Let us refer to Section 2.1 and consider the case of copper sulphate crystals placed at the bottom of a glass bottle containing water. As the water was quiescent without any disturbance, the copper sulphate crystals took unusually long time to get dissolved in water. If, on the other hand, the contents of the bottle are vigorously stirred, the copper sulphate crystals will be completely dissolved within very short time may be within a minute. Another similar example may be taken from our daily life with which most of us are well acquainted. When we add a sugar cube in a cup of tea and leave it as such, the sugar cube takes very long time to get completely dissolved. If, on the other hand, the contents of the cup are vigorously stirred with a spoon, the sugar cube gets dissolved within a very short time, may be within a minute.
The above examples distinguish two distinct types of diffusion, molecular diffusion and eddy diffusion or convective mass transfer. In Sections 2.3 and 2.4, we have already discussed molecular diffusion in fluids. In this section, we shall briefly discuss eddy diffusion or convective mass transfer.
Eddy diffusion or convective mass transfer occurs in turbulent fluids and its rate is much faster than that of molecular diffusion because of high velocity of the medium that reduces the thickness of the laminar film at the phase boundary as well as due to random motion of small lumps of fluid which physically carry the solute from one position to another.
As in case of convective heat transfer, there may be two types of convective mass transfer (i) forced convection mass transfer and (ii) natural convection or free convection mass transfer. As the names imply, in forced convection mass transfer turbulence is created in the medium by external agencies such as pump, blower, stirrer, etc. In natural convection mass transfer, on the other hand, turbulence in the medium is generated mainly due to variation in local conditions such as variation in solute concentration in the solution which causes variation in local density. For instance, when a solid gets dissolved in a liquid, the liquid adjacent to the solid surface becomes saturated almost immediately and as a result, becomes heavier than the bulk of liquid. If this phenomenon takes place at the bottom of the container, then the heavier fluid remains at the bottom and natural convection will not take place. If, on the other hand, dissolution takes place at a certain height above the bottom of the container, then the heavier fluid being above the lighter fluid, natural convection starts. The influence of natural convection on mass transfer may be considerable, particularly in the absence of forced convection.
Natural convection arising from gravitation is also of importance in mass transfer studies. Natural
convection may also occur from centrifugal effects or an electrically conducting fluid exposed to magnetic field.
Turbulence is characterized by random motion of fluid particles which are irregular with respect to both direction and time. For a fluid flowing in the turbulent motion through a duct in the axial or x-direction, the time average of velocity may be ux, at any particular instant, the actual velocity will b e ux + u′ix, where u′ix is the fluctuating or deviating velocity. Values of u′ix will vary with time through a range of positive and negative values, the time average being zero. Since flow is in x-direction only, the time average of uz is zero, but at any instant the deviating velocity in the z-direction will be u′iz.
A turbulent fluid is assumed to consist of small lumps of fluid or eddies of a wide size range (Hinze 1959, Davies 1972). Large eddies which contain about 20% of turbulent kinetic energy, produce smaller eddies to which they transfer their energy. The medium eddies make the maximum contribution to the turbulent kinetic energy. The smallest eddies which ultimately dissipate their energy through viscosity are substantially regenerated by large eddies and a state of equilibrium is established. At this stage the energy distribution becomes independent of the conditions by which the turbulence was originally produced and depends only upon the rate at which energy is supplied and dissipated.
Kolmogoroff (1941) defined the velocity u′d and length ld of the small eddies in terms of the power input per unit mass of the fluid, P/m:
u′d = (2.65)
and ld = (2.66)
Eliminating o from Eqs. (2.65) and (2.66), we get (2.67)
The kinetic theory of gases provides sound theoretical basis for the study of molecular diffusion in gases and by analogy, fairly reliable equations have been proposed for molecular diffusion in liquids.
In case of eddy diffusion, no such theoretical relations could, however, be developed because of our limited knowledge about turbulence. Some promising mathematical expressions for turbulence have recently been developed from statistical principles, which may be of use in mass transfer calculations.
Due to the deviating velocity, an eddy from layer 1, having concentration cA moves to layer 2 as shown in Figure 2.9 and looses its identity. The next eddy, having a concentration ( cA + DcA) moves from layer 2 to layer 1 and brings with it an additional amount of solute DcA.
Figure 2.9 Diffusion in turbulent stream (Eddy diffusion).
The average concentration gradient between layer 1 and layer 2 is DcA/l, where l is Prandtl Mixing length given by the relation.
u′x = Dux = -l (2.68)
The average concentration gradient is proportional to the local gradient (-dcA/dz). By analogy with eddy viscosity, an eddy diffusivity ED may be defined as the ratio of the transfer of component A to the concentration gradient.
ED = (2.69)
where, K is the proportionality constant.
From Eq. (2.69),
JA(turb) = -ED (2.70)
Equation (2.70) is very similar to Eq. (2.9) for molecular diffusion.
The total flux of component A due to both molecular and eddy diffusion may therefore be expressed as
JA = -(DAB + ED) (2.71)
where, ED is the Eddy diffusivity, L2/θ.
DAB is constant for a particular system at fixed conditions of temperature and pressure but ED depends, in addition to temperature and pressure, on the local intensity of turbulence. DAB predominates within a narrow region near solid wall or phase boundary where the flow is laminar while ED predominates in the main turbulent core.
Heat flux and momentum flux in turbulent streams can also be determined by expressions similar to Eq. (2.71).
For heat flux: q = -(a + EH) (2.72)
and for momentum flux: x = -(o + EV) (2.73) where,
EH= eddy thermal diffusivity, L2/θ EV= eddy viscosity, L2/θ
ED, EH and EV are very large, may be several hundred times larger compared to DAB, a and o respectively.
EXAMPLE 2.16 (Velocity and length scale of small eddies in a turbulent stream): Water at 25 °C is flowing through a 30 mm id pipe at an average velocity of 2.5 m/s. Determine the velocity and the length scale of the small eddies in the universal range.
For water at 25°C, viscosity is 8.937 # 10-4 kg/m$s and density is 997 kg/m3.
The value of Fanning’s friction factor, f may be calculated from the relation f = 5.62 # 10-8 Re.
Solution:
d = 30 mm = 0.03 m u = 2.5 m/s
Re = = 83670
f = 5.62 # 10-8 # 83670 = 0.0047.
The pressure drop Dp is given by the relation Dp =
Considering 1 m length of the pipe
Dp = = 1952 N/m2.
Power spent = Dp # volumetric flow rate
= Dp #
= 3.45 N$m/s for 1 m pipe
Associated mass = = 0.704 kg
= = 4.90 N$m/kg$s From Eq. (2.65), we obtain
u′d = = 0.0458 m/s.
And from Eq. (2.66), we get
ld = = 1.958 # 10-5 m.