Dimensionless state

Assuming that all the events are homogeneous in all vesicles, and using the proper dimensionless state variables and parameters, we consider the behavior for a single synaptic vesicle as described by this simple two-compartment model, where (I) and (II) denote the two compartments. 

The classical three-compartment model describes pharmacokinetics of 5-HT1A receptor agonists. By means of a sigmoidal function E (c), the 5-HT1A agonist concentration c (t) influences the set-point signal that dynamically interacts with the body temperature. By using x (t) and y (t) as dimensionless state variables for the set-point and temperature, respectively, the model is expressed by the set of two nonlinear differential equations ... 

For the case when the dimensionless state of a global function is preferred for the computation, then we can operate with dimensionless process variables and with a dimensionless mathematical model of the process. However, we can also operate with dimensional variables but with partly dimensionless functions. 

Here p and are, respectively, the mean value and the dispersion (variance) with respect to a population. These characteristics establish all the integral properties of the normal random variable that is represented in our example by the value expected for the species concentration in identical samples. It is not feasible to calculate the exact values of p and because it is impossible to analyse the population of an infinite volume according to a single property. It is important to say that p and show physical dimensions, which are determined by the physical dimension of the random variable associated to the population. The dimension of a normal distribution is frequently transposed to a dimensionless state by using a new random variable. In this case, the current value is given by relation (5.21). Relations (5.22) and (5.23) represent the distribution and repartition of this dimensionless random variable. Relation (5.22) shows that this new variable takes the numerical value of x when the mean value and the dispersion are, respectively, p = 0 and = 1. 

In Eq. (1.1.12), the following basic dimensionless state-geometric parameters of the flow are used ... 

The flowrate of oil into the wellbore is also influenced by the reservoir properties of permeability (k) and reservoir thickness (h), by the oil properties viscosity (p) and formation volume factor (BJ and by any change in the resistance to flow near the wellbore which is represented by the dimensionless term called skin (S). For semisteady state f/owbehaviour (when the effect of the producing well is seen at all boundaries of the reservoir) the radial inflow for oil into a vertical wellbore is represented by the equation ... 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

The force constants kj,kc and the dimensionless Renner parameters r, c ate defined by the adiabatic potentials for the components of the II state at pure trans (Vj, Vj) and pure cis (V, V ) bending vibrations,... 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

In the absence of body force, the dimensionless form of the governing model equations for two-dimensional steady-state incompressible creeping flow of a viscoelastic fluid are written as... 

The quantityis dimensionless and is the ratio of the strength of the transition to that of an electric dipole transition between two states of an electron oscillating in three dimensions in a simple harmonic way, and its maximum value is usually 1. 

The physical properties of argon, krypton, and xenon are frequendy selected as standard substances to which the properties of other substances are compared. Examples are the dipole moments, nonspherical shapes, quantum mechanical effects, etc. The principle of corresponding states asserts that the reduced properties of all substances are similar. The reduced properties are dimensionless ratios such as the ratio of a material s temperature to its critical... 

Only those components which are gases contribute to powers of RT. More fundamentally, the equiUbrium constant should be defined only after standard states are specified, the factors in the equiUbrium constant should be ratios of concentrations or pressures to those of the standard states, the equiUbrium constant should be dimensionless, and all references to pressures or concentrations should really be references to fugacities or activities. Eor reactions involving moderately concentrated ionic species (>1 mM) or moderately large molecules at high pressures (- 1—10 MPa), the activity and fugacity corrections become important in those instances, kineticists do use the proper relations. In some other situations, eg, reactions on a surface, measures of chemical activity must be introduced. Such cases may often be treated by straightforward modifications of the basic approach covered herein. 

The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids which equates the thermal expansion coefficient with the quantity where yis the Griineisen dimensionless ratio, C, is the specific heat of the solid, p is the density of the material, and B is the bulk modulus. For fee metals the average value of the Griineisen constant is near 2.3. However, there is a tendency for this constant to increase with atomic number. 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

From the slope and intereept of the heat absorption line, it is possible to manipulate Equation 6-117 by ehanging operating variables sueh as u, Tq, and T or design variables sueh as the dimensionless heat transfer group UA/puCp. It is also possible to alter the magnitude of the reaetion exotherm by ehanging the inlet reaetant eoneentrations. Any of tliese manipulations ean be used to vary tlie number of loeations of the possible steady states. 

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

The resolution of this paradox lies in the assumptions about standard (reference), states which are unavoidably involved in the above definitions of and /3l-l- In order to ensure that and /3l-l are dimensionless (as they have to be if their logarithms are to be used) when concentrations are expressed in units which have dimensions, it is necessary to use the ratios of the actual concentrations to the concentrations of... 

FGM at arbitrary disorder strength [27]. The average density of states, p(c), that may be obtained from their result, is plotted in Fig. 3-5 for three values of the dimensionless disorder strength g = A/(vpAo). For small disorder, one clearly observes the pseudogap. Close to the center of this pseudogap ( c A0), the energy depen-... 

Dimensionless Constants for Saturated Liquids in the Redlich-Kwong Equation of State... 

Activity is a dimensionless quantity, and / must be expressed in kPa with this choice of standard state. It is inconvenient to carry f° = 100 kPa through calculations involving activity of gases. Choosing the standard state for a gas as we have described above creates a situation where SI units are not convenient. Instead of expressing the standard state as /° = 100 kPa, we often express the pressure and fugacity in bars, since 1 bar = 100 kPa. In this case, /0 — 1 bar, and equation (6.92) becomes4... 

The need for dimensional consistency imposes a restraint in respect of each of the fundamentals involved in the dimensions of the variables. This is apparent from the previous discussion in which a series of simultaneous equations was solved, one equation for each of the fundamentals. A generalisation of this statement is provided in Buckingham s n theorem(4) which states that the number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions. In mathematical terms, this can be expressed as follows ... 

A liquid is in steady state flow in an open trough of rectangular cross-section inclined at an angle f) to the horizontal. On what variables would you expect the mass flow per unit time to depend Obtain the dimensionless groups which are applicable to this problem. 

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