Design of radiant enclosures using inverse and non-linear programming techniques

Figures & data

FIGURE 1 Boundary conditions for different design methodologies: (a) forward; (b) inverse; (c) optimization.

FIGURE 2 Parametric representation of the enclosure geometry.

FIGURE 3 Discretization of the enclosure surface.

FIGURE 4 Example of an inverse radiant enclosure problem.

FIGURE 5 Radiant enclosure design problem.

FIGURE 6 Grid refinement study.

FIGURE 7 Singular values found by decomposing the A matrix.

FIGURE 8 Distribution of q _s (u) and E _b (u) over the heater surface, found using p = 460 singular values.

FIGURE 9 Distribution of q _s (u) and E _b (u) over the heater surface, found using p = 449 singular values.

FIGURE 10 Distribution of q _s (u) and E _b (u) over the heater surface, found using p = 448 singular values.

TABLE I Heater settings corresponding to Figs. 8–10 [W/m²]

p	Φ 1	Φ 2	Φ 3	Φ 4
460	4.37 × 10^6	−1.69 × 10^6	−8.49 × 10^5	−5.50 × 10^5
449	1.87	3.78	3.87	2.44
448	0.11	−0.63	−0.67	−0.29
Φ5	Φ6	Φ7	Φ8
460	−2.57 × 10^5	−1.52 × 10^5	−3.97 × 10^4	4.83 × 10^4
449	0.90	0.13	−0.54	−0.98
448	0.11	0.31	0.48	0.65
Φ9	Φ10	Φ11	Φ12
460	−3.14 × 10^5	−3.58 × 10^5	−3.85 × 10^5	−4.00 × 10^5
449	−0.06	−0.14	0.14	0.59
448	0.49	0.37	0.23	0.05

TABLE II Values of || x ||₂ and || δ ||₂ corresponding to Figs. 8–10

Number of singular values	|| x ||2	|| δ || 2
p = 460	9.31 × 10^14	1.20 × 10^−14
p = 449	1.02 × 10^4	1.18 × 10^−25
p = 448	2.85 × 10^1	1.29 × 10^3

FIGURE 11 Distribution of q _s (u) over the design surface, found by nullifying the negative heater settings of the p = 449 solution.

FIGURE 12 Heater settings found by minimizing the non-regularized objective function, Eq. (17).

FIGURE 14 Distribution of q _s (u) over the design surface, found by minimizing non-regularized and regularized objective functions.

TABLE III Optimal heater settings for the regularized and non-regularized objective functions [W/m²]

Φ 1	Φ 2	Φ 3	Φ 4
Φ^*	1.33	2.11	2.29	−3.56
	2.54	1.17	1.25	1.24
Φ5	Φ6	Φ7	Φ8
Φ^*	2.50	−10.18	−71.25	122.8
	0.82	0.62	0.66	0.74
Φ9	Φ10	Φ11	Φ12
Φ^*	69.94	−100.0	0.10	−3.89
	0.91	0.88	0.55	0.56

FIGURE 13 Heater settings found by minimizing the regularized objective function, Eq. (32).

Table

A	=	coefficient matrix for radiosity equation
b(u)	=	see Eqs. (3) and (4)
b i	=	b(u i )
b	=	right-hand vector for radiosity equation
b ′	=	right-hand vector for 1st-order radiosity sensitivity
b ″	=	right-hand vector for 2nd-order radiosity sensitivity
C (u)	=	parametric vector defining enclosure geometry
	=	view factor from u i to an infinitely long strip centred at u j
E b (u)	=	emissive power at u, W/m^2
	=	desired emissive power distribution over the design surface, W/m^2
%EI	=	energy imbalance
F(Φ)	=	objective function
F r (Φ)	=	modified objective function, Eq. (32)
F [Φ, q s (Φ)]	=	objective function with expanded dependencies
g (Φ)	=	gradient vector
g p	=	element of the gradient vector
g(u)	=	see Eqs. (3) and (4)
g i	=	g(u i )
H(Φ)	=	Hessian matrix
H pq	=	element of Hessian matrix
k(u i , u j )	=	Kernel of Eq. (2)
N	=	number of discrete surface elements
N DS	=	number of discrete surface elements on design surface
P(u)	=	x-component of C (u)
p	=	number of singular values used to solve Eq. (16)
p k	=	search direction
Q(u)	=	y-component of C (u)
q o (u)	=	radiosity at u, W/m^2
q s (u)	=	heat flux at u, W/m^2
	=	desired heat flux distribution over the design surface, W/m^2
q s (Φ)	=	system response vector
r	=	position vector
T(u)	=	temperature at u, K
U	=	matrix obtained by singular value decompositon, Eq. (15)
u	=	parameter for enclosure representation
V	=	matrix obtained by singular value decompositon, Eq. (15)
W	=	singular value matrix
w i	=	singular value corresponding to ith diagonal element of W
x	=	solution vector for radiosity
x ′	=	1st-order radiosity sensitivity vector
x ″	=	2nd-order radiosity sensitivity vector

Table

α k	=	step size
β ij	=	blockage factor
δ	=	residual vector
ΔA	=	area of discrete surface element
ε(u)	=	surface emissivity at u
σ	=	Stephan–Boltzmann constant, 5.67 × 10^−8 W/m^2K^4
Φ p	=	pth design parameter
Φ	=	vector of design parameters (heater settings)