2.1.2. Relationship between Sensor Output and Design Variables
One Layer Model. The current output of the sensor can be related to its design parameters by using a simplified electrode model. The assumptions used for developing the model are:
Fig. 2.5. Alternative pathways of oxygen reduction at cathode surface
Fig. 2.6. One layer electrode model.
Fig. 2.7. Design variables of DO sensor.
1-D Diffusion Equation. This so-called one layer model can be extended to include the effects of other layers (such as the liquid layer) as will be shown later. According to Fick's 2nd law, the unsteady-state diffusion in the membrane is described by (using the coordinate system shown in Fig. 2.6):
where Dm is the oxygen diffusivity in the membrane and x is the distance from the cathode surface.
Boundary Conditions. The initial and boundary conditions are:
where dm is the membrane thickness and po is the partial pressure of oxygen in the bulk liquid. The second boundary condition (Eq. (3)) assumes a very fast reaction at the cathode surface, which is generally achieved when the cathode is properly polarized (with the right voltage).
Unsteady Oxygen Profile. The solution of Eq. (1) with the boundary conditions is:
The current output I of the electrode is proportional to the oxygen flux at the cathode surface:
where N, F, A, and Pm are the number of electrons per mole of oxygen reduced, Faraday constant (= 96,500 coul/mol), surface area of the cathode, and oxygen permeability of the membrane, respectively. The permeability Pm is related to diffusivity by:
where Sm is the oxygen solubility of the membrane.
Unsteady Current Output. From Eqs. (5) and (6), the current output It of the electrode is:
Steady Current Output. The pressure profile within the membrane and the current output under steady-state conditions can be obtained from Eqs. (5) and (8), respectively:
At steady state, the pressure profile is linear and the current output is proportional to the oxygen partial pressure in the bulk liquid. Eq. (10) forms the basis for DO measurement by the sensor.
Response Time. Eq. (8) shows that the rapidness of the sensor response depends on the following term:
When tis large (a thin membrane and/or high Dm), the sensor responds fast. Note that these conditions tend to weaken the assumption of membrane-controlled diffusion. Therefore, a compromise has to be made for an optimum sensor performance. Note that adjusting dm (rather than Dm) is more effective in adjusting t (because t depends on the square of dm). Eqs. (10) and (11) indicate that the design variables for a DO sensor is Pm, dm, Dm, and A (Fig. 2.7).