A nanolaminated material is fabricated with an atomic layer deposition process, resulting in a series of stacked, alternating layers of tungsten and aluminum oxide, each layer being δ = 0.5 nm thick. Each tungsten– aluminum oxide interface is associated with a thermal resistance of R"_t,i  = 3.85 x 10^–9 m² · K/W. The theoretical values of the thermal conductivities of the thin aluminum oxide and tungsten layers are k_A  = 1.65 W/m · K and k_T = 6.10 W/m · K, respectively. The properties are evaluated at T = 300 K.

(a) Determine the effective thermal conductivity of the nanolaminated material. Compare the value of the effective thermal conductivity to the bulk thermal conductivities of aluminum oxide and tungsten, given in Tables A.1 and A.2.

(b) Determine the effective thermal conductivity of the nanolaminated material assuming that the thermal conductivities of the tungsten and aluminum oxide layers are equal to their bulk values.

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TABLE A. 1 Composition. Titanium Tungsten Uranium Vanadium Zinc Zirconium Melting Point (K) 1953 "Adapted from References 1-7. 3660 2192 693 2125 1406 19070 (kg/m³) (J/kg-K) (kg/m³) 4500 19300 6100 7140 Properties at 300 K k a. 106 (J/kg K) (W/m K) (m²/s) 6570 522 132 116 489 389 278 21.9 174 27.6 30.7 116 22.7 9.32 30.5 24.5 300 465 208 68.3 12.5 10.3 41.8 100 200 12.4 186 87 122 21.7 94 35.8 258 430 25.1 205 108 31.3. 117 118 297 367 33.2 25.2 264 Properties at Various Temperatures (K) k (W/m K)/c, (J/kg-K) 400 20.4 551 159 137 29.6 125 31.3 515 111 402 21.6. 300 600 19.4 591 633 137 142 34.0 146 33.3 540 103 436 20.7 800 322 19.7 125 145 38.8 176 35.7 563 21.6 342 1000 1200 1500 2000 2500 20.7 675 118 113 148 152 43.9 49.0 180 38.2 597 23.7 22.0 24.5 620 686 107 100 95 157 167 176 362 161 40.8 645 26.0 344 44.6 50.9 714 867 28.8 344 344 33.0