Froude modeling

Froude modeling was also used to prove that a code-compliant design for atrium smoke control was faulty, and consequently the code was blamed for smoke damage to a 

Alternatively, one might satisfy convection near a boundary by invoking Il6 and Ilg where the heat transfer coefficient is taken from an appropriate correlation involving Re (e.g. Equation (12.38)). Radiation can still be a problem because re-radiation, n7, and flame (or smoke) radiation, II3, are not preserved. Thus, we have the art of scaling. Terms can be neglected when their effect is small. The proof is in the scaled resultant verification. An advantage of scale modeling is that it will still follow nature, and mathematical attempts to simulate turbulence or soot radiation are unnecessary. 

---

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. 

Osborne Reynolds identified the phenomenon of cavitation as early as 1873. By the ton of the century it had been called by its present name by R. E. Froude, the director of the British Admiralty Ship Model Testing Laboratories. 

The particle diameters in the model scale by the same factor as the bed diameter, by the ratio of the kinematic viscosities to the two-thirds power. Equating the Froude number and rearranging,... 

By satisfying both Eq. (69) and Eq. (71), the Reynolds number and the Froude numbers are kept identical between the model and the commercial bed. 

With Pf of the model set by the fluidized gas and its state, the solid density in the model follows from Eq. (76). Choosing the length coordinate of the model, Lm, which is now a free parameter, the superficial velocity in the model is determined so that the Froude number remains the same,... 

Davis (Dl) has suggested that the introduction of the Froude number into the Lockhart-Martinelli parameter. A, gives a description of gravitational and inertial forces so that this model can be applied to vertical flow. The revised parameter, X, is defined empirically for turbulent-turbulent flow as,... 

It should be noted that the frictional drop was calculated by subtracting the hydrostatic head and acceleration losses from the measured total pressure-drop where void data were lacking, a homogeneous flow model was assumed. This modification of X by use of the Froude number appears very similar to the technique used by Kosterin (K2, K3) for horizontal pipes, in which the equivalent of volume-fraction of gas flowing, with mixture Froude number as the correlating parameter. 

Due to the fact that the gravitational acceleration g cannot be varied on Earth, the Froude number (Fr) of the model can be adjusted to that of the full-scale vessel only by its velocity vm. Subsequently, Re = idem can be achieved only by the adjustment of the viscosity of the model fluid. In cases where the model size is only 10% of the full size (scale factor L /Lm = 10), Fr = idem is achieved in the model at Vm = 0.32 vj. To fulfill Re = idem, for the kinematic viscosity of the model fluid it follows ... 

We have to realize that sometimes requirements concerning physical properties of model materials exist that cannot be implemented. In such cases only a partial similarity can be realized. For this, essentially only two procedures are available (for details see Refs. 5 and 10). One consists of a well-planned experimental strategy in which the process is divided into parts, which are then investigated separately under conditions of complete similarity. This approach was first applied by William Froude (1810-1879) in his efforts to scale-up the drag resistance of the ship s hull. 

Hydrodynamic dimensionless numbers Examples are the Reynolds number, Froude, Archimedes, and Euler number. These dimensionless numbers have to be functions of identical determining dimensionless numbers of the same powers and with the same value of the other constant coefficients, so that the model and the object are similar. 

The fu st term is a modified Archimedes number, while the second one is the Froude number based on particle size. Alternatively, the first term can be substituted by the Reynolds number. To attain complete similar behavior between a hot bed and a model at ambient conditions, the value of each nondimensional parameter must be the same for the two beds. When all the independent nondimensional parameters are set, the dependent parameters of the bed are fixed. The dependent parameters include the fluid and particle velocities throughout the bed, pressure distribution, voidage distribution of the bed, and the bubble size and distribution (Glicksman, 1984). In the region of low Reynolds number, where viscous forces dominate over inertial forces, the ratio of gas-to-solid density does not need to be matched, except for beds operating near the slugging regime. 

Consider a fluidized bed operated at an elevated temperature, e.g. 800°C, and under atmospheric pressure with ah. The scale model is to be operated with air at ambient temperature and pressure. The fluid density and viscosity will be significantly different for these two conditions, e.g. the gas density of the cold bed is 3.5 times the density of the hot bed. In order to maintain a constant ratio of particle-to-fluid density, the density of the solid particles in the cold bed must be 3.5 times that in the hot bed. As long as the solid density is set, the Archimedes number and the Froude number are used to determine the particle diameter and the superficial velocity of the model, respectively. It is important to note at this point that the rale of similarity requires the two beds to be geometrically similar in construction with identical normalized size distributions and sphericity. It is easy to prove that the length scales (Z, D) of the ambient temperature model are much lower than those in the hot bed. Thus, an ambient bed of modest size can simulate a rather large hot bed under atmospheric pressure. 

In order to decide which model is relevant for the different river sections, we first calculate the roughness parameters cf (Eq. 20-36) and the element Froude Numbers FE(Eq. 20-37). 

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... 

The subscripts c and h refer to the cold and hot model, respectively. With the equality of the Froud number, U2/gl, the scale factor for the superficial gas velocity can be expressed... 

A comparison of Eqs. (10.45) and (10.46) shows that the two cannot be satisfied at the same time with a fluid of the same viscosity, as one requires that the velocity vary inversely as L, while the other requires it to vary directly as L112. If both friction and gravity are involved, it is then necessary to decide which of the two factors is more important or more useful. In the case of a ship, the towing of a model will give the total resistance, from which must be subtracted the computed skin friction to determine the wavemaking resistance, and the latter may be even smaller than the former. But, for the same Froude number, the wavemaking resistance of the full-size ship may be determined from this result. A computed skin friction for the ship is then to be added to this value to give the total ship resistance. 

It is decided to model a full-scale prototype, unbaffled, stirred vessel with a one-tenth scale model. The liquid in the prototype has a kinematic viscosity, v. of 10 7 m2 s As we have seen above, power number is a function of both Reynolds number and Froude number for unbaffled vessels. To ensure power number similarity, we need to ensure both Reynolds number and Froude number are similar from prototype to 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. 

When appropriate material systems are not available for model experiments, accurate simulation of the working conditions of an industrial plant on a laboratory- or bench-scale may not be possible. Under such conditions, experiments on differently sized equipment are customarily performed before extrapolation of the results to the full-scale operation. Sometimes this expensive and basically unreliable procedure can be replaced by a well-planned experimental strategy. Namely, the process in question can be either divided up into parts which are then investigated separately (Example 9 Drag resistance of a ship s hull after Froude) or certain similarity criteria can be deliberately abandoned and then their effect on the entire process checked (Example 41/2 Simultaneous mass and heat transfer in a catalytic fixed bed reactor after Damkohler). 

In order to verify these experimental results, the corvette Greyhound was towed by the corvette Active under the command of Froude, and the drag force in the tow rope was measured. Froude reported [19] that the observed deviations from the predictions of the model were in the range of only 7 - 10%. 

The reference [18] includes the minutes of the session of The Institution of Naval Architects in London of April 7, 1870. During this session, W. Froude presented and defended the results of his modeling with great steadfastness and conviction this represents the sidereal hour of the theory of models. 

However, he proposed a different strategy from that of Froude. In the first experiment, measurements were made with the model ship in water at Fr, = FrT, consequently Re, = Rej- p 3/2, i.e., the measurement was carried out at a correct Fr value and a false Re value. As a result, also a false value of Nc, was obtained from the relationship ... 

Two additional experiments were carried out, not with the model ship, but with a totally immersed form (Fig. 5) whose shape was given by reflecting the immersed portion of a ship s hull at the water line (at V/l3 = idem). In these experiments, the Froude number is irrelevant the friction corresponding to the surface area of the model must be divided by 2. 

Firstly, it has to be taken into account that criteria which characterize a state of flow have to be formulated in a dimensionless manner. Already W. Froude found in his experiments to determine the drag resistance of a ship s hull that the bow wave can only be reliably determined when the size of the ship model has the right proportions with respect to the travelling speed and channel width. 

The Froude number and tangential speed of the blender have been used with mixed results. Deviations from the model occur when using the Froude number if scale-up is between blenders with different geometries or if different fill volumes are used. ... 

Later Mayle, 1970 [400] continued their research by performing measurements of velocity and pressure within the fire whirl. He found that the behavior of the plume was governed by dimensionless plume Froude, Rossby, second Damkohler Mixing Coefficient and Reaction Rate numbers. For plumes with a Rossby number less than one the plume is found to have a rapid rate of plume expansion with height. This phenomenon is sometimes called vortex breakdown , and it is a hydraulic jump like phenomena caused by the movement of surface waves up the surface of the fire plume that are greater than the speed of the fluid velocity. Unfortunately, even improved entrainment rate type models do not predict these phenomena very well. 

Multiphase reactor types are highly varied. The simplest approach to analyzing and predicting their behavior is to focus on the rate limiting steps or segment the reactor and model each segment and its contributions separately. Correlations are invariably a function of phase-based Reynolds and Froude numbers. Fractional volumes and properties of the solids are factors. Where interfacial tension is an important factor, the Weber number can be added. 

---