Page 233 - Electrical Safety of Low Voltage Systems

216 Chapter Thirteen

The circulation of the fluid, by moving charges, creates a stream
3
current. If the liquid is characterized by volumetric flow rate F (m /s),
3
density d (kg/m ), and specific charge density
(C/kg), the result-
ing stream current will have a magnitude that can be expressed in
amperes as
I = Fd
(13.1)
It is important to stress that, as represented in Fig. 13.2, any object
may have either, or both, “natural” capacitance C and resistance-to-
ground R (e.g., C for a 3.6-m-diameter tank with insulating lining
equals 100 nF, as per IEEE 142–1991). The capacitance-to-ground can
store the static charge even after the process of electrification is over,
and then discharge the energy, for example, upon person’s touch.

13.3 Static Charge Energy
The accumulation of static charge over an object consists of the “de-
posit” of the charge Q, during the time t Q , therefore, the electrification
process can be analyzed by studying the stream current I = Q/t Q .
Once the charge deposits, the object’s earth potential v(t) elevates,
and coincides with the voltage to which the capacitor is charged. To
calculatesuchearthpotential,referenceismadetoFig.13.2.Thecharge
current I flowing to ground divides through C and R, hence, by ap-
plying Kirchhoff’s first law, we can write
dv(t) v(t)
I = i C (t) + i R (t) = C + (13.2)
dt R
where i C (t) and i R (t) are, respectively, the currents charging the earth
capacitance and the current through the earth resistance of the tank.
By assuming as the initial condition that the object is not charged
[i.e., v(0) = 0], the solution v(t) of the above differential equation is

t
v(t) = RI(1 − RIe − RC ) (13.3)
The product RC is dimensionally a time and is defined as the time
constant of the charging process. The time constant is the time the earth
voltage takes to reach 63.2% of its final magnitude (i.e., the product RI).
Tobetterclarifythisconcept,letusevaluatetheearthpotentialgivenin
Eq.(13.3)fort =nRC,forn = 0to3.TheresultsareshowninTable13.1.
Obviously, the greater the time constant, the longer the
capacitance-to-ground takes to reach its full charge. In theory, because
of the exponential function of Eq. (13.3), such capacitance reaches
the voltage RI after an infinite time. In practice, after three or four
times the time constant, we can deem the capacitance-to-ground fully
charged.
If the body is completely insulated from ground (i.e., R =∞), the
electric charge, as it forms, cannot be drained to earth. Equation (13.2)