Velocity work term

This indicates that large impellers running at slow speeds give a high pumping capacity and low shear rates since the impeller head or velocity work term is related to the shear rates around the impeller. 

Apply the conservation of energy, Equation (3.40). Since the control volume is fixed the pressure work term does not apply. The shear work (v x shear force) is zero because (a) the radius of the control volume was selected so that the velocity and its gradient are zero on the cylindrical face and (b) at the base faces, the velocity is normal to any shear surface force. Similarly, no heat is conducted at the cylindrical surface because the radial temperature gradient is zero, and conduction is ignored at the bases since we assume the axial temperature gradients are small. However, heat is lost by radiation as... 

In this form, each term has the dimensions of length. The terms z, P/(pg) and v2Klg) are known as the potential, pressure and velocity heads, respectively. Denoting the work terms as heads, equation 1.12 can also be written as... 

Here v is the number of ions per unit time and area transferring from the solution layer next to the electrode, to the electrode. It will at once be noticed that as long as the interfacial potential difference d< > is negative, the electrostatic work term Pde0 increases the velocity of the ion transfer reaction in an exponential way (Fig. 7.10). 

The time derivative of the displacement vector r is the velocity V, which, of course, assumes that the fluid system is moving with the fluid velocity. The left-hand side of the energy equation now represents the convective transport, and it remains to develop the heat-transfer and work terms on the right-hand side... 

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... 

The shear work due to viscous stresses also occurs at the control surfaces, the Internal work terms again being self-canceling, and consists of the product of each viscous stress (one normal and two tangential) and the respective velocity component ... 

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),... 

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 ... 

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, mweight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as... 

Fan Rating. Axial fans have the capabiUty to do work, ie, static pressure capabiUty, based on their diameter, tip speed, number of blades, and width of blades. A typical fan used in the petrochemical industry has four blades, operates neat 61 m/s tip speed, and can operate against 248.8 Pa (1 in. H2O). A typical performance curve is shown in Figure 11 where both total pressure and velocity pressure are shown, but not static pressure. However, total pressure minus velocity pressure equals static pressure. Velocity pressure is the work done just to collect the air in front of the fan inlet and propel it into the fan throat. No useflil work is done but work is expended. This is called a parasitic loss and must be accounted for when determining power requirements. Some manufacturers fan curves only show pressure capabiUty in terms of static pressure vs flow rate, ignoring the velocity pressure requirement. This can lead to grossly underestimating power requirements. 

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... 

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. 

The velocity and density have the same values at the end of the closed cycle as at the beginning. Thus, the kinetic energy is the same at the beginning and the end, and the first term in (5.36) vanishes. The work assumption (5.34) is expected to hold for any arbitrary region of the body and must therefore hold locally at every material particle, so that (5.36) reduces to... 

The term pump head represents the net work performed on the liquid by the pump. It is eomposed of four parts. They are the statie head (Hs), or elevation the pre.ssure head (Hp) or the pre.ssures to be overcome the friction head (Hf) and velocity head (Hf), which are frictions and other resistances in the piping system. These heads are discussed in Chapter 8. The head formula is the following ... 

A mechanism such as that given above provides explanations for the known effects of many process variables ". The reductive dissolution and undermining processes require access of the acid to the metal surface, hence the benefits obtained by the deliberate introduction of cracks in the oxide by cold-working prior to pickling. Also the increase in pickling rate with agitation or strip velocity can be explained in terms of the avoidance of acid depletion at the oxide-solution interface. 

This work is used to accelerate the electron, giving it kinetic energy. Also, we know that the kinetic energy of the electron can be expressed in terms of its mass and velocity ... 

In this equation the negative sign is introduced in order to maintain a consistency of sign convention when shear stress is related to momentum transfer as in Chapter 11. Since (du,/dy)v=o must be positive (velocity increases towards the pipe centre), R0 is negative. It is therefore more convenient to work in terms of / , the shear stress exerted by the fluid on the surface (= —Rq) when calculating friction data. 

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