9.110. At this point it is useful to review the concept of the macroscopic, or overall energy balance based on the first law of thermodynamics. For a control volume, we write

(Rate at which quantity enters volume element)

= (Rate at which quantity leaves volume element)

+ (Rate of quantity accumulation in volume element).

For example, a macroscopic energy balance on a volume element can be classically expressed as

A# + AZ + Aw2 = Q — Ws,

where H, Z, and и represent enthalpy, potential energy, and fluid velocity, respectively. Hence, и2 is the kinetic energy, Q is the heat added to the volume element, and Ws is the mechanical work done by the fluid, such as by means of a turbine. It is sometimes called shaft work.

9.111.

The pressure drop, or loss, in a flow system is evaluated as part of a mechanical energy balance over the system. This is often known as the extended Bernoulli equation or the mechanical energy equation. It can be derived from the equation in §9.110 as well as from a force balance on a volume element. In its simplified integral form for a unit mass in an incompressible fluid having a density p moving between two points, 1 and 2, at different heights Z with respect to a datum level, and with no shaft work or friction, the equation becomes

where p is the static pressure, и the velocity in the kinetic energy term, and g the acceleration of gravity. A term Ws may be added, expressing the work required per unit mass if a pump is included between the two points. Also, for a practical system, the term Ap^/p should be added de­scribing the friction loss in energy units per unit mass of fluid flowing. The following equation results, which can also serve as the basis for determining pumping power requirements in the system:

Therefore, the total pressure drop is the sum of terms arising from the difference in level, acceleration of the fluid, friction, and pump work. Since the kinetic energy term is usually negligible for a liquid, our review will concentrate on frictional effects.

9.112. The pressure drop Apf accompanying isothermal laminar flow at a mean velocity и is a cyclindrical pipe of diameter D and length L is the familiar Poiseuille equation

. 32 ixmL

which may be written as

where |x is the viscosity.[8] The subscript/is used in pf to indicate that the pressure loss described is due to fluid friction. The factor i! Dup on the right of equation (9.39) is the reciprocal of the Reynolds number, and so

This gives the pressure drop due to fluid friction for laminar flow in a straight pipe of circular cross section. It should be noted that replacement of D by the equivalent diameter De, as defined in §9.76, is not valid for laminar flow in channels which do not have circular cross sections, although it is a good approximation for turbulent flow (§9.115).