Paolo Baggio

Marco Baratieri

Andrea Gasparella

Giovanni A Longo

ENERGY AND ENVIRONMENTAL ANALYSIS OF AN INNOVATIVE SYSTEM BASED ON MUNICIPAL SOLID WASTE (MSW) PYROLYSIS AND COMBINED CYCLE

Keywords: Municipal Solid Waste, Pyrolysis, Heat Recovery, Combined Cycle, Environmental Impact

This paper presents the energy and the environmental impact analysis of an innovative system based on the pyrolysis of MSW which produces solid (char), liquid (tar) and gas (syngas) fuels used in a combined cycle for electric power generation. The syngas, after filtration and compression, feeds two gas turbines. In turn, the exhaust from the gas turbines, after postcombustion with char and tar, drives a steam turbine power plant. Before being discharged, the flue gas is processed in an SCR (Selective Catalytic Reduction) unit to reduce CO, VOC and NO x content and is filtered to remove particulate matter. This innovative approach to energy recovery from MSW combines high energy efficiency with a low level of polluting emissions. The estimated global efficiency of the plant, referred to the LHV of the MSW, is around 28-30%, a much higher value than ordinarily obtained in traditional waste incineration plants. The environmental analysis includes a study of the polluting emissions and the simulation of their concentration in the area surrounding the plant: the emissions of the plant have a negligible influence on the original polluting levels of the settlement area.

INTRODUCTION

The beginning of the third millennium has been characterised by a progressive increase in the demand for fossil fuels, which has caused a steep rise in oil price. At the same time, several environmental disasters have increased the sensitivity of world-wide public opinion towards the effect that environmental pollution has on human health and climate change. These conditions have fostered a renewed interest in renewable energy: solar energy, wind energy, biomass and solid wastes.

In addition, the disposal of Municipal Solid Waste (MSW) has become a critical and costly problem. The traditional landfill method requires large amounts of land and contaminates air, water and soil. As an alternative, the incineration method has the advantage of reducing waste volume, and it may also generate thermal and electric energy with low polluting emissions in the atmosphere [START_REF] Porteous | Energy from waste incineration -a state of the art emissions review with enphasis on public acceptability[END_REF]. Therefore, those countries where land resources are limited, such as in the EU and Japan, prefer burning trash, whereas countries with vast territories, such as the US and Canada, continue to apply landfill disposal.

The new MSW management strategies are based on selection, recycle and energy recovery.

The development of innovative technologies for energy recovery from MSW could contribute to the reduction of both environmental pollution and dependence on fossil fuels. The alternative options to recover energy from MSW are incineration and pyrolysis / gasification.

Incineration is a destructive process in which the hydrocarbon content of MSW is converted into flue gases at high temperature whereas pyrolysis / gasification may convert it into other hydrocarbons which may be utilized as fuels, new materials and also monomers [START_REF] Bridgwater | An overview of fast pyrolysis of biomass[END_REF][START_REF] Kiran | Recycling of plastic wastes via pyrolysis[END_REF][START_REF] Yaman | Pyrolysis of biomass to produce fuels and chemical feedstocks[END_REF][START_REF] Demirbas | Pyrolysis of municipal plastic wastes for recovery of gasoline-range hydrocarbons[END_REF][START_REF] Bebar | Analysis of using gasification and incineration for thermal processing of wastes[END_REF]. This paper analyses an innovative system exploiting the pyrolysis of MSW to produce solid, liquid and gaseous fuels used in a combined cycle for electric power generation. This system will be analysed to evaluate its global energy efficiency and its environmental impact on the surrounding area.

SYSTEM DESCRIPTION

The system, shown in figure 1, consists of two pyrolysis lines, which feed two gas turbines.

The exhaust from the gas turbine outlets, after post-combustion in a steam boiler, drives a steam turbine. The MSW, processed in a rotary kiln at 500-600°C in absence of air, produces solid (char), liquid (tar) and gaseous (syngas) fuels together with ash. Pyrolysis is an endothermic process which requires a thermal energy input obtained from the combustion of a fraction of char and tar. The syngas available at the outlet of the rotary kiln is purified in a cyclone and a scrubber to remove particulate matter and condense humidity and heavy hydrocarbon vapours. The syngas is then compressed and injected into the combustion chamber of the gas turbines.

The combustion of the syngas in the gas turbines occurs with high excess air (300-400%).

Therefore the gas exhaust from the gas turbines may be used as reactant in the recovery boiler for the combustion of char and tar in order to drive the steam turbine. Before being discharged, the flue gas is processed in an SCR (Selective Catalytic Reduction) unit integrated in the boiler to reduce CO, VOC and NO x content and is filtered to remove particulate matter.

The SCR technology combines the catalytic reduction of carbon monoxide and volatile organic compounds by noble metals with the reduction of nitrogen oxides into molecular nitrogen and water vapour by using ammonia or urea as reactants in the presence of titanium and vanadium catalysts.

The system proposed shows some innovative aspects with respect to traditional MSW plants:

1) The energy recovery on MSW is carried out by using a combined cycle instead of a simple steam turbine as in the traditional MSW incineration plants or a gas turbine or a reciprocating engine as in the MSW gasification plants. The application of a combined cycle allows a rational use of the different products resulting from the MSW pyrolysis (syngas, char, tar) increasing the conversion efficiency.

2) The system proposed shows specific plant engineering advantages: high flexibility: it is possible to modulate the power simply by turning-off one of the two pyrolysis lines. high reliability: when the pyrolysis lines are out of order it is possible to send the MSW directly to the steam boiler and the plant works as a traditional incineration one. low polluting emissions: the presence of a single line of flue gas simplifies the treatment to reduce polluting emissions

ENERGY ANALYSIS

The composition of the MSW depends upon variables such as geographical area, population density, climatic conditions and waste management policies. MSW eligible for heat recovery typically contains 60 wt% cellulosic fraction (paper, cardboard, wood), 20 wt% plastics (highdensity polyethylene (HDPE), low density polyethylene (LDPE), polypropylene (PP), polystyrene (PS), polyvinilchloride (PVC)) and 20 wt% moisture. Table 1 shows the MSW composition and heating value assumed in the present analysis: these values are typical for developed countries, as shown in [START_REF] Sørum | Pyrolysis characteristics and kinetics of municipal solid wastes[END_REF]. The MSW has a Higher Heating Value (HHV) around The thermochemical conversion of MSW into gaseous fuel may be obtained by gasification or pyrolysis.

Gasification is a high temperature (700-800°C) thermal process, requiring the addition of air and/or steam, which produces a high fraction of syngas and small quantities of char, tar and ash. However the syngas obtained can have a poor heating value due to the high content of N 2 , and CO 2 .

Pyrolysis is a thermal process evolving in absence of air at temperatures around 500-600°C which produces syngas, char and tar in fractions depending upon operating temperature, heating rate and residence time. For high tar yield, a high heating rate and short residence time are requested (Fast Pyrolysis). To maximise char and syngas yields, a low heating rate and, a long residence time should be applied (Slow Pyrolysis).

The char fraction contains carbonaceous residues derived from thermal decomposition of the organic components, unconverted organic solid and ash resulting from inorganic materials. Its heating value is comparable with those of lignite and coke.

The tar fraction is a complex liquid mixture containing resins, acids, alcohols, intermediate carbohydrates, phenols, aromatics, and aldehydes. Its heating value is comparable with those of oxygenated fuels (CH 3 OH, C 2 H 5 OH).

The composition of the syngas depends on the MSW characteristics, moisture content, operating temperature and residence time. The heating value of syngas is comparable with that of coal gas and is much lower than that of natural gas.

The accurate prediction of the syngas composition requires a complex kinetic model capable of accounting for all pyrolysis reactions taking place inside the rotary kiln. The present paper is based on a simpler equilibrium model. Clearly this approach gives only a rough estimation of the syngas composition, as normally it is not possible to achieve complete thermodynamic equilibrium inside a pyrolysis reactor. Nevertheless the predictions of equilibrium models are not far-away from the experimental results both for gasification of biomass [START_REF] Kinoshita | Chemical equilibrium computations for gasification of biomass to produce methanol[END_REF][START_REF] Wang | Experimental analysis of biomass gasification with steam and oxygen[END_REF][START_REF] Schuster | Biomass steam gasification -an extensive parametric modeling study[END_REF][START_REF] Altafini | Prediction of the working parameters of a wood waste gasifier through an equilibrium model[END_REF] and gasification of coal [START_REF] Altafini | A chemical equilibrium model of the coal gasification process based on the minimization of the Gibbs free energy[END_REF][START_REF] Li | Equilibrium modelling of gasification: a free energy minimization approach and its application to a circulating fluidized bed coal gasifier[END_REF].

The thermodynamic equilibrium model used in present paper is a two-phase (gas+solid) one, allowing to predict not only the syngas composition but also the tar+char fraction. Despite the apparent crudeness of the approach, it's nevertheless a valuable engineering aid, usually leading to errors smaller than 20% with respect to the actual results [START_REF] Baratieri | Biomass thermochemical conversion: process analysis and energy performances[END_REF]. To the authors knowledge, this approach has never before been applied to a similar plant operation analysis.

The global chemical reaction relative to MSW pyrolysis is: The equilibrium composition can be estimated minimising the total free energy of the system and verifying the conservation of mass and chemical species. The resulting set of non-linear equations is solved by iteration obtaining the moles x i of reaction products and therefore the composition of syngas.

CH α O β +yO 2 +zN 2 +wH 2 O = x 1 C+x 2 H 2 +x
A specific computer code has been developed in Matlab using the software Cantera [START_REF] Goodwin | Cantera: object-oriented software for reacting flows[END_REF] for the computation of thermochemical properties.

From the composition of the pyrolysis products it is possible to evaluate the heating value and the thermal energy input requested by the pyrolysis, i.e.:

Q PIR =[(Σ i x i h i ) PRODUCTS -(Σ j y j h j ) REAGENTS ] / M MSW (4)
where xi and hi are the moles and the specific molar enthalpy of the reaction products, yj and

hj the moles and the specific molar enthalpy of the reagents.

Table 2 gives the characteristics of the pyrolysis products at the outlet of the scrubber for a pyrolysis temperature at 500°C. The pyrolysis at 500°C of 1 kg of MSW generates around 300 g of syngas together with 700 g of char + tar and requires 1.8 MJ of thermal energy input equal to 9.2% of the MSW High Heating Value (HHV).

The processes inside the syngas compressor, gas turbines, recovery boiler and steam turbine have been simulated by assuming suitable operating conditions, efficiency and effectiveness.

Table 3 gives the main assumptions and table 4 shows the mass flow rates G, the heat flow rates q, the electric power W and the efficiency η computed for a pyrolysis temperature of 500°C.

The global efficiency of the system is equal to the ratio between the net electric power produced and the input thermal power of the MSW referred to its Lower Heating Value (LHV):

η GLOBAL =(W GAS TURB +W STEAM TURB ) / (G MSW LHV MSW ) ( 5
)
The system is fed with 34.8 MWt of MSW and produces 9.84 MWel of electric power with a global efficiency of 28.3%. The steam turbine provides 7.47 MWel (76%) with an efficiency around 29.5%, whereas the gas turbines provide 2.36 MW (24%) with an efficiency around 18.6%. The poor performance of the gas turbines is due both to operating conditions (maximum temperature around 1000°C and pressure ratio around 12) and the energy consumption for syngas compression (amounting to 17.3% of the electric power produced by the gas turbines).

A sensitivity analysis has been carried out to evaluate the effect of a temperature increase of pyrolysis from 500 to 600°C. Figure 2 shows the electric power output against the pyrolysis temperature ranging from 500 to 600°C. The syngas yield rises from 30 to 40% increasing the contribution of the gas turbines and reducing the output of the steam cycle. As a consequence, the total electric power of the system shows only a slight increase from 9.84 to 10 MWel.

ENVIRONMENTAL ANALYSIS

The polluting emissions of the plant for MSW treatment must be added to the original polluting levels of the proposed settlement area, Porto Marghera, a high density industrial area near Venice (Italy). A comparative analysis is carried out comparing the emissions of the MSW treatment plant to those of the nearest power plant, the ENEL thermo-electric power plant "Andrea Palladio" which is 1200 m distant. This power plant is instrumented with a complete set of emissions control devices: SCR unit, FGD (Flue Gas Desulphuration) unit, bag filters. The MSW plant will probably use emission control devices of a better quality level. Moreover the pyrolysis reactor will be fed with selected MSW which estimated composition is reported in table 1: only PVC plastic shows a significant chlorine content. However the percentage of PVC plastic in the MSW allows to conclude that the emission of gases containing chlorine from the MSW plant should be limited.

The concentration of polluting substances in air derives from the combined effects of the polluting emissions, storage and dispersion in the atmosphere. The storage and dispersion processes, which depend mainly on meteorological conditions, cannot be influenced; therefore the only possible positive action consists in reducing the emissions.

The analysis of polluting storage and dispersion in the atmosphere requires meteorological information (windiness, solar radiation, atmospheric stability, etc.) which can be derived from the statistical analysis of local meteorological data.

The numerical simulation of polluting emissions, dispersion and fallout was carried out by using the ISC dispersion model implemented by US EPA [START_REF] Epa | User's guide for the Industrial Source Complex (ISC3) dispersion models[END_REF]. The ISC is a three- The polluting emissions considered in the present analysis are SO x , NO x , particulate matter and CO. It must be noted that, because of the characteristics of the waste fed to the plant, i.e.

mainly cardboard and plastics from packing material, a negligible sulphur (S) content is expected, and then the assumed value of SO 2 concentration in the flue gas must be considered to be a worst case scenario.

The plant processes about 7200 kg/h of CDR and discharges about 56600 Nm 3 /h of exhaust into the atmosphere. The concentration of the polluting substances in the exhaust was derived from the measured values on the waste incineration plant of Bolzano (Italy) which is equipped with pollution control systems (including an SCR unit and bag filters) similar to the ones of the MSW plant considered [START_REF]Termovalorizzatore di Bolzano[END_REF].

The polluting emissions of the ENEL thermo-electric power plant "Andrea Palladio" were derived from the 2004 report on plant operation [START_REF] Enel S | Dichiarazione ambientale anno 2004, Impianto termoelettrico "Andrea Palladio[END_REF].

The exhaust discharge temperature is 200°C for the MSW plant and 130°C for the power plant, whereas the chimney height is around 50 m for the MSW plant and ranges between 65 and 130 m for the power plant. The meteorological data used for the simulation was derived from [START_REF]Ente Zona Industriale di Porto Marghera[END_REF]. The computational domain for the numerical simulation is a squared area 10 km Figures 3 and4 show the SO 2 concentration in the air at ground level caused by the emissions of the power plant and the MSW plant, respectively.

Figures 5 and6 show the NO x concentration in the air at ground level caused by the emissions of the power plant and the MSW plant, respectively.

Figures 7 and8 show the particulate concentration in the air at the ground level caused by the power plant and the MSW plant, respectively.

Figures 9 and 10 show the CO concentration in the air at ground level caused by the emissions of the power plant and the MSW plant, respectively.

Table 7 shows the maximum concentration of the polluting substances in the air at ground level caused by the MSW plant alone, the power plant alone and the combined effect of both plants. The polluting emissions of the MSW plant are negligible respect to those of the power plant, except for CO emission. This result is obvious, considering the difference in size between the two plants: the electrical power of the MWS plant (10 MWe) is approximately 100 times lower than that of the power plant (1 GWe). Therefore, it is useful to compare the specific polluting emissions of the plants equal to the ratio between the annual polluting emissions and the annual electric energy production: table 8 shows this comparison. The specific polluting emissions of the MSW plant are generally lower than those of the power plant. The SO 2 specific emission is exactly 6.6 times lower (worst case scenario), the NO x is 2.9 times lower, whereas the particulate specific emission is almost equal and the CO specific emission is 2.6 times higher. The polluting emissions of the MSW plant are even more negligible when compared to the total polluting emissions of all power plants in Porto Marghera (2 GWe)

Table 9 shows the comparison between the annual polluting emissions of the MSW plant and those of all power plants in Porto Marghera. The data on the polluting emissions of all power plants in Porto Marghera is derived from [START_REF] Arpav | Rapporto Ambientale d'Area di Porto Marghera -Bilancio Ambientale[END_REF].

CONCLUSION

This paper presents the feasibility study of an innovative plant based on MSW pyrolysis and combined cycle.

The coupling between pyrolysis and combined cycle ensures a conversion efficiency of MSW into electric energy of about 28-30%, a significantly higher value respect to traditional waste incineration plants.

The pyrolysis temperature only affects the distribution of the electric power production between gas and steam turbines, whereas it shows only a slight effect on the performance of the whole system.

The polluting emissions of the innovative MSW plant are negligible in term of absolute contribution and are generally lower in term of specific contribution with respect to those of the ENEL thermo-electric power plant "Andrea Palladio", except for CO emission. The polluting emissions of the MSW plant are even more negligible when compared to the polluting emissions of all power plants in Porto Marghera.

This feasibility study has been carried out with well established mathematical models which ability is recognised in literature. The integral experimental validation of the simulation required the realisation of a demonstrative plant which is out of the purpose of a feasibility study and, by the way, the possibility to build a pilot and/or full scale plant is currently under evaluation. If and when such plant will be operational, the authors will be glad to present the experimental results.

The feasibility study requires also an economic assessment of the innovative plant. However this study, due to the complexity of the plant and the different levels of electricity prices and MSW disposal tariffs, needs a specific paper to be carried out. In present paper it is possible to give only a first rough estimation of the pay-back time of the plant: around 8-10 years.

ACKNOWLEDGEMENTS

This research project was funded by Regione Veneto (Italy) DOCUP Obiettivo n.

19. 2

 2 MJ/kg, an equivalent molecular mass MMSW = 25.49 kg/ kmol and the following equivalent raw chemical formula: C H 1.515 O 0.43 + 0.283 H 2 O + ASH (1)

  3 H 2 O+x 4 CO+x 5 CO 2 +x 6 CH 4 +x 7 O 2 +x 8 N 2 (2) where CH α O β is the MSW equivalent chemical formula, y, z and w the moles of other reactants and x i the moles of reaction products. Assuming negligible presence of air, the composition of the reagents is: α = 1.515 β = 0.430 y = 0.000 z = 0.000 w = 0.283 (3)

  dimensional, steady state, gaussian model that computes the average annual concentration and the fallout of polluting substances emitted by concentrated or distributed sources. The model is based on the solution of the convective-diffusion equation modified to account for meteorological and orographical conditions.

Figure 2 .

 2 Figure 2. Electric power output vs. pyrolysis temperature.

Figure 3 .

 3 Figure 3. SO 2 concentration in the air at ground level caused by the emissions of the power plant.

Figure 4 .

 4 Figure 4. SO 2 concentration in the air at ground level caused by the emissions of the MSW plant.

Figure 5 .

 5 Figure 5. NO x concentration in the air at ground level caused by the emissions of the power plant.

Figure 6 .

 6 Figure 6. NO x concentration in the air at ground level caused by the emissions of the MSW plant.

Figure 7 .

 7 Figure 7. Particulate concentration in the air at ground level caused by the emissions of the power plant.

Figure 8 .

 8 Figure 8. Particulate concentration in the air at ground level caused by the emissions of the MSW plant.

Figure 9 .

 9 Figure 9. CO concentration in the air at ground level caused by the emissions of the power plant.

Figure 10 .

 10 Figure 10. CO concentration in the air at ground level caused by the emissions of the MSW plant.

Table 1 .

 1 Composition and heating value of MSW.

	Paper Cardboard Wood HDPE LDPE PP

Table 2 .

 2 Characteristics of pyrolysis products.

	Parameters	Values
	T PYR	500°C
	p PYR	101325 Pa
	M MSW	25.49 kg/kmol
	HHV MSW	19.2 MJ/kg
	H 2	3.16 wt% MSW	10.9 % SYNGAS
	CH 4	4.84 wt% MSW	16.6 % SYNGAS
	CO	2.05 wt% MSW	7.0 % SYNGAS
	CO 2	19.09 wt% MSW	65.5 % SYNGAS
	C	36.18 wt% MSW 51.1% CHAR+TAR
	H 2 O	32.67 wt% MSW 46.1 % CHAR+TAR
	ASH	2.00 wt% MSW	2.8 % CHAR+TAR
	GAS	0.2915 kg/kg MSW
	HHV SYNGAS	25.3 MJ/kg
	LHV SYNGAS	22.0 MJ/kg
	CHAR+TAR	0.7085 kg/kg MSW
	HHV CHAR+TAR	16.7 MJ/kg
	LHV CHAR+TAR	15.6 MJ/kg
	Q PYR	1.772 MJ/kg MSW

Table 4 .

 4 Power and efficiency.

	Parameters	Values
	G MSW	7200 kg/h
	G SynAS	2100 kg/h
	G CHAR+TAR	5100 kg/h
	W GAS TURB	2368 kW
	W STEAM TURB	7468 kW
	W GLOBAL	9836 kW
	q PYR	3544 kW
	q BOILER	25314 kW
	Q MSW	34781 kW
	η GAS TURB	18.6 %
	η STEAM TURB	29.5 %
	η GLOBAL	28.3 %

Table 5 .

 5 Polluting substance concentration in the exhaust at the chimney (dry basis, normalized to 11% O 2 content for the MSW Plant and to 6% O 2 content for the Power Plant).

		MSW Plant	Power Plant
	SO 2 [mg/Nm 3 ]	36.1	546.5
	NO x [mg/Nm 3 ]	44.5	293.9
	Partic.[mg/Nm 3 ]	3.6	8.8
	CO [mg/Nm 3 ]	18.0	16.5

Table 6 .

 6 Annual polluting emissions.

		MSW Plant	Power Plant
	SO 2 [t/year]	24.8	12562
	NO x [t/year]	30.6	6685
	Partic. [t/year]	2.5	193
	CO [t/year]	12.4	367

Table 7 .

 7 Maximum polluting substance concentration in the air at ground level.

		MSW plant Power plant MSW + Power plant
	SO 2 [µg/m 3 ]	0.350	24.00	24.00
	NO x [µg/m 3 ]	0.440	11.00	11.00
	Partic. [µg/m 3 ]	0.035	0.28	0.28
	CO [µg/m 3 ]	0.180	0.32	0.38

Table 8 .

 8 Specific polluting emissions.

		MSW Plant	Power Plant
	SO 2 [mg/kWh]	287.9	1909.4
	NO x [mg/kWh]	355.1	1016.1
	Partic. [mg/kWh]	28.8	29.3
	CO [mg/kWh]	143.9	55.8

Table 9 .

 9 Annual polluting emissions.

		MSW Plant	All Power Plants
	SO 2 [t/year]	24.8	19858
	NO x [t/year]	30.6	12996
	Partic. [t/year]	2.5	355
	CO [t/year]	12.4	573

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José Fernández-Seara

Francisco J Uhía

Jaime Sieres

Antonio Campo

A general review of the wilson plot method and its modifications to determine convection coefficients in heat exchange devices

Keywords: Wilson plot method, convection coefficients, heat exchange devices

Heat exchange devices are essential components in complex engineering systems related to energy generation and energy transformation in industrial scenarios. The calculation of convection coefficients constitutes a crucial issue in designing and sizing any type of heat exchange device. The Wilson plot method and its different modifications provide an outstanding tool for the analysis and design of convection heat transfer processes in research laboratories.

The Wilson plot method deals with the determination of convection coefficients based on measured experimental data and the subsequent construction of appropriate correlation equations. This paper is to present an overview of the Wilson plot method along with numerous modifications introduced by researches throughout the years to improve its accuracy and to extend its use to a multitude of convective heat transfer problems. Undoubtedly, this information will be useful to thermal design engineers.

The analytical treatment of convection problems require the solution of a system of mass, momentum and energy conservation equations for a body geometry and fluid properties ending with the flow and temperature fields in the fluid. Although these procedures are elaborate, analytic solutions are available for simple geometries under restrictive assumptions. Most of the convective heat transfer processes inherent to heat exchangers usually involve complex geometries and intricate flows so that the analytical solutions are not possible. Therefore, an approach based on Newtonś law of cooling, Eq. 1, provides a simple alternate avenue.

) T T ( h A q f s - ⋅ ⋅ = (1)
Newtonś law of cooling established an algebraic relation between the heat flow by convection (q), the surface area (A), an average convection coefficient (h) and the temperature difference between the solid surface (T s ) and the fluid (T f ). Under this framework, the convection problems reduce to the estimation of the convection coefficient (h).

For a given flow and surface geometry, the experimental data is usually obtained by measuring the surface area and the fluid temperatures for an imposed heat condition. Then, the convection coefficient may be calculated from Eq. (1). However, the main difficulty of this methodoly lies in the measurement of the surface temperature, because the surface temperature varies from point to point, and the flow pattern could be altered by the presence of the temperatures sensors. The problem is even more complicated if the heat transfer surface is not accessible, as usually happens with heat exchangers. Therefore, any alternative method conducive to the heat transfer calculation of convection coefficients in heat exchangers is attractive because of widespread use and practical applications.

The Wilson plot method constitutes a suitable technique to estimate the convection coefficients in a variety of convective heat transfer processes. The Wilson plot method avoids the direct measurement of the surface temperature and consequently the disturbance of the fluid flow and heat transfer introduced while attempting to measure those temperatures. Modifications of the Wilson plot method have been proposed by many researches continually to enhance its accuracy. This review paper focuses on the modifications published in the archival literature that led to the so-called modified Wilson plot method.

ORIGINAL WILSON PLOT METHOD

This Wilson plot method was developed by Wilson in 1915 [START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF] to evaluate the convection coefficients in shell and tube condensers for the case of a vapour condensing outside by means of a cool liquid flow inside. It is based on the separation of the overall thermal resistance into the inside convective thermal resistance and the remaining thermal resistances participating in the heat transfer process. The overall thermal resistance of the condensation process in a shell and tubes condenser (R ov ) can be expressed as the sum of three thermal resistances corresponding to 1) the internal convection (R i ), 2) the tube wall (R w ) and 3) the external convection (R o ), as shown in Eq. [START_REF] Wójs | Effects of the temperature interference on the results obtained using the Wilson plot technique[END_REF].

o w i ov R R R R + + = (2)
For the sake of simplicity, the thermal resistances due to the fluid fouling in Eq. ( 2) are neglected. Employing the proper expressions for the thermal resistances showing up in Eq. ( 2), the overall thermal resistance can be rewritten as Eq. ( 3).

( )

o o w w i o i i ov A h 1 L k 2 d d ln A h 1 R ⋅ + ⋅ ⋅ ⋅ + ⋅ = ( 3
)
where h i and h o are the internal and external convection coefficients, d i and d o are the inner and outer tube diameters, k w is the tube thermal conductivity, L w is the tube length and A i and A o are the inner and outer tube surface areas, respectively.

On the other hand, the overall thermal resistance can be conceived as a function of the overall heat transfer coefficient referred to the inner or outer tube surfaces and the corresponding areas. In this regard, in Eq. ( 4) the overall thermal resistance is expressed as a function of the overall heat transfer coefficient referred to the inner or outer surface (U i/o ) and the inner or outer surface area (A i/o ).

o / i o / i ov A U 1 R ⋅ = (4)
Taking into account the specific conditions of a shell and tube condenser and the equations indicated above, Wilson [START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF] theorized that if the mass flow of the cooling liquid was modified, then the change in the overall thermal resistance would be mainly due to the variation of the intube convection coefficient, while the remaining thermal resistances remained nearly constant.

Therefore, as indicated in Eq. ( 5) the thermal resistances outside of the tubes and the tube wall could be considered constant,

1 o w C R R = + (5)
Wilson [START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF] determined that for the case of fully developed turbulent liquid flow inside a circular tube, the convection coefficient was proportional to a power of the reduced velocity (v r ) which accounts for the property variations of the fluid and the tube diameter. Thus, the convection coefficient could be written according to Eq. [START_REF] Hasim | Compound heat transfer enhancement by a combination of a helically ribbed tube with twisted tape inserts[END_REF].

n r 2 i v C h ⋅ = ( 6
)
where C 2 is a constant, v r is the reduced fluid velocity and n is a velocity exponent. Then, the convection thermal resistance corresponding to the inner tube flow is proportional to 1/v r n .

Further, upon combining Eqs. ( 2), ( 5) and ( 6), the overall thermal resistance turns out to be a linear function of 1/v r n ; this is represented graphically in Fig. 1. An inspection of the correlation equation [START_REF] Zheng | Boiling of ammonia/lubricant mixture on a horizontal tube in a flooded evaporator with inlet vapor quality[END_REF] indicates that C 1 is the intercept and 1/(C 2 •A i ) is the slope of the straight line.

1 n r i 2 ov C v 1 A C 1 R + ⋅ ⋅ = (7)
On the other hand, the overall thermal resistance and the cooling liquid velocity can be obtained by experimentally measuring the inlet temperature (T l,in ), the outlet temperature (T l,out ) and the vapour condensation temperature (T v ) at various mass flow rates (m l ) of the cooling liquid under fully developed turbulent flow. Then, for each set of experimental data corresponding to each mass flow rate, the overall thermal resistance is the ratio between the logarithmic mean temperature difference of the fluids (LMTD) and the heat flow transferred between them, according to Eq. ( 8). The heat flow is determined from the enthalpy change of the cooling liquid as give in (Eq. 8).

) T T ( Cp m LMTD R in , l out , l l l ov - ⋅ ⋅ = (8)
Therefore, if a value of the velocity exponent n in Eq. ( 6) is assumed, then the experimental values of the overall thermal resistance can be represented as a linear function of the experimental values of 1/v r n . From here, the straight-line equation that fits the experimental data can be deduced by applying simple linear regression. Then, the values of the constants C 1 and C 2 are calculated from to Eq. ( 7) as indicated in Fig. 1. Once the constants C 1 and C 2 are determined, then the external and internal convection coefficients for a given mass flow rate can be evaluated from the combination of Eqs. ( 6) and ( 9).

( ) o w 1 o A R C 1 h ⋅ - = (9)
The above exposition is the original Wilson plot method [START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF]. It relies on the fact that the overall thermal resistance can be extracted from experimental measurements in a reliable manner. As a result, the mean value of the convection coefficient outside the tubes (mean value of the outside thermal resistance) and the convection coefficient inside the tubes can be estimated as a function of the velocity of the cooling liquid. Therefore, the original Wilson plot method could be applied to convection heat transfer processes when the thermal resistance provided by one An early application of the original Wilson plot method to analyse the performance of six plain and finned tube bundles forming a shell and tube heat exchanger was reported by Williams and Katz [START_REF] Williams | Performance of finned tubes in shell and tube heat exchangers[END_REF]. Experiments were conducted for water, lubricating oil and glycerine in the shell side and cooling water inside the tubes. For each of the fluids in the shell side, experiments were carried out at different temperature levels. The Wilson plot method was applied to all sets of experimental data and the outside convection coefficient were obtained. Afterwards, in a subsequent analysis the shell-side convection coefficients were correlated in form resembling the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF]. New exponents for the Reynolds and Prandtl numbers and a multiplier for each one of the tubes bundles are proposed.

More applications of the original Wilson plot method are found in the archival literature. Hasim et al. [START_REF] Hasim | Compound heat transfer enhancement by a combination of ribbed tubes with coil inserts[END_REF][START_REF] Hasim | Compound heat transfer enhancement by a combination of a helically ribbed tube with twisted tape inserts[END_REF] used the original Wilson plot to investigate the heat transfer enhancement by combining ribbed tubes with wire [START_REF] Hasim | Compound heat transfer enhancement by a combination of ribbed tubes with coil inserts[END_REF] and twisted tape inserts [START_REF] Hasim | Compound heat transfer enhancement by a combination of a helically ribbed tube with twisted tape inserts[END_REF]. The experimental apparatus consists of a double pipe heat exchanger with water as the cooling and heating fluids. The convection coefficients inside the enhanced tubes are assumed to be proportional to a power of the fluid velocity with known exponents, as indicated in Eq. ( 6). Both papers reported results of the experimental convection coefficients. Zheng et al. [START_REF] Zheng | Boiling of ammonia/lubricant mixture on a horizontal tube in a flooded evaporator with inlet vapor quality[END_REF] applied the Wilson plot method to analyze the heat transfer processes in a shell and tube flooded evaporator in an ammoniacompression refrigeration system. The study was carried out for plain tubes forming the bundle.

The ammonia-lubricant mixture evaporates outside the tubes and a heated water-glycol solution flowed inside the tubes. Results of the boiling convection coefficients were reported and a suitable correlation equation was proposed.

Fernández-Seara et al. [START_REF] Fernández-Seara | Experimental apparatus for measuring heat transfer coefficients by the Wilson plot method[END_REF] described a simple experimental apparatus that allows for the measured data required for the application of the Wilson plot method. The test section consists of a transparent methacrylate enclosure, wherein water vapour generated at the bottom condenses over a test tube cooled by circulating water inside. An asset of the experimental apparatus deals with the possibility of testing different type of tubes. Once the experimental data is recorded, the original Wilson plot method or any of the Wilson plot modifications can be applied. Also, a collection of results gathered with this experimental apparatus are reported in Fernández-Seara et al. [START_REF] Fernández-Seara | Laboratory practices with the Wilson plot method[END_REF], this time utilizing a smooth, spirally corrugated tube made of stainless steel.

MODIFIED WILSON PLOT METHODS

A m A n A A A Pr Re C Nu ⋅ ⋅ = (10)
Later on, correlation equations attributable to Petukhov [START_REF] Petukhov | Theoretical calculation of heat exchange and frictional resistance in turbulent flow in tubes of an incompressible fluid with variable physical properties[END_REF] and Gnielinski [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF] contained an unknown multiplier as part of the general functional forms for turbulent forced convection. These general functional forms are applicable to laminar forced convection as well. There are also situations where the mass flow rate of the fluid undergoing a phase change process is varied.

Moreover, there are also applications involving phase change where the convection coefficient is modified by changing the inlet flow quality (X). In these cases, appropriate correlation equations have to be constructed for the convection coefficient as a function of an unknown multiplier and the mass flow rate or the inlet flow quality. Thereby, this simple modification of the original Wilson plot method presuppossed the existence of a general functional form for the convection coefficient of the fluid whose flow conditions can be varied in the experimental analysis (fluid A). This option responds to the general form of Eq. ( 11), and a constant thermal resistance for the other fluid (fluid B) obeying Eq. ( 12). Thereafter, the overall thermal resistance can be associated with Eq. ( 13), as a function of the convection coefficient of fluid A (Eq. 11)

and the constant thermal resistance of fluid B in Eq. ( 12). The form of Eq. ( 13) is linear where the slope depends on the multiplier C A (Eq. 11) and the intercept between the regression straight line and the thermal resistance axis --the constant R B . Then, the Wilson plot permits the calculation of the unknown constant C A (Eq. 11) and the thermal resistance R B of fluid B, as represented graphically in Fig. 2. This sequence is synthesized as follows.

[ ]

, X ), v ( m f C h A A A ⋅ = ( 11
)
) t tan cons h ( C R B B = = (12) [ ] ) R R ( A , X ), v ( m f 1 C 1 R B w A A A ov + + ⋅ ⋅ = ( 13
)
Early applications of the above modification are the topic of four publications by Young et al.

[ [START_REF] Young | Effect of root wall thickness on bond resistance to heat transfer of bimetal tubes[END_REF][START_REF] Young | The investigation of heat transfer and pressure drop of 11 fin per inch tubes and coils[END_REF][START_REF] Young | Fouling of an 11-fins-per-inch coil in an internal tankless water heater[END_REF][START_REF] Young | An investigation of the fouling of 19-fin-per-inch admiralty tubes in three heat transfer units located at the Aurora Gasoline Company Refinery[END_REF]. These papers contained different ensembles such as the contact thermal resistance of bimetal tubes [START_REF] Young | Effect of root wall thickness on bond resistance to heat transfer of bimetal tubes[END_REF], heat transfer in clean finned tubes [START_REF] Young | The investigation of heat transfer and pressure drop of 11 fin per inch tubes and coils[END_REF], fouling in finned tubes and coils [START_REF] Young | Fouling of an 11-fins-per-inch coil in an internal tankless water heater[END_REF],

and fouling in shell-and-tube heat exchangers operating in a refinery [START_REF] Young | An investigation of the fouling of 19-fin-per-inch admiralty tubes in three heat transfer units located at the Aurora Gasoline Company Refinery[END_REF]. In all these works, the outside tube thermal resistance was taken as a constant and the in-tube convection coefficient was expressed by the general form of the Dittus-Boelter equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] for water as proposed by

McAdams [START_REF] Mcadams | Heat transmission[END_REF]. The application of the Wilson method was corroborated in terms of the corresponding quantities.

Recent papers devoted to the use of the modified Wilson plot method are reviewed in the forthcoming paragraphs. These papers are grouped according to the type of heat transfer process considered, but most of them are related with condensing gases. Chang and Hsu [START_REF] Chang | Single-tube performance of condensation of R-134a on horizontal enhanced tubes[END_REF] applied the modified Wilson plot method to analyse the condensation of R-134a on two horizontal enhanced tubes with internal grooves. The authors considered the functional form of the Dittus-Boelter equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] for the convection coefficient of water flowing inside the tubes Other applications of the Wilson plot method aimed at determining the thermal resistances viewed as constant, namely R w + R B in Eq. 13. The final goal could be the calculation of the convection coefficient of the corresponding fluid or any other thermal resistance that participate in the overall thermal resistance such as a contact thermal resistance. Taylor [START_REF] Taylor | Measurement of finned-tube heat exchanger performance[END_REF] articulated the Wilson plot to the determination of the air-side convection coefficient in finned-tube heat exchangers. Despite that the air-side thermal resistance was considered constant, the cooling water mass flow was allowed to vary. The Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] was linked to the water-side convection coefficient and the constant multiplier was determined. The air-side convection coefficient was obtained from the thermal resistance and later correlated in the form of Colburn j-factor. Eight fin-and-tube heat exchangers were tested and the effects of fin spacing, number of rows and fin enhancement were reported. The work of ElSherbini et al. [START_REF] Elsherbini | Experimental investigation of thermal contact resistance in plain-fin-and-tube evaporators with collarless fins[END_REF] revolved around the application of the Wilson plot method to the contact thermal resistance between fins and tubes in a plain finned heat exchanger. Using a mixture of ethylene glycol and water as coolant, the coolant flow rate was varied under constant air-side conditions. The airside thermal resistance was idealized as constant and the Gnielinski correlation equation [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF] was representative of the coolant-side convection coefficient. Experiments with unbrazed and brazed coils, both under dry and frosted conditions were conducted. The fin contact resistance was calculated from the air-side thermal resistance within the framework of the Wilson plot.

Critoph et al. [START_REF] Critoph | Contact resistance in air-cooled plate fin-tube airconditioning condensers[END_REF] embarked on an analysis similar to the one by ElSherbini et al. [START_REF] Elsherbini | Experimental investigation of thermal contact resistance in plain-fin-and-tube evaporators with collarless fins[END_REF] to address the issue of the fin contact resistance in air-cooled plate fin air conditioning condensers. In this case, the mass flow rate of refrigerant R-22 condensing inside the tubes was varied while the air-side Reynolds number was held constant. The refrigerant-side convection coefficient was paired with the Nusselt theory [START_REF] Nusselt | Die Oberflächenkondesation des Wasserdampfes[END_REF] and the air-side thermal resistance was idealized as constant. The contact fin-tube thermal resistance was calculated from the air-side thermal resistance with the Wilson plot method. Some Wilson plot applications have been identified within the platform of boiling processes.

Aprea et al. [START_REF] Aprea | Experimental evaluation of R22 and R407C evaporative heat transfer coefficients in a vapour compression plant[END_REF] determined the boiling convection coefficients of refrigerants R-22 and R-407C in a tube-in-tube evaporator. In this sense, the refrigerant flows inside the inner tube as opposed to a water-glycol solution that moves through the annulus. The inner thermal resistance was considered constant and the annulus-side convection coefficient was assumed to follow the Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF]. Not only the boiling convection coefficients for the two refrigerants were obtained, but they were compared against different correlation equations.

Finally, specialized applications have been investigated such as Hariri et al. [START_REF] Hariri | The effect of the operating conditions on a reaction calorimeter[END_REF] who by way of Wilson plot method determined the convection coefficients in the reactor and in the jacket of a calorimeter. The overall heat transfer coefficient was expressed linearly as a function of the agitation speed. The jacket convective heat transfer was calculated and the inside convection coefficient was channeled through the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] having the exponent 2/3 for the Reynolds number. Lavanchy et al. [START_REF] Lavanchy | Reaction calorimetry as a new tool for supercritical fluids[END_REF] and Fortini et al. [START_REF] Fortini | Reaction calorimetry in supercritical carbon dioxide-methodology development[END_REF] conceived a similar analysis associated with reaction calorimeters for supercritical fluids. The convection coefficient in the reactor was also channeled through with the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] having a power of the stirrer rotation speed. In addition, the thermal resistance corresponding to the jacket wall and to the coolant convection were kept constant. A further modification of the Wilson plot method has been contemplated to avoid the assumption of a constant thermal resistance for one of the fluids. This modification consists in formulating a functional form for the convection coefficient for the fluid (fluid B), instead of a constant thermal resistance. This general equation will absorb the variation of the surface temperature and will comprise an unknown multiplier, as indicated in Eq. ( 14). In this regard, the Wilson plot method envisions the calculation of the two constants embedded in the general functional forms (Eqs. 11 and 14) related to the convection coefficients of both fluids.

Determining two constants in

) T ( f C h B B B ⋅ = (14)
Utilizing the general expressions given by Eqs. ( 11) and ( 14), the overall thermal resistance is supplied by Eq. ( 15). Despite that this equation has a nonlinear character, it can be easily rearranged into a linear functional form. This is doable in such a way that the slope depends on the constant C A only and the intercept between the regression line and the thermal resistance axis delivers the constant C B as shown in Eq. ( 16). Thereafter, the application of the Wilson plot as indicated in Fig. 3 is synonymous with for the determination of both multipliers, C A and C B .

[ ]

B B B w A A A ov A ) T ( f C 1 R A , X ), v ( m f C 1 R ⋅ ⋅ + + ⋅ ⋅ = (15)
[ ]

B A A B B A B B w ov C 1 A , X ), v ( m f A ) T ( f C 1 A ) T ( f ) R R ( + ⋅ ⋅ ⋅ = ⋅ ⋅ - (16)
This systematic procedure was initially described by Young and Wall [START_REF] Young | Development of an apparatus for the measurement of low bond resistance in finned and bare duplex tubing[END_REF], who applied it to a heat exchanger owing two single-phase fluids. However, most of the references that referred to it dealt with condensation processes on tubes with cooling turbulent liquid circulating inside. The functional forms contemplated rest on the Nusselt theory [START_REF] Nusselt | Die Oberflächenkondesation des Wasserdampfes[END_REF] for the condensing side and on well-known correlation equations due to Dittus-Boelter [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF], Sieder-Tate [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] or Gnielinski [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF], all corresponding to turbulent flows inside the tubes. Kumar et al. [START_REF] Kumar | A comprehensive study of modified Wilson plot technique to determine the heat transfer coefficient during condensation of steam and R-134a over single horizontal plain and finned tubes[END_REF] extended this procedure for the calculation of the condensing convection coefficient of steam and R-134a over a plane wall and different finned cooper tubes. A suitable combination of the Nusselt theory [START_REF] Nusselt | Die Oberflächenkondesation des Wasserdampfes[END_REF] and the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] were considered for the condensing and the inner convection coefficients. Singh et al. [START_REF] Singh | Heat transfer during condensation of steam over a vertical grid of horizontal integral-fin copper tubes[END_REF] repeated the same type of analysis to study the condensation of steam over a vertical grid of horizontal integral-fin tubes. In this case, the authors modified the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] to incorporate the entrance effects in short tubes. Hwang et al. [START_REF] Hwang | Heat transfer and pressure drop characteristics of enhanced titanium tubes[END_REF] also adopted the same approach to study the heat transfer characteristics of various types of enhanced titanium tubes. These authors focused on the condensation of steam over the tubes, which are internally cooled by water.

Abdullah et al. [START_REF] Abdullah | Condensation of steam and R113 on a bank of horizontal tubes in the presence of a non-condensing gas[END_REF] and Namasivayam and Briggs [START_REF] Namasivayam | Effect of vapour velocity on condensation of atmospheric pressure steam on integral-fin tubes[END_REF][START_REF] Namasivayam | Condensation of ethylene glycol on integral-fin tubes: effect of fin geometry and vapor velocity[END_REF] paid attention to the benefits of the modified Wilson plot for the case of condensation of pure vapour over plain tubes. The ultimate goal was the construction of a correlation equation for the coolant side. On one hand, the Nusselt theory [START_REF] Nusselt | Die Oberflächenkondesation des Wasserdampfes[END_REF] and the Shekriladze and Gomelauri equation [START_REF] Shekriladze | Theoretical study of laminar film condensation of flowing vapour[END_REF] are used for the condensing vapour and on the other hand, the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] suffices for the cooling water. However, in the three papers the unknown constants are delimitated by the minimization of the sum of squares of residuals of the overall temperature difference. Rose [START_REF] Rose | Heat-transfer coefficients, Wilson plots and accuracy of thermal measurements[END_REF] discussed this procedure to calculate the unknown constants. Ribatski and Thome [START_REF] Ribatski | Nucleate boiling heat transfer of R134a on enhanced tubes[END_REF] also applied this modified scheme of the Wilson plot to resolve for the proper convection coefficient for nucleate boiling of R-134a on four different enhanced tubes that are commercially available.

The heating medium was water flowing inside the tubes. In this case, the boiling convection coefficient was assumed proportional to a power of the heat flux, being the exponent 0.7. The inner convection coefficient was expressed in the form compatible with Gnielinski correlation equation [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF]. At the end, the Wilson plot was obtained by varying the mass water flow for constant wall heat flux.

Determining more than two constants.

The most general conceptualization of the Wilson plot method involves three constants, two of them corresponding to the assumed functional form for one of the two fluids. Usually, the third unknown constant stands for the exponent of the flow parameter inherent to one of the fluids.

This unknown constant does not behave linearly, and needs to be found by an iterative procedure forcibly. Initially, Briggs and Young [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF][START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF] proposed a second linearized equation obtained by applying natural logarithms to both sides of Eq. ( 15) in such a way that this second equation gets linearized in the third unknown constant. Then, an iterative procedure engaging the two linear regression equations leads to the calculation of the three constants. Later, Shah [START_REF] Shah | Assessment of modified Wilson plot techniques for obtaining heat exchanger design data[END_REF] envisaged a new modification of the Briggs and Young method [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF] to correlate for one of the fluids when the correlation of the other fluid is unknown or is of no interest. Kedzierski and Kim [START_REF] Kedzierski | Single-phase heat transfer and pressure drop characteristics of an integral-spine fin within an annulus[END_REF], Yang and Chiang [START_REF] Yang | An experimental heat transfer study for periodically varying-curvature curved-pipe[END_REF] and Fernández-Seara et al. [START_REF] Fernández-Seara | Experimental apparatus for measuring heat transfer coefficients by the Wilson plot method[END_REF][START_REF] Fernández-Seara | Laboratory practices with the Wilson plot method[END_REF] undertook a linearization plan of the non-linear equation by means of natural logarithms. Subsequently, several authors proposed different mathematical models for the identification of the unknown parameters in the equations derivable from the Wilson method. Khartabil and Christiensen [START_REF] Khartabil | An improved scheme for determining heat transfer correlations for heat exchanger regression models with three unknowns[END_REF] looked at a nonlinear regression model based on the resolution of three equations that emanated from the least squares method. The three equations, two of them linear and the third one nonlinear, enabled the authors to evaluate the three unknown constants. Then, the thermal resistance, along with the two constants associated with the variable thermal resistance of the other fluid, were evaluated. A drawback ascribable to this approach is its validity when the thermal resistance of one of the fluids is constant. Deviating from this format, Rose [START_REF] Rose | Heat-transfer coefficients, Wilson plots and accuracy of thermal measurements[END_REF] proposed to compute the third unknown constant by way of an iterative scheme resulting from the minimization of the sum of squares of the residuals of the overall temperature difference.

Khartabil et al. [START_REF] Khartabil | A modified Wilson plot technique for determining heat transfer correlations[END_REF] scrutinized a further modification for the situation in which general forms with unknown constants are available for both fluids. This new approach ended up with the evaluation of two constants in each correlation equation together with the tube wall thermal resistance. However, this road requires the manipulation of three different sets of experimental data. Each set should be collected with a different dominant resistance (inner, outer and tube wall). Styrylska and Lechowska [START_REF] Styrylska | Unified Wilson plot method for determining heat transfer correlations for heat exchangers[END_REF] purported a unified Wilson plot method for heat exchangers for conditions wherein a Nusselt equation is tied up to one of the fluids if the other does not suffer a phase change. This proposal is potent because it permits the calculation of any number of unknown constants, treating the unknowns as observations with a covariance matrix known a priori. Recently, Pettersen [START_REF] Pettersen | Flow vaporization of CO 2 in microchannel tubes[END_REF] chose the commercial software package DataFit (based on the Levenberg-Marquardt method) to conduct a nonlinear regression analysis of the experimental data points.

Most of the references found in the specialized literature are focalized on the modified Wilson plot method proposed by Briggs and Young [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF]. Dirker and Meyer [START_REF] Dirker | Heat transfer coefficients in concentric annuli[END_REF][START_REF] Dirker | Convective heat transfer coefficients in concentric annuli[END_REF] looked into a modification of the Wilson plot to appraise a suitable correlation equation for turbulent flow in a smooth concentric annuli. These authors tested eight tube-in-tube heat exchangers with different annular diameter ratios using hot water in the inner tube and cold water in the annulus.

The format of the Sieder-Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] was the platform for the in-tube and annular convection coefficients. The multipliers present in both correlation equations in conjunction with the exponent of the Reynolds number in the annulus were determined.

Meanwhile, the Reynolds number exponent connected to the inner convection coefficient was fixed at 0.8. Results of the Reynolds number exponent and the multiplier in the annulus equation were correlated as a function of the radii ratio. The experimental-based correlation equation was also validated with a CFD (Computer Fluid Dynamics) code in Dirker et al. [START_REF] Dirker | Convection heat transfer in concentric annuli[END_REF].

Further details about the CFD analysis appeared in Van der Vyver et al. [START_REF] Van Der | Validation of CFD model of a three-dimensional tube-in-tube heat exchanger[END_REF]. An identical analysis was carried out by Coetzee et al. [START_REF] Coetzee | Heat transfer and pressure drop characteristics of angled spiralling tape inserts in a heat exchanger annulus[END_REF] to study the heat transfer behaviour of a tube-intube heat exchanger with angled spiralling tape inserts in the annulus. The analysis done in [START_REF] Coetzee | Heat transfer and pressure drop characteristics of angled spiralling tape inserts in a heat exchanger annulus[END_REF], was embraced by Louw and Meyer [START_REF] Louw | Heat transfer during annular tube contact in a helically coiled tubein-tube heat exchanger[END_REF] with the goal at quantifying the convection coefficients. Da Veiga and Willem [START_REF] Da Veiga | Characteristics of a semicircular heat exchanger used a water heated condenser pump[END_REF] picked out the same technique to figure out the convection coefficients in the hot and cold water sides of a semicircular heat exchanger.

A set of experiments was carried out by Rennie and Raghavan [START_REF] Rennie | Experimental studies of a double-pipe helical heat exchanger[END_REF] on two different tube-in-tube helical heat exchangers. Cold water with constant mass flow rate was the fluid in the tube and hot water with variable mass flow rate was the fluid in the annulus. While the inner thermal resistance was taken as constant, the convective coefficient in the annulus was assumed to follow a power law of the fluid velocity (Eq. 6). The inner thermal resistance (inner convection coefficient), the constant and the velocity exponent for the annulus convection coefficient were obtained from the modified Wilson plot. Yang and Chiang [START_REF] Yang | An experimental heat transfer study for periodically varying-curvature curved-pipe[END_REF][START_REF] Yang | An experimental study of heat transfer in curved pipe with periodically varying curvature for application in solar collectors and heat exchangers[END_REF] analyzed the convection coefficients in a tube-in-tube heat exchanger with an inner curved pipe with periodically varying curvature using water as working medium. The functional forms for the inner and annulus convection coefficients were taken as proportional to the power of the Reynolds numbers. A value of 0.8 was assigned to the Reynolds number exponent in the annulus equation. Multipliers of both equations and the Reynolds number exponent for the inner convection coefficient were determined by applying the modified Wilson plot method.

Kedzierski and Kim [START_REF] Kedzierski | Single-phase heat transfer and pressure drop characteristics of an integral-spine fin within an annulus[END_REF] also applied the Wilson plot modified as proposed by Briggs and Young [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF] to obtain the convection coefficient for an integral-spine-fin annulus in a tube-in-tube heat exchanger. However, in this case, the functional form of the annulus side includes an additional term to account for the thermally developing boundary layer. Experiments were carried out with water and 34% and 40% ethylene glycol/water mixture to determine the Prandtl number exponent of the annulus side too. Then, the modified Wilson plot was implemented in an iterative procedure that calculated the exponents of the Prandtl number and the additional term included in the equation of the annulus convection coefficient. The determination of the condensation convection coefficients of the zeotropic refrigerant mixture R-22/R-142b in a plain tube was done by Smit [START_REF] Smit | Condensation coefficients of the refrigerant mixture R-22/R-142b in smooth tubes and during enhanced heat transfer configurations[END_REF], Smit and Meyer [START_REF] Smit | Condensation heat transfer coefficients of the zeotropic refrigerant mixture R-22/R-142b in smooth horizontal tubes[END_REF] and Smit et al. [START_REF] Smit | Heat transfer coefficients during condensation of zeotropic refrigerant mixture HCFC-22/HCFC-142b[END_REF] and in micro-fin, high-fin and twisted tape insert tubes by Smit and Meyer [START_REF] Smit | R-22 and zeotropic R-22/R-142b mixture condensation in microfin, high-fin, and twisted tape insert tubes[END_REF]. These studies were performed with a horizontal tube-in-tube condenser with water as the cooling medium. Butrymowicz et al. [START_REF] Butrymowicz | Enhancement of condensation heat transfer by means of EHD condensate drainage[END_REF][START_REF] Butrymowicz | Enhancement of condensation heat transfer by means of passive and active condensate drainage techniques[END_REF] used the Wilson plot method to analyse the enhancement of condensation heat transfer by means of electrohydrodynamic enhancing technique. The modified Wilson plot method proposed by Briggs and Young [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF] was extended to determine the annulus convection coefficient with a water-to-water configuration. Relying on the modified Wilson plot proposed by Briggs and Young [START_REF] Briggs | Modified Wilson plot techniques for obtaining heat transfer correlations for shell and tube heat exchangers[END_REF], Yanik and Webb [START_REF] Yanik | Prediction of two-phase heat transfer in a 4-pass evaporator bundle using single tube experimental data[END_REF] obtained the water-side (i.e., shell-side) convection coefficient in a shell and tube evaporator with R-22 boiling inside the tubes. The refrigerant side convection coefficient was held constant by keeping the refrigerant flow rate, saturation temperature, inlet vapour quality and exit superheat constant while varying the water side mass flow rate. The functional form of the Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] was considered adequate for the water-side convective coefficient. Besides, the multiplier and the Reynolds number exponent were determined.

Benelmir et al. [START_REF] Benelmir | Identification of the heat transfer coefficient over the external area of fined tubes heat exchangers with respect to the moisture content of the air without condensation[END_REF] examined the modified Wilson plot method appended with the improved scheme recommended by Khartabil and Christensen [51]. This combination is valid when the thermal resistance of one of the fluids is constant. Then, this thermal resistance and two constants associated with the variable thermal resistance of the other fluid are calculated from a nonlinear regression analysis. In reference [START_REF] Benelmir | Identification of the heat transfer coefficient over the external area of fined tubes heat exchangers with respect to the moisture content of the air without condensation[END_REF], the influence of air humidity on the air-side convection coefficient in a plain fin and tubes heat exchanger is investigated. Here, the thermal resistance of a glycol-water mixture flowing inside the tubes is taken as constant. The air flow and humidity are varied. The Colburn j-factor versus the Reynolds number is correlated with a power law model. The modified Wilson plot enabled them to quantify the multiplier and the Reynolds exponent in the Colburn j-factor correlation equations and the inside glycol thermal resistance. The authors reports four different correlation equations that incorporate the relative air humidity.

Indirect applications of the Wilson plot method.

The Wilson plot method and its variants furnish an indirect tool to generate accurate correlation equations for convective coefficients of secondary fluids in different types of heat exchange devices. Certainly, the use of general correlation equations may lead to errors since most of them do not take into account the specific operating conditions of the experimental facilities.

Therefore, the application of the Wilson plot methods in a previous stage facilitates the determination of more accurate correlation equations that incorporate the particular features of the experimental equipment. In this section, a review of these indirect applications is addressed.

However, it is noteworthy that, if the Wilson plot method is applied indirectly, usually the authors only quote its application and do not provide detailed information on how it was implemented.

Therefore, the aim of the references to be cited lists the great variety of indirect applications that have been tackled by researchers. These references have been grouped according to the type of heat transfer process considered.

Kuo and Wang [START_REF] Kuo | In-tube evaporation of HCFC-22 in a 9.52 mm micro-fin/smooth tube[END_REF] analyzed the evaporation of R-22 in a smooth and in a micro-fin tube considering a tube-in-tube configuration with heating water in the annulus. By way of the modified Wilson plot method, the water convective coefficient in a water-to-water analysis was extracted. The Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] served as the vehicle for the convection coefficient in the annulus. The constant and the Reynolds number exponent are obtained with the modified Wilson plot method. Del Col et al. [START_REF] Del Col | Heat transfer mechanisms for condensation and vaporization inside a microfin tube[END_REF] focused on a similar application to determine the convective coefficient in the annulus in a water-to-water tube-in-tube heat exchanger. Kim and Shin [START_REF] Kim | Evaporating heat transfer of R22 and R410A in smooth and microfin tubes[END_REF] studied the evaporating heat transfer or R-22 and R-410A in horizontal smooth and microfin tubes using also a tube-in-tube configuration. These authors framed the traditional Wilson plot with three different sets of experimental data to evaluate the convection coefficient of annulus side. Brognaux et al. [START_REF] Brognaux | Single-phase heat transfer in microfin tubes[END_REF] tested the comportment of singlephase heat transfer in micro-fin tubes using a water-to-water tube-in-tube configuration. The annulus convection coefficient was calculated by means of the modified Wilson plot method considering the Dittus-Boelter equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF]. Yoo and France [START_REF] Yoo | Post-CHF heat transfer with water and refrigerants[END_REF] experimentally observed the in-tube boiling heat transfer of R-113 using a tube-in-tube configuration. The modified Wilson plot method was used in previous experiments performed with R-113 liquid inside the tube to estimate the convective coefficient of the heating water in the annulus. In this case of the Monrad and Pelton correlation equation [START_REF] Monrad | Heat transfer by convection in annular spaces[END_REF], it was taken for the annulus convection coefficient.

Wang et al. [START_REF] Wang | An experimental study of in-tube evaporation of R-22 inside a 6.5-mm smooth tube[END_REF] investigated the evaporation of R-22 inside a smooth tube in a double pipe configuration with water as the heating medium in the annulus. Within the context of the Wilson plot method, a correlation equation for the water-side convection coefficient was constructed by running separate water-to-water tests in the same experimental apparatus. Collectively, Yang

and Webb [START_REF] Yang | Condensation of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins[END_REF], Webb and Zhang [START_REF] Webb | Heat transfer and friction in small diameter channels[END_REF] and Kim et al. [START_REF] Kim | Condensation heat transfer of R-22 and R-410A in flat aluminum multi-channel tubes with or without micro-fins[END_REF] summarized the use of the Wilson plot method to determine the annulus water-side convection coefficient in an experimental facility to inspect the condensation of R-12, R-134a and R-22 and R-410A in small extruded aluminium tubes with and without micro-fins.

Chang et al. [START_REF] Chang | Performance and heat transfer characteristics of hydrocarbon refrigerants in a heat pump system[END_REF] investigated the heat transfer performance of a heat pump filled with hydrocarbon refrigerants, in which the condenser and evaporator are concentric heat exchangers. The refrigerant flows inside the inner tube and ethyl alcohol flows through the annulus. The data was channelled through the modified Wilson plot method to obtain the convective coefficient for the secondary fluid in the annulus. Bukasa et al. [START_REF] Bukasa | Heat transfer performance during condensation inside spiralled micro-fin tubes[END_REF][START_REF] Bukasa | Influence of spiral angle on heat transfer during condensation inside spiralled micro-fin tubes[END_REF] studied the condensation of R-22, R-134a and R-407C inside spiralled micro-fin tubes evaluating the effect of the spiral angle on the heat transfer rates. They used a tube-in-tube configuration in an experimental setup and with help from the Wilson plot method were able to determine a specific correlation equation that emanated from the general Sieder and Tate correlation equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF].

Sami and Song [START_REF] Sami | Heat transfer and pressure drop characteristics of HFC quaternary refrigerant mixtures inside horizontal enhanced surface tubing[END_REF], Sami and Desjardins [START_REF] Sami | Two-phase flow boiling characteristics of alternatives to R-22 inside air/refrigerant enhanced surface tubing[END_REF][START_REF] Sami | Boiling characteristics of alternatives to R-502 inside air/refrigerant enhanced surface tubing[END_REF][START_REF] Sami | Prediction of convective boiling characteristics of alternatives to R-502 inside air/refrigerant enhanced surface tubing[END_REF], Sami and Grell [START_REF] Sami | Prediction of two-phase condensation characteristics of some alternatives to R-22 inside air/refrigerant enhanced surface tubing[END_REF][START_REF] Sami | Heat transfer prediction of air-to-refrigerant two-phase flow boiling of alternatives to HCFC-22 inside air/refrigerant enhanced surface tubing[END_REF], Sami and Maltais [START_REF] Sami | Experimental investigation of two phase flow condensation of alternatives to HCFC-22 inside enhanced surface tubing[END_REF], Sami and Fontaine [START_REF] Sami | Prediction of condensation characteristics of alternatives to R-502 inside air-refrigerant enhanced surface tubing[END_REF] and Sami and Comeau [START_REF] Sami | Influence of thermophysical properties on two-phase flow convective boiling of refrigerant mixtures[END_REF][START_REF] Sami | Influence of liquid injection on two-phase flow convective boiling of refrigerant mixtures[END_REF][START_REF] Sami | Influence of gas/liquid injection on two-phase flow convective boiling of refrigerant mixtures[END_REF][START_REF] Sami | Two-phase flow convective condensation of refrigerant mixtures under gas/liquid injection[END_REF][START_REF] Sami | Influence of magnetic field on two-phase flow convective boiling of some refrigerant mixtures[END_REF] designed a sequence of experimental projects on boiling and condensation involving different refrigerants and refrigerant mixtures in an evaporator with enhanced surfaces and a condenser with coils in a vapour compression heat pump. In all the references, the data was accommodated into the Wilson plot method to eventually obtain the coils air-side convection coefficients. Sami and Desjardins [START_REF] Sami | Boiling characteristics of ternary mixtures inside enhanced surface tubing[END_REF][START_REF] Sami | Heat transfer of ternary mixtures inside enhanced surface tubing[END_REF], Sami and Song [START_REF] Sami | Flow boiling and condensation of tetrary refrigerant mixtures inside water/refrigerant enhanced surface tubing[END_REF] and Sami and Poirier [START_REF] Sami | Two phase flow heat transfer of binary mixtures inside enhanced surface tubing[END_REF] considered the same analysis cited above but in an evaporator-condenser equipment having tube-in-tube configurations. In the first two references [START_REF] Sami | Boiling characteristics of ternary mixtures inside enhanced surface tubing[END_REF][START_REF] Sami | Heat transfer of ternary mixtures inside enhanced surface tubing[END_REF], the Wilson plot was applied to isolate the water-side convection coefficient in the annulus. In contrast, in the last two references [START_REF] Sami | Flow boiling and condensation of tetrary refrigerant mixtures inside water/refrigerant enhanced surface tubing[END_REF][START_REF] Sami | Two phase flow heat transfer of binary mixtures inside enhanced surface tubing[END_REF] attention was paid to the in-tube convection coefficient since the phase change process takes place in the annulus. Yun and Lee [START_REF] Yun | Investigation of heat transfer characteristics on various kinds of fin-and-tube heat exchangers with interrupted surfaces[END_REF] experimentally analyzed various types of fin-and-tube heat exchangers. The authors also applied the Wilson plot method to obtain the air-side convection coefficient. Cheng and Van der Geld [START_REF] Cheng | Experimental study of heat transfer and pressure drop characteristics of air/water and air-steam/water heat exchange in a polymer compact heat exchanger[END_REF] experimentally studied the heat transfer of air/water and air-steam/water in a polymer compact heat exchanger. The Wilson plot method was also used to determine the air and airsteam convection coefficients.

An experimental program was devised by Han et al. [START_REF] Han | Condensation heat transfer and pressure drop characteristics of R-134a in an annular helical pipe[END_REF] to investigate the condensation heat transfer of R-134a in the annular region of a helical tube-in-tube heat exchanger. Adopting the Wilson plot method, the authors converted the data into a specific correlation equation for the water convection coefficient in the inner tube. A separate single-phase water-to-water test was performed in the same apparatus. Briggs and Young [START_REF] Briggs | The condensation of low pressure steam on horizontal titanium tubes[END_REF] and Young and Briggs [START_REF] Young | The condensing of low pressure steam on vertical rows of horizontal copper and titanium tubes[END_REF] delineated an indirect application of the Wilson plots to evaluate the convective coefficient of water inside horizontal copper and titanium tubes of a shell and tubes condenser. Later on, tthe Sieder-Tate equation [START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] was used to calculate the steam vapour condensing coefficient on the tube bundle. Belghazi et al. [START_REF] Belghazi | Condensation heat transfer of a pure fluid and binary mixture outside a bundle of smooth horizontal tubes, Comparison of experimental results and classical model[END_REF][START_REF] Belghazi | Condensation heat transfer on enhanced surface tubes: experimental results and predictive theory[END_REF][START_REF] Belghazi | Filmwise condensation of a pure and a binary mixture in a bundle of enhanced surface tubes[END_REF][START_REF] Belghazi | Experimental study and modelling of heat transfer during condensation of pure fluid and binary mixture on a bundle of horizontal finned tubes[END_REF] studied the condensation heat transfer process of R-134a and of R-134a/R-23 mixture on a bundle of horizontal smooth and finned tubes. The authors applied the Wilson plot method beforehand to compute the convective coefficient of the water inside the tubes using a water-to-water counterflow configuration in a tube-in-tube heat exchanger. They considered the functional form of the Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] in the first reference and the Gnielinski correlation equation [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF] in the last three. Gstoehl and Thome [START_REF] Gstoehl | Film condensation of R-134a on tube arrays with plain and enhanced surfaces: Part I -Experimental heat transfer coefficients[END_REF] examined the condensation of R-134a on tube arrays with plain and enhanced surfaces. The Wilson plot method was coupled with a tube-in-tube configuration to transform the inside tube convection coefficient into the Gnielinski equation [START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF]. However, a peculiarity of this case is that nucleate pool boiling was chosen for the annulus. Zhnegguo et al. [START_REF] Zhnegguo | Experimental study on heat transfer enhancement of a helically baffled heat exchanger combined with three-dimensional finned tubes[END_REF] experimental analysis of a shell-and-tube heat exchanger for the cooling of hot oil with water.

The heat exchanger consisted of a helically baffled shell and a bundle of 32 petal-shaped finned tubes. The tube-side convection coefficient was nondimensionalized into the standard Dittus-

Boelter equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF].

Additionally, plate heat exchangers were linked with the Wilson plot method. Vlasogiannis et al.

[113] analyzed a plate heat exchanger under two-phase flow conditions using air/water as the cold stream and water as the hot stream. The single-phase convection coefficient was characterized by an extension of the Wilson plot method accounting for symmetry in the two streams of the plate heat exchanger. The functional form of the Dittus-Boelter correlation equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] was instrumental for the convection coefficients of both streams. Both, the multiplier and the Reynolds number exponents were identified by applying this particular modification of the Wilson plot method. Longo et al. [START_REF] Longo | Experimental heat transfer coefficients during refrigerant vaporisation and condensation inside herringbone-type plate heat exchangers with enhanced surfaces[END_REF] investigated the vaporisation and condensation of R-22 inside herringbone-type plate heat exchangers with cross-grooved surfaces using water as heating/cooling medium. The water-side convection coefficient was obtained in a water-to-water arrangement following the footsteps of Vlasogiannis et al. [START_REF] Vlasogiannis | Air-water two-phase flow and heat transfer in a plate heat exchanger[END_REF].

The papers by Yan et al. [START_REF] Yan | Condensation heat transfer and pressure drop of refrigerant R-134a in a plate heat exchanger[END_REF] and Hsieh et al. [START_REF] Hsieh | Subcooled flow boiling heat transfer of R-134a and the associated bubble characteristics in a vertical plate heat exchanger[END_REF] refereed to the characteristics of flow condensation and flow boiling of R-134a in a vertical plate heat exchanger. Kuo et al. [START_REF] Kuo | Condensation heat transfer and pressure drop of refrigerant R-410A flow in a vertical plate heat exchanger[END_REF] and Hsieh and Lin [START_REF] Hsieh | Saturated flow boiling heat transfer and pressure drop of refrigerant R-410A in a vertical plate heat exchanger[END_REF] tackled the condensation and boiling of R-410A in a vertical plate heat exchanger also. In the four references cited above, the water convection coefficient related to a separate single-phase water-to-water test was manipulated with the Wilson plot method. Rao et al. [START_REF] Rao | An experimental and theoretical investigation of the effect of flow maldistribution on the thermal performance of plate heat exchangers[END_REF] united experimentally and theoretically effects of flow maldistribution on the thermal performance of plate heat exchangers. The convective coefficients were converted into the Dittus-Boelter equation [START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF]. The constant multiplier and the Reynolds number exponent were treated with a modified Wilson plot method.

Finally, the reference of Kim et al. [START_REF] Kim | Development of a slug flow absorber working with ammonia-water mixture: part I -flow characterization and experimental investigation[END_REF] was devoted to an experimental investigation on a counterflow slug flow absorber with an ammonia-water mixture. The absorber tested consisted of a vertical tube-in-tube with the mixture flowing in the inner tube and the cooling water in the annular part. The Wilson plot method was able to extract the convective on the side of the cooling water. In sum, the correlation equation expressed the Nusselt number as a power law of the Graetz number.

Conclusions

The general conclusions that could be drawn from the review paper is that the Wilson plot Other modifications revolve around the determination of three of more constants, encompassing a second linear regression equation and iterative schemes or even advanced mathematical tools.

The range of applications of the Wilson plot methods, either directly or indirectly, includes a large variety of convective heat transfer processes and heat exchangers encountered in industry. Owing that more than one hundred publications are reviewed, it is believed that this effort will provide useful information for future users.

Figures

  The heat transfer mechanism by convection entails to energy transfer between a solid surface and a moving fluid due to a prescribed temperature difference between the solid surface and the fluid. Heat transfer by convection is indeed a combination of conduction and fluid motion.

  of the fluids remains constant, while varying the mass flow of the other fluid within the adequate range of fully developed turbulent flow. Moreover, the exponent of the velocity in the general correlation equation considered for the convection coefficient of the fluid whose mass flow is changed should be assumed. Wilson[START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF] assumed the exponent 0.82 for the velocity. Then, the aim of the method could be, either the calculation of the convection coefficient (or thermal resistance) of the fluid that provides a constant thermal resistance, or a general correlation for the convection coefficient of the fluid whose mass flow is varied. As pointed out, the application of the Wilson plot method is based on the measurement of experimental mass flow rate and temperatures. Wójs and Tietze[START_REF] Wójs | Effects of the temperature interference on the results obtained using the Wilson plot technique[END_REF] performed an interesting analysis on the implications of the accuracy of the measured temperatures obtained by the Wilson plot method. These authors drew attention to the importance of making use of adequate experimental data to guarantee results.

3. 1 .

 1 Determining two constants in the functional form for the convection coefficient of one fluid. The other fluid has constant thermal resistance.After the formulation of Wilson[START_REF] Wilson | A basis of rational design of heat transfer apparatus[END_REF], general correlation equations for the analysis of internal forced convection based on the Reynolds analogy have appeared in the literature.Representative papers are those of Dittus-Boelter[START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF], Colburn[START_REF] Colburn | A method of correlating forced convection heat transfer data and a comparison with fluid friction[END_REF], Sieder-Tate[START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF]. These correlation equations basically relate the Nusselt number with the Reynolds and Prandtl numbers in conformity with Eq. (10). Some of these equations are sophisticated because they incorporate the variability of the fluid properties with temperature. Therefore, early modifications of the Wilson plot method assumed a general correlation for the convection coefficient in which the mass flow is varied (fluid A) as a power of the Reynolds and Prandtl numbers instead of the fluid velocity (Eq. 10). In this format, the exponents of the Reynolds number (n A ) and Prandtl number (m A ) in Eq. 10) have to be assumed.

A

  and a constant thermal resistance for the condensing fluid. Later, Da Silva et al.[START_REF] Da Silva | Electrohydrodynamic enhancement of R-134a condensation on enhanced tubes[END_REF] used the correlation obtained by Chang and Hsu[START_REF] Chang | Single-tube performance of condensation of R-134a on horizontal enhanced tubes[END_REF] for the water-side convection coefficient to study the electrohydrodynamic enhancement of R-134a condensation over the same type of enhanced tubes. Cheng et al.[START_REF] Cheng | Film condensation of HCFC-22 on horizontal enhanced tubes[END_REF] studied the condensation of R-22 on six different horizontal enhanced tubes using water flowing inside the tubes. The Dittus-Boelter correlation equation[START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] was the vehicle to obtain the water convection coefficient, whereas the condensing thermal resistance was taken as constant. Subsequently, the convection coefficients for the tested tubes were extracted by means of the Wilson plot. Kumar et al.[START_REF] Kumar | Augmentation of outside tube heat transfer coefficient during condensation of steam over horizontal copper tubes[END_REF] reported a similar analysis of[START_REF] Cheng | Film condensation of HCFC-22 on horizontal enhanced tubes[END_REF] to estimate the steam condensing convection coefficient over three different types of horizontal cooper tubes. However, in this work the Sieder-Tate correlation equation[START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] was modified to account for the entrance effects in short tubes. The functional form was associated with convection coefficient for the cooling water. McNeil et al.[START_REF] Mcneil | Dropwise condensation of steam on a small tube bundle at turbine condenser conditions[END_REF] examined the filmwise and dropwise condensation of steam and a steam-air mixture over a bundle of tubes with cooling water at turbine/condenser conditions. The thermal resistance of the shell-side was maintained constant and independent from the outside tube-wall temperature due to the vapour shear conditions generated by the high velocity of the steam. The Wilson plot method was implemented varying the cooling water mass flow and the convection coefficient was expressed by way of the Gnielinski equation[START_REF] Gnielinski | New equations for heat and mass transfer in turbulent pipe and channel flow[END_REF] with a constant multiplier. Das et al.[START_REF] Das | The use of an organic selfassembled monolayer coating to promote dropwise condensation of steam on horizontal tubes[END_REF] relied on the modified Wilson plot to determine the convection coefficient for the dropwise condensation of steam on horizontal tubes with different types of dropwise condensation promoters. It was assumed that the convection coefficient of the water inside the tubes could be accommodated into the Petukhov-Popov correlation equation[START_REF] Petukhov | Theoretical calculation of heat exchange and frictional resistance in turbulent flow in tubes of an incompressible fluid with variable physical properties[END_REF] embracing a multiplier and also that the dropwise condensation convection coefficient was constant. Both the multiplier in the inner convection coefficient equation and the condensing convection coefficient were evaluated with the help of the Wilson plot. An iterative procedure was indispensable to account for the variability of the water properties with temperature.Chang et al.[START_REF] Chang | Condensing heat transfer characteristics of an aluminum flat tube[END_REF] selected the Wilson plot method for the quantification of the condensing convection coefficients of R-134a and R-22 flowing inside four different extruded aluminium flat tubes and one micro fin tube. The tube testing was accomplished in a double-tube configuration set-up. In reference to the flat tubes, they were placed inside a shell with rectangular crosssection. In contrast, the micro-fin tube was located inside a shell with circular cross-section. The refrigerant condensation took place inside the tubes and coolant water flowed between the tube and the shell. The convection coefficient for the coolant water was held constant by controlling a steady flow rate of water and a small deviation of the mean water temperature was noticed. In this case, the condensing convection coefficient was varied by changing the quality of the inlet refrigerant; the quality was controlled by means of a pre-condenser. The condensing convection coefficient was considered proportional to a power of the inlet quality with an exponent of -0.8.The paper reported experimental results of the condensing convection coefficients as a function of the quality of inlet refrigerant. Also, this type of modification had been articulated with single-phase in-tube convection coefficients. Wang et al.[START_REF] Wang | Single-phase heat transfer and flow friction correlations for microfin tubes[END_REF] utilized a tube-in-tube heat exchanger to characterize the singlephase convection coefficients within micro-fin tubes. The authors invoked the Dittus-Boelter correlation equation[START_REF] Dittus | Heat transfer in automobile radiators of the tubular type[END_REF] for the inner convection coefficient. The thermal resistance within the annulus was held constant by holding the average fluid temperature and the Reynolds number to constant values. These authors provided suitable correlation equations for each one of the micro-fin tubes tested. Webb et al.[START_REF] Webb | Heat transfer and friction characteristics of internal helical-rib roughness[END_REF] also resorted to a tube-in-tube configuration to recommend a correlation equation for the inner convection coefficient accounting for the dimensions of the enhancement geometry. Experiments were carried out with R-12 boiling in the annulus and varying the mass flow of the cooling water inside the tubes. The Sieder-Tate correlation equation[START_REF] Sieder | Heat transfer and pressure drop of liquids in tubes[END_REF] provided the framework for the turbulent heat transfer inside the tube and a constant boiling convection coefficient in the annulus (i.e., constant thermal resistance in the annulus).

  the functional form for the convection coefficients of the two fluids. Generally, if the convection coefficient of one of the fluids (fluid A) varies, the convective thermal resistance provided by the other fluid (fluid B) is altered because the surface temperatures change, and as a consequence the fluid properties will change.

  method and the different modified versions of the Wilson plot method constitute a remarkable tool for the analysis of convection heat transfer in tubes and heat exchangers for laboratory research usage. The trademark of the Wilson plot methods is that they assist in the determination of convective coefficients based on experimental data. In sum, the collection of convection coefficients are encapsulated in concise correlation equations.The original Wilson plot method relied on two primary assumptions, a constant thermal resistance for one fluid and a known expression for the convective coefficient of the other fluid; the latter in the form of a power of the fluid velocity. The various modifications proposed by researchers all over the world have the common objective of overcoming the prevalent assumptions adhered to the original formulation of Wilson. For instance, one simple modification overcomes the assumption of constant thermal resistance and the determination of two multipliers by means of manipulating a linear regression analysis of the experimental data.

Figure

  Figure 2

2

 2 Figure 2

Fig. 1 .

 1 Figure captions

Fig. 2 .

 2 Fig. 2. Wilson plot considering a functional form for the convective coefficient of one fluid and constant thermal resistance for the other fluid.

Fig. 3 .

 3 Fig. 3. Wilson plot considering functional forms for the convective coefficients of both fluids.