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

Heat Transfer Engineering, 31(14):1125–1136, 2010
Copyright C Taylor and Francis Group, LLC
ISSN: 0145-7632 print / 1521-0537 online
DOI: 10.1080/01457631003689211

An Analysis of Heat Conduction
Models for Nanofluids
1
˜ N. N. QUARESMA,1 EMANUEL N. MACEDO,
ˆ
JOAO
HENRIQUE M. DA
2
2
FONSECA, HELCIO R. B. ORLANDE, and RENATO M. COTTA2
1

School of Chemical Engineering, Universidade Federal do Par´a, UFPA Campus Universit´ario do Guam´a, Bel´em, PA, Brazil
Mechanical Engineering Department–Polit´ecnica/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil

2

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. Different 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 dualphase-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 considers 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 independently 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 hyperbolic model for heat conduction [1–13].
Thermal conductivity of fluids plays a vital role in the development 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:

conductivity [14]. On the other hand, metals in the solid form
have thermal conductivity larger by orders of magnitude than
those of fluids. For example, the thermal conductivity of copper 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 particles
were expected to display significantly enhanced thermal conductivities relative to conventional heat transfer fluids. Numerous theoretical and experimental studies of the effective thermal
conductivity of dispersions containing particles have been conducted since Maxwell’s theoretical work on the subject was
published more than 100 years ago [14, 15]. However, early

studies of the thermal conductivity of suspensions have been
confined to those containing particles with sizes of the order of
millimeters or micrometers. In fact, conventional micrometersized particles cannot be used in practical heat-transfer equipment because of severe clogging and sedimentation problems.
In addition, recent miniaturization, leading to the increasing
practical utilization of microchannels and microreactors, also
imposed a restriction on the use of micrometer-sized particles
[14, 16–18].
Modern nanotechnology provides great opportunities to process and produce materials with average sizes below 50 nm
[14, 16–18]. Recognizing an opportunity to apply this emerging nanotechnology to established thermal energy engineering,

1125


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J. N. N. QUARESMA ET AL.

Choi and coworkers [14, 16–18] proposed that nanometer-sized
metallic particles be suspended in industrial heat transfer fluids, 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 conductivity, 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 theory. On the other hand, some studies reported that such increase 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 accounted for in the theoretical predictions for suspensions of
micrometer-sized or larger particles, which included Brownian motion, liquid layering, ballistic mechanisms, thermophoresis, 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 modeled 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 geometry 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, usually taken in the form of a heating wire, which is placed inside
the material with unknown properties. For large times, the temperature variation of the heat source is shown to be linear with
respect to the logarithm of time, so that the thermal conductivity 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 largetime solution.
heat transfer engineering

PROBLEM FORMULATION
The analysis considered here is similar to that examined
in references [12] and [13] and involves a dual-phase-lagging
model (DPLM) [5, 7]. In such a model, a finite speed of heat
propagation in the medium is taken into account through a delay
time for the establishment of the heat flux, τq , and a lag between
the heat flux vector and the temperature gradient, τT . The constitutive equation relating the heat conduction flux vector and the
temperature gradient in the dual-phase-lagging model is given
by [1–7, 12, 13]:
q + τq

∂(∇T f )
∂q
= −K ∇T f + τT
∂t
∂t

(1)

where q is the heat flux vector and K is the effective thermal
conductivity of the medium.
The energy conservation equation for a purely conducting
medium, considered in this work as a nanofluid, is written as
(Cs + C f )

∂Tf
= −∇ · q
∂t

(2)

where Cs is the volumetric heat capacity of the nanoparticles and Cf is the volumetric heat capacity of the base
fluid.
The substitution of Eq. (1) into the energy conservation equation (2) results in:
τq

∂2T f
∂(∇ 2 T f )
∂Tf
2
+
T
+
α
τ
=
α
∇
f
T
∂t 2
∂t
∂t

(3)

with the effective thermal diffusivity given by:
α=

K
Cs + C f

(4)

Therefore, the use of the constitutive Eq. (1) together with
the energy conservation Eq. (2) leads to a hyperbolic heat conduction model given by Eq. (3).
Equation (3) can be similarly obtained by considering that
the nanoparticles and the base fluid are not in local thermal
equilibrium. In this case, the energy conservation equation can
be written separately for the nanoparticles and the base fluid,
respectively, in the following form:
Cs

∂Tf
∂ Ts
= h (T f − Ts ); C f
= K ∇ 2 T f − h (T f − Ts ) (5, 6)
∂t
∂t

where h (W/m3oC) is a heat transfer coefficient between the fluid
and the nanoparticles. Note that it was considered a lumped
formulation for the nanoparticles, given by Eq. (5).
Then, by substituting Ts from Eq. (5) into Eq. (6), one obtains:
Cs C f
∂2T f
∂Tf
+
h(Cs + C f ) ∂t 2

∂t
=

Cs ∂(∇ 2 T f )
K
K
∇2T f +
(Cs + C f )
(Cs + C f ) h
∂t

vol. 31 no. 14 2010

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J. N. N. QUARESMA ET AL.

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:
τq =

Cs C f
;
h(Cs + C f )

τT =

Cs

h

(8a, 8b)

Therefore,
Cs + C f
τT = τq
Cf

(9)

τT
Cs
=1+
>1
τq
Cf

(10)

or, alternatively,

1127

θ(r,τ) = 0;

−

∂θ(R,τ)
= 0 for

∂τ

τ = 0,

∂
∂θ(R,τ)
θ(R,τ) + FoT
=1
∂R
∂τ

in A < R < 1
(13b, 13c)

at R = A,

∂
∂θ(R,τ)
θ(R,τ) + FoT
= 0 at
∂R
∂τ

for τ > 0
(13d)

R = 1, for τ > 0
(13e)

where it was considered that heat flux imposed by the probe is

constant in time.
Considering the case involving a line heat source probe of
The classical parabolic heat conduction model, which utiradius a immersed in a cylindrical medium of radius b, Eq. (3) lizes Fourier’s law as the constitutive equation that relates the
can be rewritten as
heat conduction flux vector and the temperature gradient, can
1 ∂
∂ 2 T (r, t) ∂ T (r, t)
∂ T (r, t)
∂
be directly obtained from Eqs. (13a–e) by making the relax=α
τq
+
r
T (r, t) + τT
,
2
ation times, τq and τT , equal to zero (see Eq. (1)). In terms of
∂t
∂t
r ∂r
∂r
∂t
the nonequilibrium model given by Eqs. (5) and (6), τq → 0
in a < r < b, for t > 0
(11a) and τT → 0 can be obtained with h → ∞ (see Eqs. (8a) and
(8b))—that is, the heat transfer coefficient between the fluid
which is subjected to the following initial and boundary condiand the dispersed nanoparticles becomes very large and local
tions:
thermal equilibrium is attained (Tf = Ts ).
∂ T (r, t)

T (r, t) = T0 ;
= 0 for t = 0, in a < r < b
∂t
(11b, 11c)
SOLUTION METHODOLOGY
−K

∂ T (r, t)
∂q0
∂
T (r, t) + τT
= q0 + τq
∂r
∂t
∂t
at r = a,

K

for t > 0

(11d)

∂ T (r, t)
∂
T (r, t) + τT
= 0 at r = b,
∂r
∂t

t >0

for

(11e)
where q0 is the heat flux resulting from the electrical resistance
inside the probe and T 0 is the initial temperature of the medium.
By defining the following dimensionless variables,
θ(R,τ) =
FoT =

ατq
r
a
αt
T (r, t) − T0
; R = ; A = ; τ = 2 ; Foq = 2 ;
(q0 b/K )
b
b
b
b
ατT
FoT
;β =
2
b
Foq

(12a–f)

the problem given by Eqs. (11a–e) can be rewritten in dimensionless form as:
Foq
=

∂ 2 θ(R,τ) ∂θ(R,τ)
+
∂τ2
∂τ
1 ∂
R ∂R

R

θ(R,τ) = θav (τ) + θ p (R) + φ(R,τ)

(14)

where θav (τ) is the average temperature in the medium, which
is a priori obtained from Eqs. (13a)–(13e); θp (R) is a particular
solution and φ(R,τ) is the potential to be solved with the CITT.
The average temperature θav (τ) is defined as
θav (τ) =

1
A

Rθ(R,τ)d R
1
A

Rd R

=

2
1 − A2

1

Rθ(R,τ)d R (15)
A

Now, in order to determine a solution for θav (τ), Eq. (13a) is
multiplied by [2/(1−A2)]R and integrated over the domain [A,1]
in the R-direction. The definition given by Eq. (15) is then employed and the boundary conditions (13d, 13e) are used to yield
Foq

∂
∂θ(R,τ)
θ(R,τ) + FoT
∂R
∂τ

in A < R < 1, τ > 0

For the solution of the hyperbolic heat conduction problem
given by Eqs. (13a)–(13e) we apply the classical integral
transform technique (CITT) [32, 33]. A split-up procedure [32]
is used in order to improve the convergence rate of the final

series solution. Hence, the solution for the temperature field is
written as:

d 2 θav (τ) dθav (τ)
2A
+
=
,
dτ2
dτ
1 − A2

for τ > 0 (16a)

From the initial conditions (13b) and (13c) together with the
definition (15), it results that

,
(13a)

θav (τ) = 0;

heat transfer engineering

vol. 31 no. 14 2010

dθav (τ)
=0
dτ

for τ = 0

(16b, 16c)


1128

J. N. N. QUARESMA ET AL.

Therefore, the solution for θav (τ) is obtained as
θav (τ) =

2A
τ − Foq 1 − e−τ/Foq
1 − A2

(19)

Equation (14) is now introduced into Eqs. (13a–e) and the
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), for θp (R) and φ(R,τ), respectively:
dθ p (R)
2A
1 d
R
=
,
R dR
dR

1 − A2
−

in A < R < 1 (17a)

dθ p (R)
dθ p (R)
= 1 at R = A;
= 0 at R = 1(17b, 17c)
dR
dR

The homogeneous problem for the potential φ(R,τ) is now
solved with the classical integral transform technique (CITT)
[32, 33]. For this purpose, the following auxiliary eigenvalue
problem is utilized, which shall provide the basis for the eigenfunction expansion of the potential φ(R,τ):
d i (R)
d
R
+ βi2 R i (R) = 0, in A < R < 1 (20a)
dR
dR
d

i (R)

i (R)
1

Rθ p (R)d R = 0

= Jo (βi R)Y1 (βi ) − J1 (βi )Y0 (βi R)

J1 (βi A)Y1 (βi ) − J1 (βi )Y1 (βi A) = 0,

and

1

∂φ(R,τ)
∂
R
φ(R,τ) + FoT
∂R
∂τ

in A < R < 1,

R i (R)

,

τ>0

∂φ(R,τ)
=0
∂τ

Ni =

φ¯ i (τ) =

(18e)

(18f)

R2
A
− ln(R) −
2
4(1 − A2 )2

× [4A2 ln(A) + (3 + A2 )(1 − A2 )]

(21c)

(21d)

R ˜ i (R)φ(R,τ)d R,

transform

(22a)

A
∞

˜ i (R)φ¯ i (τ),

inverse

(22b)

i=1

The additional constraints given by Eqs. (17d) and (18f)
are obtained by substituting Eq. (14) into the definition of the
average temperature θav (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
θp (R) in the form:
A
1 − A2

2 J12 (βi A) − J12 (βi )
π2
βi2 J12 (βi A)

φ(R,τ) =

A

θ p (R) =

0, i = j
Ni , i = j

1

where

˜ i (R) =

with
Rφ(R,τ)d R = 0

satisfy the

The auxiliary eigenvalue problem given by Eqs. (20a–c) allows the definition of the following integral transform–inverse
pair for the potential φ(R,τ):

∂φ(R,τ)
∂
φ(R,τ) + FoT
= 0 at R = A, for τ > 0
−
∂R
∂τ
(18d)

1

=

i (R)

where Ni is the normalization integral given by:

in A < R < 1
(18b, 18c)

∂
∂φ(R,τ)
φ(R,τ) + FoT
= 0 at R = 1, for τ > 0
∂R
∂τ

j (R)d R

A

(18a)

for τ = 0,

(21a)

i = 1, 2, 3, ... (21b)

It can be shown that the eigenfunctions
following orthogonality property [32, 33]:

∂ 2 φ(R,τ) ∂φ(R,τ)
Foq
+
∂τ2
∂τ

φ(R,τ) = −θ p (R);

i (R)

(17d)

A

1 ∂
=
R ∂R

d

= 0 at R = 1 (20b, 20c)
dR
dR
Equations (20a–c) can be analytically solved to yield the
eigenfunctions and the transcendental equation to compute the
eigenvalues respectively as [32, 33]:

with

= 0 at R = A;

i (R)/

Ni

After the definition of the integral transform-inverse pair with
the auxiliary eigenvalue problem (20a–c), the next step in the
CITT is thus to accomplish the integral transformation of the

original partial differential system given by Eqs. (18a–e). For
this 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 following
system of ordinary differential results, for the calculation of the
transformed potentials φ¯ i (τ):
1 + FoT βi2 d φ¯ i (τ)
β2
d 2 φ¯ i (τ)
+ i φ¯ i (τ) = 0,
+
2
dτ
Foq
dτ
Foq

for τ > 0
(23a)

(19)

heat transfer engineering

(22c)

vol. 31 no. 14 2010

J. N. N. QUARESMA ET AL.

d φ¯ i (τ)
=0
φ¯ i (τ) = ¯f i ;
dτ

for τ = 0

(23b, c)

× exp −

(23d)

⎧⎡
⎪
⎪
⎨⎢
× ⎢
⎣1 +
⎪
⎪
⎩

where
1

¯f i = −

R ˜ i (R)θ p (R)d R
A

for i = 1, 2, 3,. . . .
The infinite system of ordinary differential equations (23a–c)
is uncoupled and can be readily solved to yield:
⎡
⎛
⎞ ⎤
2
¯f i
1+FoT βi2
4Fo
β
q i
⎝1 − 1−
⎠τ⎦
exp ⎣−
φ¯ i (τ) =
2
2
2Foq
1+FoT β2
i

⎧⎡
⎪
⎪
⎨⎢
× ⎢

⎣1 +
⎪
⎪
⎩

⎡

⎤
4Foq βi2

1−

1+FoT βi2

⎡

1 + FoT βi2
× exp ⎣−
Foq

⎤

⎥ ⎢
⎥ + ⎢1 −
⎦ ⎣

1
2

1

1−

4Foq βi2
(1+FoT βi2 )2

⎤⎫
⎬
⎦
1−
τ
2
⎭
1 + FoT βi2
4Foq βi2

⎥
⎥
⎦

(24)

φ(R,τ) =
i=1

¯f i
˜ i (R)
2

⎡

⎛
⎞ ⎤
1+FoT βi2
4Foq βi2
⎝1− 1−
⎠ τ⎦
× exp ⎣−
2
2Foq
1+FoT βi2
⎧⎡
⎪
⎪
⎨⎢
× ⎢
⎣1 +
⎪
⎪
⎩
⎡
× exp ⎣−

⎡

⎤
1
1−

4Foq βi2
(1+FoT βi2 )2

(1 +
Foq

FoT βi2 )

⎤

⎥ ⎢
⎥ + ⎢1 −
⎦ ⎣

1
1−

4Foq βi2
(1+FoT βi2 )2

⎤⎫
⎬
2
4Foq βi
1−
τ⎦
2
⎭
1 + FoT βi2

⎥
⎥

⎦

(25)

A
1 − A2

2

R
A
− ln(R) −
2
4(1 − A2 )2
∞

× [4A ln(A)+(3 + A )(1−A )] +
2

2

2

i=1

¯f i
˜ i (R)
2

heat transfer engineering

1−
⎤

1
1−

4Foq βi2
(1+FoT βi2 )2

1−

4Foq βi2
(1 + FoT βi2 )2

τ

⎡

⎤

⎥ ⎢
⎥ + ⎢1 −
⎦ ⎣

1
1−

4Foq βi2
(1+FoT βi2 )2

4Foq βi2
(1 + FoT βi2 )
1−
τ
Foq
(1 + FoT βi2 )2

⎥
⎥
⎦

(26)

The CITT is also applied in order to obtain the solution for
the classical parabolic problem, based on Fouriers’s law. In this
case, the analytical solution is given by:
θ(R,τ) =

A
2A
τ+
2
1− A
1 − A2
−

R2
− ln(R)
2

A
[4A2 ln(A) + (3 + A2 )(1 − A2 )]
4(1 − A2 )2

+

¯f i ˜ i (R)e−βi2 τ

(27)

i=1

where the eigenquantities that appear in Eq. (27) are the same
as those defined earlier for the solution of the problem for the
potential φ(R,τ).
For large times, Blackwell [27] derived an asymptotic solution for the temperature variation at the surface of the line
heat source probe in the parabolic problem, which is shown to
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 computed
from the slope of the temperature variation [27–31]. Blackwell’s
solution [27], in terms of the dimensionless variables given by
Eqs. (12a), is
θa (τ) =

Finally, by substituting Eqs. (16d), (19), and (25) into Eq.
(14), the solution for the dimensionless temperature field is
obtained as
2A

θ(R,τ) =
[τ − Foq (1 − e−τ/Foq )]
1 − A2
+

× exp −

1 + FoT βi2
2Foq

∞

By substituting Eq. (24) into the inverse formula (22b), the
analytical solution for the homogeneous potential φ(R,τ) is obtained as
∞

1129

1
1
A ln τ + A ln
2
2

4
A2

−y

(28)

where y = 0.5772156649 is Euler’s constant. We note in Eq. (28)
that, in dimensionless terms, the slope of θa (τ) x ln τ is equal to
A/2.

RESULTS AND DISCUSSION
In this session we present numerical results for the dimensionless temperature variation at the surface of the probe, that is,
at R = A, obtained with the hyperbolic heat conduction model
given by Eq. (26). Such results are compared to those obtained
with 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 Visual
vol. 31 no. 14 2010


1130

J. N. N. QUARESMA ET AL.

Table 1

Test cases examined

0.055

Dimensions
Test
case

Alumina in water

Alumina in water
Copper in ethylene glycol
Copper in ethylene glycol

a (m)

b (m)

A

β

0.045

5 × 10−5
7.5 × 10−4
5 × 10−5
7.5 × 10−4

0.025
0.05
0.025
0.05

2 × 10−3
1.5 × 10−2
2 × 10−3
1.5 × 10−2

1.71

1.71
2.29
2.29

0.04

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, Cs = 3.430 × 106 J/m3K, Cf = 2.649 × 106 J/m3-K, β
= 1.71) and (ii) copper nanoparticles in ethylene glycol (K =
0.627 W/m-K, Cs = 2.964 × 106 J/m3-K, Cf = 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 × 10−5 m 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 temperature 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, and β = 2.29 (test case c). This figure shows
that for small dimensionless times (τ ≤ 10−5), convergence at

Analytical - Fo q = 1x10-3
FDM-Gear - Fo q = 1x10-3

Analytical - Fo q = 1x10-4

0.03

FDM-Gear - Fo q = 1x10-4
Analytical - Fo q = 1x10-5

0.025

FDM-Gear - Fo q = 1x10-5

0.02

Analytical - Fo q = 1x10-6
FDM-Gear - Fo q = 1x10-6

0.015

Analytical - Fo q = 1x10-10

0.01

FDM-Gear - Fo q = 1x10-10
0.005
0

1x10

-20

1x10

-15

1x10

-10

1x10

-5

1x10

0

τ
Figure 2 Comparison of analytical solution and finite-difference method solution using Gear’s method (referred to as FDM-Gear) for A = 1.5 × 10−2 and
β = 2.29.

the graphic scale is obtained with 5000 ≤ NT ≤ 10000. On the
other hand, for larger dimensionless times the convergence is
reached 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, we
compared its results with those obtained numerically with finite

0.011

0.011

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model - Fo q = 1x10-5

0.01
0.009

0.01

A = 2x10-3; β = Fo T/Fo q = 2.29

0.008

0.009

Analytical - NT = 100
Analytical - NT = 500
Analytical - NT = 1000
Analytical - NT = 5000
Analytical - NT = 10000

0.006

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model
A = 2x10-3; β = Fo T/Fo q = 2.29

0.008

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

0.007

θ(A,τ)

0.007

θ(A,τ)

A = 1.5x10-2; β = Fo T/Fo q = 2.29

0.035

θ(A,τ)

a
b
c
d

Nanofluid

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.05

0.005

0.006
0.005

0.004

0.004

0.003

0.003

0.002

0.002

0.001

0.001

0

0

1x10

-20

1x10

-15

1x10

-10

1x10

-5

1x10

0

τ

1.0x10-20

1.0x10-15

1.0x10-10

1.0x10-5

1.0x100

τ

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

Foq = 1 × 10−5, and β = 2.29.

heat transfer engineering

Figure 3 Comparison of the hyperbolic (Foq = 10−10) and parabolic solutions
for A = 2 × 10−3 and β = 2.29.

vol. 31 no. 14 2010


J. N. N. QUARESMA ET AL.
0.055

0.011
0.01

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.05

A = 2x10-3; β = Fo T/Fo q = 1.71

0.009

0.045

Fo q = 1x10-3

0.008

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model
A = 1.5x10-2; β = Fo T/Fo q = 1.71
Fo q = 1x10-3

0.04

-4

Fo q = 1x10-4

Fo q = 1x10

-5

0.007

0.035

Fo q = 1x10

Fo q = 1x10-6

0.006

θ(A,τ)

θ(A,τ)

1131

Fo q = 1x10-10

0.005

Fo q = 1x10-5
Fo q = 1x10-6

0.03

Fo q = 1x10-10

0.025

0.004

0.02

0.003

0.015

0.002

0.01
0.005

0.001

(a)

0
1x10-20

1x10-15

1x10-10

1x10-5

(b)

0

1x100

1x10-20

1x10-15

τ

1x100

0.055

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.05

A = 2x10-3; β = Fo T/Fo q = 2.29

0.009

0.045

Fo q = 1x10-3

0.008

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model
A = 1.5x10-2; β = Fo T/Fo q = 2.29
Fo q = 1x10-3

0.04

Fo q = 1x10-4

Fo q = 1x10-4

Fo q = 1x10-5

0.007

0.035

Fo q = 1x10-5

-6

Fo q = 1x10

0.006

θ(A,τ)

θ(A,τ)

1x10-5

τ

0.011
0.01

1x10-10

-10

Fo q = 1x10
0.005

0.02

0.003

0.015

0.002

0.01

0.001

0.005

(c)
1x10-20

1x10-15

1x10-10

1x10-5

1x100

Fo q = 1x10-10

0.025

0.004

0

Fo q = 1x10-6

0.03

(d)

0
1x10-20

1x10-15

τ

1x10-10

1x10-5

1x100

τ

Figure 4 Temperature variation at the probe surface for test cases: (a) A = 2 × 10−3 and β = 1.71; (b) A = 1.5 × 10−2 and β = 1.71; (c) A = 2 × 10−3 and β =
2.29; (d) A = 1.5 × 10−2 and β = 2.29.

differences. In this case, the problem given by Eqs. (13a)–(13e)
was discretized in the radial direction with second-order differences 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 solution (referred to in Figure 2 by FDM-Gear) for test case d (A =
1.5 × 10−2, β = 2.29) and different values of Foq . The agreement 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.

heat transfer engineering

Figure 3 presents a comparison of the hyperbolic and
parabolic solutions given by Eqs. (26) and (27), respectively,
for test case c (A = 2 × 10−3 and β = 2.29) and Foq = 1 ×
10−10. We note in this figure that for such a small value of Foq
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 and β examined in this article.
We now examine the non-Fourier effects of the probe surface temperature variation, for Foq ranging from 10−3 to 10−10.
The results obtained for the different test cases described in
Table 1 are presented in Figure 4, a–d. These figures show
vol. 31 no. 14 2010


1132

J. N. N. QUARESMA ET AL.

0.013

0.065

0.012

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.011

A = 2x10-3; β = Fo T/Fo q = 1.71

0.01
0.009

Blackwell's Solution
Fo q = 1x10-3

0.008

Fo q = 1x10-4

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.06

A = 1.5x10-2; β = Fo T/Fo q = 1.71

0.055
0.05

Blackwell's Solution
Fo q = 1x10-3

0.045

Fo q = 1x10-4

0.04

-5

Fo q = 1x10-5

θ(A,τ)

θ(A,τ)

Fo q = 1x10
0.007

-6

Fo q = 1x10

-10

Fo q = 1x10

0.006

0.035

Fo q = 1x10-6

0.03

Fo q = 1x10-10

0.005

0.025

0.004

0.02

0.003

0.015

0.002

0.01
0.005

0.001

(a)

0
-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

(b)

0
0

-20

-18

-16

-14

-12

ln(τ)
0.013

-8

-6

-4

-2

0

0.065

0.012

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.011

A = 2x10-3; β = Fo T/Fo q = 2.29

0.01
0.009

Blackwell's Solution
Fo q = 1x10-3

0.008

Fo q = 1x10-4

Hyperbolic Heat Conduction
Dual-Phase-Lagging Model

0.06

A = 1.5x10-2; β = Fo T/Fo q = 2.29

0.055
0.05

Blackwell's Solution
Fo q = 1x10-3

0.045

Fo q = 1x10-4

0.04

-5

Fo q = 1x10-5

0.007

θ(A,τ)

Fo q = 1x10

θ(A,τ)

-10

ln(τ)

-6

Fo q = 1x10

-10

Fo q = 1x10

0.006

0.035

Fo q = 1x10-6

0.03

Fo q = 1x10-10

0.005

0.025

0.004

0.02

0.003

0.015

0.002

0.01
0.005

0.001

(c)

0
-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

(d)

0
0

-20

-18

-16

ln(τ)

-14

-12

-10

-8

-6

-4

-2

0

ln(τ)

Figure 5 Comparison of hyperbolic and Blackwell’s [27] solutions for test cases: (a) A = 2 × 10−3 and β = 1.71; (b) A = 1.5 × 10−2 and β = 1.71; (c) A = 2 ×
10−3 and β = 2.29; (d) A = 1.5 × 10−2 and β = 2.29.

that non-Fourier effects are only noticeable for very small
times; as time increases, the temperature variations gradually tend to the parabolic one. In fact, even for an extremely
large value of Foq such as 10−3, the non-Fourier effects vanish for τ > 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 deheat transfer engineering

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 examine the suitability of the hyperbolic formulation to the actual
heat conduction problem in nanofluids, during thermal conductivity measurements with the line heat source probe. For this
analysis, we bring into picture the heat transfer coefficient between the base fluid and the particles considered in the thermal
nonequilibrium model given by Eqs. (5) and (6), which results
vol. 31 no. 14 2010


J. N. N. QUARESMA ET AL.
Table 2 Dimensional times
corresponding to τ = 10−4

a–d, that Blackwell’s solution would not be considered appropriate for the measurement of the thermal conductivity for τ <
1.2 × 10−4 (ln τ = –9) for test cases a and c, and for τ < 2.5
× 10−3 (ln τ = –6) for test cases b and d. In other words, the
increase generally detected with the line heat source probe for
the thermal conductivity of nanofluids cannot be attributed to
the non-Fourier heat transfer mechanisms examined earlier. In
fact, recent theoretical predictions corroborate our findings and
demonstrate that nanoparticles and the base fluid are in thermal
equilibrium in nanofluids [23, 34].

Time
τ

t (s)

10−4
10−4
10−4
10−4

0.7
2.9
1.5
5.9

Test case
a
b
c

d

1133

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 definition 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 diameters. Only test cases a and c are examined in this figure, since
they present more significant non-Fourier effects (see also Figures 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 Foq for which the system behaves hyperbolically. 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 Foq is actually smaller than 10−5. Figure 4a–d,
shows that for Foq = 10−5, non-Fourier effects are negligible
for τ > 10−4. Table 2 gives the physical times equivalent to τ =
10−4 for 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 measurement of the thermal conductivity with Blackwell’s solution for
the line heat source probe [27–31]. Indeed, notice in Figure 5,

CONCLUSIONS
In this article we presented an analytical solution based on the
classical integral transform technique for the dual-phase-lagging
heat conduction model. The physical problem examined was
representative of that used for the measurement of thermal conductivity with the line heat source probe. Results were obtained
for the temperature variation at the probe surface, for different
combinations of nanofluids and probe diameters. Such results
were compared to those obtained with the classical parabolic

heat conduction model based on Fourier’s law, as well as to the
asymptotic solution proposed by Blackwell [27] for the line heat
source probe.
The foregoing analysis reveals that non-Fourier effects are
significant only for very small times, generally in the range
where Blackwell’s solution is not valid for the measurement of
thermal conductivity. Therefore, the increase detected with the
line heat source probe for the thermal conductivity of nanofluids
cannot be attributed to the non-Fourier heat transfer mechanisms
addressed in this article.
NOMENCLATURE

10

5

Heat Transfer Coefficient (W/m 2K)

103
102

10

a
A
b
Cf
Cs
¯f i
Foq

Test-case a, 1 nm
Test-case a, 10 nm
Test-case a, 100 nm
Test-case c, 1 nm
Test-case c, 10 nm
Test-case c, 100 nm

104

1

100

FoT

10-1
10-2
10-3
10-4
10-5
1x10-10

1x10-9

1x10-8

1x10-7

1x10-6

1x10-5

1x10-4

1x10-3

Fo q
Figure 6 Heat transfer coefficient for different nanoparticle diameters.

heat transfer engineering

h
K
Ni
NT
q
q0
r
R
t
T

probe radius
dimensionless probe radius
radius of the cylindrical medium
volumetric thermal capacity of the base fluid
volumetric thermal capacity of the nanoparticles
transformed initial condition
dimensionless relaxation time associated with the heat

flux
dimensionless relaxation time associated with the temperature gradient
heat transfer coefficient
effective thermal conductivity of the nanofluid
normalization integral
truncation order in the summations
heat flux vector
heat flux at the surface of the probe
radial variable
dimensionless radial variable
time variable
temperature

vol. 31 no. 14 2010


1134

Tf
Ts
T0
y

J. N. N. QUARESMA ET AL.

temperature of the base fluid
temperature of the nanoparticles
initial temperature
Euler’s constant

[7]

Greek Symbols
α
β
βi
i

˜i
θ
θa
θav
θp
ρ
τ
τq
τT
φ
φ¯ i

thermal diffusivity of the nanofluid
ratio of relaxation times
eigenvalues
eigenfunctions
normalized eigenfunctions
dimensionless temperature
dimensionless temperature from Blackwell’s solution
dimensionless average temperature
particular solution for the dimensionless temperature field
density

dimensionless time variable
relaxation time associated with the heat flux
relaxation time associated with the temperature gradient
homogeneous solution for the dimensionless temperature
field
transformed potentials

[9]
[10]

[11]

[12]

[13]

Subscripts
i
f
s

[8]

[14]

order of the eigenvalue problem
relative to the base fluid
relative to the nanoparticles
[15]

Superscripts

∼

[16]

integral transformed quantities
normalized eigenfunctions

[17]
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Jo˜ao N. N. Quaresma received his B.Sc. in chemical engineering from the Universidade Federal do
Par´a in 1988, and his M.Sc. in chemical engineering 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 Chemical Engineering at Universidade Federal do Par´a
(FEQ/UFPA) in 1991, where he is currently an associate professor, and served for two years as graduate coordinator for the M.Sc. program in chemical

heat transfer engineering

1135

engineering. He is the author of more than 80 technical papers in major journals and conferences, and supervisor of 13 Ph.D. and M.Sc. theses. His research 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.

Emanuel N. Macˆedo received his B.Sc. in chemical 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 Engineering 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 associate 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 coordinator for the D.Sc. program in natural resources engineering for the Amazon
region.

Henrique M. da Fonseca obtained his B.S. in mechanical 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 degree between UFRJ and the Ecole de Mines d’Albi
Carmaux, in France, where his research subject is
the evaluation of the thermal signature in microfluidic 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 mechanical 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. After 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 solution of inverse heat and mass transfer problems, as
well as the use of numerical, analytical, and hybrid numerical–analytical methods 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
the period 2005–2006. He is an associate editor of Heat Transfer Engineering,
Inverse Problems in Science and Engineering, and High Temperatures–High
Pressures.

vol. 31 no. 14 2010


1136

J. N. N. QUARESMA ET AL.

Renato M. Cotta received his B.Sc. in mechanical/nuclear engineering from the Universidade Federal 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
he joined the 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 papers and 4 books. He serves in the honorary editorial boards of the International Journal of Heat and Mass Transfer and International Communications in Heat and Mass Transfer, International Journal of Thermal Sciences, International Journal of Numerical Methods in

heat transfer engineering

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 Laboratory 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).

vol. 31 no. 14 2010


Heat Transfer Engineering, 31(14):1137–1154, 2010
Copyright C 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. AHMAD1 and STANTON BORAAS2
1

ATK Space Systems, Brigham City, Utah, USA
Morton Thiokol, Inc., Space Operations, Brigham City, Utah, USA

2

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 National Oceanic and Atmospheric Administration (NOAA), the
parent organization of the National Weather Service (NWS),
responded to these criticisms by directing that a new and correct 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 announcing 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:

Each of these models has flow and thermal errors resulting from

attempts to correct or improve upon the previous model. The
Bluestein–Zecher model did not correct for the incorrect assumption of a constant facial surface temperature in the Siple
and Passel model. The Osczevski–Bluestein model incorrectly
defined the wind chill temperature in terms of an internal body
point temperature rather than the temperature of the external
skin surface. Both models attempted to account for a boundary
layer reduction of the free-stream velocity at head level. This
head-level velocity was assumed to be at a constant 33% reduction of the NWS 10 m free-stream velocity. Their inference
that this large 33% reduction is valid for all individuals in all
locations and for all ambient conditions is inaccurate. The net
effect is that these errors result in wind chill temperature predictions that not only fail to improve upon the accuracy of the
Siple and Passel values but also demonstrate that both models
lack credibility. This paper discusses, in detail, the errors in both
models, and how a continued use of either model can actually
place the listening public at a potential health risk.

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

1137


1138

R. A. AHMAD AND S. BORAAS

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 ambient 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 theoretical 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.

Ts
T sky
Ti

·q
·q

fc

r

V
Ta
Dermis

Epidermis

a) Heat losses on layer
Ts
T sky
Ti
L

·q

nc

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 Ti . The right side is exposed to the ambient
conditions where the wind velocity is V and the air temperature
•
is Ta . The wind results in a forced convection heat loss, q f c . The
ambient air temperature Ta is presumed to be less than the outer
skin surface temperature Ts . Consequently, a radiation heat loss
•
qr 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 (Ts − Twc ) 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
•

q f c = h f c (Ts − Ta )

•

s

b) Equivalent heat loss on layer when V = 0
Figure 1 Segment of skin epidermis used in the determination of the wind
chill temperature (Twc ).

where ε is the emissivity of the skin surface, σ is the
Stefan–Boltzmann constant, and Tsky is the sky temperature. To
simplify the number of variables in this analysis, the assumption
was made that the sky temperature (Tsky ) could credibly be replaced with the ambient temperature (Ta ). This is demonstrated
in the Appendix. Consequently Eq. (2a) becomes
•

q r = εσ Ts4 − Ta4

(2a)

heat transfer engineering

(2b)

The natural convection heat loss flux shown in Figure 1b in
the absence of velocity is equal to the sum of the heat losses as
shown in Figure 1a and is
•

(1)

where hfc 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
4
q r = εσ Ts4 − Tsky

T wc

•

q nc = q

•

fc

+qr

(3)

which can now be expressed as follows in terms of the previously
defined wind chill temperature (Twc )
•

q nc = h nc (Ts − Twc )

(4)

which is again in the form of the basic Newton’s law of cooling.
The difference is that hfc has been replaced by hnc , the natural
vol. 31 no. 14 2010


R. A. AHMAD AND S. BORAAS

convection heat transfer coefficient. This laminar natural convection (nc) heat transfer coefficient (hnc ) over a vertical surface
from Jakob and Hawkins [5] and Chapman [6] can be expressed
as
h nc,l,s = C1 (Ts − Twc ) L

φ=0.25

(5a)

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, hnc ,
when a value of φ is determined. The coefficient C1 is 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 (Tfilm ) which is defined as
the average of the skin surface and air temperatures, that is, Tfilm
= (Ts + Ta )/2. For a vertical plane, the expression for C1 from
Chapman [6] is
C1 = 0.59k[(gβρ2 C p )/(µk)]φ

(5b)

where g is the gravitational constant and the volumetric coefficient of thermal expansion (β) of the air is β = (Tfilm )−1 and
is given in [◦ R−1 or K−1]. Values of C1 were calculated over a
wide range of ambient temperatures (−291.4◦ F ≤ Ta ≤ 108.6◦ F
(−179.67◦ C ≤ Ta ≤ 42.56◦ C)), plotted as a function of the film
temperature, expressed in terms of the surface and ambient temperature (Ts , Ta ) and then curve fitted using TableCurve 2D [7]
to obtain the following expression:
C1 = 0.3268329 − 9.549880x10−5 (Ts + Ta )

(5c)

where the correlation coefficient r2 = 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
Twc = Ts −

Lφ
C1

h f c (Ts − Ta )
+ εσ Ts4 − Ta4

1

/(1+φ)

(6)

where C1 is calculated from Eq. (5c). Equation (6) is the most
basic and unprecedented expression for the wind chill temperature equation. As shown, it is based on the Newton’s law of
cooling and the Stefan–Boltzmann law for radiation heat exchange between two surfaces. The conductive and radiative heat
transfer quantities (k and ε) in the equation depend upon the material properties of the surface, while the forced convective heat
heat transfer engineering

1139

transfer coefficient (hfc ) does not. Therefore, by substituting an

expression for hfc into this equation, it can be made applicable to
all two-dimensional surfaces. This was done by selecting convective heat transfer coefficients for a flat surface segment from
an aerodynamic heating document by Harms et al. [8]. This
document expresses the incompressible form of the forced convection coefficient (hfc ) in terms of the Nusselt (Nu), Reynolds
(Re), and molecular Prandtl (Pr) numbers. The forms are
Nu = h f c x/k = 0.332 (Rex )0.5 (Pr)0.33

(7a)

for laminar flow and
Nu = h f c x/k = 0.0296 (Rex )0.8 (Pr)0.33

(7b)

for turbulent flow where the characteristic length (x) in the Re
number is the length L of the segment. Because the surface
temperature of the segment may differ from the free-stream
temperature and because of the absence of a pressure gradient
within the surface boundary layer, the density varies across the
layer and the flow is considered compressible. Crocco [9] developed a solution for the compressible laminar boundary layer,
while van Driest [10] developed a similar one for a turbulent
boundary layer. The document describes and uses a successful
correlation of the Crocco and van Driest results by evaluating
all of the transport and fluid properties in Eqs. (7a) and (7b) in
terms of the following reference temperature T / suggested by
Eckert [11]:
T / = 0.5 (Ts + Ta ) + 0.22 [(γ − 1)/2] M 2 Ta

(7c)

In this expression, the first term on the right side represents a
static temperature component and the second term is a recovery
temperature or a dynamic temperature component expressed as
a function of the Mach number (M). By using T / as the reference temperature, the resulting forced convection heat transfer
coefficients for laminar flow (hfc,l ) and turbulent flow (hfc,t ) are
applicable to all velocities (V) and all surface temperatures (Ts ).
For wind chill calculations, where generally M < 0.1, the dynamic term in T / can be neglected. Substituting for Re (ρVx/µ
for vertical plate or ρVD/µ for a cylinder) in the preceding equation, replacing the density in terms of pressure using the ideal
gas law (ρ = P/( /MW)T), and comparing the compressible
with the incompressible hfc , it can be shown, with some manipulations and keeping track of English units, that Eqs. (7a) and
(7b) would reduce to the following forms of the compressible
forced convection coefficient: for the laminar flow along the
segment,
h f c,l,s =

0.00963 (PV)0.5
[0.5 (Ts + Ta )]0.04 L 0.5

(8a)

and for turbulent flow,
h f c,t,s =
vol. 31 no. 14 2010

0.0334 (PV)0.8
[0.5 (Ts + Ta )]0.576 L 0.2

(8b)

1140

R. A. AHMAD AND S. BORAAS

where the subscripts fc, l, and s stand for the forced convection, 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 pressure (lbf/ft2), the ambient (Ta ) and surface temperature (Ts ) 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 hfc in the units of Btu/hr-ft2-◦ F. The resulting
value for hfc can be converted to SI units to obtain W/m2-◦ C by
multiplying by 5.6784. In this form, values of hfc can be readily
expressed in terms of the variables (P, V, Ta , and Ts ), which
are the primary variables in this study. The characteristic length
L (vertical segment length) used in the preceding equations applies to the situation where forced convection is visualized as
flow in a direction perpendicular to the vertical segment, bifurcating 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
⎧
⎤⎫1 /(1+φ)
⎡
0.00963 (P V )0.5
⎪
⎪
⎨ Lφ
(Ts − Ta ) ⎥⎬

⎢
0.04 0.5
L
Twc = Ts −
⎦
⎣ [0.5 (Ts + Ta )]
⎪
⎪
⎩ C1
⎭
4
4
+ εσ Ts − Ta
(9)
which is a detailed expression for the wind chill temperature. In
this equation, the length L is the characteristic length for natural convection parallel to the vertical segment. The coefficient
0.00963 is a dimensionless constant.
The laminar forced convection coefficient (hfc l,s ) for the twodimensional segment shown in Eq. (8a) and used in Eq. (9) for
the wind chill temperature must now be replaced with the corresponding 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 convection coefficient (hfc ,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 incompressible form of the laminar forced convection coefficient (hfc ) 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 cylinder diameter (D). This equation is a special case of the general
equation for forced convection over a cylinder, NuD = h D/k
= C (ReD )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 lamheat transfer engineering

inar flow at the stagnation point. Thus, this special case applies
to laminar stagnation heat transfer for all velocities. Neglecting
the compressibility effects and the usage of the reference temperatures (T /) of Eq. (7c), as done before, Eq. (10a) reduces to
the following form for the laminar forced convection coefficient
for a cylinder as used in this study,
h f c,l,c =

0.03238 (P V )0.5
[0.5 (Ts + Ta )]0.04 D 0.5

(10b)

where again in these equations P is the ambient pressure (lbf/ft2),
the ambient (Ta ) and surface temperature (Ts ) 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 diameter expressed in ft. The coefficient 0.03238 is a dimensionless
constant. The similarity of this expression to that for the twodimensional 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 (hfc ,l,c /hfc ,l,s ), is 3.36,
which shows that the forced convection cooling of the cylinder
with its longitudinal axis normal to the wind direction is 3.36
times greater than that for a two-dimensional surface aligned so

as to be parallel to the wind. Perhaps this explains why a person
facing into the wind on a cold wintery day may instinctively
turn his head to the side to lessen the cold sensation. Replacing
Eq. (8a) with Eq. (10b) in Eq. (9) gives
⎧
⎡
⎤⎫1 /(1+φ)
0.03238 (P V )0.5
⎪
⎪
⎨ Lφ
(Ts − Ta ) ⎥⎬
⎢
0.04
0.5
(T
)]
+
T
D
[0.5
Twc = Ts −
⎣
⎦
s
a
⎪
⎪
⎩ C1
⎭

+ εσ Ts4 − Ta4
(11a)
which becomes the analytical equation for the wind chill temperature where the coefficient 0.03238 is a dimensionless constant.
This equation correctly considers the head more like a vertical
cylinder than a vertical plane. Aside from the well-established
natural and forced convection heat transfer coefficients correlations used here, this expression is theoretically derived based
on the well-known Newton’s and Stefan–Boltzmann laws for
convection and radiation heat transfer, respectively. Therefore,
this expression for the wind chill temperature constitutes an
analytical, exact, closed-form solution.
In Eq. (11a), the surface emissivity (ε) of the human head was
determined 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. The
head 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 be
closely approximated by a 7-inch (17.78 cm) diameter cylinder
that 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
vol. 31 no. 14 2010


R. A. AHMAD AND S. BORAAS

convection heat transfer, respectively. All temperatures (Twc , Ts ,

Ta ) are expressed in ◦ F absolute (or ◦ R) and C1 is determined
from Eq. (5c). At this point, exponent φ is unknown.
It was noted in Eq. (5a) that φ 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
equation is
Twc (V = 0) = Ts
Lφ
−
εσ (Ts + Ta ) Ts2 + Ta2 (Ts − Ta )
C1

1

/(1+φ)

(11b)

Equations (11a) and (11b) are two equations in the two unknowns Twc and φ, each of which can now be determined in
terms of the only remaining variable, the air temperature Ta .
The values of φ were determined based on the initial decay
curves of the calculated Twc vsersus low wind speeds (0 < V
(mph) < 4 (0 < V (km/h) < 6.4)) for ambient temperatures considered. A plot of φ as a function of Ta was curve fitted using
TableCurve 2D [7] to obtain the following expression for φ:

φ = a + b exp[−(Ta /c)]

(11c)

where a = 0.46259934, b = 0.077254543, and c = −59.573525
and where the correlation coefficient is r2 = 0.998884458. 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. The wind chill temperature (Twc ) at a given ambient temperature (Ta ) can now be
determined from Eq. (11a) after first determining φ 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 incorrect assumption that the skin surface temperature (Ts ) remains
constant during the entire period of the skin’s exposure. The consequence of this assumption was that the Siple and Passel model
would predict wind chill temperatures (Twc ) that would be lower
or colder than the expected theoretical values. Recognizing that
the skin surface temperature cannot remain constant with increasing exposure time but rather must decrease, Bluestein and
Zecher attempted to determine this decrease through impleheat transfer engineering

1141

mentation of what they called “modern heat transfer theory.”
Such a decrease would mean that their wind chill temperatures
would 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 their
model.
In the development of their model, Bluestein and Zecher
also attempted to correct for what the JAG/TI believed to be
“overestimates” of the wind in the Siple and Passel model. The
wind 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 some
reduction in the velocity at the head due to its proximity to the
ground. In response to this, Bluestein and Zecher made the assumption 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 reduction
was immediately questioned. Their inference that this large 50%
assumption is valid for all individuals regardless of their location
and the magnitude of the wind velocity was known to be totally
inaccurate. Head-level wind reduction is the result of the head’s
immersion in either a wind-generated boundary layer or within
the flow separation region on the leeward side of an obstruction
upwind of the individual. In either case, the degree of immersion,
which in turn determines the level of wind reduction, depends
entirely upon the individual’s location relative to an obstacle and
on the magnitude of the wind velocity. Consequently, Bluestein
and Zecher’s 50% across-the-board assumption for all cases is
incorrect. This was the second error in their model.
The following subsections entitled “Surface Temperature Error” and “Wind Reduction Error” will explain in detail the basis
for each error.

Surface Temperature Error
Bluestein and Zecher were aware of the fact that Siple and
Passel’s assumption of a constant skin surface temperature (Ts )

would mean wind chill temperatures (Twc ) colder than the theoretical values. Equation (11a) confirms this as an actuality. On
the right side of Eq. (11a), the skin surface temperature (Ts )
will always be about 91.4◦ F (33.0◦ C) upon initial exposure to
ambient conditions; this is 7.2◦ F (4.0◦ C) less than the normal
body core temperature of 98.6◦ F (37◦ C). For a given set of
ambient conditions (P, V, Ta ), if the skin surface temperature
(Ts ) is 91.4◦ F (33.0◦ C), the wind chill temperature (Twc ) takes
on a specific value from Eq. (11a). Now suppose that after an
extended 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 (Twc ) that is warmer than
corresponding value of Twc when the skin surface temperature
was 91.4◦ F (33.0◦ C). Bluestein and Zecher failed to theoretically determine this warming in the wind chill temperature due
to a skin surface temperature decrease and no mention of these
results is found in their paper. Instead they added the wind
vol. 31 no. 14 2010


R. A. AHMAD AND S. BORAAS

− 0.447V + 6.6858V

32

48

64

80

97

113
27

Bluestein & Zecher Results,
Table 2 [2]

60

16

Siple & Passel Results,
Table 1 with WRF = 0 [1]
o

40

o

4

∆T wc , F ( C)
o

20

o

-7

T a , F ( C)
40 (4.4)

6.8 (3.8)

0

-18
20 (-6.7)

-20

-29

11.6 (6.4)
-40

o

16

Twc ( C)

0
80

-40
0 (-17.8)

-60

-51

-80

-62

-20 (-28.9)

-100

-73
15 (8.3)

-40 (-40)

-120

-84
0

Twc = 91.4 − 0.04544[10.45
0.5

V (km/h)

o

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 result 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 correction and the 50% wind reduction 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 obtained 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 Error,” 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:

Twc ( F)

1142

10

20

30

40

50

60

70

V (mph)

](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 Twc and Ta 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 − Ta )

(13)

With WRF = 0.33 in Eq. (13), the wind chill temperature (Twc )
was calculated for every ambient temperature (Ta ) 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 increase. Upon examination of these results, it is apparent that
the Bluestein–Zecher model shows, at most, a possible warming 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 Ta = −40◦ F (−40◦ C). This
heat transfer engineering

Figure 2 Comparison of Bluestein and Zecher [2] results with those of Siple
and Passel [1].

is incompatible with the very nature of the wind chill temperature which by definition will substantially increase (become
warmer) as the facial temperature decreases. This has already
been demonstrated above using Eq. (11a) but now a specific example will be cited. Consider winter conditions with Ta = 0◦ F
(−17.78◦ C) and V = 20 mph (32.19 km/h). Upon initial exposure 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 was
unfortunate enough to be continuously exposed to the ambient
until his skin surface temperature reached the freezing point
of 32◦ F (0◦ C). Now his wind chill temperature from Eq. (11a)
would be Twc = −40.2◦ F (−40.11◦ C). This shows a large 14◦ F
(7.78◦ C) warming of the wind chill temperature and serves to
emphasize the error in Bluestein–Zecher model, which shows,
at most, 2◦ F (1.11◦ C) of warming.
vol. 31 no. 14 2010

R. A. AHMAD AND S. BORAAS

V (km/h)

0

16

32

48

64

80

80

97 113
27

Bluestein & Zecher
Results, Table 2 [2]
Siple & Passel Results,
Table 1 with WRF = 0.33
[1]
o
o
∆T wc , F ( C)
1.8 (1)

o
o
T a , F ( C)

60

40

20

40 (4.4)

4

-7

20 (-6.7)

o

Twc ( C)

-18

o

Twc ( F)

0

16

-20

-29
1.3 (0.7)

-40

-40
0 (-17.8)

-60

-51
-20 (-28.9)

-80

-62

0 (0)
-100

-73
-40 (-40)

-120

-84

0

10

20

30

40

50

60

70

V (mph)
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)

heat transfer engineering

1143

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-m
level. Steadman then used Eq. (14) to determine the average
wind velocity (¯v) over the height of a human body and found
this average velocity to be 57% of the free-stream value. Based
on this value, Bluestein and Zecher chose to assume the average
velocity to be 67% of the free-stream value at head level. This
33% reduction (or WRF = 0.33) is another way of stating their
assumption that the free-stream velocity is 50% greater than
that at head level. There is no dispute with Buckler’s empirical
results or in Steadman’s evaluation of these results; neither is
there any dispute with Bluestein and Zecher’s subsequent 50%
assumption based upon these results. What is disputed is that
this 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-generated
boundary layer as evidenced by the fact that his boundary layer
was 33 ft (10.06 m) thick and its edge coincided with the NWS
10-m level. This is clearly shown in Eq. (14). Its thickness
certainly could be expected in the Saskatoon area where the
wind 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 fact
that the boundary-layer thickness is directly proportional to the
wind–ground surface contact length. Reduce this length and the
boundary layer thickness is reduced. Reducing the thickness
changes 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. This
would have resulted in a shortened contact length and a boundary
layer thickness less than 33 ft (10.06 m). Steadman’s calculation
would then have determined the average velocity over the body
to be greater than 57% of the free-stream velocity, that is a value
closer to the free-stream velocity. Consequently, Bluestein and
Zecher might now have concluded that the free-stream velocity
would be only slightly greater than that at head level. It is now
easy to visualize Buckler having made his measurements so far
upwind from his actual measurement station that the boundary
layer thickness he measured became equal to the height of the
base of a man’s head above the ground. This would mean no
reduction in free-stream velocity at head level or WRF = 0. Since
moving upwind is the equivalent to a shortening of the windground surface contact distance and since the starting point or
the origin of this length will always be at some point downwind

from 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 buildings
and not far removed from an obstruction such as an individual
might be on the open windswept plain of Saskatoon. As a result,
such an individual will experience a small and possibly nonexistent WRF, certainly not a WRF = 0.33. This convincingly
shows that the wind reduction depends on the location of the
vol. 31 no. 14 2010


1144

R. A. AHMAD AND S. BORAAS

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% assumption 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 individual may be immersed. Since this boundary layer is dependent
upon the magnitude of the wind velocity and the individual’s location relative to an obstruction, it must be calculated in each situation. 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 individual’s head
• the velocity profile within the boundary layer
Unfortunately this information is so dependant upon an individual’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 retarding 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
WRF = 1 − v/V

(15)

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 increase δ. 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).
heat transfer engineering

V
Ta
Turbulent Boundary
Layer
Obstruction

δ

l

x

D

Figure 4 Wind generated turbulent boundary layer.

Consider the schematic shown in Figure 4 where an obstruction
such 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 a
vortex 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 boundary

layer thickness (δ) at the individual’s location and whether or
not the individual’s head is immersed in it. If it is not, then the
head height, which is the distance of the base of the head above
the ground surface, is greater than δ and the head is exposed to
free-stream conditions so that WRF = 0. This is the situation in
Figure 4. Clearly this illustrates the importance of knowing the
values of δ for a range of velocities (V) and wind/surface contact
distances (l) that might be encountered. Without knowing δ, the
WRF cannot be determined, and if it cannot be determined,
it cannot be correctly estimated. When Bluestein and Zecher
assumed 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 analyses
that 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 layer
thickness (δ) for the turbulent flow is
δ/l = 0.37[(V l)/ν]−0.2 = 0.37(Rel )−0.2

(16)

where the kinematic viscosity (ν) of air is 150 × 10−6 ft2/s
(13.94 × 10−6 m2/s). With the velocity in the Reynolds number
(Rel ) expressed in mph, Eq. (16) becomes
δ/l = 0.0589 (V l)−0.2

(17)

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 at
the 5 ft (1.52 m) level. Each circled intersect point represents
the maximum wind/surface contact distance (lmax ) at a given
velocity 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 (lmax ) is 648 ft
(197.51 m) and the head is above the boundary layer edge and
vol. 31 no. 14 2010


R. A. AHMAD AND S. BORAAS

1145
V (km/h)

l (m)
0.0

76

152

229

305

381
2.4

8

V , mph (km/h)

32.2

48.3

80.5

WRF > 0
D , l max (ft)

25 (40.2)

112.7

128.7

1.8

100 ft
(30.5 m)

WRF > 0
D

243.8

WRF = 0

700

213.4

30 ft (9.1 m)
WRF = 0

600

40 (64.4)

96.6

274.3

800

10 (16.1)

6

64.4

900

D , lma x (m)

0

16.1

182.9

l max

500

152.4
0

10

20

30

40

50

60

70

80

90

V (mph)

Figure 6 Maximum wind/surface contact distance (lmax ) and obstruction distance (D) for turbulent boundary layer thickness (δ) of 5 ft (1.52 m).

4

1.2

2

0.6

0
0

250

500

750

1000

δ (m)

δ (ft)

70 (112.7)

0.0
1250

l (ft)

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 (lmax ), 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, lmax increases. This is more clearly demonstrated
in Figure 6 where the lmax 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 lmax value at a given velocity; it would shift the (lmax ) curve of
Figure 6 slightly downward. Lacking the information required
to correct for this slight difference, the lmax curve of Figure 6 is
presumed to be sufficiently accurate for all surfaces.
Figure 6 shows lmax 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 lmax as shown in Figure 4. The
problem here is that x is not a fixed quantity but rather increases
heat transfer engineering

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 separation 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 velocity range and adding them to the lmax distances 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 the
two 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 obstruction height is believed to be a small fraction of lmax so
that ignoring this effect should not result in any significant error. Distance D can now be considered for all conditions as an
approximate, although realistic, distance between an individual
and the obstruction that determines the flow field in his presence.
Curve D, now replacing curve lmax , 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 20
mph 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-life
situations in which the currently used 50% wind reduction is
incorrectly 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 layer
is
v/V = (y/δ)1/n

(18)

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
vol. 31 no. 14 2010


1146

R. A. AHMAD AND S. BORAAS
D (m)

0

305

610

914

1219

1524

1829

2134

0.4

2438
0.4

V , mph (km/h)

0.3

0.3

0.2

0.2
5765 ft
(1757m)

70 (112.7)

0.1

7885 ft
(2403 m)

0.0
0

1000

2000

3000

4000

5000

6000

7000

WRF

WRF

20 (32.2)

0.1

0.0
8000

D (ft)

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:

if δ > y, WRF = 1 − v/V = 1 − (y/δ)1/n

(19a)

if δ ≤ y, WRF = 0

(19b)

and

WRF can be calculated as a function of distance D in the following manner. For velocities (V) spanning the wind chill range,
select values of l and calculate δ 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 perfect 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 unobstructed 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 associated 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. Individuals 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 experience 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).
heat transfer engineering

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 generated turbulent boundary layer. These cases represent situations
that are relatively simple to analyze. In the other cases, where the
individual is within the turbulent region downstream of an obstruction, 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 the
separation region shown in Figure 4 moves away from the obstruction 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 region where WRF = 0.33. This is the only other instance where
Bluestein and Zecher’s value of WRF = 0.33 would correctly
apply.
The preceding discussion has proven the inaccuracy of a

WRF = 0.33 in rural areas. Now look at an urban area. Individuals in an urban area may be subjected to a combined effect of a
boundary layer and one or more separation regions. An individual located at some distance downstream from an obstruction on
a clear street with buildings on either side may 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 along
the street, the individual may experience an interrupted boundary 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 individually 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 value
and not the result of the individual being immersed in a wind
generated boundary layer. The first is an increase in the NWS
10-m free-stream value that could occur in a large urban area as a
result of what Schwerdt [17] referred to as “air funneling around
tall buildings.” This increase could be computed knowing the
size, number, and the layout of the buildings. The second refers
to a case where the NWS 10m free-stream value of the velocity
would be decreased. Picture the previous illustration of the wind
blowing down the 1500 ft (457.21 m) length of street, being
vol. 31 no. 14 2010

R. A. AHMAD AND S. BORAAS

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 < lmax downstream of the re-attachment
point (x), the WRF = 0 from Figure 6. Now suppose l > lmax . 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 boundary layer analyses to determine a correct wind reduction factor
(WRF).

1147

been valid for individuals in all locations and for all ambient
conditions.
In summary, the Bluestein–Zecher model was expected to
predict warmer wind chill temperatures than the Siple and Passel model as a result of the skin temperature correction and
the presumed wind reduction at head level. With its incorrect
skin temperature correction and its unrealistic 33% head level
wind reduction, 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. 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 temperature due to the skin surface temperature decrease. The balance
of 13◦ F (7.22◦ C) was due to their failure to apply a correct wind
reduction factor. Without these errors, the Bluestein–Zecher
model is essentially no different than the Siple and Passel model
it was intended to replace. When this model was being used it
misinformed the public of wind chill temperatures that were
at least 15◦ F (8.33◦ C) warmer than the theoretical values. Actions like this can lead to complacency on the part of the public
whereby they become less concerned about the possibility of
facial freezing when in reality they are at a greater risk because
freezing may be imminent.

Osczevski–Bluestein Model
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 sheltering 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 greatest benefit to the greatest 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
heat transfer engineering

The currently used Osczevski–Bluestein model as described
in the JAG/TI [3] and [4] papers was intended to improve
upon the earlier Bluestein–Zecher model. But no such improvement took place. In fact, reference [4] has identical wind chill
temperature curve fit equations and resulting wind chill temperature charts to that of reference [3]. On the contrary, the
Osczevski–Bluestein model has even less credibility than the
Bluestein–Zecher model, which, at least, reverts back to the
Siple and Passel model when its two errors are removed. This is
not the case for the Osczevski–Bluestein model, which not only
includes the same wind reduction error as the Bluestein–Zecher
model but also introduces an errant concept of the term “wind
chill temperature” that actually compounds the inaccuracy that

already exists in the Bluestein–Zecher model. This is explained
as follows.
In the Osczevski–Bluestein model, the wind chill temperature is defined as the temperature that the individual senses. This
is the temperature on the inner surface of what its authors call
the thermal resistance layer. In the human body, the location of
the sensory nerve endings, where the external skin surface temperature (Ts ) is being sensed, is at the interface of the dermis and
epidermis layers as shown in Figure 1. Therefore, the thermal
resistance layer described by the authors is the approximate 1
mm thick epidermis and the sensed temperature (Ti ) is the temperature at the interface. Although Ti is the temperature that the
individual senses, it definitely is not the wind chill temperature
(Twc ). This is visually apparent from Figure 1b and mathematically obvious from Eq. (4) where Twc is determined in terms
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R. A. AHMAD AND S. BORAAS

of Ts and not Ti . This demonstrates, without question, that the
wind chill temperature must be defined in terms of the theoretical skin surface temperature and not the temperature that the
individual senses. Furthermore, the sensed temperature is subjective 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 person.” (p. 2) Osczevski and Bluestein’s usage of a 95th percentile
of cheek thermal resistance may have averaged out the “subjectiveness” 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 unfortunate 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 ambiguous 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:
Twc,s = 35.74 + 0.6215Ta − 35.75V 0.16 + 0.4275Ta V 0.16 (20)
where the sensed wind chill temperature (Twc, s ), which is Ti , and
the ambient temperature Ta are in ◦ F and V is in mph. This wind
chill temperature (Ti ) will be much warmer than the theoretical
wind chill temperature (Twc ) as defined by Eq. (11a). It will
naturally be much warmer than the skin surface temperature
(Ts ) by virtue of the fact that it is a temperature at an internal
body point. These values of Twc, 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 Twc = −54.2◦ F (−47.89◦ C). The Bluestein–Zecher

model, from Table 2 of their paper, states that the wind chill
temperature is Twc = −27.5◦ F (−33.06◦ C). But this is 26.7◦ F
(14.81◦ C) warmer than the theoretical value because of the errors in their model. From Eq. (20), the Osczevski–Bluestein
model predicts a Twc = −22◦ F (−30◦ C). This is a value that is
32.2◦ F (17.88◦ C) warmer than the theoretical wind chill temperheat transfer engineering

ature. For the same ambient Ta = 0◦ F (−17.78◦ C) but with V =
60 mph the Osczevski–Bluestein model predicts a sensed wind
chill temperature Twc = −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 the
Osczevski–Bluestein model cannot in all seriousness be considered to have any real value. Continuing to use this model, as
is presently being done, puts the individual at an even greater
risk than with the Bluestein–Zecher model because of its much
warmer wind chill temperature predictions. This could lead to
an even greater complacency on the part of members of the
public whereby they no longer consider the possibility of facial
freezing but rather dismiss it altogether.

CONCLUSIONS
The Bluestein–Zecher and Osczevski–Bluestein wind chill
models are both afflicted with errors. Their development was
initiated in an attempt to correct for an error in the Siple and
Passel model that was then currently in use. Siple and Passel had
incorrectly assumed that the facial surface temperature would
remain constant and not decrease with increasing exposure time.
This assumption would result in the Siple and Passel model
predicting wind chill temperatures that would be colder than the
theoretical values. Warmer wind chill temperatures were desired

and would be guaranteed if the facial surface temperature was
allowed to decrease with time. Bluestein and Zecher developed
a new model that was to be an upgrade of the Siple and Passel
model in which they would account for this decrease in skin
surface temperature. They also included a 33% reduction in the
wind velocity at head level, which was intended to account for
the head’s immersion in a wind/ground surface boundary layer.
The Bluestein–Zecher model failed on both accounts. First, it
failed by determining, at most, only a 2◦ F (1.11◦ C) of warming
in the wind chill temperatures due to the skin surface temperature decrease whereas a theoretically derived equation for the
wind chill temperature shows the increase to be on the order
of 14◦ F (7.78◦ C). Second, the assumed 33% reduction in wind
velocity at head level was found to be 5 to 7 times larger than it
should 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 public
that 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 it
replaced. No improvement took place. First, it assumed the
same 33% reduction in wind speed at head level as in the
Bluestein–Zecher model. Second, it redefined the very meaning
of the term “wind chill temperature” by saying that this temperature is the “sensed” temperature (Ti ) on the inner surface of
the outer skin layer, the epidermis, or what the authors call the
vol. 31 no. 14 2010