Modeling and optimization of air-cooled condensers for low-temperaturepower cycles

Vic Van Peborgh

Abstract

This paper presents a step-wise, three zone model for air-cooled condensers for low-temperature powercycles. Through nonlinear optimization the cost optimal geometry for a number of cases is determined.The solutions of this optimization are then used to examine the influence of the condenser temperature, theambient temperature and the available land area on the optimal cost. Also, the cost as a function of cycleefficiency is examined for organic Rankine cycles.

1. Introduction

1.1. Background

Low-temperature heat sources are omnipresent.Some examples are [1]:

• industrial waste heat,

• exhaust gasses of diesel, biogas or landfill gasengines,

• combustion of waste gasses, solvents,. . . ,

• biomass incineration,

• solar thermal energy,

• geothermal energy.

The thermal energy of these sources can be con-verted to mechanical energy by a low-temperaturepower cycle, e.g. an organic Rankine cycle (ORC).

This study focuses on one component of an ORC:the condenser. More specific, only air-cooled con-densers are considered. Although heat transfer toair is less efficient than heat transfer to water or an-other refrigerant, air is often used. This is mainlybecause of the strict regulation on the usage of wa-ter in some areas, or the limited availability in thedirect surroundings of the power plant. In thesecases air can be the more economic refrigerant.

1.2. Problem statement

The efficiency of a power cycle is always limited bythe Carnot efficiency:

ηC = 1− TLTH

(1)

with TH the temperature of the (low-temperature)heat source and TL the temperature at which resid-ual heat is rejected to the environment (in the con-denser). It is obvious from equation (1) that themaximal efficiency of a power cycle increases whenthe temperature of the heat source TH rises, orwhen the heat rejection temperature TL is lowered.

When considering low-temperature heat sources,TH will be limited to about 280◦C [1], making themaximum cycle efficiency low. To compensate thelow TH as much as possible, it is very important tokeep the heat rejection temperature TL low too.

For air-cooled condensers, the absolute minimumfor the heat rejection temperature is the ambienttemperature. Because real heat transfer requiresa driving temperature difference it is important tokeep this temperature difference as small as pos-sible, i.e. to have a high-performance, optimizedcondenser.

1.3. Air cooled condenser geometry

This study presents a mathematical model for anA-frame air-cooled condenser, illustrated in figure

Figure 1: A-frame condenser module [2]

Figure 2: Flat tube section with corrugated fins [4]

1. The condenser consists of an inlet steam duct, acondensate drain, a number of fans and two bundlesof parallel finned tubes. The tube bundles consistof a large number of parallel flat tubes with cor-rugated aluminum fins, arranged in one tube row.This geometry is the most widely used geometryfor air-cooled condensers according to Putman andJaresch [3]. The type of finned tubes is illustratedin figure 2.

1.4. Scope and objectives

The scope of this study is to minimize the netpresent total cost Ctot of an A-frame air-cooled con-denser. The set of optimization variables consistsof some geometrical parameters (see table 1) andthe mass flow of air mair. Before this optimizationcan take place, a suitable mathematical model withenough degrees of freedom is to be constructed.

To realize this, the following objectives must becompleted: a) construct a mathematical model foran A-frame air-cooled condenser; b) validate this

model; c) identify a number of relevant cases to in-vestigate; and d) optimize the condenser geometryfor these cases.

The presented model is to be implemented in amuch larger model that simulates an entire ORC.This larger model is to be optimized as well, andlong computing times should be avoided. This iswhy the presented model should be as ‘fast’ as pos-sible.

1.5. Contents

The next section describes the technical model.The main principles and methodology are explainedin depth. Section 3 explains how the net presentcost Ctot is calculated. In section 4 the optimiza-tion problem is formulated and section 5 discussesthe results of this optimization.

2. Technical model

2.1. Air-cooled condenser (ACC) models in litera-ture

Several mathematical ACC models are presentedin literature. These models can always be classifiedas one of two types: computational fluid dynamics(CFD) models [5–7], or (semi-) analytic models [8–15].

Even though large-scale ACC’s are becoming moreand more popular since the late 70’s [3], all thecited models describe small-scale applications likerefrigerators, air-conditioning installations and heatpumps. Because this study considers applicationson a completely different scale and with a com-pletely different configuration, none of the citedmodels can be applied here. A new model is tobe constructed.

Although CFD-models can be very accurate, thesemodels imply a long computation time. As statedin paragraph 1.4, this model should be as ‘fast’ aspossible, ruling out the usage of CFD.

An analytic model uses correlations rather than dis-cretized conservation equations (CFD) to calculatethe heat transfer coefficient and the friction factorfor the working fluid and the cooling air. The phys-ical phenomena that take place in the condenserdiffer greatly throughout it because of the presence

2

of one- and two phase flow. This distinction in flowregime is made in the presented model.

As explained in the next section, the technical partof the model consists of two routines: one for thecondenser zones with one phase flow (desuperheatigand subcooling zone) and one for the condensationzone, where two phase flow occurs.

2.2. Technical model

2.2.1. Input

The inputs to the technical part of the model are:

• ambient pressure P◦ and temperature T◦;

• inlet condition of the working fluid;

• partial condenser geometry (table 1);

• air mass flow per unit of tubelength mair;

• desired outlet condition of the working fluid(partly defined: temperature Twf,out or qualityxwf,out).

Because the model divides the tube bundles in smallsections (see later), and the length of the finnedtubes is not an input, the air mass flow mair isindirectly calculated with the mass flow per unit oftubelength, which is a input parameter:

mair =mair

2Ltot(2)

The mass flow of air over a finned tube section withlength Lstep can now be written as

mair,step =1

2

Lstep

Ltotmair = mairLstep (3)

2.2.2. Output

The outputs of the model are:

• length of the finned tubes Ltot;

• outlet pressure of the working fluid Pwf,out;

• outlet condition of the cooling air.

With the inputs given, the model calculates howlong the finned tubes should be for the working fluidto reach the desired outlet temperature or quality.The other two outputs are also important, becausethey define the outlet condition of both the air andthe working fluid.

2.2.3. Routines

To calculate these outputs, the model divides theparallel finned tube bundles into small sections.Because the modeled ACC is a crossflow heat ex-changer, the inlet conditions of air will be the samefor every section. The heat transfer coefficient andthe friction factor for the air-side flow, respectivelyhair and fair are consequently calculated at thestart of the model.

The working fluid’s inlet condition is different forevery section, but at the first section it is known.This first inlet condition allow the calculation of theworking fluid’s outlet condition at this first section.The outlet condition at one section is the inlet con-dition at the next section. These calculations con-tinue until the desired working fluid’s final outletcondition is reached.

The technical model is based on the ε-NTUmethodology. This methodology provides a certainlevel of insight that the LMTD-method cannot givefor crossflow heat exchangers [17]. A detailed de-scription of this method can be found in any intro-ductory heat transfer textbook (e.g. Incropera andDeWitt [17]).

As stated before, the technical model consistsof two different routines: one for one phase flow(desuperheating and subcooling) and one fortho-phase flow (condensation). Both routines areexplained in the next two paragraphs.

One phase flow

Both the vapor flow during desuperheating and theliquid flow during subcooling are one phase flows.These flows abide by the same rules, and thus canbe modeled with the same routine. The one phaseflow routine divides the tube bundles in small sec-tions whose length is called Lstep. For every section,the one phase flow routine calculates the necessaryparameters in following order:

• hwf ,

• Cair, Cwf and NTU ,

• ε,• Tair,out and Twf,out,

• Pwf,out.

hwf is the heat transfer coefficient of the workingfluid flow, C is defined as C = mcp, with m the

3

Parameter Description Optimization

di [mm] finned tube inner diameter variablet [mm] finned tube wall thickness t = 1, 5mmsb [mm] distance between finned tubes variablesf [mm] distance between fins variableδf [mm] thickness of fins δf = 0, 3mmNtpr [−] number of finned tubes per bundle variableθ [◦] inclination angle of finned tubes θ = 60◦

e [mm] internal roughness of finned tubes e = 0, 046mm [16]

Table 1: Name, units and description of the geometric input parameters. The last column gives the value of the parameter inthe optimization. The geometric parameters are also explained graphically in figure 3.

mass flow and cp the specific heat capacity of theconsidered fluid.

In the desuperheating zone, these steps are repeateduntil the stopping condition is reached:

Twf ≤ Twf,sat (4)

with Twf,sat the saturation temperature of theworking fluid. In the subcooling zone, the stoppingcondition is

Twf ≤ Twf,sat −∆Twf,sub (5)

with ∆Twf,sub the specified degree of subcooling.

Two phase flow

In the condensation zone a slightly different rou-tine is used. The most important difference is thatthe finned tubes are not divided into parts with anequal Lstep. Instead, the total change in quality ∆xthat must be achieved in the condensation zone isdivided into n equal parts ∆xstep. In every step,following parameters are calculated:

• hwf ,

• NTU ,

• ε,• Lstep and Tair,out,

• Pwf,out.

These calculation steps are repeated until the stop-ping condition is reached. For a two phase flow asdesired condenser output this stopping condition isformulated as

xwf ≤ xwf,out (6)

If the working fluid’s desired outlet condition is asubcooled liquid, the stopping condition is

xwf = 0 (7)

and the working fluid is passed on to the one phaseroutine to model the one phase flow.

As stated in paragraph 2.1 the presented modeluses correlations to calculate the heat transfer coef-ficients and friction factors in each step. The nextparagraph explains which correlations are used.

2.2.4. Correlations

For the working fluid the Gnielinski [18] correla-tion was used to calculate the Nusselt number inone phase flow. From this Nusselt number the heattransfer coefficient hwf is calculated. The pressuredrop caused by friction is calculated with the fric-tion factor fwf . This friction factor is approximatedby the Churchill [19] correlation. This correlationis particularly suitable because it is fairly accurate[20] and spans the entire Rewf -domain.

In the condensation zone, the heat transfer coeffi-cient is calculated with the Shah [21] correlation.Because this correlation is only suited for condens-ing flow in horizontal tubes, it must be multipliedwith the correction factor of Wurfel et al. [22]. Thepressure drop caused by friction during condensa-tion is calculated with the Chisholm [23] correla-tion.

For the air-side heat transfer coefficient hair andfriction factor fair, no suitable correlations were

4

Figure 3: Geometry described by Wang et al. [26]. Air flowsin the z-direction.

found in literature. Although a large number ofcorrelations is available for a lot of different geome-tries [4, 24–40] none of the considered correlationsdescribes the geometry illustrated in figure 2. Parkand Jacobi [41] and Thome [42] came to the sameconclusion after their own literature studies. Be-cause no better correlation was found, the Wanget al. [26] correlations were used to calculate hairand fair. This correlation actually describes airflow in a heat exchanger such as the one illustratedin figure 3.

The used ε−NTU relationships are from Navarroand Cabezas-Gomez [43].

2.3. Validation

Because the modeled geometry (Fig. 3) does notcorrespond with the real geometry of air-cooled con-densers, no validation of the model was performed.Two datasheets of real ACC’s were obtained (fromIndaver and EDF-Luminus), both of them used thegeometry depicted in figure 2. According to litera-ture [3] this is the most widely used type of finnedtubes for ACC’s. Manufacturer’s websites back upthis statement.

2.4. Implementation

The presented model is implemented in the opensource programming language Python [44].

3. Economic model

As stated in section 1.4, a cost-optimal solution isto be found. To estimate the total cost of an air-cooled condenser, it is important to note that thetotal cost consists of two terms:

Ctot = Ccap + Cwork (8)

with Ccap the capital cost and Cwork the workingcost of the condenser installation.

The capital cost can be further decomposed:

Ccap = Che + Cfans + Cpump + Cinst + Cland (9)

with Che and Cfans the capital cost of respectivelythe finned-tube heat exchanger and the fans. Thesetwo terms are calculated using a correlation fromPeters and Timmerhaus [45]. Cpump is the capitalcost of the pump, estimated with a correlation fromSeider et al. [46]. Cinst is the installation cost andis estimated to be [45]:

Cinst = 0, 30 (Che + Cfans + Cpump) (10)

Cland is the cost of the land on which the condenseris built.

The working cost consists of two contributions:

Cwork = Cmain + Cel

= Cmain + Cel,fans + Cel,pump

(11)

Cmain is the annual maintenance cost, estimated tobe [46]:

Cinst = 0, 05 (Che + Cfans + Cpump) (12)

Cel,fans and Cel,pump are the cost of the energy con-sumed by the fans and the pump, respectively. Thelatter contribution will be a lot smaller than thethat of the fans.

The working cost over a timespan of twenty years(the estimated lifetime of the condenser installation[46]) are reduced to a net present cost that can beadded to the capital cost Ccap to form the total netpresent cost Ctot.

5

4. Optimization

A cost-optimal solution is to be found, and there-fore expression (8) is minimized. The optimizationvariables are a number of geometrical parameters(see table 1) and the mass flow of air per unit oftubelength mair. A total of nine boundary condi-tions are imposed of which five are a linear functionof the optimization variables.

The inner diameter of the finned tubes should al-ways be between 10 ≤ di ≤ 40mm. The physicallower boundary for the distance between the tubesis st ≥ di, in practice a slightly larger lower bound-ary is used to avoid numerical problems during theoptimization: 1, 5di ≤ st ≤ 5di. The upper bound-ary for st is to avoid unrealistic values from beingconsidered. The distance between the cooling finssf also has a physical lower boundary: sf ≥ δf .Again, for numerical reasons, a larger lower bound-ary is used in practice: sf ≥ 3δf .

The remaining four boundary conditions are a non-linear function of the optimization variables. Themost important boundary condition states that thepressure of the working fluid should always remainpositive: Pwf,out ≥ 0. A negative working fluidpressure has no physical relevance and causes thecalculation of the fluid properties (using RefProp[47]) to behave unexpectedly. The second non-linear boundary condition limits the length of thefinned tubes: 5 ≤ Ltot ≤ 15m. The lower boundaryfor Ltot is to avoid numerical problems. The upperboundary is a practical limit [3]. The last bound-ary condition is imposed on the occupied land areaAland. The upper limit for this parameter is variedthroughout the optimization to study the influenceof this parameter on the optimal net present cost.

As a summary, the optimization problem can bewritten as:

mindi,st,sf ,Ntpr,mair

Ctot

subject to 10 ≤ di ≤ 40mm

1, 5 di ≤ st ≤ 5 di

sf ≥ 0, 9mm

0 ≤ Pwf,out

5 ≤ Ltot ≤ 15m

Aland ≤ Aland,max

(13)

25 30 35 40 45 50

Tcond [◦C]

0

2

4

6

8

10

12

14

16

Ktot

[106e

]

Isobutan

Propane

R1234yf

R134a

R227ea

R245fa

Figure 4: Total net present cost Ktot as a function of theinitial condensing temperature Tcond (T◦ = 15◦C)

The optimization is realized in Python [44], withthe OpenOpt [48] optimization toolbox. Becausethe optimization is unstable, a number of differentsolvers (mma, scipy slsqp, ralg and scipy cobyla)was used. Manual iteration between these solverswas necessary to obtain convergence. Every solu-tion was checked with every solver.

5. Results

The optimization problem given by expression (13)is carried out for a number of different cases [49].A total of six working fluids (isobutane, propane,R134a, R1234yf, R227ea and R245fa) are consid-ered at six different condenser temperatures be-tween 25 and 50◦C. A number of different ambi-ent temperatures is considered and the influence ofAland,max is studied. In the initial cases, Aland,max

is set to +∞.

5.1. Influence of the condenser temperature

Figure 4 shows the total net present cost (equation8) as a function of the initial condensing temper-ature Tcond for the different working fluids at anambient temperature of T◦ = 15◦C.

This figure clearly shows that the cost rises rapidlywhen the condenser temperature lowers. Two ofthe points on the graph show deviant behavior:at Tcond = 25◦C, both R227ea and R245fa give

6

25 30 35 40 45 50

Tcond [◦C]

0

1

2

3

4

5

6

Ktot

[106e

]

Isobutan

Propane

R1234yf

R134a

R227ea

R245fa

Figure 5: Total net present cost Ktot as a function of theinitial condensing temperature Tcond (T◦ = 10◦C)

rise to an extremely high cost. This can be ex-plained as follows: the initial condensation tem-perature is 25◦C, but as the working fluid flowsthrough the finned tubes, the pressure Pwf declines.As the pressure declines, the condensation tem-perature also declines. In the two deviant points,the condensation temperature declines to 17,98◦Cen 16,32◦C (for R227ea and R245fa, respectively).Considering the ambient temperature in these pointis T◦ = 15◦C it is no surprise that because of thesmall temperature difference a very large heat ex-changer is needed.

Figure 5 shows the same function at an ambienttemperature of T◦ = 10◦C. Because the temper-ature difference between the air and the workingfluid is bigger, no deviant points are observed.

5.2. Influence of the ambient temperature

When comparing figures 4 and 5, it can be seenthat the cost for T◦ = 10◦C is generally lower thanthat for T◦ = 15◦C. Figure 6 illustrates this for theworking fluid R134a. Figure 6 clearly shows thatapart from being lower, the total net present costcurve also has a smaller slope when the ambienttemperature T◦ is lower. This is because both costcurves will go to +∞ when ∆T = Tcond − T◦ goesto zero. In a way, the curve for T◦ = 10◦C is ahorizontal shift to the right of the T◦ = 15◦C curve.Because of a difference of fluid properties at thedifferent condensation temperatures this horizontalshift is not exact.

25 30 35 40 45 50

Tcond [◦C]

0

1

2

3

4

5

6

Ktot

[106e

]

Tomg =15◦C

Tomg =10◦C

Figure 6: Total net present cost Ktot as a function of theinitial condensing temperature Tcond at different ambienttemperatures (R134a)

This paragraph and paragraph 5.1 can be summa-rized by stating that the net present cost grows ex-ponentially if the temperature difference betweenthe working fluid and the ambient air declines.

5.3. Influence of the available land area

In the previous cases, Aland,max was set to +∞.Because it was observed that the occupied land areabecame very big for low condenser temperatures,the influence of this parameter was studied.

Figure 7 shows how the total net present cost growsexponentially when there’s less land area available.To study this rise of the cost a bit more, the nextparagraph considers the evolution of the two com-ponents Ccap and Cwork of the total net presentcost (equation (8)) as a function of the availableland area Aland,max.

5.4. Influence of the available land area on the com-position of the total net present cost

Figure 8 illustrates the evolution of the two com-ponents of the total net present cost: the capitalcost Ccap and the working cost Cwork. It is clearthat the total cost grows exponentially when theavailable land area gets smaller. The second trendis also very clear: as this land area gets smaller thecapital cost Ccap gets smaller too. This is becausethe largest component of the capital cost is the heatexchanger cost, and a smaller heat exchanger (lessland area available) costs less. When the avail-

7

30 35 40 45 50

Tcond [◦C]

0

500

1000

1500

2000

2500

3000

3500

4000

Aland

[m2]

Aland,max = +∞Aland,max = 3000m2

Aland,max = 2500m2

Aland,max = 2000m2

(a)

30 35 40 45 50

Tcond [◦C]

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Ktot

[106e

]

Aland,max = +∞Aland,max = 3000m2

Aland,max = 2500m2

Aland,max = 2000m2

(b)

Figure 7: (a) Occupied land area Aland as a function of theinitial condensing temperature Tcond for different values ofAland,max (T◦ = 10◦C) (b) Total net present cost Ktot asa function of the initial condensing temperature Tcond fordifferent values of Aland,max (T◦ = 10◦C)

able land area gets smaller, the working cost Cwork

grows exponentially. This is because the amount ofheat to be transferred remains the same, but thetotal heat transfer area is smaller. The reductionof heat transfer area is compensated by a bigger airmass flow. A bigger air mass flow requires morepowerful fans, which consume more energy.

It should be noted that although the contributionof capital cost gets smaller when the available landarea decreases, it stays a substantial part of the to-tal net present cost. The capital cost is about 60%of the total net present cost in the case of an un-limited available land area. This 60% gradually de-

+∞ 3000 2500 2000

Aland,max [m2]

0

1

2

3

4

Ktot

[106e

]

Kwerk

Kkap

Figure 8: Evolution of the two components Ccap and Cwork

of the total net present cost Ktot as a function of the avail-able land area Aland,max (R134a, T◦ = 10◦C)

creases to 25% in the case of Aland,max = 2000m2,which still is a substantial fraction of the total netpresent cost.

5.5. Total net present cost as a function of the ORCcycle efficiency

Figure 9 shows the net present cost as a function ofthe ORC cycle efficiency of the different cases. It isclear that the total cost grows exponentially withthe ORC cycle efficiency.

This exponential growth is a consequence of the su-perposition of two trends: firstly, the ORC cycleefficiency is higher for low condenser temperatures[49]; secondly, as illustrated in figures 4 and 5, thetotal net present cost rises when the condenser tem-perature is lower.

6. Conclusion

This paper presents a new ACC model. This modelis a step-wise, three zone model. The condensergeometry was optimized for a number of differentcases. Clear trends were shown:

• the total net present cost grows exponentiallywhen the temperature difference between theworking fluid and the ambient air becomessmaller;

• the total net present cost grows exponentiallywhen the available land area gets smaller;

8

8 9 10 11 12 13

ηcycle [%]

0

1

2

3

4

5

6

Ktot

[106e

]

Isobutan

Propane

R1234yf

R134a

R227ea

R245fa

Figure 9: Total net present cost Ktot as a function of theORC cycle efficiency (T◦ = 10◦C)

• as the available land area gets smaller, theworking cost grows and the capital cost de-clines; and• a higher ORC cycle efficiency is coupled to a

much higher cost, this growth is exponential.

Although the trends predicted by this model seemplausible, the model can still be improved. Themost important improvement that can be imple-mented is a suitable correlation for the air-side heattransfer coefficient and friction factor. This corre-lation can be sought in literature (although an ex-tensive literature study was unsuccessful) or can beset up with experimental data.

References

[1] Howest Kortrijk, The Organic Rankine Cycle, URL:http://www.orcycle.be, last checked on 2013-29-04,2013.

[2] J. A. Heyns, Performance characteristics of anair-cooled steam condenser incorporating a hybrid(dry/wet) dephlegmator, Master’s thesis, Departmentof Mechanical Engineering, University of Stellenbosch,2008.

[3] R. E. Putman, D. Jaresch, The Impact of Air CooledCondensers on Plant Design and Operations, in: Pro-ceedings EPRI Condenser Technology Conference, 11–12, 2001.

[4] L. Yang, H. Tan, X. Du, Y. Yang, Thermal-flow char-acteristics of the new wave-finned flat tube bundles inair-cooled condensers, International Journal of ThermalSciences 53 (2012) 166–174.

[5] L. Borghei, R. H. Khoshkho, Computational fluid dy-namics simulation on a thermal power plant with air-cooled condenser, Proceedings of the Institution of Me-chanical Engineers, Part A: Journal of Power and En-ergy 226 (7) (2012) 837–847.

[6] X. Huang, X. Zhang, Field Test and Research on Heat-transfer Performance of Plate Air-cooled Condenser,Procedia Environmental Sciences 10 (2011) 1148–1153.

[7] M. Awad, H. Mostafa, G. Sultan, A. Elbooz, A. El-ghonemy, Performance Enhancement of Air-cooledCondensers, Acta Polytechnica Hungarica 4 (2) (2007)125–142.

[8] D. Admiraal, C. Bullard, Heat transfer in refrigeratorcondensers and evaporators, Urbana 51 (1993) 61801.

[9] C. Cuevas, J. Lebrun, V. Lemort, P. Ngendakumana,Development and validation of a condenser three zonesmodel, Applied Thermal Engineering 29 (17) (2009)3542–3551.

[10] G. Davis, T. Scott, Component, Modeling Require-ments for Refrigeration System Simulation .

[11] C. C. Hiller, L. R. Glicksman, Improving heat pumpperformance via compressor capacity control: analysisand test, Tech. Rep., MIT Energy Lab, 1976.

[12] M. Martins-Costa, J. Parise, A three-zone simulationmodel for a air-cooled condensers, Heat Recovery Sys-tems and CHP 13 (2) (1993) 97–113.

[13] S. Fischer, The Oak ridge heat pump design model:Mark III version program documentation, Tech. Rep.,ORNL Oak Ridge National Laboratory (US), 1988.

[14] J. Parise, W. Cartwright, Local analysis of three-dimensional air-cooler condenser using a two-phase flowdiagram., in: Proceedings of the 21st National HeatTransfer Conference: Heat Exchangers for Two-phaseapplications, 1983.

[15] J. Garcıa-Cascales, F. Vera-Garcia, J. Gonzalvez-Macia, J. Corberan-Salvador, M. Johnson, G. Kohler,Compact heat exchangers modeling: condensation, In-ternational Journal of Refrigeration 33 (1) (2010) 135–147.

[16] R. Fox, A. McDonald, P. Pritchard, Introduction toFluid Mechanics, John Wiley & Sons Canada, Limited,2008.

[17] F. Incropera, D. DeWitt, Fundamentals of heat andmass transfer, John Wiley, sixth edn., 2007.

[18] V. Gnielinski, Neue Gleichungen fur den Warme-undden Stoffubergang in turbulent durchstromten Rohrenund Kanalen, Forschung im Ingenieurwesen A 41 (1)(1975) 8–16.

[19] S. Churchill, Friction-factor equation spans all fluid flowregimes, Journal of chemical engineering 84 (24) (1977)91–92.

[20] D. Brkic, Review of explicit approximations to the Cole-brook relation for flow friction, Journal of PetroleumScience and Engineering 77 (1) (2011) 34–48.

[21] M. Shah, A general correlation for heat transfer duringfilm condensation inside pipes, International Journal ofHeat and Mass Transfer 22 (4) (1979) 547–556.

[22] R. Wurfel, T. Kreutzer, W. Fratzscher, Turbulencetransfer processes in adiabatic and condensing film flowin an inclined tube, Chemical engineering & technology26 (4) (2003) 439–448.

[23] D. Chisholm, Pressure gradients due to friction duringthe flow of evaporating two-phase mixtures in smoothtubes and channels, International Journal of Heat andMass Transfer 16 (2) (1973) 347–358.

[24] A. Zukauskas, R. Ulinskas, Heat transfer in tube banksin crossflow, Experimental and applied heat transferguide books, Hemisphere Pub. Corp., 1988.

[25] T. Rabas, P. Eckels, R. Sabatino, The effect of fin den-sity on the heat transfer and pressure drop performance

9

of low-finned tube banks, Chemical Engineering Com-munications 10 (1-3) (1981) 127–147.

[26] C.-C. Wang, Y.-J. Chang, Y.-C. Hsieh, Y.-T. Lin, Sen-sible heat and friction characteristics of plate fin-and-tube heat exchangers having plane fins, InternationalJournal of Refrigeration 19 (4) (1996) 223–230.

[27] M. Abu Madi, R. Johns, M. Heikal, Performance char-acteristics correlation for round tube and plate finnedheat exchangers, International journal of refrigeration21 (7) (1998) 507–517.

[28] C.-C. Wang, K.-Y. Chi, C.-J. Chang, Heat transfer andfriction characteristics of plain fin-and-tube heat ex-changers, part II: Correlation, International Journal ofHeat and Mass Transfer 43 (15) (2000) 2693–2700.

[29] G. Xie, Q. Wang, B. Sunden, Parametric study andmultiple correlations on air-side heat transfer and fric-tion characteristics of fin-and-tube heat exchangerswith large number of large-diameter tube rows, AppliedThermal Engineering 29 (1) (2009) 1–16.

[30] C. Wang, W. Fu, C. Chang, Heat transfer and fric-tion characteristics of typical wavy fin-and-tube heatexchangers, Experimental Thermal and Fluid Science14 (2) (1997) 174–186.

[31] C.-C. Wang, Investigation of wavy fin-and-tube heatexchangers: a contribution to databank, ExperimentalHeat Transfer 12 (1) (1999) 73–89.

[32] C.-C. Wang, Y.-M. Hwang, Y.-T. Lin, Empirical cor-relations for heat transfer and flow friction characteris-tics of herringbone wavy fin-and-tube heat exchangers,International journal of refrigeration 25 (5) (2002) 673–680.

[33] A. Achaichia, T. Cowell, Heat transfer and pressuredrop characteristics of flat tube and louvered plate finsurfaces, Experimental Thermal and Fluid Science 1 (2)(1988) 147–157.

[34] C.-C. Wang, C.-J. Lee, C.-T. Chang, S.-P. Lin, Heattransfer and friction correlation for compact louveredfin-and-tube heat exchangers, International Journal ofHeat and Mass Transfer 42 (11) (1998) 1945–1956.

[35] Y.-J. Chang, C.-C. Wang, A generalized heat transfercorrelation for Iouver fin geometry, International Jour-nal of heat and mass transfer 40 (3) (1997) 533–544.

[36] M.-H. Kim, C. W. Bullard, Air-side thermal hydraulicperformance of multi-louvered fin aluminum heat ex-changers, International Journal of Refrigeration 25 (3)(2002) 390–400.

[37] Y.-G. Park, A. M. Jacobi, Air-side heat transfer andfriction correlations for flat-tube louver-fin heat ex-changers, Journal of heat transfer 131 (2).

[38] W. Li, X. Wang, Heat transfer and pressure drop corre-lations for compact heat exchangers with multi-regionlouver fins, International Journal of Heat and MassTransfer 53 (15) (2010) 2955–2962.

[39] J. Dong, J. Chen, Z. Chen, Y. Zhou, Air-side thermalhydraulic performance of offset strip fin aluminum heatexchangers, Applied thermal engineering 27 (2) (2007)306–313.

[40] Y. Wang, L.-C. Wang, Z.-M. Lin, Y.-H. Yao, L.-B.Wang, The condition requiring conjugate numericalmethod in study of heat transfer characteristics of tubebank fin heat exchanger, International Journal of Heatand Mass Transfer 55 (9) (2012) 2353–2364.

[41] Y. Park, A. Jacobi, Air-Side Performance Characteris-tics of Round-and Flat-Tube Heat Exchangers: A Lit-erature Review, Analysis and Comparison, Urbana 51

(2001) 61801.[42] J. R. Thome, Engineering Data Book III, Wolverine

Tube Inc .[43] H. Navarro, L. Cabezas-Gomez, Effectiveness-ntu com-

putation with a mathematical model for cross-flow heatexchangers, Brazilian Journal of Chemical Engineering24 (4) (2007) 509–521.

[44] The Python Community, Python Programming Lan-guage - Official Website, URL: http://www.python.org,laatst nagekeken op 2013-05-07, ????

[45] M. S. Peters, K. D. Timmerhaus, Plant design and eco-nomics for chemical engineers, McGraw-Hill New York,fourth edn., 1991.

[46] W. Seider, J. Seader, D. Lewin, Product and ProcessDesign Principles: Synthesis, Analysis, and Evaluation,John Wiley & Sons Australia, Limited, second edn.,2004.

[47] E. W. Lemmon, M. L. Huber, M. O. McLinden,NIST reference fluid thermodynamic and transportproperties-REFPROP, Physical and Chemical Proper-ties Division, National Institute of Standards and Tech-nology, Boulder, Colorado 80305.

[48] Optimization department of cybernetics institute,National Academy of Sciences of Ukraine, NLP- OpenOpt, URL: http://http://openopt.org/NLP,laatst nagekeken op 2013-23-05, 2013.

[49] D. Walraven, B. Laenen, W. D’Haeseleer, Compari-son of thermodynamic cycles for power production fromlow-temperature geothermal heat sources, Energy Con-version and Management 66 (2013) 220–233.

10