Energy balance, mechanical total

Economic lot size, 350 Effective interest, 218-222, 224, 241 Efficiency, packing, 702-706 plate, 661-667 pump, 517-518, 520 Ejectors, cost of, 528 Electrical installation, cost of, 174, 807 Electricity, cost of, 815 Electromotive series of metals, 433 Emissivity of surfaces, 582-585 Energy balance, mechanical, 479-480 for reactor design, 715-716 total, 479-480... 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

Differential momentum, mechanical-energy, or total-energy balances can be written for each phase in a two-phase flowing mixture for certain flow patterns, e.g., annular, in which each phase is continuous. For flow patterns where this is not the case, e.g., plug flow, the equivalent expressions can usually be written with sufficient accuracy as macroscopic balances. These equations can be formulated in a perfectly general way, or with various limitations imposed on them. Most investigations of two-phase flow are carried out with definite limits on the system, and therefore the balances will be given for the commonest conditions encountered experimentally. 

A great deal of use has been made of the overall momentum, mechanical-energy, and total-energy balances in various forms. Some of these applications are mentioned in the succeeding sections. 

Friction is introduced into the energy balance by noting that it is a mechanical process, dWf, whose effect is the same as that of an equivalent amount of heat transfer dQf. Moreover, the total effective heat transfer results in a change in entropy of the flowing liquid given by... 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, total energy, and mechanical energy may be found in Whitaker (ibid.), Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

Equation 2.9-10 is the total differential energy balance, and it contains both thermal and mechanical energies. It is useful to separate the two. We can do this by taking the dot product of the equation of motion with the velocity vector v to get the mechanical energy balance equation ... 

For any given operating conditions involving the flow of a noncompressible fluid through a pipe of constant diameter, the total mechanical-energy balance can be reduced to the following form ... 

The various forms of energy can be related by the total energy balance or the total mechanical-energy balance. On the basis of 1 Ibmt °f fluid flowing under steady conditions, the total energy balance may be written in differential form as... 

The total mechanical-energy balance in differential form is g V dV,... 

Evaluation of the term / vdp in Eq. (4) may be difficult if a compressible fluid is flowing through the system, because the exact path of the compression or expansion is often unknown. For noncompressible fluids, however, the specific volume v remains essentially constant and the integral term reduces simply to v(p2 - Pi). Consequently, the total mechanical-energy balance is especially useful and easy to apply when the flowing fluid can be considered as noncompressible. 

LIQUIDS. For noncompressible fluids, the integrated form of the total mechanical-energy balance reduces to... 

Example 1 Application of the total mechanical-energy balance to noncom-pressible-flow systems. Water at 61°F is pumped from a large reservoir into the top of an overhead tank using standard 2-in.-diameter steel pipe (ID = 2.067 in.). The reservoir and the overhead tank are open to the atmosphere, and the difference in vertical elevation between the water surface in the reservoir and the discharge point at the top of the overhead tank is 70 ft. The length of the pipeline... 

Total mechanical-energy balance between point 1 (surface of water in reservoir) and point 2 (just outside of pipe at discharge point) ... 

From the total mechanical-energy balance, W0 = theoretical mechanical energy necessary from pump = 70 +48.4 = 118.4 ft lbf/lbm. 

GASES. Because of the difficulty that may be encountered in evaluating the exact integral of v dp and dF for compressible fluids, use of the total mechanical-energy balance is not recommended for compressible fluids when large pressure drops are involved. Instead, the total energy balance should be used if the necessary data are available. 

Because the amount of heat exchanged between the surroundings and the system is unknown, the total energy balance cannot be used to solve this problem. However, an approximate result can be obtained from the total mechanical-energy balance. 

Designate point 1 as the entrance to the pump, point 2 as the exit from the pump, and point 3 as the downstream end of the pipe. Under these conditions, the total mechanical-energy balance for the system between points 2 and 3 may be written as follows ... 

M The mechanical energy added by the pump can be determined by making a total mechanical-energy balance between points 1 and 3 ... 

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