The numerical calculation of heat transfer performance for annular flow of liquid nitrogen in a vertical annular channel

Cryogenics 41 (2001) 231±237

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The numerical calculation of heat transfer performance for annular ¯ow of liquid nitrogen in a vertical annular channel Shufeng Sun a,*, Yuyuan Wu b, Rongyi Zhao a b

a School of Architecture, Tsinghua University, Beijing, People's Republic of China School of Power and Energy Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi, People's Republic of China

Received 17 September 2000; accepted 9 March 2001

Abstract According to a separated phase ¯ow model for vertical annular two-phase ¯ow in an annular channel, the liquid ®lm thickness, distributions of velocities and temperatures in the liquid layer are predicted in the range of heat ¯uxes: 6000±12000 W=m2 , mass ¯ux: 500±1100 kg=m2 s. The pressure drop along the ¯ow channel and heat transfer coecient are also calculated. The liquid ®lm thickness is in the order of micrometers and heat transfer coecient is 2800±7800 W=m2 K of liquid nitrogen boiling in narrow annular channels. The measured heat transfer coecient is 29% higher than the calculated values. With the mass ¯ux increasing and the gap of the annular channel decreasing, pressure drop and heat transfer coecient increase. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Annular ¯ow; Heat transfer coecient; Annular channel

1. Introduction Narrow channel heat transfer enhancement technique has been successfully used in condenser±evaporator in air-separation equipment. It is also applied in gas generator of liquid oxygen. In the internal compression processes in air-separation plant the liquid oxygen from liquid pool at the bottom of upper distilling column is compressed to very high pressure up to 15 MPa by liquid oxygen pump and then is evaporated in gas generator to ®ll to the high pressure oxygen container. Narrow channel heat transfer enhancement technique has notable advantages in small temperature di€erence, high heat transfer eciency and compact con®guration without complex machining or additional surface processing. Two-phase annular ¯ow patterns accompanying with heat transfer are prevalent in many industrial processes. The reliable prediction of pressure drop and heat transfer rate associated with these processes are essential to reliable, ecient heat exchangers for energy savings through optimization of thermal processes. Annular ¯ow pattern frequently occurs in conjunction with evaporative heat transfer in tubes. Experi*

Corresponding author. E-mail address: [email protected] (S. Sun).

ments have showed that annular ¯ow and slug ¯ow are main ¯ow patterns in narrow annular and lunate channels [1,2], and vapor slug is long, liquid column is short. This kind of quasi-annular ¯ow takes up most length of the channel. Annular ¯ow is characterized by an inherently unsteady liquid ®lm ¯owing along the wall parallel to a ¯owing vapor core carrying entrained liquid droplets. The liquid ®lm structure is in¯uenced by many system variables, such as system geometry, ¯ow orientation (up, down, microgravity, or horizontal ¯ow), and liquid/vapor velocity di€erence. Evaporative two-phase ¯ow heat transfer is classi®ed by two heat transfer regimes: the convective regime and the nucleate boiling regime. In the convective regime, nucleate boiling is completely suppressed, and the heat transfer is governed by bulk turbulent motion of the liquid ®lm. Heat transfer in the nucleate boiling regime is governed by the incipience, growth, and departure of vapor bubbles along the heating surface. In two-phase ¯ow analytical model the ¯ow is usually assumed to be steady ¯ow, the geometry of the ¯ow channel is simpli®ed, and conservation law of mass, momentum, and energy are used to predict pressure drop and heat transfer rates. Separated ¯ow model has been discussed in detail by Hewitt and Hall-Taylor [3], Owen and Hewitt [4], Fu and Klausner [5]. In this paper,

0011-2275/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 1 - 2 2 7 5 ( 0 1 ) 0 0 0 5 8 - 3

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S. Sun et al. / Cryogenics 41 (2001) 231±237

Nomenclature A area d diameter E liquid mass fraction entrained f friction factor F Eq. (16) G mass ¯ux g gravitational acceleration G0 …de gql †=…fs qv u2g † h heat transfer coecient hlg latent heat of vaporization k thermal conductivity p pressure Pr Prandtl number qw heat ¯ux r radial coordinate r01 inner radius of outer tube r02 outer radius of inner tube Re Reynolds number s the gap of lunate channel T temperature u velocity 1=2 u …s0 =ql † a separated ¯ow model, its boundary conditions, and their closure relations are presented to predict pressure drop and heat transfer rate for two-phase annular ¯ow of liquid nitrogen in annular channel.

2. Model construction The ¯ow con®guration is depicted in Fig. 1, in which there exists cocurrent annular ¯ow with entrainment through an annular channel and heat input at the outer wall. The following assumptions are in the model: (1) the ¯ow is incompressible; (2) the ¯ow is steady; (3) the liquid ®lm thickness is uniform around the tube periphery; (4) the liquid/vapor interface is smooth; (5) the pressure is uniform in the radial direction; (6) evaporation occurs at the liquid/vapor interface; (7) liquid droplets entrained in the vapor core are uniformly distributed. Following Hewitt and Hewill±Taylor, a force balance on cylindrical element of the liquid ®lm on the outer tube in which acceleration is ignored leads to s 2pr dz ˆ si 2pri dz

p…r2

x y y‡ z a r em eH k l m q s

vapor quality distance from the wall yu =m axis coordinate void fraction liquid ®lm thickness eddy-viscosity eddy thermal di€usivity thermal di€usivity viscosity kinematic viscosity density shear stress

Subscripts e g i l m1; m2; m3 s sat

equivalent vapor liquid/vapor interface liquid constants solid saturated condition

du s ˆ : dy ll ‡ em ql

Eqs. (1) and (2) may be combined, which results in      du si r01 d 1 dp r01 y ˆ  ql g ‡ dy 2 dz ll ‡ em ql ll ‡ em ql r01 y " # 2 r01 d  1 ; …3† r01 y

ri2 †…dp  ql g dz†;

that is  2   r  1  dp ri r 2 i s ˆ si  ql g ‡ 2 dz r r and using the concept of eddy-viscosity,

…2†

…1† Fig. 1. Sketch of annular ¯ow with heat transfer.

S. Sun et al. / Cryogenics 41 (2001) 231±237

where s is the shear strees, dp=dz is the axial pressure gradient, em is the eddy-viscosity, d is the liquid ®lm thickness, r is the radial coordinate, y is the distance from the wall. The ``+'' applies to up¯ow and `` '' applies to down¯ow. The required boundary conditions are u ˆ 0;

y ˆ 0;

du si ˆ ; dy ll ‡ em ql

…4† y ˆ d:

…5†

A force balance on cylindrical element of the liquid ®lm on the inner tube is analyzed as above, and results in      du si r02 ‡ d 1 dp r02 ‡ y ˆ  ql g ‡ dy 2 dz ll ‡ em ql ll em ql r02 ‡ y " #  2 r02 ‡ d  1 : …6† r02 ‡ y The shear stress on the two interfaces is equal, and includes momentum exchange due to entrainment; a momentum balance on the vapor core results in qg g‰x ‡ E…1 x†Š dp …2p…ri1 ‡ ri2 †si G2 d ˆ  ‡ dz p…ri12 ri22 † x ‡ E…1 x†…qg =ql † a dz " # x2 …1 E†2 …1 x†2 x E…1 x†x  ‡ ; aqg ql …1 a† ql aE…1 x† aqg

…7†

where p…ri1 ‡ ri2 † …r01 d† ‡ …r02 ‡ d† ˆ ˆ 2 2 p…ri1 ‡ ri2 † …r01 d†2 …r02 ‡ d†2 r01

1 r02

is rational that the mass ¯uxes on two walls are equal. Eq. (9) can be written as  Z d  re y dy: …10† G…1 x†…1 E† ˆ ql u 1 r01 2 0 Assuming that the two-phase mixture is saturated, an energy balance is written as G dx Ae hlg ˆ 2pr01 dz qw ; where Ae =2pr01  de =2 ˆ re , the vapor quality change along the axis is dx qw ˆ : dz Gre hlg

…11†

There exists energy transport through the liquid ®lm. Because turbulent di€usion across the thin ®lm is typically much greater than convection downstream, a simpli®ed energy equation is   d dT …kl ‡ eH † ˆ 0; …12† dy dy where kl is the liquid thermal di€usivity, and eH is the eddy thermal di€usivity. The heat transfer coecient is de®ned as qw hˆ : …13† Tw Tsat The required boundary conditions of energy equation are qw ˆ

2d

233

kl

: T ˆ Tsat ;

dT ; dy

y ˆ 0;

…14†

y ˆ d:

…15†

According to the concept of hydraulic equivalent diameter, r01 r02 ˆ re , Eq. (7) can be written as qg g‰x‡ E…1 x†Š G2 d dp 2si ˆ  ‡ dz re 2d x ‡ E…1 x†…qg =ql † a dz " # 2 2 x2 …1 E† …1 x† x E…1 x†x  ‡ : ‡ aqg aqg ql …1 a† ql aE…1 x†

3. Empirical closure relation

…8†

A mass balance on the liquid ®lm is expressed as: Z d Z d0 G…1 x†…1 E†Ae ˆ ql u 2pr dy ‡ ql u0 2pr dy; 0

0

…9† where Ae is the cross-sectional area of annular channel. The inner wall of the annular channel is protuberant, the liquid layer thickness is thinner with the action of surface tension, whereas the outer wall of the annular channel is concave, and surface tension makes the liquid layer thicker. However, heat is input on the outer wall, the liquid layer evaporating makes the liquid layer thinner, and the annular channel is narrow, the di€erence of diameters of outer and inner tube is not large. With the above e€ects and entrainment, the assumption

The relations above are not closed, the equations about eH ; em ; si and E are needed. The interfacial shear stress is one of the most important variables that in¯uence the solution of the separated ¯ow model. Henstock and Hanratty [6] give the best results after examining many di€erent empirical correlations. The equations are as follows: 1 si ˆ fi qg u2g ; 2

…16†

( fi ˆ 1 ‡ 1400F 1 fs h Fˆ

0:707Re0:5 l


2:5

" exp

…1 ‡ 1400F † 13:2G0 F

‡ 0:0379Re0:9 l Rem2 g


2:5 i0:4

1:5

ll lg

#) ;

!m1 

…17†

qg ql

m3 ; …18†

234

S. Sun et al. / Cryogenics 41 (2001) 231±237

em 3 ˆ 0:001y ‡ ; y ‡ < 5; m    2  du y‡ 1:0 em ˆ Ky 1 exp dy 25

y 1:5 ; d

…19†

y ‡ > 5; Prt ˆ 1:07; y ‡ < 5; Prt ˆ 1 ‡ 0:855 tanhb0:2…y ‡

7:5†c;

y ‡ > 5;

…20†

where K ˆ 0:41 isp Von Karman's constant, y ‡ ˆ yu =m,   Prt ˆ em =eH , u ˆ sw =ql is friction velocity. Entrainment and void fraction are calculated as follows: " # a x ql Eˆ 1 ; …21† 2 …1 d=re † 1 x qv aˆ Fig. 2. Flow chart of solution procedure.



C0 1 ‡

1 1 x qv x ql



q

‡ Vvj Gxg

;

where ug ˆ

Gx ; qg

Reg ˆ

fs ˆ 0:046Reg 0:2 ;

Gxde ; lg

Rel ˆ

G0 ˆ

de gql : fs qv u2g

G…1

x†…1 ll

E†de

;

m1, m2 and m3 in Eq. (18) have been empirically determined, m1 ˆ 0:8, m2 ˆ 0:9, m3 ˆ 0:5. Turbulence measurements for annular ¯ow liquid ®lms are not available. Usually the wall turbulence is assumed that it is similar to that of single-phase ¯ow and single-phase ¯ow correlations for the liquid ®lm eddyviscosity are used. Following the analysis of Kays [7], the liquid ®lm eddy-viscosity and turbulent Prandtl number are, respectively, computed from

Fig. 3. Camparison between measured and predicted pressure gradients (Shearer and Nedderman data).

Fig. 4. Velocity in liquid layer.

…22†

S. Sun et al. / Cryogenics 41 (2001) 231±237

where C0 is an empirically determined distribution parameter, and Vvj is an empirically determined drift veloctiy, Klausner et al. [8] have recommended that for up¯ow C0 ˆ 0:98, Vvj ˆ 1:12 m=s.

235

5. Result and discussion

The solution procedure is shown in Fig. 2, which is described simply as: (1) an initial guess is made for the ®lm thickness, using it to calculate the pressure gradient; (2) velocity equation and its boundary conditions are used to calculate the liquid ®lm velocity pro®le; (3) mass conservation equation acts as criterion to check mass balance; (4) if mass conserving, energy equation is calculated with the temperature boundary conditions to get the liquid ®lm temperature pro®le, and the heat transer coecient. If mass equation does not balance, steps (1)±(3) are repeated.

In order to check the correction of the procedure, Shearer and Nedderman [9] data are used. Their experiment has been made with gas±water system. The comparison result is shown in Fig. 3. Average error between measured and predicted pressure gradients is 19.9%. The model is used to calculate liquid nitrogen boiling heat transfer in annular channels. The results of velocity and temperature distribution in liquid layer are shown in Figs. 4 and 5. Abscissa denotes the distance from wall, ``0'' is at the wall, and ``1'' at the liquid/vapor interface. With the e€ect of viscosity, liquid layer velocity at the wall is zero, as shown in Fig. 4. As heat ¯ux increases, more liquid evaporates, the vapor velocity becomes larger, and the shear stress on the liquid/vapor interface increases, the drag force to liquid layer is bigger, thus the liquid velocity increases. Fig. 5 shows the temperature distribution in liquid layer. It is found that the temperature decreases from wall-temperature to saturation at liquid/vapor interface. With the increasing of

Fig. 5. Temperature in liquid layer.

Fig. 6. The liquid layer thickness.

4. Equation solution

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S. Sun et al. / Cryogenics 41 (2001) 231±237

Fig. 8. Comparison between calculated and measured heat transfer coecients.

Fig. 7. Pressure drop for annular ¯ow in annular channels.

heat ¯ux, the temperature gradient in liquid layer becomes sharper. Fig. 6 shows the relation of liquid layer thickness with mass ¯ux. It can be found that the liquid layer thickness connects consanguineously with heat ¯ux and mass ¯ux. The thickness increases with mass ¯ux increasing and heat ¯ux decreasing. Mass ¯ux increases with the same heat ¯ux, more quantity of liquid is not evaporated; as the volume of vapor is unchanged, the vapor velocity corresponding to the area of whole cross-section of the channel and liquid/vapor interfacial shear stress is unchanged, too. Thus, liquid layer thickness increases. If the mass ¯ux keeps constant, with heat ¯ux increasing, volume of vapor, vapor velocity, liquid droplet entrainment increases, and liquid layer velocity increases, the liquid layer thickness becomes thinner. Pressure gradient and heat transfer coecient are shown in Figs. 7 and 8. The e€ect of mass ¯ux and heat ¯ux on pressure drop is as that of velocity. From the above analysis, thinner the liquid layer thickness, sharper the temperature gradient in liquid layer, larger the

pressure gradient and heat transfer in annular channel. In Fig. 8, dashed lines represent the calculated values, and solid lines represent the measured heat transfer coecient of nitrogen boiling heat transfer in lunate channels [10]. In the experiment, nitrogen is pumped into a lunate channel and boils in it under the electric heating. The parameter ranges are: heat ¯ux 1900±13300 2 2 W=m , mass ¯ux 600±1200 kg=m s characteristic dimension: 0.5±1.5 mm. This lunate channel has a special structure: there are sharp angles at the connected line of two tubes. Experiment [1] shows that vapor bubble is usually generated along the connected line at ®rst. It is considered that the thickness of the liquid ®lm would be thinner at the wider interspace in cross-section of the channel. Comparison between calculated and measured heat transfer coecient shows that the average error is 29%. The measured heat transfer coecient is higher than the calculate one. The reason may be the special structure of lunate channel.

6. Conclusion There are a lot of boiling heat transfer correlations of two-phase ¯ow that have been reported in the literature. Such correlations normally attempt to predict heat

S. Sun et al. / Cryogenics 41 (2001) 231±237

transfer coecient both in the convective and nucleate ¯ow boiling regimes. However, these correlations are only satisfactory in certain conditions. The present separated phase ¯ow model is applicable to annular two-phase ¯ow in the pure convective heat transfer regime in annular channel. The model is based on fundamental conservation principles and provides information on pressure gradient, heat transfer coecient velocity and temperature distributions in the liquid ®lm.

References [1] Shufeng S, Rui Z, Liufang C, et al. Visual observation and experimental study on the two-phase ¯ow boiling heat transfer of liquid nitrogen in lunate channel. Cryogenics Supercond 1999;27(4):46±52. [2] Li Bin, et al. Visual study on boiling heat transfer and ¯ow in narrow annular channel and annular channel bundle. State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an

[3] [4] [5] [6] [7] [8]

[9] [10]

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