String methods

Example 3 Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem two-dimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length,... 

In this case, both pipes and plate are long, therefore it is a two-dimensional long enclosure problem. Hotter s cross-string method can be used. 

The discussion is how would one modify Hotter s crossed-string method when the obstructing objects are not completely opaque, i.e.T 0. The procedure is a two-step one. First, the view-factor between surface 1 and surface 2 that is attributable to the obstructing objects (i.e., when x = 0 ) have to be found. The second step is to multiply this view factor by the non zero t. Example 13.5 below shows the procedure. 

By Hottel s crossed-string method, denoting the view factors by a . 

Sum of the crossed strings = 2- /9a2 +36a2 = 2a-j45 Sum of uncrossed strings = 3a + 3a = 6a By Hottel s crossed-string method, denoting the view factors by a. ... 

The Hottel crossed-string method allows us to calculate the view factor of a surface that is very long in the direction perpendicular to the cross section of the objects. This problem stretches this assumption to use the method for a space heater in a room that has a shape of a cylinder and a person at some distance away. 

Figure 1 for Prob. 13.13 shows a side view of the heater and the person. This figure is only used to understand the problem. Since the heater and the person s height are about the same and they are relatively tall compared with their cross sections, we will be able to use HottePs crossed-string method. 

Example 7 The Crossed-Strings Method Figure 5-16 depicts the transverse cross section of two infinitely long, parallel circular tubes of diameter D and center-to-center distance of separation C. Use the crossed-strings method to formulate the tube-to-tube direct exchange area and view factor s st and Ft,t, respectively. 

Note that + L is the sum of the lengths of the crossed strings, and Lj -f L4 is the sum of the lengths of the uncrossed strings attached to the endpoints. Therefore, Hotiel s crossed-strings method can be expressed verbally as... 

Determination of the view factor F - j hy the application of the crossed-strings method. 

EXAMPLE 13-5 The Crnssed-Strings Method for View Factors... 

SOLUTION The view factors between tviro infinitely long parallel plates are to be determined using the crossed-strings method, and the formuja for the view factor is to be derived. 

Analysis (a) First we label the endpoints of both surfaces and draw straight dashed lines between the endpoints, as shown fn Fig. 13-17. Then we identify the crossed and uncrossed strings and apply the crossed-strings method (Fq. 13-17) to determine the vievr factor F ... 

Which is the desired result. This is also a miniproof of the crossed-strings method for the case of two infinitely long plain parallel surfaces. 

C 7Wh.it is Ihe crossed-strings method Eor what kind of geometries is the cro.ssed-.strings method applicable ... 

String Methods. - A recently proposed approach that is closely related to the nudged elastic-band method of the preceding section is the string method of E et al.136 (see also 137). We shall here briefly outline the basics of the method and, thereby, also draw the connection to the nudged elastic-band method, following the presentation of Kanai et al.131... 

In the original approach, E et al 36 combined the string method with a molecular-mechanics method for the calculation of the total energy for a given structure, whereas Kanai et al 31 combined it with the parameter-free density-functional Car-Parrinello method. Below we shall present results from the latter study. 

In their presentation of the string method, Kanai et alP1 applied the method to the adsorption of hydrogen on the Si(100) surface, both at low and at large hydrogen coverage, in particular for the processes where the H2 molecule change adsorption site. 

E. Weinan, W. Ren, and E. Vanden-Eijnden (2005) Einite temperature string method for the study of rare events. J. Phys. Chem. B 109, p. 6688 C. Dellago, P. Bolhuis, E. S. Csajka, and D. Chandler (1998) Transition path sampling and the calculation of rate constants J. Chem,. Phys. 108, pp. 1964-1977... 

W. E, W. Ren, and E. Vanden-Eijnden (2005) Finite Temperature String Method for the Study of Rare Events. J. Phys. Chem. B, 109, pp. 6688-6693 R. Fiber, A. Ghosh, A. Cardenas, and H. Stern (2004) Bridging the gap between reaction pathways, long time dynamics and calculation of rates. Adv. Chem. Phys., 126, pp. 93-129... 

This is an identical equation for the minimum energy path [9,10] or the so-called steepest descent path. The first implementation of an algorithm to compute minimum energy paths based on the above formula was the LUP (Locally Updated Planes) method [9] that did not include the constraint on the displacement size. This is formally correct since different parameterizations of the path are possible, but may lead to numerical problems in which the distances between the intermediates grow without control. This was adjusted to produce more stable algorithms by the Nudge Elastic Band approach [11] and later by the String method [10]. 

E. Weinan, R. Weiqing, and E. Vanden-Eijnden (2002) String method for the study of rare events. Physical Review B 66, p. 52301... 

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