## The Simple Plug Flow Model

The situation is described in Fig. 5.8: all bubbles rise with the same velocity w, which is not easy to measure. Therefore, the average void velocity is used:

with Ar as the cross-sectional area of the reactor (bubble column). We can balance the gaseous oxygen in the total volume of the bubble column; and it is not necessary to determine the total gas volume, which would be nearly impossible in large open tanks.

This steady-state experiment can be carried out:

• with clean water and Na2SO3, resulting in an oxygen-free liquid system,

• with real wastewater with or without activated sludge.

Let us discuss the first case. For the steady-state oxygen gas balance, we write:

dx L

and considering Henry's law: c = H c*

as well as the boundary condition: x = 0 c = cin

the result for the concentration profile is: c K,a ln — =--= x (5.49)

Cin H w

Frequently, a mean O2 concentration of the bubbles c is calculated using:

cout which is introduced into the integral balance:

Instead of Eq. (5.50), we need to write for the case of a finite dissolved oxygen concentration (c' ( 0):

Equations (5.51) and (5.52) are approximations for two reasons:

• A totally mixed liquid system cannot be passed through by gas bubbles flowing with the same velocity and exhibit a detectable decrease in the oxygen concentration.

• A plug-style flow of bubbles cannot be realized precisely when using a swarm of bubbles.

In wastewater technology, this plug flow model is used only seldom. Instead of an exponential decrease in oxygen concentration inside the bubbles, a linear decrease is considered with an arithmetic mean value:

which is a very rough approximation for larger, actual differences (cin- cout)/cin (Zlokarnik 1979). Now, the oxygen transfer rate (OTR) can be calculated by using either:

5.4 Oxygen Transfer Rate, Energy Consumption and Efficiency in Large-scale Plants 1101 A value of further interest is the oxygen transfer efficiency (OTE):

Using Eq. (5.55), OTE can be obtained simply. We will now describe the mass transfer in more detail.

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