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Suppose the function $u(x,y,z,t)$ satisfies $$\frac{\partial u}{\partial t} = \Delta u +5$$

on a sphere of radius 2, with $\Delta$ as the Laplace operator and $u(x,y,z,0)=f(x,y,z)$. On the surface of the sphere it is given that $\nabla u \cdot \hat{n} = 6$, where $\hat{n}$ is a unit outward normal vector. Calculate the total thermal energy for this sphere as a function of time.

Solving the problem is straightforward enough (write down the integrated conservation law, use divergence theorem, etc.), but what do I do about the other physical quantities such as specific heat, density and thermal conductivity ($c,\rho ,K_0$), which are present in the original heat equation? Do I assume $c\rho =1$ and $K_0 = 1$? The entire heat equation is given by: $$c \rho \frac{\partial u}{\partial t} =K_0 \frac{\partial ^2 u}{\partial x^2} + Q(x,t)$$

Any advise would be appreciated.

Edit (this is what my instructor had to say): The only thing that I want to mention, is that if the equation is given to you, it means that specific values have already been assigned to all the problem parameters (e.g. $c, \rho , K_0$ are known).

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    $\begingroup$ Compare the parameters to the coefficients of the given equation. You indeed have $c\rho = 1$, $K_0=1$ and $Q(x,t) = 5$ $\endgroup$
    – Dylan
    Commented May 19, 2018 at 14:10

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I would prefer to think that the problem has been expressed in dimensionless units. That is to say, the units of measurement have been chosen in such a way that the numerical value of $c\rho/K_0$ is unity.

This is usual in mathematical or scientific treatments. If you want to focus on the ideas involved in the solution, it is distracting to have to worry about the units. However, in an engineering treatment, it is usually considered very important to state what units have been employed, and what constants have been assumed. Otherwise, what are you going to do with the answer?

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