Models, dynamic similarity

In the set of conservation equations described earlier, the Reynolds number and the Froude number must be the same for the model and the prototype. Since most industrial operations involve turbulent flow for which the Reynolds number dependence is insignificant, part of the dynamic similarity criteria can be achieved simply by ensuring that the flow in the model is also turbulent. For processes involving hot gases (i.e., buoyancy driving forces), the Froude number similarit) yields the required prototype exhaust rate as follows. 

Dynamic similarity requires geometric and kinematic similarity in addition lo force ratios at corresponding points being equal, involving properties of gravitation, surface tension, viscosity and inertia [8, 21]. With proper and careful application of this principle scale-up from test model lo large scale systems is often feasible and quite successful. Tables 5-... 

This is valid for any Newtonian fluid in any (circular) pipe of any size (scale) under given dynamic conditions (e.g., laminar or turbulent). Thus, if the values of jV3 (i.e., the Reynolds number 7VRe) and /V, (e/D) for an experimental model are identical to the values for a full-scale system, it follows that the value of N6 (the friction factor) must also be the same in the two systems. In such a case the model is said to be dynamically similar to the full-scale (field) system, and measurements of the variables in N6 can be translated (scaled) directly from the model to the field system. In other words, the equality between the groups /V3 (7VRc) and N (e/D) in the model and in the field is a necessary condition for the dynamic similarity of the two systems. 

Example 2-3 Scale-Up of Pipe Flow. We would like to know the total pressure driving force (AP) required to pump oil (/z = 30 cP, p = 0.85 g/cm3) through a horizontal pipeline with a diameter (D) of 48 in. and a length (L) of 700 mi, at a flow rate (Q) of 1 million barrels per day. The pipe is to be of commercial steel, which has an equivalent roughness (e) of 0.0018 in. To get this information, we want to design a laboratory experiment in which the laboratory model (m) and the full-scale field pipeline (f) are operating under dynamically similar conditions so that measurements of AP in the model can be scaled up directly to find AP in the field. The necessary conditions for dynamic similarity for this system are... 

We now turn to the electronically adiabatic ET reaction problem (cf. Sec. 2.2). There has been a spate oftheoretical papers [8,11 28,33,35,36,50] dealing with the possible role of solvent dynamics in causing departures from the standard Marcus TST rate theory [27,28] (although many of these deal with nonadiabatic reactions). The ET reaction considered is a simplified symmetric model, A1 2 A1/2 A1/2 A1/2, in a model solvent similar to CH3C1. The technical and computational... 

B. RIGOROUS MODEL. Dynamics can be included in a number of ways, with varying degrees of rigor, by using models similar to those in Sec. 3.7. Let us merely indicate how a rigorous model, like case C of Sec. 3.7, could be developed. Figure 3.8 shows the system schematically. 

Eq.(lO) represents a four-dimensional model of the reactor with external forcing disturbance, which can be used to investigate the chaotic dynamics. Similar to Eq.(4), the Eq.(lO) can be normalized. 

Jaberi, F. A., and S. A. James. 1998. A dynamic similarity model for large eddy simulation of turbulent combustion. J. Physics Fluids 10(7) 1775-77. 

Physically, this closure attempts to model the effect of SGS fluctuations on the filtered reaction rate by assuming that the largest of the SGS are dynamically similar to the smallest of the resolved scales. The influence of the smallest eddies, in particular Kolmogorov eddies, on the filtered reaction rate is assumed negligible because the large viscosity in the flame tends to rapidly dissipate these scales [19]. Further discussion and assessment of this model with comparisons to other SGS combustion closures and DNS results can be found in [17]. 

The developed model was applied to the EPS experiment (Fig.lb) to extract information on the water dynamics. Similar to the previous report [17], the EPS function decreases rapidly at a time scale of -0.5 ps, then raises again at -2 ps, and finally falls off to zero. The EPS functions acquired while keeping the delays tn (empty circles) and t23 (solid circles) fixed [20], are shifted along the vertical axis which is a consequence of the relatively short excited-state lifetime (700 fs). The peak in the EPS function around -2 ps is explained in the framework of our model as arising from interference between the chromophore and solvent responses. The delicate balance between phases of genuinely nonlinear and thermal contributions as the delay t12 between the two excitation pulses is increased, leads to the enhancement of the integrated signal that is measured in the EPS experiment. 

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. 

Then dynamic similarity between a model (in) and a prototype (p) is achieved if... 

To achieve dynamic similarity, the three dimensionless numbers for the prototype and the model must be equal, as follows ... 

It is noted here that the Froude number has changed and that dynamic similarity cannot be maintained if both, the model fluid viscosity and the model tank dimensions, are fixed because two unknowns (D and Q) are required to satisfy the two eqns. (4.64) and (4.65). Since gravity is a constant (9.81 m/s2) and p//t= 1,000 s/m2 is fixed for the model, obtaining that... 

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. 

Dimensional analysis (see drag force on a sphere example) yields CD =/ (Re). If we ensure that the value of Re is the same from prototype to model, then we will have dynamic similarity, and therefore Cd must have the same value from model to prototype. 

If we have dynamic similarity, i.e. Re = constant, then CD must be constant between model (M) and prototype (P)... 

Problem. In this example, we consider the flow around a body. Air, at atmospheric pressure, flows at 20 m s 1 across a bank of heat exchanger tubes. A l/10th-scale model is built. At what velocity must air flow over the model bank of tubes to achieve dynamic similarity ... 

Determined the fluid physical properties required for the model fluid to achieve dynamic similarity between prototype and model. 

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 complete physical law expressed as an equation between numerics is independent of the size of the system. Therefore dimensionless expressions are of great importance in problems of change of scale. When two systems exhibit similarity, one of them, and usually the smaller system, can be regarded as the "model". Two systems are dynamically similar when the ratio of every pair of forces or rates in one system is the same as the corresponding ratio in the other. The ratio of any pair of forces or rates constitutes a dimensionless quantity. Corresponding dimensionless quantities must have the same numerical value if dynamical similarity holds. 

Those that can be derived from the fundamental equations of dynamics. Engel calls these the groups that form the model laws of dynamic similarity. This is the most important category in engineering practice we come back to it later. 

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