Environ. Sci. Techno/. 1994, 28, 1367-1369
Disinfection under Dynamic Conditions: Modification of Hom’s Model for Decay Charles N. Haas’ and Josh Joffe Environmental Engineering and Science Institute and Department of Civil and Architectural Engineering, Drexel University, Philadelphia, Pennsylvania 19104
Hom’s model for microbial disinfection kinetics is frequently applied to data showing deviations from Chick’s law inactivation. However, the generally applied form of this model does not consider the possibility for disinfectant decayidemand. In this work, we derive the form of the Hom model under conditions of first-order demand; the integrated solution involves the incomplete y function. We then propose a simpler approximation to this exact solution and explore its numerical adequacy, also showing the effect of the aPProximation when applied ‘0 a real data set. This approach can be used to assess disinfection kinetics in drinking water to assure compliance with microbial reductions required under the Surface Water Treatment Rule. Introduction Under the Surface Water Treatment Rule (SWTR) and changes anticipated under the forthcoming Groundwater Disinfection Rule, drinking water treatment plants are required to achieve specified reductions of pathogens, particularly Giardia and viruses. Particularly when faced with the simultaneous challenges of minimizing the formation of disinfectant byproducts, this leads to the need to optimize disinfection processes. The SWTR describes the adequacy of disinfection in terms of “ot”values, which is the product of concentration and time yielding a given level of inactivation. In deriving these values, the adequacy of the Chick and Watson laws (1-3) was assumed. It has been shown in many studies that the universal validity of this assumption is questionable (4-12). Alternate models that can describe “shoulders” or “tailing off” in inactivation have been used by Fair et al. ( 4 )and by Hom ( 5 ) . However, these and alternate models (13-15) that can describe intrinsic deviations from ChickWatson kinetics have only been derived and utilized in systems in which the disinfection decay is small, i.e., in demand-free systems. The objective of this paper is to derive generalized forms of the Hom model (of which the model used by Fair is a special case) which are valid under conditions of disinfection decay, particularly first-order loss of residual. In a separate work, these models have been applied to describe inactivation kinetics in real waters.
In@) = -kC”t”
(1)
This can be derived from the following differential equation, under the assumptions of constant disinfectant residual, where is the concentration of viable organisms at any one time:
@! = -mkC”t“lN dt
(2)
If m equals unity, the Chick-Watson relationship is obtained. If is less than 1,tailing off occurs, i.e., a semilog plot of survival ratio versus time shows a slope of diminishing magnitude with time, If is greater than unity, shoulders occur, Le., a semilog plot of survival ratio versus time shows a slope of increasing magnitude with time. Disinfectant concentrations may change with time, due perhaps to the intrinsic demand of microorganisms (even attempts to purify organisms from associated demand often result in some irreducible minimum demand associated with the mass of the organisms themselves) or due to other components in the water. The present work focuses only upon a “pure” decay process in which the disinfectant is converted to an inert form. Also important, but a more complex case, is the situation in which a relatively potent disinfectant form (such as free chlorine) is converted to a still active but less potent form (such as combined chlorine). If the loss of disinfectant residual is first order, as has often been noted for chlorine and ozone decomposition, then the following relationship for residual versus time can be found, where COis the initial residual and k* is the first-order decay rate: C = C, exp(-k*t)
(3)
First-order disinfectant decay has often been noted to describe the loss of chlorine residuals, at least in initial stages of exposure (16-18). On this basis, the use of a first-order assumption is often appropriate. Exact Solution. By combining eqs 2 and 3 and collecting terms, the followingequation can be derived for the survival ratio:
Theoretical Development Under constant disinfection residual (C), in a homogeneous batch system, Hom proposed a model to depict the survival ratio (S-ratio of viable to initial number of organisms) after exposure at time (t). Using a slight reparameterization of Hom’s original equation, this can be written as:
This equation can be integrated analytically only by employing the incomplete y function, y(a,y), defined by the following integral:
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The final solution can then be expressed in the following form:
0013-936X/94/0928-1367$04.50/0
0 1994 American Chemical Society
Environ. Sci. Techno).,Vol. 28,
No. 7, 1894 1367
In this solution, the final result can be expressed as the product of the Hom solution in the absence of disinfectant demand and a dimensionless "efficiency factor" (q),which corrects €or the loss of disinfectant residual. It can be seen that this efficiency factor can be written in terms of the Hom exponent, m, and a dimensionless group relating contact time to the rate of decay. With the definition \k = nk*t, the relationship in eq 6 can be written in the form:
If a table of the incomplete y function is available, the explicit survival ratio can be computed. Alternatively, this function may be evaluated from one of a number of series approximations (19,20). Unfortunately, this function is not generally available in standard computer packages or curve-fittingroutines. Therefore, the authors were also interested in developing an approximation to eq 6 which relied upon readily implemented functions. ApproximateSolution. Starting from the differential eq 2, if demand is minor, eq 1can be rearranged to solve for time, and the result can be substituted for time in eq 2. This results in the following (where NOis the initial microorganism density):
Equation 3 can be substituted into this, and the variables separated to yield:
Jot
- rn(kC:)l/" exp( - y nk*t )dt (9)
This can be integrated with the following result:
which can be rewritten in terms of the dimensionless time variable and the efficiency factor as follows: In@) = -kCg"t"$
(11)
The adequacy of the approximation of eq 11 to the exact solution in eqs 6 and 7 therefore reduces to the degree to which $ approximates ?I. However, note that the former may be computed in terms of readily evaluated functions as opposed to the necessity for the evaluation of the incomplete y function for the exact solution. Evaluation of the Approximation To evaluate the goodness of the approximation, computations were conducted using a grid in two-dimensional 1388 Environ. Sci. Technol., VoI. 28, No. 7, 1994
Y
a 01
3
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
m value Figure 1. Exact value of effectiveness factor (1)as a function of m and q.
space (m and W. Values of m were 45 evenly spaced intervals between 0.3 and 2.5. Values of ?P were 71 logarithmically spaced intervals between 0.03 and 1.0. These values were chosen to cover the range of frequently encountered m values and values for disinfectant decay typically seen in disinfection experiments in laboratory systems. The exact value of the efficiency factor was computed using exact evaluation of eq 7 by employing the built-in incomplete y function in MATLAB (Mathworks, Inc.). The approximate efficiency factor was computed directly by the application of eq 11. The difference, approximate value minus exact value, was then obtained. Since the efficiency factors vary between 1.0 and 0.0, the difference as a percent provides a direct measure of the error in computing log survival ratios. Figure 1 indicates that, over the range of parameter values investigated, the numerical correction using the efficiency factor could be significant. A 9 value of 0.7 indicates that the log survival value is 70% of that which would be seen in the absence of any disinfectant decay; thus, for example, if two logs of inactivation were wanted, ignoring this correction could produce only 1.4 logs of inactivation. Therefore, the consideration of the decay process is important over much of the range of parameter values studied in this paper. Figure 2 provides a contour plot of the error of the approximation. I t can be seen that over the entire range studied, there is less than a 10%error of the approximation. As the m value deviates from unity, the error increases, with the approximation being exact for m = 1 (the ideal Chick-Watson relationship). There is also a tendency for the error to increasing at increasing values of time or disinfectant decay. In general, however, particularly given the errors inherent in conducting disinfection experiments (microbial variability, measurement of residual), it is concluded that the approximate solution provides an acceptable description of inactivation kinetics for systems in which the Hom model and first-order disinfectant decay govern. The approximation can be evaluated in terms of readily available analytical functions and, therefore, offers a useful
Conclusions
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0.6
i
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0.8
I
I
1.2
1.4
I
1.6
I
8
,
1.8
2
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I
1
2.4
m value
Figure 2. Error in approximation (approximation minus exact value) as a function of inactivation parameters. Curves of iso-error shown, with % error indicated. 1
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Acknowledgments This work has been supported by funding from the American Water Works AssociationResearch Foundation and by Drexel University. The project work was performed in cooperation with Mark Heath and Joseph Jacangelo of Montgomery-Watson Engineers, Inc. and the Portland Water Bureau.
Literature Cited (1) Chick, H.J. Hyg. 1908,8,92-157. (2) Watson, H. E.J. Hyg. 1908,8,536-542. (3) Lee, R.E.; Gilbert, C. A. J.Phys. Chem. 1918,22,348-372. (4) Fair, G. M.; Morris, J. C.; Chang, S. L.; Weil, I.; Burden, R.
.. .... ., ...... .. ... ..... .. .. ...
0.1
The Hom model for the description of kinetics of disinfection which do not follow Chick-Watson relationships can be extended to describe inactivation in batch systems subject to disinfectant decay. These systems arise in the determination of inactivation of various organisms in real waters by chemicalagents, such as chlorine or ozone. The exact solution for a system undergoing first-order decay involves the incomplete y function; however, an approximation involving only the exponential function has been developed and appears to be adequate over the commonly encountered range of inactivation conditions. This approximate relationship is suited for parameter estimation from laboratory disinfection experiments.
200
contact time(mln)
Figure 3. Comparison of exact and approximate solutions: Glardia inactivation by preformed monochioramine in Bull Run water. route to the reduction of inactivation data and the determination of inactivation parameters.
Example The adequacy of the approximation is further illustrated by the use of an example. Cysts of Giardia muris were exposed to preformed monochloramine at pH 6.9 and 18 "C in water obtained from Bull Run (Portland, OR) Reservoir. Survival was measured by excystation in procedures described elsewhere, and under the conditions of the experiment monochloramine was found to decay with aloss rate of 0.001 min-l(I6). Using the approximate equation and nonlinear least-squares analysis (21))the Hom parameters were obtained. Figure 3 compares the model fit to the optimal Hom parameters computed using the exact solution and the fit when the same parameters were used to predict survival based on the approximate solution. For this data set, m = 2.46, and the maximum 9 value was 0.3. It is noted that the computed survival predictions are indistinguishable, particularly in contrast with the results of duplicate experiments (indicated by points at the same contact time). Therefore, even beyond the parameter values noted in Figure 2, the use of the approximation provides a quite acceptable method of determining inactivation parameters of the Hom model.
P. J. Am. Water Works Assoc. 1948,40,1051-1061. (5) Hom, L. W. J. Sanit. Eng. Diu. ASCE 1972,98,183-194. (6) Severin, B. F.;Churn, C. C. In A W W A Seminar Proceedings: Assurance of Adequate Disinfection, or C-T or Not C-T; American Water Works Association: KansasCity, MO, 1987;pp 21-48. (7) Wickramanayake,G. B.; Rubin, A. J.; Sproul, 0. J. J. Am. Water Works Assoc. 1985,77,74-77. (8) Wickramanayake,G. B.; Rubin, A. J.; Sproul,0.J. J. Water Pollut. Control Fed. 1984,56, 983-988. (9) Jarroll, E. L.; Bingham,A. K.; Meyer,E. A. Appl. Environ. Microbiol. 1981,41,483-487. (10) Rice, E.W.; Hoff, J. C.; Schaefer, F. W., I11 Appl. Environ. Microbiol. 1982,43,250-251. (11) Finch, G. R.;Black, E. K.; Labatiuk, C. W.; Gyurek, L.; Belosevic, M. Appl. Environ. Microbiol. 1993, 59, 36743680. (12)Haas, C. N.;Heller, B. Water Res. 1990,24,233-8. (13) Severin,B. F.;Suidan, M. T.;Engelbrecht, R. S. J.Environ. Eng. 1984,110,430-439. (14) Haas, C. N.Environ. Sei. Technol. 1980,14,339-340. (15)Roy, D.;Engelbrecht, R. S.; Chian, E. S. K. J.Environ. Eng. Diu. ASCE 1981,107,887-899. (16) Anmangandla, U. Master of Science Thesis, Drexel University, 1993. (17) Haas, C. N.;Karra, S. B. J. Water Pollut. Control Fed. 1984,56,170-3. (18) Haas, C. N.;Karra, S. B. Water Res. 1984,18,1451-4. (19)Handbook of Mathematical Functions; Abramowitz, M., Stegun, I. A., Eds.; Dover Publications: New York, 1965. (20) Gradshteyn, I. S.;Ryzhik, I. M. Table of Integrals, Series, and Products, 4th ed.; Academic Press: New York, 1980; p 941. (21) Haas, C. N. Water Res. 1988,22,669-77. Received for review December 16, 1993.Revised manuscript received April 7, 1994.Accepted April 11, 1994." Abstract published in Aduance ACS Abstracts, May 15,1994.
Environ. Scl. Technol., Vol. 28, No. 7, 1994 1369
