mass. Another important point to be seen in Figure 15.10b is that the presence of a biofilm prevents washout of the bioreactor. The curve labeled A. = 0 represents a CSTR with no biofilm. Washout occurs at an HRT of a little more than five hours, making the substrate concentration equal to the influent. However, the presence of a biofilm allows substantial substrate removal at that HRT. Consequently, CSTRs with biofilms can remove substrate at HRTs well below those that would cause washout in a simple CSTR. Furthermore, the larger the surface area of biofilm, the shorter the HRT may be.

Figure 15.11 was prepared to show the effects of internal mass transfer resistance, again by using Eq. 15.20 with Figure 15.9 and the parameter values in Table 15.1. In this case, however, the value of kL was made very large to remove all external mass transfer resistance. As in Figure 15.10a, three curves are presented. One curve has a very large Dc value to represent the absence of internal mass transfer resistance. It is essentially the same as the curve with a k, value of 20,000 cm/hr in Figure 15.10a and can be used for comparing the relative effects of the two types of resistance. In other words, that curve represents a situation in which the overall effectiveness factor is 1.0 for all conditions. Examination of Figure 15.11 shows that the general effects of internal mass transfer resistance are similar to those of external mass transfer resistance, i.e., a reduction in the amount of substrate that can be removed by the biofilm. One difference that should be noted, however, is that whereas the external mass transfer resistance is subject to change by engineering factors such as the velocity of flow past the biofilm, the internal mass transfer resistance is not. Instead, it depends on the physical and chemical properties of the wastewater and the microorganisms in the system.

Many factors can influence the values of the mass transfer coefficients in biofilm systems. Unfortunately, space does not permit a discussion of them here. Never

Figure 15.11 Effects of internal mass transfer on the removal of soluble substrate by a CSTR containing a biolilm.

HRT, hrs

Figure 15.11 Effects of internal mass transfer on the removal of soluble substrate by a CSTR containing a biolilm.

theless, it is apparent from the above that accurate estimation of the coefficients is a requirement for proper application of mathematical models. Consequently, readers should consult the work of others for more information on this important topic* '*"

Equation 15.16 can also be used in the mass balance on substrate in a plug-flow bioreactor. The mass balance must be written around an infinitesimal section and the limit taken to get the differential equation describing the change of substrate concentration along the bioreactor length. In this case, because the substrate concentration varies along the bioreactor length, so will the overall effectiveness factor. Thus, solution of the equation requires a functional relationship for T|cl) that can be used with numerical methods to solve the problem. Applications of this method are presented elsewhere17 and the reader is referred there for more information. It is important to note, however, that the change in the effectiveness factor can be appreciable from one end of a plug-flow bioreactor to the other. Thus, assuming a constant effectiveness factor throughout the bioreactor is inappropriate.

Many advantages are associated with the use of effectiveness factors for modeling attached growth systems and they have found reasonably wide use, particularly to model fluidized bed systems."' Unfortunately, the effectiveness factor approach requires knowledge of the steady-state biofilm thickness. Thus, the biofilm and bioreactor models must be coupled with some means of obtaining the steady-state biofilm thickness. In addition, the effectiveness factor approach gets much more complex when one needs to consider dual nutrient limitation and competition for space by multiple bacterial types. These situations are more easily handled by an alternative approach. We will examine one, the pseudoanalytical approach.

15.2.3 Modeling Transport and Reaction: Pseudoanalytical Approach

Pseudoanalytical Approach. The pseudoanalytical approach uses simple algebraic expressions for the flux of substrate into a biofilm. Those expressions are based on an analysis of the results from the numerical solution of the differential equations describing transport and reaction in a biofilm. The availability of simple algebraic equations eliminates the need to repetitiously solve numerically a set of nonlinear differential equations while modeling the performance of a biofilm reactor. Pseudoanalytical solutions have been developed by several authors,2"" but the approach of Sacz and Rittmann1"is particularly accurate.

A key characteristic of the pseudoanalytical approach is that it allows calculation of the bulk substrate concentration and the biofilm thickness for a steady-state biofilm. A steady-state biofilm is one in which the gains in biofilm mass due to biomass growth are just balanced by the losses in biofilm mass due to the combined effects of microbial decay within the biofilm and detachment by shear at the liquid— biofilm interface.3'' Decay is treated in the traditional manner as presented in Chapter 5. Because of the balance between growth and loss, the biofilm attains a uniform thickness, L,. For a purely heterotrophic biofilm, that thickness is given by: s

JsYh

where b,, is the loss coefficient due to detachment caused by surface shear and the other parameters have their usual meaning. The value of the detachment coefficient varies with the shear stress on the biofilm, which depends on the hydrodynamic regime surrounding the biofilm.2",K

An important characteristic of a steady-state biofilm is the existence of a minimum bulk substrate concentration below which the biofilm cannot be maintained."'' If the bulk substrate concentration is below that value, growth cannot occur rapidly enough to replace the losses to decay and detachment and the biofilm will decrease in thickness until it ceases to exist. The minimum bulk substrate concentration, Ss,,,,,nl, is given byrh

Examination of Eq. 15.22 reveals that it is analogous to Eq. 5.14, the minimum attainable substrate concentration in a CSTR. This is because both represent the substrate concentration required to drive the growth reactions at a rate that will just balance loss by decay (and detachment, in the case of the biofilm). Since SSi„m„ is determined solely by parameters that depend on the biomass and substrate (q„, Ks, Yh, and b„) and the fluid regime (b„), it takes on special significance as a parameter in the pseudoanalytical approach.

The equations upon which the pseudoanalytical solution of steady-state biofilm kinetics is based differ somewhat from those used to develop the effectiveness factor approach and are the result of the necessity to compute the biofilm thickness, L,. They are:"'

"dT

k, where t is time. All other symbols are as defined previously. Equation 15.28 is just a rearranged form of Eq. 15.2, the flux across the hypothetical boundary layer. After the introduction of several dimensionless variables, Sâez and Rittmann1" ' solved Eqs. 15.23 through 15.28 using the numerical method of orthogonal collocation.14 This was done for 500 initial conditions, covering the entire region of feasible solutions.' The output was then used to develop the pseudoanalytical solution.

The pseudoanalytical solution is based on the flux into a deep biofilm, which is defined as one in which the substrate concentration at the biofilm-support interface is zero.1" The reason for using a deep biofilm as the reference case is because the dimensionless flux into a deep biofilm, Js.^r can t>e calculated analytically with:"

where S*. is the dimensionless substrate concentration at the liquid-biofilm interface:

Thus, for any value of Ss,, J*,!,,,, can be calculated. Once that has been done, the dimensionless flux into an actual biofilm with that value of Ss„, J*, can be computed as some function of J*,iclT:

provided an expression is available for £;. The dimensionless flux into the actual biofilm, J*, is defined as:

Consequently, once the dimensionless flux has been determined from Eq. 15.31, the actual flux, Js, associated with the liquid-biofilm interface substrate concentration, can be calculated with Eq. 15.32 since all of the parameters in the relationship are known.

In developing the pseudoanalytical approach, Sâez and Rittmann"' defined a new dimensionless group, S*,,,„,,, the dimensionless minimum bulk substrate concentration. Its value is given by:

Sshmin is important because of its physical significance.'" Recognition of the fact that Yu-q is equal to |i (Eq. 3.44), suggests that the value of S*,,m,n is an indication of the relative importance of biomass loss (by decay and detachment) and biomass growth. A large value of S5,n„„ (>1) implies that the maximum specific growth rate is not much larger than the specific loss rate of biomass from the biofilm, suggesting that the biofilm may be difficult to maintain. A small value (<1), on the other hand, suggests a potentially high net growth rate relative to losses, thereby making the biofilm easy to maintain. It is important to recognize that the value of bn, the specific detachment coefficient, depends on the flow velocity past the face of the biofilm, and thus is under engineering control. Consequently, the term Si,,m,„ represents both biological and physical factors. Because the significance of this term to the pseu-doanalytical approach is similar to the significance of the dimensionless groups in the effectiveness factor approach, we think that it should be a named dimensionless group. Therefore, we propose and use the name Rittmann number, with the symbol Ri:

Thus, the Rittmann number is the ratio of the specific loss rate of biomass from a biofilm to the net potential growth rate. As stated above, a large value means that a biofilm will be difficult to maintain, whereas a small value means that it will be easy to maintain.

Examination of the results from the 500 conditions studied by Saez and Rittmann," revealed that £ could be adequately represented by:'

Examination of Eqs. 15.35 through 15.37 shows that they all depend upon the Rittmann number, showing why it is an important parameter in the pseudoanalytical solution. Saez and Rittmann" examined the accuracy of the pseudoanalytical solution technique by using it to compute the flux into the biofilm for each of the 500 initial conditions and comparing the values with those obtained with the full numerical solution for each of the same conditions. The error depended somewhat on the value of the Rittmann number, but gave a standard error on the order of 2%, with 2.6% being the greatest observed. Thus, the pseudoanalytical solution is quite accurate.

To summarize, the calculation of the substrate flux associated with a given substrate concentration at the liquid-biofilm interface proceeds in the following manner. First, that concentration is put into dimensionless form with Eq. 15.30, allowing calculation of the dimensionless flux into a deep biofilm, Js.^vp, with Eq. 15.29. Then the Rittmann number is calculated with Eq. 15.34, allowing the parameter 4 to be determined with Eqs. 15.35-15.37. Once £ is known, the dimensionless flux into the biofilm can be calculated with Eq. 15.31, allowing the actual flux to be determined with Eq. 15.32.

Application of Pseudoanalytical Approach. The pseudoanalytical approach allows direct calculation of the flux into a steady-state biofilm associated with a given liquid-biofilm interface substrate concentration. However, what we really want to know is the flux associated with a given bulk substrate concentration, since it is the concentration that can be measured. The pseudoanalytical approach can be used to calculate it, as well as the biofilm's thickness, by combining Eq. 15.28 (in dimen-sionless form) with Eqs. 15.29, 15.31, and 15.35, giving:

tanh

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