Radiation network method

The foregoing discussions have shown the methods that may be used to calculate radiation heat transfer between surfaces separated by a completely transparent medium. The radiation-network method is used to great advantage in these types of problems. 

Conduction is treated from both the analytical and the numerical viewpoint, so that the reader is afforded the insight which is gained from analytical solutions as well as the important tools of numerical analysis which must often be used in practice. A similar procedure is followed in the presentation of convection heat transfer. An integral analysis of both free- and forced-convection boundary layers is used to present a physical picture of the convection process. From this physical description inferences may be drawn which naturally lead to the presentation of empirical and practical relations for calculating convection heat-transfer coefficients. Because it provides an easier instruction vehicle than other methods, the radiation-network method is used extensively in the introduction of analysis of radiation systems, while a more generalized formulation is given later. 

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... 

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. 

I he network method is not practical for enclosures with more than three or four surfaces, however, because of the increased complexity of the network. Next we apply the method lo solve radiation problems in two- and three-surface enclosures. 

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... 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

Elastin-mimetic protein polymers have been fabricated into elastic networks primarily via y-radiation-induced, radical crosslinking of the material in the coacervate state [10]. Although effective, this method cannot produce polymers gels of defined molecular architecture, i.e., specific crosslink position and density, due to the lack of chemoselectivity in radical reactions. In addition, the ionizing radiation employed in this technique can cause material damage, and the reproducibility of specimen preparations may vary between different batches of material. In contrast, the e-amino groups of the lysine residues in polymers based on Lys-25 can be chemically crosslinked under controllable conditions into synthetic protein networks (vide infra). Elastic networks based on Lys-25 should contain crosslinks at well-defined position and density, determined by the sequence of the repeat, in the limit of complete substitution of the amino groups. 

H.W. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, NY (1945) 6) Greenwood and Collaborators, Electronic Instruments, vol 21, MIT Radiation Laboratory Series, McGraw Hill, NY (1948) 7) R.H. Muller, AnalChem 20, 389 (1948) (Instrumental methods of analysis)... 

The first tests of this proposed method have been encouraging. On the basis of comparisons between rainfall rates measured with the differential reflectivity technique and with a network of rain gauges, Seliga et al. (1981) concluded that these first measurements of rainfall using the ZDR technique support the theoretical expectations... that rainfall rate measurements with radar can be made with good accuracy. So it may yet be possible to accurately measure rainfall with radar—provided that measurements are made with two orthogonally polarized beams. This exemplifies one of the principal themes of this book scattered polarized radiation contains information that may be put to good use. 

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