The heat content of hot dry rocks (HDR) down to depths 5 to
10 km is enormous. To extract this heat it is necessary to
create an underground water circulation system using the HDR
as a heat exchanger. To maintain the underground flow a
system of natural or artificial fractures between injection
and recovery wells shall exist. The water is injected from
above, heated in the underground heat exchanger and
recovered at elevated temperature. The water pressure
throughout the flow between the two wells should be some
higher than the saturation pressure at the water exit
temperature to prevent boiling, which can hamper the flow
and the heat exchange process. Hence the specific of HDR
systems is that the geothermal heat will appear as a
pressurized water flow at elevated temperature. To produce
power the heated water transfers its heat in a counterflow
heat exchanger to a working media circulating in a
thermodynamic cycle.
Usually defined are the water temperatures T1 at the
recovery well mouth and T0 at the exit of the power
production unit. Thus defined is the specific thermal power
carried over by 1kg/s of the water: Nth =1• cpw(T1T0),
where cpw is the water heat capacity assumed constant in the
mentioned temperature interval.
Designing the power plant it is useful to calculate the
theoretical maximum specific electrical power Nmax, which
can be produced using the available thermal power Nth.
Assuming that the geothermal water is cooled down in the
power plant heat exchanger at a constant pressure Nmax can
be calculated as an integral of works produced in a sequence
of elementary Carnot cycles operating between current water
temperatures T and T0:
,
x = T0/(T1T0)
To approach the theoretical limit it is necessary to choose
a thermodynamic cycle with a heat admission curve as close
as possible to the water cooling down isobar, and with a
heat removal curve as close as possible to the T0 isotherm.
These conditions can be to a certain extent satisfied by a
supercritical Rankine cycle.
Selecting the thermodynamic cycle one can define the work l,
which can be produced by 1 kg of the working media. The
installation power N is a product l•w, where w is the
specific flow rate of the working media. Hence to maximize N
we need to define the maximum w, which can be carried over
through the counterflow heat exchanger under the
stipulation that this flow rate is compatible with the
selected thermodynamic cycle. If along the heat admission
isobar of the cycle the working media heat capacity cpwm
would be constant, the maximum w could be calculated as
wmax = 1•cpw/cpwm
However in reality the working media heat capacity is not
constant making the definition of the maximum working media
flow rate more complicated. This issue can be clearly
demonstrated for an undercritical Rankine cycle. In this
case the heat admission curve has an isothermal evaporation
section where the heat capacity is infinite. For the
counterflow heat exchanger defined are the inlet
temperatures of the hot water T1 and of the condensed
working media Twm. Both outlet temperatures T0 and T are
functions of the flow rate w. When w is quite small T0 ≈ T1
and T = T1. With increasing w the temperature T0 decreases
however T remains equal to T1. It means that the selected
cycle remains intact. This is valid until w reaches the
value wpinch, where in some section of the heat exchanger
the water temperature becomes equal to the saturation
temperature of the working media. At w > wpinch the working
media outlet temperature becomes gradually less than T1. It
means that the thermodynamic cycle will change and in
particular its thermal efficiency ηth will decrease.
The same considerations are valid for supercritical Rankine
cycles. A supercritical isobar in the critical point
vicinity has a bend point, where the heat capacity becomes
maximum providing for formation of a pinch similar to an
undercritical isobar.
Calculations of the maximum installation power N were
carried out for undercritical and supercritical
thermodynamic Rankine cycles with hydrocarbon working media
using the TRNSYS software. When a supercritical cycle was
selected the geothermal water temperature T1 was assumed
higher than the working media critical temperature. The
calculations demonstrate that N is slightly higher for
supercritical Rankine cycles. In both cases maximum N is
obtained at working media flow rates higher than wpinch.
It is concluded that for each geothermal water temperature a
proper supercritical Rankine cycle should be selected and
for this cycle an optimum specific flow rate calculated
providing for maximum installation efficiency.
