Seperation of variables - Product solutions for unsteady heat conduction

Chard3000
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Hey guys,

I was wondering about problem 12C.1 in Transport phenomena by Bird, Stewart and lightfoot.

The problem states that a block of material initially at uniform T0 is suddenely exposed to T1 at all surfaces.

Assume a solution of T=X(x,t)Y(y,t)Z(z,t)

any help with separation of variables of this type (3D)

thanks
 
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Have you tried substituting that expression into the heat equation?
 
I have tried to to... but I do not understand how to do a separation of variables for 3 dimensions.
 
Did T = X(x,t)Y(y,t)Z(z,t) come from a suggestion in the problem or did you come up with that? T = X(x) Y(z) Z(z) W(t) is a better choice.

Either way, unless you start writing down some equations and explain what's confusing you, you won't get too far.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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