Influence of Depth and Water Pressure on the Transfer Parameters

To quantify the influence of atmospheric plus water pressure on the transfer of oxygen, the pressure situation within the tank has to be thoroughly defined and quantified. To this end, the hydraulic pressure (m water column, WC) within the tank at depth h (see Figure 4.1) is converted into the standard unit P (Pa; N/m²) and then related to the atmospheric standard pressure of P_a = 101 325 Pa = 101.325 kPa.

A bubble at depth h is exposed to an additional water pressure of ∆P (m WC) = (H_S – h), or ∆P (Pa) = 9,810⋅(H_S– h), and hence, to a total pressure of P_a + ∆P. Relating this total pressure to the atmospheric standard pressure of P_a yields the relative pressure π.

(4.2)

the conversion factor, z, being z = 9,810/101,325 = 0.0968 ≈ 0.1.

The rounded value of 0.1 reflects the rule of thumb, that 10 m of water column will double the standard pressure. In the following, the relative pressure π is the relevant pressure parameter for quantifying the influence of tank depth on oxygen transfer via the influenced parameters kL, a, and cs. These parameters, together with the water volume of the aeration tank, V, define the standard oxygen transfer rate SOTR (kg/h).

The following definitions apply.

V water volume of aeration tank [m³] A total interfacial area [m²] a specific interfacial area = A/V [m^–1] A_at bottom area of aeration tank [m²]

k_L liquid film coefficient [m/h] where k_L·a is similar to K_La₂₀ in Equation (2.42)

c_s oxygen saturation concentration [mg/l] similar to in Equation (2.42) G_s standard airflow rate [m_N³/h at STP]

As pointed out when characterizing the process of oxygen transfer in deep tanks, the first three parameters of Equation (4.3), k_L, a, and c_s, depend on water pressure and c_s, additionally on oxygen reduction within the bubble air. Since these effects are normally neglected, this equation is actually applicable for very shallow tanks (H → 0), only and should be written for these conditions with a subscript of zero.

(4.4) This approach holds also for the standard oxygen transfer efficiency SOTE (–, %) and its specific value SOTE_s (m^–1, %/m), based on the fraction or percent of oxygen absorbed per meter water depth, H. It differs slightly from per meter of bubble rise H_S, although generally reported in this latter way. Both SOTE parameters will be extensively applied in modeling. With an oxygen content of ambient air of 300 g/m_N³, the result is similar to Equation (2.51).

(4.5) More accurately for shallow tanks (H → 0), the SOTE0 is defined as follows

(4.6) Similarly, the specific oxygen transfer efficiency SOTE_s can be formulated. It has to be noticed, however, that SOTE_s is reduced during the bubble rise due to pressure changes and oxygen reduction in the air, as will be shown quantitatively later. Hence, the average value SOTE_sa over the full bubble rise is calculated by dividing SOTE by the water depth H (not by the depth of diffuser submergence H_S).

(4.7) mass of O transferred

mass of O supplied

2

average mass of O transferred

mass of O supplied water depth H of aeration tank

2 2

300 0 3.

Again, this equation can be expressed for very shallow tanks (H → 0).

(4.8)

The process of oxygen transfer in deep tanks is modeled by expressing the parameters varying with depth (kL, a, and cs) as functions of their value for shallow tanks (kL₀, a₀, and cs₀). These functions are derived based on the physical laws governing the depths dependent processes as characterized in Section 4.2.1.

The pressure influence on the bubble size is modeled by the universal gas law (P⋅V = m⋅R⋅T), to which the relative pressure π (Equation 4.2) is applied (π⋅V = m⋅R⋅T/Pa = constant). Hence, the product of the relative pressure π and the bubble volume VB is constant, and the bubble volume VB₀ is reduced inversely proportional to the relative pressure π as defined in Equation 4.2.

(4.9)

Assuming geometrically similar deformation of the bubble by compression, the bubble diameter d_B0 is changed by the 1/3-power of the volume change.

(4.10)

Finally, the total area, A, and the specific area, a, are related by the second power of the diameter. This relationship leads to the dependence of the interfacial area on pressure and on depth HS – h.

(4.11)

Next to the area parameters, the liquid film coefficient, k_L, is influenced by the pressure-dependent bubble diameter, d_B, as was shown by Mortarjemi and Jameson (1978) and Pasveer (1955). Their findings are plotted in Figure 4.2. Already in 1935, Higbie proposed the penetration theory for quantifying this interrelationship as given in Equation 2.21.

Here, vB (m/h) is the rise or slip velocity of the bubble with respect to water. As follows from Figure 4.2, this equation is valid only for bubbles greater than 2 mm.

Generally, fine bubbles have an equivalent diameter of some 2 mm, so that the Higbie theory cannot yield correct results for compressed fine bubbles of smaller than 2 mm.

By combining the results of Mortarjemi, Jameson, and Pasveer [kL = f(dB)] with Equation 4.10 [dB = f(dB0, HS-h)], an empirical relation is developed relating the liquid film coefficient to depth.

(4.13) This function proceeds from a liquid film coefficient kL₀= 0.48 mm/s, typical for an equivalent bubble diameter of dB = 3.0 mm. Figure 4.2 shows that the kL data are fitted very well by Equation 4.13. It should be noted, however, that a bubble diameter of 2 mm is reduced to only 1.55 mm in a 12 m deep tank. Hence, the liquid film coefficient is influenced only slightly under practical conditions.

The last parameter influenced by pressure is the oxygen saturation concentration.

This effect is quantified by multiplication of cs₀, the standard saturation concentration without water pressure, with the relative pressure π.

(4.14) FIGURE 4.2 Liquid film coefficient as a function of the equivalent bubble diameter after Mortarjemi and Pasveer, Higbie theory and empirical function. (From Pöpel and Wagner, 1994, Water Science and Technology, 30, 4, 71–80. With permission of the publisher, Perga-mon Press, and the copyright holders, IAWQ.)

k_L =k_Lo⋅^exp

[

(

) ]

c_s =c_so⋅ =^π c_so⋅ + ⋅

[

(

) ]

In this case, however, the parameter c_s0 is also affected by the oxygen transfer during bubble rise, decreasing the oxygen partial pressure in the bubble air. This influence is quantified via the standard oxygen transfer efficiency SOTE(h) during the bubble rise from h = 0 to h = h. In Figure 4.1, for instance, the SOTE-values for h = h₁ and h = h₂ are depicted for the purpose of illustration; quantities, which are yet unknown. With SOTE(h), as standard oxygen transfer efficiency from the level of bubble release until depth h, the saturation concentration is decreased correspondingly.

(4.15) By combining Equations 4.14 and 4.15, the final expression for the saturation concentration at any height above the diffusers, h, is obtained.

(4.16)

In summary, the influence of depth on the three basic transfer parameters, a, k_L, and c_s, can be expressed by simple mathematical functions found in Equations 4.11, 4.13, and 4.16, respectively. They include the respective values without water pres-sure, a₀, k_L0, and c_s0, and the standard oxygen transfer efficiency during bubble rise from the release level until h.