Category Archives: Particle Image Velocimetry (PIV)

SOLSTILL — A SIMULATION PROGRAM FOR SOLAR. DISTILLATION SYSTEMS

Dr. Nguyen The Bao

Hochiminh University of Technology, Vietnam.

1. INTRODUCTION

The development of SOLSTILL, a simulation program for estimating the performance of basin type solar stills is described. Models for both the standard free convection solar still and a forced convection solar still with enhanced heat recovery are included in the program. For the conventional free convection systems, the SOLSTILL program also enables simulation of more complex systems with many more parameters compared to the existing models found in the literature. A new model incorporating heat and mass transfer in forced convection solar stills with enhanced heat recovery is described in this paper. The design, fabrication and testing of an experimental system set up to validate SOLSTILL is detailed. The comparison of experimental and simulation results indicates that the program can predict distillate production at an acceptable level of accuracy.

Research on the integrated heating system of the shallow-styled solar pond

The shallow-styled solar pond is divided into the upground pond and the underground pond. It is the complete set equipment of the conservatory-modeled solar pond. It is used to heat the working water body for breeding young aquatic products in advance and cultivating bio-baits.

The concrete-brick structure is adopted to build the upground shallow-styled solar pond. The five layers of the interior surface are waterproof. Asphalt lacquer is painted and dropless translucidus membrane is adopted. The shallow-styled solar pond is multifunctional. The pond is 1m deep, 80m long and the effective volume is 750 m3. There are 12 ponds in it. The ponds of NO.1 to NO.6 are used to demonstrate the breeding, NO.8 to NO.10 are used to cultivate bio-baits and NO.11 to NO.12 are used to heat the working water body.

The upground solar pond of shallow style is in operation for 2 periods. It has an excellent effect on heat-integration. The working temperature is 5”C higher than the natural sea water temperature (NSWT). In May, the contribution rate of the solar energy of working water body achieves 100%, which shows a prominent energy-saving effect. See diagram 1, diagram 2 and diagram 3 for the temperature operation.

The underground solar pond of shallow style was brought into service in April, 2002. The building area of the pond is 300 m’ with a depth of 2m. The water level is 0.5m to 1.0m deep. Farm-oriented membranes are placed at the top of the pond and wire-steel cement pillars act as the support.

The characteristic of the underground solar pond of shallow style is that it can store a great amount of water and can make full use of soil to save up heat. The temperature varies greatly between day and night, so it is fit for the water-supply of industrialized cultivation in a large scale. The working water diagram3 Temperature curves for the temperature is 3.0"C to 5.0"C higher than that of the shallow-styled solar pond in natural one in general.

Comparison Test of Thermal Solar Systems for Domestic Hot Water Preparation and Space Heating

H. Druck, W. Heidemann, H. Mtiller-Steinhagen

Universitat Stuttgart, Institut fur Thermodynamik und Warmetechnik (ITW) Pfaffenwaldring 6, D-70550 Stuttgart Tel.: 0711/685-3536, Fax: 0711/685-3503

email: drueck@itw. uni-stuttgart. de, Internet: http://www. itw. uni-stuttgart. de

This paper presents the results on the latest comparison tests of thermal solar systems for domestic hot water preparation and space heating carried out for the German consumers’ magazine “test”. The systems were tested with regard to thermal performance, durability and reliability, environmental aspects as well as safety aspects. The test procedures as well as the results obtained are described and discussed. Possible future trends e. g. with regard to the development of the technology and the system costs will be shown.

1 Introduction

Following the last comparison test of solar thermal systems performed in 1998 a new series of test results was published by "Stiftung Warentest" in the German consumers’ magazine "test" in 2002 and 2003. The results of the test of 16 solar domestic hot water systems were published in /1/. A report on the test of 11 thermal solar systems for combined domestic hot water production and space heating, so-called combisystems, can be found in /2/.

In addition to the customer-oriented and product related results already published in the consumers’ magazine "test", this paper provides further background information. The test procedure is described in detail. Furthermore, the ranking of the results, especially with regard to the assessment of the thermal performance, is presented. Finally the long-term development of thermal solar system technology, e. g. with regard to the cost development is discussed.

2 Systems tested

The systems had to be dimensioned by the supplier or manufacturer respectively for a single-family house located at WUrzburg, Germany. The house is equipped with a 45 ° inclined, south facing roof. The daily hot water load is 200 litres (at 45 °C) and the heat insulation standard of the building with a heated living area of approx. 130 m2 fulfils the requirements according to the German Energy Saving Directive (Energieeinspar — verordung: EnEV) which is the present German directive concerning the primary energy demand of buildings. Based on this the yearly heat demand for space heating reaches 9090 kWh or 71 kWh/m2 respectively.

With regard to the 16 thermal solar systems for domestic hot water preparation (DHW) the effective collector area varied between 3.2 m2 (system H13, H15,H16) and 5.7 m2 (system H12). 12 systems are equipped with flat plate collectors and four with vacuum tube collectors (systems H13, H14, H15, H16). The effective usable storage volume of the domestic hot water stores is in the range of 268 litres (system H10) up to 419 litres (system H14). The amount of hot water that is available if only the auxiliary heated part of the store is in operation varies between 100 litres for system H6 and a maximum of 200 litres for system H11.

Concerning the system and storage concepts most of the systems are designed as the typical German „standard systems" shown in fig 1.

For all 16 systems the solar energy is transferred to the domestic hot water via a plain tube heat exchanger. The store of system H1 is the only one that is additionally equipped with a device for stratified charging. Additionally the store of system H11 is charged in a stratified way by using two solar loop heat exchangers: One located in the upper and one located in the lower part of the store. In the case of high temperatures delivered by the solar collector the flow is directed additionally via the upper heat exchanger following a special control strategy.

With regard to the solar combisystems, systems for combined domestic hot water preparation and space heating, the spectrum of investigated system concepts is much broader. Concerning space heating, for 7 of the 11 systems tested the space-heating loop is operated in the pre-heating mode. This means, that the return line of the space heating loop is only directed through the store if the temperature at the store’s space heating outlet connection is above the return temperature of the space heating loop. The energy delivered by the external auxiliary heater is only transferred to the combistore in order to heat the auxiliary part required for domestic hot water preparation. Auxiliary energy required for space heating is fed directly into the space heating loop (see figure 2a).

The other 4 systems (system C4, C5, C10, C11) use the combistore as a buffer store.

This means that the auxiliary energy is always transferred to the store and that the space heating loop is continuously supplied from the combistore (see figure 2b).

In comparison to a similar investigation of solar combisystems carried out five years ago it is remarkable that the solar combisystem technology has moved towards a higher level of integration. Five years ago when ordering a combisystem, one was supplied with a collection of separate, individual components. Today in many cases the appearance of the components already shows obviously that they belong to the combisystem of a certain manufacturer.

The positive trend towards compact systems was also documented by the fact that the manufacturers did not anymore select so-called two-store-systems for the test. The state of the art concerning maximum compactness are systems were the gas burner is already integrated into the combistore (system C9 and C11) or directly connected at the combistore (system C10).

The effective collector area of the combisystems investigated is in the range of 5.7 m2 (system C2) to 14.2 m2 (system C7). 6 systems use flat plate collectors. The other 5 systems are using vacuum tube collectors (system C2, C3, C4, C5, C10). The volume of the combistores varies from 450 litres for system C9 up to nearly 1000 litres for system C7 and C11. The usable hot water volume is in the range from 100 litres (system C2 and C3) up to a maximum of 300 litres for system C4. Concerning the systems using the combistore as a buffer for the auxiliary heater, the buffer volume for the boiler or gas burner, respectively, varies between 80 litres for system C4 and 225 litres for system C10.

Validation of measurements using additional test sequences — an extension of the test procedure for solar collectors

S. Fischer, H. Muller-Steinhagen

Universitat Stuttgart, Institut fur Thermodynamik und Warmetechnik (ITW) Pfaffenwaldring 6, D-70550 Stuttgart Tel.: 0711/685-3231, Fax: 0711/685-3503 email: fischer@itw. uni-stuttgart. de

In the European Standard EN 12975 /1/ two different approaches for the determination of the thermal behaviour of solar collectors are described: The test method under steady state conditions and the method under quasi-dynamic conditions. Although both approaches are quite different one common item exists: a test is considered complete if all required test sequences have been performed. A validation of the determined parameters, in the way done for the testing of hot water stores, is not foreseen for collectors.

The present paper describes the selection and evaluation of a suitable validation sequence. In addition, criteria are introduced that help to decide whether the found parameter set describes the thermal behaviour of the collector satisfactory or not. The procedure is introduced for the example of a CPC collector with vacuum tubes.

1. Introduction

The reason why a validation procedure may be necessary for collectors is shown in Figure 1, where the measured collector output is plotted together with the calculated collector output.

The collector parameters obtained during the test can be used for performance predictions by a dynamic system simulation /2/. The results of the performance predictions are used to compare the collectors and are often the basis for the decision which collector will be installed. In Germany the results of the performance prediction serves, in addition, as a criterion whether a collector qualifies for government subsidies or not. Taking this into account it is reasonable to introduce a validation sequence within the thermal performance testing to prove that the dynamic behaviour of the collector in operation can be described by the determined collector parameters.

Online-Visualisation

Internet visualisation of the operating state of a solar thermal system enables authorised people to fully access the system at any time from anywhere. To realise online visualisation based on an embedded system, the Java programming language is adequate. Java [5] goes back to a project to develop a simple and platform-independent programming language for consumer electronics from 1991. In 1996 Internet browsers from Netscape (Netscape Navigator/Netscape Communicator) and Microsoft (Internet Explorer) have been enabled to download Java programmes from the Internet and to execute the code within the browser. The programmes, so-called applets, have the advantage that they run locally on the machine executing the web browser. The resource­consuming and compute-intensive graphics rendering for visualisation is completely done on the client PC, so that the embedded systems resources are not used too much.

Letztes Update: 31.03.04, 0ft24:05Uhi

Fig. 4: Java applet for visualisation of system performance

Solar cooker incorporating CPC type optics and. heat pipe technology description and testing

Manuel Collares — Pereira, Joao Farinha Mendes,

Rita Costa Leal, Jose Pedro Almeida

Departamento de Energias Renovaveis, Instituto Nacional de Engenharia Tecnologia e
Inovagao, Estrada do Pago do Lumiar, 22, 1649-038 Lisboa, Portugal

1 — Introduction

In order to spread the use of solar cookers it would be desirable to provide them with some of the features usually associated with traditional ways of cooking. Actually, in most civilizations, people have some habits which will be hard to change: cooking is usually done inside the house, at any time of the day and conventional fuels can deliver high power to the cooking pan. Common solar cookers must be used outdoors, with the cooking vessel placed on the absorber plate heated by the sun. Besides, there is usually no capacity to allow cooking in the evening, i. e. when the sun is no longer shinning or at the end of the day with the sun low on the horizon when high efficiencies are impossible to be achieved.

With these considerations in mind, a device incorporating Non Imaging Optics (CPC type Optics) and heat pipe technology has been previously developed (Collares- Pereira, 2001). The energy collected by the CPC concentrator was either transmitted straight to a cooking plate for immediate use or stored for subsequent use. The heat pipe allows delivering the energy to the cooking pan, which may be at a certain distance from the CPC collector (for instance inside the house). This solution is also easily adapted to the cooking of larger quantities as is necessary, for instance, in small communities or in schools. In the present paper a new system based on the same ideas is presented. The goal of the new device, presented in this work, is to improve the thermal performance and solve some basic design problems encountered in the previous prototype, bringing us much closer to a potential industrial product.

Collector Reliability

Reliability data for the initially installed sets for each collector is presented in Tables 1 through 3. The data includes failures resulting in loss of vacuum due to causes other than the glass cracking. The set used for the Novel ICPC is for the collectors in the first half of the production run. Several distributions commonly used in reliability modelling were fit to the data in the tables. The best fits were obtained with the lognormal distribution. Survival probabilities were then calculated from the fitted lognormal distributions. These probabilities are shown in Table 4 for four year periods extending to 28 years.

Collector Performance

Novel ICPC

Prior to the start of the 1998 Sacramento demonstration, individual fourteen tube modules were tested on Sandia National Laboratory’s two-axis tracking (AZTRAK) platform. The efficiency curve for insolation of 1000 W/m2 was given in [16] as

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but with no control deficiencies are shown in figure 7. Here, daily efficiencies as high as

54.5 percent were achieved.

During 2002 the ICPC collectors were operated in the 50 to 70C temperature difference range [13, 14]. Performance closely matched the 2001 performance shown in figure 7. Allowing for some additional tubes that had lost vacuum, array performance during 2003 was essentially unchanged while operating in the lower temperature difference range [15].

Corning and Philips VTR261

During the 1982/83 period there ———- Regressed 1998 Banks Data

was only one Solarhaus tube with vacuum loss (one in the

Corning array) and no other Figure 6: 2001 Daily Collection Performance at a 90 to factors degrading performance. 110C Collector to Ambient Temperature Difference.

In 1982/83 the Corning and

Philips VTR261 collector arrays produced maximum daily solar collection efficiencies of about 56 percent and 53 percent respectively when operating in the 45C to 60C temperature difference range [17].

As of 1996/97 six tubes in the Corning array and one in the Philips array had vacuum loss and the Philips VTR261array reflectors and tubes backs had developed deposits due to growth of algae [18, 19]. The consequent reduction in performance is shown in table 5.

Daily performance in 1996/97 is shown in figures 8 and 9. The degree of degradation of performance during these years is nearly the same as for the novel ICPC operating in 2001. However, the degradation in the novel ICPC performance was characterized in a higher average daily temperature difference range (90 to 110C versus 15 to 60C) and for a higher average daily insolation level (7 versus 4 kWh/m2). Nevertheless, the two factors should offset somewhat so that the results for the novel ICPC in figure 7 may be roughly compared to those for the Corning and Philips VTR261 in figures 8 and 9.

Comparisons

Table 6 contrasts some of the important characteristics of the three different evacuated collectors that have been brought out in the paper. Some interesting observations are

• Higher survival rates are seen when the evacuated tubes are fabricated at some level of automation as opposed to being hand built.

High operating temperatures (130 to 160C) experienced by the novel ICPC was not a factor in tube failures from glass cracks

Table 5: Average Daily Performance of the Solarhaus Freiburg Collectors

Corning

Philips VTR261

1982/83

1996/97

1982/83

1996/97

Energy Delivered (kWh/m2)

2.18

1.754

2.01

1.76

Insolation (kWh/m2)

4.18

4.07

3.83

4.14

AT*Day Length (Kh)

342

275

320

237

Efficiency

0.521

0.431

0.525

0.426

Change in Efficiency

-0.17

-0.19

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Discussion of the results

The values presently analyzed were measured during the summer month of February, 2004. Although the solar absorbed energy is high, the number of showers per inhabitant

tends to increase compared to other seasons, possibly having a negative net effect on the solar fraction.

Fig. 3 shows the daily totals of solar irradiation on a horizontal surface and the daily average of ambient temperature during February 2004. The daily total of solar irradiation data and the daily average ambient temperature were measured in a site[1] located 25 km distant from the housing unit. The solar irradiation distribution as well as the ambient temperature is similar to those measured at the reference site. It can be observed in Fig 3 that the analyzed month had high solar irradiation levels, low values occurring only on two days. The daily average temperatures during the month lie between 20°C and 30°C, which are typical values for the summer season in Florianopolis.

Suspicious data, collected from vacant apartments or resulting from incorrect use of the CSDHWS, were discarded from the analysis. Measured data from a total of 44 consumers from Group A and 24 consumers from Group B were used.

The measured power of each of the showerheads was totalized in hourly intervals and divided in three subsets according to the weekdays as follows:

(a) workdays;

(b) weekends and holidays;

(c) all days.

This division is intended to verify if differences exist in the consumption profiles between workdays, weekends and holidays.

Fig. 4 shows the mean electricity consumption profile of Group B. It can be seen that the consumption profile is characterized by a very low consumption from 2 AM to 5 AM. After this time interval, the energy consumption rises to a new level that persists until the middle of the afternoon (4 PM). During this period, a small peak was found before noon, probably caused by those who work or study only during the afternoon. At the end of the afternoon, the energy consumption increases substantially, reaching its peak around 7 PM, to then decrease. The relation between the average consumption and peak demand (load factor) was 0.37. The visual inspection of Fig. 4 leads to the conclusion that there is no considerable difference between workdays and weekends.

Figure 4. Monthly average of the hourly energy of the electric showerheads of Group B

The average hourly energy consumption of Group A is compared with the values for Group B in Fig. 5. The hourly peak of energy consumption still remains in Group A, however, the energy consumption is significantly reduced compared to Group B. The load factor was 0.38, which is practically the same value obtained for Group B. Comparing the peaks of the two groups, it was found that the peak demand of Group A is 60% lower than the peak demand of Group B.

Figure 5. Monthly average of the hourly energy of the electric.

Fig. 6 shows the hourly solar fraction, considering that the hot-water consumption profile is the same for both groups. It can be observed that the solar fraction varies from 40% to 80% during most of the day, but it becomes lower in the morning periods. This is probably due to the storage tank heat losses during the night and also to those cases where stored hot-water was consumed during the previous day. From to 2 AM to 5 AM the solar fraction is not representative since hot-water consumption is low. The solar fraction obtained during the analyzed month (February 2004) was 58%.

Figure 7. Average power consumption for Group B (01-Feb-2004).

The true peak demand can be identified from instantaneous power values. In the present analysis, power is averaged in 5 minute intervals, which can be considered a good approximation of the instantaneous values. Fig. 7 shows the average power of Group B on the day during which the highest peak occurred (01-Feb-2004). The same is shown in Fig. 8 for Group A (28-Feb-2004). It can be seen in Fig. 7 that the maximum contribution of the electric showerhead to the peak of each low income consumer was 0.57 kW. On the other hand, Fig. 8 shows that for a group of consumers with the same characteristics, but owning a CSDHWS, the contribution to the peak is around 0.30 kW. Therefore, the power necessary to supply electricity to the showerheads was reduced by 47% with solar heating.

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Operational experience of the solar assisted district. heating system in Hanover-Kronsberg

Michael Bodmann, M. Norbert Fisch

Institute for Building Services and Solar Technology, Technical University of

Braunschweig,

Muehlenpfordtstr. 23, D-38106 Braunschweig, Germany

As part of the World Exposition EXPO 2000, a solar assisted district heating system for 106 residential units was put into operation in Hanover-Kronsberg in June 2000. Roof-integrated collectors with an area of 1,473 m2 were installed and a hot water storage with a volume of 2,750 m3 was erected. For the construction of the storage a nearly water diffusion tight concrete was used. In comparison with former storage constructions no inner stainless steel liner was needed.

Project description

Figure 1 shows a ground plan of solar housing estate. Residential buildings with a total living area of 7,250 m2 and a common room area of 110 m2 are connected to the solar assisted district heating. Solar collectors were mounted on the south-western and south­eastern orientated roofs. The seasonal storage was erected on a public playground in 120 m distance from the heating central, which is placed in the basement.

A simplified scheme of the solar assisted district heating is shown in Figure 2. The collected solar heat is transported by a piping network to the heating central. A heat exchanger is used to charge the storage circuit. Heat can either be directly used for covering the heating demand or be stored in the seasonal storage. If the solar heat does not have a sufficient temperature, additional heat is supplied by the heat distribution network of the entire estate Kronsberg, which is connected to a cogeneration plant.

Compared with systems of the first generation, the plant was

optimised. The storage can be charged and discharged at three levels, which allow simultaneous loading and unloading of the storage. The flow in the collector and the secondary circuit can be variably controlled by pumps and will be adjusted to the required temperature.

Hot water preparation is ensured by domestic hot water storage systems in substations. The dwellings are equipped with a low temperature heating system with radiators at 65/39°C (supply/return temperature). The heat distribution network with a maximum supply temperature of 70°C was mainly installed in the basement of the buildings.

Collector area

The collector plant is distributed in 13 partial areas. The size of the different collector fields varies between 40 and 290 m2.

The used solar roof system allows an exact adaptation to the substructure. The utilisation of the roof area was almost 90%. Therefore, the solar roof system replaced the conventional roofing. Figure 3 shows an example of collector integration in the detached buildings.

Seasonal storage

The seasonal storage was constructed as a concrete cylinder with a roof which is formed as a conical shell, see Figure 4. The internal diameter comes to about 19 m, and the largest inside height amounts to approximately 11 m.

For the construction of the storage in this project a high performance nearly water steam diffusion tight concrete was used for the first time (Reineck and Lichtenfels,

2000). Thus, it does not only carry the load but also has the function of waterproofing. In this case there was no interior stainless steel liner needed as used at the storage constructions in

Friedrichshafen-Wiggenhausen and Hamburg-Bramfeld (Fisch et al., 2001). Due to the water loss (by water steam diffusion) a water resistant insulation was necessary. Therefore, wall and
roof of the storage were insulated on the outer side with pressure resistant granulated recycling glass. The insulation thickness in the wall area increases from 30 cm at the bottom to 70 cm at the top. The thickness of the roof insulation reaches 70 cm. Above the roof insulation a protective concrete layer and an earthcover were applied. Figure 5 shows the storage after completion of the work.

Steady state analysis

The following section was carried out using the Fourier law, related to heat conduction through a material, given by eq. (1)

where T is the temperature (K), т is the instantaneous time (s), kp is the conduction coefficient of the absorber (W/mK), a is its thermal diffusivity (m2/s), p is its density (kg/m3), cp is its specific heat value (J/kg K), and qv is the internal heat generation in the absorber (W/m3).

1.1. — Mathematical model

To carry out the steady state study, the absorber surface was divided, using a mesh, in multiple similar units, all of them with same dimensions, x and y, so the problem could be considered discrete. Then, Eq.(1) was approximated by finite differences. For each (i, j) pair, derivatives from x (as well as from y) depend on the increment Ax [Ketkar, 1999]:

A sample network is shown in Fig. 1. Because pipes distribution was considered to be the same over the absorber plate, only a portion of it will be considered. The plate presented 15 equal tubes, symmetrical to the y-axis, dividing it in 30 equal zones that will present the same results. The boundary line is considered adiabatic, due to the presence of an isolator. Boundary losses are around 3% of the captured heat [Alaiz, 1981], so they could be neglected. Flow distributions estimation along the risers is the same, unless the regime is laminar [Weitbrecht et al., 2002].

After using eq. (1) and (2) in the previously mentioned mesh (see Fig. 1), it can be seen that there is a relationship between temperatures obtained form energy balance in each point (see eq. (3)). This expression will experience some changes in the boundary line, due to the previously mentioned adiabatic considerations.

Tj-ij + Tj+1,] + Tj, ]_1 + Tj, ],1 — 4Tj, j =- q AxAy

Internal heat generation, qv, includes solar radiation and heat losses by re-radiation, conduction and convection through the isolators and the crystal. Also, it is important to notice the losses of available heat that is transferred to the water that is going to be heated:

Then, unitary heat release by radiation q1 matches S (W/m3):

q1 = S = IT (та)! e (5)

where It is the incident global radiation over the inclined absorber (W/m2), та is the product transmittance-absorbance of the crystal-absorber, and e is the thickness of the plate (m). On the other hand, the unitary losses of the plate, q2, are obtained from the global losses coefficient, Ul (W/m2K), and finally, the temperature difference is calculated between the point and the environment.

• (T. — T )

?2 = Ul 1 con UL = Ub + Ut (6)

e

where Ub represents the losses behind the collector, due to the heat conduction through the isolator and the casing, and the convection to the atmosphere. Its values depend on the materials properties and wind velocity. In this way, Ut is related to front losses, and is originated from solar radiation and re-radiation, system inclination and orientation. Once calculated, both coefficients will be considered as constants in all the network points (no shadow will be considered to take place). The ambient temperature is Ta.

Finally, q3 provides the available heat that passes through the working fluid in that precise point. It can be calculated by means of the global transmission coefficient, Uc, which takes into account the heat convection between the internal wall of the pipe and the water. This available heat could be calculated as shown in eq. (7).

where h (W/m2 K) is the film coefficient (convection coefficient), R is the internal radius of the pipe (m), eq is the pipe thickness, and Kuc is a proportional coefficient depending on the pipe geometry, plate weld and position. The sub index m refers to the limit over the pipe. According to this, eq. (3) will appear as follows (eq. 8.a does not consider heat transfer to the fluid, while eq. 8.b does).

general form (8.a):

The sub index m+1 refers to the fluid at each point, related to the dimension y. Let consider m the number of distributed points along the x-axis, and n the number of points distributed over the y-axis. For each of them, there will be an equation, as eq. (8), depending on their position. This equation could be modified if adiabatic boundary is considered. Then, m x n equations will appear, for (m+1) x n unknown quantities. So, to solve this system m additional equations are needed.

These additional equations can be obtained after calculating the heat balance of the working fluid in the y direction. The working fluid will receive a certain quantity of heat from point y=j to point y=j+1, which depends on the medium temperatures of riser, Tp, and fluid, Tf, between the extremes j and j+1. The fluid flow mass, mf, the specific heat value of

water, Of, and the increment of temperature of water, AT, will also be considered (see eq. (9) and (10)).

Film coefficient, h, is calculated by means of an initial value ho, obtained from empirical relations of heat transfer theory, multiplied by a proportionality coefficient, Kh, which depends on the same factors as KUc.

Equations (8) and (11) constitute a set of (m+1) x n linear equations with (m+1) x n unknown data. In this case, however, the input fluid flow, Tff, is known, thus facilitating the equations set resolution. Moreover, this temperature serves to stabilize the system calculus, thus promoting the finding of feasible results.