Heat and Mass Transfer

, Volume 48, Issue 7, pp 1125–1134

On the modeling of aiding mixed convection in vertical channels

Original

Abstract

This paper aims at showing that to prescribe a flow rate at the inlet section of a vertical channel with heated walls leads to surprising and counterintuitive physical solutions, especially when the problem is modeled as elliptical. Such an approach can give rise to the onset of recirculation cells in the entry region while the heat transfer is slightly increased under the influence of the buoyancy force. We suggest an alternative model based on more realistic boundary conditions based on a prescribed total pressure at the inlet and a fixed pressure at the outlet sections. In this case, the pressure and buoyancy forces act effectively in the same direction and, the concept of buoyancy aiding convection makes sense. The numerical results emphasize the large differences between solutions based on prescribed inlet velocity and those obtained with the present pressure-based boundary conditions.

List of symbols

a

Thermal diffusivity (m2 s−1)

A

Aspect ratio, A = H/D

cp

Specific heat (J K−1 kg−1)

cx, cz

Stretching parameters, Eq. 13

D

Plate spacing (m)

Dh

Hydraulic diameter, Dh = 2D (m)

g

Gravitational acceleration (m s−2)

GrH

Grashof number based on H, GrH = 0ΔTH3/ν02

h

Heat transfer coefficient (W m−2 K−1)

H

Channel height (m)

k

Thermal conductivity (W m−1 K−1)

L

Channel length in the spanwise direction (m)

\( \dot{m} \)

Mass flow rate (kg s−1)

nx, nz

Numbers of grid points in x- and z-directions

p

Pressure (Pa)

ps

Pressure at the outlet section (Pa)

Pr

Prandtl number, Pr = ν0/a0

Q

Heat flux (W)

Qen

Enthalpy heat flux (W)

Q2w

Convective heat flux along the two channel walls (W)

Re

Reynolds number based on Dh, Re = w0Dh0

Ri

Richardson number, Ri = Gr/Re2

Sc

Area of the channel cross section, Sc = DL (m2)

t

Time (s)

T

Temperature (K)

u, w

Velocity components (m s−1)

x, z

Coordinates (m)

Greeks

β

Coefficient of thermal expansion, β = 1/T0 (K−1)

ΔT

Temperature difference, ΔT = (Th − T0) (K)

μ

Dynamic viscosity (Pa s)

ν

Kinematic viscosity (m2 s−1)

ρ

Density (kg m−3)

θ

Dimensionless temperature ratio, θ = (T − T0)/ΔT

τ

Dimensionless time

Subscripts

a, b

Analytical solutions

h

Hot wall

H

Quantity based on channel height

nc

Natural convection

0

Inlet section

Superscripts

Averaged quantity

*

Dimensionless quantity

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Laboratoire de Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRSUniversité Paris-EstMarne-la-ValléeFrance

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