Balance Equations for Energy Extraction and Conversion

We shall refer to the figure below in the discussion of the energy balance.

Figure 1. Drawing for Energy Balance

The well-known energy balance equation for an open system or, as I like to say, a control volume is as follows:

where U is internal energy, H = U + PV is enthalpy, W is work,
and Q is heat. The subscripts *i* and *o* refer to in and
out. Equation 1 is the First Law of Thermodynamics for a uniform-state,
uniform-flow system written as a simplified energy balance with kinetic energy
and gravitational potential energy neglected. The left-hand side is the
accumulation within the control volume (CV) with the internal energy at the end
of the experiment, U_{2}, minus the internal energy at the beginning of
the experiment, U_{1}. Energy and injection work associated with
mass entering the CV and mass leaving are represented by the enthalpy, H.
Heat and work *entering* and *leaving* the CV are represented by Q
and W.

In the case of simple steady-state extraction and conversion processes such as we are likely to encounter in the production of primary energy, Eq. 1 reduces to

or, in terms of the above drawing,

_{}

Let HR/HI = β and HN/HR = μ_{1}. Then

_{}

or

_{}

where we have used the fact that streams HR and HI have identical physical properties.

We shall refer to the figure below in the discussion of the entropy balance.

Figure 2. Drawing for Entropy Balance

The Second Law can be written as an entropy balance by
introducing the thermodynamic lost work, LW_{cv} , which is the
entropy change ∆S_{irr} due to irreversibilities within the CV
times the temperature of the environment, T_{e} . Under
reasonable assumptions, the equation for an open system or, as I like to say, a
control volume is as follows:

_{ }_{}

where U is internal energy, H = U + PV is enthalpy, W is work, Q
is heat, T is temperature, and LW_{CV} term is the irreversibility,
which is added to achieve closure on the entropy balance and which allows the
Second Law of Thermodynamics to be written for an open system. The
subscripts *i*, *o*, and *e* refer to in, out, and
environment. In our case, T_{e} = T_{o}. In the
case of simple steady-state extraction and conversion processes such as one is
likely to encounter in the production of primary energy, Eq. 1 reduces to

_{}

or, in terms of the above drawing,

_{}

Let SN/SR = μ_{2}; and, since SR/SI = β,

_{}

or

_{}

We shall refer to the figure below in the discussion of the availability balance.

Figure 3. Drawing for Availability Balance

Helmholtz availability is U – T_{e}S = A and Gibbs
availability is H - T_{e}S = E; therefore, the availability balance,
which is obtained by multiplying the entropy balance equation by T_{e}
and subtracting it from the energy balance equation, is as follows:

_{ }_{}

_{}

where U is internal energy, H = U + PV is enthalpy, W is work, Q
is heat, and T is temperature. The subscripts *i*, *o*, and *e*
refer to in, out, and environment. In our case, T_{e} = T_{o}.
In the case of simple steady-state extraction and conversion processes, such as
one is likely to encounter in the production of primary energy, Eq. 3, with T_{e}
= T_{o}, reduces to

or, in terms of the above drawing,

_{}

Let EN/ER = μ; and, since ER/EI = β,

_{}

or

_{}

The ratio of EN to ER, μ,
can be written in terms of μ_{1} and μ_{2} as
follows:

_{}

This is the Combined First and Second Laws of Thermodynamics.

We shall refer to the figure below in the discussion of the emergy balance.

Figure 4. Drawing for Emergy Balance

Let MN be the emergy of the work supplied by Nature, MR the
emergy of the immediate energy product, and MI the emergy of the energy
investment stream. Let λ_{N} be the transformity of the
energy supplied by Nature and λ_{R} the transformity of the
immediate energy product (as opposed to MR – MI, the emergy delivered to the
economy). The transformity, λ_{i}, is the number of kWhrs of
single-phase, 60 Hz, 110-volt AC electricity one can obtain from 1 kWhr of
energy source *i* by an efficient process. Thus, 1.0 kWhrs of
single-phase, 60 Hz, 110-volt AC electricity is my (arbitrary, but
well-defined) choice for the unit of emergy and λ_{i} is an electricity-based
transformity. Then, since

_{}

_{}

or,

_{}

This relation can be used to find any one of the four variables
λ_{N}, λ_{R}, μ, or β in terms of the other
three. Let us now consider two special cases and one general case:

In the case of the extraction of a naturally occurring energy source such as petroleum, HN = HR, SN = SR, and EN = ER; however, the emergy MN is less than the emergy of MR by the amount of the losses, namely

_{}

Since μ = 1,

_{}

and, in the case of petroleum from a West Texas well, λ_{N}
is slightly less than λ_{R}. If, on the other hand, the
petroleum were from the tar sands of Utah with an EROI of 2.0, say, λ_{N
}would be one-half of λ_{R}. Thus, transformity depends
crucially upon EROI.

If the product is single-phase, 60 Hz, 110-volt AC electricity,
λ_{R} = 1.0. The availability of the sunlight that must be
intercepted to produce one kilowatt hour of electricity is very large; so,
μ is large. If the EROI, β, is known,

_{}

and λ_{N} will be very small. The cost of the
sunshine is zero, which is a small number; but, the cost of the solar
collector, the area of which must be large to intercept μ kWhrs of
sunshine, is not small. Similar considerations apply to wind energy.

We shall need to deal with all four variables, λ_{N},
λ_{R}, μ, and β, to describe the conversion of biomass
to motor fuel. The general equation applies, namely,

_{}

Presumably, β, the EROI, is close to two and μ is large; therefore, the relative transformity,t, where

_{}

is very small. Thus, we expect to harvest a great deal of biomass to obtain a kWhr of single-phase, 60 Hz, 110-volt AC electricity. The effect upon greenhouse gases is close to zero over an extended period; but, the area of the land required to meet any appreciable fraction of the energy budget of the United States, say, is large.

The above equations apply to existing extraction and conversion
processes operating at steady state where β and μ are known.
Suppose the transformity of the energy provided by Nature, λ_{N},
and the transformity λ_{R’} obtained from the most efficient
competing process are known, assuming that our process in not the most
efficient. Then,

_{}

and the efficiency, η, of the process under investigation is

_{}

If our process is the most efficient, the value of the emergy of
the product is changed to λ_{R} · ER.

Thomas L Wayburn

Houston, Texas

October 15, 2006