Opimizing hea echangers Jean-Luc Thiffeaul Deparmen of Mahemaics, Universiy of Wisconsin Madison, 48 Lincoln Dr., Madison, WI 5376, USA wih: Florence Marcoe, Charles R. Doering, William R. Young (Daed: 3 Sepember 15) I. PROBLEM SETUP Consider he advecion-diffusion equaion for a passive scalar θ(, ), adveced by a seady velociy field u(), wih Dirichle boundary condiions on some domain : θ + u θ D θ, u ˆn, θ, (1) wih u. We ake θ(, ) θ (), so θ(, ). Inegraing (1) over, we have θ + u θ D θ. () The advecion erm vanishes since he walls are impenerable, and we have θ D θ ˆn ds : F [θ] (3) where ˆn is he ouward normal o. This saes ha he average θ changes according o he flu hrough he surface. Since θ(), θ poins owards he inerior of, and he inegrand on he righ-hand side of Eq. (3) is negaive (or zero). Thus hea is leaking ou of he domain, and he ulimae sae has θ everywhere. The hea flu is solely deermined by D ˆn θ a he boundary. Our problem is ha here is no velociy field in (3), so here is nohing o opimize direcly. This is a similar siuaion o he freely-decaying problem wih Neumann boundary condiions. From (1), we define he linear operaor and is formal adjoin L : u D (4) L : u D. (5) The adjoin is compued via inegraion by pars, which gives rise o hree boundary erms: f Lg f (u D )g dv fg u ˆn ds D f g ˆn ds + D g f ˆn ds + g ( u D )f dv L f g. These are noes for a alk given in he Physical Applied Mah group meeing, Madison, WI.
The firs boundary erm vanishes since u ˆn on he boundary. The ne wo will ypically vanish because of some combinaion of zero boundary condiions on f and g and heir gradiens. Here we will ypically have f g on he boundary. II. DERIVATION OF THE EXIT TIME EQUATION The Green s funcion P (,, ) saisfies he Fokker Planck equaion (also called he Kolmogorov forward equaion) P + LP, P, >, (6) wih iniial condiion P (,, ) δ( ). This gives he probabiliy densiy of finding a paricle a (, ) if i was iniially a (, ). The survival probabiliy of finding he paricle anywhere in a ime is S(, ) P (,, ) dv. (7) From his we find he firs passage ime densiy f(, ), which is he probabiliy ha a paricle has firs reached he boundary a ime : f(, ) S. (8) The epeced ei ime τ(, ) (measured from ) is hen τ(, ) ( )f(, ) d ( ) S d [( )S] + S(, ) d. S(, ) d Recall ha P (,, ) saisfies he Kolmogorov backward equaion wih respec o (, ): P + L P, P, <, (9) wih erminal condiion P (,, ) δ( ). We ac on τ wih L : L τ(, ) L S(, ) d L P (,, ) dv d P d dv S d.
3 This las erm needs o be compued carefully: { S d lim ɛ 1 S( + ɛ) d { lim ɛ 1 S( + ɛ) d + +ɛ +ɛ +ɛ τ + lim ɛ 1 S( + ɛ) d τ + S( ) τ + 1. } S( ) d S( + ɛ) d } S( ) d We hus obain τ + L τ 1, τ. (1) The ei ime τ(, ) is measured from, so if he velociy field is ime-independen hen τ does no depend on (auonomous flow), and we can drop he τ erm in (1). (For a nonauonomous flow, he siuaion is a bi more complicaed.) Iyer e al. [1] proved an ineresing fac: here eis flows ha increase τ over pure diffusion. These are animiing flows. These flows are a lile peculiar and do no concern us here. They can only eis in noncircular domains. We can relae he escape imes o he oal ime-inegraed amoun of hea in he sysem: θ d. (11) This is an inegral over space and ime, so he smaller i is, he faser hea is flued ou of he sysem (assuming he inegral converges). We have he bound θ d θ(, ) p τ(, ) q, p 1 + q 1 1, p, q 1. (1) For he special case p 1, q (and remembering ha θ ), we find θ / θ d τ. (13) This bound is sharp when θ consiss of dela funcions concenraed on poins realizing τ. I makes sense o define he lef-hand side of (13) as he cooling ime or ranspor ime. Anoher relevan form of (1) is in erms of he L 1 norm of τ, θ d τ 1 θ. (14) Hence, bringing down τ 1 amelioraes his measure of miing, as long as θ is no oo big.
4 III. OPTIMIZATION The funcional o opimize: F[τ, u, ϑ, µ, p] 1 m τ m m ϑ(l τ 1) + 1 µ( u E) p u, (15) wih m 1. Here ϑ, µ, and p are Lagrange mulipliers. This funcional is analogous o (1.11) in [], bu he boundary condiion on τ is differen. The variaions wih respec o he Lagrange mulipliers jus reurn he consrains; he oher variaions give δf δτ Lϑ + τ m 1 ; (16) δf µu τ ϑ + p. δu (17) Using L τ 1 and (16), we have τ m ϑ. From (17) we ge µ u ϑ u τ dv ϑ (1 + τ) dv ϑ In D, we use a sreamfuncion u ẑ ψ, and ake he curl of (17): τ ϑ dv. (18) µ ψ ( τ ϑ) ẑ : J(τ, ϑ). (19) For m 1, τ 1 is he inegral of τ over, since τ. We have L τ 1 and Lϑ 1. Since (4) and (5) only differ in he sign of u, we have ϑ() τ( ) : τ (), as long as he domain and boundary condiions are symmeric under inversion. (In D his is a roaion by π abou he origin.) Hence, for m 1 and a cenrally-symmeric domain (circle, square, recangle... ) we do no need o solve he ϑ equaion. From (19) we hen see ha ψ() ψ( ). To summarize, in D for m 1 we mus solve τ J(ψ, τ) + 1, τ ; (a) µ ψ J(τ, τ ), ψ, (b) wih τ () τ( ). Consider now a channel of wih 1, 1 y 1, wih period k π/l in. This is symmeric under roaion by π, so ϑ() τ () τ( ). The conducion soluion is ( τ (y) ϑ (y) 1 1 y). (1) 4 The L 1 norm of he conducion soluion is τ 1 L 1/ 1/ ( 1 1 y) dy π 4 6k. () This depends on k, since i is an inegral per period. The Pécle number is proporional o U. For small U, we can solve he opimizaion problem from he previous secion perurbaively. Le s do he case m 1. We le ε U
y 5.5.5.5 -.5..4 (a) -.5..4 (b) -.5..4 (c) FIG. 1. The opimal mean ei ime perurbaion τ 1 () a leading order, for he opimal enhancemen wavenumber k 14.3. (a) τ 1 wih ν > ; (b) τ 1 wih ν < ; (c) The sum of (a) and (b), wih (a) ou of phase by π/. and epand as τ τ + ετ 1 +ε τ +..., ϑ ϑ +εϑ 1 +ε ϑ +..., ψ εψ 1 +ε ψ +..., µ µ +εµ 1 +ε µ +.... We won give he deails here, bu he perurbaion is relaively sraighforward. In he direcion we epand in sin k, cos k, and he wavenumbers are no coupled a leading order. We hen minimize τ 1 / τ 1 over k, o find he wavenumber ha minimizes he ei ime. Numerically, we find he maimum enhancemen occurs a k 14.3, where µ.61. Even his maimal enhancemen is very small: 1 µ τ 1 1 ε τ 1 π/6k +... 1 (.83) ε + O(ε 4 ), k 14.3. (3) The wo ypes of soluions are shown in Fig. 1. These look a bi srange, since he rolls only live in half he domain. Bu linear combinaions look more sensible and have he same efficiency. Pu anoher way, we can eiher have rolls spanning he channel, or rolls in only half channel ha urn wice as fas, since he energy is fied. [1] G. Iyer, A. Novikov, L. Ryzhik, and A. Zlao s, SIAM J. Mah. Anal. 4, 484 (1). [] J.-L. Thiffeaul, Nonlineariy 5, R1 (1), arxiv:115.111.