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
\[\frac{d^{2}T}{dx^{2}}=\frac{1}{\alpha}\frac{dT}{dt}\]
\(\frac{dT}{dx}=\frac{d}{dx}\left ( 3x^{2}+3x+16 \right )
\\=6x+3
\\\frac{d^{2}T}{dx^{2}}=\frac{d}{dx}\left ( 6x+3 \right )
\\=6\)
\(\Rightarrow 6=\frac{1}{\alpha}\frac{dT}{dt}
\\\Rightarrow 6=\frac{1}{0.003}\frac{dT}{dt}
\\\Rightarrow \frac{dT}{dt}=6\times 0.003 = 0.018\;K/s\)
