Cooling Tower Modelling

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Modeling of existing cooling towers in ASPEN PLUS using an equilibrium stage method ARTICLE in ENERGY CONVERSION AND MANAGEMENT · DECEMBER 2012 Impact Factor: 4.38 · DOI: 10.1016/j.enconman.2012.03.030

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Energy Conversion and Management 64 (2012) 473–481

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Modeling of existing cooling towers in ASPEN PLUS using an equilibrium stage method João A. Queiroz a,b,c, Vitor M.S. Rodrigues b, Henrique A. Matos c, F.G. Martins a,

⇑

a

LEPAE, FEUP-DEQ, Universidade do Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal Dow Portugal, Produtos Químicos, S.U.L., R. do Rio Antuã, n.1, 3860-529 Estarreja, Portugal c CPQ, IST-DEQ, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b

a r t i c l e

i n f o

Article history:

Received 12 January 2012 Received in revised form 29 March 2012 Accepted 29 March 2012

Keywords:

Cooling tower ASPEN PLUS Simulation Equilibrium stages Murphree efficiency

a b s t r a c t

Simulation of cooling tower performance considering operating conditions away from design is typically based on the geometrical parameters provided by the cooling tower vendor, which are often unavailable or outdated. In this paper a different approach for cooling tower modeling based on equilibrium stages and Murphree efficiencies to describe heat and mass transfer is presented. This approach is validated with published data and with data collected from an industrial application. Cooling tower performance is simulated using ASPEN PLUS. Murphree stage efficiency values for the process simulator model were optimized by minimizing the squared difference between the experimental and calculated data using the Levenberg–Marquardt method. The minimization algorithm was implemented in Microsoft Excel with Visual Basic for Applications, integrated with the process simulator (ASPEN PLUS) using Aspen Simulation Workbook. The simulated cooling tower air and water outlet temperatures are in good accordance with experimental data when applying only the outlet water temperature to calibrate the model. The methodology is accurate for simulating cooling towers at different operational conditions.  2012 Elsevier Ltd. All rights reserved.

1. Introduction

Cooling towers are widely employed in many industrial applications for rejecting waste heat from the process to the environment. The principle behind a cooling tower operation is evaporative cooling which, in theory, would allow circulating water to equal ambient air wet-bulb temperature. Evaporative cooling is a process with simultaneous mass and heat transfer between air and circulating water. There are several methods and strategies related to the modeling of cooling towers with different levels of complexity. According to Jin et al. [1], the first theoretical analysis of cooling towers was performed by Dr. Fredrick Merkel in 1925. He proposed a theory relating evaporation and sensible heat transfer where there is counter flow contact of water and air. As described by Benton et al. [2] Merkel expressed the number of transfer units (NTU) as a function of the integral of the water temperature difference divided by the enthalpy gradient where, to reduce the governing relationships to a single separable ordinary differential equation, several simplifying assumptions were made: Merkel assumed that the Lewis factor, relating heat and mass transfer was equal to 1; the air exiting the tower was saturated with water vapor; and

⇑

Corresponding author. Tel.: +351 22 508 1974; fax: +351 22 508 1449. E-mail address: [email protected] (F.G. Martins).

0196-8904/$ - see front matter  2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.enconman.2012.03.030

the reduction of water flow rate by evaporation was neglected in the energy balance. Kloppers and Kröger [3] evaluated three methods used in cooling tower design, namely, Merkel, Poppe and effectiveness-NTU and gave a detailed derivation of the heat and mass-transfer equations of evaporative cooling in cooling towers. Based on Merkel equation, Picardo and Variyar [4] presented a power law that related packed height with excess air and determined equation parameters for air wet-bulb temperature between 10 and 34 C and cooling range between 40 and 20 C. They also showed that beyond a certain air flow the reduction in packed height does not justify the increase in energy utilization for air compression. Castro et al. [5] developed an optimization model for a cooling water system composed of a counter flow tower and five heat exchangers where the thermal and hydraulic interactions in the overall process were considered. They observed that forced withdrawal of water upstream of the tower is an important resource for fulfilling cooling duty requirements. Khan et al. [6] presented a fouling growth model where it was demonstrated that the effectiveness of a cooling tower degrades significantly with time, indicating that for a low fouling risk level ( p = 0.01), which is the probability of fill surface being fouled up to a critical level after which a cleaning is needed, there is about 6.0% decrease in effectiveness. Al-Waked and Behnia [7] applied computational fluid dynamics (CFD) for natural draft wet cooling tower. The difference between outlet air temperature predicted by the CFD model and


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J.A. Queiroz et al. / Energy Conversion and Management 64 (2012) 473–481

Nomenclature

F (k) G H [F (k)] I L m N P RH s T x y Y Y Y b

b Dk

sum of squared error between experimental and model prediction values humid air mass flow (kg/h) Hessian matrix of F (k) identity matrix water mass flow (kg/h) number of data points in each sub-set number of equilibrium stages pressure (kPa) relative humidity (%) search direction temperature ( C) independent variables vapor composition on equilibrium stage model output average value of the model output experimental value control coefficient in Levenberg–Marquardt method increment of k 

design results was less than 3%. Jin et al. [1] proposed a model based on heat resistance and energy balance principles where empirical parameters were introduced, avoiding the need to specify geometrical parameters. Rubio-Castro et al. [8] determined optimal cooling tower design parameters and temperature profiles across a counter flow cooling tower by applying a rigorous heat and mass transfer model. Non-linear algebraic equations were solved using a discretization approach with a fourth-order Runge–Kutta algorithm. Given a set of experimental data to train the model, Hosoz et al. [9] suggested that applying artificial neural networks (ANNs) for modeling the cooling tower performance avoided the solution of complex differential equations. Predicted and experimental values had correlation coefficients in the range of 0.975–0.994 and mean relative errors in the range of 0.89–4.64%. Pan et al. [10] presented a data-driven model-based assessment strategy to investigate the performance of an industrial cooling tower. Considering 1 month test interval and based on water mass flow rate, water inlet temperature, air dry-bulb temperature, relative humidity and fan motor power consumption the predicted water outlet temperature was within a ±5% error band and presented a mean square error of 0.29 C. Serna-González et al. [11] used mixed-integer non-linear programming (MINLP) techniques to evaluate the optimal conditions of a mechanical draft cooling tower that minimize the total annual cost for a given heat load, dry- and wet-bulb inlet air temperatures and temperature constraints on the cooling water network. Rao and Patel [12] compared the results obtained by Serna-González et al. [11] with the ones achieved when applying an artificial bee colony algorithm. Using the artificial bee colony algorithm resulted in an objective function value lower than the one achieved by Serna-González et al. [11] for all six case studies (improvement between 1.27% and 11.17%). As an alternative to the abovementioned methodologies a different approach that does not involve the solution of differential equations can be used to model a cooling tower operation by applying an equilibrium stage. While the equilibriumstage method can hardly be used for design purposes without proper correlations that allow the determination of the height equivalent to a theoretical plate (HETP) it is demonstrated in this work that both the outlet water and air temperature predicted by the model are quite accurate when compared to the experimental values.

e k rF (k)

allowable difference between two consecutive iterations model parameters gradient of F (k)

Subscripts i relative to component i j relative to stage j k kth iteration l lth experimental point test relative to test data sub-set trn relative to train data sub-set val relative to validation data sub-set Superscripts atm relative to ambient conditions a relative to air stream exp experimental data mod model predicted values w relative to water stream

2. Problem statement

When simulating a cooling water network with a process simulator such as ASPEN PLUS there is no standard procedure for setting up the cooling tower block. This means that when process alternatives are being studied or real-time optimization is being performed, there is no direct way of evaluating the impact that any change in the cooling water temperature and flow or change in ambient air conditions has on the cooling tower performance and, consequently, on the process itself. The aimof this work is therefore to describe a methodology that enables the simulation of a counter flow, induced draft, cooling tower in ASPEN PLUS. The outcome of the proposed approach is a model that simulates the behavior of a real cooling tower and is capable of working as a stand-alone model or be integrated into a larger simulation model. It should provide grounds for optimization studies, where off-design conditions such as water and air flow variations can be simulated; debottlenecking studies, where equipment limitations may be of interest as production rates are increased; and operability studies, where evolution of equipment performance can be assessed to help plant personnel troubleshoot operations. Given a set of data consisting of water and air inlet temperature, water and air inlet flow, air inlet humidity and ambient pressure the problem then consists in determining model parameters that mimics actual cooling tower performance. Although it is possible to use the model to calculate evaporation ratio and cooling tower heat duty, only the output variables which could be compared with the published data by Simpson and Sherwood [13] i.e., water and air outlet temperatures, were presented. Moreover, the model was developed considering the following assumptions: 1. Pressure drop across the cooling tower was not considered because the model was not used for design purposes and it has minor effect when compared to ambient pressure. 2. The operation of cooling tower was assumed as an adiabatic process. 3. The water stream was considered to be pure as impurities in this stream do not significantly change equilibrium properties.


475


4. The cooling tower was in steady-state operation.

Air out (T a out , Gout , RH out )

3. Modeling of a cooling tower in ASPEN PLUS

The proposed cooling tower model is implemented in ASPEN PLUS [14]. This process simulator provides several built-in model blocks that can be directly applied in process simulation. Additionally, this process simulator has an extensive physical property database where the stream properties required to model the material streams in a plant are available. For simulating the cooling tower in ASPEN PLUS an approach based on equilibrium stages is applied. For this purpose ASPEN PLUS provides the RADFRAC built-in block considering neither reboiler nor condenser, which allows the calculation of the liquid and vapor/gas equilibrium on each equilibrium stage.

Water in ( T w in , Lin)

Stage N-1

Stage N

Air in (P atm , T a in , G in , RH in)

Water out ( T w out , Lout ) Fig. 1. Representation of a cooling tower with N equilibrium stages. The air stream

3.1. Step 1 – set up the model in ASPEN PLUS

Cooling towers operate at relatively mild temperature and pressure so it can be assumed that the behavior of the thermodynamic equilibrium between water and air is close to ideality. This fact is confirmed comparing the results achieved using the cooling tower general model when applying the ideal property method and other method that takes into account non-idealities (NRTL). Moreover, considering a base case data and two scenarios with ±10% variations on RH in, T a in, T w in, L in and G in the results are identical independently of the chosen property method (Table 1). Thus, it can be concluded that assuming an ideal behavior of the system substances (water and air) is a good approximation for this process. For defining the RADFRAC block in ASPEN PLUS the following parameters were specified in ASPEN PLUS:

enters the cooling tower at the bottom stage and leaves at the top stage; it is characterized by its dry bulb temperature ( T a), flow (G) and relative humidity ( RH ). Water stream enters the cooling tower at the top stage and leaves at the bottom stage; it is characterized by its temperature (T w ) and flow (L).

parameters for a given cooling tower by calculating the number of equilibrium stages and Murphree efficiency of each stage. Heat exchangerperformance stronglydepends on thecooling mediumsupply temperature, therefore the most relevant parameter when considering a cooling tower operation is the water temperature (T w out ) that this equipment is able to provide given certain operational conditions. Model parameters values are adjusted by using a set of experimental data (l = 1, . . . , m) and by minimizing objective function (F ), which is the sum of the squared difference between exp experimental, T wout , and model outputs, T wout mod (Eq. (2)). l l ;

X m

– Calculation type set as ‘Equilibrium’. – Inlet water entered the block ‘Above-Stage’ on the first stage. – Outlet water left the block on the last stage. – Inlet air entered the block ‘On-stage’ on the last stage. – Outlet air left the block on the first stage. – Reboiler and condenser set as ‘None’. – Efficiency type set as ‘Murphree efficiencies’ on each stage. Murphree efficiency is applied when accounting for deviations from ideality, i.e., considering that in an equilibrium stage, liquid and vapor phases do not reach thermodynamic equilibrium. Eq. (1) defines Murphree vapor efficiency, Eff M i j , for component i on stage j, where yi, j and xi, j are the vapor and liquid composition, respectively, and yi j is the composition of the vapor that would be in equilibrium with the liquid leaving the equilibrium stage. Fig. 1 shows the schematic representation of an equilibrium stage in a cooling tower. yi; j  yi; jþ1 M Eff i; j ¼  ð1Þ yi; j  yi; jþ1 ;

;

F ¼

;

;exp ;mod 2 ½T wout  T wout  l l

ð2Þ

l¼1

Eq. (2) can be rewritten as Eq. (3) to give a more general formulation of the function to be minimized [15]. Each experimental measurement is described by x l, which is the independent variables vector and corresponds to the measured inlet streams values (Table 2). mod Model output, T wout ( xl, kk), specified in Table 2 depends on both l the independent variables, xl, and the model parameters, kk. The minimization of the objective function is accomplished by applying themethoddescribed in this work,which consists on an iterativeapproach thus; model parameters are adjusted for each kth iteration. mod ð xl kk Þ, equal Ideally the model would return an output, T wout l wout exp to the measured value, T l , given a certain set of inlet parameters, x l. Therefore, for m experimental data points, the goal is to find the model parameters that correspond to the minimum value of the objective function described by Eq. (3), F (kk). ;

;

;

;

X m

F ðkk Þ ¼

;exp ½T wout  T wout ;mod ð xl ; kk Þ2 l

ð3Þ

l¼1

3.2. Step 2 – determination of model parameters

Once the general model representing the cooling tower has been implemented on ASPEN PLUS it is necessary to determinethe model

Regarding the particular case of a cooling tower the parameter vector (kk) to be adjusted is the number of equilibrium stages, N , and Murphree stage efficiencies, Eff M i j . Hence, the minimization problem is subject to the following constraints: ;

Table 1

Comparison between the outputs of the model considering NRTL and IDEAL property methods. The number of equilibrium stages was set to 2 and Murphree stage efficiencies to 1. Atmospheric pressure was kept constant at 101.3 kPa.

Inputs in

RH (%)

Base case +10% 10%

80 88 71

Outputs a in

T

30 33 27

(C)

w in

T

37 41 33

(C)

in

L (kg/h)

7.5 8.3 6.7

in

G (kg/h)

10 11 9

T w out (C)

T a out (C)

NRTL

IDEAL

NRTL

IDEAL

29.5 33.3 25.6

29.5 33.3 25.6

32.4 35.9 28.6

32.4 35.9 28.6


476


Independent variables [ x]

Model variables [k]

Dependent variables [Y ]

of stages should be incremented in order to avoid an inconsistent stage efficiency value. The algorithm must then be reinitialized, taking into account the new number of equilibrium stages (N + 1).

– Inlet air temperature

– Stage 1 efficiency

– Outlet water temperature

3.2.2. Step 2.2 – Murphree stage efficiencies (Eff w j )

Table 2

List of independent, dependent and model variables.

a

– Inlet water temperature – Inlet air flow

– Stage 2 efficiency

– Inlet water flow – Inlet air humidity – Atmospheric pressure

– S tage N efficiency

..a .

The number of model variables depends on the number of equilibrium stages.

M

s:t : 0 < Eff i; j

6

1

N > 1 N is integer

To determine model parameters an algorithm was established, as shown in Fig. 2. This algorithm was implemented in Microsoft Excel, using Visual Basic for Applications, and simulations were performed in ASPEN PLUS. Information flow between ASPEN PLUS and Microsoft Excel was enabled by Aspen Simulation Workbook. Considering the set of experimental data regarding the operation of a given cooling tower the first step is to divide it into three sub-sets: training (trn), validation (val) and test (test ). Training and validation sub-sets are used to determine model parameters and are inputs of the algorithm; the test sub-set is used to confirm whether the model parameters returned by the algorithm are adequate when applied to an independent set of data. When considering only a training sub-set to adjust model parameters, each iteration would bring the difference between model and experimental values closer to zero. However, the inconvenient of this approach is that there would be a point where the model is too adjusted to that specific set of data, resulting in a model very dependent of the data set used to train the model. To avoid this situation a second set of data, the validation sub-set (val), is used to avoid the model overfitting. During the starting iterations it is expected that the error between model and experimental values decreases for both training and validation sub-sets as model parameters start to be adjusted. Despite the fact that the objective function value for the training sub-set decreases as the algorithm progresses (F trn(kk+1) < F trn(kk)), there can be point where the value of the objective function regarding the validation sub-set reaches a local minimum and then starts to increase. Beyond this point it is considered that the model becomes dependent of the training sub-set data and the inner algorithm stops (Fig. 2), thus assuming that the model parameters corresponding to this minimum is the most adequate. 3.2.1. Step 2.1 – number of equilibrium stages (N)

The problem, as described, represents a mixed integer non-linear programming (MINLP) problem. The algorithm introduced in this work transform the MINLP into a non-linear problem (NLP) model by setting a fixed value to the number of stages, eliminating the integer constraint. Fig. 2 shows howthe algorithm is structured and how the number of equilibrium stages initial estimate is confirmed at the end of inner algorithm. With this approach, the NLP minimization problem can be solved by using the Levenberg–Marquardt search method. The initial guess for the number of equilibrium stages is set to be the minimum allowed by ASPEN PLUS RADFRAC block, N = 2. Murphree efficiency for each stage is then determined using the Levenberg– Marquardt method and, if the algorithm generates a result corresponding to Murphree stage efficiencies higher than 1, the number

M ;

Levenberg–Marquardt is in nature an improved Gauss–Newton method by incorporating steepest-descent method into the iterative update scheme, using a search direction between these two methods. In the Levenberg–Marquardt method the search direction, sk, is determined by solving Eq. (4), where H [F (kk)] is the Hessian matrix of F (kk), bk is the control coefficient of the Levenberg–Marquardt method, I is the identity matrix and r F (kk) is the gradient of F (kk). When bk tends to zero Levenberg–Marquardt method approaches the Gauss–Newton method whereas when bk tends to infinity the Levenberg–Marquardt method approaches the steepest-descent method. The values of b k during the iterative process are chosen in the following way: when initializing the algorithm bk is set to a large value so that the Levenberg–Marquardt method manifests the robustness of the steepest-descent method, meaning that the initial guess can be chosen with less caution. For each iteration, if F (kk + sk) < F (kk), convergence is accelerated by decreasing bk by a certain amount set by the user; otherwise, b k is increased in order to enlarge the searching area [16]. ðH ½F ðkk Þ þ bk I Þsk ¼ rF ðkk Þ

ð4Þ

The objective function is not described by an analytical expression; therefore, the gradient is calculated using the numeric method given by Eq. (5). The step size, Dk, is an infinitesimal positive number that enables the numerical calculation of the gradient.

rF ðkk Þ ¼

ðF ðkk þ Dkk Þ  F ðkk  Dkk ÞÞ 2  Dkk

ð5Þ

Regarding the Hessian matrix, the calculation is performed by applying the expanded Taylor series truncated to the second term as shown in Eq. (6). F ðkk þ Dkk Þ ¼ F ðkk Þ þ rF ðkk Þ  Dkk þ

1 T  k  H ½F ðkk Þ  Dkk 2 k

ð6Þ

3.3. Step 3 –ASPEN PLUS and Microsoft Excel interaction

As stated in Steps 1 and 2 the model outputs – which correspond to the outlet water temperature – are generated running an ASPEN PLUS simulation model. These results generated by the process simulator are fed into the minimization algorithm implemented in Excel, which in turn will feed the simulator with new model parameters (Fig. 3). This two-way connection between the process simulator (ASPEN PLUS) and the minimization algorithm (Visual Basic for Application in Microsoft Excel) is provided by an interface between both software tools. The interface is guaranteed by Aspen Simulation Workbook [17] which allows a seamless data transfer between ASPEN PLUS and Microsoft Excel. Aspen Simulation Workbook is a tool for interfacing AspenTech’s process simulation models with Microsoft Excel worksheets. Aspen Simulation Workbook also has tools to link model variables to plant data tags imported using third-party applications. These capabilities allow modeling experts to link models and plant data and publish the resulting models as Excel worksheets [18]. 3.4. Step 4 – implementation of the proposed methodology

A step by step flowchart representing the actions that must be taken to implement the proposed methodology is shown in Fig. 3.



477

Fig. 2. Algorithmfor thedeterminationof model parameters. The inner circleof the algorithmis a NLPproblem whereas if the number of equilibrium stages is considered it is

a MINLP problem.

Due to the comprehensive experimental data as well as detailed information about the experimental setup the work of Simpson and Sherwood [13] is often used to evaluate the appropriateness of cooling tower models [6,19,20]. These authors published experimental data regarding the operation of two mechanical induced draft cooling towers, designated by tower R-1 and tower R-2. Although the ambient pressure is not mentioned in the work of Simpson and Sherwood [18], taking into account the nature of the work it is assumed that it remained approximately constant during the length of the experimental work. These two sets of published experimental data are used for validating the approach proposed in this work.

A third set of data (Table 3) was used to confirm the applicability of the proposed approach to a real industrial application, which corresponded to a set of experimental data collected from an industrial mechanical induced draft cooling tower. The industrial cooling tower with the characteristics specified in Table 4 belongs to one of the manufacturing plants of Dow Chemical Company in Portugal. Water pumps and fan motors are equipped with fixed speed drives so that the water and air flow rate are kept constant. The dry-bulb temperature and relative humidity were measured near the air entrance of the cooling tower with a data logger (Tinytag View 2 – TV4500). In- and outlet water temperature are both registered by an online process monitoring system in the


478

J.A. Queiroz et al. / Energy Conversion and Management 64 (2012) 473–481 Table 3

Collect plant data

Experimental and modeled values for Dow’s cooling tower collected during 1 month period (20 May 2011–20 June 2011). Air and water flow correspond to design. Air outlet temperature was not monitored.

(make sure equipment is operating in steady state conditions)

Build ASPEN PLUS simulation model

P atm (kPa) Step 1

Step 2

Link both tools with Aspen Simulation Workbook

Step 3

Run algorithm for determining model parameters

Model parameters Experimental inputs

Process simulator model

in ,exp

T w out ,mod

(C)

(C)

(C)

97.00 91.65 72.80 67.62 87.15 85.47 89.67 79.10 73.90 80.22 88.00 79.10 85.77 92.47 68.17 81.87 97.85 81.60 65.45 37.98 85.77 35.40 78.55 68.97 41.88 74.97 79.10 51.58 65.72 75.52

28.46 27.38 29.91 28.79 28.73 31.01 28.61 29.77 30.86 28.73 28.07 29.77 27.76 29.31 30.49 30.02 29.83 27.65 29.87 30.88 27.76 30.31 27.29 28.81 30.63 31.15 28.15 30.91 31.40 28.93

22.87 21.74 23.82 22.70 22.50 25.14 22.75 23.55 24.51 22.36 23.03 23.55 22.61 23.30 24.20 23.68 23.72 21.83 23.17 24.49 22.61 24.06 22.40 22.74 24.29 24.86 21.92 24.65 25.64 23.65

22.93 21.63 23.95 22.85 22.47 25.02 22.93 21.83 24.69 22.37 23.16 23.58 22.61 23.23 24.26 23.73 23.59 21.97 23.51 24.37 22.62 23.82 22.47 22.61 23.94 25.14 21.65 24.44 25.82 23.89

21.85 17.40 19.79 21.85 22.30 17.61 21.89 17.55 19.79 22.16

62.77 86.60 62.22 67.35 75.25 83.25 73.35 96.72 62.22 72.25

30.11 29.14 28.95 30.32 31.39 29.17 30.82 29.50 28.95 30.57

23.78 22.80 22.64 23.97 25.26 22.80 24.44 23.51 22.64 24.51

23.75 22.79 22.49 24.50 25.12 22.77 24.62 23.40 22.49 24.56

19.79 22.17 18.07 21.31 20.71 22.60 20.71 22.57 16.31 22.44

62.22 73.90 85.20 76.90 75.25 52.35 52.35 65.72 83.55 78.55

28.95 30.32 29.44 28.14 30.17 27.85 28.11 30.90 28.10 31.04

22.64 24.51 23.16 23.23 24.05 22.53 22.03 24.42 21.79 25.04

22.51 24.53 23.12 23.40 23.97 22.43 21.91 24.49 21.81 25.16

RH in

T w

(C)

(%)

17.45 15.97 21.29 20.26 17.03 21.22 18.09 19.71 21.98 17.41 19.48 19.71 18.65 17.79 22.02 19.36 17.60 17.62 21.15 27.27 18.65 26.93 19.72 19.50 25.47 22.65 16.46 24.76 25.43 21.75

in ,exp

Train

Implement minimization algorithm in Microsoft Excel

ASPEN PLUS

T w out ,exp

T a

k o Microsoft Excel o b k Minimization algorithm r o W n s t s s t o s i u u t u r t p t p t p e t a n I u u l e l O O u m a l l t a r n a e m i a e t n d p o m e S l i r m M e e i n d r p e o e x p M E x p s E A

Model outputs

Fig. 3. Overall procedure flowchart showing the interaction between the process

simulator (ASPEN PLUS) and the minimization algorithm (implemented in Microsoft Excel), being this interaction mediated by Aspen Simulation Workbook. The steps referred in this figure are described in detail in Section 3 of this work.

101.44 101.95 101.92 101.85 102.12 101.11 100.63 102.19 102.12 102.09 101.51 101.14 101.82 101.78 101.92 102.02 100.90 101.14 101.78 101.58 101.82 101.78 101.41 101.65 101.31 101.44 100.90 101.51 101.75 101.78 Validation

102.02 101.82 101.88 102.02 101.31 102.12 102.02 100.87 101.88 101.99 Test

main headers. Since the atmospheric pressure is not monitored on site this variable was retrieved from two weather stations 20 km away on opposite directions and the mean value was computed. The atmospheric pressure was obtained from Weather Underground website [21]. Due to the fact that the air outlet temperature is not monitored the objective function to be minimized only takes into account the outlet water temperature. To train, validate and test the model, experimental data was divided in three sets. Data points were randomly split for training (trn), validating (val) and testing (test ) with 60/20/20%, respectively. Data sets regarding tower R-1, R-2 and Dow’s were defined by 49, 45 and 50 experimental measurements, respectively. As stated in the algorithm shown in Fig. 2, the procedure stops when the error associated to the validation sub-set increases or, for the case when the this value is continuously decreasing, the difference between the error associated to the training sub-set of two consecutive iterations is less than a pre-set value (e). The stopping criteria that was verified for both tower R-1 and Dow’s tower was F val (kk+1) > F val (kk) and for tower R-2 F trn (kk+1)  F trn (kk) < e. Figs. 4–6 show the evolution of the objective function in regard to the number of iterations. Considering the algorithm for model identification described before it is stated that additional equilibrium stages are added only

101.88 101.78 101.85 101.82 101.78 101.75 101.78 101.99 102.02 101.07

Table 4

Dow’s cooling tower design specifications.

Type

Induced draft counter flow

Tower dimensions H W L

Packing height Design water flow Design water inlet temperature Design water outlet temperature Design air inlet wet-bulb temperature Design air flow Number of fans Nominal fan power (each)

12.85 m 13.32 m 37.58 m 5.1 m 5678 tonne/h 36.7 C 30.0 C 26.7 C 4505 tonne/h 3 75 kW


479


Fig. 4. Objective function evolution for tower R-1 for training [ j] and validation

Fig. 6. Objective function evolution for Dow’s tower for training [j] and validation

[h] sub-sets.

[h] sub-sets.

Murphree stage efficiencies for the three cooling towers were: M M =0.93, Eff M w 2 = 0.85 for tower R-1; Eff w 1 =0.89, Eff w 2 =0.74

M Eff w 1 ;

;

;

M Eff w 1

;

M Eff w 2

for tower R-2 and = 0.96, = 0.70 for Dow’s tower. The good agreement between predicted and experimental can verified in Figs. 7–9 for water outlet temperatures. Due to the fact that air outlet temperature is often unmonitored (as in Estarreja Dow’s cooling tower) the model was trained by only minimizing the error between model prediction and experimental outlet water temperature. Despite this fact, Fig. 10 shows that the model output for air outlet temperatures is in good agreement with the experimental values. ;

;

3.5. Step 5 – model performance evaluation

Fig. 5. Objective function evolution for tower R-2 for training [ j] and validation

[h] sub-sets.

when the algorithm returns stage efficiency values higher than 1. For the three case studies presented in this work, applying two equilibrium stages was sufficient to provide a good fit to experimental data, with both stage efficiencies lower than one. The algorithm stopped when k = 5, k = 4 and k = 8 for tower R-1, R-2 and Dow’s, respectively. For tower R-1 and R-2 the algorithm’s exit was activated when the objective function of the validation sub-set reached a local minimum while for Dow’s cooling tower the algorithm stopped when the difference between two consecutive iterations of the training sub-set was lower than the established margin.

To quantitatively evaluate the performance of the model the following statistical parameters were calculated: correlation coefficient (R) and root mean squared error ( RMSE ). The correlation coefficient provides a variability measure of the data reproduced in the model and the root mean squared error provides the measure of residual errors and gives a global idea of the difference between the observed and modeled values. These parameters are calculated using Eqs. (7) and (8) [22], where Y l is the model output Y , is the average value of the model output, Y l is the experimental value. b

R ¼

v ut ffiPffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiPffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi P b r ffiffi ffi ffiXffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi bffi ffi ffi ffi

RMSE ¼

m m 2 2 l¼1 ðY l  Y Þ  l¼1 ðY l  Y l Þ 2 m l¼1 ðY l  Y Þ

1 m

m

l¼1

ðY l  Y Þ2

ð7Þ

ð8Þ

Fig. 7. Water outlet temperature model predictions vs. observed value for tower R-1: (a) training [}] + validation [j] and (b) test [ N]. Assuming a constant P atm of 101.3 kPa.


480


Fig. 8. Water outlet temperature model predictions vs. observed value for tower R-2: (a) training [ }] + validation [j] and (b) test [ N]. Assuming a constant P atm of 101.3 kPa.

(a)

(b)

Fig. 9. Water outlet temperature model predictions vs. observed value for Dow’s tower: (a) training [ }] + validation [j] and (b) test [ N].

Fig. 10. Air outlet temperature model predictions vs. observed value for training [}], validation [j] and test[ N] in: (a) tower R-1 and (b)tower R-2. Assuming a constant P atm

of 101.3 kPa.

Table 5

Model performance parameters for the different data sets. The variable that is being analyzed is the water outlet temperature, T w out .

Training

RMSE R

Validation

Test

R-1

R-2

Dow

R-1

R-2

Dow

R-1

R-2

Dow

0.206 0.997

0.172 0.997

0.175 0.984

0.291 0.991

0.129 0.997

0.102 0.996

0.236 0.994

0.218 0.993

0.118 0.996



In Table 5 model performance parameters regarding outlet water temperature (T w o ut) are summarized. For the test data sub-sets of tower R-1 and R-2, respectively, RMSE test is: 0.23; 0.218 and Rtest : 0.994; 0.993, as for Dow’s industrial cooling tower the test sub-set RMSE test is: 0.129 and Rtest : 0.991. The values of the model evaluation parameters indicate a good fit between experimental and model values for both the laboratory setup (tower R1 and R-2) and the industrial application (Dow’s cooling tower). Although the model was trained using only water outlet temperatures the model performance parameters regarding air outlet temperatures (T a out ) in tower R-1 and R-2 also indicate an acceptable fit between model predictions and experimental values: RMSE test = 0.349; Rtest = 0.988 for tower R-1 and RMSE test = 0.130; Rtest = 0.997 for tower R-2. In an industrial context air outlet temperature is not a critical variable to be controlled, therefore these values are not available for Dow’s case study and the comparison between model and experimental air outlet temperatures was not possible to accomplish. All predicted values are within a margin of ±2 % from the experimental values for water outlet temperatures and ±3 % for air outlet temperatures (tower R-1 and R-2).

4. Conclusions

Cooling tower outlet temperature was predicted by applying the equilibrium stage approach with Murphree stage efficiencies. Two equilibrium stages were sufficient to provide a good fit for all three case studies. The proposed approach was validated not only with published data but also with data provided by a real industrial application. This approach can be useful when simulating a process, allowing the prompt evaluation of the impacts for different operation conditions. Although only the outlet water temperature was used to identify model parameters, predicted air outlet temperature was also in good accordance with experimental values. A critical element of process simulation is proper physical property methods, and determination of binary interaction parameters. Since ASPEN PLUS is a process simulator that provides several built-in model blocks that can be directly applied in process simulation and has an extensive physical property database, it is useful to apply the methodology described in this work to other process units where it is necessary to fit model parameters to experimental data for example in distillation towers data reconciliation. For this, model parameters such as Murphree or overall column efficiencies are adjusted to a wide variety of operating conditions to produce the best match to plant data. Model parameters that minimize the overall sum of the squared difference between experimental and simulated values are then selected. Since model parameters are adjusted to real data, once the proposed methodology is applied to determine model parameters off-design operating conditions are accurately simulated using the process simulator model. Additionally, this approach has the ability to be integrated

481

with a larger process model enabling a better process control and energy management when integrated with the distributed control system. Acknowledgments

Authors are grateful to Dow Portugal – Produtos Químicos, Sociedade Unipessoal, Lda and to Fundação para a Ciência e Tecnologia, namely for the Ph.D. student support through fellowship SFRH/BDE/51013/2010. References [1] Jin GY, Cai WJ, Lu L, Lee EL, Chiang A. A simplified modeling of mechanical cooling tower for control and optimization of HAVC systems. Energy Convers Manage 2007;48(2):355–65. [2] Benton DJ, Hydeman M, Bowman CF, Miller P. An improved cooling tower algorithm for the CoolTools™ simulation model. ASHRAE Trans 2002;108(1):AC-02-9-4. [3] Kloppers JC, Kröger DG. A critical investigation into the heat and mass transfer analysis of counter flow wet-cooling towers. Int J Heat Mass Transfer 2005;48(3):765–77. [4] Picardo JR, Variyar JE. The Merkel equation revisited: a novel method to compute the packed height of a cooling tower. Energy Convers Manage 2012;57:167–72. [5] Castro MM, Song TW, Pinto JM. Minimization of operational costs in cooling water systems. Br J Chem Eng 2000;78(2):192–201. [6] Khan JR, Qureshi BA, Zubair SM. A comprehensive design and performance evaluation study of counter flow wet cooling towers. Int J Refrig 2004;27(8):914–23. [7] Al-Waked R, Behnia M. CFD simulation of wet cooling towers. Appl Therm Eng 2006;26(4):382–95. [8] Rubio-Castro E, Serna-González M, Ponce-Ortega JM, Morales-Cabrera MA. Optimization of mechanical draft counter flow wet-cooling towers using a rigorous model. Appl Therm Eng 2011;31(16):3615–28. [9] Hosoz M, Ertunc HM, Bulgurcu H. Performance prediction of a cooling tower using artificial neural network. Energy Convers Manage 2007;48(4):1349–59. [10] Pan TH, Shieh SS, Jang SS, Tseng WH, Wu CW, Ou JJ. Statistical multi-model approach for performance assessment of cooling tower. Energy Convers Manage 2011;52(2):1377–85. [11] Serna-González M, Ponce-Ortega JM, Jiménez-Gutiérrez A. MINLPoptimization of mechanical draft counter flow wet-cooling towers. Chem Eng Res Des 2010;88(5–6):614–25. [12] Rao RV, Patel VK. Optimization of mechanical draft counter flow wet-cooling tower using artificial bee colony algorithm. Energy Convers Manage 2011;52(7):2611–22. [13] Simpson W, Sherwood TK. Performance of small mechanical draft cooling towers. ASRE J Refrig Eng 1946;52(6):543–76. [14] ASPEN PLUS, V7.1. Aspen Technology Inc.; 2008. [15] Aktepe A, Öncel Ç, Ersöz S. An artificial neural network model on welding process control of 155 mm artillery ammunition. In: 6th International advanced technologies symposium (IATS’11), Elazıg˘, Turkey; 16–18 May 2011. [16] Fang Q. Distinctions between Levenberg–Marquardt method and Tikhonov regularization. Dartmouth College Publication; 2004. [17] Aspen Simulation Workbook, V7.1. Aspen Technology Inc.; 2009. [18] Aspen Simulation Workbook User Guide, V7.1. Aspen Technology Inc.; 2009. [19] Zubair BA, Qureshi SM. A complete model of wet cooling towers with fouling in fills. Appl Therm Eng 2006;26(16):1982–9. [20] Heidarinejad G, Karami M, Delfani S. Numerical simulation of counter-flow wet-cooling towers. Int J Refrig 2007;32(5):996–1002. [21] Weather Underground; 2011 (May and June). , [retrieved 23.02.12]. [22] Sousa SIV, Martins FG, Alvim-Ferraz MCM, Pereira MC. Multiple linear regression and artificial neural networks based on principal components to predict ozone concentrations. Environ Model Softw 2007;22(1):97–103.

Cooling Tower Modelling

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