Modeling of finned flat tube heat exchangers and search of Nusselt-Reynolds numbers correlations

Using a wind tunnel, a series of model finned flat tube copper radiators was studied. All geometrical parameters of the studied radiators were fixed except for the height of the fin, which was varied in the range from 5 to 20 mm. It was found that the global water-air heat transfer coefficient weakly depends on the water flow rate and significantly depends on the air flow rate. Correlations between the Nusselt and Reynolds numbers for fins with different heights are found. Based on the obtained correlations, a procedure for scaling the heat exchanger, verified experimentally, is proposed. The experiments performed demonstrate the efficiency and time-saving potential of the proposed method for choosing the optimal finned flat tube heat exchanger for adsorption heat conversion systems.


Introduction
Due to the depressing environmental situation on the planet, there is a growing interest in energysaving technologies in the global community.Adsorption heat transformation (AHT) is an energysaving technology that allows the use of heat from both alternative energy sources and waste heat that is uselessly dissipated in industry, transport, etc. AHT is a technology that makes it possible to make available to the consumer both heat production (adsorption floor heating systems) and cold production (adsorption cooling, air conditioning and ice production systems) [12 -3 4].The main difficulty in using alternative energy sources and waste heat is their relatively low temperature potential of about 100C, so it is quite difficult to return such heat to a useful cycle.Indeed, solar collectors of the simplest designs heat the thermal fluid up to 80-100C, the temperature of a large number of industrial heat losses is in the same temperature range.Adsorption heat transformers are able to work with low temperature potential heat, but despite significant progress in recent years, commercially available AHT devices are still rare on the market and need to be optimized [56 -7].One of the most important directions of AHT optimization is the acceleration of sorption/desorption processes resulting in specific power improvement.In the last two decades, the world scientific community demonstrates increased attention to the development of adsorption methods for low-temperature heat transformation [8] as a real alternative to current compression and absorption technologies.Indeed, in recent decades, there has been a growing interest in adsorption heat transformers as an alternative to similar compression systems.The results of research in this area can be summarized as follows: -Due to the extremely low consumption of electrical energy, adsorption heat transformers are an environmentally friendly alternative to compression devices for heat and cold generation [8].To date, the various adsorption materials has been widely developed, which makes it possible to implement the working cycles of AHT with high efficiency.For the AHT systems both traditional (silica gels, zeo-lites, coals) and innovative sorbents (aluminophosphates, "salt in a porous matrix" composites) interacting with a number of working liquids (water vapor, methanol, and ammonia) can be used [9].
-Despite the development of AHT technology, due to the low specific power of heat conversion -100-300 W/(kg of adsorbent), the devices are characterized by impressive dimensions and cost, as a result, occupy a small market share [33].Previously, it was shown that low power values of AHT devices [2] are due not to the thermodynamic properties of the sorbent, but rather to the conditions of the cycle (for example, the duration of adsorption/desorption stages [10]) and the organization of heat and mass transfer [11] in the system.Thus, in order to increase the power of AHT units, it is necessary to improve the heat and mass transfer in the system under consideration.The process of heat conversion occurs in adsorber-heat exchanger (AdHex), that is why this unit should be optimized first of all [12].At the moment, generally for AHT creation different commercial radiators are used as heat exchangers.Such modules initially are specialized for processes other than AHT, in most cases for cooling engines of automobile/motorbike by dissipation of heat into the environment.At present, just first steps for development an appropriate procedure of choosing the optimal heat exchanger for AHT among commercial ones were undertaken.Recently it was shown that finned flat tubes (FFT) [131 415-16171819] heat exchangers (Hexes) are more perspective for AHT than units with a different geometry [20] (Fig. 1).On the base of numerical analysis, it was demonstrated that FFT Hexes show the best efficiency [21,22].
The main parameters required for core of FFT Hex characterization are (Figure 1): f, c -thickness of fins and channels' walls, cchannel internal thickness, Hf -fin's height, fdistance between fins, Dwidth of Hex.Other important parameter characterizing heat transition from media 1 to media 2 are: A (m 2 ) -Hexes primary surface (channels' area), Af (m 2 ) -secondary surface (fins' area), λ -thermal conductivity of Hex material (W(m٠K)), heat transfer coefficients h1 / h2 (W/(m 2 K)) between material of Hex and media 1 / media 2. The global coefficient of heat transfer UA (W/K) of the whole heat exchanger providing heat transfer between media 1 and 2, can be found according to the formula [23,24]: where E -coefficient of fins' effectiveness, K = (A + Af)/A -surface extension coefficient [23].In case of rectangular fins placed between two flat channels the E coefficient can be written [25]: For FFT Hex with thin ducts in a case of media 1 flow is laminar, Nusselt number for media 1 is close to constant Nu ≈ 8 [26].Taking into account this consideration the expression for heat transfer coefficient h1 between metal and media1 can be written as follows: where λ1 -thermal conductivity of media1, Φc -channel's hydraulic diameter of the Φc =4Dc/(2D + 2c).The same equation can be used for air.
When heat exchanger loaded with sorbent is used in AHT the adsorbent grains can be considered as the media 2 with constant heat transfer coefficient h2 between metal surface and granules.The values of h2 for typical sorbents are in a range 50 -200 W/(m 2 K) [27].From other side when Hex is used as airto-liquid radiator, and media 2 (air) flows through volume with fins, air side Nu and h2 can correlate with velocity of air uair, or, in other words, with Reynolds number of air Reair that is proportional to uair.The corelation formula between Nu and Re can be found in classical literature [28]: where Pr is Prandtl number of media 2. In case of nonideal plates, for example, perforated or wavy plates instead of (4) the correlation between Nusselt and Reynolds numbers can be presented in a more common form [29]: where a, b and c are empirical coefficients.For more complicated situations, for example, non-steady flow and presence of turbulence a huge amount of correlations was presented in literature [303 .Knowing of these corelations gives a possibility to obtain Nu and h2 for any given geometry of Hex elements.
The results of [34] confirm that if all parameters of equations (1, 2) are known the global heat transfer coefficient UA for given Hex can be found easily.In [3434] small, but representative Hex with known geometry was manufactured from core of Yamaha Aerox YQ50 radiator and tested under conditions of adsorption air conditioning cycle.The experimentally found UA and maximal power transferred from sorbent to water for the tested Hex were very close to theoretically values of the same parameters calculated according to (1,2).In [35] eight commercial radiators were considered and detailed information on their cores' geometry was comprehensively analysed.On the base of information about abovementioned Hexes' geometry UA coefficient for heat exchangers with the fixed volume were estimated.After that three small Hexes were manufactured and tested under adsorption heat storage cycle conditions.For the Hexes production the most different from each other from UA point of view commercial radiators were used.The coincidence of experimental and theoretically estimated UA values was observed again.So, there are two ways for choosing Hex the optimal from particular AHT cycle point of view among a great number of radiators produced by industry.The first possibility is direct measuring of the UA coefficient under conditions of considered AHT application.The second one is calculation of the appropriate UA coefficient on the base of Hex geometric parameters and heat transfer coefficients h1 and h2.The latest way seems much more easier, than the former one.At the same time, one should take into consideration that information about radiator's geometry and heat transfer coefficient sorbent-metal is not widly available.Indeed, in order to get data about Hexes geometry it should be dismantled.Moreover, measurement of heat transfer coefficient between metal and sorbent is complicated and time-consuming experiment.
On the other hand, it is important to remember that difference in roles (cooling of automobile engine by air/adsorption heat transformation) of commercial radiators lies in h2 coefficient nature.In case of air-to-liquid cooling of engine h2 coefficient characterizes convective heat transfer between air and metal, in the case of adsorption heat transformation the appropriate coefficient represents conductive heat transfer between sorbent and metallic support.It is evident that, the liquid side (contact with media 1) of Hex is the same for both considered roles.The question is: "Will be the Hex with highest value of UA for air-liquid engine cooling process (among a number of commercial radiators) optimal for adsorption heat transformation?".In [36] it was shown that: "Yes, testing of commercial radiators in wind tunnel can give information about their prospects for AHT".The heat exchangers considered in abovementioned article were manufactured from commercially produced radiators.That is why all geometric parameters of these Hexes differed from each other.
The purpose of this work is investigation a series of model Hexes with same geometric parameters except for the fin height by means of wind tunnel.Such fixing other parameters gives a possibility for analysis of fin's height impact on global heat transfer coefficient.Moreover, the possibility of predicting the global heat transfer coefficient when scaling the heat exchanger will be considered.

Heat exchangers
When manufacturing model heat exchangers with the same core volume (Hexes 1M-3M, Table 2), but with a different number of channels, copper was used as it is a highly heat-conducting and easily processed material (λCu = 400 W/(m٠K)).Additionally, heat exchanger 4M with geometry similar to that of Hex 3M, but twice its volume, was manufactured in order to check the correctness of the expression (1) applicability at scaling.When changing the number of channels, the height of the Hex fins will change (Table 2) too.The parameters remained the same for all exchangers were: the channel wall thickness δw 500 µm, the inner height of the channel h'c 1 mm, the distance between the fins Δf 1 mm, the fin thickness δf 50 µm.For correct comparison of volumetric power, the radiators 1M-3M were manufactured with almost the same core volume V = 160 ± 3 cm 3 .It is important, because namely volumetric conductance (UA related to volume of the unit) is one of the key parameters of adsorbers in AHT applications.

Wind tunnel set up
The main part of experimental set up for measuring of heat fluxes in tested Hexes are: wind tunnel, water circuit and data recording system (Figure 3a).The tested Hexes was situated into the wind tunnel, which is a thermally insulated channel with rectangular form with ventilator, flowmeter and flow regulator.The air circulates through this tunnel.Water circuit consists of Hexes' channels, regulator of flow, thermostat, flowmeter, pipelines and valves.The temperatures of water and air streams were controlled by 8 T-type thermocouples.Five of them were situated in the outlet part of wind tunnel with distance of 5 cm (Figure 3b).For collection data of the experiment analog-to-digital converter ADAM 4018 and personal computer were used.

Experimental procedure and data evaluation.
After setting the certain values of air and water flows the measurements were carried out.The inlet water temperature was supported almost constant Tin(w) = 40.0± 0.1 o C, however air inlet temperature was not stabilized and approximately equal room temperature.All flow temperatures both water and air were recording during measurements.Among signals from five thermocouples characterized the outlet air flow, the maximal value of temperature Tout(air) was considered.
Heat flows of air and water (Q(air) and Q(w) respectively) as well as average flow Q(av) can be calculated according to the following equations: where Cp is heat capacity, ρdensity, fflow rate of air/water [373 8-3940].The temperature driving force of the process can be characterized by Logarithm mean temperature difference LMTD: Using data about LMTD and heat flow the global heat transfer coefficient UA can be calculated: For further analysis an average value UA(av) was used.

Error analysis.
Instrumental errors (air flow meter -3%, water flow meter -2%, thermocouples and ADC ±0.1K) must be taken into account to achieve the error of the final result.So, the absolute  and relative  errors for Q, LMTD and UA can be found.The cumulative relative error in UA coefficient determination is 11.7%.

Results and discussion
Typical dependences of temperatures at inlet and outlet of the radiator, and of the logarithm mean temperature difference (LMTD) on time are presented in Figure 3.This figure evidences that in ca.500 s after beginning the experiment the temperatures approach to their stationary values and then remain constant in a range of the specified uncertainty (Table 2).The same tendency is clearly reflected in the temporal behavior of heat fluxes (Figure 4a).After the experimental run begins, the fluxes rapidly alter during first ca.200 s and then slowly approach a certain value.After approximately 500 seconds, the fluxes become constant and remain stable for a further period of time.This indicates that the system achieves a stationary state.As both the heat fluxes and LMTD values reach a constant magnitude, the values of UA also become constant, as is shown in Figure 4b.The mean value of the global heat transfer coefficient UAav, averaged over the whole steady state period (ca.600 s) were taken for further evaluation.The experimental data on the dependencies of UA on air and water flow for studied heat exchangers is summarized in Figure 6.One can see that the values of UA for radiator 1M lie in a range UA= 8.5-18 W/K while UA coefficients for radiators 2M and 3M are lower UA=6-12 W/K.Also, it can be seen that the global heat transfer coefficients for all the studied heat exchangers depend slowly on the water flow.This confirms the theoretical findings that the Nusselt number in thin planar channels is independent of the water flow velocity.At the same time, it can be seen that the heat transfer coefficients are strongly dependent on the air flow rate.This is in agreement with both theoretical considerations [33] and previously observed experimental results [36].This makes it possible to find the correlations between the Nusselt and Reynolds numbers based on the experimental data obtained for the model heat exchangers considered in this paper.Indeed, for a heat exchanger with a fixed geometry, assuming that the metal-to-liquid heat transfer coefficient is constant, the global heat transfer coefficient is a function of only one parameter -the metalto-air heat transfer coefficient h2 (Figure 7).On the other hand, the air-to-metal heat transfer coefficient, for a fixed heat exchanger geometry, is a single-valued function of the Nusselt number, as is shown by equation (3).It should also be noted that for each heat exchanger of fixed geometry, the Reynolds numbers corresponding to certain air flow rates which can be calculated using the relationship: where u is velocity of air,  is hydraulic diameter,  -is viscosity of air.Thus, the experimentally measured global heat transfer coefficients, the corresponding air-to-metal heat transfer coefficients and the corresponding Nusselt numbers can be unambiguously correlated with the Reynolds numbers for air flow.The difficulty of this procedure is that since the direct dependence between the experimentally measured global heat transfer coefficient UA and the air-to-metal heat transfer coefficient is a hyperbolic function, it is hardly to find the inverse dependence in analytical form.Therefore, the problem of finding the inverse function was solved graphically as shown in Figure 7.In such a way the one-to-one coincidences between experimental values of UA, h2 and Nu and appropriate Re values were established for all the studied radiators.These coincidences are presented in Figure 8 in double logarithmic coordinates.This kind of data presentation allows obtaining the empiric coefficients a and b of Nu-Re correlations (equation 5).The obtained coefficients are summarized in Table 3 together with the ranges of Re, Nu and appropriate h2 values where the abovementioned correlations are valid.Data presented in Table 3 demonstrates that Nu-Re correlations found for the tested heat exchangers are very close to a classical formula with a = 0.64 and b = 0.5.However, some difference between the obtained coefficients and the theoretically predicted ones may indicate, for example, the non-ideal shape of the fins in the real heat exchangers that were tested in this paper.In order to demonstrate this possibility, the double-sized radiator was manufactured with the fins and channels geometry identical to the geometry of fins and channels for heat exchanger 3M.This up-sized radiator is designated in Table 1 as 4M and is two times longer that radiator 3M whereas all the other geometric parameters remained the unchanged.The surface of 4M radiator also is two times higher than of 3M radiator (Table 1).This enlarged radiator was tested in a way how 1M-3M heat exchangers were tested before.The raw results of the direct UA measurements for this enlarged radiator (green symbols) are presented in Figure 10 and compared with data for the single-sized heat exchanger (red symbols).The red 3D surface interpolates the UA data for small heat exchanger.Multiplying the UA values of this surface by 2 gives the upscaled 3D surface (green).Figure 10a clearly evidences that a simple proportional up-scale is not a proper way for calculating the parameters of the radiator while its external sizes are scaled.Indeed, the experimental points measured for 4M radiator visibly differ from this tentative up-scaled surface.What is a true way for calculating the dependences of UA on the velocity of fluids flow for the radiator with known geometry of the core, while the dimensions of the heat exchanger are variable?Figure 10 b answers this question and gives the way for correct scaling procedure.Green 3D surface in Figure 10 b is constructed in a follows way: 1) for the selected flow rate and the corresponding Re number, using the relation (12), the appropriate number Nu is calculated; b) value of heat transfer coefficient h2 is calculated from Nu number using (3); c) the global heat transfer coefficient UA is assessed for radiator with known dimensions with use of (1-3, 5).And it is wort note that namely correlations found for small radiator 3M were used for finding the appropriate h2 and UA values (green plane in Figure 10b) for the up-scaled radiator 4M.Remember, that the core geometry for these radiators is identical (Table 1).Figure 10b reveals that the results of theoretical scaling (green 3D surface) and the results of direct UA measurements (green symbols) coincide with a reasonable accuracy.This confirms the correctness of the proposed method for calculating heat transfer rate at scaling up the radiators using the obtained Re-Nu correlations.
Another advantage of the proposed methodology of radiators testing in wind tunnel and finding Re-Nu correlations in the context of determining AdHex efficiency is that this method provides a simple and time-efficient way to experimentally measure UA for heat exchangers with a given range of sorbentto-metal heat transfer coefficient h2.Thus, it is not necessary to load adsorbent granules into the heat exchanger and perform measurements under the realistic conditions of a given adsorption cycle.It is only necessary to find the desired flow rate corresponding to the appropriate Re and Nu and corresponding to the exact h2 (sorbent-metal) value that corresponds to a specified adsorptive cycle and adsorbent.Then, at a given flow rate, UA measurements should be carried out in a wind tunnel.In other words, the change in flow rate in the wind tunnel is the analogue of the change in h2 for the adsorbent.Thus, to find the best heat exchanger for a given adsorption cycle, relatively simple and time-efficient experiments can be performed if the properties of the adsorbent are known.For example, in [35] it was shown that heat transfer coefficient h2 for desorption stage of daily heat storage cycle is as high as h2 = 125 W/(m 2 •K).Which radiator will be better for this cycle and how much higher will be its global heat transfer coefficient UA?And how to confirm it experimentally?Table 3 shows, that this value of heat transfer coefficient h2 corresponds to Re number of about Re ~ 600.And at air flow that corresponds to this value of Re radiator 3M demonstrates experimental value of UA = 10 W/K, whereas radiator 1M shows UA = 17 W/K (Figure 9).Considering that one experimental run takes no more than 30 min, it is safe to say that after 1 hour it is possible to obtain direct experimental confirmation that reducing the fin height from 2 cm to 0.5 cm allows to achieve an increase in the global heat transfer coefficient by more than 50%.This demonstrated the perspectives and time-saving potential of the proposed methodology for testing FFT heat exchangers, evaluating their conductance, finding the optimal radiator to be used as AdHex for a given application in adsorption heat transformers.The predictive ability of the proposed methodology for heat exchanger scaling is also demonstrated.

Conclusions
The paper considers a series of model water-air heat exchangers with FFT geometry.The fin height in the Hexes was varied directionally (from 0.5 to 2.0 cm) while the other geometric parameters were kept constant.The heat exchangers were investigated using a wind tunnel, the dependences of the global heat transfer coefficient UA on the flow rate of water (from 10 to 35 L/s) and air (from 0 to 0.05 L/s) were determined.Based on the data obtained, the correlations of the Nusselt numbers with the Reynolds numbers were determined, which allow scaling the heat exchangers of the considered geometry.The possibility and correctness of the scaling procedure were confirmed experimentally.A time saving procedure for choosing the optimal FFT heat exchanger for a given cycle of adsorption heat transformation is proposed and discussed.

Figure 3 .
Figure 3. (a) The wind tunnel rig scheme; (b) outlet part of wind tunnel channel.

9 Table 3 .Figure 9
Figure9demonstrates that the correlations obtained allow fitting the experimental data using the equations (1-3, 5) with a reasonable accuracy.It opens the possibility of estimation the performance of heat exchangers at their scaling up.

Table 1 .
Geometric parameters of the tested Hexes.

Table 2 .
The error analysis.