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Eq (10) Z = r = ^ = L
X Y Z
This requirement might not hold in actual scaleup practice. The model might be used to test conditions through a single element of the prototype vessel, and the prototype might contain several of these specific elements. For example, the model may be used to explore the effects of solvent flow rates on cleaning efficiency through a specific volume element. The prototype may contain several of these volume elements. For this example, the length of the prototype would not necessarily be scalable by the same factor as the vessel diameter. If all other factors are kept constant, the model and prototype might still meet the requirements for geometrical similarity even though the scale ratio along each axis might be different. This results in a more general expression of corresponding points given by the equation below:
x ó z
Eq. (11) — = X — = Y — = Z
M v ' x v z
4.2 Mechanical Similarity
Mechanical similarity in a SFPC system can only be achieved if both kinematic and dynamic similarity requirements are satisfied. In other words, kinematic similarity is achieved when corresponding particles trace out geometrically similar patterns in corresponding intervals of time. Times are measured from an arbitrary zero for each system, and corresponding times are defined as times such that t’/t = t is
constant; t is the time scale ratio (primed variables are for the pilot-scale plant). According to R. E. Johnstone and M. W. Thring:
If two geometrically similar fluid systems are kinematically similar, then the flow patterns are geometrically similar, and heat- or mass-transfer rates in the two systems will bear a simple relation to one another. Kinematic similarity in fluids entails both geometrically similar eddy systems and geometrically similar streamline boundary films. Hence, if L is the linear scale ratio, heat- and mass-transfer coefficients in the prototype will be 1/L times those in the model, from which the total quantities for heat or mass transferred can easily be calculated
Dynamic similarity is achieved when the forces retarding or accelerating the movement of mass within the system are correspondingly similar. These are the forces that define the dynamics of fluid systems in motion. In a supercritical fluids cleaning system, the forces of pressure, inertia, viscosity, and interfacial interactions are important for scaleup when it is desirable to predict either pressure drops or power consumption. For most cleaning systems, however, dynamic similarity is usually only of importance as an indirect means of establishing kinematic similarity.
4.3 Thermal Similarity
Cleaning systems are thermally similar when corresponding temperature differences can be measured by a constant ratio for kinematically similar systems. Corresponding temperature differences are the differences in temperature as measured at corresponding geometrical points at corresponding times. This requirement needs to be established for a cleaning system in which there is a flow of heat through the system. In a supercritical carbon dioxide system, for example, the solvent stream might be heated as it enters the cleaning vessel. The flow of heat from the solvent to the vessel walls and, consequently, the flow of heat from the entering solvent to the
bulk solvent are clearly dependent on solvent flow rates through the cleaning chamber. If the model and prototype are kinematically similar, and the vessels are constructed of the same material, the flow of heat through the two systems should be similar as well. This is especially important for a carbon dioxide system where solubilities are highly dependent on solvent density, and solvent density is highly dependent on temperature and pressure.
4.4 Chemical Similarity
Cleaning systems are chemically similar when corresponding concentration differences can be measured by a constant ratio for kinematically similar systems. Corresponding concentration differences are defined in the same manner as corresponding temperature differences. The differences are measured at corresponding geometrical points at corresponding times. That is, the total concentration of either the solvent or solvent/solute mixture is not important but, rather, the differences in constituent concentrations at corresponding locations and intervals.
Rate of Formation of the Solute / Solvent Mixture ^4-02) Rate of Bulk Flow
^ Rate of Formation of the Solvent / Solute Mixture
Eq. (13) -----------------------------------------------------
Rate of Molecular Diffusion
In both model and prototype, the formation time for the solvent/ solute mixture will be of the same rate or reaction order, and this requirement fixes the relative velocities in continuous-flow systems. These velocities are incompatible with the velocities necessary for kinematic similarity except at very low or very high velocities.