1. Constriction and Spreading Resistance Reference: S. Lee, S. Song, K. Moran, Constriction/Spreading Resistance Model for Electronics Packaging, ASME/JSME Thermal Engineering Conference: Vol. 4, 1995.

2. Definitions Constriction resistance – heat flows from a larger to a smaller area Spreading resistance – heat flows from a smaller to a larger area –Equations are the same for both

3. Geometry Considerations Different shapes (squares, circles) have basically the same resistance for the same square root of contact area and same area ratio (a/b) As the area ratio gets large, the geometry starts to matter more. However, at that point the constriction and spreading resistances are usually much smaller than other resistances in the system.

4. Analytical problem/ solution Ave: average resistance, based on average temperature of contact region; this is what we almost always want Max: resistance based on maximum temperature of contact region

5. Easier Approximation Including 100 terms of the infinite series results in near perfect agreement with two different numerical simulations. Approximate solutions shown below agree with infinite series solutions within 10%.

6. Easier Approximation, cont. Using the definition of Ψ, solve for R ave, which will be your constriction or spreading resistance.. Add this resistance to your 1-D thermal resistance network. For example, here is a typical resistance network: R junction-case +R contact +R spreading +R heatsink The larger the area ratio, the more important the constriction/spreading resistance is. For example

7. Terminology Other important researchers in this area: M.M. Yovanovich (1969, 76, 77, 79, 87, 92, 93) and D. P. Kennedy (1960)