The Basic Equation Of Fluid Statics

Consider a cylindrical region of arbitrary size and shape within a fluid, as shown in Fig. 4-1. We will apply a momentum balance to a ''slice'' of the

fluid that has a "z area'' Az and a thickness Az and is located a vertical distance z above some horizontal reference plane. The density of the fluid in the slice is p, and the force of gravity (g) acts in the —z direction. A momentum balance on a "closed" system (e.g., the slice) is equivalent to Newton's second law of motion, i.e.,

Because this is a vector equation, we apply it to the z vector components.

Fz is the sum of all of the forces acting on the system (the "slice") in the z direction, m is the mass of the system, and az is the acceleration in the z direction. Because the fluid is not moving, az = 0, and the momentum balance reduces to a force balance. The z forces acting on the system include the (—) pressure on the bottom (at z) times the (—) z area, the (—) pressure on the top (at z + Az) times the (+) z area, and the z component of gravity, i.e., the "weight" of the fluid (—pgA Az). The first force is positive, and the latter two are negative because they act in the —z direction. The momentum (force) balance thus becomes

If we divide through by Az Az, then take the limit as the slice shrinks to zero (Az ! 0), the result is dP , N

which is the basic equation of fluid statics. This equation states that at any point within a given fluid the pressure decreases as the elevation (z) increases, at a local rate that is equal to the product of the fluid density and the gravitational acceleration at that point. This equation is valid at all points within any given static fluid regardless of the nature of the fluid. We shall now show how the equation can be applied to various special situations.

A. Constant Density Fluids

If density (p) is constant, the fluid is referred to as "isochoric" (i.e., a given mass occupies a constant volume), although the somewhat more restrictive term "incompressible" is commonly used for this property (liquids are normally considered to be incompressible or isochoric fluids). If gravity (g) is also constant, the only variables in Eq. (4-5) are pressure and elevation, which can then be integrated between any two points (1 and 2) in a given fluid to give

This can also be written

where

This says that the sum of the local pressure (P) and static head (pgz), which we call the potential ($), is constant at all points within a given isochoric (incompressible) fluid. This is an important result for such fluids, and it can be applied directly to determine how the pressure varies with elevation in a static liquid, as illustrated by the following example.

Example 4-1: Manometer. The pressure difference between two points in a fluid (flowing or static) can be measured by using a manometer. The manometer contains an incompressible liquid (density pm) that is immiscible with the fluid flowing in the pipe (density pf). The legs of the manometer are connected to taps on the pipe where the pressure difference is desired (see Fig. 4-2). By applying Eq. (4-7) to any two points within either one of the fluids within the manometer, we see that

Figure 4-2 Manometer attached to pressure taps on a pipe carrying a flowing fluid.

When these three equations are added, P3 and P4 cancel out. The remaining terms can be collected to give

where O = P + pgz and AO = O2 — O1, Ap = pm — pf, Ah = z4 — z3. Equation (4-10) is the basic manometer equation and can be applied to a manometer in any orientation. Note that the manometer reading (Ah) is a direct measure of the potential difference (O2 — O1), which is identical to the pressure difference (P2 — P1) only if the pipe is horizontal (i.e., z2 = z1). It should be noted that these static fluid equations cannot be applied within the pipe, since the fluid in the pipe is not static.

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