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Biofilm Bottom Surface

Streamwise Velocity (cm/s):

Thickness of Boundary Layer

Thickness of Boundary Layer

-1000 -800 -600 -400 -200 0 200 400 600 800 Distance, jim

Figure 15.8 Effect of the fluid velocity past a biofilm on the thickness of the boundary layer for external mass transfer. (From T. C. Zhang and P. L. Bishop, Experimental determination of the dissolved oxygen boundary layer and mass transfer resistance near the fluid-biofilm interface. Water Science and Technology 30(1 I ):47-5S, 1994. Copyright © Elsevier Science Ltd., reprinted with permission.)

bulk fluid. Here, the material being transported is oxygen, which is being used by the biofilm as it consumes the substrate. The numbered arrows in the figure show how the fluid velocity affects the thickness of the actual boundary layer, and thus illustrates the impact on L* or k,. Many relationships arc available for relating k, to the system characteristics. They are usually defined in terms of the Reynolds number (vpw.d/(xw) and the Schmidt number (jjl„/p„D„) and are discussed in texts covering mass transfer (for example, see Weber and DiGiano4), as well as elsewhere.s Examination of common relationships reveals that many predict that kt will increase with the square root of the fluid velocity. However, because flow situations in attached growth reactors are complex, it is usually necessary to determine experimentally how k, depends on fluid velocity, or some factor affecting it, like mixing intensity in an agitated vessel or speed of rotation of a disc in a quiescent fluid.

Mass transfer within the biofilm is normally characterized by Fick's law, which for free diffusion in an aqueous solution is:

dx in which D^ is a diffusivity and dSs/dx is a concentration gradient. It is obvious from the previous discussion, however, that a biofilm is more complex than the situation of free diffusion for which Fick's law was developed. Therefore, the approach generally taken by modelers is to retain Fick's law as the governing equation, but to replace the diffusivity with an effective diffusivity, Dc:

The effective diffusivity is usually smaller than the free diffusivity due to the près-

ence of the extracellular polymer surrounding the cells in the biofilm. However, some researchers have measured effective diffusivities that are greater than the corresponding free diffusivities, ' which is consistent with the presence of advection within the biofilm as discussed in Section 15.1. Thus, while Eq. 15.5 continues to be used to describe transport within the biofilm, D, should be thought of as being due to more than diffusion alone.

Having determined how transport to and within the biofilm can be modeled, the next step is to combine transport with the reactions occurring within the biofilm to establish the relationship between the bulk substrate concentration and the rate of substrate removal by the biofilm. The resulting relationship can then be combined with the appropriate process model to simulate the performance of an attached growth bioreactor. Three techniques are in common use for combining the biofilm model with a process model: (l) the direct technique, (2) the effectiveness factor technique, and (3) the pseudoanalytical technique."' In the direct technique, the differential equations describing reaction within the biofilm are combined directly with the differential equations describing the bioreactor, giving a set of differential equations that must be solved by numerical methods. This technique is commonly employed for modeling systems involving multiple species carrying out multiple reactions. such as carbon oxidation, nitrification, and denitrification. The effectiveness factor technique pretends that the reaction rate at any point in a bioreactor can be defined by the intrinsic reaction rate expressed in terms of the bulk substrate concentration, multiplied by a factor (the effectiveness factor) that corrects for the effects of mass transport. Relationships between the effectiveness factor and the system characteristics are then coupled with the differential equations of the process model to allow simulation of bioreactor performance. The pseudoanalytical technique is similar to the effectiveness factor technique in concept, in that it develops a relationship between reaction rate and bulk substrate concentration that can then be used in the bioreactor model. In this case, however, the differential equations representing transport and reaction within the biofilm are solved numerically and the output is used to develop simplified general relationships for transport and reaction that can be solved analytically, thereby allowing them to be coupled with the process equations. In the following two sections we will investigate the effectiveness factor and pseudoanalytical techniques.

15.2.2 Modeling Transport and Reaction: Effectiveness Factor Approach17

Effectiveness Factor. The basic concept in the modeling of biofilms is that the flux of substrate to and through the liquid-biofilm interface must equal the overall utilization rate per unit of biofilm planar area. Because the local substrate utilization rate depends on the substrate concentration at that location, it is clear from Figure 15.7 that the utilization rates at various points in the biofilm will be different. The overall utilization rate by the biofilm must consider this by integrating the reaction rate over the biofilm depth. Because of this averaging and because of the requirement for substrate transport from the bulk fluid to the biofilm surface, any observed relationship between the overall substrate removal rate and the bulk substrate concentration will be different from the intrinsic reaction rate expression for substrate removal. It is often convenient, however, to express the substrate removal rate as a function of the bulk substrate concentration, Ss,„ using a correction factor that takes into account the effects of transport. The correction factor is called the effectiveness factor' and is given the symbol If the Monod equation (Eq. 3.36) expresses the intrinsic relationship between the specific substrate removal rate, q, and the substrate concentration, this can be expressed as:

in which Xluir is the mass of biomass per unit volume of biofilm and L, is the biofilm thickness. Note that the substrate concentration is expressed in terms of the bulk substrate concentration, To use Eq. 15.6 in mass balance equations for various types of biofilm reactors, information must be available about the effectiveness factor, ti,.

The nature of t^ can be determined by writing the mass balance equation for substrate for a differential element within the biofilm (Figure 15.7) and solving it to obtain the actual flux of substrate into the biofilm. The actual flux can then be used with Eq. 15.6 to deduce the effects of the system kinetic and transport parameters on r|t. If the transport parameters include transport both to and within the biofilm, the effectiveness factor is called an overall effectiveness factor, denoted as t|c„. Assume that the biofilm in Figure 15.7 has reached a steady state in which it has a constant thickness L„ a constant biomass concentration XltM1, and uses substrate at a constant rate when exposed to a bulk substrate concentration SSb. A mass balance on substrate around a differential element in the biofilm yields:

dx dx

where A, is the planar surface area normal to the direction of diffusion and x is the distance into the biofilm from the inert solid support. If DL. is constant, dividing both sides by A^ and Ax, and taking the limit as Ax approaches zero yields:

which must be solved with two boundary conditions, one at the biofilm-support interface (x = 0) and the other at the liquid-biofilm interface (x = L,). At the biofilmsupport interface there is no transfer of substrate because the solid support is inert and impermeable. Thus, the appropriate boundary condition is:

As mentioned previously, having a permeable support is possible, but that case is not considered herein. It is apparent, however, that a different boundary condition would be required. The boundary condition at the liquid-biofilm interface is more complicated. It is written by recognizing that the substrate flux at that interface must equal the substrate flux through the stagnant liquid layer. Consequently, the appropriate boundary condition is:

The development of an equation for the overall effectiveness factor, T}l(„ requires the solution of Eq. 15.8 with Eqs. 15.9 and 15.10 as boundary conditions, giving substrate flux, Js, and hence the substrate removal rate per unit of biofilm planar area, as a function of the bulk substrate concentration, Ssh. The resulting relationship can then be used in Eq. 15.6 to obtain ti,.„.

In order to develop a generalized relationship between the overall effectiveness factor and the physical and biochemical characteristics of a biofilm system, Fink et al." solved Eq. 15.8 with its associated boundary conditions. They did this by a transformation of the two-point boundary value problem into an initial value problem. In doing so they used the following dimensionless quantities:

Bi is a Sherwood number, called the Biot number."1 Recalling that a diffusivity divided by a length is equivalent to a mass transfer coefficient, it can be seen that the term D^./L, represents an internal mass transfer coefficient. Thus, the Biot number is the ratio of the external mass transfer rate to the internal mass transfer rate. This means that when Bi is large, the external mass transfer coefficient is large relative to the internal coefficient so that all resistance to mass transfer can be considered to reside within the biofilm. In other words, the external resistance to mass transfer is negligible. This situation can arise when the flow rate past the biofilm is high. Conversely, when Bi is small, all of the resistance to mass transfer can be considered to be external to the biofilm. This situation can arise when the biofilm is very thin.

<J> is a Thiele modulus. The physical significance of the Thiele modulus may be seen by squaring it, multiplying both the numerator and the denominator by (Ss„ • AO, and rearranging.

The term (qH/Ks)Ssi, in the numerator is the first order approximation of the Monod equation (Eq. 3.38) for the specific substrate removal rate. For these first order kinetics, the maximum possible removal rate will occur when the substrate concentration surrounding the bacteria is the bulk substrate concentration. Consequently, the numerator represents the maximum possible first order reaction rate. Likewise, the maximum possible diffusion rate will occur when the gradient is maximized, so that the denominator represents a maximum diffusion rate within the biofilm. Therefore the Thiele modulus is the ratio of the maximum first order reaction rate to the maximum diffusion rate. A large value of the Thiele modulus represents a situation in which the reaction rate is large relative to the diffusion rate. Such a situation is said to be diffusion limited. Conversely, a small value of <t> represents a situation in which the diffusion rate is larger than the reaction rate. Such a situation is said to be reaction limited.

<J>r is a modified Thiele modulus. The purpose of the parameter k is to take into consideration the deviation of the Monod equation from first order kinetics, which were the basis for the Thiele modulus. It will be recalled from Section 3.2.2 that when the substrate concentration is small relative to the half-saturation coefficient, the Monod equation simplifies to an expression that is first order with respect to substrate concentration, which is consistent with the basis of the Thiele modulus. Thus, when k is small, the substrate removal rate behaves in a first order manner so that cf>, equals <|>, and the Thiele modulus adequately describes the relative importance of reaction versus diffusion. On the other hand, when k is large, the substrate concentration will be large relative to the half-saturation coefficient and the Monod equation will not behave in a first order manner. In that situation, the deviation from first-order kinetics will be large and <}>f will be smaller than <t>.

The results of Fink et al.n giving the overall effectiveness factor as a function of these dimensionless groups are shown in Figure 15.9. These values of Tq1() may be used to calculate the overall flux of substrate into a biofilm of thickness L, containing microorganisms at concentration XBHr under conditions where both internal and external mass transfer resistances exist. Thus:

Modified Thiele Modulus 4>f = <j>[1/(1 + k)]1/2

Figure 15.9 Overall effectiveness factor for Monod kinetics within a flat biofilm with external mass transfer resistance. (Adapted from Fink et al.1*)

Modified Thiele Modulus 4>f = <j>[1/(1 + k)]1/2

Figure 15.9 Overall effectiveness factor for Monod kinetics within a flat biofilm with external mass transfer resistance. (Adapted from Fink et al.1*)

The two curves for Bi = ^c represent the case when the rate of external mass transfer is much higher than the rate of internal mass transfer, whereas the two curves for Bi = 0.01 represent the case where internal mass transfer is much more rapid than external mass transfer, due to a large external mass transfer resistance. Comparison of two groups of curves with different Bi values but the same (J), value demonstrates that the existence of external mass transfer resistance has a strong effect on the overall effectiveness factor. For example, when cf>, = 1.0, a ten-fold decrease in Bi (from 1.0 to 0.1) results in almost a ten-fold reduction in the overall effectiveness factor. Moreover, comparison of curves with the same Biot number but different values of the Thiele modulus shows the relative importance of reaction versus diffusion. When Bi = 0.01, the external mass transfer coefficient is much smaller than the internal mass transfer coefficient. Consequently, external mass transfer controls and the effectiveness factor is influenced little by the relative importance of reaction versus diffusion. Thus, the value of the Thiele modulus, <J>, has little effect. Under these circumstances, the effectiveness factor is dominated by external mass transfer resistance and is often called an external effectiveness factor and given the symbol x|ci... On the other hand, when Bi = there is no external resistance to mass transfer and thus the relative importance of reaction versus diffusion has a strong impact on T|c,„ as evidenced by the strong impact of the Thiele modulus. Under circumstances when Bi = the effectiveness factor is often called an internal effectiveness factor and given the symbol

Although graphical representations like Figure 15.9 are convenient for some applications, for most occasions being able to determine the effectiveness factor analytically would be better. Consequently, it is common for investigators to develop functional relationships for the limited range of conditions they are interested in and this has been done for external, internal, and overall effectiveness factors. For example, for the case in which external mass transfer resistance is negligible, i.e., Bi is very large, Atkinson and Davies" developed both complex and simplified functional relationships for the internal effectiveness factor that agree quite well with the numerical results. In the interest of brevity, their equations will not be presented here. Rather, the reader is referred to other sources.1:

Application of Effectiveness Factor. Equation 15.16 can be used with Figure 15.9 to determine the performance of a bioreactor containing a biofilm of known depth and biomass density. To illustrate how this is done, we will consider a continuous stirred tank reactor (CSTR) containing a biofilm.

Assume that steady-state conditions prevail over a reasonable time in a CSTR containing a solid surface covered by a biofilm of thickness L, containing biomass at concentration X1U„. To maintain a constant biofilm thickness, the cells generated by substrate consumption must be detached from the surface, dispersed throughout the liquid phase, and washed out in the bioreactor effluent. Because cells are in the bulk of the liquid as well as the biofilm, they are consuming substrate from both locations. Thus, the steady-state mass balance equation for substrate is:

F-Ssu - F• Ssl, - JSA, - q„ Xlillb-V = 0 (15.17)

where Js is the substrate consumption rate per unit surface area of biofilm, which is equivalent to the substrate flux. A, is the biofilm surface area in the reactor, qH ■ X^ m, is the substrate consumption rate per unit volume of reactor by dispersed bacteria, V is the bioreactor volume, F is the flow rate of influent and effluent, Ssu is the influent substrate concentration, and Ssb is the effluent or bulk liquid substrate concentration. Substitution of Eq. 15.16 for Js and the Monod equation (Eq. 3.36) for qH yields:

_ c ^ _ „ v i ( q""Ssh ^ ^ T (SS" Ssb) Tle° X"J" ' Us + SsJ V

Equation 15.19 assumes that the influent contains no biomass. It is an approximation of the biomass concentration in a CSTR without biomass recycle in which Yilob. is an observed yield that accounts for decay in both the biofilm and the dispersed bacteria. The rationale for its use is that biomass can only arise from the utilization of substrate and that the biofilm is at steady state. Therefore, Eq. 15.18 may be rewritten as:

qn-sSh

The solution of Eq. 15.20 to determine the value of Ssb associated with a given hydraulic residence time (HRT) requires an iterative approach since the value of depends on Ssb. Thus, a value must be assumed for Ssb and the value of determined from Figure 15.9 or an associated approximate equation. Equation 15.20 can then be solved for Ssb and the solution compared with the assumed value. The procedure is repeated until the assumed and calculated values of Ssb agree. An iterative solution is not required when the bulk substrate concentration is fixed at a desired value and the bioreactor HRT or biofilm surface area per unit volume (AV V) required to achieve that value is being calculated. Under that circumstance, the effectiveness factor may be determined directly from Figure 15.9 for use in Eq. 15.20.

To show the effect of external mass transfer resistance on the performance of a CSTR containing a biofilm, Figure 15.10 was prepared by using Eq. 15.20 with Figure 15.9 and the parameter values in Table 15.1.17 As shown in Table 15.1, the value of D, used to generate Figure 15.10 was extremely large X 10J) to remove all internal mass transfer resistance. Three curves are presented, one with a very large k, value to represent the absence of external mass transfer resistance, and two with kL values that might be encountered in practice. Each of these curves represents effects that might be caused by changes in the velocity of the fluid past the biofilm. Examination of Figure 15.10a shows that the effect of a decrease in k, is to reduce the activity of the microbial film, thereby making the effluent substrate concentration greater than it would be in a bioreactor with less mass transfer resistance. Figure 15.10b shows the effect of the surface area of microbial film available in a bioreactor with an external mass transfer coefficient of 20 cm/hr. There it can be seen that reactors with more biofilm area remove more substrate, but that the effects diminish as the HRT is made larger because of the effects of the bulk substrate concentration on the effectiveness factor and substrate removal by suspended bio-

Table 15.1 Kinetic Parameters, Stoichiometric Coefficients, and System Variables Used to Generate Figures 15.10 and 15.11'

Symbol

Units

Value

<1H

mg substrate COD/(mg biomass COD • hr)

0 0

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