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A numerical and experimental analysis of the methodology of thermal conductivity measurements in fluids by concentric cylinders

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Abstract

A study of thermal conductivity measurements of fluids, using the technique of steady-state heat transfer in concentric cylinders, is presented. In order to evaluate the effect of convective flow on the measured value of conductivity, numerical results, which were obtained using computational fluid dynamics (CFD) in three dimensions, are compared with experimental data and analytical results of temperature profiles. This latter was obtained considering the hypothesis of heat transfer mechanism being entirely diffusive and solely in the radial direction. An experimental design was proposed aiming to analyze the effects of glycerol mass fraction (0%, 50 and 100%), annuli size (2.525, 4.525 and 6.525 mm) and heat rate (5, 10 and 15 W) in the formation of convective streamlines. The ratio between effective and absolute conductivities (kef/k) was used as a response to evaluate the convection intensity. The results were compared with empirical equations that correlate the ratio kef/k with the dimensionless numbers of Prandtl and Rayleigh, which are in the rate 4.37 ≤ Pr ≤ 3.8 × 103 and 9.0 ≤ Ra ≤ 4.4 × 104. An evaluation of the accuracy in measuring kef is showed based on the simulated data of temperature profiles in the axial, radial and angular directions.

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Abbreviations

Symbol:

Meaning Unit (I.S.)

A :

Area of heat transfer: m2

b :

Coefficients of the multiple regression equation: dimensionless

C, m :

Correlating parameters for the Nusselt number based on Reynolds number: dimensionless

C 1 , C 2 :

Integration constants: K

c p :

Specific heat: J/(kg∙K)

D i :

Outer diameter of the inner cylinder: m

D ii :

Inner diameter of the inner cylinder: m

D o :

Inner diameter of the outer cylinder: m

D oo :

Outer diameter of the outer cylinder: m

g :

Gravity: m2/s

h :

External convective heat transfer coefficient: W/(m2∙K)

k :

Thermal conductivity: W/(m∙K)

k ef :

Effective thermal conductivity: W/(m∙K)

k eq :

Ratio kef/k: dimensionless

L :

Concentric cylinders length: m

L c :

Annuli width: m

M :

Nylon lid width: m

Nu Di :

Nusselt number for conduction or convection within the annuli: dimensionless

Nu Dicond :

Nusselt number for conduction within the annuli from the inner cylinder: dimensionless

Nu Diconv :

Nusselt number for convection within the annuli from the inner cylinder: dimensionless

Nu Doo :

Nusselt number for external cross flow in cylinders: dimensionless

P :

Pressure: Pa

Pr :

Prandtl number: dimensionless

Pr s :

Prandtl number given at cylinder external wall temperature: dimensionless

Q :

Heat rate generated per unit volume: W/m3

q :

Heat transfer rate: W

q A :

Heat flux: W/m2

q L :

Heat transfer rate per cell unit length: W/m

r,θ,z :

Cylindrical coordinates: m, rad, m

Ra :

Rayleigh number: dimensionless

Ra Di , Ra Do , Ra Lc :

Rayleigh number based on characteristics lengths Di, Do and Lc.: dimensionless

Re Doo :

Reynolds number for external cross flow in cylinders: dimensionless

T :

Temperature: K

t :

Time: s

T 0 :

Reference temperature: K

T ∞ :

External environment temperature (air): K

T fi :

Temperature at the most inner fluid layer (fluid/inner cylinder interface): K

T fo :

Temperature at the most outer fluid layer (fluid/outer cylinder interface): K

T M :

Fluid layer mean temperature: K

T room :

Room temperature: K

T si :

Temperature at the inner surface of the inner cylinder: K

T so :

Temperature at the outer surface of the outer cylinder: K

v :

Velocity: m/s.

x,y,z :

Cartesian coordinates: m

X 1 :

Coded value of glycerol mass fraction: dimensionless

X 2 :

Coded value of annuli width: dimensionless

X 3 :

Coded value of heat transfer rate: dimensionless

Y :

Mean value of kef/k for the multiple regression equation: dimensionless

α :

Thermal diffusivity: m2/s

β :

Volumetric thermal expansion coefficient: 1/K

ΔT :

Temperature difference: K

ΔT f :

Temperature difference between the most inner and outer fluid layer (Tfi-Tfo): K

μ :

Dynamic viscosity: Pa∙s

ν :

Kinematic viscosity: m2/s

ρ :

Density: kg/m3

ρ 0 :

Reference density: kg/m3

σ :

Stefan-Boltzmann constant (σ = 5.67 × 10−8 W∙m−2∙K−4): W/(m2∙K4)

τ :

Shear stress: Pa

Φ :

Energy dissipation due to viscous forces: J/(kg∙m2)

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Authors and Affiliations

  1. Department of Food Engineering, Federal University of São João del Rei, Rodovia MG 424, Km 47, 56, Sete Lagoas, MG, 35701-970, Brazil

    Henrique C. B. Costa

  2. School of Chemical Engineering, Federal University of Uberlândia, Avenida João Naves de Ávila, 2121, Bloco 1K, Campus Santa Mônica, Uberlândia, MG, 38408-100, Brazil

    Henrique C. B. Costa, Danylo O. Silva & Luiz Gustavo M. Vieira

Authors
  1. Henrique C. B. Costa
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  2. Danylo O. Silva
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  3. Luiz Gustavo M. Vieira
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Correspondence to Henrique C. B. Costa.

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The original version of this article was revised: Equations 23, 34 and Table contained an error.

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Costa, H.C.B., Silva, D.O. & Vieira, L.G.M. A numerical and experimental analysis of the methodology of thermal conductivity measurements in fluids by concentric cylinders. Heat Mass Transfer 55, 669–683 (2019). https://doi.org/10.1007/s00231-018-2448-6

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