Analysis of multi-region conduction-controlled rewetting of a hot surface with precursory cooling by variational integral method

Applied Thermal Engineering 73 (2014) 265e274

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Analysis of multi-region conduction-controlled rewetting of a hot surface with precursory cooling by variational integral method Manish Kumar Agrawal*, S.K. Sahu Discipline of Mechanical Engineering, School of Engineering, Indian Institute of Technology Indore, Indore 453441, Madhya Pradesh, India

h i g h l i g h t s
The effect of sputtering region and precursory cooling on rewetting velocity is discussed.
Closed form expression is presented for temperature distribution and wet front velocity.
Sputtering region and precursory cooling strongly influence the wet front velocity.
Neglecting the precursory cooling in the model may under predict the rewetting velocity.
Variational method solution exhibit good agreement with test data and available results.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 19 April 2014 Accepted 23 July 2014 Available online 1 August 2014

An analytical model has been proposed to evaluate rewetting velocity by employing variational integral method. The model considers three distinct regions: a dry region ahead of wet front, the sputtering region immediately behind the wet front and a continuous film region further upstream. Two different models are considered in the sputtering region for the analysis. First model considers a constant heat transfer coefficient in the sputtering region; while the other one propose a variation in heat transfer coefficient in the sputtering region. Both the models consider a constant heat transfer coefficient in the wet region and exponentially varying heat transfer coefficient in the dry region ahead of wet front. For all the cases the closed form expression is obtained for temperature field along axial direction. Relationship between various rewetting parameters such as; Peclet number, Biot number, dry wall temperature, incipient boiling temperature, sputtering length, magnitude of precursory cooling and the extent of precursory cooling has been obtained from the analysis. Present prediction obtained by employing variational integral method exhibits an excellent agreement with the previous analytical results [1e4] and test data [5,6,23]. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Rewetting Variational integral method Three-region Sputtering Precursory cooling Peclet number

1. Introduction When liquid is brought in contact with a sufficiently hot surface, a vapor blanket is formed on the solid surface that prevents the solideliquid contact. As the surface cools off, the vapor film collapses and the liquid film re-establishes contact with the hot surface. Rewetting of hot surfaces is common in many industrial applications such as: metallurgical treatments, cooling of overheated nuclear fuel rod during postulated loss of coolant accidents (LOCA), cryogenic processes, space craft thermal control and most recently cooling systems of electronic circuits.

* Corresponding author. Tel.: þ91 7324 240771; fax: þ91 0731 2364182. E-mail addresses: [email protected], [email protected] (M.K. Agrawal). http://dx.doi.org/10.1016/j.applthermaleng.2014.07.062 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

Numerous studies have been carried out in past to analyze the rewetting behavior of hot surfaces both through experimental investigation [5e7] and theoretical analysis [5,8e13]. It was observed that during rewetting of hot slab at lower coolant flow rates, the variation of temperature in the transverse direction is less compared to the longitudinal direction. Therefore, initial efforts were made to analyze rewetting problems based on onedimensional approximation [4,5]. These models were successful in correlating the test data at lower flow rates. It is observed that the temperature gradient in the transverse direction is significant at higher coolant flow rates. In a view of this several two-dimensional models were proposed [8,9,12] that correlates the test data at higher flow rates. It is observed that most of the rewetting models consider the heat transfer coefficient and rewetting temperature as an input parameter to solve the conduction equation. The solution to the above problem was obtained by Olek et al. [14] by considering

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rewetting as conjugate heat transfer problem. In their analysis, the heat transfer coefficient was not specified but was obtained as a part of the solution. The solution to a transient heat conduction problem was obtained by Dorfman [15]. The author reported that the transient cooling process is governed by a dimensionless parameter termed as Leidenfrost number, expressed as the ratio of Biot number to the square of the Peclet number. Starodubtseva et al. [16] proposed a model to study the process of rewetting of overheated surface during falling film of cryogenic liquid. Davidy et al. [17] presented solution to the rewetting problem by considering an arbitrary variation in heat flux in the dry region ahead of the wet front. The authors claimed that their model can be used for the prediction of re-flooding for a wide range of operating parameters. Recently, a review on several analytical and semi-analytical models of conduction controlled rewetting has been presented by Sahu et al. [18]. The authors [18] discussed various conduction-controlled rewetting models and summarized the closed form expressions for various rewetting parameters. A wide variety of experiments have been performed to analyze the phenomenon of rewetting [3,5e8]. Rewetting experiments demonstrate that inverse of rewetting rate increases with initial wall temperature and decreases with the coolant flow rate [5,6,8]. During experimental investigation the authors have considered various test geometry [5,19e21], different methods of venting the generated steam [22] and several modes of coolant injection system [3,23]. Tests have been carried out either for a single rod [23] or with rod bundle [24] with varied range of coolant flow rate and dry wall temperature. During experiments [25], it is observed that at higher coolant flow rates a part of the coolant sputters away from the wet front and cools the surrounding vapor. Subsequently, droplet vapor reduces the dry wall temperature ahead of the wet front. This phenomenon is termed as precursory cooling. In order to incorporate the precursory cooling in the rewetting model usually a constant heat transfer coefficient [26,27], variation of heat transfer coefficient [28,29] or variation of heat flux [30e32] ahead of the wet front was considered. From the experimental studies of Shires et al. [25], it is observed that the rate of heat removal is very high in a narrow region just behind the wet front. The region is called sputtering region and is strongly influence the wet front velocity. Therefore, adopting a constant heat transfer coefficient in the wet region may not be appropriate for the rewetting analysis. Several one-and two- dimensional, three-region rewetting models are reported that considers three different regions, namely, a wet region in the upstream direction of wet front, a sputtering region behind the wet front and a dry region in the downstream direction ahead of wet front [2,4,33,34]. Sun et al. [4] first considered a three region, one dimensional rewetting model to analyze falling film rewetting of hot surface. The two-dimensional analyses of rewetting considering a three region model in an annular geometry have been suggested by Sawan et al. [33]. Bera and Chakrabarti [35] proposed a three-region rewetting model for the cooling of a cylindrical rod by employing the Wiener-Hopf technique. Sahu et al. [1] reported the solution to the above problem valid for both Cartesian and cylindrical geometry by employing HBIM. During their investigation it is observed that the wet front velocity strongly depends on the boiling Biot number than the convective Biot number. It was reported that neglecting the boiling Biot number in the sputtering region may under predict the rewetting velocity. Recently, Sibamoto et al. [36] proposed a two-dimensional rewetting model to quantify the effect of precursory cooling and internal heat generation on the rewetting velocity and revealed the effect of wall depth conduction during prediction of rewetting velocity. The authors [37] proposed a one-

dimensional rewetting model with Precursory cooling by modifying the model presented by Sun et al. [29]. The proposed model was correlated with their test data considering proper assumptions. In actual situation, the distribution of heat transfer coefficient near the sputtering region and the dry region ahead of the wet front varies along the axial direction [38]. It is argued that accurate choice of shape of the heat transfer coefficient is important to precisely predict the wet front velocity and surface temperature distribution. It is observed that most of the three region rewetting model considers a constant heat transfer coefficient in the both liquid and sputtering region. Only a single investigation reported the one-dimensional rewetting model with large variations in the heat transfer coefficients near the wet front [39]. The authors [39] solved the one-dimensional heat-conduction equation by dividing the quenching zone into small segments of arbitrary temperature increment and heat transfer coefficient. A trial and error method is proposed to predict the wet front velocity, sputtering length and temperature profile along the hot surface. Recently, Agrawal and Sahu [13,40] and Agrawal et al. [41] solved a one-dimensional rewetting model with constant heat transfer coefficient in the wet region and an adiabatic condition in the dry region by employing Variational Integral Method (VIM). The results obtained by employing VIM exhibited good agreement with available analytical results and published test data for varied range of test parameters. In this study an attempt has been made to extend the previous work [13,40,41] to analyze the multi-region rewetting model with varying heat transfer coefficient in the sputtering and dry region ahead of the wet front. Two different rewetting models are considered for the analysis. In the first model, a constant heat transfer coefficient is considered in the sputtering region and in second model, an axially varying heat transfer coefficient in the sputtering region is considered. Both the models consider an exponentially decaying heat transfer coefficient in the dry region ahead of wet front. The present prediction is found to be in good agreement with the reported analytical results and published test data covering a wide range of coolant flow rate, dry wall temperature and test geometry. 2. Theoretical analysis 2.1. Physical model Fig. 1a schematically depicts the falling film rewetting of a onedimensional slab of infinite length. When liquid is injected from the top, the liquid cools the hot surface in the form of falling film. As the coolant moves in the downward direction a thin vapor film is formed at the solideliquid interface and prevents the contact of coolant on the hot surface. As the process continues, the temperature of the hot wall cools from initial temperature (Tw) to the wet front temperature (T0) and the vapor blanket becomes unstable and collapses (Fig. 1b). The region behind the wet front corresponds to transition boiling regime and followed by a nucleate boiling regime. Beyond this, the surface temperature drops below the temperature of incipient boiling and the heat is removed by convection to the single-phase fluid. In addition to this, the dry wall ahead of the wet front is cooled due to precursory cooling. In such a case the distribution of heat transfer along the hot object becomes non-linear and the heat transfer profile varies arbitrarily along the axial direction. Fig. 1c depicts the actual variation of heat transfer coefficient along axial direction. It is shown that the magnitude of heat transfer coefficient near the wet front location is very high and it gradually decreases in the downstream direction. In order to analyze the rewetting phenomena one can assume a constant heat

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267

Fig. 1. (aee) Schematic diagram of a three-region model of one-dimensional hot object.

transfer coefficient of different magnitude for both wet region and sputtering region. Alternatively, one can consider a constant heat transfer coefficient in the wet region and parabolic variation in heat transfer coefficient in the sputtering region. It is observed that the mechanism of heat removal rate during the precursory cooling includes the convection and radiation from the dry wall to the dropletevapor mixture. In order to model the effect of precursory cooling, one can assume an exponentially varying heat transfer coefficient in the dry region ahead of wet front. Here, the physical model is proposed that considers three different regimes, namely, dry region ahead of wet frontð0  x < þ ∞Þ, the sputtering region immediately behind the wet front ðl  x  0Þ and a continuous film region further upstream ð∞ < x  0Þ as shown in Fig. 1dee. Based on the above model, the following assumptions are made for the analysis: a) Constant heat transfer coefficient (hc) is considered in the wet region and an exponentially decaying heat transfer coefficient of the form hðxÞ ¼ ðhB =NÞeax is considered in the dry region ahead of wet front [29]. Two different models are considered to account the variation in heat transfer in the sputtering region. First model considers a constant heat transfer coefficient (hB) in the sputtering region. While, the other model considers an axially varying heat transfer coefficient of the form hðxÞ ¼ bx2 þ cx þ d in the sputtering region. b) The wet front velocity is assumed to be constant during its propagation. The end effects are neglected; therefore the problem can be reduced to a quasi-steady one [5,8]. c) In the sputtering region ðl  x  0Þ, the temperature of the hot surface varies from rewetting temperature T0 to incipient boiling temperature Tb and in the liquid region ð∞ < x  0Þ the surface temperature varies from Ref. Tb to liquid saturation temperature Ts. d) Suitable values of rewetting temperature and heat transfer coefficients are taken as input parameters to solve the conduction equation. e) Other factors such as surface finish, surface roughness, etc. are either neglected or they do not affect the process of rewetting.

2.2. One-dimensional formulation It may be noted that for a given value of Biot number and dry wall temperature, one-dimensional model predicts lower rewetting velocity compared to the two-dimensional model. In order to avoid complexity of numerical calculation, a one-dimensional model is proposed to demonstrate the effect of precursory cooling and variation in heat transfer coefficient in the sputtering region. The transfer of energy from the hot surface to the surrounding fluid may be represented by a one-dimensional control volume as shown in Fig. 2. An elemental volume of length dx will be subjected to axial head conduction. In addition the hot wall transports qv, qb and qc amount of energy to the surrounding fluid in the dry, sputtering and wet region, respectively. The heat loss from other side of the hot surface is considered to be negligible and assumed to be insulated. The one-dimensional

Fig. 2. Control volume analysis of one-dimensional hot surface.

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conduction equation for the hot solid in Cartesian coordinate system (Fig. 1a) can be written as:

v2 T

h vT K 2  ðT  Ts Þ ¼ rC d vt vx

ð  ∞ < x < þ ∞Þ

(1)

where, K, r, C and d represents thermal conductivity, density, specific heat capacity and thickness of the slab material, respectively. While, h represents the wet side heat transfer coefficient. Using the quasi-steady state assumption (vT=vt ¼ uvT=vx), Eq. (1) yields:

K

v2 T vx2

h vT ¼ 0;  ðT  Ts Þ þ rCu d vx

ð  ∞ < x < þ ∞Þ

(2)

Utilizing Eq. (4), and Eq. (6), Eq. (7a) can be expressed as:

8 ! ZL > > > d2 q dq > >  BiC q dx þ Pe > > dx > dx2 > > ∞ > > > > > ! > < Z0 d2 q dq I¼  Bi þ Pe q dx B > dx dx2 > > > L > > > > ! þ∞ > > >Z d2 q dq eax > > >  BiB q dx þ Pe > > dx N : dx2

∞ < x  L

L  x  0

(7b)

0  x< þ∞

0

Following normalized variables are defined:

x udrC T  Ts Tw  T0 T  Ts ; q¼ x ¼ ; Pe ¼ ; q ¼ ; q ¼ b ; d K T0  Ts 1 T0  Ts 2 T0  Ts l h d h d L ¼ ; BiB ¼ B ; BiC ¼ C ; a ¼ ad d K K

(5a)

At this juncture, it is necessary to assume a guess temperature profile that satisfies the boundary condition of the physical problem. Although the accuracy of the solution obtained by the variational method depends on the guess profile, there is hardly any guideline to select the best profile. Previously, Agrawal and Sahu [13,40,41] used the hybrid profile involving the exponential and polynomial functions of different orders to analyze the basic rewetting model. In this study, a hybrid profile involving both polynomial and exponential function is considered in the wet region (∞ < x  L) and the dry region (0  x < þ∞), respectively. While, parabolic polynomial function is tried in the sputtering region (L  x 0) and is expressed as.

(5b)

     q x ¼ M1 þ M2 exþL q2 þ a1 x þ L

(3)

Utilizing Eq. (3), the energy Eq. (2) is transformed into following form:

d2 q dq  Jq ¼ 0 þ Pe dx dx2

(4)

Subjected to boundary conditions given by:

qð∞Þ ¼ 0; qðLÞ ¼ q2

qðþ∞Þ ¼ 1 þ q1 qð0Þ ¼ 1

The value of J varies for different heat transfer regimes and the details of the heat transfer models are elaborated below. 2.2.1. Model (1): constant heat transfer coefficient (hB) in the sputtering region The model includes two constant and different heat transfer coefficients, namely, hC and hB in the wet and the sputtering region, respectively. While, the dry region consider an exponentially decaying heat transfer coefficient of the form (hB/N)eax. In such a case the value of J in Eq. (4) is expressed as:

8 BiC ; > > > > < BiB ; J¼ > > eax > > : BiB ; N

ð∞ < x  LÞ ðL  x  0Þ

(6)

ð0  x  þ∞Þ

In the present study, the variational principle proposed by Arpaci [42] has been employed to solve the conduction equation. The governing differential equation of the physical problem (Eq. (4)) is considered as Euler's equation of the variational formulation. Following Arpaci [42], the variational integral (I) valid for the present configuration can be expressed as:

8 > ZL > > > > Fdx  ∞ < x  L > > > > > ∞ > > > Z0 > < I¼ Fdx  L  x  0 > > > > L > > > Zþ∞ > > > > > Fdx 0  x < þ ∞ > > : 0

  q x ¼ M3 an2 x2 þ M4 a3 x þ M5 a4

 ∞ < x  L

Lx0

       q x ¼ M6 1 þ q1 1  ex þ ex 1 þ a5 x

(8a)

(8b)

0  x< þ ∞ (8c)

where, M1…M6 and a1…a5 are positive constants. It may be noted that the Eq. (8) satisfies the boundary conditions of the physical problem. In order to evaluate the constants in Eq. (8), one needs to differentiate the variational integral (I) with respect to each constant. This is expressed as:

vI ¼0 vY

(9)

where, Y ¼ a1…a5. By using Eqs. (7)e(9) and boundary conditions Eqs. (5a), (5b) one gets:

     q x ¼ exþL q2 þ a1 x þ L   q x ¼ a2 x2 þ a3 x þ a4

 ∞ < x  L

Lx0

10a

10b

(7a)        q x ¼ 1 þ q1 1  ex þ ex 1 þ a5 x

0  x< þ ∞

10c

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269

where,

q ð1 þ Pe  BiC Þ 10Peð1  q2 Þ þ 5BiB Lð1  q2 Þ  10BiB L ð1  q2 Þ ; a4 ¼ 1; ; a3 ¼ La12 þ a1 ¼  2 ; a12 ¼  1 þ BiC L 2BiB L3 þ 20L " # q1 ðPe1Þ 4

þ BiNB "

a5 ¼

q1 ð2þaÞ2

2BiB Nð2þaÞ3

 #

1þq1 ð1þaÞ2

(11)

þ 14

It may be noted that utilizing Eq. (11) and Eqs. (10aec), one can evaluate the temperature distribution in various regions, namely, wet, sputtering and dry region of the hot object. While, the nondimensional wet front velocity, Pe and the dimensionless sputtering length, L are still unknown and they can be determined by using the continuity of heat flux at the interface as below:



       dq dq dq dq ¼ and ¼ dx x¼Lþ dx x¼L dx x¼0þ dx x¼0

(12)

Combining Eqs. (10aec) and Eq. (11) with Eq. (12), the values Pe and L are determined by the simultaneous equations expressed as:

  10ð1q2 Þ q2 2BiC q2 5BiB Lð1þq2 Þ q2 1 ¼ Pe þ þ þ L 1þBiC 2BiB L2 þ20 2BiB L2 þ20 1þBiC

1  q2 ¼ q1 þ a5 L

8 Bi ; > > > C > < 2 J ¼ bx þ cx þ d; > > > eax > : d; N

ð  ∞ < x  LÞ ð  L  x  0Þ

(15)

ð0  x < þ ∞Þ

The profile bx2 þ cx þ d involves various constants. Utilizing the boundary condition (x ¼ L, BiB ¼ BiC) and (x ¼ 0, q0 ¼ 1), the constants for the heat transfer profile are obtained as below:

BiB  BiC 2ðBiB  BiC Þ ; d ¼ BiB ; c¼ L L2

(13)

b¼

(14)

The variational integral method proposed by Arpaci [42] is employed in order to solve the conduction equation. Following the procedure described in Section 2.2.1, and utilizing Eqs. (4), (15), (16) and (7a), one can express:

and

La12 þ

coefficient is considered in the dry region for the analysis. In this case the value of J in Eq. (4) is expressed as:

(16)

8 ! ZL > 2 > > d q dq > >  BiC q dx þ Pe ∞ < x  L > > dx > dx2 > > ∞ > > > > > " >   # < Z0 d2 q dq ðBiB  BiC Þ 2 2ðBiB  BiC Þ I¼  x x þ Bi þ Pe þ B q dx L  x  0 > dx L L dx2 > > > L > > > > ! > Zþ∞ 2 > > d q dq eax > > >  BiB q 0  x< þ ∞ þ Pe > > dx N : dx2

(17)

0

Eqs. (13) and (14) are of great interest as this represents the relationship among various modeling parameters, namely, Peclet number (Pe), sputtering length (L), boiling Biot number (BiB), convective Biot number (BiC), dry wall temperature (q1), sputtering temperature (q2), magnitude of precursory cooling (N) and the region for the influence of precursory cooling (a). For the case of no precursory cooling (N / ∞), the results of the present model Eqs. (13) and (14) are presented and compared with other analytical results.

Utilizing Eq. (17), Eqs. (8)e(9) and boundary condition Eqs. (5aeb) one can obtain Eq. (10), where;

2.2.2. Model (2): axially varying heat transfer coefficient(hB) in the sputtering region The model considers a constant heat transfer coefficient (hC) in the wet region, parabolic variation in the heat transfer coefficient in sputtering region and an exponentially decreasing heat transfer

ð1  q2 Þ ; a4 ¼ 1; a5 ¼ a3 ¼ La22 þ L

a1 ¼  a22 ¼

q2 ð1 þ Pe  BiC Þ ; 1 þ BiC

70Peð1  q2 Þ  7Lð1  q2 Þð3c  2b  5dÞ  7Lð3b  5c þ 10dÞ   ; L3  4b þ 7c  14d  140L " # q1 ðPe1Þ BiB þN 4

"

q1  1þq1 2 ð2þaÞ2 ð1þaÞ

#

2BiB þ 14 Nð2þaÞ3

(18)

270

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Combining Eqs. (10aec) and Eq. (18), one can evaluate the temperature distribution in various regions (wet, sputtering and dry) of the hot object. While, the non-dimensional wet front velocity, Pe and the dimensionless sputtering length, L can be evaluated utilizing Eq. (12). These yields:

  70ð1  q2 Þ q2   Pe Z 1 þ BiC 2BiC q2 7Lð1  q2 Þð3c  2b  5dÞ þ 7Lð3b  5c þ 10dÞ  ¼ Z 1 þ BiC ð1  q2 Þ  L (19) where Z ¼ L2(4bþ7c14d)  140

 and

a22 L þ

1  q2 L

 ¼ q1 þ a5

(20)

Eqs. (19) and (20) represents the relationship among various modeling parameters, namely, Peclet number (Pe), sputtering length (L), boiling Biot number (BiB), convective Biot number (BiC), dry wall temperature (q1), sputtering temperature (q2), magnitude of precursory cooling (N) and the region for the influence of precursory cooling (a). Using Eqs. (19) and (20) and by employing a suitable iterative technique, one can evaluate the Peclet number (Pe) and sputtering length (L). The present model (Eqs. (19) and (20)) are correlated with the test data for a given set of rewetting parameters and are reported in the subsequent section.

Fig. 3. Effect of convective and boiling Biot number on rewetting velocity.

used to obtain the influence of various parameters (q1, q2, L, BiB) on the rewetting velocity for the case of no precursory cooling (N / ∞) and elaborated in Figs. 3e7. While the present solution neglecting the sputtering region are compared with test data obtained for varied range of test section, coolant flow rate and dry wall temperature (Fig. 8aed). 3.1. Parametric variation

3. Results and discussion In the present analytical work, three region rewetting model is solved by employing variational method proposed by Arpaci [42]. Two different models are considered to evaluate the heat transfer in the sputtering region. For both the models, a closed form expression for the temperature field is obtained for wet, sputtering and dry regions respectively. Peclet number (Pe) is found to depend on various modeling parameters, namely, dry wall temperature, incipient boiling temperature, sputtering length, convective Biot number and boiling Biot number. The wet front velocity obtained by Model-1 and Model-2 for a given value of q2 and BiB is presented in Table 1. Previously, Duffey and Porthouse [6] have proposed the region of influence of precursory cooling (a) to be 0.00425 in their experimental investigation for water stainless steel pair. Taking a cue from their analysis, in this study the value of a is taken as 0.00425 and the ratio of convective to boiling Biot number is considered as 0.1 for the analysis. It is observed that Model-2 predicts lower wet front velocity compared to Model-1 for given set of modeling parameter. It may be noted that the influence of sputtering region, boiling Biot number and sputtering temperature on rewetting velocity for the case of no precursory cooling is reported by various researchers [1,4]. Present model (Model-1) is

The parameter q1 characterizes the difference between dry wall temperature and the wet front temperature. Here, q2 represents the incipient boiling temperature that characterizes the length of the sputtering region. The value of q2 ranges from zero to unity. It is interesting to notice that for both the extreme values of q2 the present three-region model reduces to a two region rewetting model. For, q2 ¼ 0 the wet region corresponds a uniform boiling heat transfer coefficient with Biot number Bi ¼ BiB. However, for q2 ¼ 1, the boiling region vanishes and signifies a uniform convective heat transfer coefficient (BiC) in the wet region. Fig. 3 shows the variation of Peclet number (Pe) with dry wall

Table 1 Comparison among Peclet number (Pe) between Model-1 and Model-2 for q2 ¼ 0:01; BiB ¼ 0:7; a ¼ 0:00425 and BiC ¼ 0:1BiB .

q1

0.5 1 2 3 4

Model-1

Model-2

Model-1

Model-2

Model-1

Pe

N

Pe

N

Pe

N

Pe

N

Pe

N

Model-2 Pe

N

2.14 1.34 0.89 0.72 0.63

4 4 4 4 4

2.12 1.32 0.87 0.71 0.46

4 4 4 4 4

1.37 0.86 0.55 0.44 0.37

8.5 8.5 8.5 8.5 8.5

1.34 0.84 0.54 0.42 0.24

8.5 8.5 8.5 8.5 8.5

0.80 0.49 0.29 0.20 0.16

100 100 100 100 100

0.76 0.46 0.27 0.19 0.07

100 100 100 100 100

Fig. 4. Effect of sputtering temperature on rewetting velocity.

M.K. Agrawal, S.K. Sahu / Applied Thermal Engineering 73 (2014) 265e274

Fig. 5. Effect of boiling Biot number on rewetting velocity.

temperature (q1) and BiC/BiB. For a fixed value of (q2), Peclet number decreases with the increase in the dry wall temperature. The nondimensional wet front velocity is found to increase with the boiling Biot number (BiB). For a fixed value of BiB; the velocity of wet front does not change with (BiC/BiB). This implies the significance of boiling Biot number compared to convective Biot number for evaluating the wet front velocity. This indicates that an accurate value of boiling Biot number (BiB) is needed to analyze the phenomena of rewetting. The variation of non-dimensional wet front velocity with sputtering temperature (q2) for various q1 and BiC/BiB is depicted in Fig. 4. It is observed that at lower values of q2 (q2 < 0.2), the influence of boiling Biot number vanishes and the wet front velocity approaches the limiting condition of BiC/BiB¼1.0. The magnitude of wet front velocity found to decrease significantly for the higher values of q2. This indicates that as the difference between boiling temperature and rewetting temperature decreases, the sputtering length decreases. In such a case, the wet front velocity solely depends on BiC. It is observed that the wet front velocity decreases with the increase in dry wall temperature.

271

Fig. 7. Comparison of predicted dimensionless wet front velocity with other analytical result.

The variation of wet front velocity with boiling Biot number (BiB) for various q1 and q2 is shown in Fig. 5. It is observed that wet front velocity increases with the increase in BiB. However, the wet front velocity is found to decrease with the increase in q1 and q2 This shows that at higher value of BiB, the heat removal rate from hot surface is higher leading to the increase in wet front velocity. Various researchers obtained similar observations in their analysis [1,4,12]. It is observed that the deviation of wet front velocity among various values of q2 becomes steeper at higher Biot number. The variation of sputtering length with dry wall temperature for various values of q2 and (BiC/BiB) is shown in Fig. 6. Previously, researchers have observed a short but violent sputtering zone during experiments [6,7]. It is argued that sputtering length depends on the initial dry wall temperature. In the present analysis, sputtering length is found to depend on various parameters namely, dry wall temperature, incipient boiling temperature, convective Biot number and boiling Biot number. For a given value of q2 the sputtering length decreases with the increase in q1 and BiC/BiB. For the higher value of q2 the difference of incipient boiling temperature and rewetting temperature becomes lower resulting in a smaller sputtering length. In addition, higher values of q1 represents a higher dry wall temperature and results in stronger axial temperature gradient at the wet front. Subsequently, a smaller sputtering zone is observed at higher values of q1. 3.2. Comparison with analytical results and test data

Fig. 6. Variation of sputtering length ahead of wet front.

In the past, various researchers have analyzed the rewetting behavior of hot surface considering three region rewetting model with no precursory cooling by employing suitable analytical techniques such as: Heat balance Integral Method [1] and Winer-Hopf technique [35]. Present prediction (Model-1) for the case of no precursory cooling (N / ∞) and q2 ¼ 0 is used compare with various available analytical results and are presented in Fig. 7. The results obtained by various three-region models of Sahu et al. [1], Bonakdar and McAssey [2], Casamirra et al. [3] and Sun et al. [4] are considered for comparison. Present solution obtained by employing Variational Integral Method (VIM) exhibits an excellent agreement with the results of Bonakdar and McAssey [2] and Sun et al. [4] at higher Biot number. However, at lower Biot number it over predicts the result of Bonakdar and McAssey [2] but shows good agreement with the results of Sun et al. [4]. For all the cases, the wet front

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Fig. 8. a) Comparison of predicted wet front velocity with experimental results of Duffey and Porthouse [6] with wall thickness 0.05 cmm. (b) Comparison of predicted wet front velocity with experimental results of Duffey and Porthouse [6] with wall thickness 0.085 cm. (c) Comparison of predicted wet front velocity with experimental results of Yamanouchi [5] with wall thickness 0.1 cm. (d) Comparison of predicted wet front velocity with experimental results of Sahu et al. [23].

velocity obtained from present analysis over predicts the result of Casamirra et al. [3]. The present work essentially reconfirms earlier work and demonstrates the capability of VIM for analyzing multi region rewetting model. Numerous experimental investigations have been carried out to analyze the rewetting behavior of hot surfaces in different geometries such as slab, rod, and tube [5,6,23]. Tests have been carried out either for a single rod [23] or with rod bundle [5,6] with varied range of coolant flow rate and dry wall temperature. It is observed that most of the theoretical models considered an approximate value of heat transfer coefficient (Biot number) to compare predicted models with the test data [12,43]. Most of the experimental studies use water as a coolant under atmospheric condition. In such a case, q2 may be assumed to be smaller than 0.2 as reported by Sun et al.[4]. For lower values of q2 (q2 < 0.2), the heat transfer in the continuous liquid region exerts little influence on the wet front velocity and the rewetting model can be considered as a two-region

basic rewetting model. It is observed that by reducing the present model into a two-region model (neglecting sputtering zone) both the models (Model-1 and Model-2) predict same results. The results found and correlate the test data for higher flow rates. This may be due to the higher boiling Biot number used in the entire wet region. Present prediction obtained by employing Model-2 (neglecting sputtering region) is compared with the experimental data taken from various sources with different wall thickness, test geometry and coolant flow rates and are shown in Fig. 8(aed). The experimental data are taken from a water-stainless steel pair, considering water as the coolant. It is reported that the value of N and d are obtained with the knowledge of variation of heat flux in the dry region ahead of the wet front for a nitrogenecopper pair [7]. The value of N is considered as unity and the zone of precursory cooling was considered to be 0.1 for comparing the theoretical model with the test data [37]. However, the authors [37] have varied the Biot

Table 2 Summary of correlations between flow rate per unit perimeter (j) and magnitude of precursory cooling (N). Source

N as a function of flow rate per unit perimeter (j)

Sun et al. [29] (1-D) Dua and Tien [30] (2-D slab) Sahu et al. [44] (2-D slab, cylinder) Present prediction (1-D)

N N N N

¼ 800=J1:4 ¼ ð160=JÞ þ 1 ¼ 24:01=J0:701 ¼ 18:60=J0:4947

Solution method

Model description

Analytical Winer-Hopf technique HBIM VIM

Exponential variation of heat transfer coefficient in dry region Exponential variation of heat flux parameter in dry region Exponential variation of heat flux parameter in dry region Axially varying heat transfer coefficient in sputtering region and exponentially decaying heat transfer coefficient in dry region

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number for correlating the test data at various coolant flow rates. The detailed information on the variation of heat flux in the dry region ahead of wet front for water-stainless steel pair is not reported in the literature. In the present prediction, the value of d is taken as 0.05 cm1 for comparing the experimental results. Earlier researchers have followed same procedure to compare the theoretical model with test data [29,30]. The experimental data considered in the present analysis covers a wide variation of coolant flow rate and wall thickness. The present prediction (Model-2) provides an excellent agreement with the available test data of Duffey and Porthouse [6] and Yamanouchi [5] for varied range of test geometry, coolant flow rate and dry wall temperature and is shown in Fig. 8(aec). Recently, Sahu et al. [23] reported an experimental investigation by injecting water from the top of a vertical heater tube by employing various coolant injection systems. The authors carried out tests with varied range of experimental conditions including initial surface temperature and coolant flow rate. An attempt has been made to correlate present model with the test data and is shown in Fig. 8d. The present prediction (Model-2) is found to provide an excellent agreement with the test data [23]. Based on the analysis, a correlation between magnitude of precursory cooling (N) and flow rate per unit perimeter (j) has been proposed. In past, various researchers have proposed suitable correlation between coolant flow rate and N while comparing with the test data of Duffey and Porthouse [6] and Yamanouchi [5]. A summary of these correlations are presented in Table 2. 4. Conclusion The variational integral method has been applied to analyze rewetting of hot surface by considering variation of heat transfer along the hot object. Two different models are considered to account the variation in heat transfer in the sputtering region. First model considers a constant heat transfer coefficient in sputtering region. While, the second model considers an axially varying heat transfer coefficient in the sputtering region. Both the models consider a constant heat transfer coefficient in wet region and an exponentially varying heat transfer coefficient in dry region. Closed form expressions for temperature distribution for various regions, namely, wet, sputtering and dry region have been obtained from the analysis. In both the cases, mathematical expressions have been obtained to evaluate the Peclet number and sputtering length. The sputtering length is found to be dependent onq1 and q2. Compared to convective Biot number, boiling Biot number affects significantly the wet front velocity and therefore more precise values of boiling Biot number is needed to accurately predict the wet front velocity. Additionally, it is observed that on neglecting the sputtering length, the wet front velocity reduces. At higher coolant flow rate, the wet front velocity is higher and it is necessary to include the precursory cooling in the model. Neglecting the precursory cooling in the model may under predict the rewetting velocity. The results obtained by the present prediction exhibits good agreement with available theoretical models [1e4]. The present VIM solution exhibits an excellent agreement with the test data of Duffey and Porthouse [6] and Yamanouchi [5] and Sahu et al. [23]. A correlation between coolant flow rate per unit perimeter (j) and magnitude of precursory cooling (N) of the form N ¼ 18.60/j0.4947 is proposed. Nomenclature a a1…a5 a12 ; a22 b, c, d

extent of precursory cooling constant defined in Eq. (8) constant defined in Eqs. (11) and (18) constant defined in Eq. (16)

BiC BiB C h hC hB I J K l L M1…M6 N Pe qc, qb, qv q1…q5 T Tb T0 Ts Tw t u VIM x x Z

273

Biot number with respect to convective region, hcd/K Biot number with respect to sputtering region, hcd/K specific heat, J/kg  C heat transfer coefficient, W/m2  C convective heat transfer coefficient, W/m2  C boiling heat transfer coefficient, W/m2  C variational integral defined in Eq. (7a) constant defined in Eq. (6) thermal conductivity, W/m C sputtering length, m non-dimensional distance defined in Eq. (3) constants defined in Eq. (8) magnitude of precursory cooling dimensionless wet front velocity, rCud/K heat flux from the hot wall in the wet, sputtering and dry region, respectively, W/m2 heat flux in axial direction (Fig. 2), W/m2 temperature, C incipient boiling temperautre,  C wet front temperature that corresponds to the temperature at the minimum film boiling heat flux,  C saturation temperature, C initial temperature of the dry surface, C time, sec wet front velocity, m/s variational integral method length coordinate, m dimensionless length coordinate constant defined in Eq. (19)

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