Energy incompressible flow

Here 4 = F,Jfn is the energy dissipation per unit mass. This equation has been called the engineering Bernoulli equation. For an incompressible flow, Eq. (6-15) becomes... 

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... 

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. 

As a simplification, the term in Eq. (10) that accounts for the kinetic energy of the gas jets emerging from the gas distributor is based on the expression ( 9goVl/2, which is valid for incompressible flow. Experimental investigations show [27], that for relatively low gas velocities it is possible to represent the empirically determined loss coefficients q as accurately with this simplification as by the use of expressions for compressible flow. 

Equation 6.38 is the basic form of the energy equation to be used for isothermal conditions, however it is instructive to write the equation in a slightly different form that allows easy comparison with incompressible flow. 

Frictional dissipation of mechanical energy can result in significant heating of fluids, particularly for very viscous liquids in small channels. Under adiabatic conditions, the bulk liquid temperature rise is given by AT=AP/CV p for incompressible flow through a channel of constant cross-sectional area. For flow of polymers, this amounts to about 4°C per 10 MPa pressure drop, while for hydrocarbon liquids it is about... 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

In the next section, incompressible flow with constant properties and no body forces is discussed. Under such conditions, the governing momentum equations are decoupled from the governing energy equation. Once the flow field is known, different temperature distributions may be computed with different types of thermal boundary conditions. 

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. 

Now consider the flat plate shown in Fig. 12-3. The plate surface is maintained at the constant temperature Tw, the free-stream temperature is 7U, and the thermal-boundary-layer thickness is designated by the conventional symbol 5,. To simplify the analysis, we consider low-speed incompressible flow so that the viscous-heating effects are negligible. The integral energy equation then becomes... 

The approach to calculation of the temperature field and heat transfer follows closely the hydrodynamic calculation outlined above. For incompressible flow of a fluid with constant and uniform properties, neglecting the input to the thermal field by viscous dissipation, the thermal-energy equation (obtained by a combination of the energy and momentum equations) is... 

For a flat plate, Ihe characteristic length is the distance x from the leading edge. The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. For flow over a flat plate, its value is taken to be Re = V.V ,/v = 5 X 10 The continuity, momentum, and energy equations for steady Iwo-diruensional incompressible flow with constant properties are determined from mass, momentum, and energy balances to be... 

In this section we present the governing equations for the analysis of microchannel heat transfer in two-dimensional fluid flow. For steady two-dimensional and incompressible flow with constant thermophysical properties, the continuity, momentum and energy equations... 

First of all, the density and all the thermodynamic coefficients are constants. Secondly, when the density and the transport properties are constants, the continuity and momentum equations are decoupled from the energy equation. This result is important, as it means that we may solve for the three velocities and the pressure without regard for the energy equation or the temperature. Third, for incompressible flows the pressure is determined by the momentum equation. The pressure thus plays the role of a mechanical force and not a thermodynamic variable. Fourth, another important fact about incompressible flow is that only two parameters, the Reynolds number and the Froude number occur in the equations. The Froude number, Fr, expresses the importance of buoyancy compared to the other terms in the equation. The Reynolds number indicates the size of the viscous force term relative to the other terms. It is mentioned that compressible flows are often high Re flows, thus they are often computed using the inviscid Euler (momentum) equations. 

In the incompressible energy equation only derivatives of the temperature variable occur. This means that in incompressible flow, only changes in temperature with respect to some reference state are important. As with pressure, the level of the reference temperature does not affect the solution. The actual temperature in such a incompressible flow doesn t change noticeably. 

The further special case of small temperature differences will turn out to be an incompressible flow. This flow is governed by the same equations as the adiabatic case except that the energy equation is a little different. The energy equation has only advection and conduction terms while in the adiabatic case the energy equation also includes a viscous dissipation and a pressure term. 

In this section the numerical conservation properties of finite volume schemes for inviscid incompressible flow are examined. Emphasis is placed on the theory of kinetic energy conservation. Numerical issues associated with the use of kinetic energy non-conservative schemes are discussed [158, 49, 47]. 

Felten FN, Lund TS (2006) Kinetic energy conservation issues associated with the collocated mesh scheme for incompressible flow. J Comput Phys 215 465-484... 

The frictional effects of bends and fittings are often expressed in terms of a quantity called the velocity head drop . To introduce this concept, we begin by observing that the specific energy lost to friction, F(J/kg), for incompressible flow follows from an integration of the Fanning equation (4.20) with friction factor and velocity constant over the length of pipe ... 

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