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Transient Analysis of Nonuniform Heat Input Propagation Through a Heat Sink Base

[+] Author and Article Information
Srivathsan Sudhakar

Cooling Technologies Research
Center (an NSF I/UCRC),
School of Mechanical Engineering,
Purdue University,
585 Purdue Mall,
West Lafayette, IN 47907
e-mail: ssudhak@purdue.edu

Justin A. Weibel

Cooling Technologies Research
Center (an NSF I/UCRC),
School of Mechanical Engineering,
Purdue University,
585 Purdue Mall,
West Lafayette, IN 47907
e-mail: jaweibel@purdue.edu

1Corresponding author.

Contributed by the Electronic and Photonic Packaging Division of ASME for publication in the JOURNAL OF ELECTRONIC PACKAGING. Manuscript received January 11, 2017; final manuscript received February 21, 2017; published online April 12, 2017. Assoc. Editor: S. Ravi Annapragada.

J. Electron. Packag 139(2), 020901 (Apr 12, 2017) (7 pages) Paper No: EP-17-1004; doi: 10.1115/1.4036065 History: Received January 11, 2017; Revised February 21, 2017

For thermal management architectures wherein the heat sink is embedded close to a dynamic heat source, nonuniformities may propagate through the heat sink base to the coolant. Available transient models predict the effective heat spreading resistance to calculate chip temperature rise, or simplify to a representative axisymmetric geometry. The coolant-side temperature response is seldom considered, despite the potential influence on flow distribution and stability in two-phase microchannel heat sinks. This study solves three-dimensional transient heat conduction in a Cartesian chip-on-substrate geometry to predict spatial and temporal variations of temperature on the coolant side. The solution for the unit step response of the three-dimensional system is extended to any arbitrary temporal heat input using Duhamel's method. For time-periodic heat inputs, the steady-periodic solution is calculated using the method of complex temperature. As an example case, the solution of the coolant-side temperature response in the presence of different transient heat inputs from multiple heat sources is demonstrated. To represent a case where the thermal spreading from a heat source is localized, the problem is simplified to a single heat source at the center of the domain. Metrics are developed to quantify the degree of spatial and temporal nonuniformity in the coolant-side temperature profiles. These nonuniformities are mapped as a function of nondimensional geometric parameters and boundary conditions. Several case studies are presented to demonstrate the utility of such maps.

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References

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Figures

Grahic Jump Location
Fig. 1

Schematic diagram of the chip-on-substrate domain

Grahic Jump Location
Fig. 2

(a) Plan-view locations and (b) transient heat input profiles for multiple heat sources on a single substrate. (c) The nondimensional temperature profile is shown for the coolant side at selected nondimensional time instants (indicated by the dots on the axis in (b)).

Grahic Jump Location
Fig. 3

(a) Nondimensional temperature-time plot at the center of the coolant-side surface and (b) mean steady temperature profile

Grahic Jump Location
Fig. 4

Variation of spatial nonuniformity with nondimensional thickness d* for different (a) Bi (at 1/b*= 0.5) and (b) 1/b* (at Bi = 0.1)

Grahic Jump Location
Fig. 5

(a) Isolines of spatial nonuniformity map for Bi versus d* (at 1/b* = 0.5); case studies are indicated on the map with a dot. Nondimensional temperature profiles on the coolant side are shown for (b) case #1 and (c) case #2.

Grahic Jump Location
Fig. 6

Schematic drawings of: (a) case #1 (air-cooled device); (b) case #2 (embedded microchannel cooling); and (c) case #3 (low-power device); the solution domain is outlined with a dotted line

Grahic Jump Location
Fig. 7

Variation of temporal nonuniformity with Fo for different ω (d*= 0.1, 1/b*= 0.5, Bi = 0.1)

Grahic Jump Location
Fig. 8

(a) Isolines of temporal nonuniformity map for ω versus Fo (d * = 0.1, 1/b* = 0.5, Bi = 0.1); dots on the map indicate representative cases with high (case A; ω = 0.5 and Fo = 0.02) and low (case B; ω = 0.5 and Fo = 0.75) frequency heat inputs. The evolution of the nondimensional center point temperature over one time period in the steady-periodic regime is shown for (b) case A and (c) case B.

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