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Ind. Eng. Chem. Res. 2002, 41, 2440-2447
Mathematical Models for Optimization of Industrial Tank-Washing Operations V. M. Palacios,* I. Caro, and L. Pe´ rez Chemical Engineering, Food Technology and Environmental Technology, Faculty of Sciences, Apartado 40, Polı´gono Rı´o San Pedro, s/n 11510, University of Ca´ diz, Spain
In the present work, a general mathematical model for tank-washing operations is developed. It begins with theoretical considerations for the simplest cleaning strategy, the “one tank-one wash” model, assuming several simplifications for the cleaning process. Later, we generalize the previous considerations to an industrial system of n tanks with n independent washes and discuss optimization aspects. Thereafter, we analyze more complicated strategies, such as “n tanks-one in series wash” or “one tank-i consecutive washes”. Finally, we study a mixed strategy that is the basis for a great number of different possibilities for n tanks, namely, the “consecutivein series” wash strategy. The combined strategies are shown to be a good alternative to n independent washes, resulting in a homogeneous level of cleaning and better performance. The mathematical structure of the strategies leads to theoretical optima that show the extent to which the cleaning water load can be conveniently exploited. 1. Introduction The uses of water in the food industry are the following: cleaning of surfaces in tanks or fermenters; washing of machinery; displacement of seeds, fruits, and vegetables by stem; continuous lubrication of conveyor belts and other machinery; heating or cooling reactors or other equipment; and serving as a food constituent in some formulations.1,2 Even though washing and cleaning operations produce most of the wastewater generated in a typical plant, these operations have traditionally been considered of secondary importance in the production cycle.3 At present, as a consequence of new industrial regulations, water consumption and wastewater discharges have become restrictive factors in the profitability and feasibility of certain industrial processes. Modern environmental politics force food producers to increase production without draining resources or generating residuals and induce the recovery or reuse of as much as possible of the effluents.1,2,5 In this context, reuse of washing and cleaning water is considered an important economic incentive, and this adjustment has been widely applied in the food industry.6 Water recovery operations must be thoroughly analyzed because washing and cleaning operations can have a double perspective. On one hand, these operations might be intended to achieve the maximum recovery of the fouling products, if such products are interesting from an economic point of view. On the other hand, these operations might be intended to achieve the maximum elimination of the fouling products, if they are only undesirable residuals.4,7 The optimum quality of cleaning depends on which kind of washing operation we are considering. At present, there are not many published works that develop mathematical models for optimization of the effectiveness and cost of industrial washing operations. To our knowledge, the only important work on this topic
was carried out in the galvanizing industry, to calculate the effectiveness of part washing operations in the surface treatment stage.8,9 In the present work, a general mathematical model for tank-washing operations is developed. It begins with theoretical considerations for the simplest cleaning strategy, the “one tank-one wash” model, assuming several simplifications for the cleaning process. Later, we generalize the previous considerations to an industrial system of n tanks with n independent washes and discuss optimization aspects. Thereafter, we analyze more complicated strategies, such as “n tanks-one in series wash” or “one tank-i consecutive washes”. Finally, we study a mixed strategy that is the basis for a great number of different possibilities for n tanks, namely, the “consecutive-in series” wash strategy. In particular, we develop and deeply discuss the mathematical expressions for the 2 × 2 model (two consecutive washes of two tanks in series). 2. Simplest Model: One Tank-One Wash The first washing strategy to analyze is the simplest situation, i.e., the washing of one tank by a single wash, with a given charge of water. This strategy might traditionally be the most employed in the food industry. Let us suppose a standard cylindrical tank of total volume V, with a flat base, a close flat top, and a common height/diameter ratio (H/D ) 2). Thus, the total surface area of the deposit (S) is given by
π 1/3 S ) 5 V2 2
( )
Usually, when the tank is completely empty and ready for cleaning, a very thin film of liquid is adhered to the inside surface (thickness ). Assuming that the whole internal surface of the tank is covered with this residual film, the volume of the film (F) can be calculated as
F ) S Tel.: (00) 34 956016376. Fax: (00) 34 956016411. E-mail:
[email protected].
(1)
(2)
Let us suppose that a product P is homogeneously
10.1021/ie010162j CCC: $22.00 © 2002 American Chemical Society Published on Web 04/23/2002
Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2441
dissolved or suspended in the residual liquid of the tank. If the concentration of P in this film is Cf, then, the load of this compound in the film. Lf, is
Lf ) FCf
(3)
Normally, the concentration of the compound P in the cleaning water (Cw), which is injected into the tank to carry out the cleaning operation is 0. Consequently, the load of the compound in the cleaning water (Lw) is also normally 0. Let us define the cleaning-water stream as the water that goes into a tank to do the cleaning, and the wastewater stream as the water that goes out of a tank once it has been cleaned. The total volume of water spent in a cleaning charge (W) depends on the flow of the cleaning-water stream (Q) and the injection time (t), as W ) Qt. During a cleaning session, after water injection, the total load of residual compound in the tank (Lt) is
Lt ) Lf + Lw
(4)
If complete mixing between the residual film and the cleaning water is supposed, then after the cleaning session, we will have a new homogeneous solution or suspension of P (the wastewater stream). This circumstance depends on the quality of the washing operation and the technology involved, as well as on the energy invested in the process (water pressure). Most modern washing systems produce a homogeneous stream from the tank after the cleaning session or from an additional deposit. Under these assumptions, the overall concentration of compound P in the wastewater stream (Ct) is
Ct )
Lf + Lw FCf + WCw Lt ) ) F+W F+W F+W
(5)
If we define R to be the ratio between the volume of water injected into the tank and the volume of liquid in the film, then we can evaluate the profusion of the washing operation as
W R) ) F
Qt π 1/3 5 V2 2
( )
(6)
As a consequence, the concentration of P in the outlet stream (Ct) can be expressed as a function of the washing profusion (R) in the form
Ct )
Cf + RCw 1+R
(7)
3. Generalization to n Tanks From an economic point of view, the optimization of a tank-washing process can be considered as the minimization of the ratio between the cleaning costs (Cc) and the total load of residual compound removed (Rr). This ratio is the target function for the optimization of the tank cleaning operation (Uo) and represents the cost of removing a unit of residual product P. Thus, the overall expression is
Uo )
cleaning costs (Cc) load of residual compound removed (Rr)
It follows that the total costs of the washing operation for n tanks include the cost of washing the machinery (M), the cost of the cleaning water (C), and the cost of pumping energy (E). The cost of washing the machinery includes the costs of fixed assets, maintenance, etc. It can be simply evaluated using the unitary cost concept. Thus, the machinery unitary cost (Um) is the total cost of washing the machinery divided by the total number of washes carried out during the machine’s economic life. Similarly, the cost of the cleaning water can be evaluated using the water unitary cost (Uw), that is, the price of a unit volume of cleaning water. This price normally includes subsequent depuration costs. Finally, the pumping cost can be calculated on the basis of the unitary energy cost (the cost per unit energy) (Ue) and the unitary consumption (the energy invested in injecting the volume unit of cleaning water, Uc). Thus, the total cleaning costs can be evaluated as
Cc ) M + C + E ) (nUm) + (nUwW) + (nUeUcW) (9) Moreover, the total load of residual compound removed in the cleaning operation of n tanks (n total washes) can be calculated from the concentration of P in the wastewater streams. Because the total amount of injected water in each tank (W) is normally removed after the cleaning of the tanks, we have
Rr ) (Ct - Cw)Wn
(10)
As a consequence, when we clean n tanks with n separate washes, the objective function (the unitary removal cost) acquires the value
Uo )
Um + UwW + UeUcW (Ct - Cw)W
(11)
Keeping in mind that cleaning water does not usually contain residual products (Cw ) 0) and making substitutions, eq 11 can be transformed into
Uo )
(
)( )
Um 1+R + Uw + UeUc RS Cf
(12)
Analyzing eq 12, it can be observed that the dependence of the unitary removal cost (Uo) on the washing profusion (R) is as expected, i.e., the higher the washing profusion applied in the cleaning operation, the more expensive the unitary removal cost, as R is in the numerator of the second factor. The influence of R in the denominator of the first term is less significant, because R usually takes values greater than 1, making this term negligible with respect to the others. Equation 12 also demonstrates the performance of the other variables involved in the target function. The unitary removal cost (Uo) linearly increases when the other unitary costs (Um, Uw, and Ue) and the unitary consumption (Uc) increase. However, the unitary removal cost decreases when other variables increase, such as the product concentration in the film (Cf) or the thickness of the film (). This is because, if a tank contains more residual product, a wash removes more, and the unitary removal cost diminishes. A similar argument can be made for the volume of the tank, because the unitary cost of larger tanks drops (economy of scale). 4. n Tanks-One in Series Wash Strategy
(8)
In this new washing strategy, we successively wash several tanks in series, equally stained, using the
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Figure 1. n tanks-one in series wash strategy.
Figure 2. Residual product concentration in the final wastewater steam (Ctn) for the n tanks-one in series wash strategy. Influence of the tank size (D) and number of washed tanks (n).
wastewater stream of one tank into the next (Figure 1). We suppose the same conditions as in the previous model. Starting with the general expression of eq 7, the concentration of residual product in the first tank wastewater (Ct1) is
Cw1 ) 0; Ct1 )
Cf 1+R
[
]
1 R + 1 + R (1 + R)2
(14)
The wastewater stream after washing n tanks has a concentration Ctn. Simplifying the resulting expression, we have
[ (1 +R R) ]
Ctn ) Cf 1 -
n
injection time (t), are shown. These functions can be easily derived from eqs 15 and 6. As in the previous case, the optimization strategy consists of minimizing the unitary removal cost (Uo). However, because the water is now reused, we have the cost expression
(13)
In the same way, if the wastewater of the first tank is used as cleaning water for the second tank (Cw2 ) Ct1), the concentration of residual product in the wastewater of this tank (Ct2) is
Ct2 ) Cf
Figure 3. Residual product concentration in the final wastewater steam (Ctn) for the n tanks-one in series wash strategy. Influence of the time of water injection (t) and number of washed tanks (n).
(15)
The product concentration of the wastewater stream will increase successively from the first to the nthe tank of the series. The dependence of the concentration of P in each stream (Ctn) on the number of washed tanks (n) is given by eq 15. In Figures 2 and 3, the theoretical results of Ctn versus several other variables involved in the process, such as the tank diameter (D) or the water
Cc ) (nUm) + (UwW) + (nUeUcW)
(16)
The total load of residual compound removed in this case (Rr) is calculated as the product CtnW, as the initial load W goes through all of the tanks and the entire residual product is removed in the final effluent. Uo can now be expressed as
Uo )
nUm + FRUw + nFRUeUc R n CfFR 1 1+R
[ (
)]
(17)
The minimization of this target function can be approached from two points of view: calculation of the optimum number of tanks for a given washing profusion or calculation of the optimum washing profusion for a given number of tanks. (a) Calculation of the Optimum Number of Tanks. If we apply the “singular-point” condition to eq
Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2443
17 (∂Uo/∂n ) 0), we obtain
[ ] ln
(1 +R R)
R -n G(u) 1+R ; n) ; G(n) ) R n G(n) + ln 1+R Um 1 UeUc + (18) G(u) ) UwF R Uw 1-
(
(
)
)
Equation 18 is expressed in an adequate form to make possible the calculation of n by iterative methods. The result obtained clearly indicates that, for a given set of conditions, an optimum number of tanks exists, which gives the minimum unitary removal cost. This is a consequence of the fact that the concentration of P in the wastewater stream tends asymptotically toward the initial concentration in the film as an increasing number of identical tanks are washed, increasing the load of product removed (the denominator of the target function). On the other hand, the total operation cost increases linearly with the number of tanks washed (numerator of the target function). Thus, a minimum value of the target function should exist. An analysis of eq 18 shows that the optimum value depends on only three factors: the washing profusion (R), the relative cost of energy with respect to water (UeUc/Uw), and the relative cost of fixed assets with respect to water (Um/UwF). Typical calculations might assume a washing profusion of 10, a relative energy cost of 0.01, and a relative machinery cost of 0.01. Under these conditions, the optimum number of tanks given by eq 18 is 26, a surprisingly high number that shows the extent to which a single cleaning water load can be exploited. An increase of the relative costs (i.e., an increase of the energy or machinery cost or a decrease of the water cost) decreases the optimum value. Finally, an increase of the washing profusion will increase the optimum number of tanks (e.g., for R ) 20, n ) 42). (b) Calculation of the Optimum Washing Profusion. Again applying the singular-point condition to eq 17 (∂Uo/∂R ) 0), we obtain the alternative expression
R)
[
]
n 1+R n 1 ; ; H(R) ) + + R H(u) R H(R) - 1 Um UwF H(u) ) (19) 1 UeUc + n Uw 1/(n+1)
As in the previous case, eq 19 is expressed in adequate form to make possible the calculation of R by iterative methods. There is also a given value of R that is optimum, giving the minimum unitary removal cost. Equation 19 shows, as before, that the optimum value depends on only three factors, which are now the number of tanks, the relative cost of machinery, and the relative cost of energy. A typical calculation taking the number of tanks as 10 and the relative costs as 0.01 will lead to an optimum R value of 1.8. An increase of the number of tanks increases the optimum (e.g., for n ) 20, R ) 3.9). 5. One Tank-i Consecutive Washes Strategy In this strategy, we wash only one tank, but we do it successively with different loads of fresh water (Figure
Figure 4. One tank-n successive washes strategy.
Figure 5. Residual product concentration in the wastewater steams (Cti) for the one tank-n successive washes strategy. Influence of the tank size (D) and number of washes (i).
4). If we suppose the same general conditions as in the previous strategies, we can calculate the concentration of P in the first wastewater stream (Ct1) from eq 7 as
Cw1 ) 0; Cf1 ) Cf; Ct1 )
Cf1 1+R
(20)
Once the first wastewater stream is removed, there is a new residual film in the surface of the tank, whose P concentration is assumed to be identical to that in the the wastewater. Thus, for the second wash, Cf2 ) Ct1. Following this, if we use fresh water for the new injection (Cw2 ) 0), the concentration of residual product is
Ct2 )
Cf (1 + R)2
(21)
Finally, after i washes with fresh water, the concentration of residual product in the wastewater stream is
Cti )
Cf (1 + R)i
(22)
Substituting eq 6 into eq 22, we have the variation of the concentration of P in the wastewater streams for different values of other variables, such as the tank diameter, as shown in Figure 5. As expected, the concentration decreases with the number of washes.
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Figure 6. Simplest consecutive-in series wash strategy. The 2 × 2 model.
The optimization of this strategy is based, as in previous cases, on the minimization of the target function (Uo), and the calculations are similar. The total operation costs, which include the machinery, water, and energy costs, now become
Cc ) (iUm) + (iUwW) + (iUeUcW)
(23)
After i washes of the tank, the total quantity of residual product removed will be given by the sum of the quantities in each operation, i.e. i
Rr )
∑ k)1
WCf
(1 + R)k
) WCf
[ ( )] 1
R
-
1
1
R 1+R
i
(24)
As a consequence, the target function in this case takes the form
Uo )
iUm + iFRUw + iFRUeUc 1 i CfF 1 1+R
[ (
)]
(25)
We can study the minimization of this new target function from two points of view once again: the optimum number of successive washes (i) or the optimum washing profusion (R). (a) Calculation of the Optimum Number of Washes. Analyzing eq 25 and supposing R to be constant, we can observe that, when i tends to ∞, Uo tends to ∞, but when i tends to 0, Uo tends to a constant value. Thus, if i grows, Uo will grow, and in this expression, no minimum value for i greater than 0 exists. In fact, the more washes done, he more product removed, and the higher the total operating costs. Every wash will have the same unitary cost as the last, but every wash will remove less product from the tank. As a consequence, for the one tank-i consecutive washes strategy, we can establish the optimum number of washes as 1 (the smallest possible value). Similarly, if we have n tanks and submit them to i consecutive in series washes, we will conclude that the optimum number of series to apply is 1. (b) Calculation of the Optimum Washing Profusion. In this case, we apply the singular-point condition to eq 25 for i washes, obtaining for the optimum washing profusion the expression
R ) J(R) - 1; J(R) )
[
]
iR + iJ(u) + (1 + R) 1/i ; 1+R 1 (26) J(u) ) FUw FUeUc + Um Um
Equation 26 is expressed in an adequate form to calculate R by iterative methods, as before. The optimum value depends again on only three factors, which now become the number of washes, the relative cost of water with respect to machinery, and the relative cost of energy with respect to machinery. Typical calculations, taking the number of consecutive washes as 3 and conserving the relative costs as before (FUw/Um ) 100 and FUeUc/Um ) 1), leads to an optimum R value of 0.07. This is a surprisingly low value, which indicates that, if we plan to carry out consecutive washes, it is better to minimize water use in each wash. Moreover, an increase in the number of washes decreases the optimum (e.g., for i ) 10, R ) 0.04). 6. Consecutive-In Series Wash Strategies The consecutive-in series wash strategies involve washing several tanks by simultaneously combining the two previous procedures. That is, we perform several consecutive washes of each tank and washes of several tanks in series. The simplest combined strategy of this type is the 2 × 2 model, which is shown in Figure 6. Each tank undergoes two consecutive washes, and every water load passes through two tanks in series. Thus, the second water load of each tank is always carried out with fresh water, and the second wastewater stream of each tank is used for the first water load of the following tank. We must now consider variables with two subindexes, where the first subindex refers to the number of the wash in a tank and the second refers to the number of the tank. All of the tanks start with the same concentration of P in the residual film (Cf11 ) Cf12 ) Cf1n ) Cf), but solely in the first tank is the first water load fresh water (Cw11 ) 0). Starting from the general expression 7, it can be calculated that the P concentration in the first wastewater stream of the first tank (Ct11) is
Ct11 )
Cf Cf11 + RCw11 ) 1+R 1+R
(27)
After the first wash of the first tank, the new residual film has the same P concentration as the removed
Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2445
Figure 7. Residual product concentrations in the first and second wastewater steams (Ct1n and Ct1n) for the 2 × 2 model. Influence of the number of tanks (n).
wastewater (Cf21 ) Ct11). All of the tanks are washed second with fresh water (Cw21 ) Cw22 ) Cw2n ) 0). Thus, the P concentration in the second wastewater stream of the first tank (Ct21) will be
Ct21 )
Cf Cf21 + RCw21 ) 1+R (1 + R)2
(28)
In the second tank, the first water load is the previous wastewater stream (Cw12 ) Ct21). Thus, the first wastewater stream of the second tank (Ct12) is
Ct12 )
Cf12 + RCw12 (1 + R)
)
RCf Cf + 1 + R (1 + R)3
(29)
The concentration of P in the second residual film of the second tank is equal to that in the previous tank (Cf22 ) Ct12). Thus, for the second wastewater stream of the second tank, we have
Ct22 )
Cf22 + Cw22R RCf Cf + (30) ) 2 1+R (1 + R) (1 + R)4
For n tanks, the equation corresponding to the first wastewater stream is n
Ct1n ) Cf
∑
k)0(1
Rk
)
+ R)2k+1 (1 + R)2n - Rn (31) Cf (1 + R)2n+1 - (1 + R)2n-1R
and the equation corresponding to the second wastewater stream is
(1 + R)2n- Rn ) Cf 2k+2 k)0(1 + R) (1 + R)2n+2 - (1 + R)2nR (32) n
Ct2n ) Cf
∑
Rk
In Figures 7 and 8, the influence of the number of tanks and the tank size on the P concentrations in the first and second wastewater streams are shown. As can be observed, the concentration in the first stream is
Figure 8. Residual product concentrations in the first and second wastewater steams (Ct1n and Ct1n) for the 2 × 2 model. Influence of the size of the tank (D).
much higher than that in the second (eq 31 has a smaller denominator than eq 32), similar to results obtained in the one tank-i consecutive washes strategy. Moreover, concentrations become approximately constant for the third and following tanks of the series, for practically any R value, because the first and the second tanks receive the two initial water loads with different P concentration, but all of the other tanks receiving similar loads. Thus, starting with the third tank, we can consider this strategy very similar to the first generalization studied, i.e., n tanks with n independent washes. In fact, every tank receives only one load of fresh water and releases only one stream of wastewater, and all of the streams have the same concentration. The difference is that, in this case, the P concentration in the outgoing wastewater stream corresponds to a typical wastewater from two washes. This situation makes the combined strategy much more efficient than the other does. As a consequence, the optimization study of this case can be developed in a very similar way to the generalization mentioned above, making the approximation that we have n tanks with n independents fresh loads but bearing in mind that we have two consecutive washes in each one. Thus, the total operation costs will be very similar to those obtained in the eq 9. Taking into account that the water cost is much higher than the others, if we neglect the first water load of the series, we have
Cc ) (2nUm) + (nUwW) + (2nUeUcW)
(33)
In addition, the total residual compound removed will be approximately given by the combination of eqs 10 and 31 if we neglect the ends of the series. Thus, we obtain
(1 + R)2n - Rn (34) Rr ) Ct1nWn ) CfWn (1 + R)2n+1 - (1 + R)2n-1R As a result, the form of the target function obtained here
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will be very similar to the one in eq 12
(
Um + Uw + 2UeUc Uo ) 2 RS
{
)
}
(1 + R)2n+1 - (1 + R)2n-1R Cf[(1 + R)2n - Rn]
(35)
As mentioned above, the second factor of this equation is approximately constant for n > 3. Moreover, this factor is smaller than the corresponding factor obtained in eq 12, because now approximately 10% more residual compound is removed than in that case (R ) 10). On the other hand, the first factor is now larger that the one in eq 12, because we spend twice as much in energy and machinery. However, for the values of the relative cost used, this factor is only 1% larger than the corresponding factor in eq 12. As a consequence, we can confirm that this combined strategy is better than n independent washes from the theoretical point of view. Finally, we can presume that a more exhaustive combined strategy (3 × 3, 4 × 4, ...) will lead to a more efficient washing process, as the second factor of eq 35 will be smaller at each successively higher level of combination. However, the first factor will be larger at every level, and there must be an optimum level of combination that minimizes the unitary removal cost. Analyzing the first factor of eq 35, we observe that it grows linearly with the combination level. Thus, for the values of the parameters used in this paper, this factor takes the values 1.011, 1.022, 1.033, 1.044, etc., at the combination levels 1 × 1, 2 × 2, 3 × 3, 4 × 4, etc., respectively. The rate of increment depends, of course, on the values of the relative costs used. On the other hand, the second factor of eq 35 is related to the inverse of the amount of residual compound removed in the operation. This quantity potentially decreases for successively higher levels of combination, as the P concentration in the wastewater streams grows with the number of washes in series (eq 15). Of course, the degree of increment in this case will be lower, because the combined strategies are not pure washes in series. Specifically, this factor takes the values 11.00, 10.09, 9.18, 8.73, etc., at the combination levels of 1 × 1, 2 × 2, 3 × 3, 4 × 4, etc., respectively. As a consequence, the combination of these two factors leads to an optimum combined strategy at a level of around 20 × 20 for the values of the parameters used above. Clearly, this level of combination is very complicated to install in industrial plants and does not result in a practical optimum, but once again, the surprisingly high number obtained shows the extent to which a cleaning water load can be conveniently reused.
resulting in a homogeneous level of cleaning and better performance. The mathematical structure of the combined strategies leads to a theoretical optimum level of combination around 20 for the typical conditions of the washing operation taken in this study (level of washing profusion, relative costs, etc.). This surprisingly high value is a direct consequence of the greater cost of water with respect to the other costs involved in the operation (machinery or energy), and it shows the extent to which a cleaning water load can be conveniently exploited from the theoretical point of view. Symbols C ) total water cost of a cleaning operation ($/unit) Cc ) total cleaning operation costs ($/unit) Cf ) concentration of compound P in the residual film (kg/ m3) Ct ) concentration of compound P in a tank during a washing operation (kg/m3) Cw ) concentration of compound P in a cleaning-water load (kg/m3) E ) total energy cost of a cleaning operation ($/unit) F ) volume of the residual film in the internal wall of the tanks (m3) G(n), G(u), H(R), H(u), J(R) and J(u) ) auxiliary functions i ) number of consecutive washes k ) index of the summation operator M ) total machinery cost of a cleaning operation ($/unit) n ) number of tanks to wash Lf ) load of compound P in the residual film (kg) Lt ) total load of compound P in a tank during a washing operation (kg) Lw ) load of compound P in a cleaning water load (kg) P ) any residual compound Q ) flow of cleaning water during an injection (m3/h) Rr ) total load of residual compound removed in a cleaning operation (kg/unit) S ) total internal surface area of a tank (m2) t ) time of cleaning-water injection for a wash (s) Uc ) unitary energy consumption in pumping (kWh/m3) Ue ) unitary energy cost ($/kWh) Um ) unitary machinery cost ($/wash) Uo ) unitary cleaning cost ($/kg) Uw ) unitary water cost ($/m3) V ) total volume of a tank (m3) W ) cleaning-water load in a wash (m3/wash) R ) washing profusion in a wash (cleaning water load/ volume of residual film) ) thickness of the residual film in the internal wall of the tanks (m) Subindexes n ) number of a tank in a series i ) number of a wash in a set of consecutive washes Double Subindexes
7. Final Considerations If the different washing strategies studied in this article are compared on the basis of the minimum unitary cost of cleaning, we can establish the following general considerations. For n tanks, it is logically better to apply one in-series wash than n independent washes. Moreover, it is also better to apply only one wash than more or only one series than more. However, an in-series strategy gives a different level of cleaning in each tank, and we must carry out subsequent cleaning operations. The consecutive-in series combined strategies are shown as a good alternative to n independent washes,
first ) number of a wastewater stream in a set of a tank second ) number of a tank in a series
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Ind. Eng. Chem. Res., Vol. 41, No. 10, 2002 2447 (4) Lo´pez Go´mez, A.; Virseda Chamorro, P. Contaminacio´n y economı´a del agua: Su relacio´n con la tecnologı´a enolo´gica. Vitivinicultura 1996, 72, 42. (5) Zavattoni, M. Tendencias en la limpieza industrial. Ing. Quı´m. 1996, 1, 104. (6) Martinez Nieto, L.; Garrido Hoyos, S. E. Reutilizacio´n de agua en las industrias de extraccio´n de aceite de oliva. Quı´m. Ind. 1994, 7, 13. (7) Gonza´lez, V. Minimizacio´n de efluentes en la industria alimentaria: Metodologı´a de trabajo. Aliment. Equip. Tecnol. 1993, 4, 103.
(8) Meltzer, M. Metal Bearing Waste Stream. Minimizing, Reycling and Treatment; Noyes Data Corporation: Park Ridge, NJ, 1990. (9) Shultes, I. Consideraciones sobre el lavado en la industria del tratamiento de superficies. Tecnol.Agua 1984, 12, 71.
Received for review February 20, 2001 Revised manuscript received October 26, 2001 Accepted February 11, 2002 IE010162J
