Chapter 5, Problem 5.31P

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 $D_i=2\text{mm}$ in diameter, $L_i=40\text{mm}$ long, and at a uniform temperature of $T_i=320\text{K}\text{.}$ The specific heat of the SMA varies significantly with changes in the crystalline phase. hence cvaries with the temperature of the material. For a particular SMA, this relationship is well described by $c=500\text{J/kg}\cdot \text{K}\text{+}\text{3630}\text{J/kg}\cdot \text{K}\text{×}e\left(-0.808K^{-1}\times \left|T-336\text{K}\right|\right).$ density and thermal conductivity of the SMA material are $\rho =8900{\text{kg/m}}^{\text{3}}$ and $k=23\text{W/m}\cdot \text{K,}$ respectively.
The SMA rod is exposed to a hot gas characterized by $T_{\infty }=350\text{K},h=250{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ Plot the rod temperature versus time for $0\le t\le 60\text{s}$ for the cases of variable and constant $\left(c=500\text{J/kg}\cdot \text{K}\right)$ specific heats. Determine the time needed for the rod temperature to experience $95\%$ of its maximum temperature change. Hint: Neglect the change in the dimensions of the SMA rod when calculating the thermal response of the rod.