Modeling and prototyping

The enterprise shall establish the models, simulations, or prototypes needed to support requirements definition, analyze the system architecture and design, mitigate identified risks, and thereby ensure that the final product satisfies market needs, requirements, and constraints. This effort supports the assessment of system functional and performance characteristics, producibility, supportability, environmental impact, and human systems engineering issues such as maintainability, usability, operability, and safety. 

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Steve [13] introdueed the eoneept of the aspeet ratio of a reaetor (defined as the tangent -to- tangent length of the reaetor divided by the reaetor diameter). It was shown that when the aspeet ratio (R) is the same for model and prototype reaetors, then the inerease in the... 

Important conclusions can be drawn from the general modeling Eq. (13.79). The equation shows that the required prototype flow rates are directly proportional to the model flow rates. For scaling, the equation shows that the prototype flow rate has a strong dependence on the accuracy of the model scale (5/3 power). Both of these parameters are easy to establish accurately. The flow rate is rather insensitive (varies as the 1/3 powet) to the changes in the model and prototype heat flow tates, densities, and temperatures. This is desirable because an inaccuracy in the estimate of the model variable will have a rather small effect on the tesulting ptototype flow rate. 

Figure 12.6 Model and prototype geometric similar systems...

Model and prototypes are represented by the performance curves of Figure 7.2. Comparisons are to be made at the peak efficiency, assumed to be the same for each. Data off Figure 7.2(b) are ... 

For the optimum design of a production-scale fermentation system (prototype), we must translate the data on a small scale (model) to the large scale. The fundamental requirement for scale-up is that the model and prototype should be similar to each other. 

Two kinds of conditions must be satisfied to insure similarity between model and prototype. They are ... 

Dynamic similarity of the flow fields The ratio of flow velocities of corresponding fluid particles is the same in model and prototype as well as the ratio of all forces acting on corresponding fluid particles. When dynamic similarity of two flow fields with geometrically similar boundaries is achieved, the flow fields exhibit geometrically similar flow patterns. 

This implies geometric similarity and, in addition, that the ratio of the velocities at all corresponding points is the same. If subscripts m and p denote model and prototype, respectively, the velocity ratio Vr is as follows ... 

If N is the ratio of two types of forces and is therefore a dimensionless number, then, for any two forces considered, dynamic similarity requires that N be the same for both model and prototype that is, Nm = Np. As stated before, this equality is not confined to model studies alone but is applicable to any two flow systems. 

Dynamic similarity is similarity of forces. A model and prototype are dynamically similar when all forces acting at corresponding points, on fluid elements or corresponding boundaries, form a constant ratio between model and prototype. 

Thus, if we arrange for the values of all but one of the dimensionless groups (n2 and n3, above) to be equal between model and prototype, then the remaining dimensionless group (nx above) must be equal between model and prototype. 

If we have dynamic similarity, i.e. Re — constant, then cf must be constant between model and prototype ... 

Found that use of the same fluid for model and prototype was not feasible. 

The letters R, F, and W stand for so-called Reynolds, Froude, and Weber numbers, respectively these are dimensionless numbers, as indicated. For example, if we make the Reynolds number the same in model and prototype, using the same fluid, the dimension of length is smaller in the model and hence the velocity v will have to be greater. In other words, the water would have to flow faster in the model. If we now consider the Froude number as the same in model and prototype, and that the same fluid is used in both, we see that the velocity would have to be less in the model than in the prototype. This may be regarded as two contradictory demands on the model. Theoretically, by using a different fluid in the model (thus changing p0 and p), it is possible to eliminate the difficulty. The root of the difficulty is the fact that the numbers are derived for two entirely different kinds of flow. In a fluid system without a free surface, dynamic similarity requires only that the Reynolds number be the same in model and prototype the Froude number does not enter into the problem. If we consider the flow in an open channel, then the Froude number must be the same in model and prototype. 

A test-model pump delivers, at its best efficiency point, 500 gal/min (0.03 m3/s) at a 350-ft (107-m) head with a required net positive suction head (NPSH) of 10 ft (3.05 m) and a power input of 55 hp (41 kW) at 3500 r/min, when using a 10.5-in-diameter impeller. Determine the performance of the model at 1750 r/min. What is the performance of a full-scale prototype pump with a 20-in impeller operating at 1170 r/min What are the specific speeds and the suction specific speeds of the test-model and prototype pumps ... 

For the model, Ns = 3500(500)as/350a75 = 965 S = 3500(500)°-5/10°-75 = 13,900. For the prototype, Ns = 1170(1158)a5/142.5a75 = 965 5 = 1170(1156)a5/4.06a75 = 13,900. The specific speed and suction specific speed of the model and prototype are equal because these units are geometrically similar or homologous pumps and both speeds are mathematically derived from the similarity laws. 

However, each set of factors entering in to the rate expression is also a potential source of scaleup error. For this, and other reasons, a fundamental requirement when scaling a process is that the model and prototype be similar to each other with respect to reactor type and design. For example, a cleaning process model of a continuous-stirred tank reactor (CSTR) cannot be scaled to a prototype with a tubular reactor design. Process conditions such as fluid flow and heat and mass transfer are totally different for the two types of reactors. However, results from rate-of-reaction experiments using a batch reactor can be used to design either a CSTR or a tubular reactor based solely on a function of conversion, -r ... 

When scaling the process, the model and prototype must have Reynolds numbers in the same regime (i.e., laminar or turbulent flow) in order to achieve similar results. Ideally, the two reactors would have nearly identical or identical Reynolds numbers. In order to satisfy this requirement for scaleup, the model and prototype must be similar to each other with respect to reactor design, fluid flow, and physical dimensions. According to the principle of similarity, for every process condition and point in the model, there must be a corresponding condition and point in the prototype.This principle is applied to the scaling process by observing similar requirements for several other dimensionless ratios or variables that must be treated in a similar fashion. 

Each of these four states is dependent on the other three. Mechanical similarity is impossible to achieve if the model and prototype are not geometrically similar. The requirements for chemical and thermal similarity are then impossible to achieve without the requirements for mechanical similarity being met. 

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,... 

We see that pressures are equal in both the model and prototype in geometric similarity modeling. Going a bit further, let us look at the stresses in the walls of these scaled vessels. The wall thickness was also scaled so these are Wq and vvi = Swq. The spherical hoop stress in a thin-walled ball is... 

So now we see that stresses are equal in both model and prototype. 

The velocity of the bullet is the same in both model and prototype. The time of travel from the muzzle to the target is r = Xlv, so... 

For the optimum design of a production-scale fermentation system (prototype), we must translate the data on a small scale (model) to the large scale. The fundamental requirement for scale-np is that the model and prototype should be similar. To ensure similarity, two conditions mnst be satished (1) geometric similarity of the physical boundaries and (2) dynamic similarity of the flow fields. The first requirement is obvions and easy to accomplish. The second is achieved when the values of the nondimensional parameters (such as Reynolds number and power nnmber) are the same. However, it is difficult, if not impossible, to satisfy the dynamic similarity when more than one dimensionless group is involved in a system, which creates the need for scale-np criteria. 

Obviously, the solutions of Eqs. 1.74-1.77 depend on the coefficients that appear in these equations. Solutions of Eqs. 1.74-1.77 are equally applicable to the model and prototype (where the model and prototype are geometrically similar systems of different linear dimensions in streams of different velocities, temperatures, and concentration), if the coefficients in these equations are the same for both model and prototype. These coefficients, Pr, Re, Sc, and Ec (called dimensionless parameters or similarity parameters), are defined in Table 1.10. Focusing attention now on heat transfer, from Eq. 1.14, using the dimensionless quantities, the heat transfer coefficient is given as ... 

Geometric similarity requires that the scale physical model is dimensionally similar to the prototype. Such similarity exists between the scale model and the prototype if the raho of all corresponding dimensions and all angles in the model and prototype are equal. Figure 10.1 illustrates the geometric similarity between a prototype and a scale model. 

Kinematic similarity is the similarity of fluid flow behavior in terms of time within the similar geometries. Kinematic similarity requires that the motion of fluids of both the scale model and prototype undergo similar rate of change (velocity, acceleration, etc.). This similarity criterion ensures that streamlines in both the scale model and prototype are geometrically similar and spatial distributions of velocity are also similar. 

Flows that develop a state that depends only on the local flow quantities, such as the local value of the mean velocity and the flow resistance, are said to be self-similar or self-preserving. This state of flow is present in the turbulent flow regime when sufficiently high Reynolds numbers are achieved. A majority of industrial combustion systems operate in this flow regime. When the scale model and prototype are both operating in the selfsimilar flow regime, they will manifest the same flow patterns and pressure drop coefficient despite different absolute local flow quantities. 

In summary, in two geometrically and kinematically similar systems, when the flows are in the self-similar state, the flow characteristics such as flow patterns and streamlines are similar and the pressure drop coefficients in both the scale model and prototype are the same. 

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