The temperature distribution at a certain input of time in concrete slab during curing is given by T=3x^{2}+3x+16 where x is in cm and T is in K. Find the rate of change of temperature with time. (\alpha=0.003\;cm^{2}/s)
The temperature distribution at a certain input of time in concrete slab during curing is given by T=3x^{2}+3x+16 where x is in cm and T is in K. Find the rate of change of temperature with time. (\alpha=0.003\;cm^{2}/s)
The rate of change of temperature with time is given as \frac{dT}{dt}
Here T=3x^{2}+3x+16
For a 1-D unsteady heat flow without any internal heat generation
\frac{d^{2}T}{dx^{2}}=\frac{1}{\alpha}\frac{dT}{dt}
\frac{dT}{dx}=\frac{d}{dx}\left ( 3x^{2}+3x+16 \right ) \\=6x+3 \\\frac{d^{2}T}{dx^{2}}=\frac{d}{dx}\left ( 6x+3 \right ) \\=6
\Rightarrow 6=\frac{1}{\alpha}\frac{dT}{dt} \\\Rightarrow 6=\frac{1}{0.003}\frac{dT}{dt} \\\Rightarrow \frac{dT}{dt}=6\times 0.003 = 0.018\;K/s
