9Supp%209%20Bernoulli

9Supp%209%20Bernoulli - 5 5.1 Principles of Ground Water...

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Unformatted text preview: 5 5.1 \ Principles of Ground- Water Flow just as above the earth, small drops form and these join others, till finally water descends in a body as rain, so too we must suppose that in the earth the water at first trickles together little by little and that the sources of rivers drip, as it were, out of the earth and then unite. Meteorologica, Aristotle (384—322 B.C.) INTRODUCTION Ground water possesses energy in mechanical, thermal, and chemical forms. Because the amounts of energy vary spatially, ground water is forced to move from one region to another in nature’s attempt to eliminate these energy differ- entials. The flow of ground water is thus controlled by the laws of physics and thermodynamics. To enable a separate examination of mechanical energy, we will make the simplifying assumption that the water is of nearly constant temperature. Thermal energy must be considered, however, in such applications as geothermal flow systems and burial of radioactive heat sources. There are three outside forces acting on the water contained in the ground. The most obvious of these is gravity, which pulls water downward. The second force is external pressure. Above the zone of saturation, atmospheric pressure is acting. The combination of atmospheric pressure and the weight of overlying water creates pressures in the zone of saturation. The third force is molecular attraction, which causes water to adhere to solid surfaces. It also creates surface tension in water when the water is exposed to air. The combina- tion of these two processes is responsible for the phenomenon of capillarity. When water in the ground is flowing through a porous medium, there are forces resisting the fluid movement. These consist of the shear stresses acting 5.2 MECHANICAL —————fi——— ENERGY physics. Of these, we will consider kinetic energy, gravitational potential energy, and energy of fluid pressures. A moving body or fluid tends to remain i n motion, according to Newto- nian physics. This is because it possesses ene rgy due to its motion—kinetic energy. This energy is equal to one-half the product 0 l f l f its mass and the square of l the magnitude of the velocity: ii I Ek = 1/2mv2 (5—1) t. where Ek is the kinetic energy (MLZ/TZ; slug-ftZ/s2 or kg-mz/sz) v is the velocity (L/T; ft/s or m/s) If m is in kilograms and v in meters per second, then Ek has the units of kg'mZ/s2 or newton-meters. The unit of energy is the joule, which is one newton- meter. The joule is also the unit of work. Imagine that a weightless container filled with water of mass m is moved vertically upward a distance, z, from some reference surface (a datum). Work is w done in moving the mass of water upward. This work is equal to g e( W = Fz = (mg)z (5—2) of i ha where i Wis work (MLz/Tz; slug-ftz/s2 or kg~m2/s2) i z is the elevation of the center of gravity of the fluid above the i reference elevation (L; ft or m) Ste m is the mass (M; slugs or kg) ani g is the acceleration of gravity (L/TZ; ft/s2 or m/sz) flu“ F is a force (ML/T2; slug-ft/s2 or kg-m/sz) g? The mass of water has now acquired energy equal to the work done in usef lifting the mass. This is a potential energy, due to the position of the fluid mass with respect to the datum. Eg is gravitational potential energy: resu, W = Eg = mgz (5—3) A fluid mass has another source of potential energy owmg to the pressure ‘ of the surrounding fluid acting upon it Pressure 18 the force per unit area acting on This a body f are J/ len th P = F/A (5—4) ten‘i MECHANICAL ENERGY 133 where P is the pressure (M/LTZ; slug/ft-s2 or (kg‘m/s2)/m2) A is the cross-sectional area perpendicular to the direction of the force (L2; ft2 or m2) The units of pressure are pascals (Pa), or N/mz. A N/m2 is equal to a N~m/m3, or J/m3. Pressure may thus be thought of as potential energy per unit volume of fluid. For a unit volume of fluid, the mass, m, is numerically equal to the density, p, since density is defined as mass per unit volume. The total energy of the unit volume of fluid is the sum of the three components—kinetic, gravita- tional, and fluid-pressure energy: EN = l/2pv2 + pgz + P (5—5) where EW is the total energy per unit volume. If Equation 5—5 is divided by p, the result is total energy per unit mass, E tm' 2 v P E,m=§+gz+3 (5-6) which is known as the Bernoulli equation. The derivation of the Bernoulli equation may be found in textbooks on fluid mechanics (Streeter 1962). For steady flow of a frictionless, incompressible fluid along a smooth line P —— + g2 + E = constant (5—7) Steady flow indicates that the conditions do not change with time. The density of an incompressible fluid would not change with changes in pressure. A frictionless fluid would not require energy to overcome resistance to flow. An ideal fluid would have both of these characteristics; real fluids have neither one. Real fluids are compressible and do suffer frictional flow losses; however, Equation 5—7 is useful for purposes of comparing the components of mechanical energy. If each term of Equation 5—7 is divided by g, the following expression results: 2 P v— + z + — = constant (5—8) 2g pg This equation expresses all terms in units of energy per unit weight. These are J/N, or m. Thus, Equation 5—8 has the advantage of having all units in length dimensions (L). The first term of v2/2g is (m/s)2/(m/s2), or m; the second term, z, is already in m; and the third term, P/pg, is Pa/(kg/m3)(m/s2), or 134 5.3 PRINCIPLES OF GROUND-WATER FLOW FIGURE 5.1 Piezometer measuring fluid pressure and the elevation of water. \ (N/m2)/(kg/m3)(m/sz), which reduces to m. The sum of these three factors is the total mechanical energy per unit weight, known as the hydraulic head, h. This is usually measured in the field or laboratory in units of length. HYDRAULIC HEAD A piezometer* is used to measure the total energy of the fluid flowing through a pipe packed with sand, as shown in Figure 5 .1. The piezometer is open at the top and bottom, and water rises in it in direct proportion to the total fluid energy at the point at which the bottom of the piezometer is open in the sand. At point A, which is at an elevation, z, above a datum, there is a fluid pressure, P. The fluid is flowing at a velocity, v. The total energy per unit mass can be found from Equation 5—6. i ...
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9Supp%209%20Bernoulli - 5 5.1 Principles of Ground Water...

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