Buoyancy

Either directly or indirectly, the concept of density plays an important role in a myriad of scientific operations construction of equipment, preparation of solutions, determination of volumes, accurate weighings, measuring buoyancy of objects, studying properties of gases, and so on. Density is defined as the mass per unit volume, or 

In scientific work, the densities of solids and liquids usually are expressed in grams per cubic centimeter or grams per milliliter, whereas the densities of gases usually are expressed in grams per liter. In engineering work, densities customarily are expressed in pounds per cubic foot. 

The simplest way to determine the density of a liquid is to weigh an empty vessel of known volume and then weigh it again when it is filled with the liquid. An approximate value may be determined with a simple graduated cylinder weighed on a triple-beam balance. Only a crude value can be obtained because the balance can be read only to the nearest 0.1 g and the cylinder only to the nearest 0.1 ml. 

Volumetric flask filled to the mark with a liquid. 

A clean dry 10-ml graduated cylinder weighs 37.6 g empty it weighs 53.2 g when filled to the 7.4 ml mark with an unknown liquid. Calculate the density of the liquid. 

Temperature uniformity involves improvement by movement of radiating triatomic gases as well as convection poc. (See also chap. 5 of reference 51.) Concepts of this chapter will be facilitated by the following review of the laws of gas movement concerning buoyancy, velocity head, fluid friction between gases and solids, and flow induction. 

A column of hot air (flg. 7.1) weighs less than an equally tall column of cold air, which is shown dotted to form a U-tube manometer. The dotted column corresponds to the atmosphere outside a stack or chimney. The difference in weights of the columns creates a pressure difference (AP) known as draft (see glossary), expressed in inches or millimeters of water column on a manometer. The draft is proportional to the height of the gas column and to the difference in densities of the hot and cold gas columns. The densities of air and other gases depend on their pressures and temperatures, thus density, p = p/RT, where density is pounds per cubic foot (US) or kg/m (SI), T is absolute temperature rankine (US) or kelvin (SI), and P is a constant = 53.3 fp/pound mol °R for air (US), or 287 joules-kg-mol °K for air (SI). Densities are tabulated in references 51 and 52. 

The theoretical draft (lift, suction) of a tall column of hot gas, as in a furnace, vertical duct, or stack is  

AP wc = pressure difference wc between a cold air and a hot gas column hft = height in feet of the hot gas column 

Industrial Furnaces, Sixth Edition. W. Trinks, M. H. Mawhinney, R. A. Shannon, R. J. Reed and J. R. Garvey Copyright 2004 John Wiley Sons, Inc. 

This also applies to a body submerged in a fluid that is subject to any acceleration. For example, a solid particle of volume Vs submerged in a fluid within a centrifuge at a point r where the angular velocity is on is subjected to a net radial force equal to Ap on2rVs. Thus, the effect of buoyancy is to effectively reduce the density of the body by an amount equal to the density of the surrounding fluid. 

In an isothermal and isochemical body of water under hydrostatic conditions, there is no flow of water. The potential of the water is constant, at any point in the water, and assuming the density of the water is constant, the potential can be given by (Chapter 1), 

The net forces acting on a vinit mass of separate phase oil or gas completely immersed in water under isothermal and isochemical conditions, as given by Hubbert (1953), is  

These net forces acting on oil and gas can also be expressed in terms of gravity, and the net force acting on water (Hubbert, 1953)  

From Equations 4.5 and 4.6 it follows that since g is fixed, the direction and magnitude of the net force acting on a unit mass of oil or gas immersed in water under isothermal and isochemical conditions depend upon the density of the oil or gas and the direction and magnitude of the net force acting on the 

Under hydrostatic conditions E = 0 and therefore E and Eg will be directed vertically upwards under these circumstances and Equations 4.5 and 4.6 will reduce to 

We can calculate the force exerted, by static fluids on floating and immersed bodies by integrating the vertical component of thej pressure force over the entire surface of the body. This leads to a very simple generalization, called 

Archimedes principle which is much easier to apply than the integration used in the previous section. 

Consider the floating block of wood shown in Fig. 2.11. The block is at rest, so the sum of forces in any direction on it is zero. The only forces acting on it are the gravity force and the total pressure force around its entire surface these must be equal and opposite. The vertical component of the pressure force integrated around the entire surface of a floating or submerged body is called a buoyant force. The buoyant force over the entire surface is then given by 

The statements above were worked out for a block with the axis vertical. This was convenient, because the pressure on the vertical sides did not contribute to the buoyant force. However, the result is true for any kind of body because, as shown in Fig. 2.12, any shape at all can be visualized as made up of blocks. 

Example 2.12, A helium balloon is at the same pressure and temperature as the surrounding air (1 atm, 20 C) and has a diameter of 3 m. The weight of the plastic skin of the balloon is negligible. How much paylpad can the balloon lift  

The specific heat is given in calmol K , so the right-hand side of Eq. (5.1) is in the units of cal. The mechanical work on the left-hand side of Eq. (5.1) is given in J. The conversion factor / has then the physical dimension calJ . Note that formally the conversion factor must be chosen (or inserted) in Eq. (5.1) to validate the equal sign. 

Buoyancy plays a role during the derivation of the sedimentation equilibrium and also in the broader sense with the ultracentrifuge. Someone said once with respect to the buoyancy that the most difficult problems in physical chemistry are the hydro-dynamic problems. Now, first buoyancy has to do with hydrostatics rather than with hydrodynamics. 

Question 5.1. Place a toy ship into a bathtub. Does the water level rise now the toy ship or does sink Is the force of gravity shielded by the surrounding liquid, for instance similar as the magnetic field in nuclear magnetic resonance (shielding effect) In nuclear magnetic resonance, the field is not really shielded, but rather overlaid. 

Because buoyancy problems are often misunderstood, we want it to be concerned with the problem. In fact, buoyancy is associated with the coupling of two systems. 

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In a reservoir at initial conditions, an equilibrium exists between buoyancy forces and capillary forces. These forces determine the initial distribution of fluids, and hence the volumes of fluid in place. An understanding of the relationship between these forces is useful in calculating volumetries, and in explaining the difference between free water level (FWL) and oil-water contact (OWC) introduced in the last section. 

A well known example of capillary-buoyancy equilibrium is the experiment in which a number of glass tubes of varying diameter are placed into a tray of water. The water level rises up the tubes, reaching its highest point in the narrowest of the tubes. The same observation would be made if the fluids in the system were oil and water rather than air and water. 

As well as preventing liquid carry over in the gas phase, gas carry undef must also be prevented in the liquid phase. Gas bubbles entrained in the liquid phase must be given the opportunity (or residence time) to escape to the gas phase under buoyancy forces. 

Princen and co-workers have treated the more general case where w is too small or y too large to give a cylindrical profile [86] (see also Refs. 87 and 88). In such cases, however, a correction may be needed for buoyancy and Coriolis effects [89] it is best to work under conditions such that Eq. 11-35 applies. The method has been used successfully for the measurement of interfacial tensions of 0.001 dyn/cm or lower [90, 91]. 

This method is smiple but experimentally more cumbersome than the volumetric method and involves the use of a vacuum microbalance or beam balance [22], The solid is suspended from one ann of a balance and its increase in weight when adsorption occurs is measured directly. The dead space calculation is thereby avoided entirely but a buoyancy correction is required to obtain accurate data. Nowadays this method is rarely used. 

The supporting medium was water at 298 K (p = 0.99727), and the density of latex is 1.2049 g cm . The latex particles had an average radius of 2.12 x 10 mm hence, their effective mass corrected for buoyancy is their volume times the density difference Ap between latex and the supporting medium, water... 

The table which follows gives the values of k (buoyancy reduction factor), which is the correction necessary because of the buoyant effect of the air upon the object weighed the table is computed for air with the density of 0.0012 m is the weight in grams of the object when weighed in air weight of object reduced to in vacuo = m + m/1000. 

Density of object weighed Buoyancy reduction factor, k ... 

With gravimetric methods, the magnitude of the buoyancy correction should be assessed. Particular attention must be paid to the adsorbent temperature because of the unavoidable gap between the sample and the balance case (cf. Section 6.2). 

If the buoyancy correction is ignored, the pipet s volume is reported as... 

To ensure that S eas is determined accurately, we calibrate the equipment or instrument used to obtain the signal. Balances are calibrated using standard weights. When necessary, we can also correct for the buoyancy of air. Volumetric glassware can be calibrated by measuring the mass of water contained or delivered and using the density of water to calculate the true volume. Most instruments have calibration standards suggested by the manufacturer. 

Is the failure to correct for buoyancy a constant or proportional source of determinate error ... 

What is the minimum density of a substance necessary to keep the buoyancy correction to less than 0.01% when using brass calibration weights with a density of 8.40 g/cm ... 

Bunsen-Roscoe law Bunte salt Bunte salts Bunte s salt Bunyavirus Buoyancy... 

Buoyant Effect of Air. Weighing operations performed m vacuo are not affected by buoyancy forces. An object in air, however, is subject to a buoyancy force that is equal and opposite to the gravitational force on the mass of air the object displaces (10). If the equal arm balance of Figure 1 is in balance with a test weight of mass, in one pan, and material of mass, m, in the other, m = m if they have the same density. If the densities are different, then the buoyancy forces acting on each pan affect the result. Taking moments about the center pivot point gives... 

If a 1-kg stainless weight (m = 1, OOOg, = 8,000 kg/m ) is added to one pan of the balance in Figure 1, and material with a density of 1,000 kg/m is added to the other until equiHbrium is reached, the amount of the material needed is 1001.05 g, using equation 5. Thus, it takes 1001.05 g of this material to counterbalance 1,000 g of stainless steel, because of the buoyancy effects on the dissimilar volumes. 

Drop Diameter. In extraction equipment, drops are initially formed at distributor no22les in some types of plate column the drops are repeatedly formed at the perforations on each plate. Under such conditions, the diameter is determined primarily by the balance between interfacial forces and buoyancy forces at the orifice or perforation. For an ideal drop detaching as a hemisphere from a circular orifice of diameter and then becoming spherical ... 

Variable-Area Flow Meters. In variable-head flow meters, the pressure differential varies with flow rate across a constant restriction. In variable-area meters, the differential is maintained constant and the restriction area allowed to change in proportion to the flow rate. A variable-area meter is thus essentially a form of variable orifice. In its most common form, a variable-area meter consists of a tapered tube mounted vertically and containing a float that is free to move in the tube. When flow is introduced into the small diameter bottom end, the float rises to a point of dynamic equiHbrium at which the pressure differential across the float balances the weight of the float less its buoyancy. The shape and weight of the float, the relative diameters of tube and float, and the variation of the tube diameter with elevation all determine the performance characteristics of the meter for a specific set of fluid conditions. A ball float in a conical constant-taper glass tube is the most common design it is widely used in the measurement of low flow rates at essentially constant viscosity. The flow rate is normally deterrnined visually by float position relative to an etched scale on the side of the tube. Such a meter is simple and inexpensive but, with care in manufacture and caHbration, can provide rea dings accurate to within several percent of full-scale flow for either Hquid or gas. 

The basic concepts of a gas-fluidized bed are illustrated in Figure 1. Gas velocity in fluidized beds is normally expressed as a superficial velocity, U, the gas velocity through the vessel assuming that the vessel is empty. At a low gas velocity, the soHds do not move. This constitutes a packed bed. As the gas velocity is increased, the pressure drop increases until the drag plus the buoyancy forces on the particle overcome its weight and any interparticle forces. At this point, the bed is said to be minimally fluidized, and this gas velocity is termed the minimum fluidization velocity, The bed expands slightly at this condition, and the particles are free to move about (Fig. lb). As the velocity is increased further, bubbles can form. The soHds movement is more turbulent, and the bed expands to accommodate the volume of the bubbles. 

Equations 3 to 7 indicate the method by which terminal velocity may be calculated. Erom a hydrodynamic force balance that considers gravity, buoyancy, and drag, but neglects interparticle forces, the single particle terminal velocity is... 

Transport Disengaging Height. When the drag and buoyancy forces exerted by the gas on a particle exceed the gravitational and interparticle forces at the surface of the bed, particles ate thrown into the freeboard. The ejected particles can be coarser and more numerous than the saturation carrying capacity of the gas, and some coarse particles and clusters of fines particles fall back into the bed. Some particles also coUect near the wall and fall back into the fluidized bed. 

Buoyancy. The low density, closed-ceUed nature of many ceUular polymers coupled with their moisture resistance and low cost resulted in their immediate acceptance for buoyancy in boats and floating stmctures such as docks and buoys. Since each ceU in the foam is a separate flotation member, these materials caimot be destroyed by a single puncture. 

The combination of stmctural strength and flotation has stimulated the design of pleasure boats using a foamed-in-place polyurethane between thin skins of high tensUe strength (231). Other ceUular polymers that have been used in considerable quantities for buoyancy appHcations are those produced from polyethylene, poly(vinyl chloride), and certain types of mbber. The susceptibUity of polystyrene foams to attack by certain petroleum products that are likely to come in contact with boats led to the development of foams from copolymers of styrene and acrylonitrUe which are resistant to these materials... 

Separations. Foams have important uses in separations, both physical and chemical (51,52). These processes take advantage of several different properties of foams. The buoyancy and mechanical rigidity of foam is exploited to physically separate some materials. The large volume of vapor in a foam can be exploited to filter gases. The large surface area of a foam can also be exploited in the separation of chemicals with different surface activities. 

Fig. 5. Buoyancy level controller. Courtesy of Magnetrol International, Inc.

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