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. \((\alpha=0.003\;cm^{2}/s)\)

Asked on 25th October 2019 in Propulsion.
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    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 

    \[\frac{d^{2}T}{dx^{2}}=\frac{1}{\alpha}\frac{dT}{dt}\]

    \(\frac{dT}{dx}=\frac{d}{dx}\left ( 3x^{2}+3x+16 \right )
    \\=6x+3
    \\\frac{d^{2}T}{dx^{2}}=\frac{d}{dx}\left ( 6x+3 \right )
    \\=6\)

    \(\Rightarrow 6=\frac{1}{\alpha}\frac{dT}{dt}
    \\\Rightarrow 6=\frac{1}{0.003}\frac{dT}{dt}
    \\\Rightarrow \frac{dT}{dt}=6\times 0.003 = 0.018\;K/s\)

     

     

    Answered on 25th October 2019.
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