Wastewater Pretreatment by Normal Freezing Cool Thermal Storage Process with Convective Heat Transfer Mechanism

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 14, No. 2, pp (2011) 115 Wastewater Pretreatment by Normal Freezing Cool Thermal Storage Process with Convective Heat Transfer Mechanism Chao-Ching Chang 1, Chih-Ming Chang 1, Sung-Te Jung 2 and Cheng-Liang Chang 1 * 1 Energy and Opto-Electronic Materials Research Center, Department of Chemical and Materials Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. 2 Department of Communication Engineering, Oriental Institute of Technology, Taiwan, R.O.C. Abstract The effect of convective heat transfer mechanism on the wastewater pretreatment by the normal freezing cool thermal storage process was investigated. A mathematical model considering the heat and mass transfers in a normal freezing cool-thermal storage system using wastewater as the phase change material was developed and solved for the approximate solutions. A convective transport of heat in the water phase and constant heat sinking surface temperature of solid phase were considered and incorporated with the system equations to determine the temperature profiles of solid and liquid layers. The effects of system parameters on the solidification rate and the amount of solute removal were studied. The solidification rate would increase with increasing heat transfer coefficient in overall, but an inversed trend occurred initially. The optimal amount of thermal storage and the optimal solidification thickness had been determined. Differences around 10% in the optimum operating time and the cool-thermal storage would be made if different heat transfer mechanisms were assumed in the present aqueous system studied. Key Words: Cool-Thermal Storage, Wastewater Pretreatment, Convection, Modeling, Optimization 1. Introduction *Corresponding author. chlchang@mail.tku.edu.tw The in-advance removal of contaminants from the industrial wastewater prior to its disposal in order to meet the requirements of more and more restricted regulations is demanded by the authorities. Considerable saving in the capital and operating cost could result if the wastewater is concentrated with much smaller volumn through the application of the cool-thermal storage process. The cool-thermal storage contributes to improve the load management of nuclear power supply. If the wastewater is used as cool-thermal storage media, extra benefit of preconcentration could be obtained. The coolthermal storage can be used for the comfort system during the daytime. And the chilled water resulting from ice melting in the cool-thermal discharging period can be recovered or used as cooling water. Limited number of mathematical studies on the pure water freezing or ice melting have been reported in the literature [1 10]. If the wastewater is used instead, the mathematical analysis is more complicated because the mass transfer phenomena must be included [11 14]. The mathematical analysis of cool-thermal discharge system for producing chilled air was also reported [15]. The conductive heat transfer mechanism in the water layer was assumed in the previous studies [11 14]. The effect of natural convection or forced convection in such a mathematically complicated system is not considered before. In this paper, a one-dimensional dynamic mathematical model was developed for the wastewater normal freez-

2 116 Chao-Ching Chang et al. ing cool-thermal storage system with the initial wastewater at ambient temperature and constant temperature at heat sinking surface. A convective transport of heat in the water phase was considered. The optimal solidification thickness and the optimal amount of thermal storage were then estimated. 2. Solute Removal The mathematical models commonly used in the metallurgy studies to describe solute partitioning were employed to describe the solute distribution in the ice phase [16 19]. With a convective mechanism near the interface of ice layer and wastewater layer, we may assume that the concentration in the well-mixed liquid region is homogeneous and in equilibrium with that at the freezing interface of the ice layer at any instant. As the freezing of wastewater proceeds, the contaminant in the wastewater solution is either rejected from, or accumulates in, the freezing solid, leading to water purification. The solute removal per unit area of free surface W was estimated in a previous work [12]. (1) (2) where C 0 is the initial solute concentration and H is the distribution coefficient which is the ratio of solute concentrations in the solid and liquid phase at the interface. H is generally less than one, however, it could be greater than one in the purification of indium by zone refining [19]. The optimum thickness of solidification X* for maximum solute removal was also determined as 3. Temperature Distribution in the Ice Layer Consider the freezing of a body of wastewater of depth L in a sealed insulated tank by normal freezing with a constant excess temperature s at the heat sinking surface, x = 0, as shown in Figure 1. The excess temperature of wastewater L is a function of time. The excess temperature is defined as the difference of the temperature and the freezing temperature of wastewater. It is assumed that: (1) the solute concentration of the wastewater is low such that the physical properties of the dilute aqueous solution are nearly the same as those of pure water; (2) the increase in volume of ice layer due to solidification may be neglected. With assumption (1), the freezing temperature of wastewater would be 0 C. This is an unsteady state one-dimensional heat conduction problem, the differential equation is (6) where 1 is the thermal diffusivity of ice and 1 is the excess temperature of ice. An energy balance for the wastewater layer is made as (7) where h is the convective heat transfer coefficient, is (3) When H value is equal to unity, the purification of wastewater cannot be achieved. The maximum solute removal W max is (4) (5) Figure 1. Schematic diagram of a normal freezing cool-thermal storage system.

3 Wastewater Pretreatment by Normal Freezing Cool Thermal Storage Process with Convective Heat Transfer Mechanism 117 the density of wastewater and C p is the specific heat of wastewater. The boundary conditions are: x =0 1 = s (8) x = X 1 = 0 (9) (10) where Q m is the latent heat of solidification of wastewater and k 1 is the thermal conductivity of ice. The approximate solution of Eq. (6) is represented as (11) The boundary conditions Eqs. (8) and (9) would lead to G = 0 and J = s + I. The temperature distribution of the ice layer would thus be (12) The solidification thickness X, the parameter I and the excess temperature of wastewater L are functions of time and have to be evaluated. The derivative of X can be readily obtained from Eq. (10). Substituting Eq. (13) into Eq. (7) to obtain (13) (14) By definite integration of both sides of Eq. (6) with respect to x from zero to X, the derivative of I with respect to time can be derived as (15) Eqs. (13), (14) and (15) form a system of first order differential equations which can be solved by the mathematical software Mathematica if the initial values are specified. 4. Cool-Thermal Storage The amount of thermal storage may be calculated from (16) where C p1 is the specific heat of ice. Substituting Eq. (12) into Eq. (16) and performing the integration to get (17) If the operation of cool-thermal storage by normal freezing is brought out to an end at the optimal thickness of solidification X* for maximum solute removal, and the corresponding amount of cool-thermal storage is readily obtained from Eq. (17). (18) For the purpose of illustration, let us assign numerical values for normal freezing of an aqueous solution of CaCl 2 as follows: H = 0.5, C 0 = 10 kg/m 3, L = 0.1 m, k = 8.1 kj/m hr K, = 920 kg/m, Q m = 334 kj/kg, C p1 = 1.96 kj/kg K, C p = 4.20 kj/kg K, 1 = m 2 /hr. The value of X should be zero initially. However, X appears in the denominators of Eqs. (13), (14) and (15), a very small number would be taken to avoid the error in the numerical integration. The initial value of L is 25 C. The initial value of I is C as evaluated in a previous paper [14]. Figure 2 illustrates the time history of the excess temperature of wastewater L with various heat transfer coefficient h, higher the heat transfer coefficient is, faster the temperature decreased. It takes 0.75 hr for the wastewater temperature to drop down to 0 C if the heat transfer coefficient is 1500 kj/m 2 h r K, 1.75 hr if h =500 kj/m 2 hr K and 3 hr if h = 200 kj/m 2 hr K. One would postulate that the solidification rate should increase with increasing heat transfer coefficient since the convective thermal resistance is smaller for higher h, but it is not true

4 118 Chao-Ching Chang et al. Figure 2. Temperature of wastewater layer versus time for various heat transfer coefficients (L =0.1m, s = -20 C). at the initial stage as shown in Figure 3. The higher the heat transfer coefficient is, the lower the initial solidification rate is, and which can be explained by Eq. (13). The first term on the right hand side of Eq. (13) represents the solidification rate resulted from the heat removal by the conduction through the ice layer which is inversely proportional to the ice thickness X and the second term stands for the solidification rate slowed down by the heat supplied by the convection of wastewater bulk if the wastewater temperature L is higher than 0 C. The second term will be smaller if the value of h is smaller at the initial stage which will lead to a higher solidification rate with X and L about the same initially. However, L drops faster for the case of high h, as mentioned previously, and the solidification rate will turn faster and eventually overtakes that of lower heat transfer coefficient. The second term is vanished when L drops to 0 C and the convection of the wastewater layer will have no contribution to the solidification of the wastewater. Nevertheless, the solidification rate increases with increasing heat transfer coefficient in overall as shown in Figure 4, but the effect is insignificant which Figure 3. Solidification thickness versus time for various heat transfer coefficients at initial stage (L = 0.1 m, s = -20 C). Figure 4. Solidification thickness versus time for various heat transfer coefficients (L = 0.1 m, s = -20 C).

5 Wastewater Pretreatment by Normal Freezing Cool Thermal Storage Process with Convective Heat Transfer Mechanism 119 indicates that the convective thermal resistance in the water layer is insignificant when the ice layer is getting thicker. Figure 5 presents a comparison of the solidification rates when the convective and conductive heat transfer mechanisms were assumed. The data for the conduction case was obtained in the previous study [14]. The maximum process time difference with the same solidification thickness is about 0.75 hr. It can be concluded that the convective heat transfer caused by the natural convection or agitation will speed up the solidification rate, but a vigorous agitation is not necessary since it will not make a significant difference. Figure 6 illustrates the time history of the excess temperature of wastewater L with various surface temperature s, lower the surface temperature is, faster the temperature decreases. It takes 5.7 hr to solidify 0.1 m depth wastewater when the surface temperature is -40 C, and takes 11 hr if the temperature is -20 C as shown in Figure 7. Figure 8 illustrates the time history of the excess temperature of wastewater L with various depth of wastewater L s, deeper the depth of wastewater is, slower the temperature decreases. The optimum thickness of solidification X* for maximum solute removal is m evaluated from Eq. (3) when the distribution coefficient H is 0.5 in this case. The maximum solute removal W max calculated from Eq. (4) would be 0.25 kg/m 2. Lower the surface temperature, the optimal freezing time will be less as shown in Figure 9. The optimal freezing time is about 3 hr if s is -40 C and it takes double time if s is -20 C. The optimum amount of cool-thermal storage is obtained from Eq. (18), and is indicated in Figure 10 by the dashed line. Some results of the numerical example for solidification of aqueous CaCl 2 solution in more details are shown in Table 1. With s = -20 C and h = 200 kj/m 2 hr K, the optimum operating time and amount of cool-thermal storage are 6.32 hr and kj/m 2, respectively. With h = 1500 kj/m 2 hr K, the optimum operating time and amount of cool-thermal storage are 6.12 hr and kj/m 2, respectively. The optimal values of 6.99 hr and kj/m 2 were estimated when the conductive mechanism was employed [14]. Differences of 11% Figure 5. Solidification thickness versus time for different heat transfer mechanism (L = 0.1 m, s = -20 C). Figure 6. Temperature of wastewater layer versus time for various surface temperature (h=200 kj/m 2 hr K, L = 0.1 m).

6 120 Chao-Ching Chang et al. Figure 7. Solidification thickness versus time for various surface temperature (h = 200 kj/m2 hr K, L = 0.1 m). Figure 9. Solute removal versus time for various surface temperature (C0 = 10 kg/m3, H = 0.5, L = 0.1 m). Figure 8. Temperature of wastewater layer versus time for various depth of wastewater (h = 200 kj/m2 hr K, qs = -20 C). Figure 10. Amount of cool-thermal storage versus solidification thickness (L = 0.1 m, qs = -20 C).

7 Wastewater Pretreatment by Normal Freezing Cool Thermal Storage Process with Convective Heat Transfer Mechanism 121 Table 1. Results of numerical example s =-20 C, h = 200 kj/m 2 hr K s =-20 C, h = 1500 kj/m 2 hr K X (m) (hr) W (kg/m 2 ) Q s (kj/m 2 ) 10-4 X (m) (hr) W (kg/m 2 ) Q s (kj/m 2 ) ** ** * * *** *** *Optimum value; ** Complete solidification. and 14% in the optimum operating time and 8% in the cool-thermal storage would be made when the conductive heat transfer mechanism was assumed which is generally not a proper assumption in reality. 5. Conclusion The effect of convective heat transfer on the optimal operating conditions for the cool-thermal storage coupled with the purification of waste effluent from chemical plants by normal freezing has been investigated. The theory is based on heat conduction with moving boundary. Mathematical models used in metallurgy to describe the solute partitioning are employed to the solute distribution in the solid ice layer. Mathematical models of heat transfer of the freezing process were developed. The solidification thickness and the excess temperature of wastewater L could be obtained by integrating Eqs. (13), (14) and (15) simultaneously with proper initial conditions provided. However, the solidification process should be stopped at the optimal operation time when the maximum amount of solute in the waste effluent was removed. The optimal solidification thickness to remove maximum amount of solute in the waste effluent could be found from Eq. (3). The maximum amount of solute removal could be evaluated from Eq. (4) or (5). The optimum amount of cool-thermal storage could thus be determined from Eq. (18). The convective heat transfer near the interface of the ice and water layers will speed up the solidification rate, but a vigorous agitation is not necessary since it will not make a significant difference. The solidification rate would increase with increasing heat transfer coefficient in overall, but an inversed trend occurred initially. Differences around 10% in the optimum operating time and the cool-thermal storage would be made if different heat transfer mechanisms were assumed in the present aqueous system studied. Although the case study was focusing in an aqueous system, the equations derived could be used in any other separation system involving exclusion of solute by normal freezing with the initial temperature not at the freezing point of the solution. C 0 C p G h H I J k L Q s Nomenclature initial mass concentration of solute in wastewater (kg/m 3 ) specific heat (kj/kg K) parameter defined by Eq. (11) ( C) convective heat transfer coefficient (kj/m 2 hr K) distribution coefficient of solute parameter defined by Eq. (11) ( C) parameter defined by Eq. (11) ( C) thermal conductivity (kj/m hr K) depth of wastewater (m) amount of cool thermal storage per unit area of

8 122 Chao-Ching Chang et al. free surface (kj/m 2 ) Q m heat of solidification of water (kj/kg) W solute removal per unit area of free surface (kg/m 2 ) W max maximum solute removal per unit area of free surface (kg/m 2 ) x x axis (m) X solidification thickness (m) thermal diffusivity (m 2 /hr) ice density (kg/m 3 ) excess temperature ( C) time (hr) References [1] Penner, S. S. and Sherman S., Heat Flow through Composite Cylinders, J. Chem. Phys., Vol. 15, pp (1947). [2] Landau, H. G., Heat Conduction in a Melting Solid, Q. Appl. Mat., Vol. 8, pp (1951). [3] Weiner, J. H., Transient Heat Conduction in Multiphase Media, Brit. J. Appl. Phys., Vol. 6, pp (1955). [4] Goodman, T. R., The Heat-Balance Integral and Its Application of Problems Involving a Change of Phase, Trans. ASME, Vol. 8, pp (1958). [5] Roberts, L., On the Melting of a Semi-Infinite Body of Ice Placed in a Hot Stream of Air, J. Fluid Mech., Vol. 4, pp (1958). [6] Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed. Oxford Univ. Press, New York (1959). [7] Eckert, E. R. G. and Drake, R. M. Jr., Analysis of Heat and Mass Transfer, McGraw-Hill, New York (1972). [8] Yao, L. S. and Prusa, J., Advances in Heat Transfer, Academic Press Inc., San Diego, California (1989). [9] Yeh, H. M. and Cheng, C. Y., Cool-Thermal Storage by Vacuum Freezing of Water, Energy, Vol. 16, pp (1991). [10] Yeh, H. M. and Cheng, C. Y., Cool Thermal Storage by Vacuum Freezing of Water with Constant Volume Rate of Sublimation, Energy Convers. Mgmt., Vol. 33, pp (1992). [11] Chang, C. L., Ho, C. D. and Yeh, H. M., Cool Thermal Storage Coupled with Water Treatment by Normal Freezing of Dilute Aqueous Solutions, Int. Comm. Heat Mass Transfer, Vol. 29, pp (2002). [12] Chang, C. L., Pretreatment of Wastewater by Vacuum Freezing System in a Cool-Thermal Storage Process, Sep. Purif. Technol., Vol. 26, pp (2002). [13] Chang, C. L., Chen, Y. Z. and Chang, C. M., Optimization of a Vacuum Freezing Cool-Thermal Storage Coupled with Wastewater Treatment Process, Heat Transf. Asian Res., Vol. 33, pp (2004). [14] Chen, Y. Z., Chang, C. C., Chang, C. M., Chang, H. and Chang, C. L., Modeling and Optimization for a Normal Freezing Cool-Thermal Storage with Wastewater Preconcentration System, Tamkang J. Sci. Eng., Vol. 11, pp (2008). [15] Ho, C. D., Chang, C. L., Wang, C. K. and Su, Y. S., Producing Chilled Air from Ice Melting with Air Time-Velocity Variations in Cool-Thermal Discharge Systems, Int. Comm. Heat Mass Transfer, Vol. 30, pp (2003). [16] Pfann, W. G., Zone Melting, John Wiley & Sons Inc., N.Y. (1964). [17] Davies, L. W., The Efficiency of Zone-Refining Processes, Trans. AIME, Vol. 215, pp (1959). [18] Ho, C. D., Yeh, H. M., Yeh, T. L. and Sheu, H. W., Simulation of Multipass Zone-Refining Processes with Variable Distribution Coefficients, J. Chin. Inst. Chem. Engrs., Vol. 29, pp (1998). [19] Ghosh, K., Mani, V. N. and Dhar, S., A Modeling Approach for the Purification of Group III Metals (Ga and In) by Zone Refining, J. Appl. Phys., Vol. 104, (2008). Manuscript Received: Oct. 19, 2009 Accepted: Feb. 15, 2011