Design of Radiant Enclosures using Inverse and Non-linear Programming Techniques

Traditionally, radiant enclosures have been designed using a forward " trial-and-error " methodology. Recently, however, inverse and optimization methodologies have been applied to design radiant enclosures. Both of these methodologies solve the design problem faster, and the quality of the solutions is usually better than those obtained through the forward design methodology. This paper presents forward, inverse, and optimization formulations of the infinitesimal-area method that can be used to solve design problems involving radiant enclosures. The inverse and optimization methodologies are then demonstrated and compared by using them to design a 2-D radiant enclosure containing a transparent medium. 3 NOMENCLATURE A Coefficient matrix for radiosity equation A ij Element of A matrix b(u) See Eqs. (3) and (4) b i b(u i) b Right-hand vector for radiosity equation b′ Right-hand vector for 1 st-order radiosity sensitivity b′′ Right-hand vector for 2 nd-order radiosity sensitivity C(u) Parametric vector defining enclosure geometry dF i-strip j View factor from u i to an infinitely long strip centred at u j E b (u) Emissive power at u, W/m 2 E b target (u) 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 target (u) 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 ij Element of U matrix U Parameter for enclosure representation V Matrix obtained by singular value decompositon, Eq. (15) V ij Element of V matrix W Singular value matrix w i Singular value corresponding to i th diagonal element of W X Solution vector for radiosity x′ 1 st-order radiosity sensitivity vector x′′ 2 nd-order radiosity sensitivity vector 4 Greek Symbols α k Step …