Explanation Given: The initial temperature is T i = 25 °C . The base temperature of the nano spring is T b = 50 °C . The infinity temperature is T ∞ = 25 °C . The convective heat transfer coefficient is h = 10 6 W / m 2 ⋅ K . The overall chord length is L c = 425 nm . The radius of the spring is r = 30 nm . The pitch of the spring is s = 25 nm . The diameter of the spring is D = 15 nm . The change in spring pitch with temperature is d s ¯ d T ¯ = 0.1 nm / K . Formula used: The expression for the length of the spring is given as, L = s 2 π L c r 2 + ( s 2 π ) 2 The expression for the value of m L c is given as, m L c = ( 4 h k D ) 1 2 L c Here, k is the thermal conductivity of silicon carbide. The expression for the change in base temperature of the spring is given as, θ b = T b − T i The expression for the average excess temperature is given as, θ ¯ = θ b m L c tanh ( m L c ) The expression for the average temperature is given as, T ¯ = θ ¯ + T ∞ The expression for the variation in the spring pitch is given as, s ¯ = s + d s ¯ d T ¯ ( T ¯ − T i ) The expression for the new spring length is given as, L 2 = s ¯ 2 π × L c r 2 + ( s ¯ 2 π ) 2 The expression for the change in length is given as, Δ L = L 2 − L The expression for the accuracy to which the actuation distance can be controlled is given as, R = Δ L × 1 °C T i Calculation: The length of the spring can be calculated as, L = s 2 π L c r 2 + ( s 2 π ) 2 L = 25 nm ( 10 − 9 m 1 nm ) 2 π 425 nm ( 10 − 9 m 1 nm ) { 30 nm ( 1 m 1000 mm ) } 2 + { 25 nm ( 10 − 9 m 1 nm ) 2 π } 2 L = 55.9 × 10 − 9 m ( 1 nm 10 − 9 m ) L = 55.9 nm From the table of “Thermophysical properties of selected non metallic solids” at temperature T = 300 K the thermal conductivity of silicon carbide is k = 490 W / m ⋅ K . The value of m L c can be calculated as, m L c = ( 4 h k D ) 1 2 L c m L c = ( 4 × 10 6 W / m 2 ⋅ K 490 W / m ⋅ K × 15 nm ( 10 − 9 m 1 nm ) ) 1 2 × 425 nm ( 10 − 9 m 1 nm ) m L c = 0

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ISBN: 9780470501962

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

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Chapter 3, Problem 3.138P

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The accuracy to which the actuation distance can be controlled.

Solution Summary: The author explains how the actuation distance can be controlled. The initial temperature is T_i=25°C, the base temperature of the nano spring, and the conve

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