**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.**

**Assumptions Used **

**1.**The cathode is well polished and the membrane is tightly fit over the cathode surface such that the thickness of the electrolyte layer between the membrane and the cathode is negligible.**2.**The liquid around the sensor is well agitated that the partial pressure of oxygen at the membrane surface is the same as that of the bulk liquid.**3.**Oxygen diffusion occurs only in one direction, perpendicular to the cathode surface.

**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 D_{m}* *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 d_{m} is the membrane thickness and p_{o} 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 *P _{m}* 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 P

where *S _{m}* is the oxygen solubility of the membrane.

**Unsteady Current Output. **From Eqs. (5) and (6), the current output
I_{t} 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:

and

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
D_{m}), 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
d_{m} (rather than D_{m}) is more effective in adjusting
t (because t depends
on the square of d_{m}). Eqs. (10) and (11) indicate that the design
variables for a DO sensor is P_{m}, d_{m}, D_{m},
and A (Fig. 2.7).