Heat transfer engineering an international journal, tập 31, số 14, 2010

This derivation is based on the most basic flow and heat transfer processes and is described in the following paragraphs: Using Newton’s law of cooling, the forced convection heat loss f

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Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003689211

An Analysis of Heat Conduction

Models for Nanofluids

JO ˜ AO N N QUARESMA,1 EMANUEL N MAC ˆ EDO,1 HENRIQUE M DA

FONSECA,2 HELCIO R B ORLANDE,2 and RENATO M COTTA2

1School of Chemical Engineering, Universidade Federal do Par´a, UFPA Campus Universit´ario do Guam´a, Bel´em, PA, Brazil

2Mechanical Engineering Department–Polit´ecnica/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

The mechanism of heat transfer intensification recently brought about by nanofluids is analyzed in this article, in the

light of the non-Fourier dual-phase-lagging heat conduction model The physical problem involves an annular geometry

filled with a nanofluid, such as typically used for measurements of the thermal conductivity with Blackwell’s line heat

source probe The mathematical formulation for this problem is analytically solved with the classical integral transform

technique, thus providing benchmark results for the temperature predicted with the dual-phase-lagging model

Differ-ent test cases are examined in this work, involving nanofluids and probe sizes of practical interest The effects of the

relaxation times on the temperature at the surface of the probe are also examined The results obtained with the

dual-phase-lagging model are critically compared to those obtained with the classical parabolic model, showing that the increase

in the thermal conductivity of nanofluids measured with the line heat source probe cannot be attributed to hyperbolic

effects.

INTRODUCTION

The constitutive equation that classically relates the heat flux

vector to the temperature gradient is Fourier’s law, which

con-siders an infinite speed of propagation of heat in the medium

Despite this unacceptable assumption, Fourier’s law provides

accurate results for most practical engineering applications

However, in applications involving small scales of time and

space, the use of other constitutive equations, such as those

in-dependently derived by Cattaneo [1] and Vernotte [2], may be

required Such models take into account a lag between the heat

flux vector and the temperature gradient, resulting in a

hyper-bolic model for heat conduction [1–13]

Thermal conductivity of fluids plays a vital role in the

devel-opment of energy-efficient heat transfer equipment However,

traditional fluids used in those equipments have low thermal

The authors acknowledge the financial support provided by CNPq for the

postdoctoral fellowship of Professor J N N Quaresma at the Laboratory of

Heat Transmission and Technology of the Mechanical Engineering Department

of COPPE/UFRJ This work was partially sponsored by CAPES and FAPERJ,

with major financial support provided by Petrobras S.A.

Address correspondence to Professor Helcio R B Orlande, Mechanical

Engineering Department–Polit´ecnica/COPPE, Universidade Federal do Rio de

Janeiro, UFRJ, Cx Postal 68503–Cidade Universit´aria, 21941–972, Rio de

Janeiro, RJ, Brazil E-mail: helcio@mecanica.coppe.ufrj.br

conductivity [14] On the other hand, metals in the solid formhave thermal conductivity larger by orders of magnitude thanthose of fluids For example, the thermal conductivity of cop-per at room temperature is about 700 times larger than that

of water and around 3000 times larger than that of engine oil.Therefore, fluids containing suspended solid metallic particleswere expected to display significantly enhanced thermal con-ductivities relative to conventional heat transfer fluids Numer-ous theoretical and experimental studies of the effective thermalconductivity of dispersions containing particles have been con-ducted since Maxwell’s theoretical work on the subject waspublished more than 100 years ago [14, 15] However, earlystudies of the thermal conductivity of suspensions have beenconfined to those containing particles with sizes of the order ofmillimeters or micrometers In fact, conventional micrometer-sized particles cannot be used in practical heat-transfer equip-ment because of severe clogging and sedimentation problems

In addition, recent miniaturization, leading to the increasingpractical utilization of microchannels and microreactors, alsoimposed a restriction on the use of micrometer-sized particles[14, 16–18]

Modern nanotechnology provides great opportunities to cess and produce materials with average sizes below 50 nm[14, 16–18] Recognizing an opportunity to apply this emerg-ing nanotechnology to established thermal energy engineering,

pro-1125

Choi and coworkers [14, 16–18] proposed that nanometer-sized

metallic particles be suspended in industrial heat transfer

flu-ids, such as water or ethylene glycol, to produce a new class

of engineered fluids referred to as nanofluids Experiments with

nanofluids have indicated significant increases in thermal

con-ductivity, as compared to base liquids without nanoparticles

or with larger suspended particles [14, 16–23] Generally, the

observed increase in thermal conductivity of nanofluids was

substantially larger than that predicted with the available

the-ory On the other hand, some studies reported that such

in-crease in the thermal conductivity could not be detected with

optical experimental methods [24, 25] In this context, such

a fact resulted in a search for physical phenomena not

ac-counted for in the theoretical predictions for suspensions of

micrometer-sized or larger particles, which included

Brown-ian motion, liquid layering, ballistic mechanisms,

thermophore-sis, aggregation of nanoparticles into clusters, etc [16–26]

Due to previous experimental observations that non-Fourier

effects are significant at small time and space scales [1–11],

hyperbolic heat conduction models were naturally suggested

to explain the heat transfer enhancement in nanofluids [12,

13] It was proposed [12, 13] that enhanced heat transfer in

nanofluids was caused by a heat transfer mechanism

mod-eled in terms of the dual-phase-lagging constitutive equation

[5, 7]

In this article we revisit the works of references [12] and [13]

and apply the dual-phase-lagging model to a one-dimensional

heat conduction problem in cylindrical coordinates The

geome-try examined here is that typically used for the measurements of

thermal conductivity of nanofluids with Blackwell’s heat source

probe [27] Such a technique consists of a line heat source,

usu-ally taken in the form of a heating wire, which is placed inside

the material with unknown properties For large times, the

tem-perature variation of the heat source is shown to be linear with

respect to the logarithm of time, so that the thermal

conduc-tivity can be computed from the slope of such linear variation

The temperature variation of the heat source can be measured

through the variation of the heating wire electrical conductivity

Alternatively, the heating wire and a temperature sensor, such as

a thermocouple or a PT-100, can be encapsulated in a metallic

needle that is inserted into the medium with unknown thermal

conductivity [27–31] Commercial probes are generally based

on this last construction arrangement [31]

The heat conduction problem under examination is solved

analytically by using the classical integral transform technique

(CITT) [32, 33] The controlled accuracy and analytical nature

of the solution technique developed in this work allow for the

computation of benchmark results for the temperature variation

of the probe, based on the dual-phase-lagging model Numerical

results are presented in this article for typical configurations

of probes and nanofluids, as well as for different values of

relaxation times, which result on hyperbolic effects with distinct

intensities Such results are critically compared to the classical

parabolic heat conduction model, as well as to Blackwell’s

large-time solution

PROBLEM FORMULATION

The analysis considered here is similar to that examined

in references [12] and [13] and involves a dual-phase-laggingmodel (DPLM) [5, 7] In such a model, a finite speed of heatpropagation in the medium is taken into account through a delaytime for the establishment of the heat flux,τq, and a lag betweenthe heat flux vector and the temperature gradient,τT The consti-tutive equation relating the heat conduction flux vector and thetemperature gradient in the dual-phase-lagging model is given

where q is the heat flux vector and K is the effective thermal

conductivity of the medium

The energy conservation equation for a purely conductingmedium, considered in this work as a nanofluid, is written as

(C s + C f)∂Tf

where C s is the volumetric heat capacity of the

nanopar-ticles and C f is the volumetric heat capacity of the basefluid

The substitution of Eq (1) into the energy conservation tion (2) results in:

Equation (3) can be similarly obtained by considering thatthe nanoparticles and the base fluid are not in local thermalequilibrium In this case, the energy conservation equation can

be written separately for the nanoparticles and the base fluid,respectively, in the following form:

Then, by substituting T s from Eq (5) into Eq (6), one tains:

ob-CsC f h(Cs + C f)

∂(∇2T f)

heat transfer engineering vol 31 no 14 2010

A comparison of Eqs (3) and (7) reveals the definition of

the effective thermal diffusivity given by Eq (4), as well of the

relaxation times given by:

Considering the case involving a line heat source probe of

radius a immersed in a cylindrical medium of radius b, Eq (3)

where q0is the heat flux resulting from the electrical resistance

inside the probe and T0is the initial temperature of the medium

By defining the following dimensionless variables,

the problem given by Eqs (11a–e) can be rewritten in

dimen-sionless form as:

The classical parabolic heat conduction model, which lizes Fourier’s law as the constitutive equation that relates theheat conduction flux vector and the temperature gradient, can

uti-be directly obtained from Eqs (13a–e) by making the ation times,τqandτT, equal to zero (see Eq (1)) In terms ofthe nonequilibrium model given by Eqs (5) and (6),τq → 0andτT → 0 can be obtained with h → ∞ (see Eqs (8a) and

relax-(8b))—that is, the heat transfer coefficient between the fluidand the dispersed nanoparticles becomes very large and local

thermal equilibrium is attained (T f = T s)

SOLUTION METHODOLOGY

For the solution of the hyperbolic heat conduction problemgiven by Eqs (13a)–(13e) we apply the classical integraltransform technique (CITT) [32, 33] A split-up procedure [32]

is used in order to improve the convergence rate of the finalseries solution Hence, the solution for the temperature field iswritten as:

θ(R,τ) = θ a v τ) + θp (R) + φ(R,τ) (14)whereθav(τ) is the average temperature in the medium, which

is a priori obtained from Eqs (13a)–(13e);θp (R) is a particular

solution andφ(R,τ) is the potential to be solved with the CITT.

The average temperatureθav(τ) is defined as

in the R-direction The definition given by Eq (15) is then

em-ployed and the boundary conditions (13d, 13e) are used to yield

Therefore, the solution forθav(τ) is obtained as

problem for θav(τ) given by Eqs (16a–c) is used in order to

obtain the following problems given by Eqs (17a–d) and Eqs

(18a–f), forθp (R) and φ(R,τ), respectively:

 1

A

The additional constraints given by Eqs (17d) and (18f)

are obtained by substituting Eq (14) into the definition of the

average temperatureθav(τ) given by Eq (15)

The integration of the problem given by Eqs (17a–d) can be

readily performed in order to obtain the solution for the potential

The homogeneous problem for the potentialφ(R,τ) is now

solved with the classical integral transform technique (CITT)[32, 33] For this purpose, the following auxiliary eigenvalueproblem is utilized, which shall provide the basis for the eigen-function expansion of the potentialφ(R,τ):

i (R) = J o(βi R)Y1(βi)− J1(βi )Y0(βi R) (21a)

J1(βi A)Y1(βi)− J1(βi )Y1(βi A) = 0, i = 1, 2, 3, (21b)

It can be shown that the eigenfunctions i(R) satisfy the

following orthogonality property [32, 33]:

 1

A Ri (R) j (R)d R=

˜

After the definition of the integral transform-inverse pair withthe auxiliary eigenvalue problem (20a–c), the next step in theCITT is thus to accomplish the integral transformation of theoriginal partial differential system given by Eqs (18a–e) Forthis purpose, Eq (18a) and the initial conditions (18b) and (18c)

are multiplied by R ˜ i (R), integrated over the domain [A,1] in the R-direction, and the inverse formula given by Eq (22b) is

employed After the appropriate manipulations, the followingsystem of ordinary differential results, for the calculation of thetransformed potentials ¯φi(τ):

φi(τ) = ¯f i;d ¯φi(τ)

dτ = 0 for τ = 0 (23b, c)where

The infinite system of ordinary differential equations (23a–c)

is uncoupled and can be readily solved to yield:

By substituting Eq (24) into the inverse formula (22b), the

analytical solution for the homogeneous potentialφ(R,τ) is

Finally, by substituting Eqs (16d), (19), and (25) into Eq

(14), the solution for the dimensionless temperature field is

∞

i=1

where the eigenquantities that appear in Eq (27) are the same

as those defined earlier for the solution of the problem for thepotentialφ(R,τ).

For large times, Blackwell [27] derived an asymptotic lution for the temperature variation at the surface of the lineheat source probe in the parabolic problem, which is shown to

so-be linear with respect to the logarithm of time Such solution

is convenient for the measurement of the thermal conductivity

of the medium surrounding the probe, which can be computedfrom the slope of the temperature variation [27–31] Blackwell’ssolution [27], in terms of the dimensionless variables given byEqs (12a), is

θa(τ) = 1

2A lnτ +1

2A

ln

)4

RESULTS AND DISCUSSION

In this session we present numerical results for the sionless temperature variation at the surface of the probe, that is,

dimen-at R = A, obtained with the hyperbolic heat conduction model

given by Eq (26) Such results are compared to those obtainedwith the classical parabolic problem given by Eq (27), as well

as to those obtained with Blackwell’s model given by Eq (28).Such analytical solutions were implemented under the Visualheat transfer engineering vol 31 no 14 2010

Table 1 Test cases examined

Dimensions Test

a Alumina in water 5 × 10 −5 0.025 2 × 10 −3 1.71

b Alumina in water 7.5 × 10 −4 0.05 1.5 × 10 −2 1.71

c Copper in ethylene glycol 5 × 10 −5 0.025 2 × 10 −3 2.29

d Copper in ethylene glycol 7.5 × 10 −4 0.05 1.5 × 10 −2 2.29

Fortran platform

Different test cases were examined in this work, involving

different configurations of probe diameters and nanofluids With

respect to the nanofluids, the following ones were considered

in the analysis: (i) alumina nanoparticles in water (K= 0.257

W/mK, C s= 3.430 × 106J/m3K, C f= 2.649 × 106J/m3-K,β

= 1.71) and (ii) copper nanoparticles in ethylene glycol (K =

0.627 W/m-K, C s = 2.964 × 106 J/m3-K, C f = 4.183 × 106

J/m3-K,β = 2.29) With respect to the probe geometry, it was

considered to be made of a thin resistance wire with diameter a

= 5 × 10−5m inserted into a medium with outer diameter b=

0.025 m, so that A= 2 × 10−3, such as in [12] and [13] Also

examined was another probe with diameter a= 7.5 × 10−4

m, typical of those commercially available [31] In this case,

the medium was considered with an outer diameter b= 0.05

m, so that A= 1.5 × 10−2 Table 1 summarizes the test cases

examined

Figure 1 illustrates the convergence behavior of the

temper-ature at the probe surface, obtained with different truncation

orders (NT) for the series solution in Eq (26), for A= 2 × 10−3,

Foq= 1 × 10−5, andβ = 2.29 (test case c) This figure shows

that for small dimensionless times (τ ≤ 10−5), convergence at

Figure 1 Convergence behavior of the analytical solution for A= 2 × 10 −3,

Figure 2 Comparison of analytical solution and finite-difference method

so-lution using Gear’s method (referred to as FDM-Gear) for A= 1.5 × 10 −2and

β = 2.29.

the graphic scale is obtained with 5000≤ NT ≤ 10000 On the

other hand, for larger dimensionless times the convergence isreached with approximately 500 terms in the series solution.The computation time in a Pentium Intel Dual E2160 1.8-GHz

computer was of the order of 9.7 minutes, for NT= 10000 For

the results presented next, NT= 10000 was used

In order to validate the analytical solution just presented, wecompared its results with those obtained numerically with finite

Parabolic Model Hyperbolic Model - Fo q = 1x10 -10

Hyperbolic Heat Conduction Dual-Phase-Lagging Model

differences In this case, the problem given by Eqs (13a)–(13e)

was discretized in the radial direction with second-order

differ-ences and the resulting system of ordinary differential equations

was integrated in time with Gear’s method Figure 2 shows a

comparison of the results obtained with the analytical solution

against those obtained with the finite-difference method

solu-tion (referred to in Figure 2 by FDM-Gear) for test case d (A=

1.5× 10−2,β = 2.29) and different values of Fo q The

agree-ment between such solutions is excellent, thus validating the

numerical code here developed The finite-difference solution

was obtained with 2000 nodes in the spatial grid

Figure 3 presents a comparison of the hyperbolic andparabolic solutions given by Eqs (26) and (27), respectively,

for test case c (A= 2 × 10−3andβ = 2.29) and Fo q = 1 ×

10−10 We note in this figure that for such a small value of Fo q

the hyperbolic model behaves exactly as the parabolic one, that

is, non-Fourier effects are negligible Such was also the case for

other values of A andβ examined in this article

We now examine the non-Fourier effects of the probe

sur-face temperature variation, for Fo qranging from 10−3to 10−10.The results obtained for the different test cases described inTable 1 are presented in Figure 4, a–d These figures showheat transfer engineering vol 31 no 14 2010

that non-Fourier effects are only noticeable for very small

times; as time increases, the temperature variations

gradu-ally tend to the parabolic one In fact, even for an extremely

large value of Fo q such as 10−3, the non-Fourier effects

van-ish forτ > 10−2 At small times, non-Fourier effects are more

pronounced for smaller diameters A On the other hand, the

choice of the nanofluid does not seem to affect significantly the

temperature behavior Similar conclusions can be obtained by

comparing the hyperbolic solution with the asymptotic one

de-veloped by Blackwell for the parabolic formulation, as depicted

in Figure 5a–d

The results presented in Figures 4a–d and 5a–d permit to amine the suitability of the hyperbolic formulation to the actualheat conduction problem in nanofluids, during thermal conduc-tivity measurements with the line heat source probe For thisanalysis, we bring into picture the heat transfer coefficient be-tween the base fluid and the particles considered in the thermalnonequilibrium model given by Eqs (5) and (6), which resultsheat transfer engineering vol 31 no 14 2010

ex-Table 2 Dimensional times

in the hyperbolic formulation addressed in this article Note in

these equations that such heat transfer coefficient is defined in

volumetric terms, but can be easily converted to the usual

defi-nition of the heat transfer coefficient by using the nanoparticle’s

volume to surface area ratio Figure 6 presents the heat transfer

coefficient between the base fluid and the particles for different

values of Foq, and for spherical particles of different

diame-ters Only test cases a and c are examined in this figure, since

they present more significant non-Fourier effects (see also

Fig-ures 4a–d and 5a–d) By considering a threshold value for the

heat transfer coefficient, it is possible to establish the maximum

expected value of Fo qfor which the system behaves

hyperboli-cally If we assume such a threshold value as 1 W/m2-K, which

is indeed extremely small in macroscopic means, we notice in

Figure 6 that Fo q is actually smaller than 10−5 Figure 4a–d,

shows that for Fo q = 10−5, non-Fourier effects are negligible

forτ > 10−4 Table 2 gives the physical times equivalent toτ =

10−4for the four test cases addressed in this work Notice in this

table that non-Fourier effects would have disappeared for times

much smaller than those typically considered for the

measure-ment of the thermal conductivity with Blackwell’s solution for

the line heat source probe [27–31] Indeed, notice in Figure 5,

Figure 6 Heat transfer coefficient for different nanoparticle diameters.

a–d, that Blackwell’s solution would not be considered priate for the measurement of the thermal conductivity forτ <

appro-1.2× 10−4(lnτ = –9) for test cases a and c, and for τ < 2.5

× 10−3(lnτ = –6) for test cases b and d In other words, theincrease generally detected with the line heat source probe forthe thermal conductivity of nanofluids cannot be attributed tothe non-Fourier heat transfer mechanisms examined earlier Infact, recent theoretical predictions corroborate our findings anddemonstrate that nanoparticles and the base fluid are in thermalequilibrium in nanofluids [23, 34]

CONCLUSIONS

In this article we presented an analytical solution based on theclassical integral transform technique for the dual-phase-laggingheat conduction model The physical problem examined wasrepresentative of that used for the measurement of thermal con-ductivity with the line heat source probe Results were obtainedfor the temperature variation at the probe surface, for differentcombinations of nanofluids and probe diameters Such resultswere compared to those obtained with the classical parabolicheat conduction model based on Fourier’s law, as well as to theasymptotic solution proposed by Blackwell [27] for the line heatsource probe

The foregoing analysis reveals that non-Fourier effects aresignificant only for very small times, generally in the rangewhere Blackwell’s solution is not valid for the measurement ofthermal conductivity Therefore, the increase detected with theline heat source probe for the thermal conductivity of nanofluidscannot be attributed to the non-Fourier heat transfer mechanismsaddressed in this article

NOMENCLATURE

a probe radius

A dimensionless probe radius

b radius of the cylindrical medium

Cf volumetric thermal capacity of the base fluid

Cs volumetric thermal capacity of the nanoparticles

¯f i transformed initial condition

Foq dimensionless relaxation time associated with the heatflux

FoT dimensionless relaxation time associated with the perature gradient

tem-h heat transfer coefficient

K effective thermal conductivity of the nanofluid

Ni normalization integral

NT truncation order in the summations

q heat flux vector

q0 heat flux at the surface of the probe

r radial variable

R dimensionless radial variable

t time variable

T temperatureheat transfer engineering vol 31 no 14 2010

Tf temperature of the base fluid

Ts temperature of the nanoparticles

T0 initial temperature

y Euler’s constant

Greek Symbols

α thermal diffusivity of the nanofluid

β ratio of relaxation times

θa dimensionless temperature from Blackwell’s solution

θav dimensionless average temperature

θp particular solution for the dimensionless temperature field

τ dimensionless time variable

τq relaxation time associated with the heat flux

τT relaxation time associated with the temperature gradient

φ homogeneous solution for the dimensionless temperature

field

¯

φi transformed potentials

Subscripts

i order of the eigenvalue problem

f relative to the base fluid

s relative to the nanoparticles

Superscripts

integral transformed quantities

∼ normalized eigenfunctions

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Jo˜ao N N Quaresma received his B.Sc in

chem-ical engineering from the Universidade Federal do Par´a in 1988, and his M.Sc in chemical engineer- ing in 1991 and D.Sc in mechanical engineering in

1997, both from the Universidade Federal do Rio

de Janeiro, Brazil He joined the School of ical Engineering at Universidade Federal do Par´a (FEQ/UFPA) in 1991, where he is currently an as- sociate professor, and served for two years as grad- uate coordinator for the M.Sc program in chemical

Chem-engineering He is the author of more than 80 technical papers in major nals and conferences, and supervisor of 13 Ph.D and M.Sc theses His re- search interests include the modeling and simulation of non-Newtonian fluid flow, as well as the hybrid solution methodologies in the field of heat and fluid flow He is the recipient of the Clara Martins Pandolfo Award, given by the Chemistry Council in the State of Par´a in 2007 He was a member of the Thermal Sciences Committee of ABCM–Brazilian Society of Mechanical Sciences and Engineering (a sister society of ASME), elected for the period 2004–2008 He is also an 1C researcher of CNPq, a sponsoring agency in Brazil.

jour-Emanuel N Macˆedo received his B.Sc in

chem-ical engineering from the Universidade Federal do Par´a in 1993 and D.Sc in mechanical engineering

in 1998 from the Universidade Federal do Rio de Janeiro, Brazil He worked in the Mechanical Engi- neering Department at the Universidade Federal do Par´a as a postdoctoral researcher in the period from

1998 to 2002, and joined the School of Chemical Engineering also at Universidade Federal do Par´a (FEQ/UFPA) in 2002, where currently he is an as- sociate professor and the head of the Laboratory of Processes Simulation His main research area involves the development of hybrid analytical–numerical approaches in the field of heat and fluid flow involving non-Newtonian fluids and combustion processes Currently, he is the graduate coordina- tor for the D.Sc program in natural resources engineering for the Amazon region.

Henrique M da Fonseca obtained his B.S in

me-chanical engineering from the Federal University of Rio de Janeiro (UFRJ) in 2004 After obtaining his M.S in mechanical engineering in 2007 from the same university, he started his Ph.D as a joint de- gree between UFRJ and the Ecole de Mines d’Albi Carmaux, in France, where his research subject is the evaluation of the thermal signature in microflu- idic reactors from biological medium submitted to a toxicological stress, under the supervision of Prof O Fudym and Prof H R B Orlande He is the co-author of more than 10 papers

in major journals and conferences.

Helcio R B Orlande obtained his B.S in

mechani-cal engineering from the Federal University of Rio de Janeiro (UFRJ) in 1987 and his M.S in mechanical engineering from the same university in 1989 Af- ter obtaining his Ph.D in mechanical engineering in

1993 from North Carolina State University, he joined the Department of Mechanical Engineering of UFRJ, where he was the department head during 2006 and

2007 His research areas of interest include the lution of inverse heat and mass transfer problems, as well as the use of numerical, analytical, and hybrid numerical–analytical meth- ods of solution of direct heat and mass transfer problems He is the co-author of one book and more than 160 papers in major journals and conferences He has been elected distinguished professor by the mechanical engineering classes of UFRJ in 1996 and from 1999 to 2006 He is the recipient of the Young Scientist Award given by the state of Rio de Janeiro in 2000, and of the State Scientist Award given by the state of Rio de Janeiro in 2002, 2004, and 2008 He was the secretary of the Thermal Sciences Committee of ABCM–Brazilian Society of Mechanical Sciences and Engineering (a sister-society of ASME), elected for

so-the period 2005–2006 He is an associate editor of Heat Transfer Engineering,

Inverse Problems in Science and Engineering, and High Temperatures–High Pressures.

heat transfer engineering vol 31 no 14 2010

Renato M Cotta received his B.Sc in

mechani-cal/nuclear engineering from the Universidade eral do Rio de Janeiro in 1981, and his Ph.D in mechanical and aerospace engineering from North Carolina State University, USA, in 1985 In 1987

Fed-he joined tFed-he Mechanical Engineering Department at POLI/COPPE/UFRJ, Universidade Federal do Rio

de Janeiro, where he became a full professor in

1997 He is the author of around 370 technical pers and 4 books He serves in the honorary edito-

pa-rial boards of the International Journal of Heat and Mass Transfer and

In-ternational Communications in Heat and Mass Transfer, InIn-ternational

Jour-nal of Thermal Sciences, InternatioJour-nal JourJour-nal of Numerical Methods in

Heat and Fluid Flow, High Temperatures–High Pressures, Computational Thermal Sciences, Waste and Biomass Valorization, and Advances in Heat Transfer Series, CMP He was also the editor-in-chief for the international

journal Hybrid Methods in Engineering Prof Cotta contributed as elected

president to the Brazilian Association of Mechanical Sciences, ABCM, in 2000–2001, to the Scientific Council of the International Centre for Heat and Mass Transfer (ICHMT) since 1993, to the Executive Committee of the ICHMT since 2006, as head of the Heat Transmission and Technology Labo- ratory since 1994, and as head of the Center for Analysis and Simulations on Environmental Engineering, CASEE, a research consortium involving EPRI (USA), Tetra Tech (USA), and COPPE/UFRJ (Brazil), since 2001 He is

an elected member of the National Honor Society of Phi Kappa Phi, USA (1984).

heat transfer engineering vol 31 no 14 2010

Copyright
Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003689245

Flow and Thermal Errors in Current National Weather Service Wind Chill Models

RASHID A AHMAD1and STANTON BORAAS2

1ATK Space Systems, Brigham City, Utah, USA

2Morton Thiokol, Inc., Space Operations, Brigham City, Utah, USA

The two National Weather Service (NWS) wind chill models in operation since 2001 have inherent errors The first model

attempted and failed to make a facial surface temperature correction on the existing Siple and Passel model The second

model, intended as an improvement on the first, erred by mistakenly defining the wind chill temperature in terms of an internal

body temperature rather than the facial surface temperature To account for a boundary-layer reduction in the free-stream

velocity at head level, both models incorrectly apply a constant high-percentage reduction to this velocity as measured at the

NWS 10 m level As a result of these errors, both models predict wind chill temperatures much warmer than the actual values.

These warmer temperatures can instill a public complacency whereby facial freezing is viewed as a remote possibility when

in reality it may be imminent.

INTRODUCTION

In the early 1990s, the wind chill model developed in 1945

by Siple and Passel [1] became the subject of great controversy

It was criticized by investigators as being primitive, flawed,

and lacking theoretical basis, despite the fact that the model

had already proven valuable for nearly five decades The

Na-tional Oceanic and Atmospheric Administration (NOAA), the

parent organization of the National Weather Service (NWS),

responded to these criticisms by directing that a new and

cor-rect wind chill model be developed as a replacement for the

Siple and Passel model To achieve this, NOAA assembled a

group of 22 organizations (government, academic, research)

called the Joint Action Group for Temperature Indices (JAG/TI)

to develop this new model Out of this group emerged a new

model by Bluestein and Zecher [2] in 1999 On November 1,

2001, NOAA officially implemented this model by

announc-ing it to the American and Canadian public Shortly thereafter,

sometime after 2002, a newer model was introduced Called

the Osczevski–Bluestein model [3, 4], it was intended as an

upgrade or an improvement over the Bluestein–Zecher model

ATK Launch Systems is not responsible for any errors or liabilities in this

document or in any of its future derivatives.

Address correspondence to Dr Rashid A Ahmad, 925 West 885 South,

Brigham City, UT 84302, USA E-mail: ahmadrj@msn.com

Each of these models has flow and thermal errors resulting fromattempts to correct or improve upon the previous model TheBluestein–Zecher model did not correct for the incorrect as-sumption of a constant facial surface temperature in the Sipleand Passel model The Osczevski–Bluestein model incorrectlydefined the wind chill temperature in terms of an internal bodypoint temperature rather than the temperature of the externalskin surface Both models attempted to account for a boundarylayer reduction of the free-stream velocity at head level Thishead-level velocity was assumed to be at a constant 33% re-duction of the NWS 10 m free-stream velocity Their inferencethat this large 33% reduction is valid for all individuals in alllocations and for all ambient conditions is inaccurate The neteffect is that these errors result in wind chill temperature pre-dictions that not only fail to improve upon the accuracy of theSiple and Passel values but also demonstrate that both modelslack credibility This paper discusses, in detail, the errors in bothmodels, and how a continued use of either model can actuallyplace the listening public at a potential health risk

DISCUSSION

Wind chill implies the cooling effect of the wind on an posed skin surface In winter, this surface would normally be the

ex-1137

facial surface of a fully clothed individual This cooling effect

is the result of the wind producing a forced convective heat loss

from the skin surface There is another heat loss from the skin

surface whenever the skin surface temperature is warmer than

the cold ambient air Called a radiative heat loss, it results in

an additional cooling of the skin surface In the conventional

manner of treating wind chill, both heat losses are attributed to

the “wind” effect Wind chill temperature is defined as the

ambi-ent air temperature without wind that would result in both heat

losses with the wind To clarify this definition, the following

discussion describes the theoretical derivation of a wind chill

temperature equation expressed in terms of the facial surface

temperature and the ambient conditions of air temperature and

wind velocity This equation permits the calculation of the

theo-retical wind chill temperature by which the corresponding value

in either of the empirically derived NWS wind chill models can

be compared and evaluated In addition, the equation will allow

a close examination of the thermal errors in both models but

especially the Bluestein–Zecher model Derivation of this wind

chill temperature equation was necessary because it is lacking in

current wind chill models including the Siple and Passel model

Wind Chill Temperature Equation

The determination of this wind chill temperature equation

was made for a segment of the epidermis, the skin’s outer layer,

as shown in the schematic of Figure 1 In the facial area, the

thickness of the epidermis is approximately 1 mm; thus, s ∼= 1

mm The left side of this segment is the interface with the dermis,

the adjacent inner skin layer The skin surface temperature at

this interface is T i The right side is exposed to the ambient

conditions where the wind velocity is V and the air temperature

is T a The wind results in a forced convection heat loss, q•f c The

ambient air temperature T ais presumed to be less than the outer

skin surface temperature T s Consequently, a radiation heat loss

q•r exists The bottom portion of Figure 1 shows the summation

of the two heat losses into one natural convection heat loss q•nc

when V = 0 From the temperature difference (T s − T wc) that

results in this natural convection, the wind chill temperature

Twc can be determined This derivation is based on the most

basic flow and heat transfer processes and is described in the

following paragraphs:

Using Newton’s law of cooling, the forced convection heat

loss flux is expressed as

•

where h fc is the proportionality constant and is referred to as

the forced convection heat transfer coefficient as a result of the

wind velocity V Similarly, using the Stefan–Boltzmann law, the

net rate of radiation heat exchange flux loss between the surface

and the sky is

Epidermis Dermis

b) Equivalent heat loss on layer when V = 0

Figure 1 Segment of skin epidermis used in the determination of the wind

chill temperature (T wc).

where ε is the emissivity of the skin surface, σ is the

Stefan–Boltzmann constant, and T skyis the sky temperature Tosimplify the number of variables in this analysis, the assumption

was made that the sky temperature (T sky) could credibly be

re-placed with the ambient temperature (T a) This is demonstrated

in the Appendix Consequently Eq (2a) becomes

•

which can now be expressed as follows in terms of the previously

defined wind chill temperature (T wc)

•

which is again in the form of the basic Newton’s law of cooling

The difference is that h fc has been replaced by h nc, the naturalheat transfer engineering vol 31 no 14 2010

convection heat transfer coefficient This laminar natural

con-vection (nc) heat transfer coefficient (h nc) over a vertical surface

from Jakob and Hawkins [5] and Chapman [6] can be expressed

where the subscripts nc, l, and s stand for natural convection,

laminar flow, vertical surface, respectively The characteristic

length L is the vertical dimension of a surface The exponent

φ = 0.25 for a heated vertical plane or cylindrical surface and

where the sole source of heat is within the surface itself In

instances where additional heat (metabolic) is being conducted

through the surface from inside to outside,φ is expected to take

on other values The preceding expression will then become an

“effective” or “equivalent” natural convection coefficient, h nc,

when a value ofφ is determined The coefficient C1is a function

of the air’s density, thermal conductivity, specific heat, dynamic

viscosity, and the coefficient of thermal expansion and where

L is the length of the segment The air density (ρ), dynamic

viscosity (µ), specific heat (Cp ) and thermal conductivity (k)

are calculated at the film temperature (T film) which is defined as

the average of the skin surface and air temperatures, that is, T film

= (T s + T a )/2 For a vertical plane, the expression for C1from

Chapman [6] is

C1= 0.59k[(gβρ2C p)/(µk)]φ (5b)

where g is the gravitational constant and the volumetric

coef-ficient of thermal expansion (β) of the air is β = (Tfilm)−1 and

is given in [◦R−1or K−1] Values of C1were calculated over a

wide range of ambient temperatures (−291.4◦F≤ T a≤ 108.6◦F

(−179.67◦C≤ T a≤ 42.56◦C)), plotted as a function of the film

temperature, expressed in terms of the surface and ambient

tem-perature (T s , T a) and then curve fitted using TableCurve 2D [7]

to obtain the following expression:

C1= 0.3268329 − 9.549880x10−5(T

s + T a) (5c)

where the correlation coefficient r 2= 0.995251 For potential

coding purposes, no decimal points truncations were made in

the coefficients of the curve fit correlations Furthermore, the

accuracy of these coefficients will be justified for calculating

time to freeze calculated in seconds at high wind speeds and

low ambient temperature

Substituting Eqs (1), (2), (4), and (5a) into Eq (3), and

solving for the wind chill temperature gives

a



 1/(1+φ)

(6)

where C1is calculated from Eq (5c) Equation (6) is the most

basic and unprecedented expression for the wind chill

temper-ature equation As shown, it is based on the Newton’s law of

cooling and the Stefan–Boltzmann law for radiation heat

ex-change between two surfaces The conductive and radiative heat

transfer quantities (k andε) in the equation depend upon the

ma-terial properties of the surface, while the forced convective heat

transfer coefficient (h fc) does not Therefore, by substituting an

expression for h fcinto this equation, it can be made applicable toall two-dimensional surfaces This was done by selecting con-vective heat transfer coefficients for a flat surface segment from

an aerodynamic heating document by Harms et al [8] Thisdocument expresses the incompressible form of the forced con-

vection coefficient (h fc ) in terms of the Nusselt (Nu), Reynolds (Re), and molecular Prandtl (Pr) numbers The forms are

Nu = h f cx/k = 0.332 (Rex)0.5 (Pr)0.33 (7a)for laminar flow and

terms of the following reference temperature T /suggested byEckert [11]:

T / = 0.5 (T s + T a)+ 0.22 [(γ − 1)/2] M2Ta (7c)

In this expression, the first term on the right side represents astatic temperature component and the second term is a recoverytemperature or a dynamic temperature component expressed as

a function of the Mach number (M) By using T /as the ence temperature, the resulting forced convection heat transfer

refer-coefficients for laminar flow (h fc,l ) and turbulent flow (h fc,t) are

applicable to all velocities (V) and all surface temperatures (T s)

For wind chill calculations, where generally M < 0.1, the namic term in T / can be neglected Substituting for Re ( ρVx/µ

dy-for vertical plate orρVD/µ for a cylinder) in the preceding

equa-tion, replacing the density in terms of pressure using the idealgas law (ρ = P/(/MW)T), and comparing the compressible

with the incompressible h fc, it can be shown, with some ulations and keeping track of English units, that Eqs (7a) and(7b) would reduce to the following forms of the compressibleforced convection coefficient: for the laminar flow along thesegment,

where the subscripts fc, l, and s stand for the forced

convec-tion, laminar flow, and vertical surface, respectively Similarly,

the subscripts fc, t, and s stand for forced convection, turbulent

flow, and vertical surface, respectively The preceding equations

are given in English units The coefficients 0.00963 and 0.0334

are dimensionless constants Specifically, P is the ambient

pres-sure (lbf/ft2), the ambient (T a ) and surface temperature (T s) are

both expressed in◦F absolute (or◦R), the flow velocity (V) is

expressed in ft/s, and the characteristic length L is expressed in

ft Substituting for the variables into Eq (8) in the preceding

units would yield h fcin the units of Btu/hr-ft2- F The resulting

value for h fccan be converted to SI units to obtain W/m2- C by

multiplying by 5.6784 In this form, values of h fccan be readily

expressed in terms of the variables (P, V, T a , and T s), which

are the primary variables in this study The characteristic length

L (vertical segment length) used in the preceding equations

ap-plies to the situation where forced convection is visualized as

flow in a direction perpendicular to the vertical segment,

bifur-cating and then flowing parallel to the segment (Hiemenz flow)

Flow along a portion or possibly the entire length of the

skin surface will be laminar Substitution of the laminar forced

convection coefficient in Eq (8a) into Eq (6) gives

this equation, the length L is the characteristic length for

natu-ral convection panatu-rallel to the vertical segment The coefficient

0.00963 is a dimensionless constant

The laminar forced convection coefficient (h fc l,s) for the

two-dimensional segment shown in Eq (8a) and used in Eq (9) for

the wind chill temperature must now be replaced with the

corre-sponding equation for a cylinder that simulates the human head

At this point it should be noted that for a cylinder there is no

need for an equivalent expression of the turbulent forced

con-vection coefficient (h fc,t,s) as shown in Eq (8b) since laminar

flow will extend circumferentially outward to about 80 degrees

on either side of the wind stagnation point This laminar region

is essentially the entire facial portion of the head that is directly

exposed to the wind From the mentioned aerodynamic heating

document by Harms et al [8] and Chapman [6], the

incompress-ible form of the laminar forced convection coefficient (h fc) for

this laminar stagnation region on a vertical/horizontal cylinder

is

Nu = h f c D/k = 1.14 (Re)0.5 (Pr)0.4 (10a)

where the characteristic length (D) in the Re number is the

cylin-der diameter (D) This equation is a special case of the general

equation for forced convection over a cylinder, Nu D = h D/k

= C (Re D)m (Pr) n Selecting the exponent m to be 0.5 reduces

the general relation to laminar flow Furthermore, selecting the

coefficient C to be 1.14 reduces the general relation to the

lam-inar flow at the stagnation point Thus, this special case applies

to laminar stagnation heat transfer for all velocities Neglectingthe compressibility effects and the usage of the reference tem-

peratures (T /) of Eq (7c), as done before, Eq (10a) reduces tothe following form for the laminar forced convection coefficientfor a cylinder as used in this study,

h f c ,l,c= 0.03238 (PV )0.5

[0.5 (Ts + T a)]0.04 D0.5 (10b)where again in these equations P is the ambient pressure (lbf/ft2),

the ambient (T a ) and surface temperature (T s) are both expressed

in ◦F absolute (or◦R), the flow velocity (V) is expressed in ft/s, and the characteristic length (D) is the cylinder diame-

ter expressed in ft The coefficient 0.03238 is a dimensionlessconstant The similarity of this expression to that for the two-dimensional segment as shown in Eq (8a) is striking Suppose

the diameter (D) of a cylinder is equal to the length (L) of a segment Then the ratio of the coefficients (h fc,l,c /h fc,l,s), is 3.36,which shows that the forced convection cooling of the cylinderwith its longitudinal axis normal to the wind direction is 3.36times greater than that for a two-dimensional surface aligned so

as to be parallel to the wind Perhaps this explains why a personfacing into the wind on a cold wintery day may instinctivelyturn his head to the side to lessen the cold sensation Replacing

Eq (8a) with Eq (10b) in Eq (9) gives

on the well-known Newton’s and Stefan–Boltzmann laws forconvection and radiation heat transfer, respectively Therefore,this expression for the wind chill temperature constitutes ananalytical, exact, closed-form solution

In Eq (11a), the surface emissivity (ε) of the human head wasdetermined to be 0.8 based upon a dynamic model developed

by Fiala et al [12] in which the human response to a cold,cool, neutral, warm, or hot environment was evaluated Thehead cylinder is viewed as being vertical with its longitudinal

axis normal to the wind The quantity L is the height of the head cylinder expressed in feet and D is its diameter in inches.

Because it can be demonstrated that an adult human head can beclosely approximated by a 7-inch (17.78 cm) diameter cylinderthat is 8.5 inches (21.59 cm) in length, the selected dimensions

were D = 7 in and L = 0.71 in These are essentially the

same dimensions as used by Bluestein and Zecher [2] in the

development of their wind chill model It is to be noted that L and D were used as characteristic lengths for natural and forced

heat transfer engineering vol 31 no 14 2010

convection heat transfer, respectively All temperatures (T wc , T s,

Ta) are expressed in◦F absolute (or◦R) and C1is determined

from Eq (5c) At this point, exponentφ is unknown

It was noted in Eq (5a) thatφ would be 0.25 if the source of

heat were within the skin surface itself In this instance where

the heat is being conducted through the surface from inside to

outside, as from the dermis layer through the epidermis in Figure

1,φ is expected to take on other values This value of φ was

determined by first developing a second equation for the wind

chill temperature for the case of natural heat convection in a no

wind (V= 0 mph) environment In this case, the facial surface

heats the adjacent air layer causing its upward convection This

knowns T wcandφ, each of which can now be determined in

terms of the only remaining variable, the air temperature T a

The values ofφ were determined based on the initial decay

curves of the calculated T wcvsersus low wind speeds (0< V

(mph)< 4 (0 < V (km/h) < 6.4)) for ambient temperatures

con-sidered A plot ofφ as a function of T awas curve fitted using

TableCurve 2D [7] to obtain the following expression forφ:

where a = 0.46259934, b = 0.077254543, and c = −59.573525

and where the correlation coefficient is r 2= 0.998884458 For

potential coding purposes, no decimal points truncations were

made in the coefficients of the curve fit correlations

Further-more, the accuracy of these coefficients will be justified for

calculating time to freeze calculated in seconds at high wind

speeds and low ambient temperature The wind chill

temper-ature (T wc ) at a given ambient temperature (T a) can now be

determined from Eq (11a) after first determiningφ from Eq

(11c)

Bluestein–Zecher Model

The Bluestein–Zecher [2] model was intended to correct the

Siple and Passel [1] model for four major flaws attributed to

it by the earlier investigators However, its authors recognized

that only one flaw was valid This was Siple and Passel’s

incor-rect assumption that the skin surface temperature (T s) remains

constant during the entire period of the skin’s exposure The

con-sequence of this assumption was that the Siple and Passel model

would predict wind chill temperatures (T wc) that would be lower

or colder than the expected theoretical values Recognizing that

the skin surface temperature cannot remain constant with

in-creasing exposure time but rather must decrease, Bluestein and

Zecher attempted to determine this decrease through

imple-mentation of what they called “modern heat transfer theory.”Such a decrease would mean that their wind chill temperatureswould be higher or warmer than the Siple and Passel values.But Bluestein and Zecher failed to show this expected decrease

in the skin surface temperature This was the first error in theirmodel

In the development of their model, Bluestein and Zecheralso attempted to correct for what the JAG/TI believed to be

“overestimates” of the wind in the Siple and Passel model Thewind velocity in the Siple and Passel model was the free-stream

velocity (V) measured at the NWS 10-m level This was thought

to be too large a velocity to be used, since one might expect somereduction in the velocity at the head due to its proximity to theground In response to this, Bluestein and Zecher made the as-

sumption that the free-stream velocity (V) as measured at the

NWS 10-m level would always be 50% greater than the velocity(v) at head level The magnitude of this assumed wind reductionwas immediately questioned Their inference that this large 50%assumption is valid for all individuals regardless of their locationand the magnitude of the wind velocity was known to be totallyinaccurate Head-level wind reduction is the result of the head’simmersion in either a wind-generated boundary layer or withinthe flow separation region on the leeward side of an obstructionupwind of the individual In either case, the degree of immersion,which in turn determines the level of wind reduction, dependsentirely upon the individual’s location relative to an obstacle and

on the magnitude of the wind velocity Consequently, Bluesteinand Zecher’s 50% across-the-board assumption for all cases isincorrect This was the second error in their model

The following subsections entitled “Surface Temperature ror” and “Wind Reduction Error” will explain in detail the basisfor each error

Er-Surface Temperature Error

Bluestein and Zecher were aware of the fact that Siple and

Passel’s assumption of a constant skin surface temperature (T s)

would mean wind chill temperatures (T wc) colder than the oretical values Equation (11a) confirms this as an actuality On

the-the right side of Eq (11a), the-the skin surface temperature (T s)will always be about 91.4◦F (33.0◦C) upon initial exposure toambient conditions; this is 7.2◦F (4.0◦C) less than the normalbody core temperature of 98.6◦F (37◦C) For a given set of

ambient conditions (P, V, T a), if the skin surface temperature

(T s) is 91.4◦F (33.0◦C), the wind chill temperature (T wc) takes

on a specific value from Eq (11a) Now suppose that after anextended exposure the skin surface has cooled to a temperature

of 85◦F (29.44◦C) Then for the same ambient conditions, Eq

(11a) shows a wind chill temperature (T wc) that is warmer than

corresponding value of T wcwhen the skin surface temperaturewas 91.4◦F (33.0◦C) Bluestein and Zecher failed to theoreti-cally determine this warming in the wind chill temperature due

to a skin surface temperature decrease and no mention of theseresults is found in their paper Instead they added the windheat transfer engineering vol 31 no 14 2010

reduction assumption that the free-stream velocity at the NWS

10-m level is 50% greater than that at head level By doing this,

Eq (11a) shows that this large reduction in wind velocity (V)

would lead to a sizable warming of the wind chill temperatures;

this is exactly what Bluestein and Zecher demonstrated

The Bluestein–Zecher model was expected to predict warmer

wind chill temperatures than the Siple and Passel model as a

re-sult of allowing the skin surface temperature to vary, that is,

decrease with increasing exposure time, and by introducing the

wind reduction at head level With this skin temperature

correc-tion and the 50% wind reduccorrec-tion factor, the Bluestein–Zecher

model does indeed predict wind chill temperatures that are as

much as 15◦F (8.33◦C) warmer than the corresponding Siple

and Passel values This is shown in Figure 2 where the Siple

and Passel values and the Bluestein and Zecher values were

ob-tained from Table 1 and Table 2, respectively, of the Bluestein

and Zecher paper [2] The contribution of the skin temperature

correction to the 15◦F (8.33◦C) wind chill temperature increase

was found by “removing” the effect of the wind reduction from

this increase To do this, the wind reduction at head level was

first defined in the following subsection, “Wind Reduction

Er-ror,” as a wind reduction factor (WRF) In this subsection, the

assumption that the free-stream velocity is 50% greater than

that at head level is shown to correspond to WRF= 0.33 Each

of the Siple and Passel values of velocity in Table 1 [2] was

then reduced by this value of the WRF, the corresponding wind

chill temperatures determined, and we compared them with the

Bluestein and Zecher values of Table 2 [2] This was done in

the following manner The Siple and Passel [2] results of Table

1 [2] are defined by this equation:

T wc = 91.4 − 0.04544[10.45

− 0.447V + 6.6858V0.5](91.4 − Ta) (12)

as obtained from Morgenstern [13] where it is shown as Eq

(12.4.47) In this equation V is in mph and T wc and T a are in

◦F Introducing the wind reduction factor, WRF, into Eq (12)

yields the following equation:

Twc= 91.4 − 0.04544{10.45−0.447(1 − WRF)V

+ 6.6858[(1 − WRF)V ]0.5 }(91.4 − T a) (13)

With WRF = 0.33 in Eq (13), the wind chill temperature (T wc)

was calculated for every ambient temperature (T a) in Table 1 [2]

with the results plotted in Figure 3 along with the Bluestein and

Zecher results of Table 2 [2] The difference between the two

sets of results in Figure 3 represents the contribution of the skin

surface temperature correction to the wind chill temperature

in-crease Upon examination of these results, it is apparent that

the Bluestein–Zecher model shows, at most, a possible

warm-ing of 2◦F (1.11◦C) in the wind chill temperature due to a skin

temperature correction; in fact, it shows no warming at all at

the coldest ambient temperature of T a= −40◦F (−40◦C) This

-120 -100 -80 -60 -40 -20 0 20 40 60 80

temper-ample will be cited Consider winter conditions with T a= 0◦F(−17.78◦C) and V= 20 mph (32.19 km/h) Upon initial expo-sure to the ambient, the individual’s skin temperature is 91.4◦F(33.0◦C) From Eq (11a), his initial wind chill temperature is

Twc= −54.2◦F (−47.89◦C) Now suppose the individual wasunfortunate enough to be continuously exposed to the ambientuntil his skin surface temperature reached the freezing point

of 32◦F (0◦C) Now his wind chill temperature from Eq (11a)

would be T wc= −40.2◦F (−40.11◦C) This shows a large 14◦F(7.78◦C) warming of the wind chill temperature and serves toemphasize the error in Bluestein–Zecher model, which shows,

at most, 2◦F (1.11◦C) of warming

heat transfer engineering vol 31 no 14 2010

∆T wc , o F ( o C) 1.8 (1)

1.3 (0.7)

0 (0)

Figure 3 Comparison of Bluestein and Zecher [2] results with those of Siple

and Passel (with WRF= 0.33 [1]).

Wind Reduction Error

The assumption that the wind velocity at the NWS 10 m

level will always be 50% greater than that at head level is the

more serious of the two errors in the Bluestein–Zecher model

It is incorrect in two respects First, the 50% magnitude of

this wind reduction is excessively high and will almost never

exist Second, the presumption that this reduction applies to

individuals in all locations and in all ambient conditions is totally

wrong The following discussion demonstrates this

Bluestein and Zecher correctly assumed that the velocity at

head level would be less than that at the NWS 10-m level They

chose to use a 50% reduction factor based on a 1971 study

by Steadman [14], who, in turn, determined this value based

on a study by Buckler [15] In 1969 in the area of Saskatoon,

Saskatchewan, Buckler measured wind velocity (v) as a function

of height (y) above ground level and found this relationship for

the velocity:

v= v10(y /y10)0.21 (14)

where v10 is the free-stream velocity at the NWS 10-m level

and y10 is the height of the wind sensing device at the 10-mlevel Steadman then used Eq (14) to determine the averagewind velocity (¯v) over the height of a human body and foundthis average velocity to be 57% of the free-stream value Based

on this value, Bluestein and Zecher chose to assume the averagevelocity to be 67% of the free-stream value at head level This

33% reduction (or WRF= 0.33) is another way of stating theirassumption that the free-stream velocity is 50% greater thanthat at head level There is no dispute with Buckler’s empiricalresults or in Steadman’s evaluation of these results; neither isthere any dispute with Bluestein and Zecher’s subsequent 50%assumption based upon these results What is disputed is thatthis 50% assumption cannot and does not apply to individuals

in all situations and in all ambient conditions This is explained

in the following discussion

What Buckler fortuitously measured was a wind-generatedboundary layer as evidenced by the fact that his boundary layerwas 33 ft (10.06 m) thick and its edge coincided with the NWS10-m level This is clearly shown in Eq (14) Its thicknesscertainly could be expected in the Saskatoon area where thewind can blow unobstructed for miles over flat and open prairie.What appears to have been unwittingly missed by all involved

in the determination of this wind reduction factor is the factthat the boundary-layer thickness is directly proportional to thewind–ground surface contact length Reduce this length and theboundary layer thickness is reduced Reducing the thicknesschanges the velocity (v) in Eq (14) which then reduces the

wind reduction factor (WRF) Suppose, for example, Buckler

had chosen to make his measurements at 1000 ft (304.80 m)upwind from where his measurements were actually made Thiswould have resulted in a shortened contact length and a boundarylayer thickness less than 33 ft (10.06 m) Steadman’s calculationwould then have determined the average velocity over the body

to be greater than 57% of the free-stream velocity, that is a valuecloser to the free-stream velocity Consequently, Bluestein andZecher might now have concluded that the free-stream velocitywould be only slightly greater than that at head level It is noweasy to visualize Buckler having made his measurements so farupwind from his actual measurement station that the boundarylayer thickness he measured became equal to the height of thebase of a man’s head above the ground This would mean no

reduction in free-stream velocity at head level or WRF= 0 Sincemoving upwind is the equivalent to a shortening of the wind-ground surface contact distance and since the starting point orthe origin of this length will always be at some point downwindfrom an obstruction, then moving upwind is like moving closer

to the obstruction Thus moving closer to the obstruction reduces

or eliminates the WRF In real life, most individuals will be

relatively close to some obstruction such as trees, or buildingsand not far removed from an obstruction such as an individualmight be on the open windswept plain of Saskatoon As a result,such an individual will experience a small and possibly non-

existent WRF, certainly not a WRF = 0.33 This convincinglyshows that the wind reduction depends on the location of theheat transfer engineering vol 31 no 14 2010

individual relative to an obstruction It also depends on the wind

velocity (V) which is a key factor in determining the boundary

layer thickness Therefore Bluestein and Zecher’s implication

that the head level wind reduction is independent of these factors

is unquestionably incorrect

Bluestein and Zecher obviously believed that their 50%

as-sumption was valid since it was based on test data Apparently

they did not recognize the fact that the wind reduction factor

must be determined from the boundary layer in which the

indi-vidual may be immersed Since this boundary layer is dependent

upon the magnitude of the wind velocity and the individual’s

lo-cation relative to an obstruction, it must be calculated in each

sit-uation This could have been done through a detailed boundary

layer flow analyses This procedure is described in the following

paragraphs

Head level wind reduction can be determined whenever the

head is immersed in either (a) a turbulent region on the leeward

side of a wind obstruction or in (b) a turbulent boundary layer

generated by the wind The latter, which is possibly the more

likely situation to occur, fortunately is the one that lends itself

more easily to analysis providing the following information is

known:

• the location of the boundary layer edge relative to the

indi-vidual’s head

• the velocity profile within the boundary layer

Unfortunately this information is so dependant upon an

indi-vidual’s surroundings that an exact evaluation of the head level

wind reduction may never be possible but this is no reason for

universally applying a single incorrect value of wind reduction

to all situations Head immersion in the boundary layer can be

easily determined as the following discussion shows

Wind blowing along the ground surface experiences a

re-tarding action by friction resulting in a layer called the velocity

boundary layer Within this layer, the velocity increases from

zero at the surface to its free-stream value (V) at the boundary

layer edge (δ) If an individual’s head is within the boundary

layer, it will experience a velocity (v) less than the free-stream

value (V) If a wind reduction factor (WRF) is defined as

then in this particular case, WRF > 0 represents a case where the

head is experiencing a wind reduction The thickness (δ) of this

boundary layer is a function of the free-stream velocity, the air’s

kinematic viscosity (v), and most importantly the length (l) that

the wind is in contact with the surface Another variable affecting

the thickness is the surface roughness, but this is not easily

determined What is known is that this roughness guarantees

that the flow in the layer to be turbulent and that an increase in

roughness will increaseδ Based on all this, it can be stated that

an individual exposed to a free-stream velocity will encounter

a turbulent boundary layer thickness (δ) that is dependent on

the individual’s surroundings such as the surface roughness on

his windward side and upon the wind/surface contact length (l).

l

D

x

Turbulent Boundary Layer

Obstruction

V

δ

Figure 4 Wind generated turbulent boundary layer.

Consider the schematic shown in Figure 4 where an obstructionsuch as a fence, a tree, or a group of buildings is located at a

distance (D) from an individual facing into the oncoming wind The wind at velocity V approaching the obstruction will separate

from the surface and flow over the obstruction to produce avortex separation region on its downstream or leeward side.This separated flow region will re-attach to the surface at a

distance (x) from the obstruction It is the distance l = D−x

that is the critical length in the determination of the boundarylayer thickness (δ) at the individual’s location and whether ornot the individual’s head is immersed in it If it is not, then thehead height, which is the distance of the base of the head abovethe ground surface, is greater thanδ and the head is exposed to

free-stream conditions so that WRF= 0 This is the situation inFigure 4 Clearly this illustrates the importance of knowing thevalues ofδ for a range of velocities (V) and wind/surface contact distances (l) that might be encountered Without knowingδ, the

WRF cannot be determined, and if it cannot be determined,

it cannot be correctly estimated When Bluestein and Zecherassumed that the free-stream velocity was 50% greater than

that at the head level, they were essentially assuming a WRF=(1.5−1)/1.5 = 0.33 The following paragraphs describe analysesthat could have been conducted by Bluestein and Zecher to

determine a more accurate WRF.

The analyses are based on the boundary layer concept.Schlichting [16] shows that variation in the boundary layerthickness (δ) for the turbulent flow is

δ/l = 0.37[(V l)/ν] −0.2 = 0.37(Re l)−0.2 (16)where the kinematic viscosity (ν) of air is 150 × 10−6 ft2/s(13.94× 10−6m2/s) With the velocity in the Reynolds number

(Re l) expressed in mph, Eq (16) becomes

Figure 5 shows δ as a function of V and l as expressed by

this equation Assume the base of an individual’s head is atthe 5 ft (1.52 m) level Each circled intersect point represents

the maximum wind/surface contact distance (l max) at a givenvelocity where the base of the individual’s head would be at the

boundary layer edge For example, this means that when V =

40 mph (64.37 km/h), the maximum distance (l max) is 648 ft(197.51 m) and the head is above the boundary layer edge andheat transfer engineering vol 31 no 14 2010

Figure 5 Turbulent boundary layer thickness ( δ) as a function of wind speed

(V) and wind/surface contact distance (l).

the WRF = 0 For distances greater than (l max), the individual’s

head is partially or completely immersed in the boundary layer

and although WRF > 0, its theoretical value may be very small.

As V increase, l maxincreases This is more clearly demonstrated

in Figure 6 where the l max curve is a cross plot of the intersect

points in Figure 5 At any given velocity in Figure 6, for values

of l < lmax, there is no immersion of the head and WRF= 0; for

l > lmax there is immersion and WRF > 0 It should be pointed

out that Eq (17) applies to a smooth surface such as a paved

road, sidewalk, or airport runway For other surfaces where a

roughness exists due to small objects or vegetation, values of

δ would be expected to be slightly larger This small increase

would shift the curves of Figure 5 slightly upward, thus reducing

the l max value at a given velocity; it would shift the (l max) curve of

Figure 6 slightly downward Lacking the information required

to correct for this slight difference, the l maxcurve of Figure 6 is

presumed to be sufficiently accurate for all surfaces

Figure 6 shows l max as the defining distance downstream of

an obstruction that determines whether or not the individual’s

head is immersed Because it is advantageous to reference this

defining distance to the individual’s actual distance (D) from

the obstruction, D is determined by adding the separated flow

re-attachment distance (x) to l max as shown in Figure 4 The

problem here is that x is not a fixed quantity but rather increases

500 600 700 800 900

Figure 6 Maximum wind/surface contact distance (lmax ) and obstruction

dis-tance (D) for turbulent boundary layer thickness (δ) of 5 ft (1.52 m).

with the height of the obstruction and increases as V increases.

At this point, the assumption was made that an obstruction such

as a tree or building will produce a downstream flow

separa-tion distance (x) of 100 ft and 30 ft (30.48 m and 9.14 m) at

velocities of 70 mph (112.65 km/h) and 20 mph (32.19 km/h),respectively Linearly spreading these distances over the veloc-

ity range and adding them to the l maxdistances in Figure 6 gives

curve D, the approximate distance of the individual from the obstruction This distance D is approximate because the two

separation distances (100 ft (30.48 m), 30 ft (9.14 m)) for thetwo velocities (70 mph (112.65 km/h), 20 mph (32.19 km/h))apply to one specific obstruction height and not for all heights

as assumed here However, the variation in distance x with struction height is believed to be a small fraction of l max sothat ignoring this effect should not result in any significant er-

ob-ror Distance D can now be considered for all conditions as an

approximate, although realistic, distance between an individualand the obstruction that determines the flow field in his presence

Curve D, now replacing curve l max , separates the region WRF

> 0 above it from the WRF = 0 region below From this one concludes that if the individual is within a distance D of 580 ft

(176.79 m) to 850 ft (259.08 m) from an obstruction over the 20mph to 70 mph (32.19 km/h to 112.65 km/h) velocity range, he

will still be exposed to free-stream conditions, that is, WRF=

0 It is believed that this represents a large majority of real-lifesituations in which the currently used 50% wind reduction isincorrectly applied

With reference to Figure 6, if an individual at a given velocity

(V) is at a distance greater than D from an obstruction, his

head will be partially or completely within the boundary layer

In his case, the WRF must be determined The WRF can be

computed from the velocity profile within the boundary layer.From Schlichting [16], this profile in a turbulent boundary layeris

where, as stated earlier, v is the velocity at head level, V is the

free-stream velocity at the boundary layer edge (δ), and y is the

head height above ground level, and where the exponent 1/n

depends on the surface roughness and the free-stream velocity

From its definition in Eq (15) and using Eq (18), the WRF can

heat transfer engineering vol 31 no 14 2010

5765 ft (1757m)

Figure 7 Wind reduction factor (WRF) as a function of wind speed (V) and

distance (D) using Steadman’s exponent (1/4.76) [14].

be expressed in the following manner:

ifδ > y, WRF = 1 − v/V = 1 − (y/δ)1/n (19a)

and

WRF can be calculated as a function of distance D in the

follow-ing manner For velocities (V) spannfollow-ing the wind chill range,

select values of l and calculateδ from Eq (17) Then calculate

WRF from Eq (19) using y= 5 ft (1.52 m) and Steadman’s

value (1/4.76) of the exponent The results are plotted in Figure

7 where the wind/surface contact distance (l) has been replaced

by distance D from an obstruction They show that at V= 20

mph (32.19 km/h) the individual must be at a distance D > 5765

ft (1757.19 m) from an obstruction if the WRF is to be 0.33 For

V= 70 mph (112.65 km/h), the corresponding distance would

be 7885 ft (2403.38 m) These very large distances could exist

in sparsely populated rural areas but would be not too likely

in urban areas Since the area surrounding Saskatoon certainly

qualifies as sparsely populated, these large distances are in

per-fect agreement with Buckler’s measurements of the boundary

layer edge at the NWS 10 m level

The message conveyed by Figure 7 is that for large

unob-structed distances (D) consistent with a sparsely populated rural

area like that around Saskatoon, the large value of WRF= 0.33 is

possible However, only a very few individuals might be present

to experience it For smaller unobstructed rural distances

asso-ciated with a more heavily populated area, the values of WRF

are much less Figure 7 shows that individuals within a distance

of 580 ft (176.79 m) to 850 ft (259.08 m) of an obstruction will

not experience a wind reduction; for them, WRF= 0

Individu-als at slightly greater distances will experience some reduction

although it will be small It would seem reasonable to assume

that a majority of people in a rural area might be within 1500 ft

(457.21 m) distance (approximately 1/4 mile) of an obstruction

From Figure 7, individuals at D= 1500 ft (457.21 m) will

ex-perience wind reduction factors varying from 0.111 at V= 70

mph (112.65 km/h) to 0.155 at V= 20 mph (32.19 km/h) The

average values of WRF for this rural area for distances between

580 ft (176.79 m) and 1500 ft (457.21 m) would be 0.055 at V=

70 mph (112.65 km/h) and 0.078 at V= 20 mph (32.19 km/h)

These average WRF values of 0.055 to 0.078 are so much lower

than the Bluestein and Zecher’s value of 0.33 that this value is

no longer viable

The discussion so far has dealt with the wind reduction factor

WRF in instances where the head is immersed in a wind

gener-ated turbulent boundary layer These cases represent situationsthat are relatively simple to analyze In the other cases, where theindividual is within the turbulent region downstream of an ob-

struction, the determination of the WRF is more difficult Only

when the individual is very close to the obstruction and totally

within the flow separation region (x), as shown in Figure 4, can

it be said that he is completely shielded from the wind, in which

case WRF = 1 This is the only value of WRF that is clearly

defined when considering obstructions As the individual in theseparation region shown in Figure 4 moves away from the ob-

struction and toward the wind reattachment point (x), the WRF will decrease becoming WRF= 0 somewhere before reaching

x Because the WRF varies from 0 to 1.0 within this separation

region, there is likely to be at least one location within this

re-gion where WRF= 0.33 This is the only other instance where

Bluestein and Zecher’s value of WRF= 0.33 would correctlyapply

The preceding discussion has proven the inaccuracy of a

WRF= 0.33 in rural areas Now look at an urban area uals in an urban area may be subjected to a combined effect of aboundary layer and one or more separation regions An individ-ual located at some distance downstream from an obstruction on

Individ-a cleIndivid-ar street with buildings on either side mIndivid-ay experience only

a boundary layer If the wind/surface distance (l) along the street

is 1500 ft (457.21 m) or less, which is comparable to about two

standard city blocks, then the WRF values for the two velocities

(20 mph, 70 mph (32.19 km/h, 112.65 km/h)) will be the same(0.111, 0.155) as in the above rural region If the individual is

within this 1500 ft (457.21 m) distance, the WRF might take on

the same average values (0.055, 0.78) of the rural region Now

if vehicle signs, lampposts, and other obstructions exist alongthe street, the individual may experience an interrupted bound-ary layer due to an exposure to one or a series of separation

regions Determining a WRF here would be nearly impossible.

Unless the individual manages to become completely sheltered,

in which case the WRF= 1, the theoretical wind reduction factor

WRF would very small if these turbulent regions are

individu-ally separate and none makes contact with the individual In this

case, choosing WRF= 0 would be the logical choice

There are two situations where a wind reduction at head level

is the result of a modification of the NWS 10m velocity valueand not the result of the individual being immersed in a windgenerated boundary layer The first is an increase in the NWS10-m free-stream value that could occur in a large urban area as aresult of what Schwerdt [17] referred to as “air funneling aroundtall buildings.” This increase could be computed knowing thesize, number, and the layout of the buildings The second refers

to a case where the NWS 10m free-stream value of the velocitywould be decreased Picture the previous illustration of the windblowing down the 1500 ft (457.21 m) length of street, beingheat transfer engineering vol 31 no 14 2010

deflected 90 degrees around a building and then continuing

to flow down a cross street The deflected wind including its

boundary layer would generate a turbulent region on the cross

street side of the building with a subsequent re-attachment to the

surface at a distance (x) downstream of the turning point Energy

losses incurred by the flow as a result of this turning would be

reflected as a reduced value in the free-stream velocity (V) itself

after the turn as compared with its value before the turn As

before, at distances of l < lmaxdownstream of the re-attachment

point (x), the WRF = 0 from Figure 6 Now suppose l > l max In

this case there could be a wind reduction due to boundary layer

immersion; however, some of this reduction would be attributed

to the decrease in the velocity of the re-attached flow In this

instance, the decrease in the free-stream velocity due to turning

could possibly be accounted for but its actual determination

would be difficult Combine this situation with the possibility

of the turned flow being accelerated by the already-mentioned

funneling effect and there is a possibility that the velocity of the

turned flow may return to its original NWS 10m value or even

exceed it In that case, the theoretical value of the velocity at

head level may not be too different from the NWS 10m value

Choosing WRF= 0 would again be the logical choice

The following summarizes the results of the wind

bound-ary layer analyses to determine a correct wind reduction factor

(WRF).

1 In a sparsely populated rural area where obstructions may

be separated by miles, the WRF can take on larger values

like 0.33 in the Saskatoon area Although large, this value

realistically applies to only a very small percentage of the

population

2 In a normal more populated rural area where obstructions

may be separated by distances less than 1500 ft (457.21 m),

average values of WRF vary from 0.055 to 0.078 depending

on the wind velocity These values may possibly apply to as

much as 90% of the rural population

3 In an urban area, the WRF can take on numerous values

between 0 to 1 where the latter represents complete

shelter-ing behind an obstruction The most likely scenario is wind

blowing along a street some two city blocks in length The

WRF values here would be the same as in a normal more

populated rural area These values would apply to 100% of

the urban population

Whatever wind chill model is being used by the NWS, it

is anticipated that they would use just one WRF for an entire

regional radio/TV listening area Selection of this WRF for this

region would likely be determined on the basis of the

great-est benefit to the greatgreat-est percentage of the population Based

on the already postulated distributions of the population, this

wind reduction factor should lie in the range 0.055≤ WRF ≤

0.078 This would cover almost 100% of the entire population

(100% urban, 90% rural) If Bluestein and Zecher’s values of

WRF had been within this range, not only the magnitude of

their wind reduction would have been correct but it would have

been valid for individuals in all locations and for all ambientconditions

In summary, the Bluestein–Zecher model was expected topredict warmer wind chill temperatures than the Siple and Pas-sel model as a result of the skin temperature correction andthe presumed wind reduction at head level With its incorrectskin temperature correction and its unrealistic 33% head levelwind reduction, the Bluestein–Zecher model does indeed pre-dict wind chill temperatures that are as much as 15◦F (8.33◦C)warmer than the corresponding Siple and Passel values In this

15◦F (8.33◦C) increase in wind chill temperatures, 2◦F (1.11◦C)was due to their failure to show a greater increase in this temper-ature due to the skin surface temperature decrease The balance

of 13◦F (7.22◦C) was due to their failure to apply a correct windreduction factor Without these errors, the Bluestein–Zechermodel is essentially no different than the Siple and Passel model

it was intended to replace When this model was being used itmisinformed the public of wind chill temperatures that were

at least 15◦F (8.33◦C) warmer than the theoretical values tions like this can lead to complacency on the part of the publicwhereby they become less concerned about the possibility offacial freezing when in reality they are at a greater risk becausefreezing may be imminent

Ac-Osczevski–Bluestein Model

The currently used Osczevski–Bluestein model as described

in the JAG/TI [3] and [4] papers was intended to improveupon the earlier Bluestein–Zecher model But no such improve-ment took place In fact, reference [4] has identical wind chilltemperature curve fit equations and resulting wind chill tem-perature charts to that of reference [3] On the contrary, theOsczevski–Bluestein model has even less credibility than theBluestein–Zecher model, which, at least, reverts back to theSiple and Passel model when its two errors are removed This isnot the case for the Osczevski–Bluestein model, which not onlyincludes the same wind reduction error as the Bluestein–Zechermodel but also introduces an errant concept of the term “windchill temperature” that actually compounds the inaccuracy thatalready exists in the Bluestein–Zecher model This is explained

perature (T s) is being sensed, is at the interface of the dermis andepidermis layers as shown in Figure 1 Therefore, the thermalresistance layer described by the authors is the approximate 1

mm thick epidermis and the sensed temperature (T i) is the

tem-perature at the interface Although T iis the temperature that theindividual senses, it definitely is not the wind chill temperature

(T wc) This is visually apparent from Figure 1b and

mathemat-ically obvious from Eq (4) where T wc is determined in termsheat transfer engineering vol 31 no 14 2010

of T s and not T i This demonstrates, without question, that the

wind chill temperature must be defined in terms of the

theoret-ical skin surface temperature and not the temperature that the

individual senses Furthermore, the sensed temperature is

sub-jective in nature in that the temperature sensed by one person

may be quite different from that sensed by another This is due

to the individual variations in what the authors call the thermal

resistance This was acknowledged in a quote from the JAG/TI

paper [3] where the statement was made that “Cheek thermal

resistance varies considerably among individuals In the human

studies, it varied by more than a factor of two As a result, cheek

temperatures in wind, in general, will differ from person to

per-son.” (p 2) Osczevski and Bluestein’s usage of a 95th percentile

of cheek thermal resistance may have averaged out the

“sub-jectiveness” of this sensed temperature, but it does nothing to

eliminate the error of defining the sensed wind chill temperature

as the actual or theoretical wind chill temperature as defined in

Eq (11a) Since the theoretical wind chill temperature depends

on the skin surface temperature as shown in Eq (4), it is

unfor-tunate that Osczevski and Bluestein did not make a direct use

of the measured skin surface temperatures provided to them by

DCIEM (Defense Research and Development Canada/Defense

and Civil Institute of Environmental Medicine) Chambers [3]

to develop their model Had they done so, rather than using

these measured skin surface temperatures to define the

ambigu-ous sensed temperatures through the use of an average thermal

resistance, they might have developed a wind chill model more

viable than the Bluestein–Zecher model Instead they arrived at

the following equation for the sensed wind chill temperature:

T wc,s = 35.74+0.6215T a −35.75V0.16 +0.4275T a V0.16 (20)

where the sensed wind chill temperature (T wc, s ), which is T i, and

the ambient temperature T aare in◦F and V is in mph This wind

chill temperature (T i) will be much warmer than the theoretical

wind chill temperature (T wc) as defined by Eq (11a) It will

naturally be much warmer than the skin surface temperature

(T s) by virtue of the fact that it is a temperature at an internal

body point These values of T wc, s, have been expressed in a

small chart under the heading “The Wind Chill Factor” and are

presently being disseminated by various means to the American

public Proclaimed by some meteorologists as the “warmer”

wind chill temperatures, this change in its definition will not

make an individual feel warmer The conclusion is that these

wind chill temperatures lack merit This is demonstrated in the

following discussion

Consider an individual exposed to an ambient temperature

Ta= 0◦F (−17.78◦C) and a wind velocity V= 20 mph (32.19

km/h) From Eq (11a), his theoretical wind chill temperature

would be T wc = −54.2◦F (−47.89◦C) The Bluestein–Zecher

model, from Table 2 of their paper, states that the wind chill

temperature is T wc= −27.5◦F (−33.06◦C) But this is 26.7◦F

(14.81◦C) warmer than the theoretical value because of the

er-rors in their model From Eq (20), the Osczevski–Bluestein

model predicts a T wc= −22◦F (−30◦C) This is a value that is

32.2◦F (17.88◦C) warmer than the theoretical wind chill

temper-ature For the same ambient T a= 0◦F (−17.78◦C) but with V=

60 mph the Osczevski–Bluestein model predicts a sensed wind

chill temperature T wc= −33.1◦F (−36.17◦C), which is 78.91◦F(43.84◦C) warmer than the theoretical wind chill temperature

Twc= −112.0◦F (−80.0◦C) as obtained from Eq (11a).These sensed wind chill temperatures as predicted by theOsczevski–Bluestein model cannot in all seriousness be consid-ered to have any real value Continuing to use this model, as

is presently being done, puts the individual at an even greaterrisk than with the Bluestein–Zecher model because of its muchwarmer wind chill temperature predictions This could lead to

an even greater complacency on the part of members of thepublic whereby they no longer consider the possibility of facialfreezing but rather dismiss it altogether

CONCLUSIONS

The Bluestein–Zecher and Osczevski–Bluestein wind chillmodels are both afflicted with errors Their development wasinitiated in an attempt to correct for an error in the Siple andPassel model that was then currently in use Siple and Passel hadincorrectly assumed that the facial surface temperature wouldremain constant and not decrease with increasing exposure time.This assumption would result in the Siple and Passel modelpredicting wind chill temperatures that would be colder than thetheoretical values Warmer wind chill temperatures were desiredand would be guaranteed if the facial surface temperature wasallowed to decrease with time Bluestein and Zecher developed

a new model that was to be an upgrade of the Siple and Passelmodel in which they would account for this decrease in skinsurface temperature They also included a 33% reduction in thewind velocity at head level, which was intended to account forthe head’s immersion in a wind/ground surface boundary layer.The Bluestein–Zecher model failed on both accounts First, itfailed by determining, at most, only a 2◦F (1.11◦C) of warming

in the wind chill temperatures due to the skin surface ature decrease whereas a theoretically derived equation for thewind chill temperature shows the increase to be on the order

temper-of 14◦F (7.78◦C) Second, the assumed 33% reduction in windvelocity at head level was found to be 5 to 7 times larger than itshould have been This error resulted in a 13◦F (7.22◦C) increase

in the wind chill temperature When this model was being used

by the NWS, it was actually misinforming the American publicthat the wind chill temperatures they were experiencing were

15◦F (8.33◦C) warmer than the theoretical values

The currently used Osczevski–Bluestein model was intended

to improve upon the earlier Bluestein–Zecher model that itreplaced No improvement took place First, it assumed thesame 33% reduction in wind speed at head level as in theBluestein–Zecher model Second, it redefined the very meaning

of the term “wind chill temperature” by saying that this

tem-perature is the “sensed” temtem-perature (T i) on the inner surface ofthe outer skin layer, the epidermis, or what the authors call theheat transfer engineering vol 31 no 14 2010

thermal resistance layer Although T i may be the temperature

that the individual senses, it definitely is not the actual wind

chill temperature (T wc), which is a function of the outer surface

temperature (T s) of the epidermis and not its inner surface

tem-perature (T i) This redefinition of wind chill temperature is a

major error Combined with the already errant 33% head-level

wind reduction, the Osczevski–Bluestein model yields values of

wind chill temperature that are much warmer than the incorrect

Bluestein–Zecher values

The real danger associated with using either of these wind

chill models, especially the current Osczevski–Bluestein model,

is the incorrect and unrealistically warm wind chill temperatures

associated with them When these temperatures are conveyed

to the public, especially under severe wintertime conditions,

it could lead to a substantial complacency on the part of the

recipients, who could become much less concerned about the

possibility of facial freezing when in reality they may be at a

greater risk because freezing is imminent

NOMENCLATURE

C1 coefficient in Eq (5)

Cp, Cv specific heats at constant pressure and volume,

respec-tively, J/kg-K (Btu/lbm- R)

D human head diameter or distance, m (ft)

Error possible error upon substitution of T sky with T aas

de-fined in Eqs (25) and (26)

E black body emissive power, W/m2(Btu/h-ft2)

G atmospheric sky emission, W/m2(Btu/h-ft2)

g gravitational constant, m/s2(ft/s2)

k thermal conductivity, W/m-K (Btu/h-ft-◦R)

l wind/surface contact length in Figure 4, m (ft)

L head length, cm (in.)

h convective heat transfer coefficient, W/m2- C

(Btu/h-ft2- F)

M Mach number in Eq (7c)

MW molecular weight, kg/kgmol (lbm/lbmol)

Nu Nusselt number, hx/k (plate) or hd/k (cylinder),

(di-mensionless)

P ambient pressure, atm (psia)

Pr molecular Prandtl number (dimensionless)

q• heat loss flux, W/m2(Btu/h-ft2)

Re Reynolds number, Vx/ν or Vl/ν (plate) or VD/ν

(cylin-der)

r 2 correlation coefficient

s facial thickness of the epidermis in Figure 1, mm (ft)

t time, minutes or seconds

T temperature,◦C (◦F)

T / reference temperature,◦C (◦F) in Eq (7c)

V free-stream wind velocity, km/h (mph)

v wind velocity at head level, km/h (mph)

v10 wind speed measured at 10 m (33 ft) above the ground

in Eq (14)

WRF wind reduction factor (dimensionless)

x characteristic length or separated flow re-attachment

absorp-β coefficient of thermal expansion of the air, K−1( R−1)

δ velocity boundary layer thickness, m (ft)

ε human skin’s emissivity

γ ratio of specific heats, C p/Cv

Proceedings of American Philosophical Society, vol 89,

no 1, pp 177–199, 1945

[2] Bluestein, M., and Zecher, J., A New Approach to an

Ac-curate Wind Chill Factor, Bulletin of American

Meteoro-logical Society, vol 80, no 9, pp 1893–1899, 1999.

[3] Nelson, C A., Tew, M., Phetteplace, G., Schwerdt, R.,

Maarouf, A., Osczevski, R., Bluestein, M., Shaykewich,

J., Smarsh, D., Derby, J C., Petty, R C., Berger, M.,

Quayle, R G., Santee, W R., Olenic, E., Lupo, A R.,

and Browne, K., Review of the Federal Interagency

Process Used to Select the New Wind Chill

Temper-ature (WCT) Index, 18th International Conference on

IIPS, Annual meeting, American Meteorological Society

(AMS), http://www.ofcm.gov/jagti/r19-ti-plan/pdf/entire

r19 ti.pdf and http://ams.confex.com/ams/pdfpapers/

26911.pdf, 2002

[4] Osczevski, R., and Bluestein, M., The New Wind

Chill Equivalent Temperature Chart, Bulletin of

Ameri-can Meteorological Society, vol 86, no 9, pp 1453–

1458, 2005

[5] Jakob, M., and Hawkins, G A., Elements of Heat Transfer,

3rd ed., pp 127–137, Wiley, New York, 1957.

[6] Chapman, A J., Heat Transfer, 3rd ed., Macmillan, New

York, pp 332–363 and 364–388, 1974

[7] TableCurve, Software Package Ver 5.01/2D, 4.0/3D,

SY-STAT Software, Inc., Richmond, CA, 2002

[8] Harms, R J., Schmidt, C M., Hanawalt, A J., and Schmitt,

D A., A Manual for Determining Aerodynamic Heating Of

High–Speed Aircraft, Bell Aerosystems Company, report

no 7006–3352–001, vol 1, pp 14–16, June 1959

[9] Crocco, L., The Laminar Boundary Layer in Gases,

Trans-lation, North American Aviation, Inc., Aerophysics

Labo-ratory, Los Angeles, CA, AL-684, July 1948

[10] van Driest, E R., Turbulent Boundary Layer in

Compress-ible Fluids, Journal of Aeronautical Sciences, vol 18, no.

3, pp 145–160, 1951

[11] Eckert, E R G., Survey on Heat Transmission at High

Speeds, Aeronautical Research Laboratory, report ARL

189, Office of Aerospace Research, Wright-Patterson Air

Force Base, Dayton, OH, 1961

[12] Fiala, D., Lomas, K J., and Stohrer, M., A Computer

Model of Human Thermoregulation for a Wide Range of

Environmental Conditions: The Passive System, Journal

of Applied Physiology, vol 98, 1957–1972, 1999.

[13] Morgenstern, M A., Heating, Ventilation, and Air

Con-ditioning, in Marks’ Standard Handbook for Mechanical

Engineers, eds E A Avallone and T Baumeister III, 9th

ed., McGraw-Hill, New York, pp 1261–12115, 1989

[14] Steadman, R G., Indices of Wind Chill of Clothed

Per-sons, Journal of Applied Meteorology, vol 10, no 1, pp.

674–683, 1971

[15] Buckler, S J., The Vertical Wind Profile of Monthly Mean

Winds Over the Prairies, Canada Department of Transport,

technical memo, TEC 718, Ottawa, Ontario, Canada, pp

1–16, 1969

[16] Schlichling, H., Boundary Layer Theory, 7th ed.,

McGraw-Hill, New York, pp 24–46 1979

[17] Schwerdt, R W., Letters to the Editor, Bulletin of the ican Meteorological Society, vol 76, no 9, pp 1631–1636,

Amer-1995

APPENDIX

The radiation from the skin at a surface temperature T s comes a heat loss to an infinitely large air filled enclosure where

be-the ambient air temperature is T aand where the enclosure

tem-perature, called the sky temtem-perature, is T sky Radiant emissionfrom the facial surface into the enclosure is

sky emission (G sky) is expressed as

•

Assume the skin to be diffusive-gray (α = ε) and the sky to

be isothermal and large compared to a human face; then theatmospheric irradiation is

of the sky temperatures are well defined, its specific value mostlikely would be unknown at the time when q•r is to be de-

termined This problem can be avoided by replacing the skytemperature with the ambient temperature so that Eq (23a) be-comes

but will introduce an error that may be found unacceptable.This error will be evaluated through a comparison of the ratio

(R) of the radiation heat flux ( q•r) to the convection heat flux

(q• f c ) as the change is made from the sky temperature (T sky)

to the ambient temperature (T a) With the convection heat flux(q• f c) as defined in Eq (1), this ratio when using Eq (23a)

with the ambient air temperature (T a) If the absolute value of

Tsky > Ta , then R sky > Ra and since Eq (24a) represents the

theoretical solution with no error, and Eq (24b) represents an

introduction of a possible error, then the error (Error) becomes

In Eq (24), the emissivity (ε) is assumed to be 0.8 for the

human skin and the Stefan–Boltzmann constant (σ) is 1.714 ×

10−9Btu/h-ft2- R4(5.670× 10−8W/m2-K4) The skin surface

temperature (T s) is 91.4◦F (33◦C) This is the expected skin

temperature upon its initial exposure to a cold ambient The

0 500 1000 1500 2000

0 (-17.78)

Tsky ,o F ( o C)

Very low wind speed

.

b) 0 < V (mph) < 0.6 (0 < V (km/h) < 1)

Figure 9 Convection and radiation heat fluxes as functions of h fc at T a= 0 ◦F

( −17.78 ◦C) and sky temperature (T

sky), respectively.

sky temperature (T sky) is assumed to vary from 0◦F (−17.78◦C)

to −40◦F (−40◦C) where this range lies within its limitingrange (53.3◦F to−45.7◦F) The ambient temperatures (T

a) areselected as 0◦F (−17.78◦C) and−70◦F (−56.67◦C), with thelatter temperature being near the lowest expected in wind chilltemperature calculations

The forced convection heat transfer coefficient (h fc) was culated using Eq (10b) It was calculated over a high range ofwind speeds (0< V (mph) < 150 (1 < V (km/h < 241)) and

cal-over a very low range of wind speeds (0< V (mph) < 0.6 (0 < V

(km/h< 1)) Calculated values of hfc for T a= 0◦F (−17.78◦C)are shown in Figure 8a for the high wind speed range and Figure8b for the very low wind speed range

With the earlier values of (h fc), the convection heat flux (q• f c),shown as the denominator in Eq (24) is calculated and shownheat transfer engineering vol 31 no 14 2010

b) 0 < V (mph) < 0.6 (0 < V (km/h) < 1)

Figure 10 Relative magnitudes of the convection and radiation heat fluxes as

functions of h at T a= 0 ◦ F (−17.78 ◦C) and sky temperature (T sky), respectively.

in Figure 9 for each of the velocity ranges Included in Figure 9

are the radiation heat flux (q•r) values, shown as the numerators

in Eq (24) Since Eq (24) represents the two different ratios

of (q• f c/ q•r) necessary to determine the error (E), the ratio is

determined in the following manner: From Figure 8 select a

value of h fc , which will represent a different velocity V in the

two velocity ranges With this value of h fc, determine the value

ofq• f c from Figure 9 along with values ofq•r for each of the

three sky temperatures Calculated values of (q• f c/ q•r) for each

of the sky temperatures is shown in Figure 10 over the entire

range of h fc values and for both ranges of wind speed These

values of (q• f c/ q•r ) can be used to determine the error (Error)

when T a= 0◦F (−17.78◦C).

The above calculations were repeated for a much more severe

ambient temperature of −70◦F (−56.67◦C) using Figures 11,

12, and 13 In this case, T a > Tsky and R a > Rskyso that the error

0 5 10 15 20

Calculation of this error was done in the following manner

when T a= 0◦F (−17.78◦C): In Figure 10, at each value of h

fcinboth wind speed ranges, there are three values of (q• f c/ q•r), one

for each sky temperature The inverse of these values represents

the sky ratio of Eq (24a), that is, R sky = (q• f c/ q•r)−1 The value

at T sky = T a= 0◦F (−17.78◦C) represents the ambient ratio of

Eq (24b), that is, R a = (q• f c/ q•r)−1 From Eq (25) the error(Error) is calculated and the values plotted in Figure 14a for

both wind speed ranges

Similar calculations of the error were done when T a= −70◦F(−56.67◦C) starting with the plotted results in Figure 13 In thiscase, the error was calculated using Eq (26) and the results areplotted in Figure 14b

heat transfer engineering vol 31 no 14 2010

-40 (-40) -20 (-28.89)

-40 (-40) -20 (-28.89)

0 (-17.78)

T sky ,o F ( o C)

Very low wind speed

.

b) 0 < V (mph) < 0.6 (0 < V (km/h) < 1)

Figure 12 Convection and radiation heat fluxes as functions of h fc at T a=

−70 ◦F (−56.67 ◦C) and sky temperature (T sky), respectively.

The error (Error) values in Figure 14 are shown to be the

highest at very small values of h fc and rapidly decay as h fc

values increase, i.e., as wind speeds increase We conclude by

determining the wind speeds required to reach these very high

errors These wind speeds were determined by extracting back

and plotting V versus h fcas shown in Figure 15a These high

values for the error occur for wind speed between 0 and 0.02

mph (0.03 ft/s (0.01 m/s)) and take on values of 20 and 18.5

(shown as solid circles) as shown in Figure 14a and Figure 14b,

respectively These wind speeds are extremely small and

prac-tically immeasurable in outside ambient air The lowest wind

speed used to calculate the time to freeze in our all-inclusive

model was 0.1 mph (0.15 ft/s (0.04 m/s)) At this velocity of

0.1 mph (0.16 km/h), the corresponding calculated heat transfer

coefficient is shown in Figure 11b to be 0.57 Btu/h-ft2- F (3.2

W/m2- C), which is higher than the range shown on the abscissa

in Figure 14 This slight change in the wind speed and the

corre-0 10 20 30

a) 0 < V (mph) < 150 (1 < V (km/h) < 241)

0 1 2 3 4

Figure 13 Relative magnitudes of the convection and radiation heat fluxes

as functions of h fc at T a = −70 ◦F (−56.67 ◦C) and sky temperature (T sky),

respectively.

sponding convective heat transfer coefficient result in a decrease

of this error (Error) in Figure 14a by at least 96% ((20− 0.8)/20)

= 96%) where the values of 20 and 0.8 in the bracket are the

calculated errors and are shown corresponding to h fc = 0.01Btu/h-ft2- F (0.05678 W/m2- C) and 0.25 Btu/h-ft2- F (1.4196W/m2- C), respectively Similarly, using the calculated errors of18.5 and 0.75 shown and corresponding to the same values of

hfcyields a corresponding decrease in the error of 96% ((18.5

− 0.75)/18.5) = 96%) as shown in Figure 14b

These errors (0.8 and 0.75) are relatively small for the imum wind speed of 0.1 mph (0.16 km/h) considered in ourwind chill calculations These errors will dramatically decreaseexponentially with increasing velocity Since velocities two tothree orders of magnitude greater than the minimum are possi-

min-ble in wind chill calculations, the sky temperature T skycan be

replaced with the ambient air temperature T ain Eq (23) withoutintroducing any significant error

heat transfer engineering vol 31 no 14 2010

Figure 14 Calculated possible errors for substitution of T sky by T a in the

radiation heat flux term.

Rashid A Ahmad is a senior technical specialist at

the Fluid Dynamics Section of the Design and ysis, Science and Engineering, Alliant Techsystems (ATK) Space Systems, located in Brigham City, UT.

Anal-He has B.S., M.S., and Ph.D degrees, in mechanical engineering, from the University of Illinois, Chicago.

Prior to joining ATK in January 1987, he spent one year as an Assistant Professor of Mechanical Engi- neering at the Illinois Institute of Technology West Campus, Lombard, IL His research interests include fluid mechanics, heat transfer, solid rocket motor propulsion, and recently me-

teorology He has contributed peer-reviewed articles to scientific journals and

conferences, including the Journal of Numerical Heat Transfer, the AIAA

Jour-nal of Thermophysics and Heat Transfer, Heat Transfer Engineering, JourJour-nal of

Spacecraft and Rockets, and the Joint Army-Navy-NASA-Air Force (JANNAF)

Proceedings He is an AIAA associate fellow and an ASME member He

co-authored the AIAA 1991 Outstanding Thermophysics Technical paper and was

the AIAA Utah Section Chairman for 1994–1995.

0 0.05 0.1 0.15 0.2 0.25

Is velocity measurement guaranteed?

Low or High wind speed

a) T a = 0oF (-17.78oC)

0 0.05 0.1 0.15 0.2 0.25

Is velocity measurement guaranteed?

Low or High wind speed

b) T a = -70oF (-56.67oC)

Figure 15 Required low wind speeds for the calculated possible errors for

substitution of T sky by T ain the radiation heat flux term and shown in Figure 14.

Stanton Boraas, now retired, is the former

supervi-sor of the Aero/Thermal Section at Morton Thiokol Inc./Space Operations, now Alliant Tech Systems (ATK) He received a B.S degree in aeronautical en- gineering from the University of Minnesota in 1949 and pursued graduate studies in the 1950s at the Uni- versity of Minnesota, University of Maryland, and University of Washington His specialty is contin- uum and noncontinuum fluid mechanics He has 40 years of experience in the design and performance evaluation of aircraft, missiles, ground-effect machines (GEMs), and Navy surface-effect ships (SES) Most of his experience has been in liquid/solid rocket propulsion His publications include original work in striated transonic flow, liquid propellant containment in space, practical applications of the hydraulic analogy, modeling of slag in solid rocket motors, exhaust plume contamina- tion from conventional and scarfed rocket nozzles in a space environment, and rocket nozzle contour optimization for a nonuniform gas flow field Publications

were in the Journal of Spacecraft and Rockets, AIAA Journal, and the JANNAF

Proceedings He has been an AIAA member since 1965.

heat transfer engineering vol 31 no 14 2010

Copyright
Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457631003689278

Numerical Investigation of the

Performance of a U-Shaped Pulsating Heat Pipe

S ARABNEJAD, R RASOULIAN, M B SHAFII, and Y SABOOHI

Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran

In this research the performance of a U-shaped pulsating heat pipe (PHP) was investigated using numerical methods This

heat pipe consists of two sections: The evaporator is set at the two ends of the pipe, and the middle part of the pipe comprises

the condenser section This heat pipe is a type of open looped pulsating heat pipe The governing equations are derived

analytically from the continuity, momentum, and energy equations and are solved implicitly In this model, considering the

liquid mesh, the rate of convection and boiling heat transfer in the U-shaped PHP, which has not been investigated as of yet,

are examined The effect of the evaporator temperature on the pulse amplitude and frequency, rate of convection, and boiling

heat transfer is also investigated The results show that by increasing the evaporator temperature, due to the increase in pulse

amplitude and frequency, the rate of heat transfer due to convection and boiling in the pipe will increase too Furthermore, it

is derived that by increasing the evaporator temperature, the share of boiling heat transfer will increase In order to validate

the results, the calculated heat transfer is compared to experimental and analytical results, and it is seen that the suggested

model correctly predicts the rate of heat transfer within a precise range.

INTRODUCTION

Every day, smaller electrical instruments with higher power

ranges are introduced into the market This size reduction is

possible by concentrating the electrical parts near each other,

which causes an increase in the heat flux rate Considering this

fact, the thermal management of these instruments, which have

a high power and heat flux range, creates more complications

Removing heat from such systems requires the development of

new technologies in the field of heat transfer A new type of

heat exchanger that has been receiving increasing attention in

recent years is the pulsating heat pipe (PHP), which was first

introduced by Akachi [1] in 1990 Contrary to other types of

heat exchangers, this new type of heat pipe does not need any

external force, so it does not consume energy or produce any

noise In addition, due to the phase change which occurs in the

pipe, its heat transfer coefficient is several times larger than the

ones of fins or shell and tube heat exchangers These facts lead

This research is supported by the Iran National Science Foundation under

contract 85126/68.

Address correspondence to Professor M B Shafii, Mechanical

Engi-neering Department, Sharif University of Technology, Tehran, Iran E-mail:

Behshad@sharif.edu

to PHPs with a smaller size in comparison to other types of changers, which makes them an appropriate choice for coolingsmall electrical instruments [2] Pulsating heat pipes are madefrom a long continuous capillary tube bent into many turns.These pipes are partially filled with working fluid, and because

ex-of the tube’s small diameter and also the effect ex-of surface sion, the working fluid is distributed in slug-plug order and in

ten-a rten-andom ften-ashion in the pipe (Figure 1) No wick structure isused inside these pipes, which lowers manufacturing expensesand has also facilitated the production of these pipes in smallerdiameters compared to conventional heat pipes Pulsating heatpipes consist of three general sections: (i) evaporator, (ii) con-denser, and (iii) adiabatic The adiabatic section is arbitrarilyincluded in some PHPs The adiabatic section is used in situ-ations where a distance is needed between the evaporator andcondenser and where there is no heat transfer between the PHPand its surroundings The pressure increases in the evaporatordue to high temperatures and the boiling phenomenon However,the temperature and pressure decrease in the condenser due tocondensation Therefore a constant, unsteady internal pressuredifference exists in the system which causes the oscillations ofthe working fluid between the evaporator and condenser insidethe tube In PHPs, heat is transferred from the evaporator to thecondenser through sensible and latent heat transfer, which is a

1155

result of the working fluid oscillations and phase changes In

or-der to investigate the influential factors in creating oscillations,

Tong and Wong [3] prepared a glass PHP and filmed the various

phenomena inside the pipes These investigations showed that

the important factors are as follows: The pressure increase due

to boiling and vapor generation in the evaporator section, the

intermixing of bubbles, and the nonsymmetrical distribution of

fluid in the pipe To investigate the kinematics of the liquid and

vapor in the PHP, Wong et al [4] suggested a numerical model

In this model, the effect of heat transfer and phase change on

the performance of the heat pipe was neglected In their model,

the gas section was considered as a spring and the liquid was

modeled as the mass and damper In this model, the effect of

sudden pressure changes on the movement of the liquid and

va-por, in terms of different filling ratios and different fluid lengths,

was also investigated But as mentioned earlier, neglecting heat

transfer and phase change resulted in a model that was of

lim-ited practicality Hosoda et al [5] investigated the distribution of

vapor in a meandering closed loop heat transport device They

found that when there is a large amount of liquid in the pipe,

two distinct vapor sections separated by a liquid between them

exist in the two sides of the bend, as seen in Figure 2 When one

part starts to grow larger, the other part becomes smaller and

thus the oscillation starts Hosoda et al also introduced a

sim-ple numerical model with many simplifying assumptions One

of these assumptions was neglecting gravity and boiling heat

transfer Shafii et al [6, 7] introduced a numerical model using

governing equations including the continuity, momentum, and

energy equations for looped and unlooped pulsating heat pipes

They first investigated the effect of the number of vapor plugs

and liquid slugs and found that eventually the number of

va-por plugs reduces down to the number of evava-porator sections

(Figure 3) Then, using this conclusion, they studied the

oscil-latory movement of the two liquid slugs in the condenser In

their model, the rate of convection heat transfer as a result of

oscillatory motion was considered, but the boiling heat transfer

was neglected In their second model (Shafii et al [7]), the thin

film evaporation and condensation models have been

incorpo-rated with the model to predict the behavior of vapor plugs and

liquid slugs in the PHP The results show that heat transfer in

both looped and unlooped PHPs is due mainly to the exchange

of sensible heat, and the contribution of evaporation and

con-densation heat transfer is less than 5% of the total heat transfer

rate It was concluded in this model that the total heat transfer

significantly decreased with a decrease in the heating wall

tem-perature Finally, it can be concluded from the results of their

two models that a heat pipe with two bends can be modeled as

two U-shaped pipes

Holley and Faghri [8] presented a model for a pulsating heat

pipe with capillary wick and varying channel diameter The

energy equation was solved both in the wall and wick and in

the working fluid The effects of diameter profile, gravity, filling

ratio, and heating and cooling schemes were studied with the

model In this model, the boiling phenomenon was considered

in the wick Considering the fact that the boiling heat transfer

Figure 1 Schematic of pulsating heat.

coefficient is much higher in porous surface as compared tothe case where the surface of the pipe is smooth, the heat fluxobtained in this model is quantitatively higher than experimentalresults, but qualitatively it displayed similar trends with somepublished experimental data

Zhang et al [9] investigated oscillations of the working fluidinside a U-shaped PHP In their model, the governing equa-tions, which were obtained by analyzing the conservation ofmass, momentum, and energy equations of the liquid and va-por plugs, were nondimensionalized and the pulsating flow wasdescribed by five nondimensional parameters The effects ofvarious nondimensional parameters on the amplitude and angu-lar frequency of the oscillation were investigated, but the rate ofconvection and boiling heat transfer were not studied

Zhang and Faghri [10] developed their previous model(Zhang et al [9]), and they numerically investigated the os-cillatory flow in pulsating heat pipes with arbitrary numbers ofturns They also investigated the effect of the heating and cool-ing sections length and filling ratio on the performance of thePHP In this model, the governing equations that describe the os-cillatory flow were nondimensionalized, and the parameters that

Figure 2 Schematic of U-shaped pulsating heat pipe.

heat transfer engineering vol 31 no 14 2010

Figure 3 The Shafii et al model [6].

describe the system included eight nondimensional parameters

The results show that the increase in the number of turns has no

effect on the amplitude and circular frequency of oscillations

Arabnejad et al [11] also studied liquid oscillations in a

U-shaped PHP In their model, the effects of the inner diameter and

liquid viscosity were investigated, in which an inner diameter of

1.2mm was obtained as the critical diameter However, the rate

of sensible heat transfer due to the oscillations of the liquid and

also the boiling heat transfer have not been investigated in any of

the models which have been presented for U-shaped PHPs so far

In continuation of previous studies, the oscillatory movement of

fluid in a U-shaped PHP will be studied using numerical methods

in this paper By meshing the liquid, the rate of convection and

boiling heat transfer was studied for different temperatures of

the evaporator In the end, the heat flux derived from this model

is compared to the experimental results of Khandekar et al [12]

and also of the Shafii et al [6] model

PHYSICAL MODEL

A U-shaped pulsating heat pipe is shown schematically in

Figure 2 A pipe with a small inner diameter and a length of

2L was bent into a U-shape, and a liquid section with a length

of L p is located in the bend section of the pipe This pipe

has two evaporator sections, each having lengths of L h, which

are set at the two ends of the U-shaped pipe It also includes

a condenser section with a length of 2L c which is set in the

middle The evaporator and condenser temperatures are T eand

Tc, respectively

The position of the liquid section is represented by x p, which

is considered positive if it moves to the right and negative if it

moves to the left In order to better understand the operation of

the U-shaped pipe, first consider an initial positive displacement

for x p In this case, a portion of the vapor enters the condenser

section from the left Due to heat transfer with the condenser

and the ensuing condensation that occurs, its pressure drops

and reaches P v1 Also, a part of the liquid enters the evaporator

section Therefore, a portion of the water evaporates due to

boil-ing, and the vapor pressure increases until it reaches P v2, which

is larger than P v1 This pressure difference, (P v1 − P v2)< 0,

pushes the liquid slug to the left When the liquid in the

evapora-tor moves to the left, the aforementioned reaction occurs and we

would have (P v1 − P v2)> 0, which makes the liquid move to the

right This continuous sign change in pressure eventually leads

to liquid oscillation in the pipe Convection heat transfer occursfrom the evaporator to the condenser because of liquid oscilla-tions, and warming and cooling of the fluid in the evaporatorand condenser Latent heat transfer also occurs in the evapora-tor due to boiling In general, sensible and latent heat transfersoccur simultaneously in the PHP, which are investigated in thisresearch

GOVERNING EQUATIONS

The oscillatory phenomenon in PHPs can be predicted bysolving the mass, momentum, and energy equations for eachliquid slug and vapor plug These equations are briefly pre-sented in this study The complete explanations and derivations

of equations were done by Zhang et al [9] To solve the problemanalytically, the following assumptions are made:

1 The model is considered to be one-dimensional along thelongitudinal axis of the pipe and along the direction of theflow

2 The liquid is incompressible and the vapor plugs are assumed

to behave as an ideal gas in the heating and cooling sections

3 The pressure loss due to the bend has not been considered

in this study This assumption is valid for PHPs that do notpossess many bends However, taking pressure loss into con-sideration in the bends seems necessary for PHPs with a largenumber of bends

4 In order to simplify the analysis of the momentum equationfor the liquid section, it has been assumed that the densityand viscosity of the liquid section are constant These valueshave been calculated for the average temperatures of theevaporator and condenser

5 The temperature and pressure within each vapor plug is sumed to be uniform

as-6 The friction between the vapor plug and the surrounding walland also between the vapor plug and the liquid–gas interfacehas been neglected

7 According to the results obtained in Shafii et al [7], whichstate that the effect of the thin liquid film at the end of eachliquid slug on the total rate of heat transfer is negligible,this liquid film has not been taken into consideration, but theboiling phenomenon, which had not been investigated as ofyet, has been considered in this model

The momentum equation for the liquid section is:

Ac L pρl

d2xp

dt2 = (P v1 − P v2 ) A c− 2ρl g Acxp − πDL pτp(1)The term on the left shows the change in momentum, and theright terms show the pressure force, weight, and shear stress onthe wall In this model, the shear stress on the wall is consideredheat transfer engineering vol 31 no 14 2010

for both laminar and turbulent flow regimes in order to derive

more exact results It should be noted that in the Zhang et al

[9] model, the shear stress is investigated only for laminar flow

The shear stress term is defined as:

τp= 1

2Cliρl

d xp dt

Writing the energy equation for the vapor section using the

first law of thermodynamics, we have:

ering the initial conditions that are given in the following, the

mass, temperature, and gas pressure equations can be derived as

(12)The vapor mass increases due to the boiling of the liquid slug

and decreases due to condensation When vapor enters the

con-denser, condensation occurs as a result of heat transfer between

the vapor and the condenser wall, and consequently, the por mass decreases When the liquid plug enters the evaporator,boiling occurs as a result of heat transfer between the evaporatorwall and the liquid plug Therefore, the vapor mass increases.Under certain conditions, the temperature of the vapor may riseabove the evaporator temperature due to an increase in the vapormass and also due to the work done by the liquid slug on thevapor as it moves toward the closed end of the pipe In theseconditions, condensation occurs as a result of heat transfer be-tween the vapor and the evaporator wall and consequently, thevapor mass decreases The equations given next clearly showchanges in the vapor mass according to these explanations:

˙

Qboiling hfg − ˙m cond ,1 xp < 0, Tv1 > Te

˙

Qboiling hfg − ˙m cond ,2 xp > 0, Tv2 > Te

− ˙m cond ,2 xp < 0

(14)

The calculation method for the changes in vapor mass due

to condensation and boiling is explained in the subsequent tions

sec-BOILING

Analyzing flow boiling phenomena is much more cated than analyzing nucleate boiling because of vapor genera-tion and separation of vapor from the surface, which cause two-phase flow regimes In flow boiling, the heat transfer mechanism

compli-is a combination of forced convection and boiling heat transfer

As of yet, a general way for finding the rate of heat transfer inflow boiling has not been derived Rohsenow [14] and Walley[15] were the first ones who suggested several equations for thiskind of heat transfer In Rohsenow’s method, the sum of nucle-ate boiling and forced convection heat transfer is considered asthe total rate of heat transfer So in this method, the intermixingbetween bubbles and the fluid and also the shares of forced con-vection and nucleate boiling were neglected After Rohsenow,many other researchers studied flow boiling phenomena; amongthese researchers, Liu and Winterton [16], Chen [17], Gungorand Winterton [18], and Kandlikar [19] can be mentioned Zhang

et al [20] examined the equations presented by these differentresearchers for flow boiling in small-diameter pipes He con-cluded that the coefficient of heat transfer introduced by Chen[17] corresponds to experimental data more closely than others

In Chen’s [17] equation, the coefficients of forced convection

and boiling heat transfer are corrected by the F and S coefficients

as follows:

hflow boiling= F.h sp + S.h nb (15)heat transfer engineering vol 31 no 14 2010

Figure 4 Boiling model considered for investigating boiling in PHP.

In the preceding equation, the boiling heat transfer coefficient

is calculated from the equation here [20]:

hnb = 0.00122

k0.79

l C0.45

p ,l ρ0.49 l

σ0.5µ0.29

l h0.24

fg ρ0.24 g



T0.24 sat
P0.75 sat (16)

In this equation,
Tsatis the temperature difference between

the evaporator and the saturation temperature of the liquid.
Psat

is the saturation pressure difference at the evaporator

tempera-ture and liquid pressure If the pressure distribution in the liquid

is considered to be linear using the vapor pressure on both sides

(Figure 4), then
Psat can be calculated along the liquid using

this pressure distribution, and therefore
Tsatand the saturated

temperature of the liquid can be derived as a result By

know-ing the liquid’s Reynolds number, the S coefficient is readily

calculated [20]:

(1+ 2.53 × 10−6Re1.17

By using the boiling heat transfer coefficient and by

calcu-lating S, the rate of boiling heat transfer and the rate of vapor

generation in the evaporator can be calculated from Eqs (18)

where X is the length of liquid which enters the left or right

evaporator, h nb ,x represents the boiling heat transfer coefficient

at position x of liquid, and T l ,x represents the liquid temperature

at position x.

FORCED CONVECTION HEAT TRANSFER

The convection heat transfer coefficient can be calculated as

follows with respect to the flow regime (laminar or turbulent)



Rel > 2000 Turbulent

(20)

In two-phase flow, the Dittus Boelter equation is used for Re

> 2300 [20] and empirical relations for the transient state must

be used for Reynolds numbers between 2000 and 2300 Thetransient state has been neglected in this research and the DittusBoelter equation has been used for Re> 2000.

The convection heat transfer coefficient in two-phase flow is

corrected by using the F coefficient, which is calculated using

Martinelli’s parameter [21] as shown here:

if 1Xtt > 0.1F = 2.35

1

Xtt + 0.213

0.736

if 1Xtt < 0.1F = 1

Using Eq (24), ˙mg(the vapor mass flow rate at each section

of the evaporator) is calculated first Then x qis calculated (Eq

(23)) Using x q, Martinelli’s parameter is calculated from Eq

(22) and then the coefficient F can be calculated from Eq (21).

Thus, the energy equation is found as follows in order to obtainthe temperature distribution in the liquid:

Tle = T v1 Temperature of common surface

between liquid and vapor on the left (26)heat transfer engineering vol 31 no 14 2010

Tr e = T v2 Temperature of common surface

between liquid and vapor on the right (27)After deriving the temperature distribution at each moment,

the rate of convection heat transfer in the evaporator can be

obtained from the following equation:

respectively

CONDENSATION MODEL

Considering the physical model section, condensation

hap-pens when vapor enters the condenser from the right or left

due to liquid movement Furthermore, if the vapor

tempera-ture surpasses that of the evaporator, condensation would occur

in the evaporator as well Due to curvature of the pipe in the

condenser, a centrifugal force is produced when the fluid flows

through the pipe Therefore, this centrifugal force pushes the

condensed liquid to the pipe outer surface and the vapor will

flow above the condensed liquid Therefore, it can be assumed

that the flow regime is stratified Using experimental data for

vapor flow condensation inside a tube, Jaster and Kosky [22]

suggested the formula that follows for calculating the average

condensation heat transfer coefficient for stratified flow regime:

h c = 0.728 ρl(ρl− ρg )gh fgk l3

µl (T v − T w )D

1/4

(29)

Having h c, the rate of condensed vapor mass due to contact

between the vapor and condenser can be calculated as follows:

˙

mCond ,1 = ¯h c πDx p (T v1 − T c)/hfg xp > 0 (30)

˙

mCond ,2 = −¯h c πDx p (T v2 − T c)/hfg xp < 0 (31)

Also, if the vapor temperature surpasses the evaporator

tem-perature, condensation would also occur in the evaporator and

the rate of condensed vapor mass in the evaporator would be as

For a specified time step, the equations are solved using the

implicit method through the following steps:

1 An arbitrary value is assumed for T v1 , T v2 and then P v1 , P v2

are be calculated using Eqs (10) and (12)

2 d x p

dt and x p will be calculated using Eq (1) with respect

to the flow regime It is obvious that because of the liquidmovement, the mesh will also move with it

3 Due to the vapor pressure on both sides of the liquid, the sure distribution inside the liquid is considered to be linear.Using this distribution, the saturation temperature distribu-tion in the liquid will be calculated Considering the liquid’sReynolds number,ρl d x p

pres-dt D

µl , the correction coefficient of S will

be calculated using Eq (17) Then, the boiling heat transfercoefficient calculated from Eq (16) will be corrected using

the correction coefficient of S Considering the new liquid position (x p), the rate of boiling heat transfer and also thevapor generation rate in the evaporator ( ˙mboiling) can be calcu-lated using Eqs (18) and (19), respectively In order to obtainthe boiling heat transfer, the liquid temperature distributionfrom the last time step is used

4 The average condensation heat transfer coefficient ( ¯h c) iscalculated using Eq (29) The condensed mass in the con-denser and evaporator ( ˙mCond ) can be calculated using Eqs.(30) to (33)

5 m v1, mv2will be modified for the next time step using Eqs.(13) and (14)

6 P v1, Pv2 will be calculated using Eqs (9) and (11) Then

T v1, Tv2will be derived through Eqs (10) and Eq (12)

7 T v1, Tv2will be compared to the assumed values (from step1) If these values are sufficiently close to each other, theenergy equation will be solved for the liquid section consid-

ering the boundary conditions and the new position of x p.The rate of convection heat transfer will be calculated using

Eq (28) and the time step will go forward But if T v1 , T v2

are not sufficiently close to the original values, the nal values will be replaced by the calculated values and thepreceding steps will be repeated

origi-The studies revealed that for 1000 nodes, the results will beindependent of the mesh Besides, a time step of
t = 10−5will make the results the time-step-independent solution

RESULTS

First, in order to validate our calculation software, calculationresults from the model are compared with predictions from theZhang et al [9] results for the same conditions as follows:

D = 3.34mm, L h = 0.1m, 2L c = 0.2m, L p = 0.2m,

Te = 123.4◦C,Tc= 20◦C , x p ,0 = 0.05m (34)Figure 5 shows a comparison between the displacement cal-culated by the software and the Zhang et al [9] calculations

It is obvious that the results are very close to each other whichverifies the correctness of the software In the Zhang et al [9]model, the displacement is shown as dimensionless Therefore,heat transfer engineering vol 31 no 14 2010

Figure 5 Comparison of the dimensionless position of liquid slug with

dimen-sionless time: (a) the Zhang et al [9] model, and (b) the present model.

the displacement in Figure 5 is shown as dimensionless too For

a study of how to make the results dimensionless, refer to the

Zhang et al [9] work

As stated before, the purpose of this research is to

inves-tigate the rate of heat transfer due to convection and boiling

in a U-shaped PHP One parameter that has a profound effect

on the rate of heat transfer is the temperature of the

evapora-tor Therefore, the effect of this temperature on the rate of heat

transfer is also studied In this research, thermodynamic

char-acteristics of water are considered as the working fluid and the

physical characteristics of the U-shaped pulsating heat pipe are

as follows:

D = 2mm, L h = 0.1m, 2L c = 0.2m, L p = 0.2m,T c= 20◦C

(35)Figure 6 shows the temperature distribution of the liquid plug

for three different positions, when the evaporator temperature

is equal to 70◦C In case (a) (Xp = –0.028), the liquid plug

is situated at the leftmost position and its velocity direction is

positive Also, in the second case (Xp= 0.0), the liquid is in the

middle In case (c) (Xp= 0.028), it is situated in the rightmost

position In case (a), the left end and right end of liquid are

in the evaporator and condenser, respectively Therefore, the

temperature of left end liquid is increased and the right end is

decreased In case (b), the liquid is in the middle and it has come

back from the evaporator Therefore, the left end temperature of

liquid is increased more in comparison with case (a) because the

liquid has remained in the evaporator more time In case (c), the

right end of liquid slug has moved to the evaporator Therefore,

the right end temperature of liquid slug is increased

Figure 7 shows the changes in liquid displacement, vapor

temperature, vapor pressure, and convection and boiling heat

transfer for an evaporator temperature of 70◦C As seen from

the changes in vapor pressure and temperature, it is obvious

that the pressure and temperature of the two vapor sections are

oscillating with a phase difference ofπ radians The reason for

this is that when one of the vapor parts is in the evaporator

section, its pressure and temperature increase due to boiling;

meanwhile, the other part in the condenser faces pressure and

temperature decrease due to condensation, which causes this

phase difference Figure 7 shows that at an evaporator

tem-perature of 70◦C, the rate of boiling heat transfer is less than

convection heat transfer Therefore, in this case, the forced

con-vection heat transfer has a larger share in the total rate of heat

transfer Figure 8 shows the effect of increasing the

evapora-tor temperature on the oscillation amplitude and frequency, andalso the changes in vapor temperature and pressure As seen

in the displacement graph, with an increase in the evaporatortemperature, the oscillation amplitude and frequency increase

as well, which is because of an increase in pressure As a sult, it is expected that the boiling and convection heat transferswill increase as well Figure 9 shows the rate of convectionand boiling heat transfers for different evaporator temperatures

re-It shows that with an increase in the evaporator temperature,the convection and boiling heat transfers increase considerably

An increase in the evaporator temperature causes an increase

in the temperature difference between the fluid and pipe walland, consequently, an increase in the rate of heat transfer Italso causes an increase in the amplitude and frequency; in otherwords, it increases the speed of oscillations, which is an impor-tant parameter in the increase of the convection heat transfercoefficient This speed increase also changes the flow regimefrom laminar to turbulent and vice versa The concave and con-vex regions of the heat transfer graph represent decrease andincrease in the convection heat transfer coefficient because offlow regime changes However, when the evaporator temper-ature is 70◦C, there is no regime change because of the lowoscillation speed, and therefore there are no convex or concaveregions in the heat transfer graph Figure 9b shows that when in-creasing the evaporator temperature to 100◦C, boiling becomesthe main heat transfer mechanism and the share of boiling heattransfer increases

Figure 10 shows a comparison of the transferred heat fluxbetween the studied model and the experimental results ofKhandekar et al [12] and numerical results of Shafii et al [6].The Khandekar et al [12] experimental setup and the Shafii et

al [6] model are a PHP with five and two turns, respectively.However, only one U-shaped heat pipe was considered in thismodel Thus, considering the results obtained by Zhang et al [9]

Figure 6 Liquid temperature distribution for three different positions.

heat transfer engineering vol 31 no 14 2010

Figure 7 Changes in (a) liquid displacement, (b) vapor temperature, (c) vapor

pressure, and (d) convection heat transfer and boiling at an evaporator

temper-ature of 70 ◦C.

Figure 8 Changes in (a) liquid position, (b) vapor temperature, and (c) vapor

pressure for different evaporator temperatures.

claiming that the number of turns has no effect on the mance of a PHP whenever the number of turns is less than orequal to five, the heat flux transferred by the PHP is expressed

perfor-in terms of the evaporator area and the condenser temperature isassumed to be 20◦C in order to compare the results of the stud-ied model with those of the Shafii et al [6] model and with theKhandekar et al [12] experimental results The pipe’s inner di-ameter in the Khandekar et al [12] experimental work is 2 mm,which is identical to the studied model The inner diameter in theShafii et al [6] model is equal to 1.5 and 3 mm, which is differentfrom the diameter considered in the studied model Therefore,

in order to compare the results between the Shafii et al [6]model and the model under study, the Shafii et al [6] results areinterpolated for a diameter of 2 mm Figure 10 shows that theheat flux calculated in this research has more correspondencewith the experimental results because the boiling heat transfer isconsidered in this model However, in the Shafii et al [6] model,only evaporation is considered, and therefore, due to neglectingthe boiling heat transfer, the calculated heat flux in the Shafii et

al [6] model is less than for the experimental results

CONCLUSIONS

In this research, the operation of a U-shaped pulsating heatpipe was investigated using numerical methods The governingequations, including the continuity, momentum, and energyequations, have been solved implicitly Boiling heat transferwas studied as an important heat transfer mechanism in the heatpipe Because of the considered mesh, the rate of convectionand boiling heat transfer was investigated in the U-shapedpulsating heat pipe, which had not been investigated before.The effect of the evaporator temperature on the oscillationamplitude and frequency and also on the boiling and convectionheat transfers was investigated as well Results showed that

by increasing the evaporator temperature, the rate of heat

Figure 9 (a) Changes in convection heat transfer and boiling for an evaporator temperature of 90 ◦C (b) Changes in convection heat transfer and boiling for an

evaporator temperature of 100 ◦C.

heat transfer engineering vol 31 no 14 2010

Figure 10 Comparison of heat flux derived in the corresponding model with

the Shafii et al [6] model and the Khandekar et al [12] experimental results in

different temperature differences of evaporator and condenser.

transfer increases as well, due to an increase in oscillation

amplitude and frequency Furthermore, it was shown that an

increase in the evaporator temperature causes the boiling heat

transfer to have a larger share in the total rate of heat transfer

The calculated heat flux in this model was compared with the

experimental and numerical results, which showed that this

model has closer correspondence with experimental results due

to a consideration of the boiling heat transfer

NOMENCLATURE

Ac pipe cross-sectional area (m2)

Cli friction coefficient

Cp specific heat at constant pressure (J/kg-K)

C v specific heat at constant volume (J/kg-K)

H heat transfer coefficient (W/m2-K)

h heat transfer coefficient (W/m2-K)

hc average condensation heat transfer coefficient

γ ratio of specific heats

µ dynamic viscosity (Pa-s)

1 left vapor plug

2 right vapor plug

[1] Akachi, H., U.S Patent Number 4921041, 1990

[2] Akachi, H., Polasek, F., and Stulc, P., Pulsating Heat

Pipes, Proc of the 5th International Heat Pipe Symposium,

Melbourne, Australia, pp 208–217, 1996

[3] Tong, B., and Wong, T., Closed-Loop Pulsating Heat

Pipe, Applied Thermal Engineering, ISSN 1359-4311, vol.

21/18, pp.1845–1862, 2001

[4] Wong, T., Tong, B., and Lim, S., Theoretical Modeling of

Pulsating Heat Pipe, Proc 11th Int Heat Pipe Conf., pp.

159–163, Tokyo, Japan, 1999

heat transfer engineering vol 31 no 14 2010