1、Marcio Nunes is mechanical engineer and researcher in the IPT - Institute for Research and Technology, So Paulo, Brazil. Using the Poppes Mathematical Method to Model the Thermodynamic Behavior of Evaporative Countercurrent Water Cooling Towers to Optimize Operation Marcio Nunes, MD Fellow ASHRAE AB
2、STRACT This article deals with the use of the method proposed by POPPE on the thermodynamic analysis of evaporative water cooling towers of the countercurrent type. This method was developed in the 70s and is more precise in analysis than the traditional method developed by Merkel, which is well kno
3、wn and the most used up to today. This study is a synthesis of authors mastering degree dissertation (Nunes, M., 2014). The Poppe method does not introduce the simplifications present in Merkels method, and, among the advantages it holds in, it stands out to allow a more complete analysis of heat an
4、d mass transfer processes between water and air inside the tower. On the other hand, however, its use is not so simple, because the calculation involves the solution of a system of ordinary differential equations. To solve the Merkel equation, however, a simple electronic calculator is enough. This
5、paper presents the results of seven water cooling towers performance tests of various sizes, carried out in different places and seasons, which had initially been tested using the Merkel method. The Poppe method was used after those tests, using the same measured variables collected earlier, and the
6、 results obtained by both methods were compared. These results are presented here. Due to its good precision and reliability, furnishing more information about the thermodynamic behavior of the tower and besides the availability of mathematical tools to solve the equations, this method is intended t
7、o be used to develop new geometries and materials for the fills and also to optimize the operation with respect to the energy consumption in evaporative countercurrent towers. INTRODUCTION Historically, the basic theory of cooling towers operation was proposed and published for the first time in the
8、 early 1920s, although cooling towers already existed since from the previous century. The practical use of basic differential equations, however, was first presented by Merkel (Merkel, 1925) in Germany, when he combined equations for the mathematical determination of the transfer of heat and mass i
9、nside the tower. This theory became known in 1941, when it was translated from German to English and published in the US. He adopted the model that used the enthalpy difference as the driving force to settle the exchange of sensible heat and latent heat. With the development and diffusion of the com
10、puter, however, new mathematic tools came in and other complex methods began to be known and used by the engineers. THE MERKEL METHOD The Merkel model is surely the best known and used in the evaluation of evaporative (open circuit) cooling towers. For over 50 years, it reigned supreme in the field
11、of design technologies, operation and testing of these towers. It is the most cited in the literature and is also the method used in most of technical standards for acceptance and rating tests. According to Kloppers (Kloppers and Krger, 2005), this model is based on assumptions which reduce the comp
12、utations at a relatively simple procedure and, as a result, it doesnt represent strictly the processes of heat and mass transfer occurring in an evaporative cooling tower. However, because of its simple approach, relative good accuracy and ease of use, it has long been the preferred method by manufa
13、cturers and technical standards for evaluation of the performance of towers and acceptance criteria, being used up to today. And, because it is a very popular model, it is object of study by various researchers over time, generating studies of improvements, computer programs and comparisons with oth
14、er methods. Figure 1 Countercurrent type tower It was developed for countercurrent type tower, where water and air flow in opposite directions, as can be seen in Figure 1. Merkel proposed a model that relates evaporation rates and the sensible heat exchanges that occur in direct contact between wate
15、r and air, during the water cooling, to a simple difference of enthalpy of the air. This assumption was convenient (but not limited) to various types of cooling towers. The derivation of the equations was performed for towers operating in countercurrent. To build his model, Merkel established seven
16、assumptions whose purpose was to simplify the mathematical calculations: moist air leaving the tower is considered saturated. resistance to heat transfer in the liquid film is negligible. moisture content of the air is proportional to its partial pressure in the mixture. its not considered the reduc
17、tion in the water flow due to evaporation losses in the mass and energy balances. specific heat of moist air is considered constant and equal to the dry air. Lewis factor Lef which lists the coefficients of convective heat and mass transfer is equal to 1. specific heat of water is considered constan
18、t and numerically equal to 1 (only valid for units Btu/lbmoF and kcal/kgoC). At first glance, it may seem inconsistent to consider that the water vapor present in the humid air that comes out from the tower is at saturated state, while this method doesnt consider the evaporation that occurs inside t
19、he tower. Physically, it is known that the air is able to contain water vapor in the superheated state (and therefore transparent to the human eye) at a given temperature and air pressure, and there is a maximum quantity of steam that humid air can retain. This condition is known as state of air sat
20、uration at that temperature and pressure. Whether fully or partially saturated, water vapor will have its own vapor pressure, according to Daltons Law. Considering the simplifications listed above, was developed the following equation, known as Merkel equation or Merkel number: 12TT awwe hhdTmaVKM(1
21、) The solution of this equation was initially made by graphical method, either by measuring directly the area under the curves or by the method of squaring. However, the integration method used today is the method called the Chebyshev four regions. By this method, the integral is approximated by the
22、 sum of the small areas represented by four linear segments. THE POPPE METHOD The method presented by Poppe arose in the early 1970s and doesnt behold the simplifications presented in Merkel method. It isnt a completely new method, but its calculation takes into account the evaporation of water insi
23、de the tower, and also allows that all the properties of the air and water be calculated at each point inside. It also allows considering the possibility of the moist air leaving the tower being unsaturated, saturated or supersaturated. The Poppe method is not as so simple to implement and requires
24、solving a system of multiple ordinary differential equations. It can be solved in one-dimensional form for countercurrent towers, but requires a two-dimensional calculation for other settings. The Poppe model brings more advantages over Merkel method. For example, it allows the calculation of temper
25、ature and humidity of the air at the outlet; which will permit to determine precisely the amount of water evaporated during the process, making this method suitable for the design of cooling towers. Thus, it is possible to determine the correct state of the water vapor leaving the tower. The Poppe m
26、ethod arose at a time when computers were more developed and available, and also when new materials had been developed to be used in the tower fills. Before the 70s, most of the towers had fills build up with wooden slats. With the advent of plastic contact film fills in the 70s, there was the possi
27、bility of replacing the timber by these new materials, leading to the need to develop more accurate mathematical methods to evaluate these new incoming towers. The methodology proposed by Poppe was developed for evaporative towers of the countercurrent type (Figure 1) and it is based on the followin
28、g considerations: air and water flows are in the steady state condition. tower cross-sectional area is constant throughout the fill. heat exchange through the walls of the tower to the environment is negligible. heat exchanges of fans to the air and water are negligible. water drag losses are neglig
29、ible. heat and mass exchange occurs in the transverse direction relative to the flow of air and water. heat and mass exchange occurs uniformly on a given cross-sectional area of the fill. air temperature and humidity inside are uniform in a given cross section. Figure 2 Control volumes for Poppes mo
30、del Being versatile, the user can set the differential equations in various formats, depending on the objectives to be achieved. Calculation is concentrated in the tower fill following the scheme of Figure 2, above, from bottom to the top of it. This figure shows the control volumes used for heat an
31、d mass exchange between humid air and water. This article presents the application of Poppes method to evaluate the performance of seven evaporative countercurrent type towers based on the following differential equations. These four equations were developed for unsaturated air-water mixture and tak
32、ing the water temperature as the independent variable; the dependent variables are: the enthalpy of moist air (hma), the moisture content of the air (w), the Merkel number (MeP) and the fill height (z): wpwswswvmam a s wefmam a s wswawpww TCwwwwhhhLhhwwmmCdTdw )()()1()(2) wpwswswvmam a s wefmam a s
33、wswawpwapwwwmaTCwwwwhhhLhhwwmmCmCmdTdh)()()1()(1(3) wpwswswvmam a s wefmam a s wpwweP TCwwwwhhhLhh CdTdM )()()1( (4) tdwpwswswvmam a s wefmam a s w wpww aAhTCwwwwhhhLhh mCdTdz )()()1( (5) The solution of these differential equations was performed by means of numerical integration. The method used is
34、 the fourth order Runge-Kuttas approximation method, which can be easily implemented in Matlab program using its build-up ODE45 function. COMPARISON OF BOTH METHODS IN PERFORMANCE TESTS Performance tests of cooling towers are useful not only for manufacturers in the search for greater thermal effici
35、ency, but also for owners, who often want to know how is the performance of the equipment, especially about the operational condition of the fill overtime. Testing cooling towers with respect to its thermodynamic behavior is not an easy task to accomplish, especially if the tower is of great proport
36、ions. The main difficulty is the measurement of the air flow. Fans belonging to large induced draft evaporative towers have rotors of large diameters, which turns measuring the air flow a complex task that can produce non precise results. Poppe and Merkel methods have different purposes. The applica
37、tion of Merkel method results in the determination of the Merkel number, given by equation 1 above. These coefficients give the relation between the heat exchange capacity and the water flowrate. To calculate this number, it is necessary to know the relation between the water and air flowrates, L/G.
38、 As the Merkel model was developed for humid air leaving the tower in the saturated state, the best results will be achieved for towers working close this condition. Table 1 Results of performance tests of seven towers using both methods Tower Method Cross sectional area (ft2) (m2) Overall fill heig
39、ht (ft) (m) L/G - KaV/L - 1 Merkel 861 80 8.2 2.5 1.66 1.68 Poppe 861 80 8.2 2.5 1.66 1.89 2 Merkel 861 80 5.2 1.6 1.11 1.39 Poppe 861 80 5.2 1.6 1.11 1.52 3 Merkel 1,938 180 3.6 1.1 1.58 0.87 Poppe 1,938 180 3.6 1.1 1.58 0.83 4 Merkel 1,938 180 3.6 1.1 1.57 0.79 Poppe 1,938 180 3.6 1.1 1.57 0.75 5
40、Merkel 1,615 150 9.9 3.0 0.58 3.95 Poppe 1,615 150 9.9 3.0 0.58 4.63 6 Merkel 1,615 150 6.2 1.9 1.45 1.29 Poppe 1,615 150 6.2 1.9 1.45 1.40 7 Merkel 13 1.2 3.3 1.0 2.84 0.55 Poppe 13 1.2 3.3 1.0 2.84 0.54 The use of Poppe method leads to the determination of other parameters, like air temperature an
41、d water content of the moist air leaving the tower, the temperature profile inside the fill, the flowrate of evaporated water, etc., besides the Merkel number. In order to compare both methods, Poppe model was applied to seven cooling tower previously tested for performance using Merkel method. Were
42、 used the same measured variables to feed the Poppe equations. The main results are presented in Table 1 above. The first six towers tested were installed in thermal electric plants; the seventh tower was installed in an air-conditioning system. They all were tested in steady-state condition and onl
43、y at one operational point, with the air and water flowing in constant rate. All the measured parameters were monitored for at least one hour, with the readings made at intervals of 90 seconds. The variation of each parameter was not greater than 2.5%. Figure 3 Plotted results of tests using Merkel
44、and Poppe methods. Figure 3 presents the results of Table 1 in graphical aspect. The green line is not a mean of the results, but it represents only their tendency. Figure 4 Poppe method results for tower 5 from Table 1, which fill is 9.9 ft (3 m) height Figure 4 presents an example of application o
45、f Poppes method, referred to tower 5. The mathematical model was implemented for unsaturated, saturated and supersaturated condition for the output air. Were plotted six variables throughout the tower, varying from bottom to top inside the fill, which represents the variation of each one from the co
46、ndition of the air entering the tower to its output. In this case, the coefficient KaV/L is given by the 0123450,0 0,5 1,0 1,5 2,0 2,5 3,0Coefficient KaV/LR e l ation L/ GMe rk e l Po p p ePoppes Merkel number MeP and, in order to compare it with Merkels coefficient Me given by equation (1), MeP mus
47、t be read at the top of the fill in Figure 3-(f), that corresponds to 9.9 ft (3 m) fill height in the axis for this tower. DISCUSSION This study showed that, despite the Merkel method presents many simplifications, it produced results very close to those produced by a more complex and more exact met
48、hod. Merkel model surprised by its simplicity, ease of operation and production of good results, as can be seen in the Figure 3. The best results are obtained when the output humid air is closer to the saturated condition. The development of this study showed that today it is possible to easily impl
49、ement and solve a more elaborated mathematical model, containing a system of ordinary differential equations, and use them in the survey of performance of evaporative countercurrent water cooling towers. It also showed that Poppes method is superior to Merkels, because it provides much more information about the thermodynamic behavior of the tower. Another advantage of Poppe method refers to the fact that it is possible to determin
