14th Australasian Fluid Mechanics Conference
Adelaide University, Adelaide, Australia
10-14 December 2001
An investigation on the length to diameter ratio of hot wire filament in
turbulence measurements
J.D. Li
School of the Built Environment
Victoria University of Technology, MCMC 14428, Melbourne, 8001 AUSTRALIA
Abstract
Heat generation and transfer along hot wire filament and its
supports are analyzed numerically. From this analysis, it is shown
that the ratio between the heat conduction and heat convection
depends on the length to diameter ratio of the hot wire, the length
of the wire support, the types of wire material and the Reynolds
number based on the wire diameter. By keeping this ratio to a
constant, a formula based on correlating the numerical data is
given to determine the optimum length to diameter ratio for given
Reynolds number and hot wires.
distribution along the wire depends on the balance of the heat
generation, heat losses from convection and conduction,
Reynolds number should have some effect and be considered in
determining the proper length to diameter ratio of the hot wires.
In this paper, the heat balance equation over the hot wire under
steady conditions will be solved numerically. From this the effect
of the stub length and Reynolds number will be discussed. By
specifying a relative heat conduction loss from the hot wire, the
length to diameter ratio at different Reynolds numbers can be
determined for a given stub length. The effects of using different
materials for the hot wire will also be discussed.
Introduction
Hot wires have been used to measure velocities of turbulent gas
flows for many years. Over the years, it has become a golden rule
that the wire length to diameter ratio should be above 200 as
according to Hinze [6] and Champagne, Sleicher and Wehrmann
[3]. The reason for this requirement is to have a hot wire filament
with nearly uniform temperature distribution in its central region
and the hot wire can have a high sensitivity. Because of this
requirement, the length of the hot wire in turbulence
measurements cannot in general be too short. For example, for a
5 µ m wire, the nominal hot wire length will in general be about 1
mm. On the other hand, it is also well known that the smallest
length scales of the turbulent flows decrease rapidly with
increasing Reynolds numbers, and hot wires cannot in general
resolve the smallest turbulent length scales unless the Reynolds
number is low or the experimental facility is large. In turbulence
measurements at high Reynolds numbers, hot wires of smaller
diameters (such as 2.5 and 1 µ m) are generally used. Because of
the strength requirement and difficulty of handling the wire, hot
wire of 1 µ m diameter is about the smallest that can be used.
Heat transfer analysis of the hot wire filament
Figure 1 shows a sketch of the hot wire geometry. In the figure,
2l is the length of the hot wire filament, dw the diameter of the
wire, L the length of the stub, and Ds is the diameter of the stub.
Under steady conditions, the heat generation, heat convection
and heat conduction satisfy the following equation (for detailed
derivation, see Perry [8])
d 2θ
+ ( q − ξ )θ + Z = 0
dη 2
θ −θ
θ = w a
θa
Many people have analyzed the temperature distribution along
the hot wire filament, Betchov [1], Corrsin [3], Champagne et al.
[2], Hinze [6], Freymyth [5] and Perry [8], to mention a few. In
these analyses, in order to obtain theoretical relationship between
the temperature distribution and the length to diameter ratio of
the wire, it was normally assumed that the hot wire support (the
stub, see figure 1 below for definition) is the prong of the probe
holder itself and the temperature at the end of the hot wire is that
of the ambient fluid. The exceptions to this assumption are those
of Champagne et al. [2] and Perry [8]. In Champagne et al. [2],
the experimental results for the temperature at the end of the hot
wire filament were used. In Perry [8], the temperature
distribution along the stub was assumed to be a linear function of
the stub length. Because of this, the effect of the hot wire
support on the heat balance of the hot wire has not been studied.
η =
x
l
q=
4 I 2 RxC l 2
( )
kw
d
π
Z=
4
π
(1)
I 2 Rx l 2
( )
kwθ a d
ξ = 4 Nu
kf
l
( )2
kw d
Here θ w is the temperature of the hot wire, θ a the ambient
temperature, x the distance from the midpoint of the wire, I the
electrical current in the wire, Rx the wire resistance per unit
length, kw the thermal conductivity of the wire, kf the thermal
conductivity of the fluid, C the linear temperature coefficient of
the electrical resistance, Nu the Nusselt number of the flow over
the wire, and d is the diameter of the wire which is a fuction of x.
Along the wire, d = dw while over the stub, d = Ds. For
simplicity, the change in diameter has been assumed to be a step
function and the stub is cylindrical.
It will be assumed that the wire temperature distribution is
symmetrical about its midpoint and the temperature at the
junction between the stub and the prong is that of the ambient
fluid because the prong is much bulkier in volume than the stub
and the wire. Thus the temperature distribution along the wire
satisfies the following boundary conditions,
In all these analyses, the effect of Reynolds number has not been
discussed. The Reynolds number effect is normally ‘hidden
inside’ the Nusselt number and it is normally assumed that the
Nusselt number is about 1 without regarding the flow velocity
and the diameter of the wire used. Since the temperature
513
dθ
= 0 at η = 0
dη
θ = 0 at η = 1 + L / l
For θ a = 293 K, the temperature at this junction is above 140 oC
at l/dw = 25 and 190 oC at l/dw = 200. These are much higher than
the ambient temperature. Champagne et al. [2] measured the
temperature of the hot wire with very small stub. The temperature
at the end of the hot wire was about 60 oC in the wires they used,
which is still much less than those given in figure 2. The reason
for this difference is the long stub length used in the present
simulation. Simulation using L/l = 0.1, which is close to the stub
length they used, gives the temperature at the end of the wire to
be about 60 oC. This shows that the length of the stub support for
the hot wire has a large effect on the temperature distribution
along the wire and thus a large effect on the uniformity of the
temperature distribution along the wire.
(2)
1
0.9
0.8
Figure 1 Sketch of a hot wire filament with its support and prongs.
0.7
Equation (1) together with the boundary conditions (2) were
solved numerically. Five thousand uniform grid points were used
between η = 0 and η = 1+L/l. In solving the equations, the
Nusselt number given by Fand [4]
0.6
Nu = (0.35 + 0.56 Re w0.52 ) Pr 0.3
θ
1
0.5
0.4
0.3
0.2
(3)
η
0.1
was used. Here Rew is the Reynolds number of the wire or stub
and Pr is the Prandtl number of the fluid. Equation (3) is valid
for 0.1 < Rew < 10000 as according to Holman [7]. For constant
temperature operation of the hot wire, the electrical current I
needs to be specified so that
C θ a ∫ θ dη = R − 1
decreasing l/dw
0
0
0.5
1
1.5
2
Figure 2 Temperature distribution along the hot wire and its stub support
at different l/dw.
0.7
( 4)
0
0.6
Here R is the overheat ratio of the hot wire. The increase of the
electrical resistance of the stub at temperature above the ambient
fluid has been neglected because the overall resistance from the
stub is much smaller than that from the wire. Because of (4),
equation (1) and (2) were solved using an iteration procedure.
For airflow under laboratory conditions, Pr = 0.7 and θ a = 293 K
were assumed.
0.5
L/l = 0.25
ε
0.4
L/l = 1
0.3
0.2
0.1
The effect of hot wire support
Figure 2 shows the numerical solutions for the temperature
distribution along a 5 µ m platinum wire with L/l = 0.5, Ds/dw = 5,
Rew = 3.2 based on the wire diameter (which corresponds to a 10
m/s air velocity) and R = 1.8 at four different length to diameter
ratios, l/dw = 25, 50 100, and 200.
0
0
50
100
150
200
l/dw
Figure 3 Relative heat loss from the hot wire.
Figure 3 shows the relative conduction heat loss from the hot
wire
K
ε =
(5)
K+ φ
From figure 2, it can be seen that as the length to diameter ratio
increases, the temperature distribution along the wire becomes
more and more uniform. This uniform temperature distribution is
desirable for velocity measurement of the fluid using the hot wire
and is achieved when the heat convection from the wire by the
flow is much larger than the heat conduction to the prongs.
at two different L/l with l/dw changing from 25 to 200. The wire
material and the other operating conditions for the wire are the
same as those in figure 2. Here K is the heat conduction loss at
the junction between the hot wire and the stub and φ is the heat
convection loss from the wire. They are calculated according to
Figure 2 shows that the temperature distribution over the stub is
close to a linear function near the junction between the stub and
the prong. However near the junction between the wire and the
stub, the temperature profile is nonlinear, especially at large
length to diameter ratios. Because of this, the assumption of
Perry [8] that the temperature distribution along the stub is a
linear function is not supported.
The other point that can be seen from figure 2 is that the
temperature at the junction between the hot wire and the stub is
quite high and increases with increasing length to diameter ratio.
π d w2 θ a dθ
kw
|η = 1
4
l dη
φ = π k f lNu ( R − 1) / C
K= −
(6)
Freymuth [5] shows that 1-ε is the dynamic response of the hot
wire to the ambient temperature fluctuation. It can be seen from
figure 3 that the relative conduction heat loss decreases with
increasing l/dw and L/l. At l/dw = 25, ε = 43% and 63% for L/l =
514
1.0 and 0.25, respectively. Champagne et al. [2] estimated the
heat conduction loss from the temperature measurement along
the hot wire and found that for wires at l/dw ≈ 100 with small
stubs, the end conduction loss is about 7%. Figure 3 shows that
for 7% end conduction loss, l/dw needs to be at 130 (or length to
diameter ratio of 260) for L/l = 0.25 while for L/l = 1, this can be
achieved at l/dw = 85. This shows that in using hot wire to
measure turbulent flows, a simple length to diameter ratio of 200
will not be sufficient. The length of the stub needs also be taken
into account. In using hot wires in turbulent measurements,
certain stub length is helpful in avoiding the influence of the
vortex shedding from the prongs. However this stub cannot be
too long because the cut off frequency of the dynamical response
of the stub depends on its heat capacity or the size of the stub
support. It is felt that L/l = 1∼ 2 is probably the right length to
chose for 5 µ m wire. For wires of smaller diameter, L/l could be
longer.
Material
Tensile strength (MPa)
Heat capacity (J/g-oC)
Thermal cond. (W/m-K)
Melting point (oC)
Electrical Res. (ohm-cm)
-1
Temp. Coef. (K )
Platinum
Tungsten
620
0.134
69.1
1769
750
0.134
170
3370
Titanium
Nickel
560
0.32
10
1240
1.9× 10-5
5.7× 10-6
8.2× 10-5
0.0035
0.0052
0.0038
Table 1 Mechanical, thermal and electrical properties for platinum,
tungsten and Titanium Nickel.
0.7
0.6
Tungsten wire
0.5
Freymuth [2] also suggested that the dynamic response of the hot
wire to the velocity fluctuation depend on
dφ
ε '=
(7 )
dφ + dK
Platinum wire
ε
0.4
0.3
Titanium Nickel wire
0.2
where dφ and dK are the fluctuations of heat losses from
convection and conduction, respectively, for a given velocity
fluctuation. Calculation of ε ’ vs l/dw for the same operating
condition as those of figures 2 and 3 (L/l = 1) with a small
perturbation in Reynolds number shows that this is generally
close to 1 (about 97% for 25 < l/dw < 200).
0.1
0
0
50
100
150
200
l/dw
Figure 4 Relative heat conduction loss for three different hot wire
materials.
The effect of hot wire material
The hot wire materials will also affect ε because of the different
thermal conductivity. Table 1 shows three different materials,
platinum, tungsten and titanium nickel (55% titanium and 45%
nickel). The mechanical strengths of the three materials are about
the same although the melting point temperatures are quite
different. Since the hot wire is normally operated at less than 300
o
C, hot wire made of titanium nickel will not melt. The difference
in thermal conductivity between tungsten and platinum is about a
factor a 2 while that between tungsten and titanium nickel is 17
with the tungsten having the highest thermal conductivity and
titanium nickel the lowest.
The effect of Reynolds number
So far, all the simulations have been undertaken using Rew = 3.2,
this corresponds to 10 m/s of air velocity for a 5 µ m wire. Since
the Nusselt number depends on Reynolds number, the
temperature distribution along the hot wire will also depends on
the Reynolds number. For a given wire diameter, the higher the
Reynolds number, the higher will be the Nusselt number and
hence the higher the convection heat loss. Because of this, it is
expected that the length to diameter ratio for the hot wire can be
reduced at high Reynolds numbers for a given relative
conduction heat loss, say 7%.
Figure 4 shows the ε vs l/dw for the three different hot wire
materials listed in table 1. The operating conditions for the wires
are the same as those used for figure 2 and all wires are 5 µ m and
L/l = 1. The figure clearly shows that the thermal conductivity of
the wire material has a strong effect on the relative heat
conduction loss. For tungsten wire, the length to diameter ratio
needs to be at 270 (or l/dw = 135) for a 7% relative heat
conduction loss, for platinum wire this ratio is 170 and for
titanium nickel wire this is only 64. This shows that the normally
used length to diameter ratio of 200 of hot wire is only for
platinum wires. For tungsten wire, the length to diameter ratio
needs to be nearly 60% longer than that for platinum wire while
for titanium nickel wire the length to diameter ratio needs to be
only one third of that for platinum wire. Table 1 shows that the
mechanical and electrical properties of the titanium nickel wire
are close to that of platinum and tungsten wires, and titanium
nickel could be a very good hot wire material.
Figure 5 shows l/dw for a 5 µ m platinum wire at different
Reynolds numbers with ε = 7% and L/l = 1. As expected, the
required l/dw decreases with increasing Reynolds number. At Rew
= 1, l/dw = 108 and at Rew = 100, l/dw will only be 41. The
application of this result is for turbulent measurements at high
Reynolds numbers such as experiments using hot wire in the
Superpipe at Princeton University (Zagarola and Smits [9]). In
the Superpipe, the Reynolds numbers based on the pipe diameter
Re can be varied from 3.1× 104 to 35× 106. The corresponding
Rew based on a 5 µ m wire will be from 1.2 to 1354. According to
figure 5, the length to diameter ratio for a 5 mm wire can be 210
at the lowest Re in the Superpipe and reduce to 40 at the highest
Reynolds number Re = 35× 106. This is a factor of 5 reduction in
length to diameter ratio for the hot wire. According to this, using
a smaller length to diameter ratio at high Reynolds numbers will
improve the spatial resolution of the wire without sacrificing the
sensitivity of the hot wire. The frequency response of the hot
wire will also improve as the length of the wire is reduced
because experience shows that the frequency response of short
wires is higher than that for long wires.
515
the local mean velcoity), the Kolmogorov length scale η =
(ν 3/ε )1/4 decreases from 0.37 mm at Re = 3.1× 104 to 0.003 mm at
Re = 35× 106. Here ν is the kinetic viscosity. This is a factor of 63
decrease and much larger than the factor of 5 reduction that can
be achieved for the length to diameter ratio of the hot wire at the
same Reynolds number range. However, since the length scale of
the energy containing eddies is fixed for a given pipe diameter,
the decrease of hot wire length at high Reynolds number will
help to resolve a wilder range of length scales into the inertial
range.
140
120
100
80
60
40
20
Rew
0
0.1
1
10
100
1000
Conclusions and discussion
The steady state heat balance equation in a hot wire and its
cylindrical stub support has been solved numerically. The
solutions show that the temperature uniformity in the wire
depends on the length to diameter ratio, the length of the stub,
hot wire materials and the Reynolds number of the flow. The
results show that a long stub support will result in a more
uniform temperature distribution in the wire and thus reduce the
heat conduction loss to the end. By specifying a given percentage
for the relative heat conduction loss, the length to diameter ratio
for the hot wire can be determined. It is found that for hot wires
of long stub support, small thermal conductivity, and at high
Reynolds number, smaller length to diameter ratio can be used
for turbulence measurements without sacrificing the sensitivity
and frequency response of the hot wire. The normally used length
to diameter ratio of 200 is only for platinum wire at a nominal
mean airflow flow velocity of 10 m/s. For tungsten wire, this
ratio should be increased to about 270. Based on this, it is
suggested that platinum wire should be preferred over the
tungsten wire. A better hot wire material is found to be titanium
nickel which has similar mechanical and electrical properties as
those of platinum. However, it is not clear how difficult it is in
comparison with the platinum to make very small diameter wires
using titanium nickel.
10000
Figure 5 Preferred l/dw with 7% relative conduction heat loss at different
Reynolds numbers.
The effect of Nusselt number, thermal conductivity of the hot
wire material and the stub length on the optimum length to
diameter ratio for a given relative conduction heat loss of 7% can
be correlated as
L
L
k
2l
= 3.84(1 − 0.248 ln + 0.0963(ln ) 2 )( w )0.49
dw
l
l
Nuk f
(8)
L
for
< 3.62
0.25 ≤
l
k w 0.49
L
2l
for
= 3.23(
)
≥ 3.62
dw
Nuk f
l
The effect of Reynolds number is ‘hidden’ inside the Nusselt
number and can be determined by combining (3) and (8). The
effect of L/l on the length to diameter ratio was correlated using
the numerical results from simulation at Re = 10 and 0.25 < L/l <
4.0 using platinum wire and the Reynolds number effect was
correlated using the results shown in figure 5. Equation (8) was
checked with numerical results at different kw and Reynolds
numbers, and good agreement has been found.
The normally used length to diameter ratio of 200 is based on a
Nusselt number of about 1. At low air velocity and ambient air
density this is approximately true for a 5 µ m wire. For air flow at
high velocity and high density (such as flows in the Superpipe),
the Nusselt number can be much higher than 1. For this situation,
the heat convection from the wire will be much higher than that
at low velocity, and small length to diameter ratio for the hot
wire can be used.
Although figure 5 and equation (8) both show that the length to
diameter ratio of hot wire can be reduced at high Reynolds
number, it is still recommended that hot wires of smaller
diameter should be used at high Reynolds numbers if it is
possible. However the reduction of length to diameter ratio with
wires of small diameter will not be as much as that for large
diameter wire. For a given flow, the Reynolds number based on
wire diameter will be smaller for small diameter wires than that
for large diameter wires. For example, at Re = 35× 106 in the
Superpipe, Rew = 1354 for a 5 µ m wire while Rew = 271 for a 1
µ m wire. Using (8), 2l/dw can be 40 for the 5 µ m wire with L/l =
1 but it can only be 70 for the 1 µ m wire. Choosing the length to
diameter ratio according to this, the physical length of the 5 µ m
wire at Re = 35× 106 will be 0.2 mm while it is 0.07 mm for the 1µ m wire. Thus the physical length of the wires of small diameter
is still shorter than the wires of large diameter for a specified
relative heat conduction loss and will give better spatial
resolution than large wires.
References
[1] Betchov, R. Theorie non-lineaire de l’anemometre a fil chaud,
Proc. K. Ned. Akad Wet., 52, 195-207, 1949.
[2]
Chamgagne, F. H., Sleicher, C. A. and Wehrmann, O. H.
Turbulence measurements with inclined hot wires, pt1. Heat
transfer experiments with inclined hot wire, J. Fluid Mech. 28,
pp.153-175., 1967.
[3] Corrsin, S. Turbulence: Experimental methods, Handbuch der
Physik, 9/2, pp.523-590, Springer Verlag, Berlin, 1963.
[4] Fand, R. M. Heat transfer by forced convection from a cylinder to
water in crossflow, Int. J. Heat Mass Transfer, 8, p.995, 1965.
[5] Freymuth, P. Engineering estimation of heat conduction loss in
constant temperature thermal sensors, TSI Quarterly, 3, issue 3,
1979.
It is also not claimed here that the spatial resolution problem of
using hot wire to measure the turbulent flows at high Reynolds
numbers has been solved. The smallest length scale in
turbulence, the Kolmogorov length scale, decreases much faster
with increasing Reynolds number than those shown in figure 5.
For example, in the Superpipe at y/δ = 0.1 (where y is the local
distance from the pipe surface and δ is the pipe radius) and
assuming that the energy dissipation ε equals the energy
production -uvdU/dy (-uv is the Reynolds shear stress, and U is
[6] Hinze, J. O. Turbulence, 2nd Edition, McGraw Hill, 1975.
[7]
[8]
Holman, J. P. Heat Transfer, 7th Edition, McGraw Hill, 1992.
Perry, A. E. Hot wire anemometry, Oxford University Press, New
York, 1982.
[9] Zagarola, M. V. and Smits, A. J. Mean-flow scaling of turbulent
pipe flow, J. Fluid Mech., 373, 33-79, 1998.
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