Dimensionless numbers Schmidt number

HETP = height equivalent to a theoretical plate, ft HTU = height of a transfer unit, ft L = liquid mass velocity, Ib/hr-ft m = exponent a 1.0 n = exponent 0.44 Pr = Prandtl number, dimensionless Sc = Schmidt number dimensionless U, = linear velocity of gas based on total column cross-sectional area, ft/sec... 

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

Define and interpret the following dimensionless numbers Schmidt, Prandtl, and Lewis. 

Schj.g dimensionless gas Schmidt number for H2, PgIPgDki.g Schj.l dimensionless liquid Schmidt number for H2, /Tl/PlDhi.l... 

A, B Coefficient in Equation 17 D Diameter, m D Diffusivity, mVs De Dean number. Re (D/D ) , dimensionless f Fannings friction factor, dimensionless K, Constants in Equation 52 p Pitch, m r Radius, m Re Reynolds number VDp/p, dimensionless Sc Schmidt number p/pD, dimensionless V Velocity, m/s... 

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

The film thickness 6g depends primarily on the hydrodynamics of the system and hence on the Reynolds number and the Schmidt number. Thus, various correlations have been developed for different geometries in terms of the following dimensionless variables ... 

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

Sc = Schmidt number, dimensionless Pr = Prandtl number, dimensionless Cg = gas specific heat, Btu/lb-°F a = interfacial area, fti/fti Q, = sensible heat transfer duty, Btu/hr Qj. = total heat transfer duty, Btu/hr... 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

The dynamical regimes that may be explored using this method have been described by considering the range of dimensionless numbers, such as the Reynolds number, Schmidt number, Peclet number, and the dimensionless mean free path, which are accessible in simulations. With such knowledge one may map MPC dynamics onto the dynamics of real systems or explore systems with similar characteristics. The applications of MPC dynamics to studies of fluid flow and polymeric, colloidal, and reacting systems have confirmed its utility. 

Sc = u PfDj, Schmidt number, dimensionless Sh = hDdp/Df, Sherwood number, dimensionless Sy Synergism number, dimensionless... 

Schmidt number 3 phys chem A dimensionless number used In electrochemistry, equal to the product of the dielectric susceptibility and the dynamic viscosity of a fluid divided by the product of the fluid density, electrical conductivity, and the square of a characteristic length. Symbolized SC3. shmit. nam bar thre ) Schoeikopf s acid orgchem A dye of the following types l-naphthol-4,8-dlsulfonlc acid, l-naphthylamine-4,8-disulfonicadd,and l-naphthylamine-8-sulfonicadd may be toxic. shol.kopfs, as-3d ... 

Semenov number 1 physchem A dimensionless number used in reaction kinetics, equal to a mass transfer constant divided by a reaction rate constant. Symbolized S . Formerly known as Schmidt number 2. se-m3,n6f nom-bor won ... 

Ka can be defined as a gas-phase transfer coefficient, independent of the liquid layer, when the boundary concentration of the gas is fixed and independent of the average gas-phase concentration. In this case, the average and local gas-phase mass-transfer coefficients for such gases as sulfur dioxide, nitrogen dioxide, and ozone can be estimated from theoretical and experimental data for deposition of diffusion-range particles. This is done by extending the theory of particle diffusion in a boundary layer to the case in which the dimensionless Schmidt number, v/D, approaches 1 v is the kinematic viscosity of the gas, and D is the molecular diffusivity of the pollutant). Bell s results in a tubular bifurcation model predict that the transfer coefficient depends directly on the... 

Gas constant Radius of gel particle Reynolds Number Dimensionless radius Equivalent hard sphere radius Distance within gel particle Laplace variable Schmidt Number Varicince... 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and diffu-sivity of the medium for both air and water. In air, these numbers are all of the same order of magnitude, meaning that air provides a similar resistance to the transport of momentum, heat, and mass. In fact, there are two dimensionless numbers that will tell us these ratios the Prandtl number (Pr = pCpv/kj = v/a) and the Schmidt number (Sc = v/D). The Prandtl number for air at 20°C is 0.7. The Schmidt number for air is between 0.2 and 2 for helium and hexane, respectively. The magnitude of both of these numbers are on the order of 1, meaning that whether it is momentum transport, heat transport, or mass transport that we are concerned with, the results will be on the same order once the boundary conditions have been made dimensionless. 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

Dijfusional dimensionless numbers The Peclet, Prandtl, Schmidt, Sherwood, and Nusselt number are the most common ones. 

In these electrode configurations, the solution moves past the electrodes embedded in the wall of a tube or channel. It turns out, as is to be expected, that for high Schmidt numbers (thin diffusion layer) the mass transport in the appropriate dimensionless variables is virtually identical for both electrodes. 

From our earliest example, we saw that it was advantageous to use dimensionless variables and that the characteristic quantities should be capable of being held constant. In addition, if a parametric study on the effect of varying some input quantity is to be performed, that quantity should appear in only the distinguished parameter. This is no restriction, for the others are proportional to powers of the distinguished parameter, and the proportionality constants are themselves dimensionless numbers. For example, if the viscosity is to be varied, the Reynolds and the Schmidt numbers are both functions of v, but ReSc is not so, if Sc is chosen as the dimensionless viscosity, Re = Cl Sc, where C = ReSc is independent of v. 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

The transfer coefficient can be correlated in the form of a dimensionless Sherwood number Sh(= h0dp/D). The particle diameter dp is often taken to be the diameter of the sphere having the same area as the (irregular shaped) pellet. Thaller and Thodos(38> correlated the mass transfer coefficient in terms of the gas velocity u and the Schmidt number Sc(= p/pD) ... 

Sh, Re, and Sc are the dimensionless Sherwood, Reynolds, and Schmidt numbers, dp is the relevant length scale, which here is the diameter of the particle c = 4.7 for fixed beds and laminar flows. As usual, D is diffusion coefficient and v the dynamic viscosity (both [length time-2]). 

The magnitude of Fo is often used as dimensionless time (to estimate how long diffusion has proceeded). The Schmidt number (Sc) is simply the ratio of the Pe to Re and is... 

---