Mechanical Energy Balances

The mechanical energy balance is not a fundamental principle rather, it is a corollary (Bird 1957 Bird. Stewart, and Lightfoot 2002) of the equation of motion. For constant fluid density, the macroscopic mechanical energy balance takes the form 

The term Ey is the rate of viscous dissipation of mechanical energy its estimation is described in Bird, Stewart, and Lightfoot (2002). is a reference pressure, in units of M/Lt the resulting SI unit is the Pascal. 

Equations (3.1-4) and (3.1-7) are particularly important in reactor applications. In the following examples, we show how they simplify for some common reactor systems. 

Consider a well-stirred batch reactor of constant fluid volume V in which the reactions occurring are homogeneous. As the system for our macroscopic balances, we choose the fluid in the reactor volume V then the inflows Wn and outflows Wi2 vanish and no mass-transfer surface 0 is required. The species inventories are expressed in terms of fluid concentrations as 

It is common practice to define extents of production of the various species per unit volume of a batch reactor as 

For isothermal problems of fluid dynamics, it is better to use a modified version of the energy balance called the mechanical energy energy balance. We will now develop this isothermal version of the energy balance. 

The left-hand side of equation (1.6.1) describes a change of state. The right-hand side of this equation describes the irreversible processes which contribute to the change of state. We will now define a hypothetical reversible process which can cause the same change of state as 

The term Wrev is composed of the actual rate of shaft work, Ws-, and the irreversible losses, E  

From thermodynamics Qrev is given by (Smith and Van Ness, 1959) 

The change in T5 with time is the rate of change of T5 of the contents of the macroscopic system plus the net rate of TS leaving due to the flow of material across the boundaries of the system therefore 

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Example 3 Venturi Flowmeter An incompressible fluid flows through the venturi flowmeter in Fig. 6-7. An equation is needed to relate the flow rate Q to the pressure drop measured by the manometer. This problem can he solved using the mechanical energy balance. In a well-made venturi, viscous losses are neghgihle, the pressure drop is entirely the result of acceleration into the throat, and the flow rate predicted neglecting losses is quite accurate. The inlet area is A and the throat area is a. 

Mechanical Energy Balance The mechanical energy balance, Eq. (6-16), for fully developed incompressible flow in a straight circular pipe of constant diameter D reduces to... 

The viscous or frictional loss term in the mechanical energy balance for most cases is obtained experimentally. For many common fittings found in piping systems, such as expansions, contrac tions, elbows and valves, data are available to estimate the losses. Substitution into the energy balance then allows calculation of pressure drop. A common error is to assume that pressure drop and frictional losses are equivalent. Equation (6-16) shows that in addition to fric tional losses, other factors such as shaft work and velocity or elevation change influence pressure drop. 

Equation (6-95) is valid for incompressible flow. For compressible flows, see Benedict, Wyler, Dudek, and Gleed (J. E/ig. Power, 98, 327-334 [1976]). For an infinite expansion, A1/A2 = 0, Eq. (6-95) shows that the exit loss from a pipe is 1 velocity head. This result is easily deduced from the mechanic energy balance Eq. (6-90), noting that Pi =pg. This exit loss is due to the dissipation of the discharged jet there is no pressure drop at the exit. 

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,... 

A mechanical energy balance describes the various energy forms associated with flowing fluids ... 

The modifications are substituted into the mechanical energy balance (Equation 4-1) to determine u, the average discharge velocity from the leak ... 

Because there is no pressure change and no pump or shaft work, the mechanical energy balance (Equation 2-28) reduces to... 

The mechanical energy balance (Equation 4-1) also applies to adiabatic flows. For this case it is more conveniently written in the form... 

Isothermal flow is represented by the mechanical energy balance in the form shown in Equation 4-54. The following assumptions are valid for this case ... 

This result is shown by starting with the mechanical energy balance and rearranging it into the following form ... 

For liquids stored at their saturation vapor pressure, P = Ps, and Equation 4-91 is no longer valid. A much more detailed approach is required. Consider a fluid that is initially quiescent and is accelerated through the leak. Assume that kinetic energy is dominant and that potential energy effects are negligible. Then, from a mechanical energy balance (Equation 4-1), and realizing that the specific volume (with units of volume/mass) v = 1/p, we can write... 

Use a mechanical energy balance to show that the pump work required to pump a liquid... 

Flow through spring-type reliefs is approximated as flow through an orifice. An equation representing this flow is derived from the mechanical energy balance (Equation 4-1). The result is similiar to Equation 4-6, except that the pressure is represented by a pressure difference across the spring relief ... 

Mechanical energy balance, assuming no losses in fittings, no change in elevation, and so on. 

For the same case, a mechanical energy balance gives the equation AoVo po dz + AaVe dP = tloClgFw dz - dEo (6)... 

Similarly, if Eqs. (6) and (8), are added, using the fact that F< t = V[a, an over-all mechanical-energy balance results,... 

This equation can be derived by supposing that the wall is in contact with liquid only, and that shearing forces are equal at the gas-liquid interface. Experimental pressure drops were predicted to +50% to —30% on the average, although larger individual variations existed. An attempt was made by Marchaterre (Ml) to refine this method by expressing the mechanical-energy balance in terms of... 

If it is assumed that most of the liquid flows in the form of slugs, then only one of the mechanical energy balances needs to be considered, and the cross-sectional area for slug flow becomes approximately the same as the tube area. The resulting expression is. 

When elevation head and work transfer are neglected, the mechanical energy balance equation (6.13) with the friction term of Eq. (6.18) become... 

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