Shaft work

This work can be positive or negative, depending on whether the body is being pulled up or lowered by the string. The quantity (m dv / dr) dz is an infinitesimal change of the body s kinetic energy so that the integral /(m dv/ dr) dz is equal to A k-The finite quantity of work in a process that starts and ends in equilibrium states, so that A k is zero, is therefore 

The system is the body only. In this case, is equal to (Fbuoy + Fmc -I- Fstr) which 

For a process that begins and ends in equilibrium states, AFk is zero and the finite work isw = mgAz, unaffected by the velocity v during the process. The expressions for infinitesimal and finite work in the reversible limit are 

When we compare Eqs. 3.6.3 and 3.6.5, we see that the work when the system is the body is greater by the quantity (Fbuoy + Ffric) dz than the work when the system is the combination of body and fluid, just as in the case of the freely-falling body. The difference in the quantity of work is due to the different choices of the system boundary where contact forces are exerted by the surroundings. 

Shaft work refers to energy transferred across the boundary by a rotating shaft. 

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The quantity 811 includes two kinds of work shaft work, SIF), shown in Figure 4, and work done by pressures and moving through volumes... 

Cog enera.tion in a. Steam System. The value of energy in a process stream can always be estimated from the theoretical work potential, ie, the deterrnination of how much power can be obtained by miming an ideal cycle between the actual temperature and the rejection temperature. However, in a steam system a more tangible approach is possible, because steam at high pressure can be let down through a turbine for power. The shaft work developed by the turbine is sometimes referred to as by-product power, and the process is referred to as cogeneration. 

The first integral on the right-hand side is the rate of work done on the fluid in the control volume by forces at the boundaiy. It includes both work done by moving solid boundaries and work done at flow entrances and exits. The work done by moving solid boundaries also includes that by such surfaces as pump impellers this work is called shaft work its rate is Ws-... 

A useful simphfication of the total energy equation applies to a particular set of assumptions. These are a control volume with fixed solid boundaries, except for those producing shaft work, steady state conditions, and mass flow at a rate m through a single planar entrance and a single planar exit (Fig. 6-4), to whi(m the velocity vectors are perpendicular. As with Eq. (6-11), it is assumed that the stress vector tu is normal to the entrance and exit surfaces and may be approximated by the pressure p. The equivalent pressure, p + pgz, is assumed to be uniform across the entrance and exit. The average velocity at the entrance and exit surfaces is denoted by V. Subscripts 1 and 2 denote the entrance and exit, respectively. 

Here, h is the enthalpy per unit mass, h = u + p/. The shaft work per unit of mass flowing through the control volume is 6W5 = W, /m. Similarly, is the heat input rate per unit of mass. The fac tor Ot is the ratio of the cross-sectional area average of the cube of the velocity to the cube of the average velocity. For a uniform velocity profile, Ot = 1. In turbulent flow, Ot is usually assumed to equal unity in turbulent pipe flow, it is typically about 1.07. For laminar flow in a circiilar pipe with a parabohc velocity profile, Ot = 2. 

The Bernoulli equation can be written for incompressible, inviscid flow along a streamhne, where no shaft work is done. 

With control surfaces at 1 and 2 as shown in the figure, Eq. (6-17) in the absence of losses and shaft work gives... 

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. 

Total pressure will only change in a fluid if shaft work or work of extraneous forces are introduced. Therefore, total pressure would increase in the impeller of a compressor or pump it would remain constant in the difmser. Similarly, total pressure would decrease in the turbine impeller but would remain constant in the nozzles. 

The turboexpander in combination with a compressor and a heat exchanger functions as a heat pump and is analyzed as follows In Fig. 29-44 consider the compressor and aftercooler as an isothermal compressor operating at To with an efficiency and assume the working fluid to be a perfect gas. Further, consider the removal of a quantity of heat by the tumoexpander at an average low temperature Ti-This requires that it dehver shaft work equal to Q. Now, make the reasonable assumption that one-tenth of the temperature drop in the expander is used for the temperature difference in the heat exchanger. If the expander efficiency is and this efficiency is mul-... 

High temperature due to excessive agitator shaft work resulting in high reaction rates. 

The high-veioeity gas impinges on the biade where a iarge portion of the kinetie energy of the moving gas stream is eonverted into turbine shaft work. 

The Alexandrian seientist Hero (eirea 120 B.C.) would hardly reeognize the modern gas turbine of today as the outgrowth of his aeolipile. His deviee produeed no shaft work—it only whirled. In the eenturies that followed, the prineiple of the aeolipile surfaeed in the windmill (A.D. 900-1100) and again in the powered roasting spit (1600s). The first sueeessful gas turbine is probably less than a eentury old. 

If the work term W( is expanded to breakdown shaft work done to or from the system and the work done by the system, then... 

W = shaft work in or out of the system p fluid pressure in the system V = specific volume of the fluid in the system... 

The iiitemil energy and entlialpy in Eqs. (4.5.2) uid (4.5.3), as well as in tlie other equations in tliis discussion, may be on a mass or a mole basis, or tliey may represent the total internal energy and entlialpy of tlie entire system. Most industrial facilities operate in a steady-state flow mode. If no significant mechanical or shaft work is added or withdrawn from tlie system, Eq. (4.5.3) reduces to... 

AZ = height of driving leg for thermal circulation, ft Wx = mass rate, Ib /sec, total flow, or W Wj = shaft work done by system, ft liquid A = change from one condition to another AL[ = change in equivalent length of pipe, ft, inlet piping system... 

Figure 12-37B and 12-37C show the events for an ideal constant-T compressor and engine with an atomospheric temperature, T . Wi , the gray area, raises air pressure from 1 to 2 while shaded area, flows through the compressor-cylinder walls to atmosphere. This system does not need an aftercooler. Air expands in the engine from 3 to 4 while it absorbs 0, from the atmosphere, hatched area, through the engine cylinder walls and produces shaft work hatched area. For reversible processes ... 

This contrasts with the engine cycles studied for these, net area measured shaft work output, but for compressed-air systems, net area measures work lost. Remember, completely available energy, shaft work, runs compressed-air systems higher-temperature heat runs engine cycles. 

Work interchange between a system and its surroundings can take on any of a variety of forms including mechanical shaft work, electrical work, magnetic work, surface tension, etc. For many applications, the only work involved is that of compression or expansion against the surroundings, in which case the work term in Equation 2-102 becomes... 

An inventor claims to have devised a CO. compressor that requires no shaft work. The device operates at steady state by transferring heat from a feed stream of 2 lb,/s of CO. at 150 psia and 100°F. The CO is compressed to a final pressure of 500 psia and a temperature of 40°F. Kinetic and potential energy effects are negligible. A cold source at -140°F drives the device at a heat transfer rate of 60 Btu/sec. Check the validity of the inventor s claim. 

It should be noted that the shaft work is related to the total work W by the relation ... 

The work done by the pump is found by setting up an energy balance equation. If W, is the shaft work done by unit mass of fluid on the surroundings, then —Ws is the shaft work done on the fluid by the pump. 

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