Astm stp 1534 2012

Such an approach maintains theinterface at saturation while the mixture temperatures of the liquid and vapor in the interface cells can be superheated.. KEYWORDS: phase change, boiling f

Journal of ASTM International

Selected Technical Papers STP1534

Film and Nucleate Boiling Processes

JAI Guest Editors:

K Narayan Prabhu Nikolai Kobasko

ASTM International

100 Barr Harbor Drive

PO Box C700West Conshohocken, PA 19428-2959

Printed in the U.S.A

ASTM Stock #: STP1534

Library of Congress Cataloging-in-Publication Data

Film and nucleate boiling processes / JAI guest editors, K Narayan Prabhu, Nikolai Kobasko

p cm (Journal of ASTM International Selected technical papers; STP1534)Includes bibliographical references and index

ISBN 978-0-8031-7520-4 (alk paper)

1 Nucleate boiling 2 Film boiling 3 Heat Transmission 4 Change of state (Physics)

I Prabhu, Narayan II Kobasko, N I (Nikolai Ivanovich)

QC304.F44 2012

536’.44 dc23 2012003685

Copyright © 2012 ASTM INTERNATIONAL, West Conshohocken, PA All rights reserved This material may not be reproduced or copied, in whole or in part, in any printed, mechanical, electronic, fi lm, or other distribution and storage media, without the

written consent of the publisher

Journal of ASTM International (JAI) Scope

The JAI is a multi-disciplinary forum to serve the international scientifi c and engineering community through the timely publication of the results of original research and

critical review articles in the physical and life sciences and engineering technologies These peer-reviewed papers cover diverse topics relevant to the science and research that establish the foundation for standards development within ASTM International

Photocopy Rights

Authorization to photocopy items for internal, personal, or educational classroom use, or the internal, personal, or educational classroom use of specifi c clients, is granted by ASTM International provided that the appropriate fee is paid to ASTM International, 100 Barr Harbor Drive, P.O Box C700, West Conshohocken, PA 19428-2959, Tel:

610-832-9634; online: http://www.astm.org/copyright

The Society is not responsible, as a body, for the statements and opinions expressed in this publication ASTM International does not endorse any products represented in this publication

Peer Review Policy

Each paper published in this volume was evaluated by two peer reviewers and at least one editor The authors addressed all of the reviewers’ comments to the satisfaction of both the technical editor(s) and the ASTM International Committee on Publications

The quality of the papers in this publication refl ects not only the obvious efforts of the thors and the technical editor(s), but also the work of the peer reviewers In keeping with long-standing publication practices, ASTM International maintains the anonymity of the peer reviewers The ASTM International Committee on Publications acknowledges with appreciation their dedication and contribution of time and effort on behalf of ASTM International

au-Citation of Papers

When citing papers from this publication, the appropriate citation includes the paper authors, “paper title”, J ASTM Intl., volume and number, Paper doi, ASTM International, West Conshohocken, PA, Paper, year listed in the footnote of the paper A citation is provided as a footnote on page one of each paper

Printed in Bay Shore, NYFebruary, 2012

THIS COMPILATION OF THE JOURNAL OF ASTM INTERNATIONAL

(JAI), STP1534, Film and Nucleate Boiling Processes, contains papers

published in JAI that discuss heat quenching technologies based on several developments These developments include: the mechanism of fi lm and nucleate boiling processes, calculating the duration of transient nucleate boiling mode, the accuracy of cooling curves and cooling rate measure-ments, the results of investigations based on use of noise control systems, and what are real and effective heat transfer coeffi cients

The JAI Guest Editors are Prof K Narayan Prabhu, National Institute

of Technology Karnataka, Department of Metallurigical and Materials Engineering, Surathkal, Mangalore, India and Dr Nikolai Kobasko, FASM,

IQ Technologies Inc., Akron, OH, USA

Overview vii

A Volume of Fluid Phase Change Model on Adaptive Octree Grids

M W Akhtar and S J Kleis 1 CFD-Simulation of Film Boiling at Steel Cooling Process

T Kulju, J Pyykkönen, D C Martin, E Muurinen, and R L Keiski 28 Enhancement of Heat Transfer Characteristics of Transformer Oil

by Addition of Aluminium Nanoparticles

E Rajesh and K N Prabhu 45 Modeling and Simulation of Film and Transitional Boiling Processes

on a Metallic Cylinder During Quenching

P Stark and U Fritsching 61 Correlation Between Chemical Composition of Steel, Optimal Hardened Layer,

and Optimal Residual Stress Distribution

N I Kobasko 81 Duration of the Transient Nucleate Boiling Process and Its Use for the Development

of New Technologies

N I Kobasko 103 Effect of Accuracy of Temperature Measurements on Determination of Heat Transfer

Coefficient during Quenching in Liquid Media

N I Kobasko 126 Intensive Quenching of Steels Parts and Tools in Water Salt Solutions of Optimal

Concentration

N I Kobasko, A A Moskalenko, V V Dobryvechir, and L M Protsenko 142 Microstructure and Hardness Prediction at the Core of Steel Parts of Any

Confi guration during Quenching

N Kobasko and S Guseynov 167 Experimental Investigation of the Onset of Subcooled Nucleate Boiling

in an Open-Pool Nuclear Research Reactor

A Z Mesquita, A L Costa, C Pereira, M A F Veloso, and P A L Reis 183 Boiling Heat Transfer of Butanol Aqueous Solution Augmentation of Critical

Mixtures for Boiling Flow Applications

V Srinivasan and D M Wang 285 Application of Numerical Methods to Simulate Direct Immersion Quenching Process

V Srinivasan, D M Wang, D Greif, and M Suffa 315 Modeling Vertical Subcooled Boiling Flows at Low Pressures

G H Yeoh, S C P Cheung, J Y Tu, and M K M Ho 349

Forced Convective Boiling of Ethylene Glycol/Water Mixtures Inside a Small Tube

W Yu, D M France, and J L Routbort 376 Bubble Dynamics and Heat Transfer in Pool Boiling on Wires at Different Gravity

J.-F Zhao and S.-X Wan 402 Author Index 421 Subject Index 423

Overview

The compendium on “Film and Nucleate Boiling Processes” discusses portant problems connected with the quenching processes in heat treating industry as well as boiling problems related to the nuclear power industry Both for heat treating industry and for nuclear power industry, the forma-tion of fi lm boiling should be prevented During quenching, the fi lm boiling (especially local one) results in considerable distortion, crack formation and poor mechanical properties of the material In the nuclear power industry,

im-fi lm boiling can lead to overheating of the nuclear reactor Hence, a study

of fi lm and nucleate boiling processes combined with the critical heat fl ux (CHF) densities are very important for the practice

The duration of transient nucleate boiling process is widely discussed

in STP1534 (see paper JAI103485) It has been established by the author that duration of transient nucleate boiling process is directly proportional to square of the thickness of steel parts and inversely proportional to thermal diffusivity of a material and depends on confi guration of steel parts, liquid properties and its velocity The transient nucleate boiling (self–regulated thermal process) is followed by specifi c characteristics: the surface tempera-ture during nucleate boiling is maintained at the level of the boiling point of the liquid which is used as a quenchant During this period real heat transfer coeffi cients and average effective heat transfer coeffi cients (HTCs) are consid-ered Average effective generalized Biot numbers and Kondratjev numbers can be found which remain almost the same with varying size of probes Us-ing established characteristics, the authors of the paper JAI103525 (Kobasko, Guseynov) predicted the microstructure and hardness at the core of steel parts of any confi guration Also these characteristics were used by author of the paper JAI102788 (Kobasko) to evaluate correctly optimal quenched layer which provides optimal residual stress distribution after quenching Based

on duration of nucleate boiling process, the new intensive two – step intensive quenching (IQ) technology was developed using water salt solutions of opti-

mal concentration (JAI104173, Kobasko, Moskalenko, et al.).

Rajesh and Prabhu (see paper JAI103354) investigated nanofl uids as a quenchant It has been stablished by authors that the addition of Al nanopar-ticles to the base fl uid decreases the wettability and improves its heat trans-fer capability The vapour phase stage existed for longer period of time for transformer oil than Al-transformer oil based nanofl uids The dispersion of nanoparticles in the base fl uid is believed to disrupt the vapour blanket stage

in the early stage of the cooling process The peak heat transfer coeffi cient increases with increase in the Al nanoparticle content in the oil The addition

of 0.5 vol% nanoparticles enhances the peak heat transfer coeffi cient by about

58 % Nanofl uids can also be successfully used for intensifi cation of cooling at the second step of intensive quenching to improve mechanical properties of steel Nanofl uids, as a new class of quenchants, are very promising for heat treating industry

Ravnik and Grum (JAI103386) presented the data which show that pected frequency range during nucleate boiling process depends on the bub-ble size and immersion depth and can vary within 3–18 kHz Their fi ndings are very important for designing new version of noise control system to eval-uate duration of nucleate boiling processes Some results of investigations

ex-are also discussed in the paper (JAI104173, Kobasko, Moskalenko, et al.)

Forced convective boiling of ethylene glycol/water mixtures inside a small tube were investigated by Wenhua Yu, France, and Routbort (see paper JAI103378) Equations for prediction of boiling heat transfer coeffi cients of water and ethylene glycol/water mixtures in small channels were developed

by them These equations predict the experimental data well, and most of the predicted values are within ±30% of the experimental data

Yeoh, Cheung, et al (JAI103374-10) presented the results of investigations

connected with modeling vertical subcooled boiling fl ows at low pressures A new bubble departure model including the infl uence of the Marangoni effects has also been proposed by authors, which can predict the whole observation both in microgravity and in normal gravity The value of CHF (critical heat

fl ux) in microgravity is lower than that in normal gravity, but it can be dicted well by the Lienhard-Dhir correlation

pre-Nishiguchi and Shoji (JAI103452) investigated boiling heat transfer of butanol aqueous solutions Boiling heat transfer, especially critical heat fl ux (CHF), of some aqueous solutions is enhanced by adding small amount of al-cohol such as butanol Such aqueous solutions show nonlinear surface tension dependence on liquid temperature and are sometimes called as “Self-rewetting liquid”, being applied recently to thermal devices such as heat pipes However, the heat transfer characteristics of boiling of self-rewetting liquids are not ful-

ly understood In the present research, by employing butanol aqueous solution

as a typical test solution, a fundamental boiling test is performed on a heated wire with special attention to CHF augmentation in order to observe the boil-ing phenomena and to address the fundamental issues The authors found that the dependence of CHF on liquid subcooling is peculiar With increasing subcooling, CHF decreases fi rst , reaches a minimum and then increases.CFD-Simulation of Film Boiling at Steel Cooling Process was carried

out by Timo Kulju et al (JAI103382-10) Their paper analyzes fi lm boiling

phenomena on a fl at, horizontal hot steel plate using the Volume of Fluid (VOF) method In this study, the model includes both convection and radia-tion induced mass and heat transfer, where the latter was found to be more

important to maintain the fi lm layer and fi lm boiling at high temperatures The model estimated heat and mass transfer behavior at impingement ve-locities between 1-5 m/s and temperatures between 500-1300 K The initial results obtained with the simulation suggest that CFD simulation tech-niques represent a promising alternative for studying complex and diffi cult

to measure phenomena such as high temperature fi lm boiling, and hint at a new class of experimental methods for mechanistic analysis of fl uid

Also modeling and simulation of fi lm and transitional boiling processes on

a metallic cylinder during quenching were investigated by Paul Stark, Udo Fritsching (JAI103380-10) The authors showed that the bubble crowd model

is able to investigate the separate boiling phases within one single cal model approach Simulation results are discussed for the quenching of a circular cylinder in a facing water fl ow The initial fl ow velocity and the wall superheat (at temperatures above and below the Leidenfrost point) were varied to investigate their infl uence on the vapor formation and on the local

numeri-as well numeri-as the averaged heat transfer rates

Heat and Mass Transfer Characteristics of Binary Mixtures for Boiling Flow Applications were modeled and simulated by Vedanth Srinivasan and

De Ming Wang (JAI103366-10) The authors proposed boiling mass transfer model, based on detailed empirical analysis of the heat transfer coeffi cients pertinent to binary systems and was fully implemented within the commer-cial CFD code AVLFIRE® It was used to study the heat and mass transfer characteristics of boiling fl ows inside a rectangular duct The authors under-lined that their model can be easily extended to simulate multiphase fl ow in complex systems such as a cooling water jacket for automotive applications

As a result of complex investigations (fi lm boiling, nucleate boiling and cal heat fl ux densities), it is possible to use numerical methods of calculations and make computer simulations to accurately predict heat transfer modes.Application of Numerical Methods to Simulate Direct Immersion Quench-

criti-ing Process was also investigated by Vedanth Srinivasan et al (JAI103364-10)

Authors presented the results of the numerical computations carried out to simulate the direct immersion quenching process of several test pieces using a recently developed and implemented quenching simulation methodology with-

in the commercial CFD code AVLFIRE® Numerical coupling between the ulation domains, involving the fl uid and the solid metal region, were achieved through an AVL Code Coupling Interface (ACCI) feature The computed infor-mation adjudges the presence of intense non-uniformity in the temperature distribution within the solid region which is of grave importance in evaluating the stress and fatigue patterns generated in the quenched object

sim-Experimental investigation of the onset of subcooled nucleate boiling in

an open-pool nuclear research reactor was carried out by Mesquita et al

(JAI103193-10) The investigations were done in the IPR-R1 TRIGA nuclear

research reactor at the Nuclear Technology Development Center (CDTN), in Belo Horizonte (Brazil) which is a pool type reactor According to authors, the experimental results indicated that subcooled pool boiling occurs at the clad-ding surface in the reactor core central channels at power levels in excess of

60 kW However, due to the high heat transfer coeffi cient in subcooled boiling the cladding temperature is quite uniform along most of the active fuel rod region and do not increase very much with the reactor power An operational computer program and a data acquisition and signal processing system were developed as part of this research project to allow on line monitoring of the operational parameters

In the paper (JAI104173, Kobasko), a wide discussion is provided ing an accuracy of temperature measurements during quenching of test probes and accuracy of the evaluation of effective heat transfer coeffi cients (HTC) based on these measurements It is shown by author that, due to very high cooling rate during nucleate boiling, the instrumented thermocouples should

concern-be welded to the proconcern-be to receive accurate measured experimental data.Investigations of Shekriladze (JAI103387-10) explain why during nucle-ate boiling process heat transfer coeffi cients are very high It is explained by pumping effect of growing bubble (PEGB) – thermo-hydrodynamic effect of formation of liquid jet by growing vapor bubble The author underlined that wide theoretical and experimental investigations of PEGB are necessary to clarify all important features of the effect, beginning from generation of jet

fl ow and ending with dynamic and thermal consequences

The overview shows the necessity of understanding the mechanism of fi lm and nucleate boiling as well as the assessment of critical heat fl ux density to obtain the overall picture of the process of quenching or possible occurrence

of fi lm boiling in nuclear reactor

The Editors of this compendium gratefully acknowledge the

whole-heart-ed support receivwhole-heart-ed from the ASTM whole-heart-editorial staff Their commitment and hard work in the preparation of this compendium is very much appreciated

A special thanks to Prof G.E Totten who conceived the idea and supported

us throughout The Editors also thank the reviewers of all papers for their valuable time and inputs

Prof K Narayan PrabhuNational Institute of Technology KarnatakaDepartment of Metallurgical and Materials Engineering

Surathkal, Mangalore, India

Dr Nikolai KobaskoFASM, IQ Technologies Inc

Akron, OH, USAJAI Guest Editors

Mohammad W Akhtar1and Stanley J Kleis2

A Volume of Fluid Phase Change Model on Adaptive Octree Grids

ABSTRACT: A three-dimensional phase change model using finite volumehas been developed Interface thermal conditions that are consistent with themixture formulation within the volume tracking framework are introduced Thecomputational grid is dynamically adapted near the interface using an octreebased structure Up to three levels of adaption are used to provide improvedaccuracy in regions of steep gradients of flow variables near the liquid-vaporinterface at a reduced computational cost Interface jump conditions areimposed by specifying mass, momentum, and energy sources for the mixturevariables in the interface cells Second order accurate liquid and vapor tem-perature gradients at cell faces are calculated based on the location and ori-entation of the interface (sharp interface approach) These estimates are thenused to calculate the mass source terms, which are distributed in the interfacecells consistent with the mixture formulation Such an approach maintains theinterface at saturation while the mixture temperatures of the liquid and vapor

in the interface cells can be superheated A one-dimensional cavity flow lem is used to test for the proper coupling of vapor generation rate with energyand momentum flux effects Three-dimensional bubble growth rates at 5Cuniform superheat (Ja¼ 8.6, density ratio 140, and conductivity ratio 2.3)

prob-of refrigerant FC-87 are compared with theory in the diffusion controlledregime The analytical solution of Mikic et al for bubble radius as a function oftime with a non-zero initial radius is used to validate the numerical solution.The numerical solution is validated against theoretical prediction of bubbleradius as a function of time for a more complete validation of the balances ofmass, momentum, and energy at the liquid-vapor interface

Manuscript received August 30, 2010; accepted for publication January 19, 2011; published online March 2011.

Adapt-J_ID: DOI: Date: 12-January-12 Stage: Page: 1 Total Pages: 27

Conshohocken, PA 19428-2959.

1

Reprinted from JAI, Vol 8, No 3 doi:10.1520/JAI103348 Available online at www.astm.org/JAI

KEYWORDS: phase change, boiling flows, diffuse interface models, volume

of fluid, octree grids, adaptive mesh refinement, bubble growth rates

Nomenclature

Af ¼ area side fraction

ðAinter=VcellÞ ¼ area of the interface to cell volume ratio (1/m)

c ¼ mixture (mass averaged) specific heat (J/kg/K)

n ¼ unit normal vector of the interface

p ¼ volume averaged mixture pressure ðN=m2Þ

~

q ¼ heat flux vector ðW=m2Þ

rc¼ centroid of a cell containing the interface

Senergy¼ energy source term associated with phase change ðW=m3Þ

Sm;vol¼ mass source term ðkg=m3=sÞ

~

Smom;pc¼ momentum source term associated with phase change ðN=m3Þ

~

Smom;st¼ momentum source term associated with surface tension ðN=m3Þ

T ¼ mass averaged mixture temperature (K)

Tint¼ temperature of the interface (K)

DT ¼ bulk liquid superheat

t ¼ time (s)

~

u ¼ mixture fluid velocity (m/s)

xc¼ centroid of a cell containing the interface

ds ¼ cell width

Greek Letters

a¼ volume fraction

q¼ volume averaged mixture density ðkg=m3Þ

s¼ volume averaged mixture shear stress ðN=m2Þ

Subscripts

c ¼ value at the centroid of a computational cell

cell ¼ value at the center of the computational cell

int ¼ value at the interface

l ¼ value for the liquid phase

v ¼ value for vapor phase

Introduction

Liquid-vapor phase change offers an efficient mechanism to store or releaselarge amounts of energy in the form of latent heat It is thus of importance inthe petroleum industry for oil refinement and offers an effective cooling mecha-nism for electronic devices [1] Besides being of industrial importance, thephysics of the phase change process is of fundamental interest to the heat

J_ID: DOI: Date: 12-January-12 Stage: Page: 2 Total Pages: 27

transfer community Numerous experimental studies have provided empiricalcorrelations for bubble growth rates specific to boiling modes and geometries[2] However, experiments fail to provide details of the liquid-vapor interfaceevolution and thus only general trends in bubble shape evolution can bedescribed Limitations of measurements in capturing the inherent small spatialand temporal scales of the phase change process hinder a better understanding

of the process [2] With the availability of better computing resources, fullnumerical simulations guided by theory and measurements can provide a betterunderstanding of the phase change process and help make useful predictions ofliquid-vapor interface evolution

The mechanism of bubble growth was described by Jakob in 1932 [3] as:

“…it can be imagined that during the small explosion which starts the growth of

a bubble, the interface temperature, because of the consumed heat of tion, drops immediately from the superheat temperature to the saturation tem-perature, for example from 110C to 100C As a consequence of the heattransfer from the liquid to the vapor bubble the liquid envelope is being cooledprogressively from the inside to the outer boundary; a temperature boundarylayer is created with a constantly decreasing temperature drop This thermalboundary layer increases in thickness until the thermal wave, which advancesfrom the vapor bubble interface into the liquid, has reached the outer limit ofthe hydrodynamics boundary layer The decrease in thickness of the hydrody-namics boundary layer because of the evaporation at the interface is, initially, asmall fraction of the total thickness…” [4] A detailed description of the proc-esses involved requires the solution of the complete set of mass, momentum,and energy equations with the correct initial and boundary conditions Depend-ing on whether a mixed formulation is employed, an additional equation must

vaporiza-be solved to track the motion of the interface Several studies have vaporiza-been ducted to study bubble growth under different simplifying assumptions Thefirst study on spherical bubble growth under inertia dominated regime was con-ducted by Rayleigh in 1917 [5] Subsequently, various researchers includingPlesset and Zwick [6], Prosperetti and Plesset [7], and Mikic et al [8] studiedbubble growths in thermal and surface tension dominated regimes

con-Numerical simulation of the liquid-vapor phase change process is one ofthe most challenging problems in computational fluid dynamics The presence

of discontinuities in the flow variables like pressure (due to surface tension) and

in material properties like density pose numerical challenges Multiphase flows

in general have very high density ratios, complex interface topologies, and anevolution with a wide range of spatial scales In addition, a liquid-vapor inter-face with phase change includes additional effects of mass generation/depletion.Such effects are generally included as source terms for conservation of mass,momentum, and energy It is known that inclusion of source terms can reducethe convergence rate of standard multigrid solvers [9] Thus, conducting an effi-cient full numerical solution of a phase change problem independent of gridsize is a challenging task Recently, attempts have been made to conduct directnumerical simulations of the liquid-vapor phase change process Welch [10]used a moving mesh two-dimensional (2D) finite volume method to simulatephase change to track the liquid-vapor interface Son and Dhir [11] conducted

J_ID: DOI: Date: 12-January-12 Stage: Page: 3 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 3

2D simulations of phase change using a finite difference scheme Juric andTryggvason [2] simulated liquid-vapor phase change with more complete mod-els of interface thermal boundary conditions within a front tracking framework

in two dimensions, which can handle large deformations of the interface Directnumerical simulations of multiphase flow in three dimensions using a hybrid offront tracking and a front capturing scheme was conducted by Tryggvason et al.[12] Almost all of the results presented were obtained using uniform grids Theuse of adaptive grids was limited to 2D and axisymmetric problems [13]

In this paper, a three-dimensional (3D) phase change model is presentedusing the finite volume method on an octree grid structure with adaptive meshrefinement (AMR) Using an octree grid provides the required spatial resolutionnear the interface to resolve the gradients of flow variables and material proper-ties at a reasonable computational cost The liquid-vapor interface is trackedusing the volume of fluid (VOF) technique The phase change effects areincluded by specifying mass source terms for the individual phases and momen-tum and energy sources in terms of the mixture variables An appropriate ther-mal condition is imposed in the “interface cells” (computational cellscontaining the interface), consistent with the mixture formulation This allowsthe liquid in the interface cell to be superheated while the generated vapor is atthe correct saturation temperature Surface tension effects are included usingthe model of Brackbill [14]

A one-dimensional cavity problem is constructed to check for the propercoupling of the momentum flux and energy exchange with the vapor generationrates The phase change model is then validated by applying the model to pre-dict 3D bubble growth rates with a uniform liquid superheat under zero gravityconditions The simulated growth rates are validated against theoretical predic-tions of Mikic et al [8] in the thermal (diffusion) controlled regime

Mathematical Formulation

The simulation of liquid-vapor phase change requires the modeling of masstransfer and associated momentum and energy exchanges between the twophases Instead of solving two separate sets of conservation equations for mass,momentum, and energy, a common approach is to solve a single set of equa-tions valid in both the phases This requires defining “mixture variables” to beused in such a single field formulation This “mixture variable” is defined as alinear combination of the phase variables such that they assume the phase val-ues when the location is completely within one phase or the other The weight-ing function used to define a mixture variable in a cell containing the interface

is the level set (LS) function for the LS method or the volume fraction for theVOF method Thus, a mixture property or flow variable can be expressed as

where:

P ¼ property,

a¼ volume fraction or the level set function of the vapor phase,

J_ID: DOI: Date: 12-January-12 Stage: Page: 4 Total Pages: 27

l ¼ liquid phase, and

v ¼ vapor phase

However, mixture specific heat and temperature are mass averaged asopposed to volume averaged The set of conservation equations in terms of mix-ture variables can be written as

qand c ¼ mixture density and specific heat,

~

u ¼ volume averaged mixture velocity,

T ¼ mass averaged mixture temperature,

p ¼ volume averaged mixture pressure, and



s¼ volume averaged mixture deviatoric stress

The terms ~Smom;st, ~Smom;pc, and Senergy are the volumetric source terms toaccount for surface tension, momentum, and energy exchanges between thephases, respectively The term ~q ¼ krT is the heat flux vector while the term

s :r~u, the viscous dissipation, is neglected in the numerical implementation.Surface tension effects are included in the formulation using the contin-uum-surface-force approach of Brackbill [14] as

~

Smom;st¼ rj~n (5)where:

a model is limited in accuracy for a translating bubble or droplet and needs ther advances

fur-The present phase change model is intended to be used for the simulation

of sliding bubbles of FC-87, which has a low value of surface tension (1=8th ofthat for an air-water system) Also, investigations are for bubbles with radiigreater than 1 mm Under these conditions, surface tension effects are notexpected to play a significant role in the sliding bubble phenomenon, and thus

J_ID: DOI: Date: 12-January-12 Stage: Page: 5 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 5

an advanced model of surface tension is not required Thus, for the presentstudy, more accurate schemes for interface normal and curvature are not used.The VOF scheme has been chosen for phase change implementationbecause it is volume conserving, unlike methods like LS where additional mod-eling is required to ensure mass conservation Mass conservation of the numeri-cal scheme is an important aspect for phase change problems Schemes like thecoupled level set and volume of fluid (CLSVOF) can be used, where the LSmethod can be used to get better estimates of curvature while the VOF methodcan help ensure volume conservation, but such a scheme or other hybridschemes require consistency checks between the two schemes and is expected

to be computationally more expensive than an improved VOF scheme (VOFscheme with better models for interface normal and curvature) The resultsfrom such hybrid schemes for surface tension driven flows assessed using capil-lary oscillations as a test case were compared by Popinet [9] The VOF schemewith height functions was found to be more accurate compared to CLSVOF andfront tracking at low resolutions and as accurate as Parabolic Reconstruction ofSurface Tension for higher resolutions Thus, if a more accurate scheme of sur-face tension is needed, the present model can be improved by using the heightfunction technique as described by Popinet [9]

The other source terms, ~Smom;pc and Senergy, related to phase change aremodeled to take into account the jump conditions at the interface Assumingthe interface to be thin and massless with negligible energy contribution due tointerface stretching, the jump conditions for mass, normal momentum, tangen-tial momentum, and energy can be written as [2]

 1

q2 l

uint¼ interface velocity,

L ¼ latent heat of vaporization at saturation temperature, and

~t ¼ unit tangent to the interface

For a static interface, Eq 7 reduces to pv pl¼ rj; the Young–Laplace tion for pressure jump due to surface tension For Eqs 8 and 9, no-slip condi-tions for tangential fluid velocities in the interface are assumed The second andthird terms on the right hand side of Eqs 7 and 9 are assumed to be small andthus are not included in the numerical implementation [2]

equa-J_ID: DOI: Date: 12-January-12 Stage: Page: 6 Total Pages: 27

Interface Tracking

In addition to the set of conservation equations in terms of the mixture bles, an advection equation for the vapor volume fraction is solved with geomet-rical reconstruction (piecewise-linear) to implicitly track the liquid-vaporinterface The advection equation for the vapor volume fraction is

av¼ volume fraction for the vapor and

ðAinter=VcellÞ ¼ area of the interface per unit cell volume, which can beexpressed as

Ainter=Vcell¼X~

The side fraction, Af, can be calculated by the algorithm described by Young[15] The right hand side of Eq 10 is the mass of vapor generation rate per unitvolume of the cell Once the vapor volume fraction field is solved, the liquid vol-ume fraction can be obtained by imposing the volume fraction constraint for acomputational cell

Solution of Eqs 10–12 gives the volume fraction field of the liquid and vaporphases In order to obtain the interface location and orientation from thevolume fraction data, a set of reconstruction steps as described by Welch andWilson [16] need to be followed These steps have been summarized below.(a) The orientation of the piecewise-linear interface segment in a computa-tional cell is obtained by defining the interface normal based on the vol-ume fraction gradient as

~

(b) Once the interface orientation is determined, the interface location isdetermined such that the calculated volume fraction for the phases iscorrectly represented

(c) The direction-split approach as described by Rudman [17] is then used

to calculate the amount of mass flux through the faces of the tional cell

computa-Interface Boundary Conditions

No-slip in the tangential velocities across the interface is assumed

~

ul ~t ¼ ~uv ~t ¼ ~uint ~t (14)and the liquid and vapor temperatures are assumed to be equal at the interface

J_ID: DOI: Date: 12-January-12 Stage: Page: 7 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 7

Tl¼ Tv¼ Tint (15)

The interface temperature, Tint, remains to be defined to complete the tion For macro-scale boiling problems, it may be assumed that the interfacetemperature is equal to the equilibrium saturation temperature corresponding

formula-to the system pressure

In the presence of a pressure jump across the interface, the system pressure inthe liquid and vapor phases are unequal, and thus a choice needs to be made inorder to model the interface temperature The model of Delhaye [18] assumes

TsatðpvÞ 6¼ Tv¼ Tl6¼ TsatðplÞ (17)

A second approach based on the kinetic theory of gases assumes a temperaturediscontinuity at the interface leading to the following condition

TsatðpvÞ ¼ Tv6¼ Tl¼ TsatðplÞ (18)

[19] Both of these approaches are based on the concept of interfacial resistance

to mass transfer across the interface [2]

Using the balance of entropy along with the pressure jump equation asdefined by Eq 7, a more complete equation of the interface temperature for anon-flat interface with non-reversible motion can be obtained (for details, seeRef 2) For the test cases considered in this study, the condition as given in

Eq 16 was used The effect of interfacial resistance on the saturation ture is expected to be of importance in problems involving growing bubbles in anear wall configuration like sliding bubbles The set of Eqs 2–4 and 10 can then

tempera-be solved with the boundary conditions (Eqs 6–9 and 14–16)

Grid Generation

GAMBITTM(Ansys, Inc.) was used to create the parent computational mesh fortest cases in this study In order to provide higher spatial resolution near theinterface at a reasonable computational cost, an octree based structure withAMR was used In addition to this grid structure being more efficient, thisscheme maintains the parent grid aspect ratio, which, if chosen to be unity, ismore accurate for the interface reconstruction scheme [17] The nomenclature

of Popinet [9] is used to describe the octree (3D)/quadtree (2D) discretization Aschematic of such a grid structure with its corresponding tree representation isshown in Fig 1 Each computational cell can have up to four children in twodimensions (eight in three dimensions) The base of the tree structure isreferred to as the root or the base cell and has index “0.”

The level of a cell is the level of its parent þ1 The cell without any child isknown as the leaf cell and has the smallest cell size

The parent grid is dynamically refined using the octree structure The face cells and two adjoining cells in each direction on both sides of the interface

inter-J_ID: DOI: Date: 12-January-12 Stage: Page: 8 Total Pages: 27

are refined to be leaf cells This is done in order to maintain consistent secondorder estimates of the liquid and vapor temperature gradients Figure 2 showsthe marking of the cells near the interface used for the refinement Such a pro-cedure is repeated at alternate time steps With the chosen parent grid cell sizeand simulation conditions used, up to three levels of refinement were found suf-ficient to resolve the gradient of the flow variables near the interface.

FIG 1—(a) Octree grid schematic and (b) corresponding tree representation, numbersindicate “level” of the computational cell

J_ID: DOI: Date: 12-January-12 Stage: Page: 9 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 9

Phase Change Model

The phase change model requires reconstruction of temperature gradients nearthe interface The mixture temperature gradients are neither the liquid nor thevapor temperature gradients and hence cannot be used to get the correctamount of heat flux contributions from the liquid and vapor side of the inter-face Two cells neighboring the interface both on the liquid and vapor sideswere used to construct the liquid and vapor temperature gradients at the cellfaces First, the cell center temperatures of the liquid only cells neighboring theinterface are identified Next, the distance to the effective interface location inthe adjacent interface cell is computed from the volume fraction and cell dimen-sion Finally, the liquid cell temperature, saturation temperature, and distancefrom the cell center to effective interface location are used to compute the tem-perature gradient component in that direction This process is repeated for eachcoordinate direction for both the liquid and vapor sides of the interface Such

an approach leads to an estimate of the heat flux vector at the interface for asharp interface

FIG 2—Marking of the cells near the interface for refinement (light gray–interface cells,gray–first layer, and dark gray–second layer of cells next to the interface cells)

J_ID: DOI: Date: 12-January-12 Stage: Page: 10 Total Pages: 27

In order to maintain second order accuracy for gradient calculations and tokeep the estimates consistent, all cells used for gradient calculation are refined

to the same level An alternate approach would be to use special schemes at thefaces of fine/coarse cell boundaries in order to be consistent and maintain over-all second order accuracy This has been implemented by Popinet [9] on quad/octree discretization Such an implementation would reduce the number ofrefined cells but would require additional modeling effort and was not imple-mented for the present study A one-cell thick moving interface is assumed,leading to an interface representation which is two cells wide Thus, althoughthe estimated liquid and vapor temperature gradients are calculated from cells

on the liquid and vapor sides of the interface, they are used for both sets of theinterface cells Using the interface normal and the thermal conductivities, thetotal heat flux is calculated The area of interface to cell volume ratio is thenused to distribute the volumetric mass source for the two rows of interface cells.Equation 9, after assuming the second and third terms to be small, gives theenergy source term to be included in the mixture energy equation Thus, theenergy source per unit area of the interface, S00

Senergy=L If a mass source term is specified in FluentTM, a momentum sourceterm based on the local velocity (in the present studies, this is the mixture veloc-ity) is automatically included It is seen from the constant mass source test casethat this can be corrected by including a momentum sink based on the mixturevelocity ~Smom;pc¼ m000u These source terms are returned to the Fluent~ TMsolver

to be included in the conservation equations

A mixture temperature condition is enforced at the interface in order tomaintain the (effective) interface at saturation and to ensure accurate estimates

of the heat flux at the interface Using the mixture temperature gradient,interface normal, cell width, and mixture and phase properties, the proper cellmixture temperature is calculated such that the internal energy is balanced as ifthe interface were sharp and at saturation temperature This allows the liquidand vapor temperatures in an interface cell to be above saturation while theinterface is at saturation The calculated mixture temperature is evaluated onceevery time step and is used as a thermal condition (Eq 15) for the interface cell.Since the vapor is not superheated for these studies, any leakage of heatflux not contributing to phase change would increase the vapor temperaturenear the interface A L2 norm (rms error) is constructed for the differencebetween the cell temperature and the saturation temperature for the first set ofvapor cells adjacent to the interface cells In addition to the convergence criteriaset for the mass, momentum, and energy equations, L2 norm is used as a

J_ID: DOI: Date: 12-January-12 Stage: Page: 11 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 11

measure of convergence for the imposed interface temperature condition Figure

3 shows the convergence behavior of this norm for the 3D bubble growth rates(for simulation conditions, see Fig 6) The maximum error in L2is 1.03 % for asuperheat of 5C with an average error of 0.17 % over a simulation run of 0.5 s

It should be noted that this model allows for the vapor to be superheatedwhile the interface can be maintained at the local saturation temperature Thus,the model is capable of handling dryout situations, where the heated wall canconduct heat to the interface via the superheated vapor and, in addition, canallow for a non-uniform saturation temperature along the interface For suchproblems, the error in the interface cell temperature from that of the imposedinterface temperature should be monitored

Numerical Schemes

The Navier–Stokes, energy, and the scalar transport equations for the vapor ume fraction, using a one field formulation, were solved using FluentTM 6.3.26(Ansys, Inc.) Ozkan et al [20] presented a critical evaluation of the codes CFX,FluentTM, and STAR-CD against TURBIT-VOF [21], a code that has been vali-dated against experiments of Thulasidas et al [22] Results of co-current bub-ble-train flow in a square vertical mini-channel, driven by buoyancy only and bybuoyancy and external pressure gradient, in terms of bubble velocity, mean

vol-FIG 3—L2(rms error) norm for the mixture temperature in interface cells as a function

of time for bubble growth (simulation conditions as in Fig 6)

J_ID: DOI: Date: 12-January-12 Stage: Page: 12 Total Pages: 27

liquid velocity, and bubble shape, were used for comparison purposes It wasshown that only the VOF method in FluentTMwith geometric reconstruction andTURBIT-VOF gave physically sound and consistent results Further, the geo-metric reconstruction scheme was shown to be more robust than high-order dif-ference schemes to solve the volume fraction equation Thus, we extended thecapabilities of the VOF scheme in FluentTM with user-defined functions to con-duct phase change calculations with AMR The code uses a finite volume formu-lation to convert the governing equations to a discrete form A segregated solverwith double precision was used for the solution of the discrete equations on asingle node of the 160 node Linux Cluster at the Texas Learning and Computa-tion Center ðTLC2Þ, where each node consists of two dual socket quad core 2.2GHz processors with 8 GB random access memory, at a theoretical peak per-formance of 11.3 TFLOPS.

The solution process starts with a user-defined function for initialization.Next, a set of adjust functions are called to adapt the grid dynamically and cal-culate the source terms for the mass, momentum, and energy equations Withthe source terms defined, the u, v, and w momentum equations are solved, fol-lowed by a pressure correction, update of pressure and velocity, solution of theenergy equation, and solution of the volume fraction equation FLUENTTMsol-ves the conservation equations using a finite volume formulation on a collo-cated arrangement (variables including velocity and pressure are stored at thecell centers) The conservation equation for the transport of a scalar, /, for anarbitrary control volume V in integral form is

C/¼ diffusion coefficient of the scalar and

S/¼ source of the scalar per unit volume

The integral equation, when written in discretized form, is

Of the available schemes to interpolate the face values, the QuadraticUpwind Interpolation scheme, a weighted average of second order upwind andcentral interpolation of the variable, was used [23] FLUENTTMuses a weightingfactor that is solution dependent The diffusion terms were discretized usingcentral differences Since a collocated arrangement is used to find the pressure

J_ID: DOI: Date: 12-January-12 Stage: Page: 13 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 13

values at the cell faces, the pressure staggering option scheme is used, wherein

a discrete continuity balance for a staggered control volume is used to computethe pressure at the face [23] The pressure-velocity coupling was obtained by thePressure-Implicit with Splitting of Operators (PISO) algorithm, which involves

a second correction step Details of the implementation can be found inFerzieger and Peric [24] and Issa [25] For a flow chart of the PISO algorithm,refer to Nichita [23] The resulting system of dependent linear scalar equations

is solved using a point implicit Gauss–Seidel linear equation solver in tion with an algebraic multigrid method [23] The convergence criteria forscaled residuals were set to 103 for continuity, 104 for momentum, and

conjunc-2  106for the energy equation

Solution Procedure

The solution is initialized with the volume fraction field and the mixture ture A total of 32 sets of memory locations were used to keep track of the interfacecells, neighboring liquid and vapor cells for grid adaption, and area of interface tocell volume calculation; and to store flags and intermediate values After the varia-bles are initialized, a function is called to set the user-defined memory locationsand update the flags at the beginning of each time step Next, an adjust function iscalled to mark the cells containing the interface and two neighboring liquid andvapor cells for adaption This function also creates flags for liquid and vapor tem-perature gradient calculations Another adjust function is called, which calculatesthe area of interface to cell volume ratio, sets the temperature condition in theinterface cells, and estimates the liquid and vapor temperature gradients (asdescribed in the previous section) The mass source is then calculated and is stored

tempera-in a user-deftempera-ined memory location for each cell This value is used by source tions for mass, momentum, and energy These calculated source terms are thenreturned to FluentTMto solve the mass, momentum, and energy equations

func-Results and Discussions

Model Verification

A rectangular cavity ð4:0  0:5 mmÞ was used to check (a) the vapor generationrates and (b) the momentum source effects as a result of phase change Figure 4shows the grid structure near the interface used for the cavity test cases A uniformgrid of cell size 50 lm is first generated in GAMBITTM The grid is dynamicallyadapted in FluentTM with three levels of refinement leading to a leaf cell size of6:25 lm The left half of the cavity was initialized to be FC-87 liquid while the righthalf of the cavity was initialized with the vapor phase of FC-87 The top and bottomwalls are specified as free-slip walls (zero wall shear and zero normal velocity)while the left wall is specified as a no-slip condition with no penetration The rightwall is specified as a pressure outlet, i.e., with zero gauge pressure

A one-cell thick moving interface with a linear variation in volume fractionfrom 0 to 1 is specified as an initial condition for the volume fraction field.Figure 5 shows the schematic of a one-cell thick moving interface cell across the

J_ID: DOI: Date: 12-January-12 Stage: Page: 14 Total Pages: 27

FIG 4—Adaptive quadtree grid structure near the interface for the cavity problem.

FIG 5—One-cell thick moving interface assumption

J_ID: DOI: Date: 12-January-12 Stage: Page: 15 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 15

computational grid Under this scheme, the interface would generally be spreadover two computational cells.

The volume fractions for the phases can then be written in terms of the cellwidth and the distance between the cell center and the center of the movinginterface cell The vapor volume fraction in the interface cells is the given by

!

 xi xcdx

;

Here, xi denotes the location of the center of the moving interface cell, xc

denotes the center of the cell whose volume fraction is being calculated, and dxrepresents the cell width of the interface cells The volume fraction of the liquidphase, al, can then be determined from the volume fraction constraint as 1  av

Constant Heat Flux

In order to test the phase change model, a constant heat flux of 280 W=m2wasimposed on the left wall of the cavity The liquid phase was initialized with a lin-ear temperature gradient in the liquid such that the heat flux at the interfacematches the imposed heat flux condition on the left wall The vapor phase wasinitialized at the saturation temperature for FC-87 (303:15 K at 1 atm) with azero temperature gradient

With the assumption that the heat flux at the interface would be maintained

at the initial condition, equal to the heat flux imposed at the left wall, an mate of the vapor generation rate can be calculated The theoretical vapor veloc-ity would then be

lin-In the phase change model, the total heat flux into the interface was multiplied by

a dimensionless constant h with a value of 2.75 The calculated area weightedvapor velocity had an asymptotic value of 2:34  104 m=s (within 7.1 % of thetheoretical estimate) Since the theoretical estimate is not exact as stated earlier,the numerical simulation is considered to be in good agreement with theory.The use of such a dimensionless constant for diffuse interfaces is necessarybecause although the heat flux estimates are based on a sharp interface

J_ID: DOI: Date: 12-January-12 Stage: Page: 16 Total Pages: 27

approach, the interface tracking scheme is based on a diffuse interface model.The need for this dimensionless constant has been recognized and extensivelydiscussed in other diffuse interface model studies For example, Beckermann

et al [26] showed that a theoretical value of 2.757 is appropriate when applied

to a solidification problem Further, Beckermann et al showed that the constant

of 2.757 is the correct value to be used in a general situation involving flow inboth phases as the diffuse interface thickness approaches zero [26] This valuewas used for all subsequent studies

Constant Mass Source

With the initialized volume fraction field corresponding to a one-cell thick ing interface, constant mass source strengths ranging from 10 000 to

mov-100 000 kg=m3 s were specified at the interface to check the resulting tum source effects The net force experienced on the left wall balances themomentum efflux of vapor at the right outlet within an accuracy of 2 % for thesource strengths in the specified range

momen-Bubble Growth in Superheated Liquid (Three-Dimensional)

A cubic domain of size 12 mm was used with an adaptive octree grid to simulatebubble growths in the diffusion controlled regime One of the side walls wasused as a symmetry plane while the rest of the walls were specified with pres-sure outlet conditions at bulk liquid superheat of 5C Although 1/4 of the bub-ble can be used, 1/2 of a spherical bubble of FC-87 vapor was used to studygrowths in the diffusion controlled regime This is done as we are interested instudying bubble dynamics under the action of gravity, where only one plane can

be used as a symmetry plane A vapor bubble of 1 mm radius was initialized to asaturation temperature of 303:15 K at 1 atm while the liquid was initialized at

a uniform superheat of 5C Instead of using the actual value of surface tension,

a reduced value of 104 N=m was used Since we are not studying surface sion dominated growth (large initial size), using a reduced value helps to reducethe effects of spurious currents This approach was chosen instead of employingsmoothing matrices [27] to reduce the effect of such spurious currents Thisalso allows for much larger time steps in the simulations and thus considerablesavings in CPU time The volume fraction field was initialized with a one-cellthick moving interface, as was done in the cavity test case The liquid volumefraction function in terms of the radial coordinate is then given by

!

 ri rcdr

;

J_ID: DOI: Date: 12-January-12 Stage: Page: 17 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 17

Growth Rate Comparison

Bubble growths were computed until they grew to twice their initial radius andbubble growth rates were compared against the theory of Mikic et al [8] Aspointed out by Kolev [4], the analysis of Mikic et al is a successful compromisebetween strict formulation and sound simplifying assumptions

Figure 6 shows the bubble radius as a function of time in the dimensionlesscoordinates along with the theoretical result of Mikic et al and the inertial anddiffusion limits The dimensionless coordinates are defined as

Here, al¼ kl=ðqlclÞ is the thermal diffusivity of the liquid phase with

kl¼ 0:055 W=m=K, ql¼ 1749:7 kg=m3, cl¼ 1088:9 J=kg=K, and the temperaturedifference DT is the bulk liquid superheat The vapor properties are

kv¼ 0:024 W=m=K, qv¼ 12:47 kg=m3and cv¼ 800 J=kg=K The Jakob number

is defined as

FIG 6—Comparison of growth rates (Theory, Mikic; inertial and diffusion limits(adapted from Lee and Merte [28]), simulated,) with superheat=5C and Ja=8.6

J_ID: DOI: Date: 12-January-12 Stage: Page: 18 Total Pages: 27

Ja ¼qlclDT

qvLSince the bubble was started from a non-zero initial radius, time for the simula-tions is shifted by an amount the theoretical solution takes to reach the initialradius

The agreement is excellent between theory and simulations with a superheat of 5C and Ja ¼ 8:6 The validity of the theory for this range of superheatsand Ja has been shown by the comparisons made by Lee and Merte [28] usingexperimental data of Florschuetz et al [29] (DT ¼ 3:61C, system pressure 1atm, and Ja ¼ 10:77) and Dergarabedian [30] (DT ¼ 6:4C, system pressure 1atm, and Ja ¼ 9:2)

In order to get a better view of the comparison of the simulated growthrates and the theoretical solutions, the bubble growth is plotted for tþ>105.Figure 7 shows such a comparison of the simulated growth to the asymptoticsolution of the diffusion controlled bubble growth solution The bubble growthcurve follows the asymptotic limit from the start with a minimal overshoot sincethe temperature field is initialized consistent with the theoretical solution

FIG 7—Comparison of theoretical and simulated growth rates (DT=5C and Ja=8.6;

1 % of points shown for clarity)

J_ID: DOI: Date: 12-January-12 Stage: Page: 19 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 19

(discussed in the next section on convergence studies) In the absence of such

an initialization, i.e., initializing the temperature field instead by a simple ing with the volume fraction, the growth rate curve is initially steeper, asexpected, due to the initial steep tem-perature gradients The growth rates thenslow, approaching the asymptotic solution as the solution progresses in time(data not shown)

scal-Convergence Studies

To conduct proper convergence studies, the temperature field needs to be tialized to the theoretical solution at the initial bubble radius A simple massbalance at the interface for spherical bubble growth gives

@T

@r ¼ 

Lqv2kl

A

ðtþÞ1=2

!

¼ Lqv2kl

Convergence studies are done over a time span in which the bubble grows

to twice its initial radius Time step independence is tested with bubble growthscomputed with a time step of 0.5, 0.2, and 0.1 ms Figure 8 shows that a timestep of 0.1 ms is small enough to capture the bubble growth behavior, with littledifference between the 0.2 and 0.1 ms growth rates and the theoretical results ofMikic et al

In order to check for grid independence, bubble growth rates were run on aparent grid size of 400 lm with one, two, and three levels of refinement Figure9(a) shows the grid convergence behavior for a superheat of 5C (FC-87,

Ja ¼ 8:6) on the three grids (parent cell size ¼ 400 lm with one, two, and threelevels of refinement; corresponding leaf cell size ¼ 200, 100, and 50 lm) The

50 lm grid is shown to agree well with the theoretical solution and thus is fineenough for the given conditions

Although this figure depicts grid convergence, it is not completely verifiedthat the numerical solution is independent of the dimensionless constant ðh Þ,because h can be a function of grid size (interface thickness) Thus, another set

of simulations was conducted at a superheat of 3:5C (with the same value of

h ¼ 2:75) This superheat value gives an initial temperature gradient that istwice as thick as the 5C case Figure 9(b) shows the convergence behavior ofthe solutions for DT ¼ 3:5C on the same parent grid with one and two levels of

J_ID: DOI: Date: 12-January-12 Stage: Page: 20 Total Pages: 27

refinement (corresponding leaf cell size ¼ 200 and 100 lm) The 100 lm 3:5Csolution agrees well with the 50 lm 5C case and the theoretical solution The

200 lm 3:5C solution agrees well with the 100 lm 5C case, although neither isclose to the theoretical solution The close agreement between these cases dem-onstrates that the difference with the theoretical solution is due to grid conver-gence and that the values of h for the two cases are equal Since the grids differ

by a factor of two, this demonstrates that h ¼ 2:75 is grid independent for therange of grid sizes used

It should be emphasized that the same value for the dimensionless constant

is used to compute both the one-dimensional as well as 3D bubble growths.Further, a single value for the constant gives very good agreement with thetheory independent of grid sizes and time steps (see Fig 9(b)) Thus, the con-stant is not a function of grid size, time step, heat flux at the interface, or thedimensionality of the problem

Adaptive Grid: Validation and Computational Savings

Figure 10 shows a comparison of the growths using an adaptive grid (cubicdomain, 12 mm sides) with the leaf cell size as 50 lm (three levels of

FIG 8—Simulated growth rates with time step 0.5, 0.2, and 0.1 ms (DT=5C andJa=8.6; 2 % of data points shown for clarity)

J_ID: DOI: Date: 12-January-12 Stage: Page: 21 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 21

FIG 9—Simulated growth rates; grid independence check for superheat DT=5 and3.5C; 1 % of data points shown for clarity.

J_ID: DOI: Date: 12-January-12 Stage: Page: 22 Total Pages: 27

refinement) and a uniform grid (cubic domain, 3 mm sides) with cell size as

50 lm As is seen from the plot, the growth rates using the adaptive grid is invery good agreement with the results for a uniform grid, validating the adapt-ive grid implementation For a cubic domain of size 12 mm with uniform gridspacing of 50 lm, the CPU time used would be at least 7:51  43¼ 480 hcompared to 1.16 h when using an adaptive grid (for a simulation run of 10 msfrom the initial condition of 1 mm bubble radius) Thus, for the conditionsconsidered, the solution using an adaptive grid is at least 400 times faster atearly times than a uniform grid for the same interfacial resolution and accu-racy A similar calculation at 1 s using the adaptive grid is at least 50 timesfaster than when using a uniform grid (difference due to the increased area ofthe interface)

Figure 11(a)–11(c) are contour plots of liquid fraction at 0, 0.5, and 1.0 s It

is observed that the interface remains thin and as the bubble grows, it maintains

an overall spherical shape Figure 12(a)–12(c) are contours of mixture ture field at 0, 0.5, and 1.0 s It is seen that the temperature jump from liquidsuperheat to saturation temperature occurs in a thin region and such a distribu-tion maintains spherical symmetry with bubble growth

tempera-FIG 10—Growth rates with adaptive grid (three levels of refinement, minimum ds=50 lm)and a uniform grid (ds=50 lm); 1 % of data points shown for clarity

J_ID: DOI: Date: 12-January-12 Stage: Page: 23 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 23

FIG 11—Liquid volume fraction at (a) 0 s, (b) 0.5 s, and 1.0 s (simulation conditions

as in Fig 6)

J_ID: DOI: Date: 12-January-12 Stage: Page: 24 Total Pages: 27

FIG 12—Mixture temperature at (a) 0 s, (b) 0.5 s, and 1.0 s (simulation conditions as

in Fig 6)

J_ID: DOI: Date: 12-January-12 Stage: Page: 25 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 25

A 3D phase change model within a volume tracking framework has been oped with AMR using octree/quadtree discretisation The heat flux at the inter-face is computed by using second order accurate estimates of liquid and vaportemperature gradients A temperature condition consistent with the mixtureformulation is used to enforce the saturation condition at the interface

devel-A one-dimensional cavity problem is used to test vapor generation rates andmomentum effects due to phase change 3D bubble growths are checked againsttheoretical results in the diffusion controlled regime for a uniform superheat of

5C (Ja ¼ 8:6, density ratio 140, and conductivity ratio  2:3) of FC-87 as a dation of the numerical model Grid independency and time step independenceare checked for bubble growth rates

vali-A significant improvement in computing time, a factor of 50–400, isachieved when compared to a uniform grid when using AMR for the same accu-racy The model has been verified to capture the liquid-vapor phase interactionssuitable for phase change problems involving a large bubble growing in super-heated liquid Since both liquid and vapor temperature gradients are computed,the model can be applied to dryout situations and does not assume the vaporphase to be at a uniform saturation temperature Also, a local estimate of thesaturation temperature at the interface can be made allowing for variationsalong the interface The present work can be made more robust by includinggradient calculations that would remain second order accurate across coarse/fine boundaries Further, a more accurate (at least in the zero order sense) sur-face tension model employing height functions could be included

References

Transfer Mechanisms from a Wire Immersed in Saturated FC-72 Using a Photo/LDA Method,” J Heat Transfer, Vol 118, 1996, pp 117–123.

Flow, Vol 24, 1998, pp 387–410.

Z Ver Dtsch Ing., Vol 76, 1932, pp 1161–1170.

Springer-Verlag, Berlin, Heidelberg, 2002.

Spherical Cavity,” Philos Mag., Vol 34, 1917, pp 94–98.

Liquids,” J Appl Phys., Vol 25, 1954, pp 493–500.

J Fluid Mech., Vol 85, 1978, pp 349–368.

Heat Mass Transfer, Vol 13, 1970, pp 657–666.

Flows,” J Comput Phys., Vol 228, 2009, pp 5838–5866.

Track-ing with Mass Transfer,” J Comput Phys., Vol 121, 1995, pp 142–154.

J_ID: DOI: Date: 12-January-12 Stage: Page: 26 Total Pages: 27

[11] Son, G and Dhir, V K., “Two-Dimensional Numerical Simulation of Saturated Film Boiling on a Horizontal Surface,” Proc ASME/JSME Thermal Engineering Joint Conf., HI, 1995, L S Fletcher and T Aihara, Eds., ASME, New York, pp 257–264.

Han, J., Nas, S., and Jan, Y.-J., “A Front-Tracking Method for the Computations of Multiphase Flow,” J Comput Phys., Vol 169, 2001, pp 708–759.

Cartesian, Front Tracking Method for the Motion, Deformation and Adhesion of Circulating Cells,” J Comput Phys., Vol 143, 1998, pp 346–380.

Surface-Tension,” J Comput Phys., Vol 100, 1992, pp 335–354.

Numerical Methods for Fluid Dynamics, Academic Press, New York, 1982.

Phase Change,” J Comput Phys., Vol 160, 2000, pp 662–682.

Numer Methods Fluids, Vol 24, 1997, pp 671–691.

Local Instant Formulation,” Int J Multiphase Flow, Vol 1, 1974, pp 395–409.

Univer-sity Press, New York, 1953.

CFD Codes for Interfacial Simulation of Bubble-Train Flow in a Narrow Channel,” Int J Numer Methods Fluids, Vol 55, 2007, pp 537–564.

aufsteigenden Einzelblasen und Blasenschwarmen mit einer Volume of Fluid Methode [Three-Dimenstional Numerical Simulation of Dynamics of Individual Bubbles Rising and Bubble Swarms with a Volume of Fluid Method],” Ph.D thesis, Universitat Karlsruhe (TH), Karlsruhe, Germany.

of Circular and Square Cross-Section,” Chem Eng Sci., Vol 50, 1995, pp 183–199.

Lausanne, Switzerland.

Springer-Verlag, Heidelberg, 1996.

Splitting,” J Comput Phys., Vol 62, 1986, pp 40–65.

Solid-ification,” J Comput Phys., Vol 154, 1999, pp 468–496.

Merging and Fragmentation in Multiphase Flows with SURFER,” J Comput Phys., Vol 113, 1994, pp 134.

Super-heated Liquids,” Int J Heat Mass Transfer, Vol 39, No 12, 1996, pp 2427–2447.

Bubbles in Liquids at Uniform Superheats Under Normal and Zero Gravity ditions,” Int J Heat Mass Transfer, Vol 12, 1969, pp 1465–1489.

J Appl Mech., Vol 20, 1960, pp 537–545.

J_ID: DOI: Date: 12-January-12 Stage: Page: 27 Total Pages: 27

AKHTAR AND KLEIS, doi:10.1520/JAI103348 27

Timo Kulju,1Juha Pyykko¨nen,2David C Martin,2

Esa Muurinen,1and Riitta L Keiski1

CFD-Simulation of Film Boiling at Steel

Cooling Process

ABSTRACT: This paper analyzes film boiling phenomena on a flat, horizontalhot steel plate using the volume of fluid method The influence of water deliveryvelocity and plate surface temperature on film boiling behaviour has been quan-tified for the case of a single laminar flow jet assuming two-dimensional radialsymmetry In the present study, the model includes both convection and radia-tion induced mass and heat transfer, where the latter was found to be more im-portant to maintain the film layer and film boiling at high temperatures Themodel estimated heat and mass transfer behaviours at impingement velocitiesbetween 1 and 5 m/s and temperatures between 500 and 1300 K were found to

be qualitatively consistent with available literature The initial results obtainedwith the simulation suggest that computational fluid dynamics (CFD) simulationtechniques represent a promising alternative for studying complex and difficult

to measure phenomena such as high temperature film boiling, and hint at anew class of experimental methods for mechanistic analysis of fluid mechanicaland thermal processes using purely computational methods

KEYWORDS: impinging jet cooling, film boiling, CFD, VOF, heat transfercoefficient

Manuscript received September 16, 2010; accepted for publication June 2, 2011; published online July 2011.

Laboratory, Univ of Oulu, Oulu 90014, Finland.

Oulu 90014, Finland.

Cite as: Kulju, T., Pyykko¨nen, J., Martin, D C., Muurinen, E and Keiski, R L.,

“CFD-Simulation of Film Boiling at Steel Cooling Process,” J ASTM Intl., Vol 8, No 8 doi:10.1520/JAI103382.

J_ID: DOI: Date: 12-January-12 Stage: Page: 28 Total Pages: 17

Conshohocken, PA 19428-2959.

28

Reprinted from JAI, Vol 8, No 8 doi:10.1520/JAI103382 Available online at www.astm.org/JAI

In the production of ultrahigh strength hot rolled steel plates and strips, ated cooling or direct quenching of the steel after rolling plays an importantrole in determining microstructure and final properties This cooling operation

acceler-is usually done by passing the rolled steel under banks of vertical water jets.Individual water jets impact a very small region of the steel surface, and typicalcooling systems are constructed of multiple banks, each consisting of tens orhundreds of individual jets

The metallurgical phenomena of interest in these steels typically occursover temperatures between 900C and 250C, so large temperature differencesexist between the cooling water and steel throughout the cooling process Thisimplies that thin film layer boiling phenomena [1,2] play an important role indetermining the efficiency and performance of this cooling process, particularlygiven that direct jet impact represents only a small fraction of the total surfacearea of typical steel cooling systems At these temperatures, the surface of therolled steel is typically covered with a smooth, thin, uniform layer of iron oxide.Numerical simulation of steel temperature and hardenability during cool-ing has become an important tool for both alloy and cooling process design.Accurate prediction of temperature during the cooling process requires a com-plete description of both the complex heat conduction related phenomenawithin the steel, and the heat transfer characteristics of the water jets, as well asother heat transfer processes such as radiation and contact conduction Whilethere is a considerable amount of experimental study of the heat transfer behav-iour of water jets impinging hot steel surfaces to be found in the literature[3–6], most focus on direct impingement phenomena Far less attention appears

to have been paid to film boiling heat transfer, despite recognition of the tant role it plays in the overall heat transfer efficiency of the cooling process.Unfortunately, the types of experimental studies necessary to provide data tohelp fill this knowledge gap are both expensive and complex to perform As an al-ternative, we have examined the use of computational fluid dynamics (CFD) tostudy this phenomenon, with the objective of quantifying the relationshipamong water delivery conditions, plate temperature, and the effective surfaceheat transfer coefficient in both direct impingement and film boiling regimes.For this purpose we have chosen the volume of fluid (VOF) method, which

impor-is widely used in the literature for film boiling simulations [7–10]

We have extended the basic VOF method with convection approach throughthe inclusion of radiation source terms to account for heat transfer at high tem-peratures At the temperatures of industrial interest, we expect that radiationwill play an important role in initiating and sustaining film boiling As an result,our model formulation is designed to admit a wider temperature and water inletvelocity range than other published studies using this method

A CFD Simulation for Film Boiling Phenomena

Film boiling is a phenomenon that occurs when liquid is brought into a contactwith a solid whose temperature is considerably higher than the saturation point

of the liquid In the context of cooling of rolled steel strip or plate, the steel

J_ID: DOI: Date: 12-January-12 Stage: Page: 29 Total Pages: 17

KULJU ET AL., doi:10.1520/JAI103382 29