**2.2.2. Calibration and k _{L}a Measurement **

**Liquid Phase Calibration. **Prepare an air saturated water by passing
air bubbles into a small volume (100 mL) of water. Prepare a nitrogen saturated
water in the same way. Connect the fabricated DO sensor to signal amplifying
circuit of Fig. 2.3b, and then measure the voltage output for both water
solutions. The liquids have to be agitated at high speed to obtain proper
calibration. This is so-called a 'two point' calibration.

**Gas Phase Calibration. **Perform the calibration in gas phase by
exposing the sensor to air. Do the same using nitrogen as the gas phase.
Compare the two calibrations (between liquid and gas). Should they be the
same? If not, why not?

**Measurement of Response Time. **The other important parameter of
the sensor is the response time.** **It can be measured by making a step
change in oxygen partial pressure in the measurement medium and measuring
the sensor response. The sensor can be approximated as a first order system:

where c is the oxygen concentration in the measurement sample, c_{p}
is the oxygen concentration measured by the sensor, and t_{p
}is the sensor time constant. When a step change is made in c (by transferring
the sensor from air into a nitrogen saturated, stirred water), the sensor
output decreases roughly exponentially (not exactly exponentially because
the sensor may not be a true first order system). The time constant t_{p}
is the time when the sensor response reaches 63.7% of the ultimate response
(Fig. 2.12a). The solution to Eq. (22) with the following boundary condition
is an exponential function.

Note that a normalized concentration is used: c of 1 means 100% air saturation and 0 means nitrogen saturation. The solution is:

Eq. (24) indicates that when *t* = t_{p},
c/c_{p} will be 0.64. The time constant t_{p
}can also be determined conveniently by using an integral method -
the area above the response curve is equal to t_{p
}(see Fig. 2.12b). This method is especially useful when there
is a lot of noise in the measured signal. The integration can be carried
graphically using either trapezoidal rule of Simpson's rule.

**Measurement of k _{L}a**

**Fig. 2.12. (a) Sensor response time measurement; (b)
integral method for measuring the sensor time constant. **

**Fig. 2.13. Measurement of k _{L}a by integral
method.**

where c is the oxygen concentration in the reactor and *c** is the
oxygen concentration at the gas-liquid interface. This equation can be rearranged
to:

where

Eqs. (22) and (26) can be solved simultaneously to obtain an expression
for k_{L}a. However, t_{k }can
be obtained graphically as shown in Fig. 2.13 when t_{p}
is known.

**Caution in k _{L}a Measurement. **Note that the magnitude
of t