Archimedes’ number

Arc furnaces Archery bows Archimedes number Architectural fabrics Architectural lamnates Arc-jet thrusters... 

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

The usefulness of these relationships lies in the recognition that by evaluating the Archimedes number, we can establish the theoretical settling range for the particles we are trying to separate out of a wastewater stream. This very often gives us a... 

Multiplying by a simplex composed of densities results in the Archimedes number ... 

The Archimedes number contains parameters that characterize the properties of the heterogeneous system and the criterion establishing the type of settling. The criterion of separation essentially establishes the separating capacity of a sedimentation machine. The product of these criteria is ... 

Figure for Question 2. Plot of Reynolds number, and settling number (Lyashenko number) versus Archimedes number. Use this plot for question 3. It also useful for your own design problems. 

Determine the settling velocity of spherical quartz particles in water (d = 0.9 mm) using the dimensionless plot of the Lyachshenko and Reynolds numbers versus the Archimedes number in the figure above. The Lyashenko number is the same as the dimensionless settling number. The specific weight of the quartz is 2650 kg/m and the temperature of the water is 20° C. 

This ratio represents an average between similar ratios for the laminar and turbulent regimes. In the most general case, u, = f(D, Pp, p, /r, r, w), and hence we may ignore whether the particle displacement is laminar, turbulent or within the transition regime. This enables us to apply the dimensionless Archimedes number (recall the derivation back in Chapter 5) ... 

A plot of the Archimedes number versus Reynolds number is provided in Figure 12. 

The criteria K is similar to the Archimedes number introduced in 19.30 liy Baturin and Shcpelev to characterize air jets influenced by buoyancy, or to ihe Richardson criteria used in meteorology to characterize rhe ratio of the mrbu-lence suppression by rhe buoyancy forces over the turbulence generation by the Reynolds tension, In the case of displacement ventilation, the Richardson criteria can be defined by rhe relationship -... 

Characteristics of the air jet in the room might be influenced by reverse flows, created by the jet entraining the ambient air. This air jet is called a confined jet. If the temperature of the supplied air is equal to the temperature of the ambient room air, the jet is an isothermal jet. A jet with an initial temperature different from the temperature of the ambient air is called a nonisother-mal jet. The air temperature differential between supplied and ambient room air generates buoyancy forces in the jet, affecting the trajectory of the jet, the location at which the jet attaches and separates from the ceiling/floor, and the throw of the jet. The significance of these effects depends on the relative strength of the thermal buoyancy and inertial forces (characterized by the Archimedes number). 

Using the relation between the Froude number and the Archimedes number, Atq = 1/F, the length of the linear jet zone, x, where the buoyancy forces are negligibly small can be calculated as follows ... 

To characterize the relationship between the buoyancy forces and momentum flux in different cross-sections of a nonisothermal jet at some distance x, Grimitlyn proposed a local Archimedes number ... 

Introduction of the local Archimedes criterion helped to clarify nonisothermal jet design procedure. Grimitlyn suggested critical local Archimedes number values, Ar , below which a jet can be considered unaffected by buoyancy forces (moderate nonisothermal jet) Ar, 0.1 for a compact jet, Ar, < 0.15 for a linear jet. 

The above limitation on the local Archimedes number results in the following equation for maximum temperature difference of supplied air ... 

Studies of nonisothermal mam stream and horizontal directing jet mterac-non were conducted to evaluate the maximum heat load that can be eltectively supplied by such HVAC systems. To summarize experimental data both in free and confined conditions, it was suggested that the above limiting condition is achieved when the current Archimedes number Ar ratio of rhe buoyancy forces over ineiTia forces along the resulting jet axis) does not exceed s[Pg.502]

The current Archimedes number for the resulting jet grows along the jet as it does in any nonisothermal jet. However, the consequent momentum additions by directing jets increases the inertial forces in rhe resulting jet and thus at a certain cros.s-section the current Archimedes number falls. The number of directing jets after which Ar reaches the peak can be calculated using... 

Malmstrom, T.-G. 1996.. Archimedes number and jet similarity. In Roomvent 96 Proceedings of the Sth International Conference on Air Distribution in Rooms, vol. 1, July. Yokohama, Japan. 

FIGURE a.sa Penetration depth versus Archimedes number and discharge angle. 

The theoretical analysis could also be valid for nonisothermal jets assuming that the buoyancy is negligible. Grimitlyn, as reported by Hagstrom, suggests a local Archimedes number defined as ... 

He indicates that the buoyancy is negligible for the velocity field for an Archimedes number less than a critical value equal to Ar, = 0.15. 

Ar, Re, Pr, Sc, and v] are called the Archimedes number, Reynolds number, Prandtl number, Schmidt number, and the settling velocity ratio, respectively. 

The Archimedes number may be considered as a ratio of thermal buoyancy force to inertial force, while the Reynolds number may be looked upon... 

Fully developed nonisothermal flow may also be similar at different Reynolds numbers, Prandtl numbers, and Schmidt numbers. The Archimedes number will, on the other hand, always be an important parameter. Figure 12.30 shows a number of model experiments performed in three geometrically identical models with the heights 0.53 m, 1.60 m, and 4.75 m." Sixteen experiments carried out in the rotxms at different Archimedes numbers and Reynolds numbers show that the general flow pattern (jet trajectory of a cold jet from a circular opening in the wall) is a function of the Archimedes number but independent of the Reynolds number. The characteristic length and velocity in Fig. 12.30 are defined as = 4WH/ 2W + IH) and u = where W is... 

Model experiments where free convection is the important part of the flow are expressed by the Grashof number instead of the Archimedes number, as in Eq. (12.61). The general conditions for scale-model experiments are the use of identical Grashof number, Gr, Prandtl number, Pr, and Schmidt number,, Sc, in the governing equations for the room and in the model. 

It is useful to take similarity principles and dimensionless numbers into consideration when planning experiments. Experiments may involve different levels of velocities and temperature differences. It is important to select values that give a large variation of Archimedes number (12,56) to obtain a high possibility of large physical effects in the measurements. 

The assumption of a self-similar flow (Reynolds number-independent flow) simplifies full-scale experiments and is also a useful tool in the formulation of simple measuring procedures. This section will show two examples of self-similar flow where the Archimedes number is the only important parameter. 

Figure 12.41 shows the results of three experiments with a similar Archimedes number and different Reynolds numbers. The figure shows vertical temperature profiles in a room ventilated by displacement ventilation. The dimensionless profiles are similar within the flow rates shown in the figure, although the profile may involve areas with a low turbulence level in the middle of the room. A test of this type could indicate that further experiments can be performed independently of the Reynolds numbers. 

II FIGURE 12.41 Vertical temperature profile in the room for three different experiments with identical Archimedes number. 

FIGURE 12,42 Temperature effectiveness versus airflow rate q and Archimedes number ATj/uj... 

Figure 12.42ft shows the measurements given as a function of the Archimedes number At ATqIuq. This figure is more informative than Fig. 12.42(3. The figure shows that the temperature effectiveness is a function of the Archimedes number. An identical level of j for the two diffusers A and B at the same Archimedes number implies that the temperature effectiveness is rather independent of the diffuser design and the local induction close to the diffuser. The effectiveness is probably more dependent on other parameters that are constant in the experiments, such as heat source and heat source location.

The ratio Ap/p can be replaced by AT/T. Ar relates the influence of velocity and temperature of a jet when discharged into an environment of a different temperature. In some instances the Froude number, Galileo number, or Grashof number may replace the Archimedes number. 

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