scholarly journals An Analytical Approach to Wet Cooling Towers Based on Functional Analysis

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Guo Qianjian ◽  
Xiaoni Qi ◽  
Zheng Wei ◽  
Peng Sun
Keyword(s):  
Functional Analysis ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Cooling Tower ◽  
Transfer Model ◽  
Cooling Towers ◽  
Contraction Mapping Theorem ◽  
Uniqueness Of The Solution ◽  
Linear Differential ◽  
Analytical Expressions

An analytical solution for computing the temperature distribution of air and water over the height through the cooling tower is so complex that finding the exact solution takes too much time. The purpose of this paper is to present efficient and accurate analytical expressions for the heat and mass transfer model in cooling towers. Based on the method of functional analysis, we derived an analytical solution for temperature distribution of water and air by using the method of solving linear differential equations. The error estimation, the existence, and uniqueness of the solution are given by using Banach contraction mapping theorem. The basic equation of the model on the basis of the additional assumptions on the cooling tower is solved, and the outlet parameters are also obtained.

scholarly journals Modeling of a Spray Assisted Natural Draft Cooling Tower

2008 ◽  
Vol 31 (1) ◽  
pp. 118-126
Author(s):  
O.K. Kaunde
Keyword(s):  
Momentum Transfer ◽  
Cooling Tower ◽  
Current Flow ◽  
Cooling Water ◽  
Transfer Model ◽  
Cooling Towers ◽  
Air Flow Rate ◽  
Transfer Unit ◽  
Novel Design ◽  
Gas Ratio

Cooling towers are one of the largest heat and mass transfer devices that are in common use. A novel type of cooling tower has been proposed in which air flow rate into the tower is drawn by ejector action of sprays instead of fans as is done in conventional mechanical forced or induced draft cooling towers. This novel design offers the potential of savingthe energy cost for driving the fan. The paper presents mathematical models for momentum transfer which is the driving force causing the entrainment of air. Also the heat transfer model for co-current flow of liquid and gas in the tower has been presented. The liquid to gas ratio tend to decrease as liquid rate increases. The ratio attained in the experimentallaboratory tower was 3.3, correspondingly the Momentum transfer efficiency for the tower was 60% and was the highest. Experiments for cooling water initially at 45 o C to final water temperature 27 o C showed that the cooling tower efficiency was 54% and number of transfer unit 0.8.

Response of Infinite Journal Gas Bearings to Harmonic Perturbations in the Rotational Speed

1977 ◽  
Vol 99 (4) ◽  
pp. 428-433 ◽  
Author(s):  
Y. Narkis ◽  
M. J. Cohen
Keyword(s):  
Analytical Solution ◽  
Rotational Speed ◽  
Approximate Analytical Solution ◽  
Critical Mass ◽  
Regions Of Interest ◽  
Time Dependent ◽  
Safe Operation ◽  
Linear Differential ◽  
Analytical Expressions ◽  
Gas Bearing

The dynamics of a long hydrodynamic gas bearing is investigated for periodic variations of the rotational speed. The analysis is divided into two regions of interest, namely: (1) for small eccentricities the system is represented by a pair of linear differential equations with time-dependent coefficients. Investigation for a sinusoidally varying rotational speed proves that an unloaded bearing can be stable, though it is known not to be stable at all constant speeds. An approximate analytical solution is given for the orbit of a stable journal whirling about its equilibrium position. (2) For higher eccentricities the nonlinear equations describing the motion of the journal center are derived. When the speed perturbation is small, the equations may be linearized, and analytical expressions are obtained for the calculation of journal response. At given speed and eccentricity resonance is reached at the critical mass of instability threshold, but even for smaller mass the amplitudes are liable to endanger safe operation of the system.

Case study: Theoretical calculation model and variable condition analysis research on the water-splashing noise for natural draft wet cooling towers

2020 ◽  
Vol 68 (2) ◽  
pp. 137-145
Author(s):  
Yang Zhouo ◽  
Ming Gao ◽  
Suoying He ◽  
Yuetao Shi ◽  
Fengzhong Sun
Keyword(s):  
Theoretical Model ◽  
Sound Pressure Level ◽  
Water Droplet ◽  
Water Distribution ◽  
Sound Pressure ◽  
Cooling Tower ◽  
Distribution Patterns ◽  
Calculation Model ◽  
Pressure Level ◽  
Cooling Towers

Based on the basic theory of water droplets impact noise, the generation mechanism and calculation model of the water-splashing noise for natural draft wet cooling towers were established in this study, and then by means of the custom software, the water-splashing noise was studied under different water droplet diameters and water-spraying densities as well as partition water distribution patterns conditions. Comparedwith the water-splashing noise of the field test, the average difference of the theoretical and the measured value is 0.82 dB, which validates the accuracy of the established theoretical model. The results based on theoretical model showed that, when the water droplet diameters are smaller in cooling tower, the attenuation of total sound pressure level of the water-splashing noise is greater. From 0 m to 8 m away from the cooling tower, the sound pressure level of the watersplashing noise of 3 mm and 6 mm water droplets decreases by 8.20 dB and 4.36 dB, respectively. Additionally, when the water-spraying density becomes twice of the designed value, the sound pressure level of water-splashing noise all increases by 3.01 dB for the cooling towers of 300 MW, 600 MW and 1000 MW units. Finally, under the partition water distribution patterns, the change of the sound pressure level is small. For the R s/2 and Rs/3 partition radius (Rs is the radius of water-spraying area), when the water-spraying density ratio between the outer and inner zone increases from 1 to 3, the sound pressure level of water-splashing noise increases by 0.7 dB and 0.3 dB, respectively.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Cooling tower plume abatement and plume modeling: a review

2021 ◽  
Author(s):  
Shuo Li ◽  
M. R. Flynn
Keyword(s):  
Public Health ◽  
Solar Energy ◽  
Cooling Tower ◽  
Plume Rise ◽  
Cooling Towers ◽  
Moist Air ◽  
Novel Technologies ◽  
Plume Modeling ◽  
Dry Cooling Tower ◽  
Dry Cooling

AbstractVisible plumes above wet cooling towers are of great concern due to the associated aesthetic and environmental impacts. The parallel path wet/dry cooling tower is one of the most commonly used approaches for plume abatement, however, the associated capital cost is usually high due to the addition of the dry coils. Recently, passive technologies, which make use of free solar energy or the latent heat of the hot, moist air rising through the cooling tower fill, have been proposed to minimize or abate the visible plume and/or conserve water. In this review, we contrast established versus novel technologies and give a perspective on the relative merits and demerits of each. Of course, no assessment of the severity of a visible plume can be made without first understanding its atmospheric trajectory. To this end, numerous attempts, being either theoretical or numerical or experimental, have been proposed to predict plume behavior in atmospheres that are either uniform versus density-stratified or still versus windy (whether highly-turbulent or not). Problems of particular interests are plume rise/deflection, condensation and drift deposition, the latter consideration being a concern of public health due to the possible transport and spread of Legionella bacteria.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

Predicting cooling tower plume dispersion

1997 ◽  
Vol 211 (4) ◽  
pp. 291-297 ◽  
Author(s):  
B E A Fisher
Keyword(s):  
Cooling Tower ◽  
Industrial Process ◽  
Local Environment ◽  
Meteorological Conditions ◽  
Performance Criteria ◽  
Plume Dispersion ◽  
Cooling Towers ◽  
Atmospheric Conditions ◽  
Operating Characteristics

An assessment of the effects of visible cooling tower plumes on the local environment can be a necessary part of any proposal for a new large industrial process. Predictions of the dispersion of plumes from cooling towers are based on methods developed for chimney emissions. However, the kinds of criteria used to judge the acceptability of cooling tower plumes are different from those used for stack plumes. The frequency of long elevated plumes and the frequency of ground fogging are the two main issues. It is shown that events associated with significant plume visibility are dependent both on the operating characteristics of the tower and on the occurrence of certain meteorological conditions. The dependence on atmospheric conditions is shown to be fairly complex and simple performance criteria based on the exit conditions from the tower are not sufficient for assessments.

Design of Cooling Towers by the Effectiveness-NTU Method

1989 ◽  
Vol 111 (4) ◽  
pp. 837-843 ◽  
Author(s):  
H. Jaber ◽  
R. L. Webb
Keyword(s):  
Heat Exchanger ◽  
Cooling Tower ◽  
Design Method ◽  
Design Methods ◽  
Parallel Flow ◽  
Basic Method ◽  
Cooling Towers ◽  
Flow Configuration ◽  
Heat Exchanger Design

This paper develops the effectiveness-NTU design method for cooling towers. The definitions for effectiveness and NTU are totally consistent with the fundamental definitions used in heat exchanger design. Sample calculations are presented for counter and crossflow cooling towers. Using the proper definitions, a person competent in heat exchanger design can easily use the same basic method to design a cooling tower of counter, cross, or parallel flow configuration. The problems associated with the curvature of the saturated air enthalpy line are also treated. A “one-increment” design ignores the effect of this curvature. Increased precision can be obtained by dividing the cooling range into two or more increments. The standard effectiveness-NTU method is then used for each of the increments. Calculations are presented to define the error associated with different numbers of increments. This defines the number of increments required to attain a desired degree of precision. The authors also summarize the LMED method introduced by Berman, and show that this is totally consistent with the effectiveness-NTU method. Hence, using proper and consistent terms, heat exchanger designers are shown how to use either the standard LMED or effectiveness-NTU design methods to design cooling towers.

Analysis of Heat, Mass, and Momentum Transfer in the Rain Zone of Counterflow Cooling Towers

1999 ◽  
Vol 121 (4) ◽  
pp. 751-755 ◽  
Author(s):  
E. de Villiers ◽  
D. G. Kro¨ger
Keyword(s):  
Momentum Transfer ◽  
Cooling Tower ◽  
Mechanical Energy ◽  
Performance Evaluations ◽  
Cooling Towers ◽  
Droplet Deformation ◽  
One Dimensional ◽  
Mechanical Energy Loss ◽  
Simplifying Assumptions ◽  
Semi Empirical

The rate of heat, mass, and momentum transfer in the rain zone of three counterflow cooling tower geometries is analyzed using simplifying assumptions and numerical integration. The objective of the analysis is to generate equations for use in a one-dimensional mathematical cooling tower performance evaluations. Droplet deformation is taken into account and momentum transfer is calculated from the air flow’s mechanical energy loss, caused by air-droplet interaction. A comparison of dimensionless semi-empirical equations and experimental data demonstrates the method’s capability to predict the pressure drop in a counterflow rain zone.

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