Spreading Constriction Resistance

Spreading (constriction) resistance is an important thermal parameter that depends on several factors such as (1) geometry (singly or doubly connected areas, shape, aspect ratio), (2) domain (half-space, flux tube), (3) boundary condition (Dirchlet, Neumann, Robin), and (4) time (steady-state, transient). The results are presented in the form of infinite series and integrals that can be computed quickly and accurately by means of computer algebra systems. Accurate correlation equations are also provided. 

The general expression for the dimensionless spreading (constriction) resistance 4kaRs for a circular contact subjected to an arbitrary axisymmetric flux distribution f(u) [131-133] is obtained from the series... 

The dimensionless spreading (constriction) resistance coefficient C0 is the half-space value, and the correlation coefficients C through C7 are given in Table 3.15. 

Thermal contact, gap, and joint conductance models developed by many researchers over the past five decades are reviewed and summarized in several articles [20,23,50,58,143,147,148] and in the recent text of Madhusudana [59]. The models are, in general, based on the assumption that the contacting surfaces are conforming (or flat) and that the surface asperities have particular height and asperity slope distributions [26, 116, 125]. The models assume either plastic or elastic deformation of the contacting asperities, and require the thermal spreading (constriction) resistance results presented above. 

Fig. 24. Ratio of the additional current constriction resistance to the conventional spreading resistance for different ring widths (7>nng) normalized to the diameter of the electrochemically inactive inner part (dme), indicating that for small active rings the additional resistance becomes rather important.

Spreading Resistance Within Isotropic Finite Disks With Conductance. The dimensionless constriction resistance for isotropic (k= 1) finite disks (ii < 0.72) with negligible thermal resistance at the heat sink interface (Bi = °°) is given by the following solutions. 

Lee et al. [157] recommended a simple closed-form expression for the dimensionless constriction resistance based on the area-average and centroid temperatures. They defined the dimensionless spreading resistance parameter as p = ViikaRc and recommended the following approximations. 

The contact resistance of the sphere-flat contact shown in Fig. 3.23 is discussed in this section. The thermal conductivities of the sphere and flux tube are /c, and k2, respectively. The total contact resistance is the sum of the constriction resistance in the sphere and the spreading resistance within the flux tube. The contact radius a is much smaller than the sphere diameter D and the tube diameter. Assuming isothermal contact area, the general elastoconstriction resistance model [143] becomes ... 

Steady-state and transient constriction (spreading) resistances for a range of geometries for isothermal and isoflux boundary conditions are given. Analytical solutions for half-spaces and heat flux tubes and channels are reported. 

M. M. Yovanovich, Theory and Applications of Constriction and Spreading Resistance Concepts For Microelectronic Thermal Management, in Cooling Techniques For Computers, W. Aung ed., pp. 277-332, Hemisphere Publishing Corp., New York, 1991. 

S. Song, S. Lee, and V. Au, Closed-Form Equation for Thermal Constriction/Spreading Resistances With Variable Resistance Boundary Condition, Proc. IEPS Conference, Atlanta, GA, pp. 111-121, 1994. 

S. Lee, S. Song, V. Au, and K. P Moran, Constriction/Spreading Resistance Model for Electronics Packaging, Proc. 4th ASME/JSME Thermal Engineering Joint Conference, Maui, HI, pp. 199-206, March 19-24,1995. 

---