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Heat Transfer Handbook part 35. The Heat Transfer Handbook provides succinct hard data, formulas, and specifications for the critical aspects of heat transfer, offering a reliable, hands-on resource for solving day-to-day issues across a variety of applications. | JOINT RESISTANCES OF NONCONFORMING SMOOTH SOLIDS 331 Figure 4.19 Comparison of data and model for contact between a rigid han sphere and an elastic layer on a rigid substrate. From Stevanovir et al. 2001. a z 1 c3 exp c1 Tc2 ul 4.189 with correlation coefficients c1 1.73 c2 0.734 and c3 1.04. The reference contact radius is aL which corresponds to the very thick layer limit given by 3 F p 1 3 aL L 4 E13 . t for - a 4.190 œ The maximum difference between the correlation equation and the numerical values obtained from the model of Chen and Engel 1972 is approximately 1.9 for t 0.02. The following relationship based on the Newton-Raphson method is recommended for calculation of the contact radius Stevanovic et al. 2001 a _ an - aL 1 - 1.04 exp -1.73 t a 0 734 4191 an 1 1 1.321 aL as t a 0-734 exp -1.73 t a -734 . 332 THERMAL SPREADING AND CONTACT RESISTANCES If flee first guess is a0 aL fewer than six iterations are required to give eight-digit accuracy. In the general case where the hemisphere layer and substrate are elastic the contact radius lies in the range aS a aL for E2 E1. The two limiting values of a are according to Stevanovic et al. 2002 a 3 F p 1 3 as --------- 4 3 Fp 1 3 aj --------- L 4 E13 for - 0 a 4.192 for----- œ a where the effective Young s modulus for the two limits are defined as E13 E1 v2 E3 -1 E23 1 v2 -1 E3 4.193 1 V 1 The dimensionless contact radius and dimensionless layer thickness were defined as Stevanovic et al. 2002 a T where 0 a 1 aj where a as 4.194 4.195 The dimensionless numerical values obtained from the full model of Chen and Engel 1972 for values of a in the range 1.136 a 2.037 are shown in Fig. 4.20. The correlation equation is Stevanovic et al. 2002 a as aL as 1 exp n1 4 4.196 Since the unknown contact radius a appears on both sides the numerical solution of the coreelation equation e. url e s an iterative method Newton-Raphson method to find its root. For all metal combinations the following solution is recommended Stevanovic
