CHE 52 - CHPTE Sping 2005 GENEL 2ND ODE ECTION IN TUULENT TUUL ECTOS Vassilats & T, IChEJ. (4), 666 (965) Cnside the fllwing stichiety: a + b = P The ass cnsevatin law f species i yields Ci + vci =. Di Ci + i t Upn epesenting instantaneus cncentatin by the su f the tie aveaged and fluctuatin pat Ci = C% i + Ci, we get the fllwing equatin f the tie aveaged cncentatin C % i. C ~ i ~ + vci = ( Di Ci < v Ci > ) + ~ i τ In the abve bth ~ and < > epesent enseble ( tie) aveaged values. This equies a del f the velcity cncentatin css celatin and f the ate. < vci >= Dt C % i % = k < CC >= k < ( C + C)( C + C) > ~ ~ k C C + < C C > + < C C > + < C y definitin = [ C > ] < C >= < C >= 0 a% % = k CC % % + < CC > = b hgeneus ate at lcal css celatin f ean ate = + tie aveaged cncentatin cncentatin fluctuatin ean ate < hgeneus ean ate s k < C C > C % C % t keep ate finite. Epeiental veificatin sught n tubulence effects f: - instantaneus - apid - slw eactins Cnside that in a given syste the iing ate can be established by unning instantaneus eactins at β =. Then = ()
CHE 52 - CHPTE Sping 2005 If tubulence paaetes wee nt easued, ne can deteine vs z i.e ( z/ u ). Nw the ate is due entiely t iing and ne can wite d Cu k CC dz = = (2) & ( ) C = C = C (3) whee C is the css-sectinal aveage. Thus k f the syste can be deteined ve a ange f eynlds nubes. Nw the chaacteistic eactin nube is k NM = (4) k N M >> N M > N M << instantaneus eactin apid eactin slw eactin The fllwing eactins wee cnsideed: k (L/l s) at 30 C. HCl + NaOH instantaneus 2. HCl + LiOH instantaneus 3. (COOH) 2 + 2LiOH 0 alic acid 4. HCOOH + LiOH fic acid 5. CO 2 + 2NaOH.24 0 4 6. CO 2 + n NH 3 (n=-2) 5.85 0 2 7. HCOOCH 3 + NaOH 4.7 0 ethylfate t stichietic feed f eactants, β =, and f an instantaneus eactin (such as the fist tw listed abve) f eq (2) it fllws that d ( ) 2 dz = (5) u kc z = z = 0 Integatin f (5) & (5a) yields kc = + u ( z z ) Epeiental eact with 00 tubes, ID = 0.052 inches (0.3 c) 5 in lng was used. Tubes wee pessed int hles f a /4" diaete disk that clsed the uppe end f the /4" (3.75 c) diaete (5a) (6) 2
CHE 52 - CHPTE Sping 2005 lucite tubula eact. eactants ae fed t altenate tubes with e = 3,700 in tubes and 5,000 in the eact. Jet velcity = 6 ean eact velcity. u 5.85 in/ s = 40.26 c/ s. eactant cncentatin had t be nited. Deinized wate was used. eactant caying steas ae thestated at 29.9 C in a theal bath. eactin pgess is nited by adiabatic tepeatue ise. C ρc p = H ρc p = H T T (7) (8) T T is easued by a cppe cnstant in glass shielded thecuple (30 gauge) whse efeence junctin is in ne f the feed steas just befe iing. Effective tip diaete was 0.059 in (0.5 c). Measueents wee veified with fine pbes. lind uns with pue wate wee als dne and the signal was subtacted (heat f fictin, heat lsses, etc.). Measued adial T pfiles wee flat s T can be easued as equal t the cente line tepeatue. ll fu vey apid eactins behaved as instantaneus and pduced at fied β = the sae cnvesin vs distance cuve. FIGUE : eactant cnvesin f vey apid eactins as a functin f distance f stichietic feeds (β = ) [F Vassilats and Tu, IChE J., 666 (965)] FCTIONL CONVESION, 3
CHE 52 - CHPTE Sping 2005 Least squaes fit f eq (6) at β = t the data gives 9.24( z 0.44) ) ; z 0.44 in = + > (9) Ecellent staight line is btained when 0.35. vs z is pltted ecept at vey fist few data pints z < F eq (9), based n they, we cnclude that the degee f unaccplished iing is given by a hypeblic decay law. = ; z > 0.44 + 9.24 0.44 ( z ) (0) Cpaisn f fula (6) and (9) yields k 305a L = bc ls () Nw f nn-stichietic feeds the they pedicts [eq (24) and (22) f pevius pat] = + ( β ) + g γ t (2) whee γ t is given by the slutin f β g β = ( γ ) t (3) whee g( ) = iefc (4) 2 2 ac β = (5) bc Nw we knw f eq (0). Cpaisn f data and theetical pedictins was ecellent f β in the inteval (,0). 4
CHE 52 - CHPTE Sping 2005 FIGUE 2: Factinal cnvesin vs. accplished iing, vey apid eactins. Nw ne tested the assuptin f 2nd de iing law at β F β eq (5) beces d dz b a ( )( β ) u = k C Upn integatin this yields: ( / ) k b a C ln = l nβ ( β ) ( z z) (7) u Plts f l n vs z did yield staight lines f all instantaneus eactins. The slpes f the lines evealed 305 k = ψ ( β ) (8) b C a 5
CHE 52 - CHPTE Sping 2005 whee f β =, ψ = ; and f β = 0, ψ = 0.3. ased n these findings we see that N M N M b C k = a k 305ψ β = (9) ( ) k Since k 0,000 t 20,000 L/l s, indeed all f the fist 4 eactins ae instantaneus. One shuld ecall that the 2nd de decay law f geneal law. is the peculiaity f the iing device, nt a NOTE: ial dispesin effects ae negligible and PF fulas (eq 6) pduce the desied esult. F the CO 2 + 2NaOH eactin N M = 0.5 t 0 If we assue = KC C (20) Plt f l n b K( β ) C ln n a = l β ( z z ) (2) u yields staight lines but K = K( β ). F this paticula eactin k = 2,400 (L/l s). L efe we fund f instantaneus eactins that at β = 3.88 we have k = 9, 400. l s Then, L N M =.30 f the cabn diide eactin with sdiu hydide and we get K = 2770 l s. Siilaly, at L β =.26, k = 9,700( L / l s), N M = 0.63 and we find K = 3420 l s Thus, K < k, k k 6
CHE 52 - CHPTE Sping 2005 F CO 2 + NH 3 - the sae phenenn is bseved k = 585 (L/l s) k( L/ l s) N K( L/ l s) β = 2. 24,500 0.024 588 β = 5.75 2,300 0.048 336 = and the hgeneus ate law hlds with K= k = 47. (L/l s) and z 0 = 0. Hgeneus ate eactin is bseved as 0. 3 F the Methyl fate - sdiu hydide eactin ( N M 0 ) Thughut this eact the ean ate is a fied factin, K/k, f the hgeneus ean ate even thugh the fluid is becing e hgeneus as z inceases. If PF assuptins ae O.K. then these values CC K CC = k vay f 0.72 at β =.26 t 0.78 at β = 3.88 f CO2 + 2 NaOH., and ae lwe f CO 2 + nnh 3 i.e. 0 f β = 2.t 0.43 at β = 0. F inteediate eactins thee is n clea patten egading the value f the appaent ate cnstant K t be used. The appiate epessin + is valid nly in the tw liits as K k k NM N M N 0 M The actual K, K act is less than K - estiated by abve fula f apid eactins. Thus k = + = + M K k k k k K = act + N > K M ( N ) 7