Volumetric coefficient

Volumetric Mass-Transfer Coefficients and Kia Experimental determinations of the individual mass-transfer coefficients /cg and /cl and of the effective interfacial area a involve the use of extremely difficult techniques, and therefore such data are not plentiful. More often, column experimental data are reported in terms of overall volumetric coefficients, which normally are defined as follows ... 

Employing wood chips, Cowan s drying studies indicated that the volumetric heat-transfer coefficient obtainable in a spouted bed is at least twice that in a direct-heat rotaiy diyer. By using 20- to 30-mesh Ottawa sand, fluidized and spouted beds were compared. The volumetric coefficients in the fluid bed were 4 times those obtained in a spouted bed. Mathur dried wheat continuously in a 12-in-diameter spouted bed, followed by a 9-in-diameter spouted-bed cooler. A diy-ing rate of roughly 100 Ib/h of water was obtained by using 450 K inlet air. Six hundred pounds per hour of wheat was reduced from 16 to 26 percent to 4 percent moisture. Evaporation occurred also in the cooler by using sensible heat present in the wheat. The maximum diy-ing-bed temperature was 118°F, and the overall thermal efficiency of the system was roughly 65 percent. Some aspec ts of the spouted-bed technique are covered by patent (U.S. Patent 2,786,280). 

Note that the product of the mass-transfer coefficient and the interfacial area is a volumetric coefficient and obviates the need for a value of the interfacial area. While areas for mass transfer on plates have been measured, the experimental contacting equipment cuffered significantly from that used for commercial distillation or gas absorption, and the reported areas are considered unreliable for design purposes. 

The volumetric coefficient h a from the combination of Eqs. (14-178) and (14-179) is useful in defining the effect of variable changes but is limited in value because of its dependence on D. The prodiicl of area and coefficient obtained from a given mass of hqiiid is proportional to (1/D ) for small diameters. The prime problem is that droplet-size estimating procedures are often no better than 50 percent. A secondary problem is that there is no that truly characterizes either the motion or transfer process for the whole spectrum of particle sizes present. See Eqs. (14-193) and (14-194). 

The experimental data usually give the specific heat at constant pressure cP. Theories usually refer to the specific heat at constant volume cv. The specific heat cP is greater than cv by a factor (1 + jgT), where f5 is the volumetric coefficient of thermal expansion and yG is the so-called Griineisen parameter ... 

In many types of equipment used for gas—liquid reactions, the interfacial area available for mass transfer cannot be determined. The experimentally determined rates of mass transfer are therefore usually reported in terms of transfer coefficients based on unit volume of apparatus rather than on unit interfacial area. These volumetric coefficients are denoted by Kia, Kia, k fi and k[a where a is the interfacial area per unit volume of the equipment. 

Here u and v are the x and y components of the velocity, x is the distance up the plate, y is the distance to the plate, T is the temperature, T is the ambient temperature, g is the acceleration due to gravity, and / is the volumetric coefficient of expansion. It has already been argued that convective momentum and energy transport dominate diffusive processes in the flow direction. 

The mass transfer coefficients considered so far - namely, kQ,kj, KQ,andKj - are defined with respect to known interfacial areas. However, the interfacial areas in equipment such as the packed column and bubble column are indefinite, and vary with operating conditions such as fluid velocities. It is for this reason that the volumetric coefficients defined with respect to the unit volume of the equipment are used, or more strictly, the unit packed volume in the packed column or the unit volume of liquid containing bubbles in the bubble column. Corresponding to /cg, Kq, and we define k a, k, a, K, /i, and K a, all of which have units of (kmol h m )/(kmol m ) - that is, (h ). Although the volumetric coefficients are often regarded as single coefficients, it is more reasonable to consider a separately from the Ar-terms, because the effective interfacial area per unit packed volume or unit volume of liquid-gas mixture a (m m ) varies not only with operating conditions such as fluid velocities but also with the types of operation, such as physical absorption, chemical absorption, and vaporization. 

Thus, the required packed height Z can be calculated using Equation 6.38 with the given values of r and the volumetric coefficient Kq a oi a. 

Yeast cells are cultivated under aerobic conditions at 30 C in an aerated stirred-tank fermentor where air (21 vol% of oxygen) is supplied at a superficial gas velocity of 50 mh . The overall volumetric coefficient for oxygen transfer based on the liquid phase concentration is 80 h T... 

A 30 cm-diameter bubble column containing water (clear liquid height 2 m) is aerated at a flow rate of 10 m h . Estimate the volumetric coefficient of oxygen transfer and the average bubble diameter. The values of water viscosity = 0.001 kg m s , density p = 1000 kgm , and surface tension cr = 75 dyne cm" can be used. The oxygen diffusivity in water is 2.10 X 10 ... 

The packed density of the bed, the void fraction of the particle bed, and the density ofthe feed solution are 386 kg m , 0.5, and 1000 kg m , respectively. The averaged overall volumetric coefficient of mass transfer is 9.2 h , and a constant pattern of the adsorption zone can be assumed in this case. 

Thus, when deahng with gas transfer in aerobic fermentors, it is important to consider only the resistance at the gas-liquid interface, usually at the surface of gas bubbles. As the solubihty of oxygen in water is relatively low (cf. Section 6.2 and Table 6.1), we can neglect the gas-phase resistance when dealing with oxygen absorption into the aqueous media, and consider only the liquid film mass transfer coefficient Aj and the volumetric coefficient k a, which are practically equal to and K a, respectively. Although carbon dioxide is considerably more soluble in water than oxygen, we can also consider that the liquid film resistance will control the rate of carbon dioxide desorption from the aqueous media. 

The volumetric coefficient A a for oxygen absorption into oil-in-water emulsions is of interest in connection with fermentation using hydrocarbon substrates. Experimental results [7] have shown that such emulsions can be categorized... 

Estimate the liquid-phase volumetric coefficient of oxygen transfer for a stirred-tank fermentor with a diameter of 1.8 m, containing a viscous non-Newtonian broth, with consistency index K = 0.39, flow behavior index n = 0.74,... 

Estimate the liquid-phase volumetric coefficient of oxygen transfer for a bubble column fermentor, 0.8 m in diameter 9.0 m in height (clear liquid), containing the same liquid as in Problem 12.2. The superficial gas velocity is 150 m h . ... 

Estimate the residual capacity, when the interstitial velocity is doubled. It can be assumed that the averaged overall volumetric coefficient increases with the interstitial velocity to the power of 0.2. 

In most types of mass-transfer equipment, the interfacial area, a, that is effective for mass transfer cannot be determined accurately. For this reason, it is customary to report experimentally observed rates of transfer in terms of mass-transfer coefficients based on a unit volume of the apparatus, rather than on a unit of interfacial area. Calculation of the overall coefficients from the individual volumetric coefficients is made practically, for example, by means of the equations ... 

The transition at 19° C involves an expansion of 0.0058 cm3/g (Clark and Muus). Sincethe transition temperatureincreaseswith pressure by about 0.013° C per atmosphere (Beecroft and Swenson), the latent heat is about 3.2 cal/g. These values are for the crystal and would be reduced in proportion to the crystalline content. The transition at 30° C is only about one-tenth as large. The over-all increase in entropy at these transitions is about 0.0108 cal deg-1g-1. The portion due to the increase in volume is (a// ) A V, where a is the volumetric coefficient of thermal expansion and / is the compressibility. Since the compressibility of the crystal is not known, this quantity is somewhat uncertain. Using the average of the values of a (Quinn, Roberts, and Work) and p (Weir, 1951) for the whole polymer above and below the transitions, it appears that (a/P)A V is about 0.0041 cal deg 1g 1. The entropy of the transition corrected to constant volume is, therefore, about 0.0067 cal deg g-1. 

A volumetric coefficient of expansion is based on the following formula ... 

To express the change in fluid density as a result of the fluid temperature, we can define the volumetric coefficient of thermal... 

Determination of the Volumetric Coefficient kLaL Using an Instantaneous, Irreversible chemical reaction For an irreversible instantaneous reaction,... 

Natural convection occurs when a fluid is in contact with a solid surface of different temperature. Temperature differences create the density gradients that drive natural or free convection. In addition to the Nusselt number mentioned above, the key dimensionless parameters for natural convection include the Rayleigh number Ra = p AT gx3/ va and the Prandtl number Pr = v/a. The properties appearing in Ra and Pr include the volumetric coefficient of expansion p (K-1) the difference AT between the surface (Ts) and free stream (Te) temperatures (K or °C) the acceleration of gravity g(m/s2) a characteristic dimension x of the surface (m) the kinematic viscosity v(m2/s) and the thermal diffusivity a(m2/s). The volumetric coefficient of expansion for an ideal gas is p = 1/T, where T is absolute temperature. For a given geometry,... 

---