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Nanofins as a Means of Enhancing Heat Transfer: Leading Order Results

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The porous media approach is being adapted to a system of nanoparticles that are attached to a solid surface (a metal wire) embedded into a stagnant fluid, forming by design nanofins around the wire. The analyzed system resembles the Transient Hot Wire experimental method used in evaluating the thermal conductivity of a fluid or nanofluid suspensions. Since the attachment of the nanoparticles to the wire is done by design (as distinct from uncontrolled agglomeration around the wire), one major objective in such a design is attempting to enhance the heat transfer from the wire. The latter objective is analyzed via a short times approximation of the solution. Preliminary results based on the leading order solution shows that such a heat transfer enhancement is indeed possible and presents major advantages compared to commonly used macro-fins.

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A :

Cross sectional area of the wire (m2)

A sf :

Solid–fluid interface heat transfer area (m2)

c p :

Fluid’s specific heat for the single phase (J/kg K) (dimensional)

c p,f, c s :

Fluid and solid phases specific heat, respectively (J/kg K) (dimensional)


Exponential integral function

h :

Integral heat transfer coefficient for the heat conduction at the the solid–fluid interface, (W/m3 K) (dimensional)

i :

Electric current passing through the wire (A)

i s :

Electric current supplied by the source (A)

k :

Thermal conductivity of the fluid for the single phase (W/m K) (dimensional)

k s :

Effective thermal conductivity of the solid phase (W/m K) (dimensional)

\({\tilde {k}_{\rm s}}\) :

Thermal conductivity of the solid phase (W/m K) (dimensional)

k f :

Effective thermal conductivity of the fluid phase (W/m K) (dimensional)

\({\tilde {k}_{\rm f}}\) :

Thermal conductivity of the fluid phase (W/m K) (dimensional)

k eff :

Effective thermal conductivity of the medium (W/m K) (dimensional)

\({k_{{\rm eff}}^\ast }\) :

Dimensionless effective thermal conductivity of the medium

L :

The length of the wire (m)

\({\dot {q}_L }\) :

Amount of heat generated per unit length of wire (W/m)

\({q_{r^{\ast}}}\) :

Radial component of the heat flux (W/m2)

r * :

Radial coordinate (m) (dimensional)

r :

Dimensionless radial coordinate, equals \({{r_\ast }/{\sqrt{{k_{\rm f}}/h}}}\)

\({r_{{\rm w}^{\ast}}}\) :

Wire’s radius (m) (dimensional)

R :

Electrical resistance (Ω) (dimensional)

R w :

Electrical resistance of the wire (Ω)

R o :

A reference value of the electrical resistance (Ω)

R 1, R 2 :

Fixed electrical resistances in the Wheatstone bridge (Ω)

R 3 :

Variable electrical resistance [Ω], (potentiometer) in the Wheatstone bridge

R th,eff :

Effective thermal resistance (m K/W)

\({R_{\rm th,eff}^\ast }\) :

Dimensionless effective thermal resistance

t * :

Time (s) (dimensional)

t :

Dimensionless time, equals t * h/γ f

T :

Temperature, equals (T *T o) (K) (dimensional)

T C :

Coldest boundary temperature (K) (dimensional)

V :

Voltage drop across the wire (V)

z * :

Vertical coordinate (m) (dimensional)

α :

Fluid’s thermal diffusivity for the single phase, equals k/ρ c p (m2/s) (dimensional)

α f :

Fluid phase effective thermal diffusivity, equals k f/γ f (m2/s) (dimensional)

α s :

Solid phase effective thermal diffusivity, equals k s/γ s (m2/s) (dimensional)

χ :

Heat capacity ratio, equals γ f/γ s

γ s :

Solid phase effective heat capacity (dimensional)

γ f :

Fluid phase effective heat capacity (dimensional)

γ Eu :

Euler–Mascheroni constant, equals 0.5772156649

\({\varepsilon }\) :

Solid fraction, equals (1 − φ )

θ s :

Dimensionless solid phase temperature, equals \({{(T_{\rm s^{\ast}} -T_{\rm o} )}/{\left( {{\dot {q}_L }/{2\pi k_{\rm f}}}\right)}}\)

θ f :

Dimensionless fluid phase temperature, equals \({{(T_{\rm s^{\ast}} -T_{\rm o} )}/{\left( {{\dot {q}_L }/{2\pi k_{\rm f}}}\right)}}\)

φ :

Porosity/fluid fraction (dimensionless)

\({\phi }\) :

Dependent variable following the transformation \({\phi =\eta \left( {{\partial T}/{\partial \eta }}\right)}\)

ρ e :

Electrical resistivity of the wire (Ω m)

ρ :

Fluid’s density for single phase (kg/m3)

ρ f :

Fluid phase density (kg/m3)

ρ s :

Solid phase density (kg/m3)

σ :

Thermal conductivity ratio, equals k f/k s (dimensionless)

η :

Similarity variable, equals \({{r_{\ast}^{2}}/{4\alpha t_{\ast}}}\) (dimensionless)

κ :

Thermal diffusivity ratio, equals α f/α s = σ /χ (dimensionless)


Corresponding to dimensional values, except for cases where there is no ambiguity as listed in this nomenclature


Related to the solid phase


Related to the fluid phase


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Correspondence to Peter Vadasz.

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Vadasz, P. Nanofins as a Means of Enhancing Heat Transfer: Leading Order Results. Transp Porous Med 89, 165 (2011).

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  • Nanofins
  • Transient hot wire
  • Porous media
  • Nanofluids
  • Effective thermal conductivity