Energy equation flow work

The enthalpy term h includes both internal energy and flow work terms. Since dw=0 between compressor stations, the equation becomes ... 

From the steady-flow energy equation, the work output in an actual (irreversible) flow through a control volume CV, between states X and Y in the presence of an environment at To (Fig. 2.2), is... 

In Equations (2.21) and (2.22), the left-hand side represents accumulation of energy within the system. There is no flow work associated with this term hence, the appropriate property is U. On the other hand, the first two terms on the right-hand side account for energy flowing in and out of the system, respectively. These terms must account for both the internal energy and flow work of the flowing streams. In this case, h is appropriate. It is worthwhile for you to take a moment and reconcile the use of U and h above. It will save you many mistakes down the road ... 

In practice, the loss term AF is usually not deterrnined by detailed examination of the flow field. Instead, the momentum and mass balances are employed to determine the pressure and velocity changes these are substituted into the mechanical energy equation and AFis deterrnined by difference. Eor the sudden expansion of a turbulent fluid depicted in Eigure 21b, which deflvers no work to the surroundings, appHcation of equations 49, 60, and 68 yields... 

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. 

The flow in an axial-flow eompressor is defined by the eontinuity, momentum, and energy equations. A eomplete solution to these equations is not possible beeause of the eomplexity of the flow in an axial-flow eompressor. Considerable work has been done on the effeets of radial flow in an axial-flow eompressor. The first simplifieation used eonsiders the flow axisym-metrie. This simplifieation implies that the flow at eaeh radial and axial station within the blade row ean be represented by an average eireumferen-tial eondition. Another simplifieation eonsiders the radial eomponent of the veloeity as mueh smaller than the axial eomponent veloeity, so it ean be negleeted. 

In defining the thermal efficiency of the closed gas turbine cycle, such as the one shown in Fig. 1.2, we employed the first law of thermodynamics (in the form of the steady-flow energy equation round the cycle), which states that the heat supplied is equal to the work output plus the heat rejected, i.e. 

For a real (irreversible) flow process through the control volume CV between fluid states X and Y (Fig. 2.4), with the same heat rejected at temperature T [Q x = [0rev]x)> the work output is [WcvJx. Heat [Qq x Iso be transferred from CV directly to the environment at Tq. From the steady-flow energy equation,... 

The restriction of applying equation (5.47) to an isolated system seems to seriously limit the usefulness of this equation since we seldom work with isolated systems. But this is not so the pre-eminent example of an isolated system is the universe, since neither mass nor energy can flow in or out of the universe. Thus, the isolated system shown in Figure 5,6 can be made the universe, with A the system of interest, and B the surroundings. When we designate the combined system as the universe, we can drop the subscript A in... 

In order to maintain isothermal flow it is necessary for heat to be transferred across the pipe wall. From equation 6.7, for flow in a section with no shaft work and negligible change in elevation, the energy equation takes the form... 

For steady flow of an ideal gas between points 1 and 2, distance L apart, in a pipe of constant cross-sectional area in which no shaft work is done, the energy equation is given by equation 6.19. For the general case of polytropic flow, from equation 6.27, the equation of state can be written as... 

For adiabatic flow with negligible change of elevation and no shaft work, the energy equation reduces to... 

Example 2.7. To show what form the energy equation takes for a two-phase system, consider the CSTR process shown in Fig. 2.6. Both a liquid product stream f and a vapor product stream F (volumetric flow) are withdrawn from the vessel. The pressure in the reactor is P. Vapor and liquid volumes are and V. The density and temperature of the vapor phase are and L. The mole fraction of A in the vapor is y. If the phases are in thermal equilibrium, the vapor and liquid temperatures are equal (T = T ). If the phases are in phase equilihrium, the liquid and vapor compositions are related by Raoult s law, a relative volatility relationship or some other vapor-liquid equilibrium relationship (see Sec. 2.2.6). The enthalpy of the vapor phase H (Btu/lb or cal/g) is a function of composition y, temperature T , and pressure P. Neglecting kinetic-energy and potential-energy terms and the work term,... 

By subtracting the mechanical-energy contributions from the total energy equation, a thermal energy equation can be derived. It is this equation that proves to be most useful in the solution of chemically reacting flow problems. By a vector-tensor identity for symmetric tensors, the work-rate term in the previous sections can be expanded as... 

What is the relevance of pV-V here. Discuss the trade-offs in introducing enthalpy as the dependent variable in the energy equation. Discuss the notion of flow work and how it is involved in going from the internal energy to the enthalpy formulation. 

As illustrated in Fig. 4.18, a uniform pressure gradient causes flow between a central rod and an outer guide. Assume the purely axial flow of an incompressible fluid, v = w = 0. The axial velocity varies as a function of r alone, u r). Assume that dp/dz is a constant, and that dp/dr = 1 /r(dp/dO) = 0. For this one-dimensional parallel-flow situation, develop an expression for the work term in the total energy equation that is, for this special case, expand... 

Assume that the water contains a radioactive substance that causes internal volumetric heat generation q (W/m3). With the objective of deriving a total energy equation for the annular flow, state the first law. Be careful with the signs and the definitions of positive heat transfer and work. 

Consideration of Eqs. (29) and (32) shows that the mechanical energy equation involves only the recoverable or reversible work. In order to calculate this term on the average, however, it is necessary to compute the total work done W and subtract from it the part lost due to friction or the irreversible work F. If Eq. (60) is applied to steady flow in a pipe and divided by the mass flow rate, the following per unit mass form is obtained,... 

The Steady Flow Energy Equation relates the changes in potential and kinetic energy, and enthalpy of streams flowing in and out of a control volume, to the rate of heat transfer to, and the rate of shaft work from, the control volume. Here the generation term is zero. If a chemical reaction were occurring within the control volume, the heat of reaction would need to be accounted for in a generation term. Note that if we are... 

The prediction of convective heat transfer rates will, however, always involve the solution of the energy equation. Therefore, because of its fundamental importance in the present work, a discussion of the way in which the energy equation is derived will be given here [2],[3],[5],[7]. For this purpose, attention will be restricted to two-dimensional, incompressible flow. 

It seems reasonable to assume that similar temperature profiles will also exist when viscous dissipation is important. Attention will first be given to the adiabatic wall case. If the wall is adiabatic and viscous dissipation is neglected, then the solution to the energy equation will be T = Ti everywhere in die flow. However, when viscous dissipation effects are important, the work done by the viscous forces leads to a rise in fluid temperature in the fluid. This temperature will be related to the kinetic energy of the fluid in the freestream flow, i.e., will be related to u /2cp. For this reason, the similarity profiles in the adiabatic wall case when viscous dissipation is important are assumed to have the form ... 

In the absence of any work interactions (such as electric resistance heating), the conservation of energy equation for the steady flow of a fluid in a tube can be expressed as (Fig. 8-10)... 

The term Wj, often referred to as the sha/t work, could be produced from such things as a stirrer in a CSTR or a turbine in a PFR. In most instances, the flow work term is combined with those terms in the energy balance that represent the energy exchange by mass flow across the system boundaries. Substituting Equation (8-4) into (8-3) and grouping terms, we have... 

The source term S in the energy conservation equation in general includes contributions from flow work (Ujdp/dxj), viscous dissipation (z dujdx ), and any external heat source. For simplicity, only conduction and radiation are included in further discussions. 

Applying the steady energy equation to low speed liquid flow in pipes with no shaft work and negligible viscous work, 3delds ... 

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