Capacitor Electrode Dimensions

The correct choice o f the cylinders length, L and the radii, a and b (see figure 4.6) involves taking into account the general requirements for a capacitive concentration transducer (see section 4.4) particularly those relating to its proposed application for measuring vehicular particulate emissions.

Firstly and most important, the transducer’s capacitance should be within the measurement range of the commercial LCR meter (Hewlett Packard 4263B) employed in this study which is between 1 pico-Farad and 1 Farad. Measurements outside this range will require specially constructed LCR meters, which will add to the system’s complexity and cost. Equation 4.26 indicates that the measured capacitance o f the coaxial transducer increases with the following:

a) an increase in the length, L o f the cylinders

b) a decrease in the radii ratio, b/a or the gap between the two coaxial cylinders

Furthermore, the transducer should be sensitive enough to detect very small changes in the solids concentrations. This is especially so in case o f monitoring vehicular

Chapter 4 Capacitance Technique fo r Concentration Measurements

Also, the effective dielectric constant o f the air/solids mixture is expected to be small bearing in mind that 75-85% o f the exhaust particulate mass is carbon which has a low dielectric constant o f 5.7 (Frey et al., 1967; Bagyi, 1998). Therefore, it is essential to increase the system’s sensitivity in terms of maximising the effective dielectric constant for a given solids concentration in the mixture.

Considering, the fact that the dielectric constant o f a particle-gas suspension is proportional to the polarisation o f its elementary dipole units, which is in turn proportional to the imposed electrical field (see sections 4.3.1-4.3.2), then an increase in the latter will result in an increase in the dielectric constant o f the material. Also since the electric field between two electrodes at a constant voltage is inversely proportional to the separation distance between them, then the electric field will be increased by decreasing the separation distance between the electrodes (see Chapter

6). In addition, in order to further increase the electric field the applied voltage used

throughout the experimental series is 1 Volt, which the maximum output signal provided by the specific LCR meter.

Another important feature required for a capacitance sensor, is uniform measurement sensitivity over the whole cross sectional area o f the sensing volume. This can be achieved by ensuring that the electric field between the capacitor’s electrodes is as uniform as possible. In theory, a capacitor with completely uniform electric field is impossible to be constructed. This is because there will always be a small distortion of the field right on the edges of the electrodes (fiinging) due to a small increase in the charge density on these areas (Scott, 1959; Feynman, 1964). In practice though, this error can be effectively by reducing the separation distance between the electrodes in comparison with their length (Scaife, 1989; Jonassen, 1998). Xie et al. (1990) suggest that in practical measuring systems, electrode lengths about twice the separation distance will be adequate to give high sensitivity and to minimise firinge effects. In other applications (Dotson et al. 1949; Sami et al., 1980; Irons et al., 1983) electrodes lengths varying from 1.8 to 6 times the separation distance have

Chapter 4 Capacitance Technique fo r Concentration Measurements

In case of the coaxial capacitor, the electric field is symmetrical and directed radially outward from the central axis (Jonassen, 1998). Its strength is inversely proportional to the distance, r from the centre and is given by:

where:

Q is the total charge on the electrodes r is the distance firom the centre L is the electrode length

The electric field even though well defined, is not uniform along the cross-section of the capacitor (Jonassen, 1998). The non-uniformity of the field is characterised by the denser field lines in the centre o f the capacitor. However this non uniformity becomes less profound when the separation distance becomes smaller. Therefore, in the case of coaxial capacitors, it is necessary to have the smallest possible separation distance, which will increase the uniformity of the electric field at the main body of the capacitor as well as at the edges.

In conclusion, it is clear that it is advantageous for the coaxial capacitor electrodes to have a small separation distance with relatively long length. However, the minimum separation distance is dictated by two factors. These include the breakdown electric strength o f air and the build up o f high back pressures.

Air, under normal circumstances, is a good insulator. However, there is a critical value o f electric field strength that air can withstand before electrical breakdown occurs and a violent discharge (spark/flashover) takes place. The theoretical breakdown strength o f air at atmospheric pressure is 3x10® V/m (Bolton, 1980; von Hippel, 1995). For a given electrode separation distance, it is therefore possible to calculate the maximum voltage difference that can be applied before electrical

Chapter 4_________________________Capacitance Technique fo r Concentration Measurements

Small electrode separations may cause unacceptably high back pressures during flow measurements. In the specific application of monitoring vehicular emissions for example, ISO 10054:1998(E) requires that the maximum back pressure generated by the measuring instrument should not exceed 1 kPa so that the engine’s performance will not be affected (ISO 10054,1998).

Long electrode lengths are desirable since this increases the measured signal (see equation 4.26) as well as reducing fringing effects. The maximum electrode length is however limited by practical considerations such as portability and more importantly, to ensure the homogeneity o f the test suspension along the measurement zone (Louge et al., 1996).

Taking all the above mentioned factors into account, the following dimensions were selected for the construction o f the coaxial capacitor transducer (see figures 4.7 and 4.8 and plates 4.1-4.3):

• Inside radius o f the outer cylinder, b = 1.62 cm • Outside radius o f the inner tube, a = 0.90 cm • Length o f the cylinders, L = 12 cm

The inter-electrode separation distance (i.e. b-a), between the two electrodes is 0.72 cm. This is almost 16 times smaller than the electrode length. Furthermore, at atmospheric pressure the capacitor can withstand a voltage o f 33 kV before electrical discharge occurs. In the present study a voltage o f maximum 1 Volt is used which is well below the critical value mentioned above. Finally, direct measurements indicate that the resulting back pressure for this design configuration is minimal even at high sampling rates (0.05 kPa for 850 1/min air flowrate).