Heat transfer correlation for film boiling in vertical upward flow

International Journal of Heat and Mass Transfer 107 (2017) 112–122

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Heat transfer correlation for film boiling in vertical upward flow Lokanath Mohanta a,⇑, Faruk A. Sohag a, Fan-Bill Cheung a, Stephen M. Bajorek b, Joseph M. Kelly b, Kirk Tien b, Chris L. Hoxie b a b

The Pennsylvania State University, Department of Mechanical and Nuclear Engineering, University Park, PA 16802, United States U.S. Nuclear Regulatory Commission, Office of Nuclear Regulatory Research, Washington, D.C. 20555-0001, United States

a r t i c l e

i n f o

Article history: Received 12 April 2016 Received in revised form 1 September 2016 Accepted 7 November 2016

Keywords: Rod bundle Subcooled film boiling Inverted Annular Film Boiling Inverted Slug Film Boiling

a b s t r a c t There is no heat transfer correlation available in the Inverted Slug Film Boiling (ISFB) regime. In this study, a new correlation for film boiling in a vertical channel is proposed using a semi-empirical model. This correlation is applicable in the Inverted Annular Film Boiling (IAFB) heat transfer regime as well as ISFB heat transfer regime. A semi-empirical model is developed to establish the functional dependence of the Nusselt number on various controlling parameters. Test data obtained from the transient reflood tests carried out in a full length 7
7 vertical Rod Bundle Heat Transfer (RBHT) facility are used to develop the correlation. The predicted Nusselt number agrees within 15% with the RBHT data set and other data set available in the literature. This heat transfer correlation is applicable up to a void fraction of 90%, i.e., both in the IAFB and ISFB regimes. The proposed correlation is also applicable to a tube geometry using a modified coefficient in the correlation with an error less than 10%. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Film boiling heat transfer occurs in the post critical heat flux (CHF) regime. In film boiling, the vapor envelopes the heating surface completely, thus preventing direct liquid contact with the heating surface. When film boiling occurs in the presence of forced convective flow in tubes or channels, it is referred to as flow film boiling. The latter is further categorized into two different flow regimes, namely, Inverted Annular Film Boiling (IAFB) and Dispersed Flow Film Boiling (DFFB) depending upon the void fraction. When the void fraction is less than 0.5 or so [1–3], the flow regime is known as IAFB, and known as DFFB when the void fraction is larger than 0.9 [4,2]. In DFFB, the void fraction can be close to unity and liquid is present in the form of dispersed droplets. When the void fraction range is between 0.5 and 0.9, the heat transfer regime is called Inverted Slug Film Boiling (ISFB). The IAFB heat transfer is characterized by a continuous liquid core enveloped by a vapor film separating it from the heated wall downstream of the quench front. The heat transfer takes place from the wall to the vapor and then to the interface by convection, also some heat is transferred from wall to interface by radiation. As a result of continuous vaporization of liquid in the IAFB regime, higher void fraction and higher vapor flow cause breakup of the ⇑ Corresponding author. E-mail address: [email protected] (L. Mohanta). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.11.018 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

liquid core into slugs thus resulting in the transition region known as Inverted Slug Film Boiling (ISFB) regime. This regime contains liquid slugs along with droplets. Further downstream, liquid slugs breakup into droplets and the void fraction increases significantly, leading to a droplet flow regime known as DFFB. The IAFB regime finds importance in safety analysis of nuclear reactors during Loss-of-Coolant Accidents (LOCA) for Pressurized Water Reactors and during the transient operation of Boiling Water Reactors. This flow regime is also observed in other areas of engineering such as cooling of rocket engines, hydrogen-fueled automobiles, quenching of metals, and cryogenic applications. 1.1. Film boiling models Film boiling has been extensively studied by many researchers in the past. Bromley [5] presented one of the earliest models for pool film boiling on external surfaces based on Nusselt’s film condensation theory and using experimental data on horizontal tubes. The heat transfer coefficient was expressed as

h¼C

" # ~ g q k3 ðq  q Þ 1=4 h fg g g l g ; DT l g D

ð1Þ

~ fg is the modified latent heat of vaporization given by where h

 fg þ 0:5cp;g ðT W  T sat Þ: ~fg ¼ h h

ð2Þ

L. Mohanta et al. / International Journal of Heat and Mass Transfer 107 (2017) 112–122

113

Nomenclature A cp C D Dh f g G h  h  hfg Ja k L Lc m _ m n Nu p P Pr q q00 Re T u yþ v z

constant specific heat constant diameter hydraulic diameter friction coefficient acceleration due to gravity mass flux heat transfer coefficient enthalpy latent heat of vaporization Jakob number thermal conductivity length characteristic length scale exponent mass flow rate exponent Nusselt number pressure perimeter Prandtl number exponent heat flux Reynolds number temperature velocity non-dimensional distance specific volume length in flow direction

Bromley [5] model which is applicable for saturated film boiling in a liquid pool, does not account for liquid subcooling. Bromley et. al. [6] also studied film boiling with forced upward convection over horizontal tubes. The effect of convection was found to be impor pffiffiffiffiffiffi tant for Froude number u= gD > 2. Since then many analytical and experimental work has been carried out for film boiling on flat plate for both horizontal as well as vertical orientation. A boundary layer type analysis was carried out by Koh [7] for film boiling on an isothermal vertical surface in a liquid pool using the similarity method. Frederking [8] presented an integral analysis for natural convection film boiling. The effect of subcooling on laminar film boiling has been discussed by Sparrow and Cess [9]. The process of saturated film boiling on a vertical surface was studied experimentally by Suryanarayana and Merte [10]. The heat transfer coefficient was found to decrease immediately downstream of the leading edge of the film due to film growth, but further downstream the heat transfer coefficient increases due to interfacial disturbances. Bui and Dhir [11] theoretically and experimentally investigated the natural convection saturated film boiling on a vertical surface. Vijaykumar and Dhir [12] studied subcooled pool film boiling on a vertical surface. The heat transfer coefficient was found to decrease from the leading edge and found to be dependent on the degree of subcooling. Results obtained from their analysis were found to compare well with the experimental data for a vertical surface but underpredicted the experimental results for a vertical cylinder. All of the studies mentioned above assumed a smooth interface between liquid and vapor and their results only agreed near the quench front. Cess and Sparrow [13] analyzed the film boiling in a forced convection boundary layer flow on a horizontal flat plate. Meduri et al.

Greek letters void fraction d vapor film thickness d ; dþ non-dimensional vapor film thickness DðÞ difference of ()  emissivity U two-phase friction multiplier kc critical wave length l dynamic viscosity m kinematic viscosity mt turbulent viscosity q density r Stefan–Boltzmann constant s shear stress

a

Subscripts g gas or vapor phase exp experimental i interface l liquid phase QF quench front r radiation sat saturation sub subcooling sup superheat v vapor w wall

[14] experimentally investigated wall heat flux partitioning and interfacial heat transfer during the subcooled flow film boiling in a vertical surface. They proposed a correlation for the Nusselt number for the wavy interface region. Their model accounts for the subcooling in the flow as well as the liquid flow effects. However, it cannot be applied near the quench front when the interface is smooth. This model fails to predict film boiling in a cylindrical geometry [15]. The works discussed so far are for flat plate geometries. Furthermore, researchers have investigated film boiling in tube and annulus outside the cylindrical rods. Dougall and Rohsenow [16] theoretically and experimentally investigated film boiling in a vertical tube. Their analyses using turbulent integral method yielded the following expression for the Nusselt number

Nu ¼

 1 q ðq q ÞgD3 3  þ 13 0:794 g l l2 g d R dþ 0

g

dyþ m 1þPrmgt

:

ð3Þ

Dougall and Rohsenow’s analysis was restricted to saturated liquid inlet temperatures. They assumed that interfacial shear was equal to wall shear stress. Due to this approximation, the effect of liquid flow is not accounted for in the model. Greitzer and Abernathy [17] studied the film and transition boiling in a vertical tube using methanol flow. They identified the effect of vapor bulge at interface. Experimental work showed the effects of liquid velocity and subcooling in the flow, but they did not present any quantitative analysis. Using a scaling analysis, they proposed a relation for the heat transfer coefficient. However, this model is restricted to natural convection saturated film boiling. Sudo [18] investigated film boiling in a single rod experiment

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using a steel tube. For saturated boiling, he presented a correlation similar to Bromley model with a different length scale, that is the distance from the quench front and with a modified coefficient. As this model is similar to the model by Bromley, it does not account for the effect of the liquid flow. Sudo [18] suggested a linear relation to account for the enhancement due to subcooling at the quench front as shown below

hsub ¼ 1 þ 0:025DT sub hsat

ðDT sub

in  CÞ:

ð4Þ

Fung et al. [19,20] studied subcooled and low quality film boiling using water in a vertical inconel tube. They observed that the heat transfer coefficient decreases from the leading edge of the film but after L=D > 30, it starts increasing. Mosaad and Johannsen [21] experimentally studied subcooled film boiling in a vertical tube for an upward water flow. They included the subcooling effect by using a a non-linear relation, but did not account for the effect of liquid flow. Hammouda, Takenka, Nakla [1,4,2] have also investigated the inverted annular flow experimentally using refrigerant fluids. They have discussed about the heat transfer regimes and heat transfer coefficient variations in different regimes. These authors modeled the heat transfer in IAFB using two-fluid model method. However, they did not present any correlation for the Nusselt Number. Shiotsu and Hama [22] proposed two different relations for the heat transfer coefficient for flow film boiling in a vertical cylinder; one for velocity below 1 m/s and the other for velocity higher than 1 m/s. Available literature on film boiling is either confined to flat plate and single tube or rod experiments. Mohanta et al. [23] experimentally investigated subcooled and saturated film boiling in transient reflood experiments obtained from a 7
7 rod bundle heat transfer (RBHT) experimental facility. They presented a Nusselt number correlation by extending the correlation used in the TRACE code [3]. Their correlation can be applied in the ISFB regime as well, with void fraction up to 80%. However, this model is purely empirical in nature and does not account for the effect of subcooling and the effect of liquid flow. 1.2. Heat transfer models in best estimate codes IAFB heat transfer is frequently encountered in the nuclear reactor safety analysis. In one of the system code for reactor analysis, the COBRA-TF code, the criterion for IAFB to exist is that the void fraction is less than 0.4. The heat flux in the IAFB is computed from the larger values among the correlation for DFFB and modified Bromley correlation which is given below [24]

q00 ¼ 0:62

#1=4 0:172 "~ hfg g qg k3g ðql  qg Þ Dh ðT w  T sat Þ: kc DT lg D

ð5Þ

The US NRC’s system code TRACE [3] uses the following empirical Nusselt number during IAFB for void fraction below 0.6

Nu ¼

  hd ¼ 1 þ 1:3 0:268d0:77  0:34 ; kg

ð6Þ

rod bundle. A scaling method has been introduced to obtain a functional relation of the Nusselt number with the non-dimensional film thickness and fluid properties. The new correlation in the current form can be applied to IAFB as well as ISFB heat transfer regimes, that is up to a void fraction of 0.9. At the void fraction of 0.9, the present correlation is comparable to the existing correlation for DFFB. This new correlation developed in this study can also be applied to the tube geometry with a modified correlation coefficient, which is discussed in the paper. 2. Experimental method 2.1. Experimental facility The RBHT test facility was developed to provide new data for models and correlations development in support of the TRACE code development efforts. Various types of reflood and two-phase heat transfer experiments have been performed in the facilities as reviewed in the dissertation [25]. Fig. 1 shows an isometric view of the test facility with some of the components labeled. The test section consists of a full length rod bundle arranged in a 7
7 rectangular array housed in a square channel of 90.2 mm. The flow housing which is connected to the lower and upper plenum, is made up of Inconel 600 having a wall thickness of 6.4 mm to withstand higher temperature and pressure. There are six quartz windows in the flow housing which are used for flow visualization and droplet measurement. The flow housing has 23 pressure taps at different elevations through which differential pressure (DP) cells are connected to measure the pressure drop. The flow housing also has 13 taps at various elevations to accommodate traversing steam probes. The diameter of the heater rod is 9.5 mm, the heated length is 3.657 m and the pitch is 12.6 mm. The heater rods are electrically connected to a nickel plate by means of a Morse taper at the top end and through a low-melt reservoir (which accommodates expansion of heater rods) at the bottom end [26]. The power ratio (ratio of the local power at a given location to the average power) is 0.5 at the top and bottom of the bundle and 1.5 at the peak power location which is 2.73 m from the bottom. The rod bundle contains 256 thermocouples, embedded on the inside surface of the cladding along the entire length of the test section. There are seven mixing vane spacer grids in the rod bundle. The spacer grid design is similar to the spacers used in commercial fuel bundles, having dimples and mixing vanes. Spacer grids are made with Inconel 600 having 0.508 mm thickness. The grid height is 44.45 mm, including the height of mixing vane. Spacer grids are placed at different elevations such that equal distance of 522 mm between two adjacent grids is maintained except for Grids 1 and 2. These two grids are separated by a 588 mm distance. There are other dedicated instrumentation to measure other test parameters such as system pressure, injection flow rate, exhaust flow rate, and steam temperature. More details about the test facility and instrumentation can be found in the facility description report [27].

where d is the non-dimensional vapor film thickness.

2.2. Test procedure

1.3. Objectives of the present work

To prepare for an experiment, the low-melt reservoir is heated to the melting temperature. The data acquisition system is turned on to monitor the temperatures during preheating (by steam) before power is supplied to the heaters. Water is first deionized and then heated up to the prescribed inlet temperature. After that, preheated steam from the steam generator is allowed to pass through the upper plenum to remove any liquid water present in the system. Heater rods and the housing walls are preheated to 644 K using the steam. The lower plenum is filled with water from

Most of the heat transfer models available in the literature are restricted to either flat plate or cylindrical tubes. The existing models are only applicable in the IAFB regimes with a smooth interface, when the void fraction is less than 0.6. In this study a semiempirical correlation is developed for rod bundles in the film boiling regime. The data used for this correlation development are obtained from the transient experiments conducted in a full length

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EXIT FLOWMETER PRESSURE OSCILLATION DAMPING TANK

EXHAUST MUFFLER

STEAM SEPERATOR TANK

EXHAUST VALVE UPPER PLENUM

HEATED WATER SUPPLY TANK

CARRY OVER TANK 7X7 ROD BUNDLE FLOW HOUSING

LOWER PLENUM

INLET FLOWMETER

LOW MELT RESERVOIR AND CONNECTOR PLATE

Fig. 1. Isometric view of the RBHT test facility.

the supply tank until the injection line. Data is recorded from this time onward. The upper plenum pressure is set to a desired value. The power to the heater rods are applied and cladding temperatures are monitored. When the maximum cladding temperature in the bundle reaches to a desired temperature, water injection is started from the lower plenum. Next, the power level, reflood temperature, scram temperature, and flooding rate are maintained throughout the test by applying appropriate controls. Once the entire rod bundle is quenched, power and coolant injection to the bundle are terminated.

l

is 0.95 [21]. Radiation heat transfer to the vapor is neglected. The Nusselt number for convection heat transfer is defined as

Nu ¼

hd ; kg

ð10Þ

where d is vapor film thickness and kg is the thermal conductivity of vapor at the film temperature. It should be noted that to facilitate direct comparison of experimental data with the existing film boiling models by Bromley, Sudo and Mosaad, the hydraulic diameter of the subchannel is used as the length scale in the Nusselt Number.

where q00w is the wall heat flux, T w the cladding wall temperature and T sat the saturation temperature of water at the system pressure. The measured heat transfer coefficient consists of convection and radiation components, i.e.,

2.3.2. Void fraction In the RBHT facility the void fraction is not measured directly. Instead, it is deduced from the DP cell measurements. The DP cells measure two-phase pressure drop in the span of the DP cell. The two-phase pressure drop consists of three different components including the pressure drop due to gravity, friction, and acceleration. The pressure drop due to friction is modeled using the Friedel model [28] and the acceleration pressure drop is estimated using the homogeneous model. Both of these models require vapor mass quality data. The latter is calculated using the measured heat fluxes and mass flow rates of liquid and vapor exiting the test facility. The quality calculation method can be found in the dissertation work of Mohanta [25]. The local void fraction is the fraction of vapor head to the span length. The pressure drop due to elevation is calculated by subtracting the frictional and acceleration pressure drop from the measured pressure drop by the DP cells.

hexp ¼ h þ hr :

Dpelev ¼ Dpmeasured  Dpfric  Dpaccel :

2.3. Data reduction method 2.3.1. Heat transfer coefficient The cladding temperatures are measured using the thermocouples embedded inside the cladding. The outer cladding temperature and the surface heat flux are calculated by solving a transient one-dimensional inverse heat conduction problem. The experimental film boiling heat transfer coefficient is calculated as

hexp ¼

q00w ; T w  T sat

ð7Þ

ð8Þ

The heat transfer coefficient due to radiation is calculated as

  rðT w þ T sat Þ T 2w þ T 2sat   ; hr ¼ 1l 1 D Dþ2d w þ 

ð9Þ

l

where w and l are the emissivity of wall and liquid, respectively. To estimate the radiation heat transfer, w is taken to be 0.8 and

ð11Þ

The elevation pressure drop in the span of the DP cell is essentially due to the liquid head as the gas density is three orders of magnitude lower than the liquid density. The local void fraction is obtained as follows,

a¼1

Dpelev

ql gzspan

;

where zspan is the span length of the DP cell.

ð12Þ

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2.3.3. Frictional and acceleration pressure drop model The frictional pressure drop is estimated using the Friedel model [28]. The Friedel model is based on two-phase friction multiplier by considering 25,000 data points including pressure measurements in single as well as two-phase flow. This model accounts for the flow direction and geometry. The frictional pressure drop is calculated by Friedel model [28] as

Dpfric

" # 2f lo G2 ml ¼ DzU2l : Dh

ð13Þ

The friction factor f lo , for liquid only is calculated using Blasius correlation and U2l is the two-phase friction multiplier. The pressure drop due to acceleration in the span of a DP cell is calculated using the homogeneous flow model, as given below

Dpaccel ¼

00 4Gv fg q ðzout  zin Þ:  Dh hfg

ð14Þ

The relative contribution of hydrostatic, acceleration and frictional pressure drop varies with the void fraction. When the void fraction is low, the contribution due to gravity head is very high. At a higher void fraction the frictional pressure drop contributes the most.

3. Development of a semi-empirical model The Nusselt number in a convective system depends on the Reynolds number of the flow and Prandtl number of the fluid. We have chosen a similar form for the Nusselt number in studying the Inverted Annular Film Boiling. The Reynolds number is usually calculated based on the fluid properties and velocity. However, in IAFB it is extremely difficult to measure the velocity of the vapor. Note that, none of the previous studies has reported any vapor velocity measurement in the IAFB regime. So to estimate the vapor velocity we will use the method elaborated by Cachard [31]. He has presented a method to obtain the Reynolds number for laminar vapor flow in the film using the fluid properties and void fraction information. In this paper we extend that relation for Reynolds number to turbulent flow by using a general friction factor. We have also substituted the Prandtl number by film Prandtl number to account for the vapor superheat in the correlation. In addition to that, we have accounted for the effects of liquid core velocity and liquid subcooling using asymptotic expressions. The Nusselt number for heat transfer in a convective flow is generally given by, n Nu ¼ CRem g Pr ;

where for flow film boiling under consideration, Reg is the vapor Reynolds number. According to Cachard[31], the vapor Reynolds number in the laminar film can be expressed in terms of the nondimensional film thickness, as follows,

2.4. Uncertainties in measurements Uncertainties involved in the experimental data are due to two possible error sources, namely the uncertainties due to experimental measurements and data reduction. The uncertainties due to direct measurements of temperature, pressure, flow rate, and power measurement, are listed in Table 1. The uncertainty in the heat transfer coefficient is contributed by many sources including the uncertainty in thermocouple measurement, properties of different material of the heater rod, geometry of heater rod, location of the thermocouple, power and pressure measurements [29,30]. It was observed that the highest uncertainty in the heat transfer coefficient is early in the transient, particularly at the beginning of reflood (20s). But the early transient data was not used in the present study for model development. The percentage uncertainty is below 5% after initial 20 s and it stays below 2% for most of the duration of the test. Uncertainty in the cladding temperature is less than 0.5%. The heat loss from the insulation at the flow housing is approximately 1% of the total bundle power [27]. The uncertainty involved in the calculation of void fraction is primarily due to the uncertainty in the pressure measurement by the DP cells. The uncertainty of the DP cell having a span of 76 mm is 0.13 mm of water column, which is less than 0.2%. In addition to the uncertainty of the DP cells, the uncertainty in the void fraction also includes the uncertainties associated in the models for the frictional pressure drop and acceleration pressure drop.

ð15Þ

Reg ¼

  d3 g qg ql  qg 6l2g

¼

d3 : 6

ð16Þ

Details for the Reynolds number estimation has been described in the Appendix. In obtaining the above expression for Reg , Cachard assumed that the shear stress at the interface is equal to the shear stress at the wall. This will be true when the interface is stationary. However, the velocity at the interface would be at least equal to or more than the velocity of the liquid core. Hence with increasing liquid velocity, the interfacial velocity would increase. So the interfacial shear stress would deviate from the wall shear stress with change in the liquid velocity. This effect is not accounted for in the expression of Reynolds number. To include the effect of liquid core velocity, the inlet liquid Reynolds number is considered in the Nusselt number correlation along with the vapor Reynolds number. The above equation obtained by Cachard [31] shows that the vapor Reynolds number is a function of the non-dimensional film thickness for a laminar vapor flow. Extending his work, a general form of the friction factor for both laminar and turbulent flow can be expressed as

f w ¼ AReq g ;

ð17Þ

where

q ¼ 1;

for Laminar flow;

ð18Þ

¼ 0:25 for Turbulent flow:

Using the general expression for the wall friction, the Reynolds number can be expressed as

Table 1 Uncertainty in the experimental measurements. Measurement

Unit

Range

Uncertainty

Temperature (Thermocouple) Pressure (DP Cell)

C mm H2 O mm H2 O g/s m3 =min kW

10–1371 0–76 0–6350 0–75 0–7.1 0–750

±1.1 ±0.1 ±10.8 ±0.1 ±0.09 ±1.2

Injection mass flow rate Steam flow rate Power

 1 Reg  d3 2q :

ð19Þ

In film boiling, instead of the Prandtl number of vapor, the film Pr

Pr h

Prandtl number, Pr g;film ¼ Ja g ¼ C p DfgT is more relevant, as shown by sup

many researchers [7,9,13,32]. Substituting the expression for Reynolds number and film Prandtl number in Eq. (15), gives

L. Mohanta et al. / International Journal of Heat and Mass Transfer 107 (2017) 112–122

Nu 

 

d3

1 2q

m Prng;film :

ð20Þ

In a simplified form, the above equation can be written as 0

Nu  dm Prng;film :

ð21Þ

117

compared with four different film boiling correlations. A new correlation is developed using the data so obtained and the semiempirical model presented in the preceding sections. The performance of the new correlation is evaluated with data from other facilities.

Further expanding, Eq. (21) leads to

 3m0 2 3  d g qg ql  qg lg hfg n 5 Nu  4 : l2g kg DT

4.1. Visualization

ð22Þ

At the limit when the values of m0 and n are the same, the above equation becomes,

  3m0 2 3 d g qg ql  qg hfg 5 : Nu  4 lg kg DT

ð23Þ

The above form of the Nusselt number is similar to the expression obtained by Bromley [5] for pool boiling with the value of the exponent m0 ¼ 0:25, and with a different length scale. The length scale used by Bromley was the diameter of the tubes, however, in this case the length scale is the vapor film thickness. This expression is also similar to the work by Berenson [33] having a length scale of the diameter of the bubble. Similar expressions using the distance from the quench front as the length scale, have been reported by Sudo [18], Mosaad and Johannsen [32] and Meduri [14]. This justifies the use of film Prandtl number instead of the Prandtl number of vapor. To obtain the Bromley-equivalent correlation, one of the assumption is that the exponents m0 and n are the same. However, even for single phase flow this is not true. In this study, the exponents m0 and n are treated to be different and Eq. (21) will be considered to correlate the data. 4. Results and discussions In this section visualization of time progression of an inverted annular film is presented. The data obtained from the tests is

(a) t= 6 s

(b) t=110 s

Fig. 2 shows the photographs of film boiling at different times, taken at the viewing window near Grid 6 for a test with 40 psi (276 kpa) and 6 in/s (15.24 cm/s), inlet subcooling of 83 K and peak power of 2.31 kW/m per rod. Fig. 2a shows the photograph of heater rods immediately after the injection starts (6 s after injection). At this time, cooling happens due to steam flow only. This is the time when the vapor is visually observed for the first time at the viewing window at Grid 6. Fig. 2d shows that the quench front is upstream of the Grid 6, at t = 135 s after start of the reflood. A smooth film is observed downstream of the quench front. The lighter color is the vapor film enveloping the heater rods showing the IAFB heat transfer regime with a smooth interface. Qualitatively, it can be seen that even downstream of the spacer grid the film is smooth. This explains that spacer grids do not significantly contribute to heat transfer enhancement in the IAFB regime [25]. At t = 125 s after start of reflood in Fig. 2c, the film appears more wavy compared to Fig. 2d. It indicates instability growth at the interface. At t = 110 s after start of the reflood (Fig. 2b), the pinch in the liquid core and higher vapor thickness are observed. This can be noticed by the appearance of lighter shade in the photograph. Periodically alternating liquid core expansion and contraction is observed from the figure. This shows that pinching is higher when the quench front is far away from the observing location. At this time the void fraction is higher compared to the void fraction at the time instant shown in Fig. 2c and d. The appearance of vapor bulge is the precursor to the liquid slug formation and shows the transition from IAFB to ISFB regime. It can be concluded that when the void fraction is low (near the quench front, Fig. 2d) the instability wave growth is slow and at a higher void fraction

(c) t=125 s

(d) t=135 sec

Fig. 2. Photographs showing reflood process at various times after injection and IAFB with smooth and wavy interfaces.

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(far from the quench front, Fig. 2b) the instability waves grow faster. A similar conclusion has been discussed by Mohanta et al. [34] in a theoretical stability analysis of co-axial jets involving heat and mass transfer at the interface. 4.2. Comparison with existing models The present experimental data are compared with four different film boiling models including the models by Bromley model [5], Sudo [18], Mosaad and Johannsen [21], and the correlation in the TRACE code [3]. Comparisons of these models with the RBHT experimental data are shown in Fig. 3a–d. In Fig. 3a–c, higher Nusselt numbers correspond to the data points near the quench front and lower values represent data points far downstream of the quench front. However, in Fig. 3d small Nusselt numbers correspond to location near quench front. The RBHT data includes both saturated as well as subcooled data. Fig. 3a compares the Bromley model [5] with the RBHT data. The Bromley model over-predicts the data near the quench front and under-predicts the data at a higher void fraction i.e., far away from the quench front. Fig. 3b and c compare the film boiling models by Sudo [18] and Mosaad [21] with the RBHT data, respectively. The prediction using these models leads to an error of nearly 50%. Fig. 3d compares the TRACE correlation [3] with the RBHT data. Among the four film boiling models, the prediction using the TRACE correlation is closer to the data. However, it

under-predicts the Nusselt number. Comparing Fig. 3a–d, it can be concluded that the Bromley model and TRACE correlation capture the heat transfer behavior better than the other two models. This is the reason why the data points are more scattered in Fig. 3b and c. Comparing all the data points from RBHT with the TRACE correlation, the root mean square error is found to be 30%. Table 2 compares the Mean Absolute Percentage (MAP) error and Root Mean Square errors in predicting the RBHT data using different film boiling models. 4.3. New correlation The semi-empirical model described in the previous section provides the basis for correlation development. Eq. (21) shows that the Nusselt number depends on the Reynolds number and the film Prandtl number. This form is obtained with several simplifications, one of them is that the interfacial shear stress is equal to the wall shear stress. In reality the interfacial shear stress would depend on the relative velocity between the liquid core and the vapor film. Similarly local subcooling in the flow also affects the interfacial mass transfer. Using the RBHT test data, the effects of liquid Reynolds number and local subcooling are examined. Fig. 4 shows the variation of Nusselt Number with the nondimensional vapor film thickness, d at various liquid Reynolds numbers. The non-dimensional film thickness represents the vapor Reynolds number, as given in Eq. (19). For the experiments having

200

200 +50%

150 -50%

100

150

Nu, Predicted (Mosaad)

Nu, Predicted (Bromley)

+50%

50

100

-50% 50

0

0

0

50

100 150 Nu, Measured (RBHT)

200

0

(a) Comparison with Bromley Model [5]

50

100 150 Nu, Measured (RBHT)

200

(b) Comparison with Mossad Model [21] 30

200 +50%

+50% Nu(δ), Predicted (TRACE)

Nu, Predicted (Sudo)

25

150

20

-50%

100

-50%

15 10

50

0

5 0

0

50

100 150 Nu, Measured (RBHT)

(c) Comparison with Sudo Model [18]

200

0

5

10 15 20 25 Nu(δ), Measured (RBHT)

30

(d) Comparison with TRACE Correlation [3]

Fig. 3. Comparison of the RBHT data with models from the literature.

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119

Table 2 Error in predicting the data using existing film boiling models. Model

Length scale

Mean absolute % error

RMS error

Bromley Sudo Mosaad TRACE

Hydraulic diameter Hydraulic diameter Hydraulic diameter Vapor film thickness

33 67 41 26

41 104 49 30

16

Nusselt Number

14

12 10 8 6

Rel,in

4

4400

2

8700

0

0

20 40 60 80 Non-dimensional Film Thickness (δ*)

Fig. 4. Effect of the inlet liquid Reynolds number on the Nusselt number.

inlet liquid Reynolds number Rel;in ¼ 4400, the film Prandtl number varies from 2.4 to 4, whereas for the experiments with Rel;in ¼ 8700, the film Prandtl number varies from 2.5 to 3.7. It is evident from Fig. 4 that there are parameters apart from the vapor Reynolds number and the film Prandtl number affecting the Nusselt number. Clearly the liquid Reynolds number is one of these parameters. It can be observed that the Nusselt number is higher for a test with higher liquid Reynolds number for a given nondimensional film thickness. The effect of liquid Reynolds number can also be ratified using the experimental data by Fung [19,20] for a single tube experiment. The effect of liquid Reynolds number using Fung’s data [19,20] is shown in Fig. 5. In addition to the liquid Reynolds number, the effect of local subcooling of the liquid core is also examined using the RBHT data. Fig. 6 shows the variation of the Nusselt number with the nondimensional film thickness having different degrees of subcooling. The subcooling in the flow is represented by the subcooled Jakob number. This figure shows that for a given non-dimensional film thickness, the Nusselt number is higher for a higher subcooled Jakob number. The effects of inlet Reynolds number and subcooling have been previously observed by many researchers [14,32,2] and are consistent with the present results. To include the effect of inlet Reynolds number and local subcooling, asymptotic forms are chosen as enhancement factors. So that when there is no liquid motion, i.e., pool boiling, the correlation would be able to predict the pool boiling Nusselt number. Similarly in the absence of local subcooling in the flow, the correlation would predict the saturated flow film boiling behavior. Accordingly, the form of the correlation is chosen as



3 5 Nu ¼ b1 ðd Þ 2 Pr bg;film 1 þ b4 Rebl;in

b



1 þ b6

Jasub 7 JabSup

!

:

ð24Þ

Meduri et al. [14] observed that in IAFB on a vertical surface the Nusselt number linearly depends on the local subcooling and the slope of the linear function decreases with wall superheat. A linear dependence on subcooling was also observed by Sudo et al. [18].

Fig. 5. Effect of inlet liquid Reynolds number on the Nusselt number in single tube experiment by Fung [19,20].

Fig. 6. Effect of local subcooling on the Nusselt number.

The term



1 þ b6 Jabsub has been chosen to account for the effect 7 JaSup

of subcooling. This form is similar to previous works by Meduri et al. [14] for a vertical plane surface and by Dhir and Purohit   [35] for flow film boiling on spheres. The term 1 þ b4 Reb5 accounts for the effect of liquid Reynolds number, this form is consistent with correlations by [21,14,35].

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L. Mohanta et al. / International Journal of Heat and Mass Transfer 107 (2017) 112–122

4.4. Comparison of the model with other test data

Fig. 7. Comparison of the predicted Nusselt number with the Nusselt number measured in the RBHT experiments.

To check the applicability of the present correlation to other data, two different sets of data are chosen to compare with the new correlation. First comparison is made with the FLECHTSEASET experiments conducted in a rod bundle with 161 heater rods [29,30]. Two of the high flooding rate tests having flow rate of 0.076 m/s and 0.152 m/s (FLECHT -SEASET test no: 31302 and 31701) are chosen for comparison. The comparison is done for these tests when the quench front is at 1.67 m and thermocouples chosen are up to a elevation where void fraction is less than 0.9. Fig. 8 shows the comparison of predicted data. Experimental data are predicted within 20% error margin. Fig. 9 compares the predicted data with IAFB experiments in a single tube conducted by Fung [19] for different mass fluxes. The legend of the figure shows the mass fluxes of the tests. It is noted that the Reynolds number for these tests are well beyond the limits in which the correlation is developed. The predicted results show a biased error. However, the functional relationship is captured well. Generally the Nusselt numbers even in single-phase flow, tend to be higher for non-circular channels. To reduce the biased error the coefficient b1 is modified to 0.2 for tubes as compared to the value of 0.262 for a rod bundle. The correlation for tube becomes  0:74

Nu ¼ 0:2ðd Þ

Pr 0:43 g;film

!   Jasub 0:53 1 þ 0:0016Rel;in 1 þ 4:42 1:23 : JaSup

ð26Þ

The predicted results with modified coefficient b1 ¼ 0:2 is shown in Fig. 10. This figure shows that most of the data points are predicted within 20% error margin with the modified coefficient. The mean absolute error in predicting Fung [19] data with the present correlation is 9% and the root mean square error is 12%. The TRACE correlation predicts the data with root mean square error of 16%. 4.5. Comparison of the model with DFFB correlation Fig. 8. Comparison of the predicted Nusselt number with the Nusselt number measured in the FLECHT-SEASET rod bundle experiments [29,30].

The values of all the coefficients are calculated using a nonlinear regression method in MATLAB. The coefficient of determination (R2 ) value for the data fit is 0.77. The final form of the correlation obtained using nonlinear regression is

Nu ¼ 0:262ðd Þ

Pr0:43 g;film

!   Jasub 0:53 1 þ 0:0016Rel;in 1 þ 4:42 1:23 : JaSup

12

ð25Þ The above correlation is developed using the test data with pressure ranging from 138 to 414 kPa, reflood rate of 0.076 m/s to 0.152 m/s and inlet subcooling of 11–83 K. The range of void fraction included in this correlation is 0 to 0.9. That means this correlation is valid both in the IAFB as well as ISFB regimes. This is a significant improvement compared to the TRACE correlation for IAFB which is applicable only up to a void fraction of 0.6. The range of the controlling parameters in terms of non-dimensional numbers are, 4400 6 Rel;in 6 9800; 0 6 JaSub 6 0:046; 0:18 6 JaSup 6 0:43; 2:1 6 Pr g;film 6 5:3 and 0 6 d 6 86. Fig. 7 shows a comparison of the predicted data with the experimental data. Note that, 90% of the predicted data fall within the 20% error band. The mean absolute percentage error is 11% and the root mean square error is 15%. It is to be noted that for the TRACE correlation, the root mean square error is 30%.

+20% 10

Nu (Predicted)

 0:74

In the post CHF regime, the DFFB regime occurs when the void fraction is more than 0.9 [36,1,37]. The present correlation is valid up to a void fraction of 0.9. In this section, the results from the new correlation is compared with the predicted results using correlations for DFFB proposed by Polomik et al. [38] and Dougall and Rohsenow [16] in Fig. 11. Nusselt number presented in Fig. 11 are calculated based on the hydraulic diameter of the subchannel. To compare the DFFB correlations, data points were chosen when the void fraction is 0:88 < a < 0:92. The predicted data using the

8 -20% 6 4 2

100

200

300

350

400

500

0 0

2

4

6

8

10

Nu (Measured, Fung) Fig. 9. Comparison of the predicted Nusselt number with the Nusselt number measured in single tube experiments by Fung [19,20].

L. Mohanta et al. / International Journal of Heat and Mass Transfer 107 (2017) 112–122

10

+20%

9 -20%

Nu (Predicted)

8 7 6 5 4 3

121

tion showed a stable film near the quench front and wavy film far downstream of the quench front. A new correlation is developed for film boiling in a rod bundle for void fraction up to 90%, covering the IAFB and ISFB flow regimes. The new correlation has been compared with other data sets available in the literature and data are predicted within 15% error. The correlation can also be used for the tube geometry using a modified coefficient. At the void fraction of 90%, the present correlation is compared with the correlations for DFFB. It is found that the present correlation predicts better than the Dougall and Rohsenow, and Polomik correlations.

2 1

100

200

300

350

400

500

Acknowledgment

0 0

2

4

6

8

10

Nu (Measured, Fung) Fig. 10. Comparison of the predicted Nusselt number with the Nusselt number measured in single tube experiments by Fung [19,20] with modified coefficient.

The work performed at the Pennsylvania State University was supported by the U.S. Nuclear Regulatory Commission under Contract # NRC-HQ-60-15-T-0001. Appendix A The vapor velocity is not measured in the current experimental study. Note that, none of the previous studies has reported any vapor velocity measurement in IAFB. Nevertheless, to use the above form of Nusselt number, the Reynolds number needs to be estimated using known parameters. Cachard [31] presented a method to compute the vapor velocity. The steady state governing equations for two-phase separated flow are given as.

Nu(D), Present Data/Correlation

80 +20% 60 -20% 40

(i) Continuity Liquid:

20

RBHT Data

@ ðq al ul Þ ¼ m000 l : @z l

Correlation 0 0

20

40 Nu(D), Polomik

60

80

Vapor:

 @  q ag ug ¼ m000l : @z g

(a) Comparison with Polomik Correlation

Nu(D), Present Data/Correlation

80

60

ql al ul

@ul @p si P i ¼ ql a l g   al  m000 l ðul  ui Þ: @z @z A

qg ag ug 20 RBHT Data 0

80

si P i

(b) Comparison with Dougall and Rohsenow Correlation Fig. 11. Comparison of the predicted Nusselt number with DFFB correlations.

present correlation are compared well with the Polomik correlation except for the low flooding tests. This is because the Polomik correlation under-predicts the experimental data. The Dougall and Rohsenow correlation consistently under-predicts the experimental data, hence the correlation predictions are higher in the DFFB regime.

A

þ al

sw Pw A

  ¼ al ag ql  qg g:

ð31Þ

Cachard [31] assumed that in IAFB the liquid fraction is close to unity, the liquid velocity is much lower than the vapor velocity and there is a smooth vapor–liquid interface. These assumptions lead to the following approximations

Pi ’ Pw ;

ag ¼

ð32Þ

Pi d ; A

5. Conclusions A semi-empirical model has been proposed for the Nusselt number during film boiling in a vertical channel. Flow visualiza-

  @ug @p si Pi sw Pw ¼ qg ag g þ  ag þ þ m000 l ug  ui : @z @z A A ð30Þ

Combining Eqs. (29) and (30) by multiplying with ag and al , respectively and neglecting the convective terms, the following equation is obtained

Correlation

20 40 60 Nu(D), Dougall and Rohsenow

ð29Þ

Vapor:

-20%

0

ð28Þ

(ii) Momentum Liquid:

+20%

40

ð27Þ

si ’ sw ’ f w qg

ð33Þ u2g : 2

Using the above approximations, Eq. (31) becomes

ð34Þ

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L. Mohanta et al. / International Journal of Heat and Mass Transfer 107 (2017) 112–122



si ¼

 ql  qg gd 2

:

ð35Þ

For a laminar flow, Cachard [31] assumed the friction factor for smooth wall to be

fw ¼

24 : Reg

ð36Þ

The vapor Reynolds number can be calculated as

Reg ¼

qg ug 2d : lg

ð37Þ

Substituting the vapor velocity from Eq. (37) into Eq. (34) the interfacial shear is obtained as

si ¼

3l2v Reg

qg d2

:

ð38Þ

Using Eqs. (35) and (38), the Reynolds number reduces to

Reg ¼

  d3 g qg ql  qg 6l

2 g

¼

d3 : 6

ð39Þ

The above equation obtained by Cachard [31] shows that the vapor Reynolds number is a function of the non-dimensional film thickness in laminar vapor flow. References [1] N. Hammouda, D. Groeneveld, S. Cheng, ‘‘An experimental study of subcooled film boiling of refrigerants in vertical up-flow, Int. J. Heat Mass Transfer 39 (18) (1996) 3799–3812. [2] M.E. Nakla, D.C. Groeneveld, S.C. Cheng, ‘‘Experimental study of inverted annular film boiling in a vertical tube cooled by R-134a, Int. J. Multiph. Flow 37 (1) (2011) 67–75. [3] US NRC, TRACE V5.0 Theory Manual: Field equations, Solution methods, and Physical models, 2007. [4] N. Takenaka, T. Fujii, K. Akagawa, K. Nishida, ‘‘Flow pattern transition and heat transfer of inverted annular flow, Int. J. Multiph. Flow 15 (5) (1989) 767–785. [5] L.A. Bromley, Heat transfer in stable film boiling, Chem. Eng. Prog. 46 (5) (1950) 221–228. [6] L.A. Bromley, N.R. Leroy, J.A. Robbers, Heat transfer in forced convection film boiling, Ind. Eng. Chem. 45 (12) (1953) 2639–2646. [7] J.C.Y. Koh, Analysis of film boiling on vertical surfaces, J. Heat Transfer 84 (1) (1962) 55–62. [8] T.H.K. Frederking, Laminar two-phase boundary layer in natural film boiling, ZAMP 14 (1963) 207–218. [9] E.M. Sparrow, R.D. Cess, The effect of subcooled liquid on laminar film boiling, J. Heat Transfer 84 (2) (1962) 149–155. [10] N.V. Suryanarayana, J.H. Merte, Film boiling on vertical surfaces, J. Heat Transfer 94 (4) (1972) 377–384. [11] T.D. Bui, V.K. Dhir, Film boiling heat transfer on an isothermal vertical surface, J. Heat Transfer 107 (4) (1985) 764–771. [12] R. Vijaykumar, V.K. Dhir, An experimental study of subcooled film boiling on a vertical surface-thermal aspects, J. Heat Transfer 114 (1992) 169–178. [13] R.D. Cess, E.M. Sparrow, Film boiling in a forced-convection boundary layer flow, J. Heat Transfer 83 (3) (1961) 370–375.

[14] P.K. Meduri, G.R. Warrier, V.K. Dhir, Wall heat flux partitioning during subcooled forced flow film boiling of water on a vertical surface, Int. J. Heat Mass Transfer 52 (1516) (2009) 3534–3546. [15] S. Yi, G.R. Warrier, V.K. Dhir, Heat transfer and hydrodynamics of subcooled inverted annular flow film boiling on a single rod heater, in: Proceedings of the ASME 2009 International Mechanical Engineering Congress and Exposition, 2009. [16] R. Dougall, W. Rohsenow, Film boiling on the inside of vertical tubes with upward flow of the fluid at low qualities, Tech. Rep. 9079-26, M.I.T., 1963. [17] E. Greitzer, F. Abernathy, Film boiling on vertical surfaces, Int. J. Heat Mass Transfer 15 (3) (1972) 475–491. [18] Y. Sudo, Film boiling heat transfer during reflood phase in postulated PWR Loss-of-Coolant accident, J. Nucl. Sci. Technol. 17 (7) (1980) 516–530. [19] K. Fung, S. Gardiner, D. Groeneveld, Subcooled and low quality flow film boiling of water at atmospheric pressure, Nucl. Eng. Des. 55 (1) (1979) 51–57. [20] K.K. Fung, Subcooled and Low Quality Film Boiling of Water in a Vertical Flow at Atmospheric Pressure (Ph.D. thesis), The University of Ottawa, 1981. [21] M. Mosaad, K. Johannsen, ‘‘Experimental study of steady-state film boiling heat transfer of subcooled water flowing upwards in a vertical tube, Exp. Thermal Fluid Sci. 2 (4) (1989) 477–493. [22] M. Shiotsu, K. Hama, Film boiling heat transfer from a vertical cylinder in forced flow of liquids under saturated and subcooled conditions at pressures, Nucl. Eng. Des. 200 (12) (2000) 23–38. [23] L. Mohanta, M.P. Riley, F.B. Cheung, S.M. Bajorek, J.M. Kelly, K. Tien, C.L. Hoxie, Experimental investigation of inverted annular film boiling in a rod bundle during reflood transient, Nucl. Technol. 190 (3) (2015). [24] A. Sridharan, Modeling of Inverted Annular Film Boiling using an Integral Method (Ph.D. thesis), The Pennsylvania State University, 2005. [25] L. Mohanta, Theoretical and Experimental Study of Inverted Annular Film Boiling and Regime Transition During Reflood Transients (Ph.D. thesis), The Pennsylvania State University, 2015. [26] L.E. Hochreiter, F.B. Cheung, T.F. Lin, J.P. Spring, S. Ergun, A. Sridharan, A. Ireland, E.R. Rosal, RBHT Reflood Heat Transfer Experiments Data and Analyis, Tech. Rep. NUREG/CR-6980, US Nuclear Regulatory Commission, 2012. [27] E.R. Rosal, T.F. Lin, I.S. McClellan, R.C. Brewer, Rod bundle heat transfer (RBHT) test facility description report, no. NUREG/CR-6976, 2010. [28] L. Friedel, Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow, in: European Two-phase Flow Group Meeting, Paper E 2, pp. 1979, 1979. [29] M. Loftus, L. Hochreiter, C. Conway, C. Dodge, A. Tong, E. Rosal, M. Valkovic, S. Wong, PWR FLECHT SEASET unblocked bundle, forced and gravity reflood task data report. volume 1, Tech. Rep., Westinghouse Electric Corp., Pittsburgh, PA (USA). Nuclear Energy Systems Div., 1981. [30] M. Loftus, L. Hochreiter, C. Conway, C. Dodge, A. Tong, E. Rosal, M. Valkovic, S. Wong, PWR FLECHT SEASET unblocked bundle, forced and gravity reflood task data report. volume 2. appendix c, Tech. Rep., Westinghouse Electric Corp., Pittsburgh, PA (USA). Nuclear Energy Systems Div., 1981. [31] F. Cachard, Development, implementation and assessment of specific closure laws for inverted annular film-boiling in a two-fluid model, Tech. Rep. NUREG/ IA-0133, U.S. Nuclear Regulatory Commission, 1996. [32] M. Mosaad, K. Johannsen, An improved correlation for subcooled and low quality film boiling heat transfer of water at pressures from 0.1 to 8 MPa, Nucl. Eng. Des. 135 (3) (1992) 355–366. [33] P.J. Berenson, Film-boiling heat transfer from a horizontal surface, J. Heat Transfer 83 (3) (1961) 351–358. [34] L. Mohanta, F.-B. Cheung, S.M. Bajorek, Stability of coaxial jets confined in a tube with heat and mass transfer, Physica A 443 (2016) 333–346. [35] V. Dhir, G. Purohit, Subcooled film-boiling heat transfer from spheres, Nucl. Eng. Des. 47 (1) (1978) 49–66. [36] N. Takenaka, K. Akagawa, T. Fujii, K. Nishida, Experimental study on flow pattern and heat transfer of inverted annular flow, Nucl. Eng. Des. 120 (23) (1990) 293–300. [37] M. Andreani, G. Yadigaroglu, Prediction methods for dispersed flow film boiling, Int. J. Multiph. Flow 20 (1994) 1–51. [38] E. Polomik, S. Levy, S. Sawochka, Film boiling of steam-water mixtures in annular flow at 800, 1100, and 1400 psi, J. Heat Transfer 86 (1) (1964) 81–87.