Shape memory alloys (SMAs) are metals that undergo a change in crystalline structure within a relatively narrow temperature range. A phase transformation from martensite to austenite can induce relatively large changes in the overall dimensions of the SMA. Hence, SMAs can be employed as mechanical actuators. Consider an SMA rod that is initially D_i = 2 mm in diameter, L_i = 40 mm long, and at a uniform temperature of T_i = 320 K. The specific heat of the SMA varies significantly with changes in the crystalline phase, hence c varies with the temperature of the material. For a particular SMA, this relationship is well described by c = 500 J/kg · K + 3630 J/kg ·
K X e^{(–0.808 K–1X|T–336K|)}. The density and thermal conductivity of the SMA material are p = 8900 kg/m³ and k = 23 W/m ·
K, respectively.
The SMA rod is exposed to a hot gas characterized by T_∞ = 350 K, h = 250 W/m² · K. Plot the rod temperature versus time for 0 ≤ t ≤ 60 s for the cases of variable and constant (c = 500 J/kg · K) specific heats. Determine the time needed for the rod temperature to experience 95% of its maximum temperature change.