Unsteady conjugate problem of a dissipative fluid in a horizontal channel with a periodic variation temperature

Abstract

The unsteady two-dimensional transient heat transfer problem referring to a fully laminar flow developing in a parallel-plane channel exposed to a periodic variation surface temperature with distance is numerically studied. The effects of channel thickness, Péclet number, wall-to-fluid conductivity ratio, thermal diffusivity ratio, angular frequency and the viscous dissipation parameter are determined in the solutions. The non-linear equations are discretized by means an implicit finite difference scheme and the electric analogy to the resulting system is applied to convert these equations into a network-electrical model that was solved using a computer code (electric circuits simulator). In this scheme, only spatial discretization is necessary, while time remains as a real continuous variable, and its programming does not require manipulation of the sophisticated mathematical software that is inherent in other numerical methods. The network simulation method, which satisfies the conservation law for the heat flux variable and the uniqueness law for temperature, also permits the direct visualization of the local and/or integrated transport variables at any point or section of the medium.

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Abbreviations

A :

Ratio of diffusivities, α s/α f

B :

Dimensionless angular frequency

Br :

Brinkman number

c e :

Specific heat

C :

Capacitor

k :

Thermal conductivity

G :

Control-voltage current-source

L 0 :

Half distance between the channel walls

L 1 :

Thickness of the pipe

N :

Number of cells

Nu :

Nusselt number

Pe :

Péclet number

q :

Heat flux

R :

Resistor

t :

Time

T :

Temperature

u :

Velocity

U :

Dimensionless velocity

x :

Axial co-ordinate

y :

Vertical co-ordinate

α :

Diffusivity

β :

Angular frequency

ΔT :

Oscillation amplitude temperature

ΔX :

Axial thickness of the cell

ΔY :

Vertical thickness of the cell

Γ :

Dimensionless geometric parameter

θ :

Dimensionless temperature

μ :

Dynamic viscosity

ρ :

Density

τ :

Dimensionless time

f :

Associated to fluid

i, j :

Associated with i,j nodal point

i, i−Δ, i+Δ:

Associated to the centre, left and right position on the cell

m :

Medium value

mea :

Associated to measurement

s :

Associated to solid

w :

Solid-fluid interface

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Correspondence to Joaquín Zueco.

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Zueco, J. Unsteady conjugate problem of a dissipative fluid in a horizontal channel with a periodic variation temperature. Meccanica 43, 37–46 (2008). https://doi.org/10.1007/s11012-007-9092-4

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Keywords

  • Transient heat conduction
  • Conjugate problem
  • Viscous dissipation
  • Network method
  • Heat transfer