed model predicts the film condensation on vertical twisted tubes within _+_5%. The comparison to experimental data of Brodov et al. [12] showed close agreement for a l p = 0 . 2 .
4. Conclusions
1. A simple ,:odel has been developed that may predict the film condensation on both horizontal and vertical spirally fluted tubes. 2. The proposed model shows that a maximum enhancement ratio for horizontal finned tubes is about 5, while for vertical tubes it is about 3. 3. The results of the proposed model agree with experimental results for integral rimmed tubes within +20% while the agreement with the results for vertical tubes is much better (+5%).
5. Symbols a g h ho k l
-- flute depth, m -- acceleration of gravity, m/s 2 - - condensation heat transfer coefficient, kW/m 2 °K -- condensation heat transfer coefficient for plain robe, kW/m 2 °K -- thermal conductivity, kW/m °K -- tube length, m
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301 Nu P p R Re u,v We X x,y y,z Yt, Ys At
--------------
Nusselt number pressure, bar pitch o f flute, m radius o f curvature, m Reynolds number velocity components, m / s wetted perimeter, m length o f helix, m natural coordinates Cartesian coordinates half length o f the valley region, m half length of the flute, m temperature difference between v a p o r and condensing surface, °K
5b
---
~max 0 3, /x # a g,
--------
thickness o f liquid film, m a x i m u m film thickness region, m m a x i m u m film thickness helix ant~le specific latent heat, J/kg viscosity, K g / m s density, K g / m 3 surface tension, N / m azimuth angle measured tube
Greek m in the crest in the flute, m
f r o m top o f
301
References [1] P.J. Marto et ai., An experimental comparison of enhar, ced heat transfer condenser tubes, ASME-Advances in Enhanced Heat Transfer, 1979, pp. 1-9. [2] V.R. Gregorig, Zeit Math. Phys., 5 (1954) 36. [3] S. Hirasawa et al., Proc., 6th Int. Heat Transfer Conf., Vol. 2, Toronto, 1980, pp. 413-418. [4] S. Hirasawa et al., Int. J. Heat Mass Tranffer, 23 (1980) 1471. [5] R.L. Webb, S.T. Keswani and T.M. Rudy, Proc., 7th Int. Heat Transfer ConL, Munich, 1982, pp. 175-180. [6] K. Fathalah et al., Desalination, 65 (1987) 25. [7] H. Honda et al., A theoretical model of film condensation in a bundle of horizontal low finned tubes, SDME-HTD-85, 1987, pp. 79-85. [8] H. Honda et al., Proc., 2nd ASME-JSME Thermal Engineering Joint Conf., Vol. 4, 1987, pp. 385-392. [9] T.M. Rudy and R.L. Webb, Proc., ! st ASME-JSME Thermal Engineering Joint Conf., Vol. 1, 1983, pp. 373-377. [10] R.L. Webb, T.M. Rudy et al., J. Heat Transfer, 107 (1985) 369. [11] T. Adamek and R.L. Webb, Int. J. Heat Mass Transfer, 33 (1990) 1721. [12] Y.M. Bradov et al., Thermal Engineering, 34 (1987) 58 (in Russian). [13] K.K. Yau et al., J. Heat Transfer, 108 (1985) 377. [14] A.S. Winniaracchi, P.J. Marto and J.W. Rose, J. Heat Transfer, 108 (1986) 960. [15] A.S. Winniaracchi et al., Film condensation of steam on horizontal finned tubes: Effect of fin spacing, thickness and height, ASME-HTD-47, 1986, pp. 9399.
Desalination 101 (1995) 295-301
Enhanced film condensation of steam on spirally timed tubes N.H. Aly, S.D. Bedrose Reactors Department, Heat Transfer Laboratory and Desalination Laboratory, Atomic Energy Authority, Cairo, Egypt
Received 5 April 1993; accepted 20 December 1994
Abstract Film condensation of stagnant water vapor on both vertical and horizontal spir~ly fluted robes was studied theoretically. A sine flute was taken as a typical example. In the analysis two regions of the flute, namely the valley and the crest, are treated separately. The condensation process takes place on the crest of the flute and runs down to the valley by surface tension aud gravity. In the valley region no heat transfer occurs, and the condensate moves down stream by gravity to be drained. The equation of motion and energy balance are used to conclude relations between condensate flow rate and film thickness for both regions. By equating the condensation rate in the crest region to the change of flow rate in the valley region, the film thickness and 'he local heat transfer coefficient are calculated using numerical integra~on. The average heat transfer coefficient and Nusselt cumbers are calculated and compared to that of smooth robes. The results show that the enhancemem due to fluting may reach live times in the case of horizontal tubes, while for vertical tubes it is much .lower. The the~retical results are compared to the available experimental results of film condensation on ll~rizontal finned tubes and twisted vertical tubes, which are very similar to spirally fluted tubes, and the:; ~how good agreement. Keywords: Film condensation; Spirally fluted tubes
1. Introduction The cost of tubing in heat exchangers represents a substantial proportion of the capital cost. In the case of desalination plants, the cost of tubes may reach 15-20% of the total cost of the plant. A considerable reduction of the capital cost can be achieved by using enhancement heat transtbr surfaces. The required heat transfer surfaces may be reduced by 50% [1]. These enhancement surfaces SSD100lI-9164(95)00033-X
may be finned tubes or profded tubes such as roped, twisted or fluted ones. Important applications of fluted robes are the long-robe vertical desalination plants, MSF preheaters and naval condensers. These enhanced surfaces may break the condensate f'dm and a reasonable part of the surface, having a very thin f'dm thickJmss, which will result in higher condensation rates. Gregorig [2] described the phenomenon of
296
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
increased condensing side heat transfer coefficients of fluted surface by the effect of surface tension. Thereafter, some researches were conducted to confirm the importance of surface tension in film condensation on finned tubes [3-5]. Several attempts were made to solve the twodimensional flow problem of film condensation on fll~ted and finned horizontal tubes. Solutions for horizontal helically fluted tubes showed a large effect of helix angle [6]. Honda et al. [7,8] developed theoretical models to predict the condensation rate on horizontal low-finned tubes assuming a rectangular fin cross-section. Webb et al. [9,10] developed a model for prediction of the condensation coefficient on horizontal in~!egral fin tubes, neglecting the condensation rate in the condensate flooded region. They assumed a continuous profile shape of the fins. Recentlv Adamak and Webb [11] developed a model for prediction of film condensation on horizontal integral fin tubes, either the fins having continuous profile shapes or having a rectangular cross-section. The film condensation in this mod~l was divided into three distinct zones: the unflooded, the flooded and the drop-off. Also, the film surface was divided into segments, and the total condensation rate was calculated as the sum of condensation rates on these segments. The surface tension and gravity forces were taken into consideration in their model. Many experimems were conducted pertaining to condenser applications. However, few experiments were conducted on spirally fluted tubes. Often these experiments measured the overall heat transfer coefficients. Marto et al. [1] showed that the twisted horizontal tubes improved the overall heat transfer coefficient to reach about three times that of smooth tubes. Brodov [12] correlated the data for vertical twisted tubes and showed maximum enhancement of 40% for film condensation in this c~se. Yau et al. [13] investigated the effect of fin spacing on the condensation of steam on horizontal integral fin tubes. Winniaracchi et al. [ 14,15] conducted a set of experiments on condensation of steam on horizontal spacing, thickness
and height. Their results showed best performance of the finned tubes when the value of fin spacing was nearly equal to the fin ti~ickness (1-2 mm) and the maximum enhancement ralio wa3 about 5. In the present work a simple model was .'l~,,eloped based on Nusselt assumptions to predict heat transfer coefficients on both horizontal and vertical fluted tubes. The results of the proposed model were compared to both the previous models and the experiments results where available.
2. The proposed model The film condensation on both horizontal and vertical spirally fluted tubes was studied. A sine wave flute was guggest.~a as a typical example. The tube is helical!) grooved with helix angle 0. The flute dept~, is a and its pitch is p. The flute geometry and used coordinates are shown in Fi~ --i. The film of condensate is divided into two regions, namely the crest and valley. In the cre~t region the film thickness is minimum and he~t transfer takes place. The condensate is drained from the crest to valley region by surface tension and gravity forces. In the valley region no heat transfer occurs, and the condensate moves down stream by gravity in the direction of the helix. Thus, the condensate velocity is considered in the direction of the flute axis [6]. This motion happens whether the tube is placed horizontally or vertically. The system is described by a set of equations. These equations are solved to determine the film thickness distribution and hence the heat tra,sfer coefficient. The film thickness is calculated in the crest and valley regions and matched in a suitable point where 6b=O. 1 5max [4,6]. 2.1. The valley region The equation of motio~ in the valley region 02// + Oq2U P g / ~ . . . . =
0y 2
0z 2
/~
(1)
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
297
Fig. 1. Flute geometry and coordinates system.
where/3 may be approximated as: cos 0 for vertical tubes; cos 0 sin • for horizontal tubes. The boundary conditions of Eqn. (1) are: u,:--0 at Y=Yb; u = 0 at z = 0 ; (#u/#y)=O at y=O; (au/az)=0 at z=& To determine the velocity distribution, the film thickness should be known. Using the geometrica2 relations of the flute and approximating the tilm surface as a circular arc of radius R [7,8,11], the following relations are derived:
a (X) = amax(X) + R - ~
- y2
(2)
Yb = Yscos-I
(4) /a
~x
+
6---~=-- - 1 + l
where Ys is taken on natural coordinates. Eqs. (1)=(4) are solved using finite difference techniques to determine the velocity distribution in the valley region. The mass flow rate in the x direction can be expressed as
- a / 2 t_1-cos~r y Yb ~i
~a(x) = 2P l I p u dy dz
(3)
=P/2 I II+ [--~sin-~]2 dy
(5)
where x = e/cos 0 for vertical tubes and x ='I'r/cos 0 for horizontal tubes.
2.2 The crest regioa In this region the heat transfer takes place and the velocity of condensate has two components - u,v in x,y directions. Thus, the equations of
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
298 motion are:
Nu=hP
,.~
2a2u-'°g#;/~ Oz az 2a2v-IIaP+'°g~' ] ~ .
(6)
k 4rn Re = - ~ Ix w e
where P=f(o,tS) and may be expressed as [4,6]
a2a j
r
A simple computer program has been developed to solve the model equations and to calculate the required variables.
-o'__
aP _ a
ay2
3. Results and discussion where 3,=sin0 sin0 for vertical tubes; and 3,=cos0 cos 0 cos ~I, and 0 = t a n - l pl2a for horizontal tubes. The boundary conditions of Eq. (6) are: u - - v = 0 at z--0; and (aulaz)-- (avlaz) at z=a. The rate of condensation along a distance dx is given by Ys
dth(.-¢) = k A t I
1
dydx
(7)
Eqs. (6) and (7) are solved to give the film thickness distribution as a function of geometric and heat transfer parameters. By equating the rate ot condensation in the crest region to the increase of condensate flow rate in the valley region, the following equation is obtained: th(x+dx) - n~(x) = dth(x)
(8)
Thus, using the relations between flow rates and film thickness in Eqs. (5) and (7) to solve Eq. (8), the film thickness distribution A(y) is calculated. The heat transfer coefficients, Nu and Re numbers are calculated from the film thickness and velocity distributions using numerical integration, where x Ys
h=
k IIb"gdydx
The proposed model has been used to calculate the film thickness distributions and hence the heat transfer coefficient in film condensation on fluted tubes. By comparing average heat transfer coefficients on fluted tubes to that on smooth tubes, the enhancement ratio is predicted. The helix angle, the flute depth and pitch are varied to predict their effect. In Fig. 2 the two positions of the fluted tubes under investigation are shown. In Fig. 3 the depth of the flute is varied from .75-2 mm. The heat transfer coefficient decreases as the heat flux increases while the rate of this decrease is affected by variation of heat flux. However, it is noticed that thi~ rate of decrease is very small for flute depths between 1.5-2 mm. These results are very near to the experimental results on finned tubes [13,14]. The helix angle of horizontal fluted tubes largely affects the enhancement ratio. This effect is shown in Fig. 4 for different values of depth (.52 mm). The results show an increase of enhancement ratio as the helix angle increases. The best performance is obtained for a = 1.5-2 mm to give a maximum enhancement ratio of about 5. This result is very near to that of integral fin tubes
[14,15]. The variation of flute pitch may slightly increase the enhancemem ratio as shown in Fig. 5. This increase is due to the increase of the thin film area if the ratio between pitch and depth increases. Thus, more condensate is produced for the same length of the tube. However, this increase does not
N,H. Ai~, S.D. Bedrose /Desalination 101 (1~;95) 295-301
Rim
Rim
thinning
thickening
299
I Vertical t u b e
Horizontal tube Fig. 2. Horizontal and vertical spirally fluted tubes. ~0.00
,~000
..--"
.......................
°°
.~3.00 ~ a = 2 m m
J~"
--a - 1.5 m m ........ o = | ='rim ---o = .7,5 m m
10,00.
~.oo
°'°°o.=~'"'~:;~'""6:~'""~:~'"";;~"'"~:~"'";:~o
~'~o.~
Q, MW/m 2 Fig. 3. Variation of condensation heat transfer coefficient of horizontal tube with heat flux. -"
o
mm 1.5 mm 1ram .5 mm
2
_ _ _
o
==
........
o o
=
. . . . . . ~ . b ~ . . . . . . ~ . b o . . . . . . 8o.bb .....
"~:oo
Helix angle O Fig. 4. Effect of helix angle on enhancement ratio of horizontal abe.
6.00 I ~.0O ~
-"
5O.00
0 " =ram - ~.5"
¢1 0
•
4
, "~="%
40.00
,j 4,00 I
~
°i
~
o
O
("4 E 30.00
.3.0O
2.0O
:~ 1.00 ~
~ ¢.~llete¢ mod~ (~COW4nnklr~hCl~ ~tul(HI86).o-.5 mm ~ a ~ w~m'~mcm:n~ eto~tg86).o- t mm 44~441~ a c h c n , ~t41(Ig86).0-- I.~rnm ~w4~4cnclti ¢ta~19861,0,,, 2 mm
0.0O ! . . . .
,i~ . . . . . . . ~ i ~ . . . . . . i.~0 Pitch, mm Fig. 5. Variation of enhancement ratio with flute pitch on horizontal fluted tubes. o,0o
o ~o ......
.2o.oo J~ 10.00
~
fo
m
!
V~Q~r~you el ¢1l(1~05) ...... ~mc~hi( t 9a6)
°'00o.o~...... ~ : ~ ) " ' " ' ~ : ~ ...... ~:6o...... ;3o ...... ~,~6o Pitch, m m
Fig. 6. Effect of flute pitch on condensation heat transfer coefficient of horizontal finned and fluted tubes.
300
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301
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o 4.00
o - 2 .
.
.
.
o
.
•
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.¢ ;. ~.-.o...~
71 Z 0.20
2.00
"~
~
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% ,%N
1.~ _ __
o.oo
~
1 .....
,.oo
0.10 ...... ..n,. ococo rr"rv'~
hoeizontol tube raft,col tube
"~'6.b'ci.... ¥6.~6 .... ~6.~;~ . . . . "8"6.b'6.... ~~o'.oo Helix angle ®
Fig. 7. Ettect of helix angle on enhancement ratio of condensation on tluted tube. exceed 10% for different values of depth. For a further increase of pitch, the enhancement ratio will decrease since the surface of the tube approaches a smooth tube surface. In Fig. 5 a sample of experimental results [ 14,15] are presented and show that the proposed model agrees with the results for integral fin tubes within +20%. The comparison between the results of the Adamek model [11] and the experimental results for integral fin horizontal tubes [13,14] is shown in Fig. 6. The proposed model agrees with these results within +20% as shown in the same figure. Also, the maximum enhancement is obtained for p = 1.5-2 mm. The comparison between the vertical and horizontal cases of spirally fluted tubes proves the advantage of a horizontal position. The enhancement ratio against a helix angle was plotted for various ~,alues of depth in the two cases as shown in Fig. 7. The maximum values of enhancement ratio is about 3 using vertical tubes while it exceeds 5 in using horizontal tubes. The helix angle increase has an opposite effect in the two tubes due to its definition as the angle between the flute and tube axes. To coml,are the results of the proposed model to experimental data for film condensation on vertical profiled tubes, N u and R e numbers are calculated for different values of a / p . The relation
O¢OgoHd moet,tl.,.L'2.. I e,ar'~se~t . - ~ e l . u , D=.15 c . o o o l c o mo~e,.a,'a=,." Beoao~ * t a l ( t g S T ' a ! o - . ! Broaov eto~( ~IJoT~.o/p..; ~ 8tO4ov etol( l g a - ) , o / p - . 2
Fie Fig. 8. Variation of Nu with Re for vertical fluted tubes. between N u and R e is shown in Fig. 8. The proposed model predicts the film condensation on vertical twisted tubes within _+_5%. The comparison to experimental data of Brodov et al. [12] showed close agreement for a l p = 0 . 2 .
4. Conclusions
1. A simple ,:odel has been developed that may predict the film condensation on both horizontal and vertical spirally fluted tubes. 2. The proposed model shows that a maximum enhancement ratio for horizontal finned tubes is about 5, while for vertical tubes it is about 3. 3. The results of the proposed model agree with experimental results for integral rimmed tubes within +20% while the agreement with the results for vertical tubes is much better (+5%).
5. Symbols a g h ho k l
-- flute depth, m -- acceleration of gravity, m/s 2 - - condensation heat transfer coefficient, kW/m 2 °K -- condensation heat transfer coefficient for plain robe, kW/m 2 °K -- thermal conductivity, kW/m °K -- tube length, m
N.H. Aly, S.D. Bedrose /Desalination 101 (1995) 295-301 Nu P p R Re u,v We X x,y y,z Yt, Ys At
--------------
Nusselt number pressure, bar pitch o f flute, m radius o f curvature, m Reynolds number velocity components, m / s wetted perimeter, m length o f helix, m natural coordinates Cartesian coordinates half length o f the valley region, m half length of the flute, m temperature difference between v a p o r and condensing surface, °K
5b
---
~max 0 3, /x # a g,
--------
thickness o f liquid film, m a x i m u m film thickness region, m m a x i m u m film thickness helix ant~le specific latent heat, J/kg viscosity, K g / m s density, K g / m 3 surface tension, N / m azimuth angle measured tube
Greek m in the crest in the flute, m
f r o m top o f
301
References [1] P.J. Marto et ai., An experimental comparison of enhar, ced heat transfer condenser tubes, ASME-Advances in Enhanced Heat Transfer, 1979, pp. 1-9. [2] V.R. Gregorig, Zeit Math. Phys., 5 (1954) 36. [3] S. Hirasawa et al., Proc., 6th Int. Heat Transfer Conf., Vol. 2, Toronto, 1980, pp. 413-418. [4] S. Hirasawa et al., Int. J. Heat Mass Tranffer, 23 (1980) 1471. [5] R.L. Webb, S.T. Keswani and T.M. Rudy, Proc., 7th Int. Heat Transfer ConL, Munich, 1982, pp. 175-180. [6] K. Fathalah et al., Desalination, 65 (1987) 25. [7] H. Honda et al., A theoretical model of film condensation in a bundle of horizontal low finned tubes, SDME-HTD-85, 1987, pp. 79-85. [8] H. Honda et al., Proc., 2nd ASME-JSME Thermal Engineering Joint Conf., Vol. 4, 1987, pp. 385-392. [9] T.M. Rudy and R.L. Webb, Proc., ! st ASME-JSME Thermal Engineering Joint Conf., Vol. 1, 1983, pp. 373-377. [10] R.L. Webb, T.M. Rudy et al., J. Heat Transfer, 107 (1985) 369. [11] T. Adamek and R.L. Webb, Int. J. Heat Mass Transfer, 33 (1990) 1721. [12] Y.M. Bradov et al., Thermal Engineering, 34 (1987) 58 (in Russian). [13] K.K. Yau et al., J. Heat Transfer, 108 (1985) 377. [14] A.S. Winniaracchi, P.J. Marto and J.W. Rose, J. Heat Transfer, 108 (1986) 960. [15] A.S. Winniaracchi et al., Film condensation of steam on horizontal finned tubes: Effect of fin spacing, thickness and height, ASME-HTD-47, 1986, pp. 9399.