# Mixed convection heat transfer from a horizontal channel with protruding heat sources

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## Abstract

A numerical investigation is carried out to study fluid flow and heat transfer characteristics of conjugate mixed convection from a two dimensional horizontal channel with four protruding heat sources mounted on one of the finite thick channel walls. The flow is assumed as laminar, hydrodynamically and thermally developing. Water and FC70 are the fluids under consideration. The geometric parameters such as spacing between the channel walls (*S*), size of protruding heat sources (*L*_{h}×*t*_{h}), thickness of substrate (*t*) and spacing between heat sources (*b*) are fixed. Results are presented to show the effect of parameters such as Re_{ S }, Gr _{ S } ^{*} , Pr, *k*_{p}/*k*_{f} and *k*_{s}/*k*_{f} on fluid flow and heat transfer characteristics. Using the method of asymptotic expansions, correlations are also presented for the maximum temperature of heat source.

## Keywords

Heat Transfer Mixed Convection Heat Transfer Characteristic Horizontal Channel Fluid Flow Characteristic## List of symbols

*A*Aspect ratio,

*L*/*S**b*Spacing between the heat sources, m

*c*Specific heat, J/kg K

*g*Acceleration due to gravity, 9.81 m/s

^{2}- Gr
_{S}^{*} Modified Grashof number, based on volumetric heat generation,

*g*β Δ*T*_{ref}*S*^{3}/ν^{2}- Gr
Grashof number defined with wall temperature,

*g*β(*T*_{H}−*T*_{C})*L*_{c}^{3}/ν^{2}*k*Thermal conductivity, W/mK

*L*_{h},*t*_{h}Width and height of the protruding heat source, respectively, m

*L*_{c},*t*_{c}Height and thickness of the cavity wall, respectively, m

*L*Length of the channel walls, m

- Nu
_{avg} Average Nusselt number based on

*L*_{c}, \(\int_0^1 {{\text{Nu}}_Y {\text{d}}Y} \)- Nu
_{Y} Local Nusselt number based on

*L*_{c}, −(∂θ/∂*X*)_{X=0}*p*Pressure at any location in the computational domain, Pa

*P*Non-dimensional pressure at any location in the computational domain

- Pe
_{S} Peclet number based on

*S*,*u*_{∞}*S*/α- Pr
Prandtl number, ν/α

*q*_{v}Volumetric heat generation from the protruding heat sources, W/m

^{3}- Re
_{S} Reynolds number based on

*S*,*u*_{∞}*S*/ν- Ri
_{S}^{*} Modified Richardson number based on

*S*, (Gr_{ S }^{*}/Re_{ S }^{2}) or \((g\beta \Delta T_{{\text{ref}}} S/u_\infty ^2 )\)*S*Spacing between the channel walls, m

*t*′Time, s

*t*Thickness of the plate, m

*T*Temperature, K

*u*Horizontal velocity, m/s

*U*Non-dimensional horizontal velocity,

*u*/*u*_{∞}*v*Vertical velocity, m/s

*V*Non-dimensional vertical velocity,

*v*/*u*_{∞}*x*,*y*Horizontal and vertical distances, respectively, m

*X*,*Y*Non-dimensional horizontal and non-dimensional vertical distances,

*x*/*S*,*y*/*S*, respectively

## Greek symbols

- α
Thermal diffusivity of the fluid, m

^{2}/s- β
Isobaric cubic expansivity of the fluid, −1/ρ(∂ρ/∂

*T*)_{ P }, 1/K- δ
Convergence criterion, in fractional form, |(φ

_{new}−φ_{old})/φ_{new}|- ɛ
_{1},ɛ_{2},ɛ_{3},ɛ_{4},ɛ_{5} Parameters in the asymptotic expansion

- φ
Any variable (

*U*,*V*or θ), over which convergence is being tested for- ν
Kinematic viscosity of fluid, m

^{2}/s- θ
Non-dimensional temperature at any location in the computational domain, (

*T*−*T*_{∞})/Δ*T*_{ref}or (*T*−*T*_{C})/Δ*T*_{ref}- ρ
Density of the fluid, kg/m

^{3}- τ
Non-dimensional time,

*u*_{∞}t’/*S*- ψ
Non-dimensional stream function,

*U*=∂ψ/∂*Y*,*V*=−∂ψ/∂*X*

## Subscripts

- c
Cavity

- C
Cold

- f
Fluid

- h
Heat source

- H
Hot

- max
Maximum

- new, old
Values of the dependent variables (

*U*,*V*, θ) obtained from the present and previous iterations- p
Protruding heat source

- ref
Reference value

- s
Substrate

- ∞
Ambient

## Miscellaneous symbol

- Δ
*T*_{ref} Reference temperature difference, (

*q*_{v}*L*_{h}*t*_{h}/*k*_{f}) or (*T*_{H}−*T*_{C}), K

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