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Frontiers of Mechanical Engineering

ISSN 2095-0233

ISSN 2095-0241(Online)

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Front. Mech. Eng.    2015, Vol. 10 Issue (3) : 263-276    https://doi.org/10.1007/s11465-015-0345-y
RESEARCH ARTICLE
An invariant descriptor for conjugate forced convection-conduction cooling of 3D protruding heaters in channel flow
Thiago ANTONINI ALVES(),Paulo H. D. SANTOS,Murilo A. BARBUR
Department of Mechanical Engineering, Federal University of Technology, Paraná, Brazil
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Abstract

In this research, the temperatures of three-dimensional (3D) protruding heaters mounted on a conductive substrate in a horizontal rectangular channel with laminar airflow are related to the independent power dissipation in each heater by using a matrix G+ with invariant coefficients, which are dimensionless. These coefficients are defined in this study as the conjugate influence coefficients (g+) caused by the forced convection-conduction nature of the heaters’ cooling process. The temperature increase of each heater in the channel is quantified to clearly identify the contributions attributed to the self-heating and power dissipation in the other heaters (both upstream and downstream). The conjugate coefficients are invariant with the heat generation rate in the array of heaters when assuming a defined geometry, invariable fluid and flow rate, and constant substrate and heater conductivities. The results are numerically obtained by considering three 3D protruding heaters on a two-dimensional (2D) array by ANSYS/FluentTM 15.0 software. The conservation equations are solved by a coupled procedure within a single calculation domain comprising of solid and fluid regions and by considering a steady state laminar airflow with constant properties. Some examples are shown, indicating the effects of substrate thermal conductivity and Reynolds number on conjugate influence coefficients.
Keywords channel flow      conjugate forced convection-conduction cooling      conjugate influence coefficients      discrete heating      invariant descriptor      thermal management     
Corresponding Author(s): Thiago ANTONINI ALVES   
Online First Date: 07 September 2015    Issue Date: 23 September 2015
 Cite this article:   
Thiago ANTONINI ALVES,Paulo H. D. SANTOS,Murilo A. BARBUR. An invariant descriptor for conjugate forced convection-conduction cooling of 3D protruding heaters in channel flow[J]. Front. Mech. Eng., 2015, 10(3): 263-276.
 URL:  
https://academic.hep.com.cn/fme/EN/10.1007/s11465-015-0345-y
https://academic.hep.com.cn/fme/EN/Y2015/V10/I3/263
Fig.1  Basic configuration used for the tests
Fig.2  Domain of the mathematical model
Fig.3  Three-dimensional non-uniform mesh (3D perspective)
Fig.4  Three-dimensional non-uniform mesh. (a) xy plane; (b) yz plane; (c) xz plane for y=0.16H
Fig.5  Streamlines around an array of 3D protruding heaters for Re=100 (perspective 3D view)
Fig.6  Streamlines around an array of 3D protruding heaters for Re=100. (a) xy plane for z=0; (b) xz plane for y=0.15H
Fig.7  Velocity profile around an array of 3D protruding heaters for Re=100. (a) xy plane for z=0; (b) xz plane for y=0.15H
Re Lrec/H
100 1.19
150 1.60
200 2.00
250 2.43
300 2.84
Tab.1  Length Lrec of the recirculation downstream Heater #3
Fig.8  Length of the recirculation downstream Heater #3
Fig.9  Average adiabatic Nusselt number for an adiabatic substrate and a single active heater
Re N u ad , 1 ‾ N u ad , 2 ‾ N u ad , 3 ‾
100 11.20 9.78 9.99
150 13.59 11.59 11.88
200 15.66 13.14 13.56
250 17.43 14.52 15.09
300 19.02 15.79 16.53
Tab.2  Average adiabatic Nusselt number for a single active heater
Heater C m
#1 1.2083 0.4833
#2 1.3095 0.4358
#3 1.2080 0.4575
Tab.3  Coefficients of Eq. (15) for ks/k=0
Fig.10  Isotherm maps for an adiabatic substrate and a single active heater. (a) Heater #1; (b) Heater #2; (c) Heater #3
Fig.11  Isotherm maps for an adiabatic substrate with all active heaters (1-1-1)
Test Active heater T1/°C T2/°C T3/°C
1 #1 (1-0-0) 151.75 44.95 27.80
2 #2 (0-1-0) 1.33 174.73 49.26
3 #3 (0-0-1) 0.00 1.11 171.50
4 All (1-1-1) 156.92 225.30 252.98
5 All (0.4-0.3-0.3) 62.63 72.20 78.74
6 All (0.5-0.3-0.2) 78.19 76.69 64.16
Tab.4  Temperature increase of each 3D protruding heater for Re=100 and ks/k=0
Fig.12  Conduction fraction (qcd/q) for Heater #2
Fig.13  Isotherm maps for a conductive substrate and single active heater. (a) Heater #1; (b) Heater #2; (c) Heater #3
Fig.14  Isotherm maps for a conductive substrate and all active heaters (1-1-1)
Test Active Heater ΔT1/°C ΔT2/°C ΔT3/°C
1 #1 (1-0-0) 62.60 22.49 13.40
2 #2 (0-1-0) 9.23 66.33 23.40
3 #3 (0-0-1) 1.35 9.62 66.47
4 All (1-1-1) 73.24 98.54 103.57
5 All (0.4-0.3-0.3) 28.24 31.81 32.41
6 All (0.5-0.3-0.2) 34.36 33.10 27.08
Tab.5  Temperature increase of each 3D protruding heater for Re=100 and ks/k=84
A Heat transfer area/m2
C Coefficient
cp Specific heat
g* Discrete superposition kernel function
g+ Conjugate influence coefficient
g−1 Element of G−1/(K·W−1)
G+ Matrix of conjugate influence coefficients
G−1 Matrix of inverse discrete Green’s function
h Convective heat transfer coefficient/(W·m−2·K−1)
H Channel height/ m
Hh Heater height/m
k Air thermal conductivity/(W·m−2·K−1)
kh Heater thermal conductivity/(W·m−2·K−1)
ks Substrate thermal conductivity/(W·m−2·K−1)
L Total channel length/m
Lh Block heater length/m
Ld Channel downstream length/m
Lu Channel upstream length/m
Ls Spacing between heaters/m
m Exponent
m ˙ Mass flow rate/(kg·s−1)
N Total number of heaters in the array
Nu Nusselt number
P Pressure/Pa
q Heat transfer rate/W
qcd Conductive heat transfer rate/W
qcv Convective heat transfer rate/W
qL Smaller heat generation rate/W
Re Reynolds number
S Source term/(W·m−3)
t Substrate thickness/m
T Temperature/K or °C
u x-velocity component/(m·s−1)
v y-velocity component/(m·s−1)
w z-velocity component/(m·s−1)
W Channel width/m
Wh Heater width/m
Ws Channel spanwise width/m
x Cartesian coordinate/m
y Cartesian coordinate/m
z Cartesian coordinate/m
ρ Specific mass/(kg·m−3)
µ Dynamic viscosity/(Pa·s)
Tab.6  Nomenclature
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