International Journal of Heat and Mass Transfer 68 (2014) 332–342

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Heat transfer mechanisms in pool boiling

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.037

⇑ Mobile: +1 201 566 1205; fax: +1 212 650 8013.E-mail address: hroh@ccny.cuny.edu

Heui-Seol Roh ⇑Department of Mechanical Engineering, City College of New York, New York, NY 10031, USA

a r t i c l e i n f o

Article history:Received 1 May 2013Received in revised form 16 September2013Accepted 16 September 2013Available online 13 October 2013

Keywords:Pool boilingHeat transferNatural convectionNucleate boilingTransition boilingFilm boiling

a b s t r a c t

Pool boiling is a heat transfer mechanism carrying a phase transition from liquid to vapor. However, theexact characteristics of pool boiling are obscure because of the lack of the theoretical approach method.We have proposed a statistical thermodynamic heat transfer theory which is applicable to the heat trans-fer mechanisms of both conduction and internal convection. We here apply the heat transfer theory topool boiling as an explicit illustration to understand the kinetics of phase transition mechanisms atthe interface of two different phase materials. Three variable separation constants stand for particle num-ber constants and play the key roles in exploiting the distinct boiling mechanisms. The theory accountsfor the four boiling mechanisms of natural convection, nucleate boiling, transition boiling, and film boil-ing in pool boiling. It is able to match heat fluxes in wide ranges of temperature, time, and space. Partic-ularly, the heat flux curve is sketched as a function of excess temperature between wall temperature andsaturation temperature in the four regimes. Three limiting heat fluxes and four activation temperaturesare employed as input parameters. Theoretical heat flux–temperature curves agree with experimentalheat flux profiles in a significantly broader range of temperature than those from any existing theories.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

A phase transition is a vital subject in science and engineeringsince it is central to matter in the universe. Heat transfer is a driv-ing force for a phase transition which is the transformation of athermodynamic system from one phase of matter to another.

Pool boiling heat transfer has recently had various applicationsin reactors, rockets, distillation, air separation, refrigeration, andpower cycles. Boiling is a phase transition process of heat transferwhose area is undergoing accelerating expansion. However, thecomplicated characteristics of pool boiling remain incompletelyunderstood. In this context, a systematically unified theory for heattransfer in non-equilibrium, quasi-equilibrium, and equilibriumhas been proposed [1].

We here apply the heat transfer theory [1] to pool boiling toprove the validity of the theory and to understand the detailedcharacteristics of pool boiling. The theory has a close analogy tothe statistical thermodynamic theory for electrochemical reactions[2,3]. It can exactly account for heat flux profiles for the four mech-anisms of natural convection, nucleate boiling, transition boiling,and film boiling. In other words, the heat transfer theory producesheat fluxes in pool boiling as a function of temperature, time, andspace in the whole range of boiling regimes.

Since the pioneering investigation by Nukiyama’s experiment[4], heat transfer in pool boiling has attracted great attention, froman empirical approach to theoretical understanding of boiling phe-nomena. The four mechanisms in pool boiling have been observed.Several physical mechanisms have been proposed for heat transferoccurring during the saturated boiling process [5]: microlayerevaporation [6], reflooding transient conduction [7,8], natural con-vection [9,10], microconvection [11–13], and combined mecha-nism models [14,15]. Nevertheless, the heat flux as a function oftemperature has not been theoretically evaluated with completesatisfaction. The heat flux in a wide range of temperature has beenonly determined experimentally since theoretical correlations arevalid only in the narrow regimes they were developed for. Existingtheories such as the Rohsenow equation in nucleate boiling [11] re-flect the heat fluxes in a narrow temperature regime of boiling.They do not predict both the limiting heat flux and the activationtemperature of different boiling regimes. They cannot demonstratethe S-shape behavior of heat flux versus excess wall temperature.

There is also a heat flux or temperature excursion at the initia-tion of nucleate boiling. The mismatch at the end of a natural con-vection regime and the inception of the nucleate boiling regimehas been reported in Refs. [15–21] and an interpolation formulafor the boiling curve has also been suggested by Bergles and Rohse-now [22]. Therefore, the reason for the mismatch deserves to be animportant issue in understanding boiling, and a rigorous approachis required to clarify this phenomenon.

Nomenclature

h heat transfer coefficientk spatial particle numbern0 particle number per unit volumeq non-equilibrium heat flux vectorq heat fluxq0 exchange heat fluxq1 limiting heat fluxqe external heat fluxqC heat flux in natural convectionqN heat flux in nucleate boilingqT heat flux in transition boilingqF heat flux in film boilingT0 global equilibrium temperatureT0 local equilibrium temperatureTB excess temperaturetB excess timerB excess displacementn unit vectorTw wall temperatureTb activation temperaturetb activation timerb activation distanceTnc activation temperature in natural convectionTnb activation temperature in nucleate boilingTtb activation temperature in transition boilingTfb activation temperature in film boilingq1C limiting heat flux in natural convectionq1N limiting heat flux in nucleate boilingq1T limiting heat flux in transition boilingq1F limiting heat flux in film boilingTC excess temperature from activation in natural convec-

tionTN excess temperature from activation in nucleate boilingTT excess temperature from activation in transition boilingTF excess temperature from activation in film boilingtC excess time from activation in natural convectiontN excess time from activation in nucleate boiling

tT excess time from activation in transition boilingtF excess time from activation in film boilingzC excess distance from activation in natural convectionzN excess distance from activation in nucleate boilingzT excess distance from activation in transition boilingzF excess distance from activation in film boiling

Greek symbolsa thermal particle numberb temporal particle numberk total mean free pathk0 mean free paths total reaction times0 mean reaction timerc reaction cross sectionac thermal particle number in natural convectionan thermal particle number in nucleate boilingat thermal particle number in transition boilingaf thermal particle number in film boilingbc temporal particle number in natural convectionbn temporal particle number in nucleate boilingbt temporal particle number in transition boilingbf temporal particle number in film boiling

Subscripts and superscriptsC natural convectionN nucleate boilingT transition boilingF film boilingc natural convectionn nucleate boilingt transition boilingf film boiling1 limiting valueb activation value

H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342 333

2. Heat transfer theory for thermal non-equilibrium

A total system containing reactants and products is thermallyisolated from its surroundings and is in thermal non-equilibriumwhile the two phases of reactants and products are in mechanicaland chemical equilibrium. The system in thermal non-equilibriumis considered under the assumption that the physics of energytransfer can be described in terms of physical parameters in localthermodynamic equilibrium [23,24].

We propose a postulate that the heat flux can be described as afunction of temperature, time, and space:

q ¼ qðT; t; rÞ

or, equivalently,

q ¼ qðT; @T=@t;rTÞ;

where T, @T/@t, and rT are the generalized coordinates and veloci-ties. It is anticipated that the three variables are independent andorthogonal. Furthermore, the expression of the heat flux can be writ-ten as a product of temperature, time, and space dependent terms:

q ¼ qðTÞBðtÞAðrÞ:

In this case, we can establish the Euler–Lagrange equation underthe assumption of L / q:

r½@L=@ðrTÞ� þ @=@t½@L=@ð@T=@tÞ� � @L=@/ ¼ 0:

Here, we may parameterize the variables during non-equilibriumprocesses like

rT ¼ T0=s;

@T=@t ¼ T0=k:

T0 is the equilibrium temperature. s is the total reaction time whichis defined by the mean reaction time s0 multiplied by the particlenumber of the system, and k is the total mean free path which is de-fined by the mean free path k0 multiplied by the particle number ofthe system.

From the Euler–Lagrange equation leads to the partial differen-tial equation for heat transfer [1]:

kr � qþ s@q=@t � T0 @q=@T ¼ 0; ð1Þ

where q = qTn = qn is the non-equilibrium heat flux vector with theunit vector n. The first and the second terms on the left hand side ofEq. (1) originate from the kinetic energy densities and the thirdterm comes from the thermal potential density. In the absence ofthe third term, Eq. (1) leads to the conservation of heat energy.

Using the method of separation of variables for the heat flux

qðT; t; rÞ ¼ qðTÞBðtÞAðrÞ; ð2Þ

334 H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342

we obtain three differential equations from Eq. (1):

dq=dT þ ða=T0ÞqðTÞ ¼ 0; ð3Þ

@B=@t þ ðb=sÞBðtÞ ¼ 0; ð4Þ

r � Aþ ðk=kÞAðrÞ ¼ 0: ð5Þ

The three constants a, b, and k in the separation of variables areintroduced, and their relation becomes

aþ k� b ¼ 0: ð6Þ

The separation constants represent

a: the thermal particle number,b: the temporal particle number,k: the spatial particle number.

The relation among the separation constants (6) is one for non-relativistic dynamics, and the following is one for relativisticdynamics:

a02 þ k02 � b02 ¼ 0;

where a = a02, k = k

02, and b = b02. the thermal particle number con-

stant in Eq. (3) is connected to the entropy change by a = lnX = S/kB where X is the number of microstates in the microcanonicalensembles. kr = b/s = 1/s0 in Eq. (4) stands for the rate constant inheat transfer and is the inverse of the mean reaction time s0.k = b � a = k/k0 in Eq. (5) is connected to the reaction cross sectionrc = V/kk0, and n0 = k/V is the particle number per unit volume.

When we solve Eqs. (3)–(5), it is required to set the boundaryand initial conditions. To apply the differential equations to poolboiling with the given conditions, it is convenient to convert theproperty variables to their excess forms. The temperature, time,and space variables represent the excess quantities, respectively:

TB ¼ jT � T0j;

tB ¼ t � t0;

rB ¼ jr � r0jn;

where n is the unit vector in the diffusion direction of the heat flux.As for solutions of the internal convection and external conduc-

tion heat flux (qT and qd) in the non-equilibrium process of nonzeroT, t, and r, we derive the three integrated solutions from (3)–(5),respectively:

qðTBÞ ¼ q0 expð�aTB=T0Þ; ð7Þ

BðtBÞ ¼ B0 expð�btB=sÞ; ð8Þ

AðrBÞ ¼ A0 expðk � rB=kÞ; ð9Þ

where kb = kbn = (b � a)n. The heat flux thus has the final form:

qðTB; tB; rBÞ ¼ q00 expð�aTB=TÞ expð�btB=sÞ expðk � rB=kÞ; ð10Þ

where q00 = q0B0A0.

3. Heat transfer mechanisms of pool boiling

The heat transfer theory described above is here applied toobservable heat fluxes in pool boiling which has four distinctmechanisms of natural convection, nucleate boiling, transitionboiling, and film boiling. These heat fluxes with thermal, temporal,and spatial dependence in (7)–(9) are not directly observable inpool boiling experiments. We need to modify the expressions of

the heat fluxes so that they are consistent with pool boiling mea-surement conditions with external heat fluxes.

In the presence of an external heat flux, the heat flux, Eq. (3), atan interface of two different phases becomes

qe ¼ qþ ðT0=aÞdq=dT; ð11Þ

where qe is the externally applied heat flux and q is the heat flux tobe evaluated. Eq. (11) is generally applicable for evaluating the heatflux for boiling, condensation, solidification, and melting at theinterface of two phases.

The heat transfer process at the interface of solid and liquid, asillustrated in Fig. 1(a), can be examined in terms of Eq. (11). Theboiling and solidification processes are the typical examples. Inthe cylindrical coordinates of r, /, and z, as shown in Fig. 1(b),the heat flux q and the temperature T depend on r and z since qand T are symmetric in the azimuthal angle /. The origin is locatedat the interface of solid and liquid.

The heat potentials at the solid–liquid interface are depicted inFig. 1(c). The heat potential is defined by Q = U + PV � lN, and Q⁄ isthe activation energy. Note that the heat potential is defined byQ = U + PV � lN = TS, and the heat transfer DQB can be defined byDQB = DQ � DQ0 = �akBTB in the heat canonical ensemble.TB = jT � T0j is the excess temperature between the wall tempera-ture and the relative equilibrium temperature T0. The boiling pro-cess depends on the applied heat flux or temperature whichchanges the heat potentials, as drawn in Fig. 1(d). There is discrep-ancy between heat fluxes in the absence (the black line) and in thepresence (the red line) of an external heat flux.

Fig. 2 illustrates the heat transfer profiles as a function of excesstemperature during equilibrium and non-equilibrium processes inthe presence of external temperature. The reversible and irrevers-ible heat fluxes are respectively expressed as

q0ðT0Þ ¼ q0 expðaT0=T0Þ;

qðTBÞ ¼ q1½1� expð�aTB=T0Þ�:

An applied external temperature is at the beginning utilized tomake an equilibrium process proceed until the temperature attainsthe activation temperature. Before the external temperature is ap-plied, the system is in the equilibrium process with the exchangeheat flux q0. At the activation temperature Tb, the equilibrium pro-cess reaches the limiting heat flux q1 as illustrated in Fig. 2. The heatflux gap between the equilibrium and non-equilibrium processesdepends on the entropy constant a. Above the activation tempera-ture, a non-equilibrium process proceeds from the exchange heatflux q0 toward the limiting heat flux q1. The non-equilibrium pro-cess is an irreversible process, and the limiting heat flux in equilib-rium plays the role of an internal heat flux so that the non-equilibrium process eventually reaches the limiting heat flux asthe excess temperature increases. The limiting heat flux q1 is deter-mined by the equilibrium process caused by the heat potential dif-ference DQ0 ¼ ðQp

0 � Qr0Þ between thermal reactants and products.

The heat flux gap or the heat potential gap is illustrated in specificheat profiles of phase transition phenomena which include the kpoint [25,26]. Note that the limiting heat flux q1 becomes equiva-lent to the external heat flux qe in Eq. (11).

The activation temperature Tb, activation time tb, and activationdistance rb are respectively defined by the maximum values inmagnitude:

Tb ¼ jTp0 � Tr

0jmax ¼ jT � T0jmax;

tb ¼ jtp0 � tr

0jmax ¼ jt � t0jmax;

rb ¼ jrp0 � rr

0jmax ¼ jr � r0jmax:

Fig. 1. Boundary and coordinates at the solid–liquid interface and heat potentials at the vapor–liquid interface. (a) Boundary at the interface of solid and liquid. (b) Spatialcylindrical coordinates and the spatial particle number constant k. (c) Heat potentials in the absence of an external heat flux. (d) Heat potentials in the presence of an externalheat flux (the red line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Heat transfer profiles as a function of excess temperature during theequilibrium process of T0 < T < (T0 + Tb) and the non-equilibrium process ofT > (T0 + Tb) in the presence of external temperature. The activation temperatureis measured from its global equilibrium T0.

Fig. 3. Heat fluxes in the four layers formed by the four polarization mechanisms asfunction of temperature, time, and space.

H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342 335

The particle number constants lead to

a ¼ ðT0=TbÞ lnðq1=q0Þ;

b ¼ ðs=tbÞ lnðB1=B0Þ;

k ¼ ðk=rbÞ lnðA1=A0Þ;

where q0 is the exchange heat flux and q1 is the limiting heat flux.The relation among the particle number constants is given by

aþ k� b ¼ 0. When the restriction is applied to pool boiling, thefollowing three non-equilibrium regimes may be classified:

a ¼ b > 0 and k ¼ 0 for natural convection;a ¼ b� k > 0 for nucleate boiling;a ¼ b� k < 0 for transition boiling:

ð12Þ

Moreover, the non-equilibrium process proceeds to the equilibriumprocess in transition boiling:

a ¼ b� k > 0 for transition boiling;a ¼ b� k < 0 for film boiling:

ð13Þ

During these processes, the stable regimes satisfy

a > 0 for thermal stability;

b > 0 for temporal stability;

k > 0 for spatial stability:

According to Eqs. (12) and (13), we expect the four layers in poolboiling mechanisms. The four layers are schematically demon-strated in Fig. 3. Each layer has a boiling process depending ontemperature, time, and space. Its layer thickness also depends onthem.

The microlayer [6] corresponds to the natural convection layerin pool boiling. There are two opposite heat fluxes in nucleate boil-ing and transient boiling. The heat flux by liquid is larger than theheat flux by vapor in nucleate boiling while the heat flux by vaporis larger than the liquid heat flux in transient boiling. In naturalconvection, the heat flux by liquid is leading while in film boiling,the heat flux by vapor is leading. Simple and reasonable solutions

can be tried by considering an effective heat flux in nucleation boil-ing. As illustrated in Fig. 3, only one effective heat flow in nucleateboiling will be taken into account instead of two opposite heatfluxes. In transient boiling, the unstable effective flow has areversed direction and the flow is mixed with the initiation of

336 H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342

another flow with the same flow direction and an opposite sign inits amplitude.

There are a maximum limiting heat flux in natural convection, amaximum heat flux in nucleate boiling, and two maximum andminimum limiting heat fluxes in transient boiling, and a minimumlimiting heat flux in film boiling.

We here concentrate on the application of Eq. (11) to pool boil-ing. In pool boiling, four control mechanisms are involved: naturalconvection, nucleate boiling, transition boiling, and film boiling. Inthis case, the differential equation reduces to

qe ¼ qþ T0½ð1=acÞdqC=dT þ ð1=anÞdqN

=dT � ð1=atÞdqT=dT

� ð1=af ÞdqF=dT�: ð14Þ

The second term on the right hand side represents the heat flux pro-duced through the area which actively participates in the four boil-ing mechanisms.

The second term on the right hand side of (14) contains the fourmechanisms of boiling which are based on Newton’s law in whichthe heat flux is proportional to temperature difference. The fourmechanisms of natural convection, nucleate boiling, transient boil-ing, and film boiling are additive in the heat flux. Thus, the totalheat flux is the sum of the parallel heat fluxes:

q ¼ qC þ qN þ qT þ qF : ð15Þ

Under the assumption that each boiling mechanism is separablefrom each other, Eq. (14) can be solved in the following, eventhough the four mechanisms are more or less coupled. Theassumption is justified since each boiling mechanism has a domi-nant contribution to the heat flux in a different temperatureregime.

4. Four control mechanisms of pool boiling

4.1. Natural convection: the process of (a = ac = bc > 0 and kc = 0)

In a process dominated by natural convection, the differentialequation for the heat flux is

qeC ¼ qC þ ðT0=acÞdqC=dT; ð16Þ

where qe is the applied heat flux to the solid wall. From Eq. (16), theheat flux in natural convection, qC, is given as a function of the ex-cess temperature TC:

qC ¼ q1C ½1� expð�acTC=T0Þ�; ð17Þ

where q1C = qeC. The limiting heat flux at the activation temperatureof natural convection Tnc is determined at the end of the equilibriumprocess:

q1C ¼ q0 expðacTnc=T0Þ:

Here, the excess temperature TC measured from the absolute activa-tion temperature (T0 + Tnc) in natural convection is defined by

TC ¼ Tw � ðT0 þ TncÞ;

where T0 is the saturation temperature, Tw is the wall temperatureand Tnc is the activation temperature measured from the saturationtemperature T0 in natural convection.

The heat flux qC in natural convection [4] can be given by

qC ¼ h0CTC ¼ 0:14ðjl=LÞðGr PrÞ1:3TC ð18Þ

and from Eq. (17) the heat flux is approximately expressed in theform

qC ¼ q1CacTC=T0:

The limiting heat flux can be thus approximated as

q1C ¼ h0CT0=ac ¼ 0:14ðjl=LÞðGr PrÞ1:3ðT0=acÞ;

where h0c is the heat transfer coefficient, jl is the thermal conduc-tivity of liquid, L is the solid length, Gr is the liquid Grashof number,and Pr is the liquid Prandtl number. Rewriting Eq. (18), we find theheat flux in natural convection

qC ¼ 0:14ðjl=LÞðGr PrÞ1:3ðT0=acÞ½1� expð�acTC=T0Þ�: ð19Þ

4.2. Nucleate boiling: the process of (a = an = bn � kn > 0)

In the dominant process of nucleate boiling, the heat flux equa-tion is similarly given by

qeN ¼ qN þ ðT0=anÞdqN=dT�; ð20Þ

where qeN is the applied heat flux to the solid wall. From Eq. (20),the heat flux in nucleate boiling, qN, becomes

qN ¼ q1N½1� expð�anTN=T0Þ�; ð21Þ

where q1N = qeN. Similarly, the limiting heat flux at the activationtemperature of nucleate boing Tnb is expressed as

q1N ¼ q0 expðanTnb=T0Þ:

The excess temperature TN measured from the activation tempera-ture Tnb is defined by

TN ¼ Tw � ðT0 þ TnbÞ: ð22Þ

Using Eq. (22), the wall temperature has the form

Tw ¼ T0 þ Tnb � ðT0=anÞ ln ZNq ;

where ZNqe ¼ 1� qN=q1N .

Since the maximum heat flux by Kutateladze [27] and Zuber [9]leads to

qmax ¼ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4; ð23Þ

the limiting heat flux in nucleate boiling can be approximately ex-pressed as

q1N ¼ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4;

where r is the surface tension of liquid in contact with its vapor andg is the gravitational acceleration. Rewriting Eq. (21) in terms of Eq.(23), we get the heat flux in nucleate boiling

qN ¼ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4½1� expð�anTN=T0Þ�: ð24Þ

4.3. Transition boiling: the coupled process of(a ¼ �aa

t ¼ �ðbat � ka

t Þ < 0 for non-equilibrium anda ¼ ab

t ¼ ðbbt � kb

t Þ > 0 for equilibrium)

The limiting heat flux in nucleate boiling experiences the resis-tive transition process in transition boing. Two opposite heat fluxesexist in this regime. The dominant processes in this regime respec-tively hold the negative particle number constant ða ¼ �aa

t < 0Þ fornon-equilibrium and the positive particle number constantða ¼ ab

t > 0Þ for equilibrium.The heat flux is presented as the differential equation

0 ¼ qT þ ðT0=anÞdqT=dTT � ðT0=atÞdqT

=dT: ð25Þ

We may choose the following form of the heat flux in this regime:

qT ¼ q1N expðaat TT=T0ÞHðTtr � TÞ þ q1F expð�ab

t TF=T0ÞHðT � TtrÞ;ð26Þ

where H is the Heaviside step function. The transition temperatureis given by

H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342 337

Ttr ¼ ½1=ðabt � aa

t Þ�½ðabt Tfb � ab

t TtbÞ þ lnðq1N=q1FÞ�: ð27Þ

Eq. (26) can be rewritten as

qT ¼ q1N expðaat TT=T0Þ if Ttb < T < Ttr;

and

q1F expð�abt TF=T0Þ if Ttr < T < Tfb:

The maximum heat flux is given by Eq. (23) and the minimum heatflux [9] known as the Leidenfrost point is expressed as

q1F ¼ qmin ¼ 0:09hlvqvðrgðql � qvÞ=ðql þ qvÞ2Þ

1=4: ð28Þ

The activation temperature for transition boiling takes the form

TT ¼ Tw � ðT0 þ TtbÞ; ð29Þ

where Ttb is the activation temperature in transition boiling.

4.4. Film boiling: the process of (a = �af = �(bf � kf) < 0)

In the governing process of film boiling, the equation for theheat flux leads to

0 ¼ qF � ðT0=af ÞdqF=dT: ð30Þ

The heat flux in film boiling, qF, is determined as

qF ¼ q1F ½expðaf TF=T0Þ�: ð31Þ

Around the temperature TF, the heat flux is rewritten as

qF ¼ 0:09hlvqvðrgðql � qvÞ=ðql þ qvÞ2Þ

1=4expðaf T

F=T0Þh i

; ð32Þ

where the minimum heat flux [9] known as the Leidenfrost point isgiven in Eq. (28). The temperature for film boiling is given by

TF ¼ Tw � ðT0 þ TfbÞ; ð33Þ

where Tfb is the activation temperature in film boiling.

5. Heat flux as a function of temperature, time, and space

The individual four boiling mechanisms and their coupledmechanisms are described from the view points of heat flux versusexcess temperature between wall temperature and saturation tem-perature. The total heat flux can approximately be the sum of heatflux in the four boiling regimes:

q ¼ 0:14ðjl=LÞðGr PrÞ1:3ðT0=acÞ 1� expð�acTC=T0Þh i

þ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4 1� expð�anTN=T0Þh i

þ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4 expðaat TT=T0ÞHðTtr � TÞ

h

þ0:09hlvqvðrgðql � qvÞ=ðql þ qvÞ2Þ

1=4

� expð�abt TF=T0ÞHðT � TtrÞ

i

þ 0:09hlvqvðrgðql � qvÞ=ðql þ qvÞ2Þ

1=4½expðaf TF=T0Þ�; ð34Þ

where the first term on the right hand side represents the heat fluxdue to natural convection, the second the heat flux due to nucleateboiling, the third the heat flux in transition boiling, and the fourththe heat flux in film boiling without radiation.

The excess temperatures from the absolute activation tempera-tures in the four mechanisms are indicated as

TC ¼ Tw � ðT0 þ TncÞ;TN ¼ Tw � ðT0 þ TnbÞ;TT ¼ Tw � ðT0 þ TtbÞ;TF ¼ Tw � ðT0 þ TfbÞ;

ð35Þ

where T0 is the saturation temperature and Tw is the wall tempera-ture. The activation temperatures of Tnc, Tnb, Ttb, and Tfb relative tothe global equilibrium temperature T0 are positive quantities inpool boiling.

In terms of (34), the excess temperature T in each boiling mech-anism can be expressed as

T ¼ T0 þ Tnc � ðT0=acÞ ln ZCqe;

T ¼ T0 þ Tnb � ðT0=anÞ ln ZNqe;

T ¼ T0 þ Ttb þ ðT0=atÞ ln ZTqe;

T ¼ T0 þ Tfb þ ðT0=af Þ ln ZFqe;

ð36Þ

where Zqe = 1 � qT/q1T in natural convection or nucleate tempera-ture boiling, Zqe = qT/q1T in transition or film temperature boiling,and the limiting heat flux q1T depends on temperature.

In the presence of the external heat flux qe, the temporal heatflux Eq. (4) in boiling becomes

qe ¼ qþ ðs=bÞdq=dt; ð37Þ

where s is the total reaction time. Solving (37), we obtain the heatflux:

qðtBÞ ¼ q1t 1� expð�btB=sÞ� �

: ð38Þ

Combining (34) and (38), we derive the time dependent heat flux innatural convection and nucleate boiling:

qðT; tÞ ¼ 0:14ðjl=LÞðGr PrÞ1:3ðT0=acÞ½1� expð�acTC=T0Þ�� 1� expð�bctC=sÞ� �

þ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4

� 1� expð�anTN=T0Þh i

1� expð�bntN=sÞ� �

; ð39Þ

where the temporal particle number constant bc in natural convec-tion is less than the temporal particle number constant bn innucleate boiling, and the diffusion process in natural convection isslower than that in nucleate boiling.

The excess times in the four mechanisms indicate

tC ¼ t � ðt0 þ tncÞ;tN ¼ t � ðt0 þ tnbÞ;tT ¼ t � ðt0 þ ttbÞ;tF ¼ t � ðt0 þ tfbÞ;

ð40Þ

where t0 is the global equilibrium time. tnc, tnb, ttb, and tfb are theactivation times for the four boiling mechanisms. The time in eachboiling mechanism is stated as follows:

t ¼ t0 þ tnc � ðs=bcÞ ln MCe ;

t ¼ t0 þ tnb � ðs=bnÞ ln MNe ;

t ¼ t0 þ ttb þ ðs=btÞ ln MTe ;

t ¼ t0 þ tfb þ ðs=bf Þ ln MFe ;

ð41Þ

where Me = 1 � qt/q1t in natural convection or nucleate temporalboiling, Me = qt/q1t in transition or film temporal boiling, and thelimiting heat flux q1t depends on time. Since the temporal particlenumber constant bc in natural convection is less than bn in nucleateboiling, the heat transfer process in natural convection is slowerthan that in nucleate boiling.

To understand spatial behavior for the heat flux in the presenceof an external source qe(r = 0) = q1r, we modify Eq. (5):

qe ¼ r � qþ ðk=kÞqðrÞ: ð42Þ

Eq(42) has the integrated solution

qðrBÞ ¼ q1r ½1� expðk=kÞn � rB�¼ q1r ½1� expððb� aÞ=kÞrB cos hÞ�; ð43Þ

Fig. 4. Schematic diagrams of heat fluxes in the four regimes of pool boiling. (a) As afunction of excess temperature. (b) As a function of time. (c) As a function ofdistance.

338 H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342

where h is the angle between the vectors of n and z: (b � a)/k)n � rB = (k/k)zB cosh and tanh = kr/kz.

We expect three types of solutions depending on the value of(a � b)cosh in the direction perpendicular to the boundary. For(a � b)cosh = 0, this process takes place in stable natural convec-tion. For (a � b)cosh < 0, this process occur in stable nucleate boil-ing with a positive spatial gradient of temperature field in liquid.For (a � b)cosh > 0, this process is known as transient boiling inan unstable process with a negative spatial gradient of tempera-ture field in liquid.

The excess distances along the z axis in the four boiling mech-anisms indicate

zC ¼ jz� ðz0 þ zncÞj;zN ¼ jz� ðz0 þ znbÞj;zT ¼ jz� ðz0 þ ztbÞj;zF ¼ jz� ðz0 þ zfbÞj;

ð44Þ

where z0 is the boundary position in global equilibrium. znc, znb, ztb,and zfb are the activation distances for the four boiling mechanisms.The distance in each boiling mechanism is denoted as

z ¼ z0 þ znc;

z ¼ z0 þ znb þ ðk=knÞ ln LNe ;

z ¼ z0 þ ztb � ðk=ktÞ ln LTe ;

z ¼ z0 þ zfb � ðk=kf Þ ln LFe ;

ð45Þ

where Le = 1 � qz/q1z in nucleate spatial boiling, Le = qz/q1z in transi-tion or film spatial boiling, and the limiting heat flux q1z depends ondistance. The layer in natural convection is uniform spatially, com-pared with the other layers.

The direction of the heat flux is relatively perpendicular to theboundary between the solid wall and liquid when the heat flux in-creases as the temperature increases before the heat flux reachesthe limiting heat flux: cosh � 1. This regime corresponds withthe columnar growth of bubble in boiling. Likewise, this schemeis applied to the columnar heat flux in natural convection regime.On the other hand, the direction of the heat flux becomes parallelto the boundary when the heat flux reaches the limiting heat flux:cosh � 0. In the regime, the heat flux is dominated by the circleheat flux whose surfaces are parallel to the boundary. This regimeis associated with the equiaxial growth of boiling.

According to the temperature, spatial, and temporal behavior ofheat fluxes, it is stated that natural convection is a slow, stable, andirreversible process, nucleate boiling is a fast, stable, and irrevers-ible process, and transient boiling before the transition tempera-ture Ttr is a fast, unstable, and irreversible process. At theboundary, the temperature is the local equilibrium temperature.In the transition regime, the temperature profile is inverted com-pared with them in natural convection and nucleate boiling. Thisindicates a spatial instability which is a different type of instabilityin addition to the Helmholtz instability and the Taylor instability[4,5,15,17].

Fig. 4 demonstrates the schematic diagrams for heat fluxes as afunction of excess temperature, time, and distance in the four re-gimes of pool boiling. In the heat flux curve of Fig. 4(a), the threelimiting heat fluxes of natural convection, nucleate boiling, andtransition boiling, and the four activation temperatures of naturalconvection, nucleate boiling, transition boiling, and film boilingare used as input parameters. Fig. 4(b) and (c) outline predictionsof the heat fluxes as the function of excess time and excess dis-tance. The main point is that the temporal and spatial dependenceof the heat fluxes are governed by their corresponding particlenumber constants. For example, there is no spatial dependence in

the natural convection regime of Fig. 4(c) since the spatial particlenumber is zero in the regime.

6. Comparison results of heat flux

In Fig. 5(a), the comparison curves of heat flux versus excesstemperature in pool boiling are demonstrated to evaluate the heattransfer theory. The experimental data are adapted from Collier[16] and Bui and Dhir [28,29] among numerous supporting exper-imental data [30–33].

The heat transfer theory thus predicts the heat flux as a functionof time in nucleate boiling. To complete the heat flux curve, themean reaction time becomes an input parameter. Therefore, infor-mation of two mean reaction times for natural convection andnucleate boiling are required in the theory. Heat flux comparisonsbetween theory and measurement are shown as a function of time,as illustrated in Fig. 5(b). The measurement data by Yeoh et al. [34]are adapted as an example. Another comparison profile is shown inFig. 5(c) where the experimental data are adapted from Fauser andMitrovic [35].

From Eq. (39), the heat flux in nucleate boiling becomes

Fig. 5. Comparison results between theoretical calculations and experimental data in pool boiling. (a) Heat flux as a function of temperature or excess temperature. Theexperimental data are adapted from Collier [16] and Bui and Dhir [28,29], respectively. (b) Heat flux as a function of time. The experimental data are adapted from Yeoh et al.[34]. (c) Temperature profile as a function of time. The experimental data are adapted from Fauser and Mitrovic [35].

H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342 339

qðTN ; tNÞ ¼ q1N ½1� expð�anTN=T0Þ�½1� expð�bntN=sÞ�: ð46Þ

When a limiting heat flux q1N(TN, tN) at t0 = 0 is applied, the temper-ature becomes a function of time:

T ¼ T0 þ Tnb � ðT0=anÞ ln 1� q= q1N 1� exp �bnt=sð Þ½ �� �� �

: ð47Þ

For a limiting heat flux q1T(T, t) at t = 0, the temperature in transientboiling is expressed as

T ¼ T0 þ Ttb � ðT0=atÞ ln 1� q= q1T 1� expið�btt=sÞ½ �� �� �

: ð48Þ

7. Applications of the heat transfer theory

The heat transfer theory predicts several parameters which canbe verified by experimental results.

7.1. Natural convection

When the Taylor’s expansion is utilized, Eq. (17) becomes

qC ¼ q1C ½acTC=T0 � ðacTC=T0Þ2=2þ ðacTC=T0Þ

3=6

� ðacTC=T0Þ4=24þ ðacTC=T0Þ

5=120 . . .�: ð49Þ

If temperature is near the activation temperature, T � Tnc, the heatflux qC is proportional to TC: qC / TC.

At T = 3Tnc, the contributions of second and third term are thesame. If temperature is larger than the temperature of T = 3Tnc,the heat flux qC may be proportional to (TC)5/4 up to higher naturalconvection temperature regime according the experimental powerlaw behavior in natural convection [17,36]. The contribution of the

remaining terms and the first term makes the power law behaviorqC / (TC)5/4, which has a higher heat flux than qC / TC produced bythe sole contribution of the first term.

In the first order approximation, Eq. (19) yields

qC ¼ ðq1Cac=T0ÞTC ¼ ðq0Cac=T0ÞðT � TncÞ ¼ hðT � TncÞ; ð50Þ

where the heat transfer coefficient h is defined by.

h ¼ ðq1Cac=T0Þ: ð51Þ

7.2. Nucleate boiling

The Rohsenow equation in nucleate boiling [11] is written as

q ¼ llhlvðgðql � qvÞ=r1=2½cpðTw � T0Þ=PrsCsf hlv �3; ð52Þ

where hlv is the latent heat, cp is the constant pressure specific heat,and ll is the viscosity of liquid. Csf is an empirical constant whichaccounts for the particular combination of liquid and solid material,and the exponent s of the Prandtl number Prs differentiates only be-tween water and other liquids. q denotes the density, and the sub-scripts l and v in the densities reflect saturated liquid and saturatedvapor.

Eq. (52) shows the (TN)3 power law behavior in the heat flux innucleate boiling. To illustrate this in the heat transfer theory, theTaylor’s expansion for Eq. (21) is used:

qN ¼ q1N ½anTN=T0 � ðanTN=T0Þ2=2þ ðanTN=T0Þ

3=6

� ðanTN=T0Þ4=24þ ðanTN=T0Þ

5=120 . . .�: ð53Þ

Fig. 7. Schematic diagram to present the activation temperatures, times anddistances during pool boiling processes. All the activation temperatures, times, anddistances are measured from their global equilibriums of T0, t0, and z0, respectively.

340 H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342

At T = 2Tnb, the contributions of first and second terms are the same.If temperature is near the two times of the activation temperatureTnb, T � 2Tnb, the heat flux qN is proportional to (TN)3: qN / (TN)3.The contribution of the third term is dominant and the power lawbehavior of qN / (TN)3 is justified around T � 2Tnb.

Therefore, the Rohsenow equation (52) is valid near T = 2Tnb,according to the heat transfer theory, and the energy carried awayby vaporizing bubbles on the solid surface becomes

q ¼ qvðp=6ÞD3b hlvð1=f Þðn=AÞ; ð54Þ

where Db is the bubble departure diameter and (n/A) is the numberof active sites per unit area. The bubble departure diameter [36] isestimated as

Db � ðr=gðql � qvÞÞ1=2:

The ð1=f Þ is the bubble growth time or the waiting time for the va-por to reemerge from a surface cavity [36],

1=f ¼ Db=vb � Dbðrgðql � qvÞ=q2l Þ�1=4

:

Around the temperature of T = 2Tnb, two types of bubble growth areexpected. The first phase of bubble growth is the inertia growthwhich controls the large pressure difference at the interface of li-quid–vapor. The second phase of bubble growth is the thermalgrowth in which the rate of energy transfer from the liquid to thevapor–liquid interface maintains the pressure.

7.3. Film boiling

Eq. (33) yields, at small range of TF/T0, the similar power lawbehavior with the heat flux used by Bromley [20]:

qF ¼ 0:62 ½k3vqv ðql�qv Þðhlv þ0:4Cpv ðTw�T0ÞÞ=lv DðTw�T0Þ�

1=4ðTw �T0Þ; ð55Þ

where D is the diameter of sphere and hlv is the latent heat betweenliquid and vapor.

In Eq. (33), radiation contribution to the heat flux is not in-cluded. If it is included, the heat flux has the form based on exper-iments [15–17]

qF ¼ 0:09hlvqvðrgðql � qvÞ=ðql þ qvÞ2Þ

1=4½1þ expðaf TF=T0Þ�

þ ð3=4Þqrad: ð56Þ

The heat flux due to radiation is

qrad ¼ rresðTFÞ4; ð57Þ

where rr is the Stefan-Boltzmann constant and es is the emittance.

8. Discussion

From the definition of the internal convection heat transfercoefficient and the expression for the heat flux in the linear

Fig. 6. Heat flux contribution profiles of internal convective and conductive boilin

approximation, the (internal convection) heat transfer coefficientcan be determined:

h ¼ q1ða=T0Þ; ð58Þ

where T0 is the equilibrium temperature of the convection flow inliquid. The heat transfer coefficient in natural convection is thus de-rived from (17):

h ¼ 0:14 ðkf =LÞðGr PrÞ1:3:

The heat transfer coefficient in nucleate boiling is also obtainedfrom (23) and (24):

h ¼ 0:149hlvq1=2v ðrgðql � qvÞÞ

1=4ðan=T0Þ:

Fig. 6(a) presents a schematic heat flux profile in the internal con-vective and external conductive mechanisms of pool boiling as afunction of temperature. Before the applied temperature reachesthe activation temperature, the equilibrium process is dominant.After the initiation of the internal convective boiling mechanisms,the entropy changing process takes place from the nearly constanttemperature to the nearly constant heat flux process with the con-stant higher entropy. The heat flux change is prevailing beforereaching the limiting heat flux while the temperature change isdominant in the regime near the limiting heat flux in natural con-vection and nucleate boiling. Internal heat convection due to tem-perature change is the leading process in natural convection whileinternal heat convection due to entropy change is the prevailingprocess in nucleate boiling. During the film boiling process, radia-tion appears.

Fig. 6(b) illustrates the heat flux profile of internal convectiveand external conductive boiling mechanisms as a function of dis-tance. The isentropic process does not contribute to the heat fluxas a function of temperature which includes only the internal con-vection contribution, but it does contribute to the heat flux as afunction of distance which includes both conductive and internalconvective contribution. The spatial particle number constant k

g mechanisms. (a) As a function of temperature. (b) As a function of distance.

Fig. 8. Schematic diagrams of spontaneous symmetry breaking. (a) At the activation temperature of transition boiling. (b) Transition from non-equilibrium to equilibrium atthe transition temperature. (c) At the activation temperature of film boiling.

H.-S. Roh / International Journal of Heat and Mass Transfer 68 (2014) 332–342 341

contains the thermal particle number constant a representing theinternal convective heat transfer and the temporal particle numberconstant b representing the temporal dependence.

Fig. 7 depicts a schematic diagram to present the activationtemperatures, times, and distances during pool boiling processes.The pool boiling processes are separated by four distinctive mech-anisms of natural convection, nucleate boiling, transition boiling,and film boiling.

As shown in Fig. 2, the activation temperatures of natural con-vection and nucleate boiling, phase transitions are of first order atthe activation temperatures, while phase transitions are continu-ous at the activation temperatures of transition and film boiling.Fig. 8 demonstrates the schematic diagrams of spontaneous sym-metry breaking at the activation temperatures of transition andfilm boiling, respectively. As depicted in Fig. 8(a), at the continuousphase transition temperature of transition boiling, non-equilib-rium spontaneous symmetry breaking occurs. However, as shownin Fig. 8(c), equilibrium spontaneous symmetry breaking takesplace at the continuous phase transition temperature of film boil-ing. The thermal particle numbers in Eq. (13) change the sign atthese temperatures. As drawn in Fig. 8(b), the transition processfrom non-equilibrium to equilibrium proceeds at the transitiontemperature Ttr during transition boiling.

Three kinds of instabilities take place in the transition regime ofpool boiling. They are thermal instability, spatial instability, andtemporal instability in a static fluid. In the instability regimes,the temperature profiles are inverted, and the heat flux profilesare inverted as a function of temperature, space, and time. Theseinstabilities are the extension of the Helmholtz instability andthe Taylor instability [4,15,17].

9. Conclusions

The statistical thermodynamic heat transfer theory for non-equilibrium, quasi-equilibrium, and equilibrium [1] has been pro-posed based on thermal energy conservation. Here we apply it topool boiling to examine the characteristics of boiling preciselyand to validate the theory. In pool boiling, the four boiling regimesof natural convection, nucleate boiling, transition boiling, and filmboiling are considered and the heat flux curve is drawn as a func-tion of excess temperature between wall and saturation tempera-ture. The S-shape heat flux curve is logically produced and themismatch around the onset of nucleate boiling is well explainedin this scheme. Natural convection is a slow, stable, and irrevers-ible mode and the nucleate boiling is a fast, stable, and irreversiblemode. The former is the process with a liquid phase while the lat-ter is the process with the two phases of liquid and vapor. Theyhave different initiation temperatures depending on internal con-vective heat transfer processes. To verify the heat transfer theory,comparisons between theoretical calculations and experimentalobservations have been carried out.

Three limiting heat fluxes and four activation temperatures areemployed as input parameters. The limiting heat flux in boiling isconnected to the constant heat flux (or constant entropy) process.The theoretical heat flux–temperature curves agree with experi-mental heat flux profiles in a wide temperature regime as well asthose from existing theories useful only in a specific temperatureregime. The theoretical calculations clearly demonstrate the powerlaw behaviors in certain regimes of natural convection and nucle-ate boiling and show the maximum heat flux and the minimumheat flux. The relationship between the maximum and the mini-mum heat flux is determined, and the relationship between theactivation temperature of natural convection and the onset tem-perature of nucleate boiling is analyzed.

The heat transfer theory sheds light on specific characteristicsin each pool boiling regime. It can also be extended to flow boilingwhen the fluid motion in pool boiling is combined with the dy-namic fluid flow induced by mechanical external force. Moreover,the heat flux mechanisms of boiling are applicable to understandthe other phase transitions such as condensation, solidification,and melting [37]. The heat transfer theory will have widespreadapplications in science and engineering including boiling, refriger-ation, nuclear reactor, and chemical fuel cells. It will play a crucialrole in enhancing the heat transfer of thermodynamic devices.

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