**2.1.3. Effect of Liquid Velocity and Properties on Sensor
Performance**

**Real Situation. **In reality, the assumption 2 made earlier is not
entirely satisfactory. A stagnant liquid film always exists outside the
membrane even at high liquid velocity. Actually, when a DO sensor is used
to measure DO concentration in two different liquids at the same partial
pressure of oxygen, the readings are not the same. This indicates that the
sensor output depends, to a certain extent, on the properties of the liquid.
One layer model cannot explain this behavior. A more realistic model has
to consider both the membrane and the liquid film as shown in Fig. 2.8.

**Two Layer Model. **The effect of liquid layer on sensor current
output can be estimated by expanding the "one layer" model. At
steady state, the oxygen flux *J* through each layer in Fig. 2.8 should
be the same:

where K is the overall mass transfer coefficient and k_{L} and
k_{m }are mass transfer coefficient for liquid film and membrane,
respectively.

**Ohm's Law Analogy.** The inverse of the mass transfer coefficient
can be termed as the mass transfer resistance. From Eq. (12), it can be
shown that:

Equation (13) says that the overall*l* mass transfer resistance,
1/K, is the sum of the liquid phase mass transfer resistance, 1/k, and the
membrane phase mass transfer resistance, 1/k_{m}. The derivation
is based on Ohm's law analogy - J is consid*e*red the current and p
as the voltage. The individual resistances can be replaced by:

where d_{L} and P_{L }are liquid film thickness and the
oxygen permeability of the liquid film, respectively. Note that a stagnant
liquid film was assumed here, although it is more accurate to use the convective
mass transfer coefficient.

**Fig. 2.8. Two layer model for DO sensor.**

**Fig. 2.9. Flow sensitivity of DO sensor.**

**Two Layer Model. **When the individual mass transfer resistances
are considered the steady state sensor output becomes:

where is defined by:

Alternatively, I_{s} can be written as:

The time constant t of Eq. (11) can be modified to:

where d is defined by:

**Flow Sensitivity. **The DO sensor, when placed in a stagnant liquid,
produces a** **diffusion gradient extending outside the membrane and
farther into the liquid.. When the liquid is stirred, the diffusion gradient
can no longer be extended beyond the liquid film around the membrane. Since
the diffusion gradient becomes steeper with decreasing liquid film thickness,
the current output of the sensor increases with increase in liquid velocity
(Fig. 2.9a). Note also that the response time of the sensor increases as
the liquid velocity decreases (Fig. 2.9b). This so-called "flow sensitivity"
is greater for a sensor with a larger cathode because the size of the stagnant
diffusion field is proportionally greater with a larger cathode.

**Condition for Membrane Control of Diffusion. **From Eq. (17), the
condition for a membrane-controlled** **diffusion becomes:

To achieve this condition, a relatively thick membrane with a low oxygen permeability have to be used. Note that this contradicts the requirement for a fast sensor response. When this condition is achieved, the oxygen sensor output depends only on membrane properties as given by Eq. (10) and the sensor calibrated in one liquid can be used in other liquids without recalibration. In reality, however, there is always a liquid film (however thin it may be) and this causes variations in calibration in different liquids.