Scaling analysis dimensionless groups

Dimensional Analysis. Dimensional analysis can be helpful in analyzing reactor performance and developing scale-up criteria. Seven dimensionless groups used in generalized rate equations for continuous flow reaction systems are Hsted in Table 4. Other dimensionless groups apply in specific situations (58—61). Compromising assumptions are often necessary, and their vaHdation must be estabHshed experimentally or by analogy to previously studied systems. 

Turbomachines can be compared with each other by dimensional analysis. This analysis produces various types of geometrically similar parameters. Dimensional analysis is a procedure where variables representing a physical situation are reduced into groups, which are dimensionless. These dimensionless groups can then be used to compare performance of various types of machines with each other. Dimensional analysis as used in turbomachines can be employed to (1) compare data from various types of machines—it is a useful technique in the development of blade passages and blade profiles, (2) select various types of units based on maximum efficiency and pressure head required, and (3) predict a prototype s performance from tests conducted on a smaller scale model or at lower speeds. 

Conventional dimensional analysis uses single length and time scales to obtain dimensionless groups. In the first section, a new kind of dimensional analysis is developed which employs two kinds of such scales, the microscopic (molecular) scale and the macroscopic scale. This provides some physical significance to the exponent of the Reynolds number in the expression of the Sherwood number, as well as some bounds of this exponent for both laminar and turbulent motion. 

Conventional dimensional analysis employs single length and time scales. Correlations are thus obtained for the mass or heat transfer coefficients in terms of the minimum number of independent dimensionless groups these can generally be represented by power functions such as... 

In situations where a complete description of the physical behavior of a system is unknown, scale-up approaches often involve the use of dimensionless groups, as described in Chapter 1. Unlike flow behavior in a blender, the flow behavior of powder through bins and hoppers can be predicted by a complete mathematical relationship. In light of this, analysis of powder flow in a bin or... 

Perform a dimensional analysis on the thin wall injection molding scaling problem of Example 4.8 and find the dimensionless groups used in the example. 

Fluid flow is often turbulent, and so heat transfer by convection is often complex and normally we have to resort to correlations of experimental data. Dimensional analysis will give us insight into the pertinent dimensionless groups see Chapter 6, Scale-Up in Chemical Engineering, Section 6.7.4. 

In broad terms scale-up is an engineering technique for translating performance in a small system to performance in a large system. A useful review of the formal material available on theories of models, similitude, dimensional analysis, etc., written from the chemical engineering point of view, is available in a recent book (J4). The practical applications of these theories involve the use of dimensionless groups, such as Reynolds number, in correlations which describe the performance of a system in terms equally applicable to large or small systems. This method of scale-up is familiar to all engineers. 

Scale-up implies a change from a small configuration to a larger one. To successfully perform the scale-up of a chemical process, one must first establish the categories for which similarity must be ensured. The difficulty that arises is that techniques based on the governing differential equations or dimensional analysis provide only a means for identifying pertinent dimensionless groups. Their absolute relationships in complex processes must be developed from small-scale experiments which usually... 

As long as one is aware of such limitations, excellent results are obtained with the scaling technique. Another problem comes up, however, when one cannot keep all material properties constant. Perhaps a different metal must be used or a different explosive, or some other property changed. In that case, a different scaling technique must be used, and for that we turn to dimensional analysis and scaling by means of dimensionless groups. 

The model (1) has four parameters R, K, A, and B. As usual, there are various ways to nondimensionalize the system. For example, both A and K have the same dimension as A, and so either N/A or N/K could serve as a dimensionless population level. It often takes some trial and error to find the best choice. In this case, our heuristic will be to scale the equation so that all the dimensionless groups are pushed into the logistic part of the dynamics, with none in the predation part. This turns out to ease the graphical analysis of the fixed points. 

Friction in pipes. For straight horizontal pipes resistance to flow arises because of viscous shear or friction at the wall. Dimensional analysis leads to the conclusion that for smooth pipes the pressure drop (scaled to make it dimensionless) is a function only of the Reynolds number. The dimensionless group containing the pressure drop is known as the friction factor ... 

As R(j tends to infinity, b becomes negligible in (115) and 6/41 tends to a constant value for each type of aggregate. This could be expected from a dimensional analysis argument. Since Rc/a is the only dimensionless group in the problem involving a length scale, with the exception of Ka, which is kept constant here, its influence can vanish only when it tends to infinity. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

The dimensional analysis gives basic relationships between the above-mentioned dimensionless groups, which are useful to describe hydrocyclone operation and are the basis for scale-up calculations aimed at adapting results from laboratory experimentation to an industrial scale for a number of processes in diverse industries. For example, for feed suspensions up to 10% by volume Medronho and Svarovsky (1984) proposed the following relations, for hydrocyclones following Rietema s optimum proportions and treating inert solids suspensions ... 

Relationships as those shown above can be useful in guiding the experimentation required by scale up and process transfer. However, as the number of variables, and thus the possible dimensionless groups that can be proposed, are quite large, simple dimensional analysis may not be very effective, and inspectional analysis based on the understanding of the underlaying mechanisms may be more appropriate. 

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