Page 64 - Geothermal Energy Renewable Energy and The Environment

Thermodynamics and Geothermal Systems 47

and the limitations that must be recognized when considering in more detail reservoir assessment
(Chapter 7) and geothermal power production (Chapter 9).

synopsIs
Thermodynamic principles have long established that heat and work are directly related by simple
functions. These functional relationships establish that: all substances possess some quantity of heat;
the availability of that heat to do work depends upon the temperature difference between the substance
and its surroundings; that the maximum amount of heat that can be converted to work is independent
of the pathway; and the only thermodynamically significant determinant of the amount of work that
can be done is the temperature difference between the initial and final states of the system. The tem-
perature difference also establishes the thermodynamic efficiency of the process being employed to
extract the heat for work. The Gibbs function defines the available heat by considering the attributes
and state of the system. The parameters that determine the Gibbs energy are enthalpy, entropy, heat
capacity, and the temperature and pressure of the system. From these parameters and through knowing
the initial and final pressure and temperature states of a system, it is possible to completely character-
ize the thermodynamic properties of a material. Since all systems are stable only in their lowest energy
state, the Gibbs function allows determination of the stable configuration of a system by providing the
means to compare the Gibbs energy of the various possible combinations that a chemical system can
take. Water, for example, can exist in solid, liquid, or vapor forms; the Gibbs function allows compu-
tation of the Gibbs energy at any pressure and temperature for each of the possible phase states, thus
identifying the lowest possible Gibbs energy at any set of pressure and temperature conditions.

Problems
3.1 In a Carnot cycle involving an ideal gas that has a molar volume of 40,000 cm /mole,
3
assume that the initial pressure of the gas phase is 1 bar and the volume is 1 cubic
meter. What is the initial temperature of the Carnot cycle?
3.2 In order to accomplish an isothermal expansion from the initial condition in problem 1
to a pressure of 0.25 bars, and a volume 3 m , how much heat must be added (assume
3
C p = 1.02 kJ/kg-K and the density of the gas is constant at 1.2 kg/m )?
3
3.3 What volume of water would be required to cool 1 m of a rock composed of 100%
3
potassium feldspar from 300°C to 295°C, if the water is allowed to increase in tem-
o
perature by only 1 (assume the heat capacities for water and potassium feldspar are
their respective values at 25°C and 300°C)? The gram formula weight for potassium
feldspar is 278.337 gm/mole and is 18.0 gm/mole for water. The molar volume for
potassium feldspar is 108.87 cc/mole and is 18.0 cc/mole for water.
3.4 A borehole is drilled into a geothermal reservoir and encounters 300°C water at a depth
of 3000 meters. As the fluid rises to the surface, at what depth would it flash to steam if a
lithostatic pressure gradient (see sidebar) were maintained throughout the column? What
would be the depth of flashing if a hydrostatic pressure gradient existed in the well?
3.5 If an ideal gas isothermally expands from a volume of 1 m to 2 m against a pressure
3
3
of 1 MPa, how much work is performed and what is the efficiency?
3.6 If 10 kg of liquid water completely flashes to steam at a pressure of 10 bars, what is the
total enthalpy available for work in the steam?
3.7 How much of the enthalpy in Problem 3.6 would be used for work if the end point for
the cycle is 50°C?

reFerences

Bowers, T. S. 1995. “Pressure-Volume-Temperature Properties of H 2 O-CO 2 Fluids.” In Rock Physics and Phase
Relations, ed. T. J. Ahrens, 45–72. Washington, DC: American Geophysical Union.
Helgeson, H. C., J. M. Delany, H. W. Nesbitt, and D. K. Bird. 1978. “Summary and Critique of the Thermodynamic
Properties of Rock-Forming Minerals.” American Journal of Science 278-A:1–229.