Thermal Spreading Resistance for Square and Rectangular Entities Alok Bhatt and Jinny Rhee San Jose State University One Washington Square San Jose, CA 95192-0087 [email protected] , [email protected] Abstract A systematic study was performed using numerical modeling to understand the effects of thermal spreading resistance on the total resistance from junction to ambient for square and rectangular entities. Simultaneously, a literature survey was conducted to understand effects of thermal spreading resistance on circular, square and rectangular heat sources and heat spreading plates. Simulation from numerical model (Flotherm) was then compared with results from analytical solution derived by Lee [1] and Ellison [2] for square and rectangular heat sources respectively as a benchmarking process for the numerical model. The model presumes a heat source on a large cooling plate beneath which has a constant heat transfer coefficient on the sink side. Numerical simulations were performed for steady-state conditions with different range of dimensions for source to plate edge length ratio and dimensionless thickness. Results for dimensionless spreading resistance as a function of relative contact size, plate thickness and Biot number were derived using Flotherm and compared with the analytical models mentioned above. Results show that numerical modeling using software like Flotherm can offer excellent results within acceptable percentage difference when compared to analytical solutions. Keywords Thermal spreading resistance, constriction resistance, dimensionless spreading resistance, high power LEDs Nomenclature Symbols a

x-direction, rectangular plate dimension

b

y-direction, rectangular plate dimension

t

z-direction, rectangular plate thickness

h

Total heat transfer co-efficient, W/ area. K

Biot.τ

Biot number and τ product, ht/k,

k

Thermal conductivity of plate, W/ length. K

R

Total maximum thermal resistance from source to ambient, R = RU + RSP

RSP

Internal conduction spreading resistance, K/W

RU

1-D thermal resistance, RU = RC + RE, K/W

T

Plate temperature, K

Greek Symbols α Dimensionless source width, ∆x/a Β

Dimensionless source length, ∆y/a

∆x

Source width, x2-x1

∆y

Source length, y2-y1

ρ

Substrate aspect ratio a/b

τ

Dimensionless rectangular plate thickness, t/a

Ψ

Dimensionless total resistance, Ψ = Ψ U + Ψ SP

ΨU

Dimensionless 1-D resistance

Ψ SP

Dimensionless spreading resistance

Subscripts U One-dimensional SP Spreading Max Maximum

1. Introduction Thermal spreading resistance is a problem commonly known in the thermal analysis of electronic packages. Spreading resistance occurs whenever heat flows from a source to the sink with different cross-sectional areas. There is also a similar term called as constriction resistance[1] which quite often understood similar to the spreading resistance. Constriction resistance occurs when heat flows from a larger area to a narrow region and the spreading resistance occurs when heat flows from a small area to a larger area in contact. Many times constriction and spreading resistances become very important that they play a significant role in calculating the thermal performance of a device. The early work was started by Kennedy (1960) who investigated thermal resistance problem and derived analytical solutions for axi-symmetric models with uniform heat-flux source on a finite cylinder. However, he assumed isothermal boundary conditions over the sink side. Wide ranges of design charts are provided where isothermal boundary conditions are identified over different surfaces. The assumption of isothermal boundaries made by Kennedy fail to apply to many practical problems, and for certain problems, spreading resistance value that are under predicted by order of magnitude [1]. Nelson and Sayers (1992) utilized finite difference method for control volume to derive spreading resistance (with maximum temperature) for both axisymmetric and planner

models. They compared their results with 3-D spreading resistances calculated for rectangular models with various aspect ratios. They used TAMS (Thermal Analysis of Multilayer Systems) for their three dimensional model for comparison [2]. Negus and Yovanovich (1987) developed a thermal analysis procedure for calculating the temperature distribution of a semiconductor die with multiple heat sources. Isothermal boundary conditions are also assumed here along the bottom of the die. A solution for constriction resistance due to heat flux applied on the top surface of a two-layer compound disk with a film coefficient over the lower surface was derived by Yovanovich, et al.(1979)[6]. They also further took their cylindrical geometry to rectangular sources and plate for up to two layers [4]. Lee et al. (1995) derived formulas to calculate spreading resistance for circular and square source and place with constant heat transfer coefficient on the sink side. This was the initial work to take account of both internal and external resistances. The conduction effects are commonly known as internal resistance and convection/radiation effects are known as external resistance [4]. Lee used a two-dimensional heat conduction equation in a cylindrical coordinate system to derive dimensionless spreading resistance. Lee also provide an approximated and simple formulas for maximum and average spreading resistance that matches within 10% with his analytical solutions over the range of parameters found in the microelectronics industry [1]. Thus, the work by Lee assumes a model where the source and the sink plate are circular. However, Lee’s work can be applied to square and rectangular sources and plates by calculating the radius with areas equivalent to that of square sources and spreader plates. The formulas developed by Lee et al. offers a very important tool for predicting thermal spreading resistance for square source and square spreader plate. However, the percentage error between simplified formulas and exact solution becomes larger with large aspect ratios for the rectangular heat source and spreader plate [9]. The most recent work in investigating the thermal spreading effects on non-circular source and spreader plate is conducted by Ellison (2002). In his excellent paper Ellison derived dimensionless expressions in form of infinite series to compute maximum thermal spreading resistance and surface temperature fields. He used 3-D partial differential equation for steady-state heat conduction equation and constant heat transfer coefficient on the sink side to calculate maximum dimensionless spreading resistance. Ellison has also provided a wide variety of graphs with maximum dimensionless spreading resistance as a function of projected heating area, dimensionless thickness and Biot number.

Figure 1: Geometry and boundary conditions for thermal spreading resistance problem The figure shows the constant heat flux Q at the top and rest of the boundaries are insulated except a constant heat transfer coefficient on the sink side. The thermal resistance network includes three parts: One-dimensional material resistance, spreading resistance as heat move from a narrow region to a larger cross-sectional spreading plate and lastly the external resistance. It should be noted that the plate is insulate everywhere except on the sink side and at contact area of heater with the plate. 2.1 Analytical Model for Benchmarking Numerical model was benchmarked with two different analytical solutions derived by Lee et al. and Ellison. Lee in his work derived simple approximated formulas using twodimensional heat conduction equation in a cylindrical coordinate system. Lee derived simple approximated formulas for the dimensionless spreading resistances for circular and square heat entities.

(Eq. 1)

As an extension to Lee’s work Ellison derived formulas for maximum spreading resistance using 3-D partial differential equation for steady-state heat conduction equation. Ellison also explains that the total resistance consists of one-dimensional conduction resistance and spreading resistance.

2. Methodology Numerical modeling software, Flotherm, was used to perform analysis on thermal spreading resistance of square and rectangular entities. A numerical model was build similar to what shown in the figure below. The numerical model developed has the capability to perform analysis for wide range of heating plate and spreading plate dimensions.

(Eq. 2) In his paper Ellison also furnish graphs for maximum spreading resistance as a function of heater contact area ratio, dimensionless thickness and Biot number. For benchmarking

purposes a similar kind of graph was utilized for benchmarking with the Flotherm model. The graph with source (heater) dimensions half that of the spreader plate was utilized as a benchmarking case. Also, the aspect ratio for source as well for spreader plate is one. It means the source and spreader plate are square. Here, square plate and source are chosen as Lee’s formulas are derived for square source and plate. Therefore, the Flotherm can be compared to Lee and Ellison’s results at the same time.

In the equation 2 above, R is the total thermal resistance calculated, RU is one directional conduction resistance and RSP is the thermal spreading resistance. RU was calculated from formula given below.

(Eq. 4) Where,

2.2 Numerical Model using Flotherm 3. Results Analysis Initial modeling with Flotherm yields the following results where arrows give the heat flux vector distribution and the temperature distribution is given by color contours. These initial results shown how the heat flows from region of higher temperature (i.e., near the junction) to the sink and was per expectations.

Figure 3: Heat flux and temperature distribution obtained from initial modeling from Flotherm Figure 2: Boundary conditions: as adiabatic at all boundaries except constant h at z=t. For square plate ∆X=∆Y and a=b. Instead of calculating spreading resistance for each Biot.τ (ht/k), three Biot.τ were chosen as 0.01, 0.1 and 1.0. Similarly, five values of dimensionless thickness τ were chosen as 0.02, 0.03, 0.1, 0.35 and 1.0. Different cases with values of heat transfer coefficient were determined depending upon value of Biot.τ. Likewise, different values of dimensionless thickness (τ) were determined and for value of τ, dimensions of length were calculated. The definition of τ is defined as τ = t/a, where thickness was fixed as 0.02m. After setting up the dimensional parameters for the given case Flotherm model was run for each case under steady-state conditions. Finally, junction temperature was determined for each case. From the values of junction temperature, total thermal resistance from junction to case was predicted using formula given below. Rja = (Tj – Ta) / W (Eq. 3) Where, Rja = Total thermal resistance from junction to case (°C/W) Tj = Junction temperature (°C) Ta = Ambient temperature (°C) W = Total power supplied from the heater (Watts)

Further, graphs were drawn similar to shown in figure 2 to support and benchmark Flotherm model. Results for dimensionless thermal spreading resistance obtained from Flotherm were utilized to replicate graphs from figure 2. When results from numerical model were compared to analytical model from Ellison and Lee, it was observed that Flotherm results matches to the Ellison’s analytical solution within 5% and with Lee’s analytical model within 10% difference. One possible reason of the greater percentage difference for Lee’s solutions is that Lee’s formulas are derived for cylindrical coordinates and an equivalent area was used to calculate thermal spreading resistance for a square source and spreading plate in our case. Results obtained for combination of τ and Bi.τ are listed below in table 1. From the results obtained above it was determined that numerical modeling using software like Flotherm can offer excellent results within acceptable percentage difference.

Table 1: Results from numerical modeling

Figure 4: Dimensionless spreading resistance as a function of Biot.τ and τ derived from Flotherm. 4. Summary Numerical modeling was performed using Flotherm to calculate dimensionless thermal spreading resistance for difference set of cases. Further, obtained results were compared to analytical solutions derived in past by Ellison and Lee. Square source and square spreader plate were chosen as such aspect ratios can be commonly found in microelectronics industry. From the numerical analysis performed, its was concluded that numerical simulation software like Flotherm have capability to perform such analysis and yields results within acceptable percentage difference as compared to analytical and exact solutions. Current numerical model is well qualified to perform parametric studies. It was observed that the spreading resistance effect become more dominant when the source size is much less then spreader plate size. Current study on thermal spreading resistance becomes important in applications like LED (Light Emitting Diodes) industry where LEDs dissipating heat with high power density equivalent to 10 W/mm2 and heat source to heat spreading plate aspect ratio of more than 1:10. Moreover, the current work can also be applicable to electronics cooling industry where heat sink base dominates the thermal spreading resistance effects when in contact with high power processor underneath and can lead to exploration of different packaging materials with higher range of conductivity that can significantly control spreading resistance effects. Acknowledgements Authors acknowledge Rockwell Collins for their support and encouragement to conduct the study at San Jose State University.

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