A Universal Engineering Model For Cooling Towers

This paper presents a universal engineering model, which can be used to formulate both counterflow and crossflow cooling towers. By using fundamental laws of mass and energy balance, the effectiveness of heat exchange is approximated by a second order polynomial equation. Gauss-Newton and Levenberg-Marquardt methods are then used to determine the coefficients from manufactures data. Compared with the existing models, the new model has two main advantages: (1) As the engineering model is derived from engineering perspective, it involves fewer input variables and has better description of the cooling tower operation; (2) There is no iterative computation required, this feature is very important for online optimization of cooling tower performance. Although the model is simple, the results are very accurate. Application examples are given to compare the proposed model with commonly used models. NOMENCLATURE Cpw: specific heat of water under constant pressure; Cs: derivative of saturated ha with respect to Tw; c0-c5: curve-fitting constants; dk: search direction of optimization method; Fdata: computation value of εa; ha: enthalpy of air; hs,w: saturation air enthalpy at Tw; hs,wb: saturation air enthalpy at Twb; J(uk): Jacobian matrix of uk; ma: mass flow rate of air; me: mass flow rate of evaporation; ma: mass flow rate of air; mm: mass flow rate of makeup water; mw: mass flow rate of water; NTU: number of transfer units; m: ratio of air to water effective capacitance rate; Qe: heat evaporation rate of loss water; Qrej: heat rejection rate of cooling tower; Tdb: dry-bulb temperature of air; Tm: temperature of makeup water; Tw: temperature of water; Twb: wet-bulb temperature of air; uk: the value of c0-c5 of the kth iteration; εa: heat transfer effectiveness of Braun’s model; ∆h: enthalpy difference with respect to ∆T; ∆T: approach of cooling tower; λk: control coefficients; Subscript i: inlet; o: outlet; INTRODUCTION Cooling towers are commonly used to dissipate heat from heat sources to heat sink (ambient environment). Their applications are typically in Heating Ventilation and Air Conditioning (HVAC) systems and power generators, etc. Heat rejection of cooling towers is accomplished by heat and mass transfer between hot water droplets and ambient air. Although cooling towers are relatively inexpensive and normally consume around ten percent of the whole system energy, their operation has significant effect to the energy consumption of other related subsystems (RMIRA 1995; Michel 1995). Therefore, optimizing cooling tower performance will not only increase the tower efficiencies but also has direct effect to other subsystems. As such, there has been some research interest in this area. Austin (1997) recommended regression methods to create the models of each component in air conditioning systems for predicting and optimizing the system performance. Flake (1997) utilized a different regression technique to determine parameters of the cooling tower model developed by Braun (1989) and to build a predictive model for optimal supervisory control strategies. However, due to the lacking of an effective and precise model for cooling towers, which is essential to estimate and verify the energy savings by different optimization strategies, the research on optimization of cooling tower performance is still in the primary stage. Attempts to develop the cooling tower models have a relative long history, the first such work may trace back to 1925, when Merkel developed a practical model for cooling tower operation, which has been the basis for most modern cooling tower analyses. In his model, the water loss of evaporation is neglected and the Lewis number is assumed to be one in order to simplify the analysis. However, as evaporate water cannot be neglected in cooling tower operation, Merkel’s model is not accurate enough and not suitable for real applications. A more rigorous analysis of a cooling tower model that relaxed Merkel’s restriction was given by Sutherland (1983). In 1989, Braun developed “effectiveness models” for cooling towers, which utilized the assumption of a linearized air saturation enthalpy and the modified definition of number of transfer units. The models were useful for both design and system simulation and has been adopted by the simulation software TRNSYS (SEL 2000). However, Braun’s model needs iterative computation to obtain the output results and is not suitable for online optimization. Bernier (1994) reviewed the heat and mass transfer process in cooling towers at water droplet level and analyzed an idealized spray-type tower in one-dimension, which is useful for cooling tower designers, but no much information is provided to plant operators. Soylemez (1999) presented a quick method for estimating the size and performance of forced draft countercurrent cooling towers and experimental results were used to validate the prediction formulation. Unfortunately, this model also need iterative computation and not suitable for online optimization. In this paper, a universal engineering model, which can be used to formulate both counterflow and crossflow cooling towers, is proposed. Extending the methods provided by Merkel and Braun and using fundamental laws of mass and energy balance, the effectiveness of heat exchange is approximated by a second order polynomial equation. Gauss-Newton and Levenberg-Marquardt methods are then used to determine the coefficients from manufactures data. Compared with the existing models, the new model has two main advantages: (1) As the engineering model is derived from engineering perspective, it involves fewer input variables and has better description of the cooling tower operation; (2) There is no iterative computation required, this feature is very important for online optimization of cooling tower performance. Although the model is simple, the results are very accurate. Application examples are given for both counterflow and crossflow to compare the proposed model with commonly used models. COOLING TOWER MODEL ANALYSIS The mechanism of heat and mass transfer between ambient air and condenser water inside a cooling tower is illustrated in Figure 1. Figure 1. Schematic representation of heat and mass transfer in the cooling tower AIR WATER i a m , i a h , o a h , o a m ,