# Find the rate of change of temperature with time.

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\)