Heat equation

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Can anyone hep me understand this solution

I have attached both question and solution

I can see the application of Fourier's law to the right hand side of the solution, but why is right hand side multiplied by area?, also to the left, what has happened to Ko.

Thanks

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Since the volume of of fluid is taken as constant

The left hand side tells you that the time rate of heat flow into the fluid in the bath

time rate of heat flow =( mass of fluid times specific heat)  * ( the time rate of change of temperature of the fluid)

time rate of heat flow =(volume * density * specific heat) * (the time rate of change of temperature of the fluid)

time rate of heat flow = $\left( {V{c_f}{\rho _f}} \right)\left( {\frac{{\partial u}}{{\partial t}}} \right)$

This is not part of the conduction equation of the bar.

But the conduction equation of the bar does give the quantity of heat passing through any cross section of the bar, as a function of the distance along the bar.

Formally it says that the time rate of heat flow through a section is proportional to the area of cross section and the temperature gradient at that section or

time rate of heat flow through a section = (a constant) * (area) * (temperature gradient)

time rate of heat flow through a section =  $= \left( { - {K_0}} \right)*\left( A \right)*\left( {\frac{{\partial u}}{{\partial x}}} \right)$

The constant K0 is called the thermal conductivity and taken as positive, from its use in other situations.
Since heat flows from a higher temperature to a lower one this introduces the negative sign.

Equating these rates of heat flow leads to the equation shown.

The hint tells you to find a solution to the for the temperature u at x = L.

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