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At the start of the ite ration a rather lar ge value is chosen for the Levenberg p a rameter v , and v is gradually reduced when ll £ + _1!. T hus the method start s as a steepest- descent method and it may en d as a Gaus s - Newton method in the last iterations . It is rob ust and often works well , although it is not spectacularly fast . A good , recent description of Levenber g-Marquardt methods is Fletcher 47 N umerical Methods in Reac t io n E ngineering ( 1 980 , Chap s . In the followin g w e assume that one of t he 5 and 6) .
The lar ge temp e r ature gradients which o cur radially reactor ) can be studied with sufficient accuracy in the u be s of a re fo r min without too many details in the pelle t - p hase desc ription . I f the r a e of e a t o n is moderate , the radial diffusion terms are small comp ared to the axial gradients , and one m ay r e u c e Eqs . ( 1 1 6 ) and ( 1 17) to a one - dimensional model . T here are c e rt ainly situation s w here this approximation may lead to loss o f major fe atures of the model , but the potential re d c tion in computer expenditure is so s i i ic an t that we feel it n ec s s a y to comment on p rop er aver a gi n techniques for t he radial gradients before we discuss suitab le numerical techniques for solution o f he fu ll od e l .
1 + R. F . 1 1 of F i , and � E;, i until = � s at t he given £ :::: E o · followin g procedure wit h a diagonal calculation to � Gradient of z: for bot h reactor m o de l s C ! t �. " Finally , use Eq . 12.. d r,; 50 Vil ladsen and Miche lsen S till , a close analo gy exists i f t he differential equation model is approx imated by a discrete model from T = 0 to T m ax · T his can be done , for example , by global colloc ation . I f the colloc ation order N is hi gh enough , t he di sc rete model is an adequate representation of the original model , and the solution :l.
Chemical Reaction and Reactor Engineering (Chemical Industries) by Carberry