Electronics Cooling and Heat Sinks: Method

Electronics Cooling and Heat Sinks

The tool follows the heat of an electronic part from its junction to the air: through the package, across the mounting interface, into the base of a heat sink, and from the fins to the air by convection and radiation. The framework is the electrical analogy of MIL-HDBK-251 (Reliability/Design Thermal Applications, 1978, chapters 6, 8 and 9): heat rate is current, temperature is voltage, and each step of the path is a thermal resistance in °C/W. The junction part of the tool is plain series arithmetic; the sink part computes the last resistance, sink to ambient, from the geometry. Temperatures are in °C and power in watts in both unit systems, as the handbook writes them; lengths, velocity, flow and pressure follow the unit switch.

Junction temperature network

For a part dissipating P on a sink that carries the total power ΣP of all its parts,

T_J = T_A + P (θ_JC + θ_CS + R_sp) + θ_SA ΣP,

where θ_JC is the junction-to-case resistance of the data sheet, θ_CS the case-to-sink (interface) resistance, R_sp the conduction and spreading resistance of the sink base under the part and θ_SA the sink-to-ambient resistance. The parts on one sink are coupled through the common term θ_SA ΣP: every part raises the sink temperature for all the others. A part on its own sink has T_J = T_A + P (θ_JC + θ_CS + θ_SA,own). The margin is T_J,max − T_J, a plain comparison with the rating you enter (use the derated value your reliability rules require). The required sink resistance is the value of θ_SA that puts the junction on its limit; for a shared sink the smallest requirement among its parts governs. The handbook's Example 8-9 is this calculation: 15 W, case limit 75 °C in 50 °C air and a 0.4 °C/W interface leave 5/3 − 0.4 = 1.27 °C/W for the sink.

Interface. A layer of thickness t, conductivity k and area A has R = t/(k A) (handbook eq. 8-13); a joint described by a contact conductance h has R = 1/(h A). The handbook's Table VI gives typical case-to-sink values for TO-3 and TO-66 cases: no washer 0.20 °C/W dry; 3 mil Teflon 1.45 dry and 0.80 with silicone grease; 2 mil mica 0.80 and 0.40; 20 mil anodized aluminium 0.40 and 0.35; its Figure 25 and Appendix D give contact resistance against pressure.

Spreading. A source smaller than the base forces the heat to spread, which adds resistance beyond the one-dimensional t_b/(k A_base). The source and the base are replaced by concentric circles of equal areas, radii a and b; the base has thickness t_b and conductivity k, its rim is adiabatic and its far face is cooled through the sink resistance R₀ = θ_SA as a uniform film h = 1/(R₀ π b²). With ε = a/b, τ = t_b/b and Bi = h b/k, separation of variables gives the exact series the tool uses, with δ_n the zeros of J₁:

ψ_avg = ετ/√π + (4/(√π ε)) Σ J₁²(δ_nε) φ_n / (δ_n³ J₀²(δ_n)),   ψ_max = ετ/√π + (2/√π) Σ J₁(δ_nε) φ_n / (δ_n² J₀²(δ_n)),

φ_n = (tanh δ_nτ + δ_n/Bi) / (1 + (δ_n/Bi) tanh δ_nτ),   R = ψ / (√π k a).

The first term is the one-dimensional conduction through the base; ψ_avg refers to the mean temperature of the source and ψ_max to its centre. Lee, Song, Van Au and Moran (1995) replaced the series by one term with λ = π + 1/(√π ε): ψ_avg = ετ/√π + ½ (1 − ε)^3/2 φ(λ) and ψ_max = ετ/√π + (1 − ε) φ(λ)/√π. The tool prints both. On a grid of 120 cases (ε 0.05 to 0.8, τ 0.02 to 1, Bi 0.01 to 100) the closed form as implemented here is within 5 % of the series for τ = 1 and within about 11 % in the average value for τ ≥ 0.1, but it reaches +24 % (average) and +51 % (centre) for thin bases, τ = 0.02, under large sources; 90 of the 120 cases are within 10 %. The series was confirmed by a finite-volume solution of the disc, thin bases included. The source is taken at the centre of the base; a source near an edge spreads less well and runs hotter.

Plate-fin sink in natural convection

The fins are vertical, N of them of thickness t and height H on a base of width W and length (vertical height) L; the spacing is S = (W − N t)/(N − 1). The air rises through the N − 1 channels. For symmetric isothermal plates Elenbaas (1942) measured the channel Nusselt number as a function of El = g β ΔT S⁴/(ν α L) and fitted (El/24)(1 − e^−35/El)^3/4. Bar-Cohen and Rohsenow (1984) joined the two analytical limits, the fully developed channel El/24 and the isolated plate 0.59 El^1/4, in the composite relation the tool uses:

Nu_S = h S/k = [576/El² + 2.873/El^1/2]^−1/2.

The coefficient is referred to the difference between the plate and the entering air. Maximising the heat from a given width of thin fins, proportional to Nu_S/S², gives the optimum spacing S_opt = 2.714 (ν α L/(g β ΔT))^1/4; a numerical maximisation of the composite relation returns 2.7155. The air properties are those of the site's Convection Correlations tool (ideal gas, Sutherland viscosity and the conductivity formula of the 1976 Standard Atmosphere, NIST Shomate heat capacity), re-implemented in this tool's library and evaluated at the film temperature and the pressure of the entered altitude.

Each inner fin is a straight fin with the channel coefficient on both faces, efficiency tanh(m H_c)/(m H_c) with m = √(2h/(k t)) and the tip allowed for by H_c = H + t/2 (the Harper–Brown corrected length of the Fins and Extended Surfaces tool). The two end fins have the channel coefficient inside and the isolated vertical-plate coefficient of Churchill and Chu outside, m = √((h_ch + h_o)/(k t)). The base strips between the fins take the channel coefficient and the two base edges the outer one. The mounting face of the base is taken as covered by the parts and the chassis.

Radiation. A fin channel is a cavity. Its interior (two fin faces and the base strip, area A_ch, emissivity ε) exchanges with the surroundings only through its openings: the tip plane S × L and the two ends S × H. Treating the interior as one grey surface and the openings as a black surface at T_A gives q = σ (T_s⁴ − T_A⁴)/[(1 − ε)/(ε A_ch) + 1/(A_chF)], with the exchange area from reciprocity and the enclosure rule: the openings see either the channel or each other, so A_chF = A_O − (4 A_tipF_tip–end + 2 A_endF_end–end). F_tip–end is the closed-form view factor of perpendicular rectangles with a common edge and F_end–end that of opposite parallel rectangles (Hamilton and Morgan, NACA TN 2836). A deep narrow channel radiates almost as a black opening whatever its surface; the fins add little radiating area beyond the envelope of the sink, which is why radiation favours wide spacing and a high emissivity. The outer faces, fin tips and base edges radiate freely. All surfaces are taken at the base temperature.

Iteration. Convection and radiation both depend on the sink temperature, so the tool solves T_s by bisection until their sum equals the power on the sink, and reports θ_SA = (T_s − T_A)/ΣP for that power; the residual of the energy balance is printed. A chart repeats the solution for every fin count that fits the width, which shows the optimum.

Forced convection through the fins

The approach flow, less the entered bypass fraction, passes through the channels: V = (1 − f_bp) Q/((N − 1) S H). The air heats up along the channel, so the sink is a heat exchanger with one stream at uniform temperature: with ṁ c_p the capacity rate of the channel flow, NTU = η_o h A/(ṁ c_p) and

θ_SA = 1/(ṁ c_p (1 − e^−NTU)),

referred to the inlet air. The resistance cannot fall below 1/(ṁ c_p), reached when the air leaves at the sink temperature. For laminar flow the mean coefficient (on the log-mean difference) is the thermal-entry relation between isothermal parallel plates, Nu = 7.54 + 0.03 Gz/(1 + 0.016 Gz^2/3) with Gz = (2S/L) Re Pr on the hydraulic diameter 2S of the gap; a marching solution of the Graetz problem reproduces it within 2 % for Gz from 2 to 1,000. It assumes a developed velocity profile, which is slightly conservative for air. Above a channel Reynolds number of 2,300 (hydraulic diameter 2SH/(S + H)) the Gnielinski relation with Petukhov's friction factor is used. The fins are straight fins with an adiabatic tip (a shroud or duct wall at the tips), and the overall surface efficiency η_o = 1 − (A_f/A)(1 − η_f) includes the base strips. For comparison the tool prints the composite model of Teertstra, Yovanovich and Culham, which joins the fully developed limit Re* Pr/2 (all the channel air at the wall temperature) to the developing flat-plate limit in one Nusselt number on the inlet temperature difference.

Pressure drop. Δp = (K_c + K_e + 4 f_app L/D_h) ρV²/2. The entrance is a sudden contraction to the area ratio σ = (N − 1) S/W, with K_c from the handbook's Table XV (0.46 at σ = 0.1 falling to 0 at σ = 1; 0.5 is taken at σ = 0); the exit is a sudden expansion, K_e = (1 − σ)², the handbook's eq. 9-65. These replace the Kays and London charts, which are not reproduced. Laminar friction carries the developing-flow allowance f_app Re = [(3.44/√L⁺)² + (f Re)²]^1/2, L⁺ = L/(D_h Re), with the fully developed f Re of the rectangular channel from the Shah and London polynomial 24 (1 − 1.3553 α + 1.9467 α² − 1.7012 α³ + 0.9564 α⁴ − 0.2537 α⁵), which the exact series reproduces within 0.06 %.

Fan. A quadratic through three entered points of the fan curve stands for the fan; the operating point is the flow at which its pressure equals the pressure drop of the sink, found by bisection. With a bypass fraction the fan delivers the whole flow and the sink sees its share; the pressure across the bypass path is taken equal to that across the sink. The fin-count chart is redrawn at the operating point of each count: more fins add area but cut the flow, and the best count lies between.

Sealed enclosure

The enclosure estimate is the handbook's method for equipment cooled by natural means (paragraphs 8.1.2 and 8.1.3). Free convection from each face: q/A = C k (a ΔT/L)^1/4 ΔT (eq. 8-28), with the convective modulus a = g β ρ² c_p/(μ k) (eq. 8-26), the characteristic length of Table III (vertical faces: the height, limited to 24 in; horizontal faces: length × width/(length + width)) and the constant of Table IV (vertical plates 0.55, horizontal plates facing upward 0.71, facing downward 0.35). For sea-level air the handbook condenses this to q/A = 0.00394 C ΔT^1.25/L^0.25 W/in² (eq. 8-30); the tool evaluates eq. 8-28 with the air properties at the film temperature and the altitude pressure, which reproduces the k a^1/4 column of the handbook's Table V within 1 % from 10 to 260 °C (3.8 % at −18 °C, where the table is irregular). Radiation: q = F_e F_a σ A (T_s⁴ − T_A⁴) (eq. 8-33) with F_e = ε and F_a = 1 for a box small against its surroundings. The surface temperature is solved so that the two together equal the power. The handbook's Examples 8-4 (64 W from the top of an enclosure), 8-5 (58 W radiated by a 6 × 12 × 6 in box) and 8-6 (36.7 W convected by the same box) are reproduced by the test script. The handbook's guides are a heat flux of no more than 0.25 W/in² and a surface rise of no more than 10 °C where the surface can be touched.

Altitude. The pressure comes from the 1976 U.S. Standard Atmosphere (three layers to 32 km). At a given temperature the density is proportional to the pressure while viscosity, conductivity and specific heat are not affected, so free convection weakens roughly as the square root of the density and a fan moves the same volume but less mass (handbook paragraph 8.1.2.2). Radiation is unchanged, which is why it dominates at altitude.

Board conduction and thermal vias

A board clamped to a cold wall is a strip whose copper planes and laminate conduct in parallel, conductance per unit width k t = k_cu t_cu c + k_lam t_lam with c the fraction of the width the copper covers (the handbook's Examples 8-7 and 8-8 add the layers the same way). With the power uniform over the board, the peak rise above the wall is Q L/(8 k t w) when both ends are cooled and Q L/(2 k t w) when one is. Where heat leaves a lead and fans out, the handbook integrates along a wedge; for a sector of angle φ, R = ln(r_o/r_i)/(φ k t), which with φ = 1 rad gives its 772 °C/W for the copper of Example 8-7 and with φ = 2π the radial fin resistance of Example 8-9. A thermal via is a copper tube of section π (d² − (d − 2t_p)²)/4 and length the board thickness; n vias are n such tubes in parallel, plus the fill if any.

Transient

The sink as one lumped mass has capacitance C = ρ c V and time constant τ = θ_SA C: T(t) = T_A + ΣP θ_SA (1 − e^−t/τ), 90 % of the rise after 2.3 τ. For repetitive rectangular pulses of peak power P_pk and duty D, superposing the average power on the steady resistance and the last pulse on the single-pulse impedance Z_th (read from the data-sheet curve at the pulse width) gives the peak junction rise above the case, P_pk [D θ_JC + (1 − D) Z_th].

Checks and a correction to the handbook

The test script reproduces seven worked examples of MIL-HDBK-251 to the printed digits (the composite bar of p. 52 and Examples 8-4 to 8-9) and its Tables IV, V and XV, and compares every numerical part with an independent numpy/scipy implementation. One printed result is not reproduced on purpose: in Example 8-9 the value 0.045 read from the free-convection nomograph is the heat flux q/A in W/in² at ΔT = 15 °C, but the example uses it as a coefficient and multiplies by ΔT again, which gives a fin 3.76 in in diameter; the consistent diameter is 14.6 in. The handbook's Stefan–Boltzmann constant, 36.8 × 10⁻¹² W/(in²⋅K⁴), is 0.6 % above the present value.

Limits

The default sink (100 × 100 × 40 mm, twelve 2 mm fins, 25 W) is illustrative, not a published example. The sink is isothermal apart from the spreading resistance under one centred source size; the channel relations are for smooth parallel plates and do not cover pin fins, staggered or louvred fins, horizontal bases in natural convection, or flow across the fins; mixed convection is not modelled; makers' measured θ_SA curves take precedence where they exist. The Fins and Extended Surfaces, Convection Correlations, Radiation View Factors, Pump and Fan Performance and Standard Atmosphere and Airspeeds tools treat the separate pieces in more detail.