When the kidneys fail to remove sufficient impurities from the blood, a device for removing the impurities is used, which is termed a hemodialyzer or just a dialyzer. Basically it transfers the impurities from the blood to another fluid termed the dialyzate by mass transfer through a membrane. A schematic diagram of a dialyzer is given in Fig. 5.1.

We now consider the derivation of a PDE model based on mass conservation.

1D PDE model

The configuration of a 1D hemodialyzer model is explained in Fig. 5.1, primarily with words.

We can note the following details about the model represented in Fig. 5.1:

1. • The model is one dimensional (1D) with distance along the dialyzer, z, as the spatial (boundary value) independent variable. Time t is an initial value independent variable.

2. • Two PDE-dependent variables, u1(z, t),u2(z, t), represent the impurity concentrations in the blood and dialyzate, respectively.The PDEs that define these dependent variables are derived subsequently.

3. • Blood enters the left end at concentration u1L(t). This BC is not designated as u1(z = 0, t) because of a header volume at the left end (explained next).

4. • Similarly, the exiting blood concentration at the right end is designated as u1R(t) rather than u1(z = zL, t) (again, because of a header volume).

5. • The entering and exiting dialyzate concentrations are u2 (z = zL, t) and u2 (z = 0, t), respectively.

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