Question 10.12: KNOWN: Chips on a ceramic substrate operating at power level......
KNOWN: Chips on a ceramic substrate operating at power levels corresponding to 50% of the critical heat flux.
FIND: (a) Chip power level and temperature rise of the chip surface, and (b) Compute and plot the chip temperature \mathrm T_{\mathrm s} as a function of heat flux for the range 0.25 ≤ \mathrm q_{\mathrm s}^{\prime\prime}/\mathrm q_{\max}^{\prime\prime} ≤ 0.90.
ASSUMPTIONS: (1) Nucleate boiling, (2) Fluid-surface with \mathrm C_{\mathrm{s,f}} = 0.004, n = 1.7 for Rohsenow correlation, (3) Backside of substrate insulated.
PROPERTIES: Table A-5, Refrigerant R-113 (1 atm): \mathrm T_{\text{sat}} 321 K = 48°C, ρ_{\ell} = 1511 kg/m³ , ρ_{\mathrm v} = 7.38 kg/m³, \mathrm h_{\mathrm{fg}} = 147 kJ/kg, σ = 15.9 × 10^{-3} N/m; R-113, sat. liquid (given, 321 K): \mathrm c_{\mathrm p,\ell} = 983.8 J/kg⋅K, μ_{\ell} = 5.147 × 10^{-4} N⋅s/m², \mathrm{Pr}_{\ell} = 7.183.
SCHEMATIC:
ANALYSIS: (a) The operating power level (flux) is 0.50 \mathrm q_{\max}^{\prime\prime}, where the critical heat flux is estimated from Eq. l0.7 for nucleate pool boiling,
q _{\max }^{\prime \prime} = 0.149 h _{ fg } \rho_{ v }\left[\sigma g \left(\rho_{\ell}-\rho_{ v }\right) / \rho_{ v }^2\right]^{1 / 4}
q _{\max }^{\prime \prime} = 0.149 \times 147 \times 10^3 \frac{ J }{ kg } \times 7.38 \frac{ kg }{ m ^3}\left[15.9 \times 10^{-3} \frac{ N }{ m } \times 9.8 \frac{ m }{ s ^2}(1511 – 7.38) \frac{ kg }{ m ^3} /\left\lgroup 7.38 \frac{ kg }{ m ^3}\right\rgroup^2\right]^{1 / 4}
q _{\max }^{\prime \prime} = 233 kW/m^2.
Hence, the heat flux on a chip is 0.5 × 233 kW/m² = 116 kW/m² and the power level is
q _{\text{chip}}= q _{ s }^{\prime \prime} \times A _{ s }=116 \times 10^3 W / m ^2 \times 25 mm ^2\left(10^{-3} m / mm \right)^2=2.9 W.
To determine the chip surface temperature for this condition, use the Rohsenow equation to find \Delta\mathrm T_{\mathrm e} = \mathrm T_{\mathrm s} – \mathrm T_{\text{sat}}~\text{with}~\mathrm q_{\mathrm s}^{\prime\prime} = 116 × 10³ W/m². The correlation, Eq. 10.5, solved for \Delta\mathrm T_{\mathrm e} is
\Delta T _{ e }=\frac{ C _{ s , f } h _{ fg } \operatorname{Pr}_{\ell}^{ n }}{ c _{ p , \ell}}\left\lgroup \frac{ q _{ s }^{\prime \prime}}{\mu_{\ell} h _{ fg }}\right\rgroup^{1 / 3}\left[\frac{\sigma}{ g \left(\rho_{\ell}~ – ~ \rho_{v}\right)}\right]^{1 / 6}=\frac{0.004~ \times~ 147 ~\times ~10^3 J / kg (7.18)^{1.7}}{983.8 J / kg \cdot K } \times
\left\lgroup \frac{116~ \times~ 10^3 W / m ^2 \cdot}{5.147~ \times ~10^{-4} \frac{ N \cdot s }{ m ^2} ~\times~ 147 ~\times~ 10^3 \frac{ J }{ kg }}\right\rgroup^{1 / 3}\left[\frac{15.9~ \times ~10^{-3} N / m }{9.8 \frac{ m }{ s ^2}(1511 ~- ~ 7.38) \frac{ kg }{ m ^3}}\right]^{1 / 6}=19.9^{\circ} C.
Hence, the chip surface temperature is
T _{ s }= T _{\text{sat}}+\Delta T _{ e }=48^{\circ} C +19.9^{\circ} C \approx 68^{\circ}C.
(b) Using the IHT Correlations Tools, Boiling, Nucleate Pool Boiling — Heat flux and Maximum heat flux, the chip surface temperature, \mathrm T_{\mathrm s}, was calculated as a function of the ratio \mathrm q_{\mathrm s}^{\prime\prime}/\mathrm q_{\max}^{\prime\prime}. The required thermophysical properties as provided in the problem statement were entered directly into the IHT workspace. The results are plotted below.
COMMENTS: (1) Refrigerant R-113 is attractive for electronic cooling since its boiling point is slightly above room temperature. The reliability of electronic devices is highly dependent upon operating temperature.
(2) A copy of the IHT Workspace model used to generate the above plot is shown below.
| // Correlations Tool – Boiling, Nucleate pool boiling, Critical heat flux q”max = qmax_dprime_NPB(rhol,rhov,hfg,sigma,g) // Eq 10.7 g = 9.8 // Gravitational constant, m/s^2 /* Correlation description: Critical (maximum) heat flux for nucleate pool boiling (NPB). Eq 10.7, Table 10.1 . See boiling curve, Fig 10.4 . */ // Correlations Tool – Boiling, Nucleate pool boiling, Heat flux qs” = qs_dprime_NPB(Csf,n,rhol,rhov,hfg,cpl,mul,Prl,sigma,deltaTe,g) // Eq 10.5 //g = 9.8 // Gravitational constant, m/s^2 deltaTe = Ts – Tsat // Excess temperature, K Ts = Ts_C + 273 // Surface temperature, K Ts_C = 68 // Surface temperature, C //Tsat = // Saturation temperature, K /* Evaluate liquid(l) and vapor(v) properties at Tsat. From Table 10.1. */ // Fluid-surface combination: Csf = 0.004 // Given n = 1.7 // Given /* Correlation description: Heat flux for nucleate pool boiling (NPB), water-surface combination (Cf,n), Eq 10.5, Table 10.1 . See boiling curve, Fig 10.4 . */ // Heat rates: qsqm = qs” / q”max // Ratio, heat flux over critical heat flux qsqm = 0.5 // Thermophysical Properties (Given): Tsat = 321 // Saturation temperature, K Tsat_C = Tsat – 273 // Saturation temperature, C rhol = 1511 // Density, liquid, kg/m^3 rhov = 7.38 // Density, vapor, kg/m^3 hfg = 147000 // Heat of vaporization, J/kg sigma = 15.9e-3 // Surface tension/ N/m cpl = 983.3 // Specific heat, saturated liquid, J/kg.K mul = 5.147e-4 // Viscosity, saturated liquid, N.s/m^2 Prl = 7.183 // Prandtl number, saturated liquid |
