Apparent Emissivity in the
Base of a Cone
Cosmin DAN, Gilbert DE MEY
University of Ghent
Belgium
Overview
1.
2.
3.
4.
5.
6.
7.
Introduction
Description of the problem
Configuration factors computation
Apparent emissivity computation
Genetic algorithms (GA)
Results
Conclusions
Introduction
•
-
Why to calculate the apparent emissivity?
Build a black-body
What kind of shapes has usually a black body
Find the optimal shape
Solve the radiative energy balance equations
Write a computer program for apparent
emissivity computation
- Use a genetic algorithm optimization method
Description of the problem
• Circular cavities
– Diffuse-gray surfaces
– Known distribution of the temperature
– Only radiative heat transfer
– Known inner walls emissivity not depending
on temperature
– Known geometrical dimensions
– Unknown apparent emissivity
Description of the problem
εapp
εapp
εapp
εapp
Description of the problem
• The boundary integral equation
q( r )
1   ( r )
cos cos 
4
4
 T ( r )   [T ( r ) 
q( r )]
dS 
2
S
 (r )
 ( r )
 r  r
q(r )
q(r )
r
r

| r  r |

Description of the problem
• The net radiation method
  ij
1  j 
    Fi  j  q j 
j 1 j
j 
N
R
N

T j4 ( ij
j 1
1
 ij  
0
 Fi  j ) 
N
4
4
F

(
T

T
 i j i
j )
j 1
when i  j
when i  j
j
i
z
Description of the problem
• The net radiation method
Aij qi   Ci 
for i  j

aij  
for i  j

1  (1  εi ) Fi  j

εi
(1  ε j ) Fi  j
εj
and
– Gauss elimination method
N
ci  εi (Ti   T j4 Fi  j )
4
j 1
Configuration factors computation
A2
r2
h
r1
F12 
1
2

R2 2 
2
X

X

4
(
)


R1


A1
X  1
1  R22
R12
r1
r2
with R1 
and R2 
h
h
Configuration factors computation
Aj
R
Ai
x
Ai1

Ai2
Aj1
Aj2
1
Fi  j 
A j1 ( F j1i 2  F j1i1 )  A j 2 ( F j 2i 2  F j 2i1 )
Ai

Apparent emissivity
– Radiative heat flux leaving the cavity
– Radiative heat flux that would leave the cavity
if it were a blackbody
– Computation of the total heat flux for different
particular cases with specified temperature
– Find the highest value for apparent emissivity
• Same length
• Same open area
• Different shapes for cavity
Genetic algorithms
– System for function optimisation
– Adaptive search – iterative procedures
– Variables = structures xi, population
– Vector of parameters to the objective function
f ( p1 , p2 ,... pn )
p1  1000
x1 (t )  p1 p2 ... pn  100011001110
P(t )  x1 (t ), x2 (t ),... x N (t ) 
Genetic algorithms
Initialize population P(t)
Evaluate objective function
Complete terminate criteria
No
Select the best members
Mate and mutate
Replace old members
Yes
Printout results
Genetic algorithms
• Two steps to create a new population
– Selection of the best members for replication
– Alteration of the selected members using
genetic operators:
• Crossover
– Two parents → two offspring's
– 100011001111 and 011100110011 → 100011000011 and
011100111111
• Mutation
– Alteration of one ore more bites in parent structure
– 100011001111 → 100010001111
Genetic algorithms
Genetic Algorithm
Generate initial population
Software
Calculate the objective function
(εapp)
Generate the new population
according with performances
Check the terminate criteria
Return results
Return the value of the objective
function for each structure
Genetic algorithms
R
’
’
’
P1 (z1 ,R1 )
P2(z2,R2)
P1(z1,R1)
P0(z0,R0)
P3(z3,R3)
•Termination criteria:
Acceptable approximate solution
Fixed total number of evaluations
z
Results
7.00E-10
Summation -relative error (%)
6.00E-10
5.00E-10
4.00E-10
3.00E-10
2.00E-10
1.00E-10
0.00E+00
1
51
101
151
201
Number of the surface (-)
N
 Fij  1
j 1
where i  1...N
251
301
Results
• Configuration factors from each surface to
the open area:
– Disk-to-disk method
– Monte Carlo integration method
• Configuration factors from each surface to
all the others surfaces
– Formula for differential configuration factors
between two ring elements
– The approximated formula e-2z
Results
0.6
0.8
0.5
0.6
Fij
cone angle 60°
0.4
0.3
0.4
cone angle 15°
0.2
0.2
0.1
0
0.00
0
0.20
0.40
z/L
0.60
0.80
1.00
Results
Configuration factors [-]
0.025
0.02
differential
formula
exponential
aproximation
disk-to-disk
0.015
0.01
0.005
0
0
0.5
1
1.5
2
2.5
Cylinder length [m]
3
3.5
4
Results
Relative difference [%]
6.00E-05
disk-to-disk and
differential
formula
5.00E-05
4.00E-05
3.00E-05
2.00E-05
1.00E-05
0.00E+00
0
0.5
1
1.5
2
2.5
Cylinder length [m]
3
3.5
4
Results
Configuration factors [-]
0.03
disk-to-disk middle
area
disk-to-disk last
area
disk-to-disk first
area
0.02
0.02
0.01
0.01
0.00
0
0.2
0.4
0.6
Cylinder length [-]
0.8
1
Results
60
q[W/m^2]
cone
40
cylinder
cone-cylinder
20
0
0
0.2
0.4 z/L 0.6
0.8
1
Absolute difference [W/m^2]
Results
7
6
5
4
cylinder and conecylinder
3
cylinder and cone
2
1
0
0
0.2
0.4 z/L 0.6
0.8
1
Results
1
Apparent emissivity
εwall=0.75
εwall=0.50
0.8
εwall=0.25
0.6
cone
cylinder
0.4
0.2
0
2
4
6
8
10
L/R
12
14
16
18
20
Results
Apparent emissivity
0.975
0.970
0.965
0.960
0.955
0.950
0.945
0
500
1000
Trials number
1500
Results
R
R
’
P1 (z1’,R1’)
P2(z2,R2)
P1(z1,R1)
P0(z0,R0)
•
•
•
•
•
•
P3(z3,R3)
L=z3=30 cm;
0 ≤ z1 ≤ L; εwall=0.9
R1=R2=15 cm
εapp=0.971 for z1=0.91 cm
R1=R2=5 cm
εapp=0.975 for z1=0.01 cm
z
P0(z0,R0)
•
•
•
•
P1’(z1’,R1’)
P2(z2,R2)
P3(z3,R3)
P1(z1,R1)
z
P4(z4,R4)
L=z2=z3=z4=30 cm;
0 ≤ z1 ≤ L; εwall=0.9
R1=R2=10 cm; R3=2.5 cm
εapp=0.9986 for z1=0.01 cm
Conclusions
• New tool for determination of the maximum
value of the apparent emissivity
• Results for configuration factors were verified
and compared
• Good agreement between the results
• A software was written for the computation of the
apparent emissivity
• The software was combined with an optimisation
genetic algorithm routine
• The cylindrical cavities have higher apparent
emissivities than the conical cavities for the
same length and the same open area