AdAptive reliAbility structures of heAt exchAnge surfAce in turbine condenser

The aim of the paper is to prove adaptive reliability structures of heat exchange surface, which stems from the matter of regulating the surface in order to maintain effective process of heat exchange by sustaining the requested pressure of steam condensation in variable exploitation conditions, which determines changes in the pipe system of the condenser and involves assessment of reliability of its surface of heat exchange. Adaptive reliability structures of surface of heat exchange (pipe subsystem) are the reliability structures, which are altered in the course of adjusting the pipe system to the actual exploitation conditions of the condenser in the power system. Having delved into the current state of the art with IT data bases (Science Direct, Knovel, Nauka Polska, BazTech, google) it was concluded that there has been no algorithm for assessing reliability of surface of heat exchange of steam turbine condenser which would include regulation of the surface in order to exchange the heat effectively and sustain the requested pressure of steam condensation in variable exploitation conditions, which has a significant influence on the quality of the technical power system exploitation, in which the condenser is a part. Publication [10] shows that sustaining given pressure of steam condensation in the condenser in variable conditions is vital for maintaining requested power efficiency of the technical power system. The aforementioned publication puts forward a particular technical soluKrzysztof ŁuKAszewsKi


Introduction
The aim of the paper is to prove adaptive reliability structures of heat exchange surface, which stems from the matter of regulating the surface in order to maintain effective process of heat exchange by sustaining the requested pressure of steam condensation in variable exploitation conditions, which determines changes in the pipe system of the condenser and involves assessment of reliability of its surface of heat exchange.
Adaptive reliability structures of surface of heat exchange (pipe subsystem) are the reliability structures, which are altered in the course of adjusting the pipe system to the actual exploitation conditions of the condenser in the power system.
Having delved into the current state of the art with IT data bases (Science Direct, Knovel, Nauka Polska, BazTech, google) it was concluded that there has been no algorithm for assessing reliability of surface of heat exchange of steam turbine condenser which would include regulation of the surface in order to exchange the heat effectively and sustain the requested pressure of steam condensation in variable exploitation conditions, which has a significant influence on the quality of the technical power system exploitation, in which the condenser is a part.
Publication [10] shows that sustaining given pressure of steam condensation in the condenser in variable conditions is vital for maintaining requested power efficiency of the technical power system.The aforementioned publication puts forward a particular technical solu-

AdAptive reliAbility structures of heAt exchAnge surfAce in turbine condenser
AdAptAcyjne struktury niezAwodnościowe powierzchni wymiAny ciepłA skrAplAczA turbiny pArowej* In this paper adaptive reliability structures of heat exchange surface in turbine condenser was proved from the angle of effective heat exchange in variable conditions of its exploitation.Then, determinant factors for design and exploitation in assessment of reliability of pipe subsystem in turbine condenser were suggested.The influence of change of scheme of the pipes, constituting the surface of heat exchange, which stems from the matter of regulating the surface in an attempt to both condense the given amount of steam and maintain the given pressure in the condenser in variable conditions of its exploitation on the reliability of the pipe subsystem was determined.The surface of heat exchange is regulated by enabling and disabling the flow of cooling water through given amount of pipes, in a given way, that is by enabling or disabling possible combination of given pipes in given exploitation conditions.An algorithm to assess the reliability of the pipe subsystem in the condenser in given exploitation conditions, means of regulating the surface and up-to-date technical condition was put forward.The reliability of pipe subsystem has a significant influence either on reliability of the condenser while exploited or in the further course, indirectly on sustaining the requested reliability in the power system therein.Effective operation of the condenser in technical power system is performed by sustaining the given pressure of steam condensation, which is vital in maintaining the required energy efficiency of technical power system in variable exploitation conditions.The exemplification of the aspects put forward in the paper pertains to steam turbine condensers.tion, which comes to proper division of the heat exchange surface of the condenser at the stage of its design (a particular number of non-adjustable parts of surface and one part of regulated surface of heat exchange), as well as the setup of the part while being exploited.Such technical solution while the condenser is exploited in the steam power system enables effective regulation of the flow of the water cooling the condenser.The regulation then not only allows for a particular heat exchange between the fluids, but also considers relations among the velocity of the cooling water flow, the erosion and deposition of pollutants on the surface of heat exchange as well as the costs of pumping the cooling water.

Keywords
In publications [2,8] the influence of exploitation conditions of turbine steam condensers on power plant efficiency was proved.
Publication [11] includes the problems of assessing the reliability of the exchanger and heat exchangers.It is possible to determine the models of reliability heat exchangers structures on the basis of the models of basic reliability structures of technical objects, included i.a [6,14].
Sources lack the presence of adaptive structures of reliable surfaces of heat exchange in steam turbine condensers, which may be caused by the means of regulating these surfaces to maintain the given pressures of condensation of the steam in variable exploitation conditions due to exploitation of technical power systems, of which they are a part.
As concerns the aim of the paper and the research into the current state of art the following problem recurs: how to sustain the requested reliability of the steam turbine condenser while it is exploited to a given time?

Designing process of reliability structures of heat exchange in the condenser
At the stage of designing of the steam turbine condenser, its reliability model is created ( ) wc R t , taking into consideration applica- tions in technical power system, possible kinds of damages to it as well as the construction of the condenser in accordance with the method included in publication [12].That is, ( ) wc R t reliability model for the condenser depicted in figure 1 may be referred to as a serial structure of reliability of subsystems of given elements, i.e. each of tube sheets ( )

R t (system of adjusting the valves shutting off the flow of cooling water through given pipes of the condenser).
The reliability model , () ps r R t of pipe subsystem, which refers to the algorithm in figure 1, is determined with serial reliability structure of nth number of pipes: Model ( 1) is defined within given exploitation conditions: maximal value of the heat stream 1, m a x Q  of condensation of the steam in the condenser, minimal value of the overall heat transfer coefficient ,min i k (through the surface of heat exchange with depositions), maximal value of temperature 2,max ' T of cooling water on the input of the condenser.In these conditions while the condenser is being exploited, the flow of cooling water through all the pipes is enabled.

Fig. 1. The algorithm of assessing expected reliabilities of elements of the condenser due to assumed reliability of the condenser (the formula in blocks 1.4 and 1.5 is due to transformation of reliability of the condenser into desirable pipes reliabilities -publication [12] involves the description).
The next significant stage of condenser design is the division of heat exchange surface with regard to anticipated, typical exploitation conditions as present in paper [10].Both insights allow to assume the following reliability model of pipe subsystem , () in which () R R t stands for the reliability model of pipes subsystem of an adjustable number of pipes, and () NR R t stands for the model of reliability of pipe subsystem, consisting of a number of mth pipe systems with a particular number of enabled and disabled pipes in these systems.
In the first row, it is considered how to divide the surface of heat exchange of the condenser in terms of typical, anticipated states of exploitation of the condenser due to maximize heat exchange efficiency.Thus, in this way a particular way of regulating the heat exchange surface is implicated, that is for given exploitation conditions, the flow of cooling water (with optimal value of flow velocity) is enabled through minimal number of pipes so as to sustain requested and constant pressure of steam condensation.The next implication revolves around creating particular reliability structures of pipe subsystem and implementing them into the , () ps r R t model.As a result, () R R t model is determined with a serial-parallel structure and hence is a part of n R number of pipe subsystem.
The subsystem is a proper combination of the structure as for the grading of enabling and disabling a given nth number of pipes out of n R number in given systems in given exploitation conditions, in which p=n R -n: sciENcE aNd tEchNology ,1, , while n= n R,1 then: and, while p= n R,1 : ,1, 1 in case the pipe subsystem is adjusted by enabling single pipes, then function , The () NR R t model may also be determined with the serialparallel structure and it makes the pipe subsystem of n NR number of pipes.Consequently, the subsystem makes a proper combination of the structure as for grading of enabling and disabling given mth pipe systems, where k=m NR -m in given exploitation conditions: ,1, , while m= m NR, or, while k=m NR : ,1, 1 in case if mth systems of n-numbered pipes are adjusted than functions , The next step is to consider the division of the surface of heat exchange with regard to typical, assumed exploitation states of the condenser due to maximum reliability , () ps r R t of pipe subsystem in random configuration of enabling and disabling particular pipe systems while sustaining the requested pressure in the condenser.In such approach, the models () R R t and () NR R t are defined with a threshold reliability structure of k-out-of-n type since there is no need to retain the grading to enable and disable particular pipe systems (assuming identical reliability functions of elements of the structure): ( ) ( )

The reliability structures of heat exchange surface in exploitation
Figure 2 below presents the algorithm to assess reliability R ps,r (t i ) of pipe subsystem in a given time t i of exploitation of the condenser, which includes the following change of values in given time spans [t i,min, t i,max ]: stream of heat i Q  transferred in the condenser, tempera- ture of T' 2,i cooling water on the input of the condenser, the number of pipes or change of the number of enabled pipes ) ) from the exploitation ("jammed"), the pollution of pipe surface as well as the possible air mass content in condensation of steam by calculating the value of overall heat transfer coefficient k e,p,i in given time (problems of ridding of the air in the condenser are not discussed in the paper and hence treated as background problems).This allows for assumed regulation of the heat surface with regard to a given effective transfer heat i Q  in given time spans t i considering the assessment of pipe subsystem reliability , , , , , , ps r e i i ps r e i ps r i i in these time spans basing on actual reliability nth pipes ( ) , implemented at the stage of design, or provide opportunity to alter (update) the reliability function at the stage of the condenser exploitation.While the condenser is being exploited in technical power system, the following values are monitored: pressure p 1 of steam condensation in the condenser and average velocity w 2 of the flow of cooling water through the condenser pipes, which indicate the efficiency of heat transfer with regard to both assumptions as for the steam turbine operation and economic reasons (the cost of pumping the cooling water).This gives ground, according to the algorithm from figure 2., to assess the exploitation surface A e,i of heat transfer and mass cooling water flow volume 2,i m  through particular system of pipes.Subsequently, the electrical conductivity of the condensate Γ is being constantly monitored.In case the value of the conductivity is below the admissible value, reliability structure of pipe subsystem R ps,r (t i ).needs to be redefined.
Otherwise, if Γ value is higher than admissible, it may cause damage to the pipe (a burst).In such circumstances, different system of pipes needs to be implemented urgently: , , , ) . Newly designated value of the surface of heat exchange A e,i is then examined whether it provides effective heat transfer in given exploitation conditions.It must be stressed that only systems with given number of pipes do have an influence on the process of heat transfer.The velocity of the cooling sciENcE aNd tEchNology water flow through the pipes may be increased from above the optimal value to the maximum admissible value 2, 2, 2,max i i w w w + ∆ ≤ and is performed in case bigger number of pipes needs to be enabled than this would result from sustaining optimal cooling water flow after damage in a particular number of pipes.This aims to sustain requested pressure in the condenser , which is the result of the decrease in the heat stream transferred in the condenser 1, m i n Q  and the decrease of effective power Ne,p,i in the steam turbine, results from the assumed condition of the seal flow of the cooling water through particular pipe system.Ultimately, the condenser should be excluded from exploitation and either include another one or shut down the power system and thus cease to exploit it.The enumerated actions are determined by functioning of a power system in given time of the condenser operation.
It is assumed that experimental researches of ith pipes have been conducted in order to estimate the reliability function of ith pipes.This allows to introduce and implement the values of reliability to models of particular reliability systems of pipe subsystem and calculate reliability of the subsystem in given time and given exploitation conditions.In case that ith number of pipes have been damaged, they are replaced with ones of the same kind.In case the difference among their real value of reliabilities and the values obtained from the implemented functions , out of the group (interval) of a given reliability structures R ps,r (t i ), results in a feedback, while estimating the exploitation of heat exchange surface A e,i as for current monitoring purpose and, having reconsidered the condition suggesting that the reliability value R ps,r (t i ) calculated when the condenser is exploited is equal or higher than assumed admissible reli-ability R ps,r,dop (t i ) in a given time interval t i .Subsequently it results also in monitoring current exploitation conditions and forecasting these conditions in further time intervals t i .

Exemplification of the adaptive characteristic of the heat exchange surface in the steam turbine condenser
The calculation example pertains to empirical studies of the damage to the condenser pipes, included in publication [1,15], on the basis of which, normal distribution has been assumed.The parameters of the distribution m=15,7, and σ=6,2 as for 100 pieces of condenser pipes were included into the calculation on the basis of studies of damages to condenser pipes of power units 225MW (publication [15]).According to paper [10] the overall number of pipes (12000) was assumed.Calculations and diagrams were generated with the use of BlockSim software by HBM Prenscia (BlockSim -integrated software allowing for analysis of RBD reliability structures).The example illustrates the calculations of reliability of pipe subsystem with regard to the contents of the paper, in case they include reliability function of pipes made out on the basis of empirical studies of power units condensers.The abridged method for designing heat exchangers of technical power systems with regard to their requested reliability is included in papers [12,13], where means of increasing the reliability of heat exchangers, if necessary, were highlighted.
The pipe subsystem of the steam turbine condenser of 12000 pipes consists of the following pipe systems: n R,1 =2000, means 20 pipe system 100 pipes each, where the function of reliability of a system may be determined as . There is lack of damaged pipes ("jammed"), n NR,u,i =0, n R,u,i =0.

sciENcE aNd tEchNology
The function of reliability , () ps r R t of pipe subsystem may be de- fined with the formula (11) if the current exploitation conditions W e,i determine enabling the cooling water flow through 12000 pipes: The reliability function , () ps r R t may be defined with formula (12) if current exploitation conditions W e,i determine enabling the cooling water flaw through 10100 pipes, and 900 pipes are a backup to streams of transferred fluid heat: ) in serial structure The function of reliability may be defined , () ) in parallel structure, and the latter in serial structure The function of reliability may be defined , () ps r R t with the formula ( 14) if the current exploitation conditions W e,i determine enabling the cooling water flow through 8100 pipes and 3900 pipes are a backup to streams of transferred fluid heat: ( ) {1 ) in serial structure flow), which has an influence on heat transfer effi- producers.The researches include the , e i CI characteris- tics, identifying ith pipes of the condenser, exploitation conditions of the and value of damage is determined and hence the pipe is considered damaged if the determined values are exceeded).Values of reliability of pipes ( ) { } t z are read with the use of reliability function

Fig. 2 .
Fig. 2. The algorithm of assessing exploitation reliability of pipe subsystem of steam turbine condenser in given time and conditions

Fig. 7 .
Fig. 7. Flowchart of reliability structure of pipe subsystem -formula (14), where box R,1,i means 20 systems ( 100 ,1, 4, ( ) [ ( )] in parallel structure, and sub diagrams NR,1-NR,2 pipe systems ( structure, and the latter in serial structureThe function of reliability may be defined , () ps r R t with the for- mula (15) if the current exploitation conditions W e,i determine enabling the cooling water flow through 8000 pipes and 4000 pipes are a backup to streams of transferred fluid heat: