ENHANCEMENT OF BOILING HEAT TRANSFER AT ...

ENHANCEMENT OF BOILING HEAT TRANSFER AT NEAR-ZERO QUALITIES: A THERMODYNAMIC NON-EQUILIBRIUM SLUG FLOW MODEL Jader R Barbosa, Jr. Department of Mechanical Engineering Federal University of Santa Catarina Florianópolis – SC – 88040900 Brazil E-mail: [email protected]

Geoffrey F Hewitt Department of Chemical Engineering and Chemical Technology, Imperial College, London Prince Consort Road London, SW7 2BY, UK Email: [email protected]

ABSTRACT This paper presents a calculation methodology to predict the peaks in heat transfer coefficient at near zero equilibrium quality observed in forced convective boiling in vertical conduits. The occurrence of such peaks is typical of low latent heat, low thermal conductivity systems (such as refrigerants and hydrocarbons) and of systems in which the vapour volume formation rate for a given heat flux is large (low pressure water). The methodology is based on a model that postulates that the mechanism behind the heat transfer coefficient enhancement is the existence of thermodynamic non-equilibrium slug flow, i.e., a type of slug flow in which rapid bubble growth in subcooled boiling leads to the formation of Taylor bubbles separated by slugs of subcooled liquid. Results are compared with experimental data for forced convective boiling of pure hydrocarbons and show considerable improvement over existing correlations. 1. INTRODUCTION Several heat transfer equipment operate at moderate exit qualities and low to moderate mass and heat fluxes [1, 2]. Under these conditions, intermittent flow patterns such as slug flow and churn flow are more likely to prevail than annular flow. Wadekar and Kenning [1] and Wadekar [2] explored the reasons for the lack of experimental data and of suitable predictive methods, despite the industrial significance of such flows. As pointed out by Collier and Thome [3] and by Hewitt [4, 5], the classical representation of forced convective boiling under low wall heat flux conditions implies that the heat transfer coefficient rises in the subcooled boiling region and then is roughly constant until the point where the convective boiling dominant region is reached. Nevertheless, a number of studies demonstrate that, under some conditions, a very distinctive departure from the classic behaviour takes place. For example, recent experimental work on boiling of pure hydrocarbons (n-pentane and iso-octane) in a vertical electrically heated test section [6,7] showed that the heat transfer coefficient is locally enhanced in the region near zero equilibrium quality. The objective of this paper is to examine the mechanisms which can lead to this local heat transfer enhancement and to present a methodology capable of predicting the heat transfer coefficient behaviour near zero quality [6]. The postulated mechanism is the existence of local thermal nonequilibrium instabilities leading to an abrupt flow pattern transition in the subcooled region [8,9]. As will be seen, conditions of the experimental data against which the present method will be compared are favourable to the occurrence of such types of instability. This paper is organised as follows. In Section 2, an account of the experimental observations of the peaks in heat transfer coefficient is made together with possible explanations for the heat transfer enhancement. The proposed calculation methodology is described in detail in Section 3. Results are presented in Section 4, where the present method is compared with experimental data for two hydrocarbons and with predictions from existing correlations. Finally, conclusions are drawn in Section 5. -1-

2. HEAT TRANSFER COEFFICIENT PEAKS NEAR ZERO QUALITY 2.1. Experimental observations A detailed compilation of experimental data showing near zero equilibrium quality heat transfer coefficient peaks has been presented by Hewitt [4,5]. Cheah [10] observed peaks in heat transfer coefficient in boiling of water in vertical tubes at sub-atmospheric pressures (250 mbar). The peaks were not observed at atmospheric pressure. Similar behaviour has been reported by Thome [11] for boiling of refrigerants in horizontal tubes.

q& w . Tw − Tb

(1)

0.2

6000

30

0.0

4000

20

TEMPERATURE DIFFERENCE [deg C]

α =

Equilibrium subcooling

QUALITY

Here, the time averaged local heat transfer coefficient is defined as

HEAT TRANSFER COEFFICIENT [W/m2.K]

The first systematic investigation of the heat transfer enhancement near zero qualities was made by Kandlbinder [6], who conducted experiments on flow boiling of pentane and iso-octane in a (0.0254 mm ID, 8.5 m long) vertical tube. Kandlbinder’s experiments covered ranges of mass flux from 140 to 510 kg/m2s, of heat flux from 10 to 60 kW/m2, of inlet subcooling from 40 oC to 10 oC and of pressure from 2.4 to 10 bar. A typical example of the heat transfer coefficient enhancement phenomenon observed by 8000 40 Kandlbinder is depicted in Heat transfer coefficient (Kandlbinder, 1997) Figure 1. Equilibrium quality

Figure 1 also shows the profiles of local thermodynamic equilibrium 2000 10 quality and of local equilibrium subcooling (the -0.2 difference between the saturation temperature corresponding to the local 0 0 pressure and the 0.00 2.00 4.00 6.00 8.00 10.00 thermodynamic equilibrium DISTANCE [m] bulk temperature). Figure 2 shows the axial Figure 1. Heat transfer coefficient enhancement near-zero quality. distribution of temperatures Experimental conditions: fluid: n-pentane, inlet pressure: 6.0 bar, measured at the centre of the total mass flux: 377.4 kg/m2s,wall heat flux: 49.9 kW/m2, inlet pipe by Kandlbinder [6] for temperature: 67.5 oC. the same case depicted in Fig. 1. Profiles of calculated thermodynamic equilibrium bulk temperature and of saturation temperature are also presented. As can be seen, the existence of thermal non-equilibrium (characterised by the difference between the equilibrium temperature and the experimental values) persists over quite a considerable length (roughly up to 6 m in this particular case) downstream of the theoretical Point of Net Vapour Generation (NVG) [12]. More recently, Urso et al. [7] extended Kandlbinder’s database by conducting experiments using iso-octane over a lower range of mass fluxes (70 to 300 kg/m2s). The objective of Urso et al.’s work was to obtain a wider range of qualities over which the sub-annular regimes (bubble, slug and churn) would persist. The heat transfer coefficient peaks at near-zero quality were also observed consistently.

-2-

TEMPERATURE [K]

2.2. Prediction efforts 110.00 As yet, no calculation method satisfactorily predicts the trends of heat transfer coefficient leading to an 100.00 enhancement near zero quality. Kandlbinder [6] compared his data for the saturated boiling region 90.00 ( Tb = Tsat ) with the correlations of Chen [13], Shah [14], Kandlikar [15] and Steiner and Taborek [16]. Although the correlations performed 80.00 NVG (theoretical) satisfactorily over the whole saturated data range (average errors less than Experimental (Kandlbinder, 1997) 25% and standard deviations of the 70.00 Saturation temperature order of 20% to 30%), they have Equilibrium bulk temperature severely underpredicted the Slug temperature experimental heat transfer coefficient 60.00 in the region near-zero quality. Urso 0.00 2.00 4.00 6.00 8.00 10.00 et al. [7] were the first to include some DISTANCE [m] mechanistic insight into the calculation procedure. They have Figure 2. Profiles of experimental (centre line), saturation, made use of a modified version of a equilibrium bulk and liquid slug temperature profiles. model for slug flow heat transfer [1, Experimental conditions: fluid: n-pentane, inlet pressure: 2]. No attempt, however, was made to 6.0 bar, mass flux: 377.4 kg/m2s, wall heat flux: 49.9 kW/m2, take account of thermal noninlet temperature: 67.5 oC equilibrium effects (see Section 2.3. below) in the methodology. Their model provided results which generally agreed with their own data, but failed to predict the heat transfer coefficient behaviour near zero vapour qualities. 2.3. Discussion of mechanisms Hewitt [4,5] suggested that the most probable explanation for the occurrence of the near-zero quality peak in heat transfer coefficient was the existence of a local thermal instability of the type observed by Jeglic and Grace [8] and by Ishii [9]. Basically, in a situation where the conditions for bubble nucleation at the wall are poor, the layer of fluid adjacent to the wall becomes highly superheated. Therefore, once a bubble is nucleated, it grows rapidly, suddenly releasing the thermal energy stored in the surrounding liquid. Jeglic and Grace [8] showed that the rate of change in void fraction in the subcooled region was high and abrupt and that it was associated with the formation of a vapour slug. In summary, there are four aspects that favour the occurrence of the observed enhancement mechanism. These are as follows: (1) Large vapour formation for a given superheat. This implies high liquid-to-gas density ratios and/or low latent heat of vaporisation; (2) Low liquid thermal conductivity leading to high wall superheat. This explains the occurrence of this type of enhancement in flows of organic fluids and refrigerants; (3) High subcooling. In light of their experimental results, Jeglic and Grace [8] concluded that the abrupt transition associated with the formation of a vapour plug took place when the subcooling was significant. One, therefore, expects that, at low subcooling, nucleate boiling at the wall initiates early in the test section, thus preventing excessive superheats in the liquid; (4) Low mass transfer resistance to bubble growth. In mixture systems, the less volatile material concentrates on the liquid side of the interface and offers an additional resistance to rapid bubble growth. This is probably the reason for near zero-quality enhancement having been observed by Kandlbinder [6] only during the single component boiling experiments. -3-

3. THE METHODOLOGY 3.1. Slug flow and thermal non-equilibrium The basic structure of the present model is a slug unit that consists of a Taylor bubble surrounded by a falling liquid film and a liquid slug, as seen in Fig. 3. Theoretical models for the prediction of adiabatic slug flows [17-20] rely on solutions of time and area averaged mass and momentum conservation equations for the Taylor bubble and liquid slug regions using appropriate closure relations. A similar approach will be pursued here taking account of non-equilibrium effects on the structure of the flow.

Lb VGB

r g

It is assumed that the abrupt vapour growth in the subcooled region gives rise to the formation of Taylor bubbles. As they are formed, the Taylor bubbles become separated by regions of subcooled liquid (slugs). The falling film surrounding the Taylor bubble is presumed saturated. Phase change and gas hold-up within the liquid slug are assumed negligible.

VLB

Ls VLS VGS

An energy balance over the slug unit gives

dmG dT ∆hv + m S c pL S = Q& dt dt

z

(2)

where dmG is the mass of vapour generated in the Taylor bubble region and m s is the mass of the liquid slug. As Q& = π q& w d T (LB + LS ) and m S = π ρ L LS d T2 4 , Eq. (2) becomes dmG π d T  dT  dT ρ L c pL S  . =  q& w (LB + LS ) − dt 4 ∆hv  dt 

Figure 3. A schematic representation of real slug flow.

(3)

The masses of vapour and liquid within the slug unit are given by

mG = ρ G L B mL = ρ L

π (d T − 2δ )2

{L d 4

π

S

4 2 T

[

+ LB d T2 − (d T − 2δ )

(4) 2

]}.

(5)

By combining the overall mass conservation dmG = −dm L

(6)

with Eqs. (3) to (5) assuming that (i) the thickness of the liquid film surrounding the Taylor bubble is small compared with the pipe diameter, (ii) the mass of the liquid film is small compared with that of the slug and (iii) that the densities do not vary strongly with time/distance, one obtains dT  dLB 1  4 q& w (LB + LS ) − ρ L c pL LS S  =  ρ G ∆hv  dt dT dt  dLS dTS 4 q& w (LB + LS )  1  = −  c pL LS . dt dt ρ L dT ∆hv  

(7) (8) -4-

As a system, the slug unit moves at a velocity equal to the rise velocity of the Taylor bubble through the pipe. Therefore, for a stationary observer dt dz = 1 VGB , and Eqs. (7) and (8) become dT  dLB 1  4 q& w (LB + LS ) = − ρ L c pL LS S   ρ G ∆hv  dz d T VGB dz  dLS dTS 4 q& w (LB + LS )  1  . = −  c pL LS dz dz ρ L d T VGB  ∆hv 

(9) (10)

The average slug temperature profile dTS dz is determined through an energy balance in the liquid slug. This can be written as follows (see Fig. 4.a)

d Q& S = (eS mS ) − m& in hin + m& out hout dt

(11)

where eS is the internal energy of the liquid slug and m& in and m& out are the mass flow rates entering and leaving the liquid slug, respectively. It is assumed that compressibility effects in the liquid slug are negligible (c p ≈ cv ) . In light of previous hypotheses, combination of overall mass conservation and mass conservation in the Taylor bubble region gives

d d m S = − mG = m& in − m& out . dt dt

(12)

Preliminary calculations, of the same nature of those performed in [18], showed that the fraction of the liquid slug cross section from which liquid is drawn into the falling film is relatively thin. Given that the superheat in the fluid layers adjacent to the wall is considerably high, the liquid in the fraction of the slug cross section that supplies liquid to the film (region “A” of Fig. 4.b) is likely to have an average temperature close to the saturation temperature. Thus, since saturation has been assumed in the Taylor bubble region, it is hypothesised that hin ≈ hout . Following the above simplifications, and substituting Q& = q& L π d and m = π ρ L d 2 4 , Eq. (11) becomes S

w

S

T

G

G

B

T

ρ 1 dLB dTS 4 q& w (Tsat − TS ) = − G dt d T ρ L c pL ρ L LS dt

(13)

Since the liquid slug travels at a velocity VLS , the above equation can be written as

dTS q& w ρ 1 dLB 4 (Tsat − TS ) . = − G dz d T ρ L c pLVLS ρ L LS dz

(14)

In the thermal non-equilibrium region, i.e. TS ( z ) < Tsat , the liquid slug temperature gradient, the Taylor bubble growth rate and the liquid slug contraction rate are given by the simultaneous solution of Eqs. (9), (10) and (14). A stepwise solution of this system of equations gives local values of LB , LS and TS along the thermal non-equilibrium region. Calculation of additional slug flow parameters and determination of boundary conditions are addressed in the next section.

-5-

3.2. Modelling 3.2.1. Slug flow parameters In this work, ideal slug flow (no vapour entrainment in the liquid slug) is assumed. Although this is an oversimplification of the real situation and may lead to some discrepancies, it does not alter qualitatively the main results and conclusions of the present formulation.

A m& in

VLB , 0

q& w

LS

r g

As can be seen from several models available in the literature [17-20], under the ideal slug flow assumption, the overall vapour mass balance and the mass conservation equations for the mixture in the Taylor bubble and liquid slug regions become U GS = βε BVGB ε BVGB + (1 − ε B )VLB = U M V LS = U M

VGB

δ

m& out VLB

z wall superheat max

(a) (15) (16) (17)

zero

(b)

Figure 4. (a) A schematic representation of the transfer of energy into the liquid slug; (b) Illustration of the wallsuperheat and the liquid shedding into the film region.

where U M = U GS + U LS = G[(1 − xG ) ρ L + xG ρ G ] is the mixture velocity, β = LB (LB + LS ) and the

film thickness and the void fraction in the Taylor bubble region are related by δ = d T (1 − ε 1B 2 ) 2 . The Taylor bubble velocity is given by the expression of Nicklin et al. [21] VGB = 1.2 U M + V0

(18)

where V0 = γ ( g d T ) is the rise velocity of a Taylor bubble in quiescent liquid given by Wallis [22]. The parameter γ was correlated by 1/ 2



 0.01 N f  0.345

γ = 0.345 1 − exp − 

  3.37 − Bo   1 − exp  m    

(19)

where  10 when N f > 250  −0.35 m = 69 N f when 18 < N f < 250 .  25 when N f < 18 

(20)

The falling film velocity is given by [22]

-6-

V LB

  g d T2 ρ L  2   1 − ε 1B 2 , when Re f < 750 − 0 . 333    =  ηL   − 11.2 g d 1 − ε 1 2 1 2 , when Re > 750 T B f 

(

[

(

)

)]

(21)

where Re f = VLB ρ L δ η L is the falling film Reynolds number. In the present methodology, Eqs. (15)-(21) are solved together with Eqs. (9), (10) and (14). At each step ∆z , a value of β is calculated and introduced together with the physical properties in Eqs. (15)-(21) to give the local value of the real quality, xG , and also other slug flow parameters. 3.2.2. Boundary conditions Boundary conditions at the onset of slug flow are required for Eqs. (9), (10) and (14). In the present thermal non-equilibrium model, based on the visual observations of Jeglic and Grace [8], the onset of slug flow is associated with the point at which vapour is initially formed. Therefore, it is postulated that the distance from the liquid inlet up to this point can be calculated using a correlation to predict the NVG point, such as that of Saha and Zuber [12]. Saha and Zuber’s model also provides the liquid slug temperature at the onset of slug flow, TS ,o .

The specification of the lengths of the liquid slug and Taylor bubble regions at the onset of slug flow, LS ,o and LB ,o , is somewhat arbitrary. Values of LS in equilibrium slug flow reported in the literature range from 12d T to 20d T [23, 24]. Equilibrium slug flow models usually employ an intermediate value, i.e. 16d T [25, 26]. Here, for consistency, it has been assumed that LS ,o = 16d T .

The length of the Taylor bubble region at the onset of slug flow was specified based on an estimation of the minimum stable Taylor bubble length, LB ,min , which would enable bubble growth with distance. LB ,min was determined by a search for the local minimum of Eq. (9) through d 2 LB dz 2 = 0. In the present calculations, the value of LB ,o was taken as 1.2 LB ,min . As no direct

experimental data on the initial bubble length were available to confirm this hypothesis, several values of LB ,o were tested and the value which provided the best agreement in terms of calculated heat transfer coefficient was selected. Nevertheless, results showed that the sensitivity of the calculated heat transfer coefficient to the choice of LB ,o was marginal and even variations as high as 50% of LB ,min did not affect the results significantly. This is, however, an aspect of the methodology which certainly deserves further investigation. 3.3. Time averaged local heat transfer The time averaged local heat transfer coefficient has been defined by Eq. (1). In the slug flow regime, the local, time averaged wall temperature is given by t sp

Tw =

1 Tw dt . t sp ∫0

(22)

Taking into account the existence of a Taylor bubble region and a liquid slug region one can write t sf 1   Tw dt + Tw = t sp  ∫0 

 1 T dt ∫ w  = t sp (t sf Tw, f + t ssTw,S ) . t sf 

t sf + t ss

-7-

(23)

As mentioned before, as a system, the slug unit moves at a uniform velocity V GB . Thus t ss = L S V GB , t sf = L B V GB , and t sp = (L S + L B ) V GB . In terms of the length fraction of the slug unit occupied by the Taylor bubble region, Eq. (23) can be written as Tw = β Tw, f + (1 − β )Tw, S .

(24)

3.4. Heat Transfer in the liquid slug Heat transfer in the liquid slug region is calculated using a superposition model [13], assuming slug body subcooling. Thus q& w = α fc , S (Tw, S − TS ) + α nb , S (Tw, S − Tsat )

(25)

or, in terms of Tw, S Tw, S =

q& w + α fc ,S TS + α nb , S Tsat

α fc ,S + α nb , S

.

(26)

The forced convective heat transfer coefficient in the liquid slug is given by

α fc , S = Fα c , S where

α c , S = 0.023

λL dT

(27)

Re 0LS.8 PrL0.4

(28)

and Re LS = ρ LVLS d T η L is the Reynolds number associated with the liquid slug. F is the Chen [13] two-phase enhancement factor. In the subcooled slug region, F is set equal to unity [3, 27], since the slug void fraction is disregarded. The nucleate boiling heat transfer coefficient was calculated from α nb , S = Sα n , S , where

λ0L.79 c 0pL.45 ρ L0.49 (Tw, S − Tsat ) ( p w, S − p sat )0.75 = 0.00122 0.24 σ 0.5η L0.29 ρ G0.24 hLG 0.24

α n,S

(29)

is the Forster and Zuber [28] correlation and

(

S = 1 + 2.53 × 10 −6 Re1LS.17

)

−1

(30)

is the Chen [13] nucleate boiling suppression factor. 3.4. Heat Transfer in the Taylor bubble Heat transfer in the Taylor bubble region is modelled assuming superposition of heat transfer mechanisms [13] and saturation throughout this region. Thus q& w = (α fc , f + α nb , f )(Tw, f − Tsat )

(31)

or

-8-

Tw, f = Tsat +

q& w . α fc , f + α nb, f

(32)

In the film region, the heat transfer coefficient is calculated using the correlation of Chun and Seban [29]

α fc , f λL

 η L2   2   ρL g 

1/ 3

= 0.0038(4 Re f

)

0.4

PrL0.65 .

(33)

In the falling film region, the nucleate boiling coefficient is calculated through the Forster and Zuber [28] correlation ― Eq. (29) using Tw, f ― and the suppression factor, S , is estimated using Re F in Eq. (30).

4. RESULTS Figure 5 exhibits, together with the heat transfer coefficient distribution, the difference between the equilibrium bulk temperature, Tb , and the calculated liquid slug temperature, TS . In this particular


QUALITY


case, the difference increases from zero up to approximately 3oC at the point where Tb = Tsat and then decreases in the equilibrium saturated region ( z > z sat ) . The slug temperature profile is illustrated in Fig. 2, 8.00 together with the 0.4 8000 Heat transfer coefficient (Kandlbinder, 1997) saturation and Tbulk - Tslug equilibrium bulk Equilibrium quality temperature profiles. The two 6.00 vertical lines in Fig. 6000 5 represent the NVG Point [12] (which in the present model 4.00 0.0 4000 corresponds also to the point of onset of slug flow) and the point of transition from slug flow to 2.00 2000 churn flow. The NVG S/C transition from slug flow to churn flow was calculated -0.4 using the model of 0.00 0 Jayanti and Hewitt 0.00 2.00 4.00 6.00 8.00 10.00 [23], who DISTANCE [m] associated this transition to Figure 5. Variation of the difference between the equilibrium and slug flooding in the temperatures as a function of distance. Coincidence of temperature Taylor bubble difference and heat transfer coefficient peaks. Experimental conditions: 2 region. fluid: n-pentane, inlet pressure: 4.9 bar, total mass flux: 376.0 kg/m s, 2 o It is interesting to wall heat flux: 50.0 kW/m , inlet temperature: 60.7 C. note that the peak in the temperature difference distribution somewhat coincides with that of the heat transfer coefficient. This result has been observed consistently in our simulations and goes a long way towards -9-

explaining the near-zero quality heat transfer coefficient enhancement. Once the large nucleated bubble fills the pipe cross-section, it continues to grow (as will be shown below), increasing the fluid velocity and reducing the residence time in the channel of the liquid slugs between successive bubbles. This reduced residence time implies that the liquid in the slug remains subcooled at any given position and, therefore, the actual quality (and hence the heat transfer coefficient) is much higher than that calculated on a thermodynamic basis (see Fig. 1). As can be seen from the liquid slug temperature profile in Fig. 2, since the slug velocity is ever increasing (as will be shown below), the subcooling is reduced much more slowly as a function of distance than would be the case for an equilibrium flow [7]. Moreover, Fig. 5 shows that the decay of the heat transfer coefficient back towards the classical behaviour of forced convective boiling [3-5] corresponds to the region of breakdown of slug flow into churn flow (S/C), predicted by the model of Jayanti and Hewitt [23]. Since this transition is related to the collapse of the slug unit, the mixing of the phases would result in attainment of thermodynamic equilibrium, thus eliminating remaining local subcooling effects. A further example of the variation of the difference between the equilibrium and slug temperatures as a function of distance is given in Fig. 6. In this figure, contrary to example of Fig. 5, the transition to churn flow takes place before the liquid slug reaches saturation, i.e., the point at which Tsat − TS = 0 . It is interesting to note that, as the slug-to-churn (S/C) transition takes place, the heat transfer coefficient decreases sharply, reiterating the fact that the subsistence of liquid subcooling is responsible for the local heat transfer enhancement. 12.00 Heat transfer coefficient (Kandlbinder, 1997)

6000

0.4

Equilibrium quality

0.0

4.00

-0.4

4000

QUALITY

8.00

NVG

2000

S/C

-0.8

0.00 0.00

2.00

4.00

6.00

8.00


Tbulk - Tslug


Figure 7 shows the variation with distance of the length of the Taylor bubble and liquid slug regions and of the Taylor bubble length fraction, β . As can be seen, the length of the liquid slug remains approximately constant whilst the Taylor bubble length virtually doubles along the distance over which the slug flow regime prevails. As expected, the Taylor bubble length fraction somehow follows the trend of the Taylor bubble length. It is the increase in vapour mass fraction resulting from the growth of the Taylor bubble shown in Fig. 7 that may trigger an early transition to churn flow.

0 10.00

DISTANCE [m]

Figure 6. Variation of the difference between the equilibrium and slug temperatures as a function of distance. Experimental conditions: fluid: iso-octane, inlet pressure: 2.2 bar, total mass flux: 296.7 kg/m2s, wall heat flux: 60.1 kW/m2, inlet temperature: 55.3 oC.

- 10 -

G (1 − xG ,eq )

ρ L (1 − ε G ,eq )

(34)

where xG ,eq is the equilibrium quality

0.3

0.2

0.3

Bubble length fraction

BUBBLE LENGTH FRACTION

VL =

LENGTH OF SLUG AND BUBBLE REGIONS [m]

Calculated Taylor bubble and liquid slug velocities are plotted in Fig. 8 as function of distance in the region of occurrence of slug flow, i.e., between the NVG point and the slug-to-churn transition (S/C). Both velocities increase as a result of the increase in real quality associated with the increase in Taylor bubble length depicted in Fig. 7. The velocity of the liquid phase in a hypothetical situation in which complete thermodynamic equilibrium takes place is also shown in Fig. 8 (“Liquid equilibrium velocity”). In the region 0.6 0.6 upstream of the point at which the equilibrium quality is zero (marked by X in the figure), this velocity is equal to 0.5 0.5 the liquid superficial velocity, G ρ L . As equilibrium phase change takes place, the liquid velocity is estimated 0.4 0.4 through

0.2

and ε G,eq is the void fraction calculated Length of Taylor bubble region 0.1 0.1 based on the equilibrium quality. Here, 2.80 3.20 3.60 4.00 4.40 4.80 ε G ,eq was estimated using the ChexalDISTANCE [m] Lellouche [30,31] correlation. The Figure 7. Variation of LS , LB and β . Experimental equilibrium quality (also shown in Fig. conditions: fluid: n-pentane, inlet pressure: 4.9 bar, 1) was calculated through a heat balance. total mass flux: 376.0 kg/m2s, wall heat flux: 50.0 As can be seen, in the region of kW/m2, inlet temperature: 60.7 oC. occurrence of slug flow, the velocity of 4.0 the subcooled slug is considerably Taylor bubble velocity higher than the velocity of the liquid Liquid slug velocity phase in a thermodynamic equilibrium Liquid equilibrium velocity situation. 3.0 An illustration of the local heat transfer coefficient prediction capability of the model is provided in Figs 9 and 10 for typical n-pentane and iso-octane runs. 2.0 The trend of the experimental data is well picked up by the correlation in both cases. Although it systematically underpredicts the data in the whole 1.0 subcooled region (the point of saturation is marked by X) and in part S/C NVG X of the equilibrium saturated region of the case shown in Fig. 9, the correlation 0.0 exhibits an excellent agreement with 0.00 2.00 4.00 6.00 8.00 10.00 DISTANCE DISTANCE[m/s] [m] the data in the subcooled region of Fig. 10. Figure 8. Taylor bubble and liquid slug velocity profiles as a function of distance in the region of occurrence of slug flow. The equilibrium liquid velocity is shown for Analogously to the Tb − TS profile comparison. Experimental conditions: fluid: n-pentane, (see Fig. 5), the peaks in the nlet pressure: 4.9 bar, total mass flux: 376.0 kg/m2s, wall calculated heat transfer coefficient heat flux: 50.0 kW/m2, inlet temperature: 60.7 oC. profiles somewhat coincide with VELOCITY [m/s]

Length of liquid slug region

- 11 -

8000 Experimental (Kandlbinder, 1997) Present model (Non-equilibrium slug flow)


those in the experimental data. Results obtained using the Chen [13] correlation are also shown in Figs. 9 and 10. In the subcooled region, the corrections suggested by Butterworth and coworker [3, 27] were applied to the Chen [13] correlation. As can be seen, the non-equilibrium slug model performs better than the Chen model in this region in both cases. In the equilibrium saturated region (downstream of the X point), the Chen model predicts a sharp increase in the heat transfer coefficient, in contrast with the decreasing trend exhibited by the non-equilibrium slug flow model. In short, as has been observed by Kandlbinder [6], the Chen [13] model, together with other correlations [14-16], underpredicts the data in the near-zero quality region and fail to identify a point of maximum in the heat transfer coefficient profiles. The general heat transfer behaviour illustrated in Figs. 9 and 10 is typical of the conditions investigated here for both hydrocarbons.

Chen (1966)

6000

4000

X 2000 3.20

3.60

4.00

4.40

4.80

5.20

DISTANCE [m]

Figure 9. Local heat transfer coefficient prediction. Experimental conditions: fluid: n-pentane, inlet pressure: 6.0 bar, total mass flux: 377.4 kg/m2s, wall heat flux: 49.9 kW/m2, inlet temperature: 67.5 oC.

6000 Experimental (Kandlbinder, 1997) Present model (Non-equilibrium slug flow)


Figure 11 presents a general comparison between experimental and calculated heat transfer coefficients for both n-pentane and iso-octane over the whole range of parameters assessed by Kandlbinder [6]. The performance of the proposed nonequilibrium slug flow model is superior to that of the Chen [13] correlation and 95% of the data fall the +/- 18% relative error band. The largest deviations between the Chen correlation and the experimental data take place in the equilibrium saturated region as observed in Figs. 9 and 10.

Chen (1966)

5000

4000

3000

2000

X 1000

As far as the local heat transfer 3.00 3.50 4.00 4.50 5.00 5.50 DISTANCE [m] coefficient prediction is concerned, the performance of the non-equilibrium model could have been improved by selecting a Figure 10. Local heat transfer coefficient prediction. Experimental conditions: fluid: iso-octane, inlet more appropriate and up-to-date set of pressure: 3.1 bar, total mass flux: 200.8 kg/m2s, wall relationships to replace, for example, Eqs. heat flux: 19.5 kW/m2, inlet temperature: 117.4 oC. (28)-(30) and (33). In particular, it has been shown [32] that the Forster and Zuber [28] correlation for the nucleate boiling contribution (Eq. 29) is ‘subject to considerable uncertainty’. Nevertheless, we opted for an adaptation of the Chen [13] model to better demonstrate the effect of the proposed thermal non-equilibrium phenomenon in comparison with the original - 12 -

and well accepted Chen formulation. The aim of the paper is not to put together the more accurate correlation, but to point out and predict the underlying phenomena through a set of physically consistent conservation equations. Limitations and inconsistencies related to the definition and the application of the two-phase enhancement and nucleate boiling suppression factors in several widely used correlations have been addressed in a recent paper by Webb [33].

CALCULATED HEAT TRANSFER COEFFICIENT [W/m2.K]

5. CONCLUSIONS 8000 This paper presented a model for +18% predicting the heat transfer coefficient peaks observed in the near-zero quality region in boiling of hydrocarbons in a 6000 vertical pipe [6,7]. The principle of the -18% model is the occurrence of thermal nonequilibrium effects [8, 9] associated with the formation of a type of slug flow 4000 in which the Taylor bubbles are separated by subcooled liquid slugs. The main conclusions arising from this work are as follows: 2000 (1) It was shown that the heat transfer coefficient peaks Present model (Non-equilibrium slug flow) coincide with the peaks in the Chen (1966) calculated difference between 0 the equilibrium bulk and average 0 2000 4000 6000 8000 slug temperatures, Tb − TS . EXPERIMENTAL HEAT TRANSFER COEFFICIENT [W/m2.K] Thus, slugs remain subcooled for distances longer than would be Figure 11. Overall comparison of local the case for equilibrium flow heat transfer coefficient predictions. situations. A consequence of this effect is that the wall temperature in the liquid slug region is lower than that in the equilibrium case. As the heat transfer coefficient is defined in terms of an equilibrium relationship (Eq. 1), the net result is an increase in the heat transfer coefficient;

(2) In addition to the apparent increase described in (1) above, the increasing velocity of the subcooled liquid slugs resulting from the acceleration of the Taylor bubbles predicted by the model may contribute to the local enhancement of the heat transfer coefficient in the near-zero quality region; (3) This is the first model to predict the local heat transfer enhancement in the near-zero quality region. In the portion of the pipe over which non-equilibrium slug flow takes place (between the NVG point and the point of transition to churn flow [23]), the heat transfer coefficient was predicted within error limits lower than those by established methodologies [6, 13]. (4) Further experiments are needed to provide support for future developments in the proposed model, such as the determination of boundary conditions at the onset of non-equilibrium slug flow and the inclusion of a model to predict the gas hold-up in the subcooled liquid slug. ACKNOWLEDGEMENTS JRB thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FUNCITEC (Fundação de Ciência e Tecnologia do Estado de Santa Catarina) for financial support.

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NOMENCLATURE Roman (ρ L − ρ G )g d T2 Bo = - Bond number

σ

c p - Specific heat capacity – [J/kg.K] d T - Pipe diameter – [m] g - Acceleration due to gravity – [m/s2] h - Enthalpy – [J/kg] L S - Length of the liquid slug region – [m] L B - Length of the Taylor bubble region – [m] m - Mass – [kg] m& - Mass flow rate – [kg/s]

Nf

[d =

3 T

g ( ρ L − ρ G )ρ L

]

1/ 2

ηL

- Viscosity number

p - Pressure – [Pa] η cp Pr = - Prandtl number

λ

q& w - Wall heat flux – [W/m2] t sp - Time period during which a slug unit is seen at a fixed location – [s] t sf - Time period during which a Taylor bubble (falling film) is seen at a fixed location – [s] t ss - Time period during which a liquid slug is seen at a fixed location – [s] Tb – Equilibrium bulk temperature – [K] Ts – Liquid slug bulk temperature – [K] Tw – Wall temperature – [K] Tw, f - Local wall temperature averaged over t sf - [K] Tw, S - Local wall temperature averaged over t ss - [K]

U GS - Superficial velocity of the vapour – [m/s] U LS - Superficial velocity of the liquid – [m/s] VGB - Rise velocity of a Taylor bubble in slug flow – [m/s] V LB - Falling film velocity – [m/s] VLS - Rise velocity of a liquid slug – [m/s] xG - vapour mass fraction (quality) – [-] z - Distance along the pipe – [m] Greek α - Heat transfer coefficient – [W/m2.K] α fc, S - Heat transfer coefficient due to forced convection in the liquid slug – [W/m2.K]

α nb, S - Heat transfer coefficient due to nucleate boiling in the liquid slug – [W/m2.K] β - Length fraction of the slug unit occupied by the Taylor bubble region – [-] δ - Falling film thickness – [m] ∆hv - Latent heat of vaporisation – [J/kg] λ - Thermal conductivity – [W/m.K] η - Viscosity – [kg/m.s] ε - Void fraction – [-] - 14 -

ρ - Density – [kg/m3] σ - Surface tension – [N/m] Subscript B - Taylor bubble f - Falling film G - Vapour L - Liquid S - Slug sat - Saturation Superscript - Time averaging REFERENCES [1] Wadekar, V.V. and Kenning, D.B.R., Flow Boiling Heat Transfer in Vertical Slug and Churn Flow Region, Proc. 9th International Heat Transfer Conference, Jerusalem, Vol. 3, pp. 449-454, 1990. [2] Wadekar, V.V., Vertical Slug Flow Heat Transfer with Nucleate Boiling, ASME HTD-Vol. 159 Phase Change Heat Transfer, 1991. [3] Collier, J.G. and Thome, J.R., Convective Boiling and Condensation, 3rd ed., Oxford University Press, 1994. [4] Hewitt, G.F., Challenges in Boiling Research, Keynote Lecture, Boiling 2000: Phenomena and Emerging Applications, UEF, April 30-May 5, Anchorage, AK. [5] Hewitt, G.F., Deviations from Classical Behaviour in Vertical Channel Convective Boiling, Multiphase Science and Technology, vol. 13, no. 3 & 4, pp. 341-371, 2001. [6] Kandlbinder, T.K., Experimental Investigation of Forced Convective Boiling of Hydrocarbons and Hydrocarbon Mixtures, Ph. D. thesis, Imperial College, University of London, 1997. [7] Urso, M.E., Wadekar, V.V. and Hewitt, G.F., Flow Boiling at Low Mass Flux, Proc. 12th International Heat Transfer Conference, Grenoble, pp. 803-808, 2002. [8] Jeglic, F.A. and Grace, T.M., Onset of Flow Oscillations in Forced Flow Subcooled Boiiling. NASA Technical Note TN D-2821, Lewis Research Center, Cleveland, Ohio, USA, 1965. [9] Ishii, M., Wave Phenomena and Two-Phase Flow Instabilities. In: Handbook of Multiphase Systems (Ed. G. Hetsroni), Hemisphere Publishing Co, Washington DC, USA, 1982. [10] Cheah, L.W., Forced Convective Evaporation at Sub-atmospheric Pressure, Ph.D. thesis, Imperial College, University of London. [11] Thome, J.R., Flow Boiling in Horizontal Tubes: A Critical Assessment of Current Methodologies. Proc. 1st Symposium on Two-Phase Flow Modelling and Experimentation Conference, Vol. 1., pp 41-52, 1995. [12] Saha, P. and Zuber, N., Point of Net Vapor Generation and Vapor Void Fraction in Subcooled Boiling, Proc. 5th International Heat Transfer Conference, Tokyo, Paper B4.7, 1974. [13] Chen, J.C., A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow. Industrial and Engineering Chemistry: Process Design Develop., vol. 5(3), pp. 322-329, 1966. [14] Shah, M.M., Chart Correlation for Saturated Boiling Heat Transfer: Equations and Further Study. ASHRAE Transactions, vol. 88(1), pp. 185-196, 1982. [15] Kandlikar, S., A General Correlation for Saturated Two-Phase Flow Boiling Heat Transfer Inside Horizontal and Vertical Tubes. Journal of Heat Transfer, vol. 112, pp. 219-228, 1990. [16] Steiner, D. and Taborek, J., Flow Boiling Heat Transfer in Vertical Tubes Correlated by an Asymptotic Model. Heat Transfer Engineering, Vol. 13(2), pp. 43-69, 1992. [17] Fernandes, R.C., Semiat, R. and Dukler, A.E., Hydrodynamic Model for Gas-Liquid Slug Flow in Vertical Tubes. AIChE Journal, vol. 29(6), pp. 981-989, 1983.

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[18] Orell, A. and Rembrand, R., A Model for Gas-Liquid Slug Flow in a Vertical Tube. Industrial and Engineering Chemistry: Fundamentals, vol. 25, pp. 196-206, 1986. [19] Sylvester, N.D., A Mechanistic Model for Two-Phase Vertical Slug Flow in Pipes. Journal of Energy Resources Technology, vol. 109, pp. 206-213, 1987. [20] De Cachard, F. and Delhaye, J.M., A Slug-Churn Flow Model for Small-Diameter Airlift Pumps, International Journal of Multiphase Flow, vol. 22(4), pp. 627-649, 1996. [21] Nicklin, D.J., Wilkes, J.O. and Davidson, J.F., Two-Phase Flow in Vertical Tubes. Transactions of the Institution of Chemical Engineers, vol. 40, pp. 61-68, 1962. [22] Wallis, G.B., One Dimensional Two-Phase Flow, Mc-Graw-Hill, New York, 1969. [23] Jayanti, S. and Hewitt, G.F., Prediction of the Slug-to-Churn Transition in Vertical Two-Phase Flow, International Journal of Multiphase Flow, vol. 18(6), pp. 847-860, 1992. [24] Costigan, G., Flow Pattern Transitions in Vertical Gas-Liquid Flows, D.Phil. Thesis, Oxford University, 1997. [25] Taitel, Y., Barnea, D. and Dukler, A.E., Modelling Flow Pattern Transition for Steady Upward Gas-Liquid Flow in Vertical Tubes, A.I.Ch.E. Journal, vol. 26, pp. 345-354, 1980. [26] Barnea, D. and Yacoub, N., Heat Transfer in Vertical Upwards Gas-Liquid Slug Flow, International Journal of Heat and Mass Transfer, vol. 26(9), pp. 1365-1376, 1983. [27] Butterworth, G. and Shock, R.A.W., Flow Boiling, Proc. 7th International Heat Transfer Conference, Munich, Vol. 1, pp. 11-30, 1982. [28] Forster, H.K. and Zuber, N., Dynamics of Vapor Bubbles and Boiling Heat Transfer, AIChE Journal, vol. 1(4), pp. 531-535, 1955. [29] Chun, K.R. and Seban, R.A., Heat Transfer to Evaporating Liquid Films, Journal of Heat Transfer, vol. 93, pp.391-396, 1971. [30] Chexal, B., The Chexal-Lellouche Void Fraction Correlation for Generalized Applications, Electric Power Research Institute Report (EPRI) NSAC/139, 1991. [31] Levy, S., Two-Phase Flow in Complex Systems, Wiley Interscience, NY, 1999. [32] Aounallah, Y. and Kenning, D.B.R., Nucleate Boiling and the Chen Correlation for Flow Boiling Heat Transfer, Experimental Heat Transfer, vol. 1, pp. 87-92, 1987. [33] Webb, R.L., Commentary on Correlations for Convective Vaporization in Tubes, Journal of Heat Transfer, vol. 125, pp.184-185, 2003.

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