CORRECTION TO THE METHOD OF TALMADGE AND FITCH Arcadio P. Sincero Morgan State University

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1 CORRECTION TO THE METHOD OF TALMADGE AND FITCH Arcadio P. Sincero Morgan State University ABSTRACT The method of Talmadge and Fitch used for calculating thickener areas was published in Although in the United States, this method has largely been superseded by the solids flux method, there are other parts in the world that use this method even up to the present 2. The method, however, is erroneous and this needs to be known to potential users. The error lies in the assumption that the underflow concentration, C u, and the time of thickening, t u, in a continuousflow thickener can be obtained from data obtained in a single batch settling test. This paper will show that this assumption is incorrect. Keywords: continuous-flow thickening, batch thickening, Talmadge and Fitch, underflow concentration, thickening zone INTRODUCTION Before presenting the derivation of the method of Talmadge and Fitch, it is important to learn the basic concept of batch thickening. This will put the reader on a sound footing for understanding the materials that follow. Figure 1 shows the schematic of the concept. A sample is poured into a graduated cylinder at initial time t o and allowed to settle. Various zones are then formed As a result of settling, zone A is cleared of solids; hence, it is called clarification zone. In B, the solids are settling in a uniform velocity; hence, the zone is called uniform settling zone. The concentration of the solids in this zone is constant. In zone C, the solids concentration increases or thickens from the value at the interface B-C to the value at the interface C-D. Hence, this zone is called thickening zone; this is where thickening of the solids starts. Finally, in D, the solids are compressed and the matrix is compacted and consolidated. Hence, this zone is called compression zone. In this zone, the solids are further thickened by the compression, and compaction, and consolidation processes. Let us consider the process in more detail. Assume that initially at time t o, the initial concentration is C o. Some time later at t 1, the four zones are formed. B retains the initial concentration and settles at a rate characteristic of this concentration. Since the concentration is constant at C o throughout this zone, B settles at a constant velocity, hence the name uniform settling (velocity) zone. This velocity can be determined by following the interface A-B with time. D is formed by particles piling on top of each other producing the largest concentration of the zones. There will be a gradation of concentration from D to B. Hence, zone C will have this gradation forming a concentration gradient in the zone. As mentioned above, this is where the thickening process starts. The gradient will be constant. As the particles pile up one on top of 1 W. P. Talmadge and E. B. Fitch (1955). Ind. Eng. Chem. 7, Yanqiu, Zhang, etc. (1988). Amendment of Design of Secondary Settling Tank, Environmental Engineering. Vol 16, No 6, Ministry of Industry, Beijing, China. 1

2 CSCE/EWRI of ASCE Environmental Engineering Conf Niagara 2002 another, zone D lengthens. To maintain the concentration gradient, the concentration immediately below interface B-C and the concentration immediately above interface C-D must remain constant for a constant length of zone C. Since D is lengthening and to maintain the length of C and hence the gradient, the interface B-C must move up at the same speed as the Figure 1. Schematic of thickening lengthening of D. This means that zone B is eroded both at its top and its bottom. Hence, eventually, this zone must disappear. This happens at time t 3. After time t 3, zone C starts to diminish and totally disappear at time t 5 after which time pure compression commences. Pure compression continues until t 7 where the slope of the curve now exhibits the tendency to become horizontal. After a very long time, the slope will, indeed, be horizontal. The time t, somewhere between t 3 and t 5, is called the critical time and the corresponding concentration of solids is called the critical concentration. As evidenced from this batch analysis, various degrees of sludge thickening can occur. DERIVATION OF THE TALMADGE AND FITCH METHOD Having discussed the basics of batch thickening, we are now ready to address the derivation of the method of Talmadge and Fitch. It will be noted that this derivation would seem very convincing, unless the reader is very keen in analyzing the derivation. Figure 2 is a settling curve derived from Figure 1. A straight line is drawn tangent to the critical point at (t, H ). This line intersects the ordinate at a height of H 1. The height at the 2

3 Sincero 2002 Wastewater Treatment 1 critical point is designated as H. H u is the height of the sludge at the desired underflow concentration, corresponding to a time of thickening t u. H o is the height of the sample at beginning of the experiment. Referring to Figure 1, these heights pertains to the interface A-B as it moves down with time in the process of settling. Figure 2. Determination of t u (Talmadge and Fitch) H 1 H is the settling velocity at the critical point at time t. In Figure 2, locate the point t (H u, t u ). The slope of the tangent line at this point represents the settling velocity at the point. H Hu This velocity is given by. By virtue of the straight line construction, this is also the tu t settling velocity at the critical point. Equating these two settling velocities produces 3 H 1 H t = H H tu t u (1) Solving for t u results in t u = H H 1 1 H H u (2) 3 R. S. Ramalho (1977). Introductioin of Wastewater Treatment Processes. Academic Press, New York. 3

4 CSCE/EWRI of ASCE Environmental Engineering Conf Niagara 2002 Figure 3. Determination of critical point (Talmadge and Fitch) With t u known, the crossectional area, A, of the thickener can be calculated. Let Q be the influent flow to the thickener, C o be the influent solids concentration, and C u be the underflow concentration. (C u corresponds to t u.) Then, And QC o t u = AH u C u A = QC t o u HuCu Equation () shows that it is important to determine the time of thickening t u accurately, since this determines the cross-sectional area of the thickener. The parameters on the right-hand side of Equation (2) are determined through a graphical procedure in the method of Talmadge and Fitch 5. As shown in the above derivation, a geometric construction is required to locate the critical point (t, H ), at concentration C. Once this point is located, further geometric construction may proceed to determine H 1, H u and H. The value of t u can then be calculated from Equation (2). Or, as is often used in practice, can simply be R. S. Ramalho (1977). Introductioin of Wastewater Treatment Processes. Academic Press, New York. 5 Metcalf & Eddy, Inc. (1991). Wastewater Engineering, Treatment, Disposal, and Reuse. McGraw-Hill, Inc., New York. (3) ()

5 Sincero 2002 Wastewater Treatment 1 determined from the geometric construction, itself. In the method of Talmadge and Fitch, t u, as mentioned above, determines the size of the thickener. It is therefore the single most important parameter in this method. Figure 3 is an extension of Figure 2, which shows the geometric construction for the location of the critical point. A straight line H o c is drawn tangent to the upper limb of the settling curve. Another straight line de is then drawn tangent to the lower limb, intersecting the previous line at point g. Accordingly, these two lines form the angle H o ge. Bisecting this angle, locates the critical point (t, H) at concentration C on the settling curve. DISCREPANCY OF THE METHOD AND ITS CORRECTION The slope of the tangent line in Figure 2 is a measure of settling velocity. The construction of this tangent and Equation (1) infer that the velocity corresponding to H u at time t u is equal to that at the critical point at H. By a material balance, the underflow concentration C u corresponding to H u can be calculated. Since C u is at H u and C is at H and, since the velocity at H u is equal to that at H, these results indicate that concentrations C u and C have equal corresponding settling velocities. Refer to Figure 3. The straight H o h corresponds to zone B in Figure 1. The slope of this line is the settling velocity of this zone. Sine zone B corresponds to a solids concentration, this solids concentration would have the settling velocity of this zone. A number of experiments of differing solids concentration could be conducted yielding plots similar to Figure 3. In other words, a table could be prepared indicating that a sample of a given solids concentration would correspond to a definite settling velocity. In other words, these experiments would conclude that sludges of equal solids concentrations settle at the same rate. As seen from the geometric construction and derivation in the method of Talmadge and Fitch, C u and C are shown to have equal settling velocities. Hence, the method implies that C u and C are equal. This is not, however, true. From Figure 3, C is at a height of H and C u is at a height of H u. By material balance and since H u is less than H, C u must be greater than C. Since this results in a contradiction, the method of Talmadge and Fitch has a serious discrepancy. Let us formally show that C u is greater than C. Let A be the cross-sectional area of the thickener. Although in Figure 3, C is indicated to be the concentration at height H, the actual concentration encompassed by this height would be greater as a result of the compression of the layers below height H. However, for the sake of showing that C u is greater than C, it is sufficient and safe to assume the value C. Hence, by material balance between height H and height H u, And, AH C = AH u C u (3) C u = () H C Hu 5

6 Now, since H u is much less than H, it follows from this equation that C u is much greater than C. This is the conclusion we had set out to do. CSCE/EWRI of ASCE Environmental Engineering Conf Niagara 2002 It is certainly not possible that C u can be obtained through a single settling curve such as Figure 2, as assumed by the method. It cannot be assumed that the settling velocity at the critical point is equal to the settling velocity at the underflow concentration, as the method is trying to show. Instead of only one settling curve, a correction to obtain the value of C u would be to perform several settling curves on different sludge concentrations. The solids fluxes can then be calculated from where the limiting solids flux is obtained. Obtaining the limiting solids flux is equivalent to obtaining C u. Of course, this correction is the solids flux method, which is not the topic of the paper and, hence, will not be pursued further. CONCLUSIONS C u and t u are two critical parameters used in the design of thickeners using the method of Talmadge and Fitch. The method assumes that these parameters can be obtained from a single batch settling test. This paper, however, has shown that this is not possible. Although, it might not have been noticed at the time when the method was made known, the concentration at the underflow is being made equal to that at the critical point. This is a very serious discrepancy of the method. When this method appeared in 1955, it was good news to the design engineers who always look to avoid doing plenty of laboratory work. It is now, however, evident that the method of Talmadge and Fitch has a very big discrepancy. The one settling curve (Figure 3) that engineers only like to perform needs to be augmented and several batch settling curves need to be obtained. This, in fact, is the essence of the solids flux method. As discussed in some textbooks, the concept behind the method is very difficult to understand; in fact, it is not even explained in such a way as to be easily understood. The mechanics of applying it, however, is very, very simple. It is this simplicity of application that makes it attractive as compared to the solids flux method. However, since discrepancy exists, it is just prudent that this error be made known. It is worthy to note, that in some parts of the world, this method is still in use. Yanqiu (1988), for example, reported that this method is used in China. Symbols C o C u H o H 1 H H u t t u References original sludge concentration thickener underflow sludge concentration original sludge height intercept on ordinate by tangent to the critical point sludge height at critical point height of thickened sludge at underflow concentration time to critical point time of thickening 6

7 Metcalf & Eddy, Inc. (1991). Wastewater Engineering, Treatment, Disposal, and Reuse. McGraw-Hill, Inc., New York. Ramalho, R. S. (1977). Introduction to Wastewater Treatment Processes. Academic Press. Sincero 2002 Wastewater Treatment 1 Talmadge, W. P. and E. B. Fitch (1955). Ind. Eng. Chem. 7, 38. Yanqiu, Zhang, etc. (1988). Amendment of Design of Secondary Settling Tank, Environmental Engineering. Vol 16, No 6, Ministry of Industry, Beijing, China. 7