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 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 with correlation coefficients c1 c2 and c3 . 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 œ The maximum difference between the correlation equation and the numerical values obtained from the model of Chen and Engel 1972 is approximately for t . 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 - exp t a 0 734 4191 an 1 1 aL as t a 0-734 exp 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 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 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 The dimensionless numerical values obtained from the full model of Chen and Engel 1972 for values of a in the range a are shown in Fig. . The correlation equation is Stevanovic et al. 2002 a as aL as 1 exp n1 4 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