The Network Method

We do not go into any detail of the integration methods here, as it seems that direct finite difference methods are preferable. 

Since about 1989, Homo and coworkers have published a series of papers on their network thermodynamic method of simulation. Only a few of these will be cited here. In the first, the 1989 work, the method is described [309], and again in 1992-4 [271,305,306], adding cyclic voltammetry. In the 1994 paper [305], there is a good description of the method, and an indication how it can be adapted to a multitude of different electrochemical systems. A Chinese group has also used this method [205,208,209,210]. 

Very briefly, basing the description on [305], the diffusion-reaction equation, of the form of (9.75) is semidiscretised as in MOL, to 

Then both sides are multiplied by the spatial interval h, and the result expressed, term by term, as 

The method has not taken on. One paper [168] reports the use of PSPICE, but for simulating actual resistance in an electrolyte, modelled as a resistance network. This is quite a different application, and much more directly relevant. 

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In 1987, He and He used the network method to investigating P-V path structures of benzenoid and coronoid system [11],... 

At this point we introduce a very useful interpretation for Eq. (8-38). If the denominator of the right side is considered as the surface resistance to radiation heat transfer, the numerator as a potential difference, and the heat flow as the current, then a network element could be drawn as in Fig. 8-24 to represent the physical situation. This is the first step in the network method of analysis originated by Oppenheim 120]. 

A problem which may be easily solved with the network method is that of two flat surfaces exchanging heat with one other but connected by a third surface which does not exchange heat, i.e., one which is perfectly insulated. This third surface nevertheless influences the heat-transfer process because it absorbs and re-radiates energy to the other two surfaces which exchange heat. The network for this system is shown in Fig. 8-28. Notice that node is not connected to a radiation surface resistance because surface 3 does not exchange energy. Notice also that the values for the space resistances have been written... 

The network method which we have used to analyze radiation problems is an effective artifice for visualizing radiant exchange between surfaces. For simple problems which do not involve too many surfaces the network method affords a solution that can be obtained quite easily. When many heat-transfer surfaces are involved, it is to our advantage to formalize the procedure for writing the nodal equations. For this procedure we consider only opaque, gray, diffuse surfaces. The reader should consult Ref. 10 for information on transmitting and specular surfaces. The radiant-energy balance on a particular opaque surface can be written... 

A square room 3 by 3 m has a floor heated to 300 K, a ceiling at 290 K, and walls that are assumed perfectly insulated. The height of the room is 2.5 m. The emissivity of all surfaces is 0.8. Using the network method, find the net interchange between floor and ceiling and the wall temperature. 

Oppenheim— Radiation Analysis by the Network Method, in Hartnett et al.—Recent Advances in Heat and Mass Transfer, McGraw-Hill. 

The systematic approach described above for solving radiation heal transfer problems is very suitable for use with today s popular equation solvers such as lili.V, Mathcad, and Matlab, especially when there are a large number of surfaces, and is known as the direct melhod (formerly, the matrix method, since it resulted in matrices and the solution required a knowledge of linear algebra). The second method described below, called the network method, is based on Ihe electrical network analogy. 

Consider an enclosure consisting of two opaque surfaces at specified temperatures r, and T2, as shown in Fig. 13-24, and try to determine llie net rate of radiation heat transfer between the two surfaces with the network method. Surfaces 1 and 2 have emissivities c, and and surface areas /1 and A2 and are maintained at uniform temperatures T, and T, respectively. There are only two surfaces in the enclosure, and thus we can write... 

Oppenheim, K.A. Radiation analysis by the network method. Trans. Amer. Soc. Mech. Engrs. 78 (1956) 725-735... 

L6pez-Garcia, J.J., Grosse, C., and Homo, J., Numerical study of colloidal suspensions of soft spherical particles using the network method. 1. DC electrophoretic mohrUty, J. Colloid Interface Sci., 265, 327, 2003. 

Moya AA, Hayas A, Homo J (1995) Study of electrical migration in electrochemical cells by the network method. Ber Bunsenges Phys Chem 99 1037-1042... 

Moya AA, Homo J (19%) Simulation of nonstationeiry diflusion-migtation processes in electrochemical cells using the network method. Electrochim Acta 41 285-290... 

Homo J, Gonzalez CF, Hayas A (1995) The network method for solutions of oscillating reaction-diffusion systems. J Comput Phys 118 310-319... 

Garcia-Hernandez MT, Castilla J, Gonzalez-Femandez CF, Homo J (1997) Apphcation of the network method to simulation of a square scheme with Butler-Volmer charge transfer. J Electroanal Chem 424 207-212... 

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