Several Thermodynamic Aspects Of Hydro Thermal System

Several Thermodynamic Aspects of Hydrothermal Systems Toni Alchofino (10205013) Physics Department, Faculty of Mathematics and Natural Sciences, Institute Technology Bandung E-mail: [email protected] Abstract There are some several thermodynamic aspects which are involved in geothermal. And some of them can be described mathematically such as natural geothermal gradients, hydrothermal convection system, fluid flow and mass transport. However, another aspects are even complicated to be described and need more specific advance study. In hydrothermal system where the availibility of fluid and heat source form the system, not only heat transfer but also fluid flow is tangled. This report provide how the aspects derived by considering of some approximations. Most of them derived assuming that the fluid of hydrothermal system is single phase., liquid.

1. Introduction Heats in the earth are naturally come from two major source. Internal source and external source. Where the external source is the sun’s heat and the internal source come from depth of the earth.

1.2 Geothermal In Shallow Subsurface (picture from www.geothermal.marin.org)

2. Natural Geothermal Gradients

1.1 Earth Interior (picture from www.geothermal.marin.org)

The fundamental heat source in the earth is radioactive decay beneath the earth. The heat propagates to surface through heat transfer mechanisms such as conduction, convection, radiation and advection. But dominantly by conduction and convection. Some of hot melted rock beneath the earth could arise to shallow subsurface of the earth through thin geologic layer in a regional. The availibility of this heat source, liquid, and the permeability, form a typical geothermal systems called hydrothermal systems. There, in this system heat flow and fluid flow would occur.

Naturally, the heat in the earth will increase gradually to depth. The ratio between the increasing heat (∆T) and the increasing of the depth (∆Z) is called thermal gradient ∇T . Assuming T is a linear function of ∆Z, expression of ∇T = (∆T)/ (∆Z) will depend on the magnitude of interval (∆Z). Applying differential calculus for limit of (∆Z) Æ 0, the ∂T expression can be written as ∇T = or ∂Z ∂T ∂T ∂T , , . in 3D ∇T = ∂x ∂y ∂z Based on Fourier’s law of heat conduction, the heat flux q (W/m2, watt per square meter) assosiated with temperature gredient ∇T is given by :

q = − K∇T …………………………...(2.1)

Where K is thermal conductivity and minus-sign occurs because heat flow from higher to lower temperature. In typical continental crust, temperatures will increase by ≈ 3 0C for every 100 m depth, so ∇T ≈ 0.030C/m. A typical value for rocks is K ≈ 2 W/m0C.

10 m/s2, the pressure increase in a column of water for 1 m increase in depth is approximately 104 kg/ms2. (1 Newton = 1Kg m/s2) so it can be written as 104 N/m2 or 10000 Pa per meter depth or 0.1 bar per meter depth. P vs Z 250

Chart Title

P( bar)

200

y = 33.333x - 666.67 R2 = 1

100 50

7000

0

6000

0

5000 Z(m)

150

5

10

15

20

25

Z(m)

4000 3000

3.1 Pressure Gradient In Static ρ value

2000 1000 0 0

50

100

150

200

250

T(0C)

2.1 Thermal Gradient in Typical Continental Crust

So, according to equation 2.1, the heat flux in a typical continental crust with a typical thermal conductivity is 0.06 W/m2.

Fluids density generally vary with depth as functions of temperatures, pressure, and salinity. For liquids (Pruess 2002), the temperature and pressure dependence of density can be approximated as :

ρ = ρ 0 [1 − α (T − T0 ) + β ( P − P0 )] …..(3.2) Α is the expansivity and β is the compressibility of the fluid. When fluid density varies with depth, because of generally increasing temperatures and pressure, Eq. 3.1 is no longer applicable. We should calculate pressure increase in every interval which spesific density for each interval. N

N

i =1

i =1

∑ ΔP = ∑ ρ i gΔz When we let Δz Æ0 we obtain the integral z

2.2 A Sample Of Temperature Measurement In Geothermal Field (picture from www.geothermal.marin.org)

3. Convective Systems of Hydrothermal Under static condition which no flow, the pressure of reservoir fluid with density ρ is given by weight of the fluid column per unit area and increases with depth z according to :

P( z ) = ∫ ρgdz ………………………..(3.2) 0

Warm water is less dense than cold water. According to this reality, we’ve got thermal bouyancy phenomenon which can be described below

Ph ( z ) = ρgz ………………………….(3.1) For water at ambient conditions of temperature and pressure we have ρ ≈ 1000 Kg/m3. Acceleration of gravity is g=

3.2 Thermal Bouyancy flow (Pruess, 2002)

Then we can get the “Darcy Velocity” as ……………………..(4.4) 5. Conclusions 3.3 Hydrothermal convetion in a fluid layer heated from below (Pruess, 2002)

4. Fluid Flow Fluids move in response to forces, the most important of which are pressure forces and force of gravity and also sometimes capillary pressures. The rate of fluid flow not depend on the magnitude of the pressure but the intensity of pressure change. According to Darcy’s experiment (1856), the fluid mass flow rate F per unit cross-sectional area of the medium is proportional to the pressure gradient.

F = −k

ρ ΔP ……………………. (4.1) μ ΔX

or in 3D it can be showed as below : ⎛ ΔP ⎞ ⎜ ⎟ ⎛ Fx ⎞ Δ X ⎜ ⎟ ρ ⎜ ΔP ⎟ ………..(4.2) ⎜ Fy ⎟ = − k ⎜ ⎟ Δ Y μ ⎜ Fz ⎟ ⎜ ⎟ Δ P ⎝ ⎠ ⎜ Δz − ρg ⎟⎠ ⎝

Naturally the temperature of the earth will increase proportionally to the increasing of the depth. The same as temperature, fluid pressure beneath the earth also increase proportionally to the increasing of the depth. In geothermal area where the magma may arise through the thinned or fractured crusts, the gradient of temperature and pressure coul be significantly high even in relatively shallow subsurface. Because of the difference pressure and density of the reservoir’s fluid, thermal bouyancy occurs in convective system of hydrothermal. Fluid moves esspecially in response to force. The fluid’s flow rate esspecially depend on the pressure intensity change. So if the pressure gradien is high, the fluid’s flow rate will high. But, the pressure gradient is not the only major factor. A fluid will flow wether there is a permeability. The higher permebality and porosity, fluid’s flow velocity will increase. 6. References

A minus-sign occur because fluid flows from higher pressure to lower pressure, k is absolute permeability and μ is viscosity.

Fowler.1990.The Solid Earth.Cambridge University Press.

Deviding the mass flux by the fluid density gives the vollumetric flux (amount of fluid volume crossing a unit cross sectional area per unit of time).

Pruess. Lectures on Mathematical Modelling of Fluid Flow and Heat Transfer In Geothermal Systems. Lawrance Barkeley National Laboratory University of California.

……….(4.3)

Zemansky, M.W., and Dittman,R.H.1986. Kalor dan Termodinamika,penterjemah Liong, T.H Bandung:Penerbit ITB.