Liquid, cross section

Figure 1. Flotation of an elongated cylinder (fiber) by surface forces. Density of fiber greater than density of liquid. Cross sectional view of fiber at equilibrium in the air-liquid interface...

Liquid surge volume (VL) is calculated u g Eq. (4c). The required liquid cross sectional area (AL, in fl2) is calculated as follows, by selecting a vessel length (L) ... 

Vessel total cross sectional area (AT) is the sum of the vi or and liquid cross sectional area ... 

AH vessel heavy liquid cross sectional area, fr2. 

One more application area is composite materials where one wants to investigate the 3D structure and/or reaction to external influences. Fig.3a shows a shadow image of a block of composite material. It consists of an epoxy matrix with glass fibers. The reconstructed cross-sections, shown in Fig.3b, clearly show the fiber displacement inside the matrix. The sample can be loaded in situ to investigate the reaction of matrix and fibers to external strain. Also absorption and transmission by liquids can be visualized directly in three-dimensions. This method has been applied to the study of oil absorption in plastic granules and water collection inside artificial plant grounds. 

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

Perhaps the best discussions of the experimental aspects of the capillary rise method are still those given by Richards and Carver [20] and Harkins and Brown [21]. For the most accurate work, it is necessary that the liquid wet the wall of the capillary so that there be no uncertainty as to the contact angle. Because of its transparency and because it is wet by most liquids, a glass capillary is most commonly used. The glass must be very clean, and even so it is wise to use a receding meniscus. The capillary must be accurately vertical, of accurately known and uniform radius, and should not deviate from circularity in cross section by more than a few percent. 

A jet emerging from a nonciicular orifice is mechanically unstable, not only with respect to the eventual breakup into droplets discussed in Section II-3, but, more immediately, also with respect to the initial cross section not being circular. Oscillations develop in the Jet since the momentum of the liquid carries it past the desired circular cross section. This is illustrated in Fig. 11-20. 

The oscillating jet method is not suitable for the study of liquid-air interfaces whose ages are in the range of tenths of a second, and an alternative method is based on the dependence of the shape of a falling column of liquid on its surface tension. Since the hydrostatic head, and hence the linear velocity, increases with h, the distance away from the nozzle, the cross-sectional area of the column must correspondingly decrease as a material balance requirement. The effect of surface tension is to oppose this shrinkage in cross section. The method is discussed in Refs. 110 and 111. A related method makes use of a falling sheet of liquid [112]. 

Derive, from simple considerations, the capillary rise between two parallel plates of infinite length inclined at an angle of d to each other, and meeting at the liquid surface, as illustrated in Fig. 11-23. Assume zero contact angle and a circular cross section for the meniscus. Remember that the area of the liquid surface changes with its position. 

According to the simple formula, the maximum bubble pressure is given by f max = 27/r where r is the radius of the circular cross-section tube, and P has been corrected for the hydrostatic head due to the depth of immersion of the tube. Using the appropriate table, show what maximum radius tube may be used if 7 computed by the simple formula is not to be more than 5% in error. Assume a liquid of 7 = 25 dyn/cm and density 0.98 g/cm. ... 

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

Slurry or slip casting provides a relatively inexpensive way to fabricate unifonn-thickness, thin-wall, or large cross section shapes [4o, 44, 45, 46, 42 aiid 48]. For slip casting, a slurry is first poured into a porous mould. Capillary suction then draws the liquid from the slurry to fonn a higher solids content, close-packed, leather-hard cast on the inner surface of the mould. In a fixed time, a given wall thickness is fonned, after which the excess slurry is drained. 

Since capillary tubing is involved in osmotic experiments, there are several points pertaining to this feature that should be noted. First, tubes that are carefully matched in diameter should be used so that no correction for surface tension effects need be considered. Next it should be appreciated that an equilibrium osmotic pressure can develop in a capillary tube with a minimum flow of solvent, and therefore the measured value of II applies to the solution as prepared. The pressure, of course, is independent of the cross-sectional area of the liquid column, but if too much solvent transfer were involved, then the effects of dilution would also have to be considered. Now let us examine the practical units that are used to express the concentration of solutions in these experiments. 

Equation (9.28) describes the velocity with which a cylindrical shell of liquid moves through a capillary under stationary-state conditions. This velocity times the cross-sectional area of the shell gives the incremental volume of liquid dV which is delivered from the capillary in an interval of time At. The total volume delivered in this interval AV is obtained by integrating this product over all values of r ... 

Tetrakis(cumylphenoxy)phthalocyanine (5) a PC derivative, having liquid—crystalline-like substituents (38—43) was studied because the cross-section area of the substituents is much larger than that of a normal alkyl chain, and therefore, the requirement of minimized free volume in the assembly may be easier to accomplish. 

S s, s Cross-sectional area S for minimum cross-sectional area between rows of tubes, flow normal to tubes 5,, for tube-to-baffle leakage area for one baffle for shell-to-baffle area for one baffle for area for flow through window S, g for gross window area S, for window area occupied by tubes Slope of rotary shell Specific gravity of fluid referred to liquid water m fft... 

For gradual changes in channel cross section and hquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x. 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Figure 6-32, taken from Govier and Aziz, schematically indicates four flow pattern regions superimposed on a plot of pressure gradient vs. mixture velocity = Vl -t- V5 = Qj + ( s)/A where and Vs are the superficial liquid and solid velocities, Qi, and ( 5 are liquid and solid volumetric flow rates, and A is the pipe cross-sectional area. is the transition velocity above which a bed exists in the bottom of the pipe, part of which is stationary and part of which moves by saltation, with the upper particles tumbling and bouncing over one another, often with Formation of dunes. With a broad particle-size distribution, the finer particles may be fully suspended. Near V 4, the pressure gra-... 

Amplitude of controlled variable Output amplitude limits Cross sectional area of valve Cross sectional area of tank Controller output bias Bottoms flow rate Limit on control Controlled variable Concentration of A Discharge coefficient Inlet concentration Limit on control move Specific heat of liquid Integration constant Heat capacity of reactants Valve flow coefficient Distillate flow rate Limit on output Decoupler transfer function Error... 

The capiUaiy rise in a smaU vertical open tube of circular cross section dipping into a pool of liquid is given by... 

Volute casings take the form of a spiral increasing uniformly in cross-sectional area as the outlet is approached. The volute efficiently converts the velocity energy imparted to the liquid by the impeller into pressure energy. 

Packed columns must be provided with good initial distribution of liquid across the column cross section and redistribution of liquid at various height intei vals that decrease with increasing column diameter. A wide variety of distributors and redistributors are available. Packed columns should be considered when ... 

Information on the liquid- and gas-handling capacity of the contacting device chosen for the pariicular separation problem. Such information includes pressure drop charac teristics of the device, in order that an optimum balance between capaital cost (column cross section) and energy requirements might be achieved. Capacity and pressure drop charac teristics of the available devices are covered later in this Sec. 14. 

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