Energy Balance, Bernoulli Equation

A typical piping application starts with a specified flow rate for a given fluid. The piping system is then designed with the necessary valves, fittings, etc. and should be sized for the most economical pipe size, as discussed in Chapter 7. Application of the energy balance (Bernoulli) equation to the entire system, from the upstream end (point 1) to the downstream end (point 2) determines the overall net driving force (DF) in the system required to overcome the frictional resistance ... 

If there is no relative motion within the fluid, the differential form of the energy balance (Bernoulli equation) reduces to... 

If the friction loss is negligible, the energy balance (Bernoulli s equation) becomes... 

When the energy balance on the fluid in a stream tube is written in the following form, it is known as Bernoulli s equation ... 

Consider a section of uniform cylindrical pipe of length L and radius R, inclined upward at an angle 0 to the horizontal, as shown in Fig. 6-2. The steady-state energy balance (or Bernoulli equation) applied to an incompressible fluid flowing in a uniform pipe can be written... 

Equation 1.13 is simply an energy balance written for convenience in terms of length, ie heads. The various forms of the energy balance, equations 1.10 to 1.13, are often called Bernoulli s equation but some people reserve this name for the case where the right hand side is zero, ie when there is no friction and no pump, and call the forms of the equation including the work terms the extended or engineering Bernoulli equation. 

In the earlier days of the petroleum age, many pipe experiments were conducted. In the quest for the magic formula, one was found to be the closest to utopia even to this day, called the Darcy formula. The Darcy formula is derived manually from the Bernoulli principle, which simply describes the energy balance between two points of a fluid flowing in a pipe. This energy equation is also applicable to a static condition of no flow between the two points. The classic Bernoulli energy equation [1] is ... 

The Z factor is simply the static head due to elevation height above a referenced elevation point. The 144 PJp factor is the pressure factor noted as the pressure head. Consider again Eq. (5.52) from Chap. 5. This equation is the basic form of the Bernoulli energy balance equation. 

The Macroscopic Energy Balance and the Bernoulli and Thermal Energy Equations, 54... 

THE MACROSCOPIC ENERGY BALANCE AND THE BERNOULLI AND THERMAL ENERGY EQUATIONS... 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

The most important relationship in designing flow systems is the macroscopic mechanical-energy balance, or Bernoulli s equation. Not only is it required for calculating the pump work, but it is also used to derive formulas for sizing valves and flow meters. Bird, et al. [6] derived this equation by integrating the microscopic mechanical-energy balance over the volume of the system. The balance is given by... 

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

Given fluid conditions (pressure, flow rate, velocity, elevation) at the inlet and outlet of an open system and values of friction loss and shaft work within the system, substitute known quantities into the mechanical energy balance (or the Bernoulli equation if friction loss and shaft work can be neglected) and solve the equation for whichever variable is unknown. 

Another equation that resembles the Bernoulli equation very much is derived from the total energy balance [168]. For one component fluids the total energy equation (1.96) can be written ... 

This relation is seemingly identical to the classical Bernoulli equation (1.238) along a given streamline. It is emphasized that the Bernoulli equation (1.251), as derived from the energy balance (1.96), is restricted to steady-, incompressible- and isentropic flows. 

In addition, real process units are neither completely inviscid, irrotational or isentropic, thus several extended forms of the Bernoulli equation are used in practice constituting various forms of energy balances containing loss and/or work terms. Two extremely important variants of the original relation are outlined in the following paragraphs. 

MECHANICAL-ENERGY EQUATION. The Bernoulli equation is a special form of a mechanical-energy balance, as shown by the fact that all the terms in Eq. 

The power supplied to the vessel contents, which is responsible for the agitation and creation of large interfacial area, is derived from the gas flow. Bernoulli s equation, a mechanical energy balance written for the gas between location o (just above the sparger orifices) and location 5 (at the liquid surface), is... 

The energy balance for steady, incompressible flow, which is called Bernoulli s equation, is probably the most useful single equation in fluid mechanics. 

In this chapter we concentrated on problems most easily solved by the energy balance (of which Bernoulli s equation is a restricted form). The... 

Bernoulli s equation is the energy balance equation for steady flow of constant-density fluids. 

In Chap. 51 we derived Bernoulli s equation from the energy balance equation. Since the energy balance has no onc-dimensional restriction on it, the same approach must apply to two- and three-dimensional flows. However, in our derivation bf Bernoulli s equation, we restricted our attention to systems with only one flow in and out. How can we apply this idea to a two-dimensional flow field in which there is a continuously varying velocity over some region of space In Fig. 10.15 such a region is shown with no sources, sinks, or solid bodies, but with streamlines. 

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