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Frequency response and instantaneous temperature profile of cold-wire sensors for fluid temperature fluctuation measurements

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Abstract

The information of the wire response is necessary for the estimation of corrections and uncertainty of temperature measurements. This paper describes the theoretical response of cold-wire sensors to temperature fluctuations in a fluid flow. Existing transfer functions of cold wires are approximate and implicit functions of frequency. We present the exact solutions of heat conduction equations for a cold wire and stubs taking account of the prong effect. Because the solutions have simple forms of elementary functions, we can easily calculate the frequency response of cold wires. Sample calculations are given under several typical conditions. Also, the instantaneous temperature profiles of a cold wire are obtained for the first time.

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Abbreviations

c :

specific heat

d :

wire diameter, 2r

h :

heat transfer coefficient

i :

probe current

K :

(r/r s )2 (λ/λ s )

L :

dimensionless wire length, l/r

l :

wire length

L c :

dimensionless cold length, l c /r

l c :

cold length, \(\{ 2h/r\lambda ) - i^2 {\text{ }}R_0^* \beta /(\pi r^2 {\text{ }}\lambda {\text{)\} }}^{ - {\text{ }}1/2}\)

M :

time constant, ϱ cr/(2 h)

R :

dimensionless wire radius, hr/λ

R * :

electrical resistance per unit length

R Emphasis>/* a :

electrical resistance per unit length at temperature θ a

R *0 :

electrical resistance per unit length at temperature θ 0

r :

wire radius

T :

dimensionless time, h 2 κ t/λ 2

t :

time

U a :

fluid velocity

X :

dimensionless coordinate, x/r

x :

coordinate along wire

Z :

radius ratio between stub and wire, r s /r

α :

electrical resistivity

β :

thermal coefficient of electrical resistance

κ:

thermal diffusivity, λ/(ϱ c)

λ :

thermal conductivity

ϱ :

density

η :

gain

η t :

plateau value of gain

Θ :

dimensionless temperature, \((\theta - \theta _\infty )/(\overline {\theta _p } - \overline {\theta _a } )\)

Θ b :

\((i^2 /\pi ){\text{ \{ (}}l_{cs} /r)^2 {\text{ }}(R_{as}^* /\lambda _s ) - (l_c /r)^2 (R_a^* /\lambda )\} /\overline \theta _p )/(\overline {\theta _p } - \overline {\theta _a } )\)

Θ c :

dimensionless temperature of junction of wire and stub, \((\theta _c - \theta _\infty )/(\overline {\theta _p } - \overline {\theta _a } )\)

Θ m :

dimensionless mean temperature of wire, \((\theta _m - \theta _\infty )/(\overline {\theta _p } - \overline {\theta _a } )\)

θ :

temperature

θ a :

fluid temperature

θ c :

junction temperature between wire and stub

θ m :

mean temperature of wire

θ 0 :

standard temperature

θ ∞ :

wire temperature of infinite length without fluid temperature fluctuations, \(\overline \theta _a + i^2 (l_c /r)^2 {\text{ }}R_a^* /\pi {\text{ }}\lambda )\)

ΔΘ a :

dimensionless amplitude of fluid temperature fluctuations, \(\Delta \theta _a /(\overline \theta _p - \overline \theta _a )\)

Δθ a :

amplitude of fluid temperature fluctuations

τ p :

time constant ratio of stub to wire, M p /M

φ :

phase angle

Ω :

dimensionless frequency, λ 2 ω/(h 2 κ)

ω :

frequency of temperature fluctuations

p :

prong

s :

stub

References

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Authors and Affiliations

  1. Dept. of Mechanical Engineering, Nagoya Institute of Technology, 466, Gokiso-cho, Showa-ku, Nagoya, Japan

    T. Tsuji, Y. Nagano & M. Tagawa

Authors
  1. T. Tsuji
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  2. Y. Nagano
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  3. M. Tagawa
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Tsuji, T., Nagano, Y. & Tagawa, M. Frequency response and instantaneous temperature profile of cold-wire sensors for fluid temperature fluctuation measurements. Experiments in Fluids 13, 171–178 (1992). https://doi.org/10.1007/BF00218164

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  • DOI: https://doi.org/10.1007/BF00218164

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