Volumetric flux density

A general expression for average void fraction in an adiabatic two-phase flow or in saturated boiling was suggested by Zuber and Findlay (1965). Defining the local superficial velocities, VSG(= Qc/A) and VSL (= QJA), as volumetric flux densities, they assumed the velocity and void distributions to be... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Here v,. is the volumetric flux density of the fluid in m /(m s) which is equal to the superficial velocity w of the fluid above the bed in the particle-free column. The pressure drop of the fluid in the bed is given by... 

Soil has a substantial volumetric heat capacity, but it does not have a high thermal conductivity coefficient, Ks°l1. Heat is therefore not readily conducted in soil, where the heat flux density by conduction is... 

This volumetric water flux density directed upward at the soil surface equals (1 x 10-8 m3 m-2 s 1)(l mol/18 x 10-6 m3), or 0.6 x 10-3 mol mT s-1 (= 0.6 mmol m-2 s-1). When discussing water vapor movement in the previous section, we indicated that Jm> emanating from a moist shaded soil is usually 0.2 to 1.0 mmol m-2 s-1, so our calculated flux density is consistent with the range of measured values. The calculation also indicates that a fairly large gradient in hydrostatic pressure can exist near the soil surface. 

A steady state is often not achieved in soils, so a steady rate is sometimes used, where the rate of volumetric water depletion is constant, leading to relations considerably more complicated than are Equations 9.8 and 9.9.3 We note that the equations describing water flow in a soil for the onedimensional case (Eq. 9.7), for cylindrical symmetry (Eq. 9.8), and for spherical symmetry (Eq. 9.9) all indicate that the volume flux density of water is proportional to LSGl1 times a difference in hydrostatic pressure (see Table 9-2). 

The scalar product of the vectors m and w yields a volumetric woik or volumetric energy which has the dimension of a pressure. Let us assume a horizontal flow of an ideal ( 7 = 0) fluid. According to the law of conservation of momentum, the differential of the momentum flux density is equal to the differential of the pressure ... 

Figure 26 (A) A 25 L prototype of the solar volumetrically illuminated photobioreactor DiCoFluV (Cornet, 2010). (B) EDStar geometric structure both the reactor 1Z) and the 979 light-diffusing optical fibers [T) are cylinders 1 m high the reactor s diameter is 16.5 cm, the distance between two fiber axes is djr = 4.8 mm, and the fiber radius is ryr = 1.2 mm.Tl and T are diffuse-reflective with uniform reflectivity p and pT, respectively.. is Lambertian emitting with the uniform surface flux density qn,v (Q Two-dimensional hexagonal lattice fiber arrangement an optical-path example in the culture medium V.

This relation can be used, for example, to calculate volumetric productivity of a photobioreactor at different biomass concentrations when assuming a constant incident photon flux density Iph(O)- In Fig. 26 results are shown for a photobioreactor with an optical depth of 3 cm. The shape of this relation will be discussed later after having introduced a similar approach based on the model of Jassby and Platt. 

The outcome shows that at the point of maximal volumetric productivity the photon flux density at the backside of the photobioreactor /ph(d) is equal to the compensation point of photosynthesis fph,c which was derived earlier ... 

Also when employing more realistic and more complex models, such as Jassby and Platt with spectral resolution the same rule applies volumetric productivity of a photobioreactor is maximal when the biomass density is such that the photon flux density at the darkest zone of the reactor is equal to the compensation point of photoautotrophic growth. In practice it has also been confirmed that photobioreactors can be operated at maximal productivity when the biomass concentration is maintained at those levels that almost aU light is absorbed within the microalgal culture but at the back still a minimal amount of light is left to compensate for maintenance purposes (Barbera et al, 2015 Takache et al, 2012 Zijffers et al, 2010). 

Volumetric Productivity and Biomass Yield on Photons vs Photon Flux Density... 

Clearly, no simple analytical solution of the differential equation derived from the biomass balance can be obtained. Adopting numerical methods based on this differential equation the increase in biomass in time in a batch culture can stiU be calculated. In Fig. 28 batch growth is simulated based on the model ofjassby and Platt including spectral resolution. This simulation is based on C. sorokiniana grown under a constant photon flux density of 1.5 X 10 molph m s. The biomass concentration increases in time, but the rate of increase slows down. This is more apparent in the yield of biomass on photons which is maximal after about 24 h after which it decreases again. At 24 h the biomass concentration is optimal in the sense that at this point in time the volumetric productivity, as well as the biomass yield on light, is maximal. 

That is, the larger and/or heavier particles are assumed to move through a pseudofluid having the properties, and flowing with the superficial velocity (or volumetric flux), of the combined pure liquid and smaller and/or lighter particles. The free settling terminal velocity, C/qi, is thus based on a pseudofluid density of... 

In Eqs. (6) and (7) e represents the internal energy per unit mas, q the heat flux vector due to molecular transport, Sh the volumetric heat production rate, ta, the mass fraction of species i, Ji the mass flux vector of species i due to molecular transport, and 5, the net production rate of species i per unit volume. In many chemical engineering applications the viscous dissipation term (—t Vm) appearing in Eq. (6) can safely be neglected. For closure of the above set of equations, an equation of state for the density p and constitutive equations for the viscous stress tensor r, the heat flux vector q, and the mass flux vector 7, are required. In the absence of detailed knowledge on the true rheology of the fluid, Newtonian behavior is often assumed. Thus, for t the following expression is used ... 

Denitrification rates in the other two main ODZs, the ETNP and ETSP appear to be about 25 Tg a each Codispoti et al. (2002). In the ETNP denitrification was estimated by the geostrophic flux of nitrate deficit out of the ODZ area, with nitrate deficit determined from nutrient-density relationships (Codispoti, 1973 Codispoti and Richards, 1976). Volumetric estimates of denitrification rate based on ETS activity are in concert with this rate (Codispoti and Richards, 1973 Devol, 1975). The ETSP the denitrification rate is based on measurements of ETS activity (Codispoti and Packard, 1980). As with the Arabian Sea these estimates are again for canonical denitrification. It is likely that processes identified in the Arabian Sea are also occurring in the ETNP and ETSP. Thus, the rate of 25 Tg a might be raised to 50 Tg ar. Given a denitrification rate between 30 and 50 Tg a for each of the major ODZ s, global water-column denitrification would appear to between 90 and 150 Tg a. ... 

The conservation law for mass may be applied to a fluid device with one input and one output. For a fluid of constant density p and a uniform velocity v at each cross section A, the mass rate m, the volumetric rate q, and the mass flux G, are... 

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