Dimensional analysis definition

The definition for 6y, must be developed. The definition requires dimensional consistency the gap cannot be calculated as for 6c- The dimensional analysis in the y direction involves 6c, and the remaining solid bed at any position z down the channel. The following equation was developed to meet these criteria. 

Dimensional analysis shows that, in the treatment of natural convection, the dimensionless Grashof number, which represents the ratio of buoyancy to viscous forces, is often important. The definition of the Grashof number, Gr, is... 

Just as process translation or scaling-up is facilitated by defining similarity in terms of dimensionless ratios of measurements, forces, or velocities, the technique of dimensional analysis per se permits the definition of appropriate composite dimensionless numbers whose numeric values are process-specific. Dimensionless quantities can be pure numbers, ratios, or multiplicative combinations of variables with no net units. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

FIGURE 13.2. Principal component analysis definition of PCI and PC2 for a two-dimensional data set. 

Next, we proceed to dimensional analysis of the same problem by the method of physical similitude. From the definition of... 

We can use dimensional analysis to convert a measurement from one metric unit to another. We recognize that by definition 1 dollar = 100 cents, and also that 1 cent = 0.01 dollar. We can use a factor corresponding to either of those equalities. Note that cent is related to centi-, the metric prefix for 0.01. We can simply substitute... 

For the single-reaction cases, we performed dimensional analysis and found a dimensionless number, the Thiele modulus, which measures the rate of production divided by the rate of diffusion of some component. A complete analysis of the first-order reaction in a sphere suggested a general approach to calculate the production rate in a pellet in terms of the rate evaluated at the pellet exterior surface conditions. This motivated the definition of the pellet effectiveness factor, which is a function of the Thiele modulus. 

A dimensional analysis of the perfonnance index shows that it has the dimension of an inverse viscosity. With a little algebra, using the equations for permeability [Eq. (2.51)3, die definitions of the flow-resistance parameter [Eq. (154)3, linear velocity [l s. (110 and (111)3. retention factor [Eq. (161)3 and the reduced plate height [Eq. (113)3, we can lit the performance index into its components ... 

Equation 6.10 is a definition of the Reynolds number based on power dissipation using a velocity term derived from dimensional arguments. This prompted Middleman (1965) to comment that derivation of Equation 6.11 using Kolmogorov s theory is a sophisticated form of dimensional analysis. Even with this oversimplification. Equation 6.10 still needs to be modified for gas-liquid mass transfer for which has a wide variation, and hence, an average is difficult to define. In view of this, most investigators resorted to correlating the volumetric mass transfer coefficient,. The correlations proposed for stirred tank reactors were therefore of the form (Hickman 1988 Middleton 2000)... 

Careful measurements and the proper use of significant figures, along with correct calculations, will yield accurate numerical results. But to be meaningful, the answers also must be expressed in the desired units. The procedure we use to convert between units in solving chemistry problems is called dimensional analysis (also called the factor-label method). A simple technique requiring little memorization, dimensional analysis is based on the relationship between different units that express the same physical quantity. For example, by definition 1 in = 2.54 cm (exactly). This equivalence enables us to write a conversion factor as follows ... 

Table 2.1 presents a non-exhaustive list of expressions of the characteristic times corresponding to the most commonly used phenomena involved in chemical reactors. The previous definitions unfortunately do not always enable one to build the expressions presented in Table 2.1. Various methods can be used such as a blind dimensional analysis, similar to the Buckingham method used for dimensionless numbers [7], which can be applied to the list of fundamental physical and chemical properties. Nevertheless, the most relevant method consists in extracting the expressions from a mass/heat/force balance.

Insightful understanding of the granulation processes is essential for the identification of key variables and parameters for the dimensional analysis and scale-up considerations. While development of definitive mathematical models for the granulation processes is incomplete, the scaling approaches recommended in this chapter help reduce uncertainty during new product development and transfer to industrial sites. 

Heat and mass transfer coefficients are usually reported as correlations in terms of dimensionless numbers. The exact definition of these dimensionless numbers implies a specific physical system. These numbers are expressed in terms of the characteristic scales. Correlations for mass transfer are conveniently divided into those for fluid-fluid interfaces and those for fluid-solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. The formulation of the correlations can be based on dimensional analysis and/or theoretical reasoning. In most cases, however, pure curve fitting of experimental data is used. The correlations are therefore usually problem dependent and can not be used for other systems than the one for which the curve fitting has been performed without validation. A large list of mass transfer correlations with references is presented by Perry [95]. 

Unit for temperature The SI unit system defines a base unit for temperature kelvin with the dimension ( ). The physical origin of this definition is the thermodynamic temperature scale where the temperature unit is defined as being proportional to the kinetic mean energy of the molecules in a system of matter. Therefore, for dimensional analysis it is admissible to use the same unit for temperature quantities as for energy, if this is expedient. In this case, the dimension for temperature is (L MT ). 

This chapter discusses mass transfer coefficients for dilute solutions extensions to concentrated solutions are deferred to Section 9.5. In Section 8.1, we give a basic definition for a mass transfer coefficient and show how this coefficient can be used experimentally. In Section 8.2, we present other common definitions that represent a thicket of prickly alternatives rivaled only by standard states for chemical potentials. These various definitions are why mass transfer often has a reputation with students of being a difficult subject. In Section 8.3, we list existing correlations of mass transfer coefficients and in Section 8.4, we explain how these correlations can be developed with dimensional analysis. Finally, in Section 8.5, we discuss processes involving diffusion across interfaces, a topic that leads to overall mass transfer coefficients found as averages of more local processes. This last idea is commonly called mass transfer resistances in series. 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

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