Dimensionless exchange

The dimensionless limiting current density N represents the ratio of ohmic potential drop to the concentration overpotential at the electrode. A large value of N implies that the ohmic resistance tends to be the controlling factor for the current distribution. For small values of N, the concentration overpotential is large and the mass transfer tends to be the rate-limiting step of the overall process. The dimensionless exchange current density J represents the ratio of the ohmic potential drop to the activation overpotential. When both N and J approach infinity, one obtains the geometrically dependent primary current distribution. 

At this point we can be more explicit about what we mean by NMR time scale. We will define the dimensionless exchange ratio (R) as the total rate constant (k, in reciprocal seconds) divided by the difference in frequency between two exchanging signals (Av0, in hertz) ... 

If equation 1.2 corresponds to the reference exchange half reaction in a system, then Ka x may be defined as unity by convention and equation 1.5 indicates plainly that the component X represents the dimensionless exchanger phase potential. 

On the other hand, the dimensionless exchange rate if/, is given by ... 

Dimensionless exchange current density A ecd (for reactions that are activation controlled in the linear region)... 

Fig. 11 Interaction parameter /sans for ethylene-norbornene copolymer blends computed from Eqs. 3 lb and 32c as a function of the composition of component 1 (N Ei x). Different curves correspond to various compositions / of component 2 (N Ei ). The dimensionless exchange energy e is chosen as s/kT = 10 , and z = 6. Note that the computed XSANS is independent of blend composition within the BLCT...

The Ft correction factor is usually correlated in terms of two dimensionless ratios, the ratio of the two heat capacity flow rates R and the thermal effectiveness P of the exchanger ... 

Heat Exchangers Since most cryogens, with the exception of helium 11 behave as classical fluids, weU-estabhshed principles of mechanics and thermodynamics at ambient temperature also apply for ctyogens. Thus, similar conventional heat transfer correlations have been formulated for simple low-temperature heat exchangers. These correlations are described in terms of well-known dimensionless quantities such as the Nusselt, Reynolds, Prandtl, and Grashof numbers. 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

Separation Factor By analogy with the mass-action case and appropriate for both adsorption and ion exchange, a separation factor / can be defined based on dimensionless system variables [Eq. (16-10)] by... 

Perdew and Wang (PW86) " proposed modifying the LSDA exchange expression to tliat shown in eq. (6.23), where x is a dimensionless gradient variable, and a, b and c being suitable constants (summation over equivalent expressions for the a and P densities is implicitly assumed). 

Heat exchanger efficiency, dimensionless. = Mass flow rate (tube or cold side), Ib/hr. = Mass flow rate (shell or hot side), Ib/hr. 

LMTD = log mean temperature difference, °F M = mass flow rate, Ib/hr Ntu = number of heat transfer units, dimensionless N = number tubes/row in direction of air flow n = number tubes/row, per ft of exchanger width, 1 /ft Q = total exchanger heat load (duty), Btu/hr R = = heat capacity ratio, dimensionless... 

In order to consider chromatographic processes in a more universal manner as processes in which film control of heterogeneous mass-exchange is also possible, dimensionless criteria for the conditions of formation of sharp zone boundaries may be represented by the parameter A [124,125]. The evaluation of this parameter is carried out on the basis of dynamic (chromatographic) and kinetic experiments ... 

Example 8.9 Find the temperature distribution in a laminar flow, tubular heat exchanger having a uniform inlet temperature and constant wall temperature Twall- Ignore the temperature dependence of viscosity so that the velocity profile is parabolic everywhere in the reactor. Use art/P = 0.4 and report your results in terms of the dimensionless temperature... 

Derive a dimensionless form of the equations and thus obtain the important dimensionless groups governing the dynamic behaviour of the heat exchanger. 

Here ia is the exchange current density of the electrode reaction based on the bulk concentration aa and ac are the anodic and cathodic charge transfer coefficients, respectively and y is a dimensionless kinetic parameter. 

At last, the Fermi coupling potential may be expressed beyond the exchange approximation by introducing the dimensionless Fermi coupling parameter A ... 

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