What is the difference between a first order compartment and diffusion Biologists use Compartment models to represent the flow and storage of fluids in an animals body. In tissue (like muscle) the diffusion of blood is more accurately represented by the diffusion equation: $dB/dt = -\operatorname{Div}(\operatorname{Grad}B)$. A first order compartment model would look like $dB/dt = - k  B$ where $k$ is the reciprocal "diffusion" time. 
Are these two equations equivalent and in what limiting cases?   
 A: $\frac{\partial B}{\partial t} =- \operatorname{div}\nabla B$ is a partial differential equation: $B$ is a function of time and space coordinates, representing the concentration of blood per volume at various parts of tissue. 
$dB/dt=-kB$ is an ordinary differential equation: $B$ is a function of time only, representing the total amount of blood within a compartment at time $t$. It says that the compartment loses some percentage of amount per unit time. 
These are quite different models: one pays attention to the spatial distribution of blood and studies diffusion, the other works only with the total amount, and studies inflow/outflow. There is no direct way to get one from the other. 
However, if  you consider many compartments with amounts $B_1,B_2,\dots$ which exchange blood according to the above (each giving up some $kB_n$ amount to its "neighbors"), then this system of ordinary equations begins to look like a PDE when the number of compartments is large (indeed, this is a space-discretization of the diffusion PDE).
