Modern Engineering Thermodynamics

4.7 Nonmechanical Work Modes of Energy Transport 117
4.7.1 Electrical Current Flow Work
Electrical current flow work occurs whenever current-carrying wires (pure conductors) cross the system boundary.
This is the most common type of nonmechanical work mode encountered in thermodynamic system analysis.
The generalized force here is the intensive property voltage (the electric potential) ϕ, and the extensive property
generalized displacement is the charge q. 7 Then, assuming the voltage is applied to the system,
ðdWÞ electrical
current
= − ϕdq
and
Electrical current i is defined as
so dq = idt, and
ð 1 W 2 Þ electrical
current
= −
i = dq
dt
Z 2
1
ϕ dq (4.45)
ðdWÞ electrical
current
= − ϕidt (4.46)
Then, electric current work is
Electrical current work
ð 1 W 2 Þ electrical
current
=
Z 2
1
ϕidt
(4.47)
From Ohm’s law, the instantaneous voltage ϕ across a pure resistance R carrying an alternating current, described
by i = i max sin(2πft), is
ϕ = Ri = Ri max sin ð2πftÞ
where f is the frequency and ϕ max = Ri max . Thus, Eq. (4.47) gives the electrical current work of n cycles of an
alternating electrical current applied to a pure resistance from time 0 to time t = n/f as
ð 1 W 2 Þ electrical
Z t = n/f
= − ϕ max i max sin 2 ð2πftÞ dt
current
0
= − ϕ e i e t = − ϕ 2 e ðt/RÞ = − i2 e
p Rt ffiffiffi
p ffiffiffi
where ϕ e and i e are the effective voltage and current defined by ϕ e = ϕ max / 2 and ie = i max / 2 :
= − ϕ max i max ðt/2Þ
(4.48)
Electrical work can exist in either open or closed systems (we do not consider the flow of electrons across a system
boundary to be a mass flow term). When the electron supply is going into a finite system, such as a battery
or a capacitor, Eq. (4.45) or (4.47) is convenient to use. But, when an essentially infinite supply of voltage and
current is used, it is more convenient to use the instantaneous rate at which electrical work is done, or the electrical
power, defined as
ð _WÞ electrical
current
= dW dt
= − ϕi (4.49)
OHM’S LAW
This law was discovered experimentally by George Simon Ohm (1787–1854) in 1826. Basically, it states that, for a given
conductor, the current is directly proportional to the potential difference, usually written as ϕ = Ri, where R is the electrical
resistance in units of ohms, where 1 ohm = 1 volt/ampere.
7 The electrical potential ϕ and the electric field strength vector E are related by E = −∇(φ), where ∇( ) is the gradient operator.