Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study

Applied Thermal Engineering xxx (2016) xxx–xxx

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research Paper

Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study Baojin Qi a,b,⇑, Jinjia Wei a,b, Xiang Li a a b

School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

h i g h l i g h t s
Coupled non-linear governing equations for heat and mass transfer are developed.
Film thickness, heat flux and velocity are influenced by grooves shape parameters.
Involute groove can provide a better heat transfer performance than trapezoid groove.
The results of present model are in good agreement with the experimental data.

a r t i c l e

i n f o

Article history: Received 30 March 2016 Revised 24 August 2016 Accepted 15 October 2016 Available online xxxx Keywords: Numerical analyses Involute groove Film condensation Heat transfer enhancement

a b s t r a c t The present paper reported numerical analyses and experimental studies on fluid flow and heat transfer for laminar film condensation on various grooved surfaces. Different from the previous literatures, the coupled non-linear governing equations for the fluid flow, mass transfer and two-dimensional thermal conduction were developed base on some reasonable assumptions. Through analyzing the influences of groove shape parameters on film thickness and heat flux, we found that both of them were significantly influenced by the pitch, height and profile radius of the groove. The velocity distributions on the condensate-vapor interface and distributions of wall temperature on various grooves were also studied systematically. The calculations indicated that the surface tension gradient of the liquid film drove the liquid to flow from the crest into the trough. The horizontal velocity decreased first and then increased gradually on the crest and reached a maximum value on profile, and then decreased to zero in the trough region. On the contrary, the condensate flowed at very low vertical velocities on the crest and profile of groove, but in trough region these velocities increased rapidly to a relatively high level. The results of comparison showed that both the distributions of wall temperature and heat flux on involute groove were more favorable to reduce thermal resistance and enhance heat transfer than other grooves. In order to validate the feasibility and reliability of the present analyses, experiments were carried out with some involute grooved surfaces for various physical dimensions, a trapezoid grooved surface and a smooth surface respectively. Both the numerical results and experimental data indicated that involute groove can provide a better heat transfer performance, and the heat flux of involute grooved surface was at least 20% and 50% higher than that of trapezoid grooved and rectangular grooved surfaces. The present analyses were feasible, and could be used in the parameter design and heat transfer calculation of involute grooved surfaces as well as trapezoid grooved surfaces. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Condensation of vapor have been widely applied in various industrial applications that require high heat transfer rates such as desalination, chemical engineering, refrigeration and power generation owing to its high heat transfer coefficient. If the liquid ⇑ Corresponding author at: 28 West Xianning Road, Shanxi, Xi’an, China.

phase fully wets a cold surface in contact with a vapor near the saturation conditions, the conversion of vapor to liquid will take the form of filmwise condensation. During this process, the condensation takes place at the interface of a liquid film covering the solid surface, and film removed from the surface under the action of the gravity and shear stresses. The film renders a high thermal resistance to heat transfer and therefore, a relatively large temperature gradient prevails across it.

E-mail address: [email protected] (B. Qi). http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

Nomenclature g h hfg k M P p ps Qt q q(z) Ri Rp S s,n T

acceleration of gravity, m/s2 height of the groove, m latent heat of vaporization, J/kg slope mass flow rate of condensate, kg/s pitch of groove, m pressure, Pa saturation pressure of the vapor, Pa heat transfer rate of the condensing plate, W heat transfer rate per unit length in z-direction, W/m heat transfer flux, W/m2 heat transfer rate per unit length in z-direction, W/m radius of curvature of the condensate-vapor interface, m profile radius of curvature, m area, m2 coordinates along and normal to the grooved surface, m temperature, K

In order to reduce the thermal resistance and enhance the heat transfer of filmwise condensation, many methods were developed, and several excellent researches of this topic have been reported both in experimental results and numerical simulations [1–4]. Among them, utilizing the effect of surface tension to enhance laminar film condensation on a vertical grooved surface was one of the most effective methods [5]. During filmwise condensation, the thickness of liquid film strongly affected the heat transfer performance. It was expected that the heat transfer during filmwise condensation would be enhanced if the liquid film was thinner. The adoption of grooved surface could make the liquid film thinner locally by utilizing surface tension, and therefore was recognized as an effective method for enhancing the heat transfer of filmwise condensation. An equation to calculate the film thickness on the crest using a graphical procedure was first proposed by Gregorig [6], but the procedure for design of the trough was not specified. Thereafter, much research of filmwise condensation on grooved surfaces has been conducted, and several shapes of grooves, such as triangular [7], sinusoidal and rectangular [8,9], were developed. Mori and co-workers [7] investigated the condensation on triangular, wavy and flat bottomed grooves with theoretical analysis and experiments. The thickness equation of liquid film on the crest and the momentum equation in the trough were established. By solving the equations, average heat transfer coefficient was obtained. The experimental results for triangular grooves verified the accuracy of the analyses. Park and Choi [8] investigated the condensation on triangular grooves. A three-dimensional heat transfer procedure for the condensing film flowing down along a vertical grooved tube was developed. The Navier-Stokes and energy equations were solved using a SIMPLE-type finite volume method. The calculations were in excellent agreement with the data from Ref. [6]. Bilen et al. [9] performed an experimental study on surface heat transfer for different geometric groove shapes (circular, trapezoidal and rectangular). Among the grooved tubes, heat transfer enhancement was obtained up to 63% for circular grooves, 58% for trapezoidal grooves and 47% for rectangular grooves, in comparison with the smooth tube at the highest Reynolds number (Re = 38,000). Garg et al. [10,11] established the equations of fluid mechanics and thermal conduction for condensation on sinusoidal grooves by setting up curvilinear orthogonal coordinate system and employing the boundary layer theory. The linear extrapolation of the results obtained by solving the equations with finite difference method showed a fairly good agreement with experimental data. Fujii and Honda [12,13] studied the condensation on sinusoidal

Ts Tc Tf Tw u x,y z

ai

l c

hi

q qv d k kf

saturation temperature of the vapor, K temperature of the cooling surface, K temperature of the liquid film, K temperature of the grooved surface, K velocity, m/s coordinates along and normal to the cooling surface, m vertical coordinate, m opening angle of involute equation, ° dynamic viscosity, Pas surface tension of the condensate, N/m pressure angle of involute curve, ° density of the condensate, kg/m3 density of the vapor, kg/m3 thickness of the condensate film, m thermal conductivity, W/(mK) thermal conductivity of the condensate, W/(mK)

grooves. The basic equations of liquid film both on crest and in trough coupled with two-dimensional thermal conduction equation were setup and solved with numerical method, and their numerical results agreed well with the experimental data. Heat transfer of laminar film condensation on a vertical sinusoidal grooved tube was studied numerically by Zhu and co-workers [14]. Their calculation results were in good agreement with a maximum deviation of 18%, and can be used in the parameter design of sinusoidal grooved tubes and other types of grooved tube such as triangular shape and rectangular shape. Park [15] conducted a three-dimensional numerical investigation of the flow, heat and mass transfer characteristics of the grooved tube with films flowing down on the outside tube walls. The velocity and temperature fields were successfully predicted for various groove shapes by using the moving-grid technique. Present simulations show that sinusoidal grooved surfaces have good performances in laminar film condensation [11–14]. However, because of the complicated curve cutting edge, it is very difficult to form sinusoidal grooves even with advanced NC machine tools. So far, sinusoidal grooves can only be precisely machined by wire-electrode cutting, but this processing method is hard to form high-quality and large-scale productions due to its poor surface finish, low production rate and high cost. Thus the applications of sinusoidal grooved surfaces in heat transfer enhancement are severely constrained. It is highly necessary and also an important task to develop some new shapes of grooves by numerical analyses and experimental verifications. These new grooves can enhance condensation heat transfer as good as sinusoidal groove, but the machining processes should be implemented easily. The outlines of involute grooves and trapezoid grooves are similar to those of sinusoidal grooves, and they can be easily formed by involute cutters on milling machine. However, their condensation heat transfer has not been studied deeply and systematically. In the current study, we firstly design involute groove (with infinite profile radius of curvature) and trapezoid groove as models. Secondly, theoretical analyses are carried out for laminar film condensation on involute groove and the trapezoid groove, and the distributions of film thickness and temperature, the heat flux and flow of liquid film are systematically analysed. Then, the comparisons of numerical simulations between our study and literatures were conducted. Finally, experiments are performed on the two grooved plates. The experimental data are compared with analytical results to investigate the feasibility of the analyses and the condensation performance of involute grooved surfaces.

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

2. Theoretical analyses 2.1. Physical model The physical models of the involute and trapezoid grooves are schematically shown in Fig. 1. Several isosceles grooves are machined vertically on the condensing surface, and the cooling surface is flat. For involute grooves, the pitch and the height of the groove are P and h, and the root fillet radius of the groove is 1 r, r ¼ 10 P. The profile radius of curvature is Rp, Rp = f (hi, ai), where the parameters hi and ai represent the opening angle and pressure

3

angle of involute equation respectively. Because the cross section of the grooves is of the symmetric shapes, only a half of the cross sections and coordinate systems are considered for simplification, as shown in Fig. 2. The grooved surface is denoted by A-H; the condensate-vapor interface by I-M; the cooling surface by ON. IAO and MHN are symmetric boundary. The bottom of the groove, denoted by GH, is transition arc and line shaped to facilitate the machining. The crest is connected smoothly with the trough by an arc denoted by B-E. The liquid will flow in the horizontal direction due to the difference of the grooved surface curvature. We assume the liquid flows in opposite directions from the starting point D, the central point of arc B-E, to both sides, and therefore the liquid film can be divided into four regions: region I surrounded by DFLK, II surrounded by CDKJ, III surrounded by FHML, and IV surrounded by ACJI. The liquid film in regions I, II and IV is thin, while thick in region III. The condensate-vapor interfaces in adjacent regions are connected smoothly with each other. We setup a curvilinear orthogonal coordinate system(s, n, z) where s is arc length along the grooved surface, n is perpendicular to the grooved surface, and z is the vertical direction (the same direction as gravity, g). In the analysis, the following assumptions are made.

Involute grooves

Trapezoid grooves Fig. 1. Physical model of the grooves.

Involute grooves

(1) The vapor velocity is very small, so the interfacial shear between the liquid and vapor is negligible. (2) The condensate flows slowly so that the inertia term in the momentum equations is negligible. (3) The condensate flows in a steady and laminar condition. (4) The curvature of the liquid film in the gravitational direction is far less than that in the horizontal direction, so variation of the curvature in the gravitational direction is negligible. (5) The condensate-vapor interface in regions III and IV is assumed to be a circular arc shape. (6) The vertical thermal conduction within the liquid film is neglected.

Trapezoid grooves

Fig. 2. Cross sections and coordinate systems of grooves.

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

of liquid that goes into the element per unit time and per unit area is given by

2.2. Fundamental equations 2.2.1. Regions I and II Consider the condensate as viscous liquid, incompressible Newtonian fluid. Based on the theory of hydrodynamics, the differential form of mass conservation equation can be expressed as

ZZ

!

!

ZZ

!

ðn1  u ÞdA ¼

!

ðn2  u ÞdA

A1

ð1Þ

un dsdz ¼

ð5Þ

Substituting Eqs. (5) into (4), we obtain the continuity equation in the integral form

A2

where A1 and A2 are the flowing areas of inlet and outlet respectively. We consider the equilibrium of a small element cut out of the liquid film, as shown in Fig. 3. So the liquid flowing into the element is equal to the volume flowing out.

  kf dsdz @T qhfg @n n¼d

@ @s

Z

d

us dn þ

0

Z

@ @z

d

uz dn ¼

0

  kf @T qhfg @n n¼d

ð6Þ

The momentum equations in the curvilinear orthogonal coordinate system are

h   i 8  2 2 s s n > s-direction :  H11 @p þ Hl1 @s@ H11 @u  Hun2 @s@ 1R  H21 R @u  HuRs 2 ¼ 0 þ H1 @@nu2s þ H1 @@zu2s  1R @u > @s @n @s @s > 1 1 > < h   i  @p @ 2 un @ 2 un us @ 1 1 @un n s þ Hl1 @s@ H11 @u þ H  þ  HunR2 ¼ 0 n-direction :  @n þ H þ H21 R @u 1 1 2 2 2 R @n @s @s @n @z H1 @s R 1 > > h   i > > : z-direction : ðq  q Þg  @p þ l @ 1 @uz þ H @ 2 uz þ H @ 2 uz  1 @uz ¼ 0 1 @n2 1 @z2 V R @n H1 @s H1 @s @z

The volume flow into the element can be written as

8 > < s-direction : n-direction : > : z-direction :

Rd dz 0 us dn un dsdz Rd ds 0 uz dn

where H1 ¼

ð2Þ

: z-direction :

Rd us dn þ ds @s@ ðdz 0 us dnÞ   Rd Rd @ ds 0 uz dn ds 0 uz dn þ dz @z

dz

ð3Þ

The simultaneous equations above can be developed into a set of partial differential equation

 Z d   Z d  @ @ dz ds ds us dn þ dz uz dn ¼ un dsdz @s @z 0 0

The thickness of liquid film in this region is very thin so that boundary layer theory can be applied to simplify the momentum @ @ @ @  @n , @z  @n , and equations. Recognizing un  us , un  uz , @s @  @n , and using the assumptions above, the inertia is relatively small term and can be neglected. The momentum equations are simplified as

Rd 0

Rp n . Rp

@2 @n2

The volume flow out of the element are given by

8 < s-direction :

ð7Þ

ð4Þ

This equation indicates that the flow increment from s-direction and z-direction equals volume flow in n-direction. The total amount

8 s-direction : > > < n-direction : > > : z-direction :

@ 2 us @n2 @p @n

¼ l1

@p @s

ð8Þ

¼0

@ 2 uz @n2

¼

ðqqv Þg

l

where us, un, and uz are the velocity components in the s, n, and zdirections, respectively, p is the pressure in the condensate film, q and l are the density and dynamic viscosity of the condensate, and qv is the density of the vapor. Integration of Eq. (8) with boundary conditions

(

us ¼ uz ¼ 0 at n ¼ 0

ð9Þ

l @u@ns ¼ S1 ; l @u@nz ¼ S3 at n ¼ d Yields

8 < ux ¼ S1 n þ RP l l

dp ds

h

n ln



RP þn RP þd



þ R ln



RP þn RP



i n

: u ¼ S3 n þ g ðq  q Þnð2r  nÞ z v l 2l

ð10Þ

Here S1 and S3 are the shear stress in the s-direction and z-direction on the condensate-vapor interface, respectively. For incompressible viscous flow, S1 and S3 can be regarded as infinitely small quantities, that is S1 = S3 = 0 [16,17]. In addition, d(s, z) is the condensate film thickness. Integrating momentum equations with boundary conditions, and substituting the solutions into the continuity Eq. (6), the liquid film thickness equation is obtained as follows.



  kf T s  T w 1 @ @p ðq  qv Þg @d3 þ d3 ¼ 3l @s @s 3l @z qhfg d

ð11Þ

The Eq. (8) shows that there is no pressure variation in n direction of the liquid film, so the pressure can be replaced by pressure at the interface. The liquid pressure at the interface is given as follows. Fig. 3. Schematic diagram of a small element cut out from the liquid film.

p ¼ ps 

c Ri

ð12Þ

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

The second term on the left-hand side of (11) is much smaller than its first term, so we can neglect the second term during calculation, and the error is less than 2%. Substituting (12) into the simplified equation of (11), we get



 

c @ 3d 1 d ds Ri 3l @s

¼

kf T s  T w d

ð13Þ

qhfg

Eq. (13) for the determination of condensate film thickness d   d 1 . Referring to the Ref. [7], the equation involves the value of ds Ri to determine the curvature of interface is written as

1 @ 2 d=@s2  Ri ð1 þ @d=@sÞ3=2

ð14Þ

Substituting the Eqs. (14) into (13), the equation for the thickness of the condensate film in region I and II is obtained:

"

c d 3d @ 2 d=@s2 d ds ð1 þ @d=@sÞ3=2 3l ds

!#

¼

kf T s  T w d

qhfg

ð15Þ

The boundary conditions are



d ¼ d0 ;

@d @s

¼ 0 at s ¼ 0 ðpoint DÞ

ð16Þ

k ¼ kL at s ¼ sF ðpoint FÞ

where kL denotes the slope of interface at point L in the oxy coordinate system. 2.2.2. Regions III and IV Using the assumption (5), the interface is approximately of a circular arc shape under the action of surface tension, so the curvature of the interface approaches a constant. Therefore, the driving force in horizontal direction decreases sharply, so the liquid flows mainly along the vertical direction. Assuming there is only z-direction flow in region II, the momentum equation is given by

@ 2 uz @ 2 uz g þ 2 þ ðq  qv Þ ¼ 0 @x2 @y l

@T ðsÞ

f of the liquid film at A-H can the temperature gradient @n be calculated. The heat flux per unit length along z-direction is

Z

D 0

ð18Þ

Thermal conduction equation: According to assumption (6), only two-dimensional thermal conduction within the grooved plate as well as within the liquid film will be considered. The thermal conduction equation is

¼0

ð19Þ

kf @T f ðsÞ ds AG @n

ð21Þ

where D is the length of the curve A-H, AG is the heat transfer area per unit length. (6) Solving Eq. (17) subject to boundary conditions (18) by the finite element method, we obtain uz, and the mass flow rate in regions III and IV can be calculated as

Z

MIII ¼

!

SIII

Z

quz dS; MIV ¼

SIV

quz dS

at x ¼ 0; x ¼ p=2

ð22Þ

where SIII , SIV is the area of region III, IV respectively. The total mass flow rate of regions III and IV is

M ¼ M III þ M IV

The boundary conditions are

8 @T > < @x ¼ 0 T ¼ Ts > : T ¼ Tc

j1  T w ðsÞ=T 0w ðsÞj < 1  106 is satisfied. (5) With the temperature distribution in liquid film obtained,

qðzÞ ¼

8 > < uz ¼ 0 at FL; CJ; F  H; and A  C @uz ¼ 0 at HM; AI @x > : @uz ¼ 0 at IJ; LM @n

@2T @2T þ @x2 @y2

(2) Solving the Eq. (15) subject to boundary conditions of region II at point D determined based on the result of region I, the interface of kJ is obtained. (3) Because the interface in regions III and IV is smoothly connected with the interface in regions I and II respectively, and is approximately of the arc shape, the interface LM and IJ in regions III and IV can be determined respectively. (4) With the determined interface, Eq. (19) was integrated with the boundary conditions (20) by using the finite element method. The calculations produce the temperature distribution T 0w ðsÞ of the grooved surface. Let T w ðsÞ = T 0w ðsÞ and then repeat (1–3), until the convergent condition

ð17Þ

The boundary conditions are

k

0

the interface KL of region I is obtained. kL is the last calculation of kL . As shown in Fig. 4, kL can be determined as follows: the angle between y-axis and the tangent lines of grooved surface at point F is a, and the angle between this line and the tangent line of interface at point L is b, so we   obtain kL ¼ tan p2  a þ b . We define the thin film in this way - the thickness of film is thinner than one-tenth of the groove height. FL is the boundary 1 between regions I and III, so it satisfies the equation jFLj ¼ 10 jOAj. The film thickness in the trough region will increase with the increasing of z values (see details in Section 4), and the location of points F and L will move towards crest of the groove. During this case, the value of b can increase from 0° to 30°. In the present analysis angle b is taken as 15° for various a. The calculations show that the difference of the results between taking the angle b from 0° to 30° is no more than 5%. Therefore, the change of b gives little effect on the results.

ð23Þ

(7) When the length of the groove is z, the amount of heat flux is

ð20Þ

at I  M at ON

q¼

hfg M ¼ AG

Z

z

qðzÞdz

ð24Þ

0

Differentiating the above equation with respect to sF, s-coordinate of point F, we obtain

2.3. Solving procedure (1) Define the temperature difference between the vapor and the cooling surface is DT ¼ T s  T c . Assuming the temperature distribution of the grooved surface A-H is T w ðsÞ, the solution of the liquid film thickness Eq. (15) subject to the boundary conditions (16) is obtained by using Runge-Kutta method. The value of d0 is corrected iteratively, until the 0

convergent condition j1  kL =kL j < 1  105 is satisfied. Then

dz 1 dq ¼ dsF qðzÞ dsF

ð25Þ

The boundary conditions are

z ¼ z0 ; q ¼ q0 at sF ¼ sF0

ð26Þ

Because values of sF0, q(z) and q0 in boundary conditions (26) are all already known, we can find the relationship between the amount of

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

Involute grooves

Trapezoid grooves Fig. 4. Method of defining kL.

heat flux q and z by solving Eq. (25) subject to (26) with the RungeKutta method. z0 in boundary conditions (26) denotes the length of the initial portion of the groove, in which the liquid film both on crest and in trough are thin, and is the key to solving Eq. (25). As an analogy to the Nusselt solution, the relation between q and z can be expressed as q = a . . . zb. Giving an arbitrary value to z0, we may obtain q = a . . . zb + c. In order to make c equal 0, the value of z0 is modified iteratively so as to satisfy the convergent condition c < 1  106 by solving Eq. (25). 3. Experiments for condensation heat transfer 3.1. Experimental apparatus and heat transfer plates The experimental apparatus, shown schematically in Fig. 5, consists of four parts: the steam generating system, cooling system, steam condensing system, and data acquisition/control system. Steam at about 100 °C was supplied from a steam generator, and then passed through a superheater to remove mist. The dry steam was led into the condensing chamber and condensed on the surface of the test condensing plate. The steam velocity in the duct was obtained by dividing total flow rate of steam through the duct by the cross-sectional area and was maintained at approximately 5 ± 0.5 m/s. The steam temperature was measured by a platinum resistance thermometer (Pt 100), with an accuracy of ±0.1 °C. The steam pressure was monitored by a manometer, and maintained at atmospheric pressure by adjusting the valve installed at the outlet of the condensing chamber. The condensing chamber is shown in Fig. 6. This condensing chamber can freely adjust angle (from horizontal to vertical) by rotating device to satisfy experiment requirement. A baffle was set at top of the condensing plate to eliminate the sweep effect of steam flow to the condensation droplets. The condensate on the test surface was collected by a funnel and then flowed into a measuring tube. A window was mounted at the chamber for visual observation of the condensation process. The chamber was insulated by rock

wool with thickness of 50 mm to diminish the condensation that occurred at other place. By doing this, the influence of the unwanted condensation was very limited and can be neglected in the experiments. The excessive steam and the condensate condensed on the other surfaces of the condensing chamber flowed into an auxiliary condenser and were collected by another measuring tube. Water was used to cool the test surface. It was kept at a constant temperature in a cooling-water tank and was sprayed on the backside of the test condensing plate by means of a pump and a nozzle. Heat flux through the heat transfer plate was regulated continuously by adjusting the pressure and flux of cooling water. The inlet and outlet water temperature were also measured by two platinum resistance thermometers (Pt 100). The uncertainty of temperature measurement was estimated as ±0.1 °C. Schematic diagram of the condensing plates were shown in Fig. 7. Each plate was made of 4.0 mm-thick low carbon steel. The condensing surface had an area of 50  50 mm2, and 20 grooves with pitch P = 2.5 mm, the height is variable for various grooves. The radius of the arcs B-E and G-H were both 0.3 mm. Two holes of 1 mm in diameter were drilled at the condensing plate. Their depths were 15 mm and 25 mm, respectively. Two sheathed copper-constantan thermocouples (type T, Omega Engineering Inc.) were inserted into the two holes to measure the wall temperature. The uncertainty of temperature measurement was estimated as ±0.1 K. Experimental data were collected using Agilent 34970A data acquisition unit, and all the data including the pressure, thermocouple readings, surface subcooling temperature, heat flux, condensation heat transfer coefficients can be calculated by data reduction software and the real-time data profiles can be displayed on computer monitor. 3.2. Experimental data reduction The heat transfer coefficient h is defined as

h¼

q q ¼ DT 0 ðT s  T w Þ

ð27Þ

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

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6 4 7

9

2

5

3 12 11

10

8

1

15

14

13

Fig. 5. Schematic diagram of the experimental apparatus.

and the heat transfer rate calculated from the flow rate and temperature increment of the cooling water was obtained from the following equation

Q 2 ¼ G  C pl  ðT out  T in Þ

ð29Þ

The difference between Q1 and Q2 was less than 5%, and therefore it is reasonable to calculate the heat flux through the grooved surfaces using the following equation.

q¼

x

Q1 þ Q2 2Ag

ð30:1Þ

where Ag is the area of the grooved surface. For the smooth surface, the heat transfer can be obtained from

z

q¼

Q1 þ Q2 2As

ð30:2Þ

The temperature of the condensing surface Tw was calculated from Eq. (31) by using the heat flux q obtained in Eq. (30).

Tw ¼ Ti þ

Fig. 6. Photograph of the condensing chamber.

In order to calculate the heat transfer coefficient with Eq. (27), it is necessary to obtain the heat flux q and surface temperature Tw. In the current research, the heat produced by condensation on the condensing surface was transferred to the cooling water through the heat transfer plate. The heat transfer rate calculated from the amount of the condensate per unit time can be expressed as

Q1 ¼

M  hfg t

ð28Þ

q  Dl k

ð31Þ

where Dl is the distance between the condensing surface and the measuring points of the sheathed thermocouples inserted into the heat transfer plate. 4. Results and discussion 4.1. Analyses of influencing factors The comparisons of film thickness for grooves are displayed in Fig. 8. The calculations of the present grooves (involute groove and trapezoid groove) were obtained from the theoretical models

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

Fig. 7. Schematic diagram of the condensing plate.

0.08

0.1

0.06

Film thickness δ / mm

Film thickness δ / mm

0.07

Involute groove Trapezoid groove Sinusoidal groove [18]

0.05 0.04 0.03 0.02

h/P = 0.3 h/P = 0.4 h/P = 0.5 h/P = 0.6

0.01

h/P = 0.5, h/R p= 1/5

0.01 0.0

h/R p = 1/5

0.2

0.4

0.6

Coordinate s

0.8

0.0

1.0

0.2

0.4

0.6

Coordinate s

0.8

1.0

Fig. 9. Thickness of condensate film for various values of h/P. Fig. 8. Comparisons of film thickness for various grooves.

0.1

Film thickness δ / mm

in Section 2, while the results of sinusoidal groove were based on the model reported in literature [18]. As shown in the figure, the thickness of liquid on present grooves decreased first and then increased with increasing the values of s, but the thickness of liquid on sinusoidal groove kept decrease. The reason for this was that: (a) the maximum curvature was obtained at point D for the present grooves, so the liquid started to flow in opposite directions from here under the action of surface tension; (b) unlike the these two grooves, the liquid start to flow from the vertex of the sinusoidal groove (at point A, s = 0). Through the comparison between the involute groove and trapezoid groove, it could be found that a larger thin film area can be obtained on involute groove, and more import, the adoption of grooved surface made the liquid film thinner than trapezoid groove on the crest. These numerical analyses indicated that involute groove can produce larger surface tension and lower flow resistance in horizontal direction owing to its sleek contour. So we could expect to achieve a lower thermal resistance and a better heat transfer performance on involute groove.

h/P = 0.5 h/Rp= 1/20 h/Rp= 1/10 h/Rp= 1/5 h/Rp= 1/3

0.01

0.0

0.2

0.4

0.6

0.8

1.0

Coordinate s Fig. 10. Thickness of condensate film for various values of h/Rp.

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

0.50 0.45

us

0.18 0.16

0.40 0.35

0.14 0.12 0.10

0.30 0.25

0.08

0.20

0.06

0.15 0.10

uz

0.04 0.02

0.05 0.00 Involute groove Trapezoid groove -0.05 Sinusoidal groove

0.00 -0.02 -0.04 0.0

0.2

0.4

0.6

Coordinate s

0.8

Vertical velocity uz / ·

0.20

Horizontal velocity us / ·

Figs. 9 and 10 show the thickness of condensate film on involute groove for various values of h/P and h/Rp (taking z = 10.0 mm for example) on semilog coordinates respectively. As the profile radius of curvature was a function of the opening angle and pressure angle, we took hi = 30°and ai = 20° as a typical example during analyses, the similar conclusions could be obtained by choosing the values of hi and ai from reasonable range. The nondimensional number h/P and h/Rp were proportional to the slope of tangent line of measured point and the change rate of curvature respectively. As expected, the film thickened quite rapidly in the trough region while remaining thin over the crest, forming an obvious uneven distribution. However, the thickness of liquid film significantly decreased with the increase of h/P and h/Rp in regions I and II. It was mainly because that the larger values of h/P and h/Rp could both lead to greater variations in curvature, which can cause a local larger surface tension gradient along the interface of the groove and liquid. This surface tension gradient drove the liquid into the trough region effectively and also played the roles on thinning film thickness and enhancing heat transfer. After the two ratios reaching certain extent, their effects on the film thickness became very limited due to the limitation impact on the profile radius of curvature. Fig. 11 shows the thickness of liquid film at various z positions. When z approached zero, the film thickness was extremely thin and uniform. As the value of z increased, the film thickness increased sharply in the trough region while remaining almost unchanged over the crest. This was mainly because that the condensate on crest flowed into the trough region under the action of the surface tension, so it can keep thin even at the low part of the plate (for larger values of z). However, the condensate would accumulate in the trough region, not only from the horizontal flow on the crest but also vertical flow from the top of the plate. Moreover, we found that the rate of accumulate was gradually declined owing to the reduction of the thin film region on groove surface. The velocity distributions at the condensate-vapor interface are shown in Fig. 12. The horizontal components of the velocity us on present grooved surfaces approached zero at point D and the liquid started to flow in opposite directions. So when the s values increased, the velocity us decreased gradually in regions IV and II to zero and then started to rise with increasing the values of s, reached a maximum value in region I and then decreased to zero again in the trough region. But for sinusoidal groove, the values of us reached zero at the center of crest (s = 0), and thus the velocity us increased with increasing the values of s in regions I, II and IV and also reached a maximum value in region I, and then decreased to zero again in the trough region. Comparing with trapezoid groove, we found that the condensate flowed faster in most regions

1.0 Involute

Fig. 12. Velocity distributions at the condensate-vapor interface.

of involute groove. These results totally corresponded with the conclusions of thickness distribution of condensate film. It was clear from Fig. 12, which also showed the vertical components of the velocity uz, that the condensate flowed at very low vertical velocity (uz  0) in regions I, II and IV, but in region III the value of uz increased rapidly to a relatively high level. The calculation results suggested that the flow rate in drainage channel was much greater than that on crest, and almost all of condensate discharged through the drainage channel. Therefore, the flow process of condensate included the two steps: in the beginning, the horizontal flow played a dominate role, and the condensate was transfer from crest to trough; then the gathered liquid was discharged under the action of gravity in trough, in this case, the vertical velocity was far larger than the horizontal velocity. @p The above phenomena can be explained by the ratio of @n to @p caused by surface ðq  qv Þg from the momentum Eq. (8), where @n tension of liquid was the causation and motivity of the horizontal flow; ðq  qv Þg caused by gravity drove the condensate to flow @p and ðq  qv Þg were both external forces acting on vertically. @n the per unit volume of condensate. Fig. 13 shows the values of @p j @n  ðq1q Þg j along s axis in logarithmic coordinates. As shown in

v

@p the figure, in region I, II and IV, @n was 1–2 orders of magnitude higher than ðq  qv Þg, so it was a surface tension dominated flow @p was much smaller than phenomenon. While in regions III, @n (q  qv)g, less than 10%, so the gravity played the leading role for this flow phenomenon.

0.18 0.16

h/P = 0.5, h/Rp = 1/5

Film thickness δ / mm

0.14

z = 20 mm

0.12 0.10 0.08

z = 10 mm

0.06 0.04

z = 5 mm

0.02

z = 0.1 mm

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Coordinate s Fig. 11. The thickness of liquid film at various z positions.

Fig. 13. The values of

@p @n

 ðq1q

v Þg

along s axis in logarithmic coordinates.

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B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

374

5.0

372

4.0

371 370 369 368 367

Involute groove Trapezoid groove Sinusoidal groove [18]

366 365 0.0

0.2

0.4

0.6

3.5 3.0 2.5 2.0 1.5 1.0

0.8

1.0

Coordinate s

5

10

15

20

25

30

35

40

Subcooled temperature T / K

Fig. 14. Distributions of wall temperature on various grooves.

Heat flux q / W m -2

Experimental data of Involute grooved surface Calculations of Involute grooved surface Experimental data of Trapezoid grooved surface Calculations of Trapezoid grooved surface Experimental data of Smooth surface Nusselt model [19] Zhu [20]

4.5

Heat flux q / W m-2

Wall temperature Tw / K

373

10 5

Fig. 16. Comparisons of heat flux between calculations and experimental values.

6

4.2. Comparisons between calculations and experiments

5

In order to verify the above analytical method, taking involute and trapezoid grooved condensing surface as models. We first calculated the amount of heat flux at various temperature differences, and then compared the solutions with the experimental results. Fig. 16 illustrated the comparisons of heat flux between the calculations and experimental data for various grooved condensing plates. Besides, calculated results based on Nusselt model in Ref. [19] and the experimental results of smooth surface were also illustrated in this figure, and the calculations agreed well with the experimental results. This indicated that the test system was stable and reliable. As shown in Fig. 16, the experimental data demonstrated that the heat transfer of grooved surfaces was enhanced by 1.5–2.5 times higher than smooth surface. For the grooved surfaces, the experimental data of heat flux increased with the rise in subcooled temperature, and also the calculations followed the same trend. The calculations and the experimental values showed in good agreement with a maximum deviation of 15%. It was proved that the present theoretical analyses about heat transfer of filmwise condensation both on involute and trapezoid grooved surfaces were feasible. Through the comparison of data from experiment and literature [20], we found that the heat transfer performance of involute grooved surface was at least 20% and 50% higher than that of trapezoid grooved and rectangular grooved surfaces respectively, and the enhancement of heat transfer became more obvious in high subcooling. The experimental results above were the strong evidence to support the conclusion of film thickness and heat flux distribution.

4 3 2 Involute groove Trapezoid groove Sinusoidal groove [18]

1 0 0.0

0.2

0.4

0.6

0.8

1.0

Coordinate s Fig. 15. Distributions of heat flux on various grooves.

Figs. 14 and 15 show the distribution of wall temperature Tw and heat flux q on various grooves respectively. Through analyzing the trends of temperature and heat flux curves of grooves, it was found that the distributions of surface temperature and heat flux presented obvious lateral inhomogeneity. Both temperature and heat flux curves of present grooves increased in regions IV, II and I firstly, and then decreased in regions I and III, and the maximum wall temperature and heat flux were both obtained in regions I, owing to its thinnest of liquid film. But for sinusoidal groove, the both curves kept decrease resulting from the gradually thickened liquid film along with s axis. Moreover, wall temperature in regions I, II and IV was higher than that in region III, the similar trend was also followed by the heat flux. It was mainly because that the liquid film was much thicker in region III, and the thermal resistance of this region was larger than other regions, so the heat flux was mainly transferred through the groove in regions I, II and IV. By comparing the wall temperature and the heat flux between the involute groove with trapezoid groove, we found that both the wall temperature and the heat flux in regions I, II and IV were slightly higher for involute groove. The proposed reason for this was that the surface shape and the variations in curvature of involute groove caused a greater surface tension to the condensate film on the shoulder of the crest, so that thinner liquid film, higher wall temperature and larger heat flux could be easily obtained in these regions. While in region III the wall temperature and heat transfer achieved from the two groove were almost equal, probably owing to the similar outlines in this region for the two grooves.

5. Conclusions Heat transfer and fluid flow for laminar film condensation on various grooves has been studied using numerical analyses. The coupled fluid flow and two-dimensional thermal conduction equations were established. By numerically solving equations, the calculations of heat flux and fluid flow for grooved plates were obtained. The influences of non-dimensional number h/P and h/Rp on film thickness and heat flux were discussed. The velocity distributions at the condensate-vapor interface and distributions of wall temperature on various grooves were also studied systematically. Experimental studies were conducted on involute and trapezoid grooved surfaces, and the experimental data of heat flux were obtained and compared with numerical results under various temperature differences. The following conclusions were obtained from the above studies.

Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207

B. Qi et al. / Applied Thermal Engineering xxx (2016) xxx–xxx

(1). The gradient of surface tension of the liquid film drove the liquid to flow from the crest into the trough, forming a thin liquid film on the crest and thick film in trough. The thickness of liquid film was significantly influenced by the nondimensional number h/P and h/Rp especially in regions I and II owing to the greater variations in curvature and larger surface tension gradient along the interface of the groove and condensate. (2). The horizontal velocity us decreased first and then increased gradually as the s values increased, and reached a maximum value in region I and then decreased to zero in the trough region. On the contrary, the condensate flowed at very low vertical velocity in regions I, II and IV, but in trough region the values of uz increased rapidly to a relatively high level. (3). The distributions of surface temperature and heat flux presented obvious lateral inhomogeneity. The wall temperature and heat flux in regions I, II and IV were higher than those in region III, and the both curves increased in regions IV, II and I firstly, and then decreased in regions I and III, and the maximum wall temperature and heat flux were both obtained in regions I. Moreover, the distributions of wall temperature and heat flux on involute groove were more favorable to reduce thermal resistance and enhance heat transfer than trapezoid and rectangular grooves. (4). The comparisons of experimental data indicated that the grooved surfaces could enhance the heat transfer at least 1.5 times higher than smooth surface. Moreover, the heat transfer performance of involute grooved surface was at least 20% and 50% higher than that of trapezoid grooved and rectangular grooved surfaces respectively. (5). The calculations followed the same trend as experimental data. The results were in good agreement with a maximum deviation of 15%. The present analyses were reliable, and can be used in the parameter design and heat transfer calculation of trapezoid grooved surfaces. Acknowledgement The authors are grateful to the financial supports by the National Natural Science Foundation of China under the Grants of 51306141 and the Specialized Research Fund for the Doctoral Program of Higher Education under the Grants of 20120201120068.

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Please cite this article in press as: B. Qi et al., Enhancement of condensation heat transfer on grooved surfaces: Numerical analysis and experimental study, Appl. Therm. Eng. (2016), http://dx.doi.org/10.1016/j.applthermaleng.2016.10.207