A Simple Mathematical Model for Refrigerating Compressor Optimization

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Scientific Journal of Maritime Research 32 (2018) 146-151 © Faculty of Maritime Studies Rijeka, 2018

Multidisciplinary Multidisciplinarni SCIENTIFIC JOURNAL OF znanstveni časopis MARITIME RESEARCH POMORSTVO

A Simple Mathematical Model for Refrigerating Compressor Optimization Darko Glujić, Predrag Kralj, Dragan Martinović University of Rijeka, Faculty of Maritime Studies, Studentska 2, 51000 Rijeka, Croatia, e-mail: [email protected]; [email protected]; [email protected]

ABSTRACT ARTICLE INFO

A marine equipment optimization depends on the determined criteria. Optimal equipment production Preliminary communication and its usage instead of the standard one, could increase the investment costs resulting in the Received 1 March 2018 increase of the payback period and the important characteristic of the marine equipment should be KeyAccepted words: 4 June 2018 in accordance with the technical rules of the classification society and the international conventions. In some cases the optimization of marine equipment is possible and desirable. The paper deals with the refrigerating systems in general and especially with the refrigerating compressor, the most Optimization expensive part of the system. The optimization criterion is the minimization of the total costs of the Refrigerating compressor refrigerating system. The model should show the way to decrease refrigerating system costs and give Thermal insulation an information tool for a quicker selection of refrigerating system elements. Costs

1 Introduction

In case of provision store refrigerating system on board a ship, exact thermal loads could not be deter- Marine refrigerating systems application on board mined, unless there has been a measurement performed ships is standard for a number of years, with the provi- on an actual ship: thermal load with intentionally or un- sion being the most applied system on board ships [1]. intentionally introduced air and with human work could Although this system could seem to be rather complex, for not be determined exactly because it strongly depends the purpose of such basic modelling, as used in this paper, on the way the cook uses the chamber; the thermal load it could be simplified. with metabolic respiration depends on the fact wheth- The provision system usually consists of several cooling er the food is being loaded fresh or frozen; the thermal chambers, each of them having different temperature, vol- load with electrical equipment, namely fans, lights and ume, lights, air circulation etc. The simplified model used defrosting heaters also varies a lot because there could in this paper will have only one chamber having mediate be a chamber with natural evaporation, the number of temperature, insulation, thermal coefficients and volume. lights depends on the size of the chamber and perhaps Such a simplification could be made even for a number of the chamber could be entered into without switching the compressors. Namely, even if there are two of them, usually lights on etc. That is the reason why the model in this only one is working, while the other is ‘on standby’. A math- paper deals with the thermal load through the chamber ematical model of the refrigerating system having just one walls only. chamber has been presented by Brito et al. [2]. Although the The criterion for the optimization model in this paper paper tries to calculate each and every thermal load of the would be equipment costs, the one mostly appraised and system, some simplifications are made. Simplified mathe- used by shipowners. Obviously, one could select cheap matical models are generally used in the thermal processes equipment with a shorter operating life, while the other analysis: certain parameters could be neglected; sometimes could choose the most expensive equipment with low op- mean values are used, as has been done in the papers by erating costs and long operating life. The approach used in Milošević et al. [3, 4] and Kralj et al. [5]. this paper is to minimize the total costs. D. Glujić et al. 147

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The process is simple enough but the problem rises compressor is determined and from it the capacity of ev- with the equipment standardization. Namely, most of the ery other systems element. Hence, the capacity and price countries with a strong shipbuilding industry have de- of every refrigerating system’s element is proportional to veloped certain standards of production, as have the ma- the capacity of the compressor, the equation (2) can be jor equipment producers. In this respect, the optimal but simplifiedI Ccomp andA gives· Ccomp B · Ccomp C · Ccomp D · Ccomp nonstandard product could increase the investment costs E · Ccomp F · Ccomp Cinsul in such a manner that the installation of a standard model = + + + + + (3) seems a more profitable solution. However, the model pro- + + + posed in this paper could, at least, point-out which of the I C A B C D E F C standard equipment is the closest to the optimal one. or, if simplifiedcomp even further insul The optimal equipment would, furthermore, result in = (1 + + A, +B, C,+ D,+ E + ) F+ (4) the reduced energy consumption and consequently, reduced environment pollution as suggested in the papers where coefficients and stand for linear de- [6, 7, 8]. Additionally, the optimized equipment could be- Kpendence of the element price on the compressor price. If come a part of an integrated ship managing system as sug- the sum of the coefficients in brackets are denominated as gested by Martinović et al. [9, 10] and Kralj [11] and could theI equationKCcomp C (4)insul becomes be easily integrated in the ship cogeneration system [12]. Furthermore, if there is an integrated managing system, = + (5) advanced fault avoiding information systems could be im- In the expression (5) the compressor price results from plemented [13, 14]. 2 A Simplified Mathematical Model the expression of the compressor’s real cooling capacity given with

The standard model of any equipment total costs is giv- t (6) en inU theI expressionE The value [h] stands for the real operation time of the Qo = + U (1) compressor during one day (usually the 24 hour period is I selected) and [W] stands for the theoretical capacity of where, in case of the refrigerating equipment, [$] stands the Qcompressork · A · θ in accordance with the expression bellow for total costs, [$] stands for investment costs of the com- o pressor, condenser, expansion valves, evaporators,E piping = k Δ (7) with all the necessary equipment and, insulation or in -1 -2 A short term, the costs of equipment, and [$] stands for ex- Here [WK m ] stands for2 θ the heat transfer coefficient ploitation costs or the costs of the equipment operation. through the insulation, [m ] stands for the outer surface The price is given in USD because it is a standard in ma- of the cooling chamber and, Δ [K] stands for the tempera- rine transportation. ture difference between the chamber temperature and the The simplified model consists of a single refrigerating surrounding temperature. In the expression (7) the heat chamber, the influence of intentional or unintentional ven- transfer coefficient is well known and is given as tilation, heat dissipation from human work in the chamber, lights, ventilator dissipation are all neglected and only (8) the heat gain through the insulation is considered. Such a simplification is used because for any other model validat- The provision plant usually consists of several cham- ing each heat gain the exact design of the provision system bers, two of them (meat and fish) having temperatures and its usage should be known. Some of the actual heat of approximately -20°C, and the others with the operat- gains are unpredictable: number of entrances, would the ing temperatures above 0°C (from +4 till +13°C). The person switch on the lights or not, how often would the surrounding temperature is set as a tropical one, namely food be loaded etc. Using such a simplification the invest- 35°C. In accordance with the above simplification, the me- I C C C C C C C ment costscomp arecond given TXVin theevap expressionpipe refrig insull diate temperature of a single chamber is set as 0°C. Hence, the temperature difference in the equationc (7) is deter- = C + + + + + + (2)C comp comp cond mined as 35 K. C TXV If the specific price of the compressor is [$/W], the where [$] stands for the price of theC compressor, evap C Q · c [$] stands for the price of the condenser, [$] stands totalcomp price ofreal thecomp compressor is given in the expression Cpipe for the price of the expansion valve(s), [$] stands for = (9) the price of the evaporator(s), [$] stands for the price Crefrig of the piping including all controlling and regulating ele- The total price of the insulating material is given in the Cinsul ments, [$] stands for the price of the refrigerant and, expressionCinsul m belowinsul · cinsul [$] stands for the price of the insulating material. By refrigerating capacity determination, the capacity of the = (10) D. Glujić et al.

148 / Scientific Journal of Maritime Research 32 (2018) 146-151 cinsul

minsul where [$/kg] stands for the specific price of the insu- The implementation of the equations (6) to (8) into lating material and, [kg] stands for the total mass of the (15) one and with a few simplifications results in the it, calculatedminsul A · inδinsul accordance · ρinsul with equation

≈ A (11) (17) 2 δinsul ρinsul Here the [m ] is the outer surface of the chamber, If the equations (13) and (17) are implemented into [m]3 stands for the thickness of the insulation and, the Ubasicf δequation (1) it could be given as [kg/m ] stands for the density of the insulating material. insul The implementation of the equations (6) to (11) into = ( ) (18) the equation (5) gives If the first derivative of function (18) is equalized to 0, the value of the optimal insulation layer thickness is given (12) in the following expression

The simplest way of writing the equation (12) is (19) (13) and, from the optimal insulation layer thickness the optimal refrigerating compressor’s capacity is achieved in where the constants in the equation are equations (6) to (8). 3 Numerical Solution and Graphical Interpretation

(14) Matlab

has been used to create the algorithm for the nu- The exploitation costs is given in the expression below merical solution of the problem. Obviously, the analytical solution could not predict the changes of some parameters ε (15) included, i.e. price of energy or system elements. That is why t several calculations have been made with different prices where Gis the coefficient of the performance of the com- and payback periods. But the other values are used as con3 - pressor, represents again the operating hours of the com- stant, i.e. volume of the cooling chamber, which is 200 m . pressor, is a constant considering the electric motor and Figure 1 represents results for the system with all of cEE the compressor’s efficiency and the price of the electric the elements being twice as costly as the compressor it- energyG ηcomp [$/Wh], · ηem · c Egiven in the equation self, with the price of the energy 20 c/kW and, with the amortization period of 10 years. Optimal insulation thick- = (16) ness results in 11.85 cm.

Figure 1

Presentation of the Results with a Presumed Set of Values 1 D. Glujić et al.

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Figure 2

Presentation of the Results with a Presumed Set of Values 2

Figure 3 P

resentation of the Results with a Presumed Set of Values 3

Figure 4

Presentation of the Results with a Presumed Set of Values 4 D. Glujić et al.

150 / Scientific Journal of Maritime Research 32 (2018) 146-151

Figure 5

Presentation of the Results with a Presumed Set of Values 5

4 Conclusion For the same price of the system (compressor and other elements), the same amortization period and the price of the energy half of the price from the first model, the algo- The energy efficiency of complex power plants should rithm has given the optimal insulation thickness of 8.1 cm. always be considered. The ship’s power plant is no ex- The numerical solution is given on Figure 2. ception, quite the opposite. However, in the past, there The third solution has been created for the energy were some prevailing factors used during the process of price of 30c/kW, the same price of the system and the design, such as safety and availability. Not that those are same amortization period. The optimal insulation thick- not important any more, but the efficiency of every ship’s ness has reached 14.7 cm. component has become important because a reduced The next model has used the energy price of 20 c/kW, efficiency usually means more pollutants in the marine the same price of the system as in the first one, but with environment. the oversized compressor because of other heat losses The paper deals with a small, but important com- from the refrigerated chamber (10 times the compressor ponent on board every ship – refrigerating compressor. from the first model) and, with the amortization period Namely, it deals with its optimization. Those compressors shorten to three years only. The optimal insulation thick- emerge in the provision refrigerating plant, air condition- ness has resulted in 19.6 cm. The numerical solution is ing plants, cargo hold refrigeration and so on. The opti- shown on Figure 4. mization method applied in this paper is quite simplified The last model has used the energy price of 20 c/kW, and could be used for any kind of refrigerating compres- the price of all other system elements together being 1.2 sor, provided that some actual values of the refrigeration the price of the compressor, the compressor’s capacity 10 plant are known. The shortcoming of the presented model, times of the compressor from the first model and the am- using the thermal load through the walls only, should be ortization period of three years. The solution is presented overpassed in the future, knowing the actual refrigerating on Figure 5. The optimal insulation resulted 20.25 cm. system design. Using the iterative method through expressions (6) to In this paper, an imaginative cooling space has been (8) and the same computer, the algorithm optimal com- used. The mathematical model has calculated an optimal pressor capacities have been calculated. Those values are insulation layer thickness and, through it, the optimal re- Tablepresented 1 in Table 1. frigerating compressor capacity. Several results have been obtained depending on energy prices and amortization OptimalOptimal insulation Compressor layer Capacities Refrigerating compressor periods. The main purpose of the model however is the thickness [cm] optimal capacity [kW] indication of the one way of solving the refrigerating compressor optimization. References 11.85 21.5 8.1 31.4 Brodski sustavi mikroklime – automatizacija i opti- mizacija 14.7 17.3 [1] Kralj, P.: 19.6 13 , Zbornik Pomorskog fakulteta u Rijeci, Rijeka, god. 20.25 12.6 12 (1998), pp. 197-203. D. Glujić et al.

/ Scientific Journal of Maritime Research 32 (2018) 146-151 151 Simulation and op- Appendix timization of energy consumption in cold storage chambers [2] fromBrito, theP. – horticulturalLopes, P. – Reis, industry, P. – Alves, O.:

Simplified IntMathematical J Energa environmentModel of Va- Listing of Matlab computer application: Eng (2014), doi: 10.1007/s40095-014-0088-2. porization in a Fresh Water Vacuum Distillation Generator clear [3] Milošević, Š. – Kralj, P.: , clc Proceedings from the SymposiumVacuum distillation “Energy fresh and water Environ ge- d = 0.01 %početna debljina izolacije ment”, Opatija, 1996, pp. 237-244. nerator application on board ship t = 24 %sati rada kompresora u danu [4] Milošević Š. – Kralj, P.: , ELMAR ’98, Zadar, 1998., L = 10 %koeficijent topliske vodljivosti zraka pp. 196-200. Lm = 0.026 %koeficijent toplinske provodljivosti [5] Kralj, P. – Martinović, D. – Tudor, M.: Analysis of thermo- izolacije dynamic and technological basics of the marine fresh wa- A = 280 %površina komore ter generatorPrilog raspravi model, Desalinationo zaštiti morskog and okoliša water Treatment, (2017) 1-6, pp. 180-185, doi: 10.5004/dwt.2017.21552. q = 35 %razlika unutarnje i vanjske [6] Kralj, P.: Minimization of, Zbornik energy and ra- temperature exergydova Pomorskog consumption fakulteta, by application Rijeka, Vol. of combined 11 (1997), plant pp. 119-128. on mari- Ck = 0.1135 %cijena kompresora (€/W) [7] Martinović, D. – Kralj, P. – Bukša, A.: times isolated communities K = 2 %koeficijent cijene armature u odnosu Razvoj brodskih rashlad- na cijenu kompresora nih sustava – utjecaj propisa, ELMAR o zaštiti ’98, okoliša Zadar, 1998, pp. 22-25. [8] Kralj, P. - Bukša, A. - Martinović, D.: R = 40 %gustoca izolacije Stanje, Pomorstvo, i razvoj inte Ri- Ci = 8 %cijena izolacije (€/kg) jeka, Vol. 13 (1999), pp. 211-222. gralnog upravljanja brodom Cs = 0.0002 %cijena elektricne energije (€/Wh) [9] Martinović, D. - Tireli, E. – Kralj, P.: , Zbornik radova Međunarodnog G = 10 %broj godina na kojem se promatra znanstveno-stručnog simpozija o prometnimIntegration znanostima, Control and e = 3 %COP Portorož, 1997, pp. 129-134. Supervision into an Intelligent Machinery Management Sys- for i = 2 : 51 [10] temMartinović, D., Tireli, E., Kralj, P.: d(i) = d(i-1)+0.01 , 2nd Congress; Transport Traffic Logistics, Proceedings, y(i) = ((24*A*q*Ck*K)/t)*(1/((1/L)+ Portorož,Prijedlog Slovenia, sustava 2000, pp.upravljanja 349-353, vakuumskog ISBN 86-435-0352- genera- (d(i)/Lm)+(1/L))) + (A*d(i)*R*Ci) tora5. slatke vode [11] Kralj, P.: x(i) = (t*A*q*0.94*0.98*Cs*365*G)* (1/(e*((1/L)+(1/L)+(d(i)/Lm)))) , Zbornik radova PomorskogTechno-economic fakulteta, Ri- analysisjeka, Vol. of 10 the (1996), cogeneration pp. 83-90. process on board ships end [12] Brozičević, M. – Martinović, D. - Kralj, P.: figure , Journal plot (y, ’linewidth’, 2) os Sustainable Development of TransportMaintenance and Logistics, of ship’s sysVol.- tems2, No. 1, 2017, doi: 10.14254/jsdtl.2017.2-1.1. hold [13] Tudor, M. – Bukša, A.Utjecaj – Kralj, rizika P.: kvara na računalni oda- plot (x, ’linewidth’, 2) bir pristupa, Pomorstvo, održavanju Rijeka, brodskih Vol. 18 (2004), sustava pp. 29-42. [14] Tudor, M. - Kralj, P.: xlabel (‘Insulation thickness [cm]’) , Pomorstvo, Rijeka, ylabel (‘Cost [€]’) Vol. 14 (2000), pp. 43-52. legend (‘initial cost’, ’operational cost’)