Viscous work

Neglect the effects of viscous work (i.e., viscous dissipation), and compare with the steady-state form of the thermal-energy equation for an incompressible fluid. Are they the same equation If so, why If not, why ... 

Energy convected in left face + energy convected in bottom face + heat conducted in bottom face + net viscous work done on element = energy convected out right face + energy convected out top face... 

The convective and conduction energy quantities are indicated in Fig. 5-6, and the energy term for the viscous work may be derived as follows. The viscous work may be computed as a product of the net viscous-shear force and the distance this force moves in unit time. The viscous-shear force is the product of the shear-stress and the area dx,... 

Energy convected in + viscous work within element... 

In the actual case of a boundary-layer flow problem, the fluid is not brought to rest reversibly because the viscous action is basically an irreversible process in a thermodynamic sense. In addition, not all the free-stream kinetic energy is converted to thermal energy—part is lost as heat, and part is dissipated in the form of viscous work. To take into account the irreversibilities in the boundary-layer flow system, a recovery factor is defined by... 

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

To be able to apply the ID equations one often has to estimate terms that cannot be calculated directly without further empirical information, or we might simply neglect the insignificant terms for each particular application. The work associated with the term ((T) (v) ) e, can be split into two contributions. In chemical reactor analysis (and many other low Mach number flows in chemical engineering [13]) the viscous work ((g) A (v)a) ez is normally neglectable relative to the pressure work (p)a vz)a-... 

For all parts of the control surface which are solid walls, v = 0 from the viscous no-sllp condition. Hence, kFviscous stresses = 0 Identically. If parts of the control volume surface denote the surface of a machine, the viscous work Is contributed by the machine and should rather be absorbed In the term, Shatt- At an Inlet or outlet, the flow Is approximately normal to the element da. Thus, the only viscous work term comes from the normal stress, ann ndo-Since viscous normal stresses are extremely small In most cases. It Is customary to neglect viscous work at Inlets and outlets of the control volume. 

It is important to note that Eq. 5.2.9 is applicable only when all the assumptions made in its derivation are valid negligible kinetic, potential and electric energies, negligible effect of pressure, and negligible viscous work. 

For most chemical processes, the kinetic and potential energy of the streams are negligible in eomparison to the enthalpy, and the viscous work is usually small hence, Eq. 5.2.45 reduces to... 

Since Eqs. 7.1.5 and 7.1.15 contain another dependent variable, 0, flie dimensionless temperature, we should solve the energy balance equation simultaneously with the design equations. For plug-flow reactors with negligible viscous work, the energy balance equation (derived in Section 5.2) is... 

When the gas velocity approaches the sound velocity, the kinetic energy and viscous work terms in the energy balance equation are not negligible (as assumed in Chapter 5). For these cases, we write the general energy balance equation for a differential plug-flow reactor with length dL (see Eq. 5.2.44),... 

For flow in cylindrical conduits, the viscous work per unit mass of the fluid is expressed in terms of a friction factor and a specific kinetic energy by... 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

For the simple geometry of flow in a tapered tube (Figure 1.2), the energy balance of equation (1.4.8) can be simplified. First, = 0 because the velocity on the surface of the control volume is zero. It should be noted that the shear force is not zero at the surface, S. However, the viscous work is zero since it is the product of the system velocity times the viscous force. For the viscous work to be nonzero, the velocity at the control boundary surface, 5, must be nonzero. However, the no-slip condition states that the velocity at the surface S is zero. A second simplification occurs because surface area term is nonzero only at the entrance and exit of the tapered tube. The tapered tube surface area term therefore becomes... 

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