The Crossed-Strings Method

Solution The circumferential area of each tube is At = itD per unit length in the infinite dimension for this two-dimensional geometry. Application of the crossed-strings procedure then yields simply 

5-15 Distribution of radiation to rows of tubes irradiated from one side. Dashed lines direct view factor F from plane to tubes. Solid lines total view factor F for black tubes backed by a refractory surface. 

5-14 View factors for a system of two concentric coaxial cylinders of equal length, (a) Inner surface of outer cylinder to inner cylinder. (b) Inner surface of outer cylinder to itself. 

5-16 Direct exchange between parallel circular tubes. 

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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,... 

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. 

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

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. 

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. 

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... 

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

E, W., Ren, W., Vanden-Eijnden, E. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys. 126(16) (2007). doi 10. 1063/1.2720838... 

To illustrate the validity of the models presented in the previous section, results of validation experiments using lab-scale BSR modules are taken from Ref. 7. For those experiments, the selective catalytic reduction (SCR) of nitric oxide with excess ammonia served as the test reaction, using a BSR filled with strings of a commercial deNO catalyst shaped as hollow extrudates (particle diameter 1.6 or 3.2 mm). The lab-scale BSR modules had square cross sections of 35 or 70 mm. The kinetics of the model reaction had been studied separately in a recycle reactor. All parameters in the BSR models were based on theory or independent experiments on pressure drop, mass transfer, or kinetics none of the models was later fitted to the validation experiments. The PDFs of the various models were solved using a finite-difference method, with centered differencing discretization in the lateral direction and backward differencing in the axial direction the ODEs were solved mostly with a Runge-Kutta method [16]. The numerical error of the solutions was... 

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