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Optimisation of egg-crate type evaporators in domestic refrigerators
Applied Thermal Engineering 21 (2001) 751±770
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Optimisation of egg-crate type evaporators in domestic refrigerators P.K. Bansal *, T. Wich, M.W. Browne Department of Mechanical Engineering, The University of Auckland, Private Bag-92019, Auckland, New Zealand Received 1 December 1999; accepted 15 June 2000
Abstract This paper presents a parametric heat transfer study of the new Ôegg-crateÕ type evaporators in domestic refrigerators/freezers by varying ®n density, ®n height and ®n thickness, using a steady state simulation model of Bansal et al. [1]. Simulations were carried out over a wide range of ®n parameters with ®nning factors ranging from 3.5 to 9. The simulation model predicted the evaporator capacity, UA and temperature drop of air across the evaporator for the varying parameters. A geometrically improved evaporator is proposed having an optimum heat transfer performance, in terms of maximum evaporator capacity per unit weight. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Evaporator; Heat transfer; Fins; Simulation; Model; NTU-e; Optimization; Forced ¯ow; Parametric study
1. Introduction The eect of ®ns on the heat exchanger performance has been studied experimentally by numerous researchers, but the results are often dierent and contrary. Turaga et al. [2] experimentally determined the eects of the coil geometry and ¯uid ¯ow parameters on the performance of the plate ®n-and-tube heat exchangers. The heat exchangers varied in the ®n density and the tube rows. Other parameters, such as the outside tube diameter, transversal and longitudinal tube spacing were constant. Tests were done under dry and wet surface conditions in a closed loop duct with chilled water as the working ¯uid in the evaporator tubes. Increasing ®n density increased the total surface area that led to an improvement in the total heat transfer, but this resulted in the decrease of the j-Colburn factor. Chen and Ren [3] tested two-row plate ®n-and-tube heat *
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1359-4311/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 0 ) 0 0 0 7 7 - 6
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Nomenclature
A d fc fo fw Hf hc hi ho j k m Nu Pr Q r R Re Rc
area, m2 diameter, m evaporator capacity factor optimisation factor evaporator weight factor ®n height, cm contact conductance, W/m2 K inside heat transfer coecient, W/m2 K outside heat transfer coecient, W/m2 K Colburn factor thermal heat conductivity, W/m K mass ¯ow rate, kg/s Nusselt number Prandtl number heat transfer rate, W radius, m ®nning factor Reynolds number thermal contact resistance between ®n and tube, m2 K/W
Rw Sf UA mmax w Greek b d g l q
thermal wall resistance, m2 K/W ®n density, 1/m overall heat transfer conductance, W/K air velocity at minimum ¯ow area, m/s weight, kg letters parameter thickness, mm eciency dynamic viscosity, kg/ms density, kg/m3
Suxes f ®n h hydraulic HTF heat transfer ¯uid i inside max maximum min minimum o outside opt optimisation t tube
exchangers with varying ®n density. They observed a decreasing airside heat transfer with increasing ®n density. Vortices were formed around the tubes due to air velocity. Increasing ®n density reduced the development of these vortices. Seshimo and Fujii [4] experimentally investigated the airside heat transfer of cross ¯ow heat exchangers with varying geometric parameters. They observed increased airside heat transfer coecients with increasing ®n density, while with increasing ®n heights, the airside heat transfer coecient decreased. The heat transfer performance improved with increasing air¯ow rates. Kayansayan [5] investigated the eects of the outer surface geometries on the performance of 10 geometrically distinct plate ®n-and-tube heat exchangers in cross-¯ow con®guration with tube rows in staggered arrangement to each other. All geometric properties were integrated in the ®nning factor R, de®ned as the ratio of the total outside surface area to the outside tube surface area, which varied from 11 to 23. Hot water was used as the working ¯uid inside the tubes and the inlet temperatures of the heat transfer ¯uid (air) varied from 7°C to 19.5°C. Correlations of the j-Colburn factor showed a strong dependency of the sensible heat transfer coecient on the ®nning factor. A major parameter in the ®nning factor was the ®n density. The tests showed that j-factors decreased with increasing ®n density; this eect became prominent at high Reynolds numbers. Horuz et al. [6] tested a four-row plate ®n-and-tube heat exchanger with staggered tube rows. The evaporator worked under dehumidifying conditions
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with R-134a as the working ¯uid and moist air as the heat transfer ¯uid. It was found that the heat transfer increased with increasing ®n density. Less ®n spacing decreased the overall heat transfer coecient U, but the in¯uence of the increasing total outside surface area (A) led to an improvement in the total UA-value and hence a better heat transfer. In this paper, the in¯uence of ®n parameters (density, thickness and height) on the heat transfer performance of new Ôegg-crateÕ type evaporators has been investigated analytically. For this, a computer simulation model developed by Bansal et al. [1] has been used to predict the steady state heat transfer performance of this evaporator. The objective of this study was speci®cally to optimize the geometry of the current evaporators to achieve maximum evaporator capacity per unit material weight. This oers two advantages ± improved heat transfer performance of the evaporator and reduced material costs to the manufacturer. 2. Details of the evaporator ÔEgg-crateÕ type evaporators are new forced ¯ow evaporators developed for domestic refrigerators/freezers that replace the older roll bond type natural convection evaporators. They are made of aluminum with continuous rectangular ®ns while the ®n layers are press ®tted onto the serpentine bent evaporator tube (Fig. 1). These evaporators work in counter/parallel-cross ¯ow con®guration with refrigerant R-134a as the working ¯uid inside the tubes. The evaporators are manufactured in various sizes (Fig. 2) and with dierent front staging at the air entrance area. Staging is a method to improve the air¯ow through the evaporator coils and decrease the airside pressure drop by varying the ®n space at the leading edge of the frontal area (front staging). This is achieved by widening the ®n space or cutting the ®ns (Fig. 3), which generally decreases the total heat transfer area. The evaporator with less staging is referred as the base model and the one with more staging as the modi®ed or staged model. For the base model, the ®n space in the last tube row is twice compared with the non-staged tube rows after it. This eect is achieved by removing every second ®n in that tube row. The modi®ed model has a wider ®n staging in the last tube row Np (four times bigger ®n space than the non-staged tube rows). The staging scheme of both the evaporators is shown in Fig. 3 while the geometrical dimensions of both the evaporators are given in Table 1. In this study, the optimization of a modi®ed evaporator having three ®n layers and eight tube rows per layer was performed by varying its ®n density (spacing), ®n height and ®n thickness. 3. Description of the model The model simulates the steady state heat transfer performance of the evaporator following an elemental NTU-e approach. It discretises the evaporator into ®n and tube elements of equal geometrical size (see shape of an element in the right bottom corner of Fig. 1). The size of one ®n and tube element is de®ned as the product of ®n spacing, ®n height and ®n depth (equals to longitudinal tube pitch). Referring to Fig. 1, the number of elements depend on the geometrical dimensions of the evaporator e.g. number of tube layers (Nl ), tube rows per layer (Np ) and ®ns per tube row (Nf ). For each element, an energy balance is carried out for both the ¯uids (air and refrigerant) with an iterative scheme until the steady state outlet conditions are found from the
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Fig. 1. Schematics of the egg-crate evaporator (with 3 tube-and-®n layers having 8 tube rows per layer).
inputs. The model accounts for the refrigerant pressure drops inside the tubes, which leads to a temperature change of the refrigerant along the tube during evaporation. The model assumes that the circulating air is dry. Details of the model are given by Bansal et al. [1] and Wich [7]. The main inputs to the model include
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Fig. 2. Dierent sizes of the base model of the Ôegg-crateÕ type evaporator.
Fig. 3. Scheme of the ®n staging for the entering air at the ®rst layer of the base and the modi®ed evaporator.
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Table 1 Geometrical dimensions of the tested egg-crate type evaporators Evaporator dimensions
Base model 3 8 72
Modi®ed model 3 8 72
Number of layers, Nl Number of tube passes, Np Number of ®ns per pass, Nf Fin layer width, Wl (m) Fin layer depth, D (m) Fin height, Hf (m) Fin thickness, df (m) Fin spacing, Wf (m) Transversal tube pitch, St (m) Longitudinal tube pitch, Sl (m) Overall tube length, Lt (m) Tube outside diameter, do (m) Tube inside diameter, di (m) Hydraulic diameter, dh (m) Frontal area of evaporator, Afr (m2 ) Tube outside surface area, At;o (m2 ) Total evaporator outside surface area, Ao (m2 ) Finning factor, R
3 8 72 0.45 0.235 0.02 3 10 4 6:25 10 3 0.022 0.03 12.094 8 10 3 6:4 10 3 6:7 10 3 0.0297 0.316 2.306 7.30
3 8 72 0.45 0.235 0.02 3 10 4 6:25 10 3 0.022 0.03 12.094 8 10 3 6:4 10 3 7:3 10 3 0.0297 0.316 2.118 6.70
· the inlet temperatures and mass ¯ow rates of both the heat transfer ¯uid and the refrigerant, · the refrigerant quality at evaporator inlet and pressure drop of the refrigerant across the evaporator, and · geometrical dimensions of the evaporator (e.g. tube diameter, ®n parameters, number of ®n layers, number of tube rows) and thermal conductivity of the material. Main outputs of the model include the evaporator capacity, the UA, the outlet temperatures of both the ¯uids and geometrical dimensions (e.g. hydraulic diameter, total heat transfer surface area, minimum air¯ow area) of the evaporator. The model can account for both the two-phase and superheated regions. Following Bansal and van Neuren [8], the outside heat transfer coecient is expressed in terms of the dimensionless j-Colburn factor. For this type of evaporator, it is given as j 0:669Reo 0:419 R
0:407
;
1
where the application range of the above correlation is 350 < Re < 1850 and 7 < Nu < 20. The airside Reynolds number is given as mmax qHTF dh m_ HTF dh ; 2 Reh lHTF Amin lHTF and the ®nning factor (R) as Ao ; 3 R At;o where do is used as the typical geometrical length as per the recommendations of Kays and London [9].
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The airside heat transfer coecient can be expressed as: ho
Nuo kHTF 0:669Re0:581 R h dh
0:407
1=3
Pr
kHTF : dh
4
Therefore, the overall heat transfer conductance (UA) of the evaporator can be expressed as 1 1 1 ; 5 Rc Rw UA ho At;o gf Af hi At;i where Rc is the contact resistance between outside tube surface and ®n collar and Rw is the heat conductance resistance of the tube wall. Following Nho and Yovanovich [10], the contact resistance between ®n collar and tube is given by Rc
1 : hc Ac
6
The heat conductance hc is given as h i hc exp 8:6379 0:1844 dIfpi=do 0:75 df fpi1:25 ;
7
where d is Vickers indentation for 25 g load [lm] and fpi is ®ns per unit length [1/m]. The tube expansion interference (I ) is an industrial term and is given by Ernest et al. [11] as I do 2dt
di ;
8
where dt is the tube wall thickness. The rectangular ®n design of the egg-crate type evaporator diers from common ®n-and-tube heat exchangers and no ®n eciency correlation could be found in the literature for this special ®n type. Therefore, the ®n eciency is calculated with the sector method for annular ®ns as proposed by Mills [12], using Bessel functions of the ®rst and second kind. The correlation is given in the appendix. 4. Experimental validation of the model Bansal et al. [1] performed the validation of the simulation model with the experimental data for both the base and the modi®ed evaporators at three mass ¯ow rates of air. The computer program converged for all the data points and the results were satisfactory. The simulated evaporator capacities and the UA values agreed with the experimental values to within 5%.
5. Results and discussion The parametric study of the heat transfer performance of the Ôegg-crateÕ evaporators with varying geometry was investigated in view of the following objectives: (i) improved evaporator capacity and (ii) less cost (i.e. less material) in achieving the same or higher capacity compared to the present design.
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At ®rst, the in¯uence of the ®n parameters (®n density, ®n height, ®n thickness) was assessed individually. Then, the three ®n parameters were varied together to ®nd the most ecient evaporator design (in view of (i) and (ii) above). Other dimensions of the present staged evaporator design (i.e. tube diameter, longitudinal tube pitch, number of tube rows, number of ®n layers, ®n layer width etc.) were unchanged. Other simulation parameters included 1. air mass ¯ux varied from 0.51 to 1.01 kg/m2 s (the values of 0.51, 0.67, 0.84 and 1.01 kg/m2 s correspond to air velocities of 0.4, 0.53, 0.66 and 0.8 m/s, respectively), 2. air inlet temperature 13°C, refrigerant inlet temperature 20°C 3. Refrigerant mass flow rate 1:5 10 3 kg/s 4. Refrigerant inlet quality 0:2. The model was compared with 51 experimental data points of two Ôegg-crateÕ type evaporators ± one base and the other staged evaporator. However, the parametric variation has been studied only for the staged evaporator. Following relationships may be deduced from Eqs. (4) and (2): 1=R0:407 ; and Reh 1=Amin : h0 1=dh Re0:581 h It may be inferred from above relationships the outside heat transfer coecient (ho ) is proportional to the Reynolds number (Re) but inversely proportional to both the hydraulic diameter (dh ) and the ®nning factor (R), while Re is inversely proportional to the minimum ¯ow area (Amin ). Therefore, the in¯uence of the ®n parameters (®n density, ®n height, ®n thickness) was assessed individually ®rst for the improved evaporator design. (a) Eect of ®n density: The present evaporator design has a ®n density of 160 and ®n space of 6.25 mm. In the simulation, ®n density was varied in the range of 40 to 220 (i.e. ®n spacing from 5 to 50 mm). Since dh decreases with ®n density, increasing ®n density leads to decreased Amin and hence increased Reynolds numbers. This, therefore, results in marginal improvement in ho but both the UA and the evaporator capacity improve appreciably as shown respectively in Figs. 4±6. It is evident that low hydraulic diameters and high air mass ¯ux rates can improve the heat transfer rates signi®cantly. This result agrees with the experimental results of Turaga et al. [2], Seshimo and Fujii [4] and Horuz et al. [6], who have also observed increased heat exchanger performance with increasing ®n density and increasing air¯ow rates. These results also agreed with Kayansayan [5] who found that the j-Colburn factor decreased with increasing ®nning factor. However, these results disagreed with Chen and Ren [3] who noted that ®n density is ineective as the ratio of ®n space to tube outside diameter increased from 0.57 upwards. (b) Eect of ®n height: The present design has a ®n height of 2 cm and 2.2 cm transversal tube pitch. Therefore, the ®n height was varied in the range from 1 to 10 cm. Increasing the ®n height results in increased hydraulic diameter, increased minimum ¯ow area and increased ®nning factor. As the minimum ¯ow area increases, the Reynolds number decreases and consequently, the outside heat transfer coecient decreases. The corresponding results of ho , UA and the evaporator capacity can be viewed in Figs. 7±9, respectively. Higher air mass ¯ux always results in the improved heat transfer due to more turbulence. Fig. 10 shows that increasing the total heat transfer surface area Ao even by a factor 6 (as a result of increase in the ®n height) would not improve the UA appreciably. It has been shown by Bansal et al. [1] that the hydraulic diameter is an important
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Fig. 4. Variation of the outside heat transfer coecient ho with ®n density for varying air¯ow rates.
Fig. 5. Variation of UA with ®n density at various air¯ow rates.
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Fig. 6. Variation of evaporator capacity with ®n density at various air¯ow rates.
Fig. 7. Variation of outside heat transfer coecient with ®n height at various air ¯ow rates.
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Fig. 8. Variation of UA with ®n height at various air¯ow rates.
Fig. 9. Variation of evaporator capacity with ®n height at various air¯ow rates.
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Fig. 10. Variation of UA with evaporator outside surface area at various air¯ow rates.
parameter in the heat transfer behaviour of the evaporator. ho , UA and evaporator capacity decrease with increasing hydraulic diameter (i.e. with increasing ®n height) because of the worsening air¯ow conditions due to reduced Reynolds numbers. Shorter ®ns are better for improved heat transfer but they lead to increased airside pressure drop. These results agree with the study of Seshimo and Fujii [4], whose experimental results indicate that the outside heat transfer coecient increases with shorter ®ns. (c) Eect of ®n thickness: The present design uses 0.3 mm thick ®ns. Thus, the ®n thickness was varied from 0.1 to 3 mm. The general trend shows that thicker ®ns lead to increased outside heat transfer (Fig. 11), increasing both the UA-value (Fig. 12) and the evaporator capacity (Fig. 13) due to the reduced outside heat transfer resistance. Since all other parameters remained unchanged except the ®n thickness (so ®n space was 6.25 mm), the minimum air¯ow area (Amin ) decreased with thicker ®ns and thus the Reynolds number increased by a factor 4. Fig. 14 shows the variation of ®n eciency with ®n thickness. It can be observed that the eciency does not improve appreciably after ®n thickness of 0.5 mm, although higher air mass ¯ux improves the ®n eciency slightly. It may be inferred here that the present design of 0.3 mm is nearly an optimum in terms of the ratio of high ®n eciency and low material usage. The outside heat transfer coecient ho , the UA and the evaporator capacity decrease with larger hydraulic diameters (i.e. thinner ®ns) but improve with higher air mass ¯ux rates. There was no study in the open literature, where the in¯uence of the ®n thickness was investigated but the simulation results agreed with the experimental results of Wich [7]. In these test data, the evaporator capacity decreased with thinner ®ns (from 0.3 to 0.2 mm) for constant ®n space of 6.25 mm.
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Fig. 11. Variation of outside heat transfer coecient with ®n thickness at various air¯ow rates.
Fig. 12. Variation of UA with ®n thickness at various air¯ow rates.
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Fig. 13. Variation of evaporator capacity with ®n thickness at various air¯ow rates.
Fig. 14. Variation of ®n eciency with ®n thickness at various air¯ow rates.
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Now the three parameters (®n density, ®n height, ®n thickness) were varied simultaneously with the intention of maximizing the evaporator capacity but using less material (i.e. cost savings to the manufacturer). Following ®n parameters were used to maximise the evaporator performance: · Fin density (Sf ) varying from 160 to 220 ®ns/m · Fin height varying (Hf ) from 1.4 to 2 cm · Fin thickness (df ) varying from 0.2 to 0.7 mm The simulation model predicted the evaporator capacity and UA of the present design (three ®n layers with 8 tube rows, Sf 160/m, Hf 2 cm, df 0:3 mm) 153.5 W and 33.8 W/K respectively. 5.1. Evaporator with maximum capacity (Case I) The results of the parametric study revealed that the highest evaporator capacity corresponded to the highest ®n density (220), the smallest ®n height (1.4 cm) and the thickest ®ns (0.7 mm). In this case, the capacity increased by 14.5% to 178.0 W (compared with 153.5 W of the present design) but this also required 70.5% additional material weight. These results are shown in Fig. 15
Fig. 15. Comparison of evaporator capacity and weight for the present design of egg-crate evaporator (3 tube layers, 8 tube rows) with Cases I (maximum capacity but more weight) and II (optimized capacity per unit weight).
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where the left two adjacent columns represent the capacity and the weight of the present design (both shown as 100%) so that the corresponding improved design performance can be compared with this (as the base case). The middle columns of the ®gure represent the results for this case where the comparative capacity has increased to 114.5% while the evaporator weight has also increased to 170.5%. This obviously does not conform to the interests of the manufacturer where the material savings are as important as the increased evaporator capacity for the process to be economically viable. 5.2. Evaporator with maximum capacity per unit weight (Case II) To ®nd a more economical solution for an improved evaporator design, a new parameter was introduced to yield a design with maximum evaporator capacity per unit weight, called the optimization factor fo . This is de®ned by Eq. (9) as the ratio of the evaporator capacity factor ( fc ) and the evaporator weight factor ( fw ). The evaporator capacity factor fc was de®ned as the ratio of evaporator capacities of the optimized geometry and the present design ( Q100% ). The evaporator weight factor fw was the ratio of the material weights of the optimized design and the present design ( w100% ). fo
fc Q_ opt =Q_ 100% : fw wopt =w100%
9
The evaporator capacity factor should be P 1 (or P 100%), which represents either increased or equal capacity for the geometrically optimized evaporator. The evaporator weight factor should be 6 1 (or 6 100%) to achieve material savings. This means that a geometrically improved evaporator should yield a factor fo P 1. The highest optimization factor fo was found to be 1.5 for an evaporator with 0.2 mm thick ®ns, a density of 160 ®ns/m and ®n height of 1.4 cm. For this, the evaporator capacity increased by 4.7% (to 162.7 W) and the corresponding material weight reduced to 72% of the present evaporator. The capacity and the weight information of this optimized evaporator design is shown in column 3 of Fig. 15. The geometrical dimensions of the optimized and present evaporators are given in Table 2. Fig. 16 reproduces the airside j-Colburn and friction factors as a function of the Reynolds number for the present and the optimized evaporator design. Both the j- and f-factors for the optimized evaporator are higher than the present design. For the same air mass ¯ux rates, the simulated airside heat transfer coecient ho increases by a factor of 2 but the corresponding air
Table 2 Geometrical dimensions for the optimised and the present evaporator Evaporator (3 8 72)
dh (mm)
Amin (m2 )
Ao (m2 )
Finning factor
Optimized Present design
4.6 7.3
0.007571 0.015017
1.76 2.12
5.6 6.7
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Fig. 16. Comparison of j- and f-factors for the optimized (Case II) and present evaporators vs. Reynolds number.
pressure drop also increases by a factor of 6 (compared with the present design). This design obviously oers some potential bene®ts to the manufacturers.
6. Conclusions The hydraulic diameter, which is related to the minimum air¯ow area, has a signi®cant in¯uence on the heat transfer performance of the evaporator. For the same air¯ow rates, the smaller the minimum air¯ow area, the higher the airside Reynolds numbers. This results in increased Nusselt numbers due to the turbulent ¯ow conditions and hence better heat transfer performance. Consequently, both the UA and evaporator capacities increase. With increasing ®n heights, the UA decreases (despite increasing outside surface area Ao ) due to the worsening air¯ow conditions (laminar region) with increasing hydraulic diameter and minimum ¯ow area. It shows that an increase in Ao does not necessary lead to an improved heat transfer. The parametric study for an optimum design of the evaporator in terms of maximum capacity per unit weight, revealed two conclusions: (i) thinner (<0.3 mm) and shorter (<2 cm) ®ns yielded maximum heat transfer performance of the evaporator and (ii) higher ®n density increased the evaporator capacity but simultaneously also increased the material weight (i.e. additional cost to the manufacturer). An optimum solution with a combination of ®n density of 160 ®ns/m, ®n height of 1.4 cm and ®n thickness of 0.2 mm, yielded increased evaporator capacity by about 4.7% but with a consequent 72% weight of the material. The j-Colburn and friction factors for the improved new design are
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higher than in the present design. The outside heat transfer coecient is about two times higher than the present design while the UA also improved by 18%. Acknowledgements The authors are thankful to Fisher & Paykel Ltd. for sponsoring the project and providing all necessary support. Appendix A A.1. Fin eciency correlation
The rectangular ®n design of the Ôegg-crateÕ type evaporator (Fig. 17) diers from the common ®n-and-tube heat exchangers and no correlations could be found in the open literature for this type of ®n. Therefore, a correlation proposed by Mills [12] for an annular ®n using Bessel functions of the ®rst and second kind was used here: gf
2r1 =b K1 br1 I1 br2 I1 br1 K1 br2 ; r22 r12 K0 br1 I1 br2 I0 br1 K1 br2
A:1
where K and I are the Bessel functions, r1 is the tube outside radius and r2 is the equal radius of the ®n area (Fig. 17). The ®n parameter b is given as b
ho : kf df
A:2
Fig. 17. Scheme of the ®n dimensions.
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The equal radius is derived from the area of one ®n: r Af : requal p
769
A:3
The surface area of one ®n can be calculated as Af Sl Hf
Wc Hc Wf Sl :
A:4
A.2. Correlation for the hydraulic diameter
The hydraulic diameter for ¯ow inside a non-circular duct is given by dh 4Ac ; L Pw
A:5
where Ac is the air ¯ow cross-sectional area (m2 ), Pw is the wetted perimeter (m) and L is the eective ¯ow length (m). Accounting for the characteristic heat exchanger geometry, the hydraulic diameter may be de®ned as dh 4rL
Afr ; Ao
A:6
where Afr is the frontal area (m2 ) and Ao is the evaporator outside surface area (m2 ). Following Idem et al. (1990) [13], the parameter r is given as ``the ratio of minimum heat exchanger face area per unit length to the frontal area per unit length'' and can be arranged for the Ôegg-crateÕ type evaporator as: r
Hf
do
Hf Wf Hf
do df Sf
:
A:7
The eective ¯ow length of the air through the evaporator and is given as L Np Sl :
A:8
The frontal area is the entrance surface area for the air¯ow and is given as Afr Wl St Nl ;
A:9
where the ®n layer width Wl is Wl
Nf : Sf
A:10
The ®nned area per unit length is given as the product of the surface area of one ®n and the ®n density Sf : Af;u 2Sl Hf
Wc Hc 2Wf Sl Sf ;
A:11
where factor 2 accounts for heat exchange on both sides of the ®n. The total outside area per unit length consists of the ®nned area and the tube outside area minus the tube area, which is bonded with the ®n:
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Ao;u Af;u p 1
df Sf do :
A:12
The total evaporator outside surface area is then given as Ao Ao;u Wl Nl Np
Aloss; staging ;
A:13
where Aloss; staging is the reduced area at the leading edge due to staging and is given by For the base evaporator Aloss;
staging
1 Nf Af;u Wl Nl 2Wf Sl Nl : 2 2
For the staged evaporator 3 1 Nf Aloss; staging Af;u Wl Nl 2Wf Sl Nl : 4 2 4
A:14
A:15
The minimum ¯ow area of the Ôegg-crateÕ type evaporator is given by Amin bWf Hf
do
df Wf Hf
do cNl Wl Sf ;
A:16
where all the parameters are given in Table 1.
References [1] P.K. Bansal, T. Wich, J. Chen, M. Browne, A simulation model of new egg-crate type forced ¯ow evaporators in domestic refrigerators and freezers, ASHRAE Transactions, submitted for publication. [2] M. Turanga, N. Tanaka, M. Takeshita, Performance of direct expansion plate ®nned tube coils for air cooling and dehumidi®cation, Int. J. Refrig. 11 (3) (1998) 78±86. [3] Z.Q. Chen, J.X. Ren, Eect of ®n spacing on the heat transfer and pressure drop of a two-row plate ®ne and tube heat exchanger, Int. J. Refrig. 11 (1988) 356±360. [4] Y. Seshimo, M. Fujii, An experimental study on the performance of plate ®n and tube heat exchangers at low Reynolds numbers, ASME/JSME Ther. Engng. Proc. 4 (1991) 449±454. [5] N. Kayansayan, Heat transfer characterization of plate ®n-tube heat exchangers, Int. J. Refrig. 17 (1) (1994) 49±57. [6] I. Horuz, E. Kurem, R. Yamankaradeniz, Experimental and theoretical performance analysis of air-cooled plate®nned-tube evaporators, Int. Comm. Heat and Mass Transfer 25 (6) (1998) 787±798. [7] T. Wich, An NTU-eectiveness model for new forced ¯ow evaporators in domestic refrigerators and parametric study of the evaporator geometry, Diplomarbeit thesis, The University of Auckland, New Zealand, 1999. [8] P.K. Bansal, O. van Neuren, An experimental study on new egg-crate type evaporators in domestic refrigerators and freezers, ASHRAE Trans. 104 (2) (1998). [9] W.M. Kays, A.L. London, Compact Heat Exchangers, third ed., Mc Graw-Hill, New York, 1984. [10] K.M. Nho, M.M. Yovanovich, Eect of oxide layers on measured and theoretical contact conductances in ®nnedtube heat exchangers, in: R.K. Shah, A.D. Kraus, D. Metzger (Eds.), Compact Heat Exchangers, A Festschrift for A.L. London, Hemisphere Publishing Corporation, New York, 1990, pp. 397±420. [11] T.L. Ernest, J.W. Sheeld, H.J. Sauer, Finned tube contact conductance: characterizing the integrity of the mechanical bond, ASHRAE Trans. 91 (2a) (1985) 85±95. [12] A.F. Mills, Heat Transfer, Richard D. Irwin, USA, 1992. [13] S.A. Idem, A.M. Jacobi, V.W. Goldschmidt, Heat transfer characterisation of a ®nned tube heat exchanger (with and without condensation), J. Heat Transfer 112 (1990) 64±79.
www.elsevier.com/locate/apthermeng
Optimisation of egg-crate type evaporators in domestic refrigerators P.K. Bansal *, T. Wich, M.W. Browne Department of Mechanical Engineering, The University of Auckland, Private Bag-92019, Auckland, New Zealand Received 1 December 1999; accepted 15 June 2000
Abstract This paper presents a parametric heat transfer study of the new Ôegg-crateÕ type evaporators in domestic refrigerators/freezers by varying ®n density, ®n height and ®n thickness, using a steady state simulation model of Bansal et al. [1]. Simulations were carried out over a wide range of ®n parameters with ®nning factors ranging from 3.5 to 9. The simulation model predicted the evaporator capacity, UA and temperature drop of air across the evaporator for the varying parameters. A geometrically improved evaporator is proposed having an optimum heat transfer performance, in terms of maximum evaporator capacity per unit weight. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Evaporator; Heat transfer; Fins; Simulation; Model; NTU-e; Optimization; Forced ¯ow; Parametric study
1. Introduction The eect of ®ns on the heat exchanger performance has been studied experimentally by numerous researchers, but the results are often dierent and contrary. Turaga et al. [2] experimentally determined the eects of the coil geometry and ¯uid ¯ow parameters on the performance of the plate ®n-and-tube heat exchangers. The heat exchangers varied in the ®n density and the tube rows. Other parameters, such as the outside tube diameter, transversal and longitudinal tube spacing were constant. Tests were done under dry and wet surface conditions in a closed loop duct with chilled water as the working ¯uid in the evaporator tubes. Increasing ®n density increased the total surface area that led to an improvement in the total heat transfer, but this resulted in the decrease of the j-Colburn factor. Chen and Ren [3] tested two-row plate ®n-and-tube heat *
Corresponding author. Tel.: +649-373-7599, ext.: 8146; fax: +649-373-7479. E-mail address: [email protected] (P.K. Bansal).
1359-4311/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 4 3 1 1 ( 0 0 ) 0 0 0 7 7 - 6
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Nomenclature
A d fc fo fw Hf hc hi ho j k m Nu Pr Q r R Re Rc
area, m2 diameter, m evaporator capacity factor optimisation factor evaporator weight factor ®n height, cm contact conductance, W/m2 K inside heat transfer coecient, W/m2 K outside heat transfer coecient, W/m2 K Colburn factor thermal heat conductivity, W/m K mass ¯ow rate, kg/s Nusselt number Prandtl number heat transfer rate, W radius, m ®nning factor Reynolds number thermal contact resistance between ®n and tube, m2 K/W
Rw Sf UA mmax w Greek b d g l q
thermal wall resistance, m2 K/W ®n density, 1/m overall heat transfer conductance, W/K air velocity at minimum ¯ow area, m/s weight, kg letters parameter thickness, mm eciency dynamic viscosity, kg/ms density, kg/m3
Suxes f ®n h hydraulic HTF heat transfer ¯uid i inside max maximum min minimum o outside opt optimisation t tube
exchangers with varying ®n density. They observed a decreasing airside heat transfer with increasing ®n density. Vortices were formed around the tubes due to air velocity. Increasing ®n density reduced the development of these vortices. Seshimo and Fujii [4] experimentally investigated the airside heat transfer of cross ¯ow heat exchangers with varying geometric parameters. They observed increased airside heat transfer coecients with increasing ®n density, while with increasing ®n heights, the airside heat transfer coecient decreased. The heat transfer performance improved with increasing air¯ow rates. Kayansayan [5] investigated the eects of the outer surface geometries on the performance of 10 geometrically distinct plate ®n-and-tube heat exchangers in cross-¯ow con®guration with tube rows in staggered arrangement to each other. All geometric properties were integrated in the ®nning factor R, de®ned as the ratio of the total outside surface area to the outside tube surface area, which varied from 11 to 23. Hot water was used as the working ¯uid inside the tubes and the inlet temperatures of the heat transfer ¯uid (air) varied from 7°C to 19.5°C. Correlations of the j-Colburn factor showed a strong dependency of the sensible heat transfer coecient on the ®nning factor. A major parameter in the ®nning factor was the ®n density. The tests showed that j-factors decreased with increasing ®n density; this eect became prominent at high Reynolds numbers. Horuz et al. [6] tested a four-row plate ®n-and-tube heat exchanger with staggered tube rows. The evaporator worked under dehumidifying conditions
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with R-134a as the working ¯uid and moist air as the heat transfer ¯uid. It was found that the heat transfer increased with increasing ®n density. Less ®n spacing decreased the overall heat transfer coecient U, but the in¯uence of the increasing total outside surface area (A) led to an improvement in the total UA-value and hence a better heat transfer. In this paper, the in¯uence of ®n parameters (density, thickness and height) on the heat transfer performance of new Ôegg-crateÕ type evaporators has been investigated analytically. For this, a computer simulation model developed by Bansal et al. [1] has been used to predict the steady state heat transfer performance of this evaporator. The objective of this study was speci®cally to optimize the geometry of the current evaporators to achieve maximum evaporator capacity per unit material weight. This oers two advantages ± improved heat transfer performance of the evaporator and reduced material costs to the manufacturer. 2. Details of the evaporator ÔEgg-crateÕ type evaporators are new forced ¯ow evaporators developed for domestic refrigerators/freezers that replace the older roll bond type natural convection evaporators. They are made of aluminum with continuous rectangular ®ns while the ®n layers are press ®tted onto the serpentine bent evaporator tube (Fig. 1). These evaporators work in counter/parallel-cross ¯ow con®guration with refrigerant R-134a as the working ¯uid inside the tubes. The evaporators are manufactured in various sizes (Fig. 2) and with dierent front staging at the air entrance area. Staging is a method to improve the air¯ow through the evaporator coils and decrease the airside pressure drop by varying the ®n space at the leading edge of the frontal area (front staging). This is achieved by widening the ®n space or cutting the ®ns (Fig. 3), which generally decreases the total heat transfer area. The evaporator with less staging is referred as the base model and the one with more staging as the modi®ed or staged model. For the base model, the ®n space in the last tube row is twice compared with the non-staged tube rows after it. This eect is achieved by removing every second ®n in that tube row. The modi®ed model has a wider ®n staging in the last tube row Np (four times bigger ®n space than the non-staged tube rows). The staging scheme of both the evaporators is shown in Fig. 3 while the geometrical dimensions of both the evaporators are given in Table 1. In this study, the optimization of a modi®ed evaporator having three ®n layers and eight tube rows per layer was performed by varying its ®n density (spacing), ®n height and ®n thickness. 3. Description of the model The model simulates the steady state heat transfer performance of the evaporator following an elemental NTU-e approach. It discretises the evaporator into ®n and tube elements of equal geometrical size (see shape of an element in the right bottom corner of Fig. 1). The size of one ®n and tube element is de®ned as the product of ®n spacing, ®n height and ®n depth (equals to longitudinal tube pitch). Referring to Fig. 1, the number of elements depend on the geometrical dimensions of the evaporator e.g. number of tube layers (Nl ), tube rows per layer (Np ) and ®ns per tube row (Nf ). For each element, an energy balance is carried out for both the ¯uids (air and refrigerant) with an iterative scheme until the steady state outlet conditions are found from the
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Fig. 1. Schematics of the egg-crate evaporator (with 3 tube-and-®n layers having 8 tube rows per layer).
inputs. The model accounts for the refrigerant pressure drops inside the tubes, which leads to a temperature change of the refrigerant along the tube during evaporation. The model assumes that the circulating air is dry. Details of the model are given by Bansal et al. [1] and Wich [7]. The main inputs to the model include
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Fig. 2. Dierent sizes of the base model of the Ôegg-crateÕ type evaporator.
Fig. 3. Scheme of the ®n staging for the entering air at the ®rst layer of the base and the modi®ed evaporator.
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Table 1 Geometrical dimensions of the tested egg-crate type evaporators Evaporator dimensions
Base model 3 8 72
Modi®ed model 3 8 72
Number of layers, Nl Number of tube passes, Np Number of ®ns per pass, Nf Fin layer width, Wl (m) Fin layer depth, D (m) Fin height, Hf (m) Fin thickness, df (m) Fin spacing, Wf (m) Transversal tube pitch, St (m) Longitudinal tube pitch, Sl (m) Overall tube length, Lt (m) Tube outside diameter, do (m) Tube inside diameter, di (m) Hydraulic diameter, dh (m) Frontal area of evaporator, Afr (m2 ) Tube outside surface area, At;o (m2 ) Total evaporator outside surface area, Ao (m2 ) Finning factor, R
3 8 72 0.45 0.235 0.02 3 10 4 6:25 10 3 0.022 0.03 12.094 8 10 3 6:4 10 3 6:7 10 3 0.0297 0.316 2.306 7.30
3 8 72 0.45 0.235 0.02 3 10 4 6:25 10 3 0.022 0.03 12.094 8 10 3 6:4 10 3 7:3 10 3 0.0297 0.316 2.118 6.70
· the inlet temperatures and mass ¯ow rates of both the heat transfer ¯uid and the refrigerant, · the refrigerant quality at evaporator inlet and pressure drop of the refrigerant across the evaporator, and · geometrical dimensions of the evaporator (e.g. tube diameter, ®n parameters, number of ®n layers, number of tube rows) and thermal conductivity of the material. Main outputs of the model include the evaporator capacity, the UA, the outlet temperatures of both the ¯uids and geometrical dimensions (e.g. hydraulic diameter, total heat transfer surface area, minimum air¯ow area) of the evaporator. The model can account for both the two-phase and superheated regions. Following Bansal and van Neuren [8], the outside heat transfer coecient is expressed in terms of the dimensionless j-Colburn factor. For this type of evaporator, it is given as j 0:669Reo 0:419 R
0:407
;
1
where the application range of the above correlation is 350 < Re < 1850 and 7 < Nu < 20. The airside Reynolds number is given as mmax qHTF dh m_ HTF dh ; 2 Reh lHTF Amin lHTF and the ®nning factor (R) as Ao ; 3 R At;o where do is used as the typical geometrical length as per the recommendations of Kays and London [9].
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The airside heat transfer coecient can be expressed as: ho
Nuo kHTF 0:669Re0:581 R h dh
0:407
1=3
Pr
kHTF : dh
4
Therefore, the overall heat transfer conductance (UA) of the evaporator can be expressed as 1 1 1 ; 5 Rc Rw UA ho At;o gf Af hi At;i where Rc is the contact resistance between outside tube surface and ®n collar and Rw is the heat conductance resistance of the tube wall. Following Nho and Yovanovich [10], the contact resistance between ®n collar and tube is given by Rc
1 : hc Ac
6
The heat conductance hc is given as h i hc exp 8:6379 0:1844 dIfpi=do 0:75 df fpi1:25 ;
7
where d is Vickers indentation for 25 g load [lm] and fpi is ®ns per unit length [1/m]. The tube expansion interference (I ) is an industrial term and is given by Ernest et al. [11] as I do 2dt
di ;
8
where dt is the tube wall thickness. The rectangular ®n design of the egg-crate type evaporator diers from common ®n-and-tube heat exchangers and no ®n eciency correlation could be found in the literature for this special ®n type. Therefore, the ®n eciency is calculated with the sector method for annular ®ns as proposed by Mills [12], using Bessel functions of the ®rst and second kind. The correlation is given in the appendix. 4. Experimental validation of the model Bansal et al. [1] performed the validation of the simulation model with the experimental data for both the base and the modi®ed evaporators at three mass ¯ow rates of air. The computer program converged for all the data points and the results were satisfactory. The simulated evaporator capacities and the UA values agreed with the experimental values to within 5%.
5. Results and discussion The parametric study of the heat transfer performance of the Ôegg-crateÕ evaporators with varying geometry was investigated in view of the following objectives: (i) improved evaporator capacity and (ii) less cost (i.e. less material) in achieving the same or higher capacity compared to the present design.
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At ®rst, the in¯uence of the ®n parameters (®n density, ®n height, ®n thickness) was assessed individually. Then, the three ®n parameters were varied together to ®nd the most ecient evaporator design (in view of (i) and (ii) above). Other dimensions of the present staged evaporator design (i.e. tube diameter, longitudinal tube pitch, number of tube rows, number of ®n layers, ®n layer width etc.) were unchanged. Other simulation parameters included 1. air mass ¯ux varied from 0.51 to 1.01 kg/m2 s (the values of 0.51, 0.67, 0.84 and 1.01 kg/m2 s correspond to air velocities of 0.4, 0.53, 0.66 and 0.8 m/s, respectively), 2. air inlet temperature 13°C, refrigerant inlet temperature 20°C 3. Refrigerant mass flow rate 1:5 10 3 kg/s 4. Refrigerant inlet quality 0:2. The model was compared with 51 experimental data points of two Ôegg-crateÕ type evaporators ± one base and the other staged evaporator. However, the parametric variation has been studied only for the staged evaporator. Following relationships may be deduced from Eqs. (4) and (2): 1=R0:407 ; and Reh 1=Amin : h0 1=dh Re0:581 h It may be inferred from above relationships the outside heat transfer coecient (ho ) is proportional to the Reynolds number (Re) but inversely proportional to both the hydraulic diameter (dh ) and the ®nning factor (R), while Re is inversely proportional to the minimum ¯ow area (Amin ). Therefore, the in¯uence of the ®n parameters (®n density, ®n height, ®n thickness) was assessed individually ®rst for the improved evaporator design. (a) Eect of ®n density: The present evaporator design has a ®n density of 160 and ®n space of 6.25 mm. In the simulation, ®n density was varied in the range of 40 to 220 (i.e. ®n spacing from 5 to 50 mm). Since dh decreases with ®n density, increasing ®n density leads to decreased Amin and hence increased Reynolds numbers. This, therefore, results in marginal improvement in ho but both the UA and the evaporator capacity improve appreciably as shown respectively in Figs. 4±6. It is evident that low hydraulic diameters and high air mass ¯ux rates can improve the heat transfer rates signi®cantly. This result agrees with the experimental results of Turaga et al. [2], Seshimo and Fujii [4] and Horuz et al. [6], who have also observed increased heat exchanger performance with increasing ®n density and increasing air¯ow rates. These results also agreed with Kayansayan [5] who found that the j-Colburn factor decreased with increasing ®nning factor. However, these results disagreed with Chen and Ren [3] who noted that ®n density is ineective as the ratio of ®n space to tube outside diameter increased from 0.57 upwards. (b) Eect of ®n height: The present design has a ®n height of 2 cm and 2.2 cm transversal tube pitch. Therefore, the ®n height was varied in the range from 1 to 10 cm. Increasing the ®n height results in increased hydraulic diameter, increased minimum ¯ow area and increased ®nning factor. As the minimum ¯ow area increases, the Reynolds number decreases and consequently, the outside heat transfer coecient decreases. The corresponding results of ho , UA and the evaporator capacity can be viewed in Figs. 7±9, respectively. Higher air mass ¯ux always results in the improved heat transfer due to more turbulence. Fig. 10 shows that increasing the total heat transfer surface area Ao even by a factor 6 (as a result of increase in the ®n height) would not improve the UA appreciably. It has been shown by Bansal et al. [1] that the hydraulic diameter is an important
P.K. Bansal et al. / Applied Thermal Engineering 21 (2001) 751±770
Fig. 4. Variation of the outside heat transfer coecient ho with ®n density for varying air¯ow rates.
Fig. 5. Variation of UA with ®n density at various air¯ow rates.
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Fig. 6. Variation of evaporator capacity with ®n density at various air¯ow rates.
Fig. 7. Variation of outside heat transfer coecient with ®n height at various air ¯ow rates.
P.K. Bansal et al. / Applied Thermal Engineering 21 (2001) 751±770
Fig. 8. Variation of UA with ®n height at various air¯ow rates.
Fig. 9. Variation of evaporator capacity with ®n height at various air¯ow rates.
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Fig. 10. Variation of UA with evaporator outside surface area at various air¯ow rates.
parameter in the heat transfer behaviour of the evaporator. ho , UA and evaporator capacity decrease with increasing hydraulic diameter (i.e. with increasing ®n height) because of the worsening air¯ow conditions due to reduced Reynolds numbers. Shorter ®ns are better for improved heat transfer but they lead to increased airside pressure drop. These results agree with the study of Seshimo and Fujii [4], whose experimental results indicate that the outside heat transfer coecient increases with shorter ®ns. (c) Eect of ®n thickness: The present design uses 0.3 mm thick ®ns. Thus, the ®n thickness was varied from 0.1 to 3 mm. The general trend shows that thicker ®ns lead to increased outside heat transfer (Fig. 11), increasing both the UA-value (Fig. 12) and the evaporator capacity (Fig. 13) due to the reduced outside heat transfer resistance. Since all other parameters remained unchanged except the ®n thickness (so ®n space was 6.25 mm), the minimum air¯ow area (Amin ) decreased with thicker ®ns and thus the Reynolds number increased by a factor 4. Fig. 14 shows the variation of ®n eciency with ®n thickness. It can be observed that the eciency does not improve appreciably after ®n thickness of 0.5 mm, although higher air mass ¯ux improves the ®n eciency slightly. It may be inferred here that the present design of 0.3 mm is nearly an optimum in terms of the ratio of high ®n eciency and low material usage. The outside heat transfer coecient ho , the UA and the evaporator capacity decrease with larger hydraulic diameters (i.e. thinner ®ns) but improve with higher air mass ¯ux rates. There was no study in the open literature, where the in¯uence of the ®n thickness was investigated but the simulation results agreed with the experimental results of Wich [7]. In these test data, the evaporator capacity decreased with thinner ®ns (from 0.3 to 0.2 mm) for constant ®n space of 6.25 mm.
P.K. Bansal et al. / Applied Thermal Engineering 21 (2001) 751±770
Fig. 11. Variation of outside heat transfer coecient with ®n thickness at various air¯ow rates.
Fig. 12. Variation of UA with ®n thickness at various air¯ow rates.
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Fig. 13. Variation of evaporator capacity with ®n thickness at various air¯ow rates.
Fig. 14. Variation of ®n eciency with ®n thickness at various air¯ow rates.
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765
Now the three parameters (®n density, ®n height, ®n thickness) were varied simultaneously with the intention of maximizing the evaporator capacity but using less material (i.e. cost savings to the manufacturer). Following ®n parameters were used to maximise the evaporator performance: · Fin density (Sf ) varying from 160 to 220 ®ns/m · Fin height varying (Hf ) from 1.4 to 2 cm · Fin thickness (df ) varying from 0.2 to 0.7 mm The simulation model predicted the evaporator capacity and UA of the present design (three ®n layers with 8 tube rows, Sf 160/m, Hf 2 cm, df 0:3 mm) 153.5 W and 33.8 W/K respectively. 5.1. Evaporator with maximum capacity (Case I) The results of the parametric study revealed that the highest evaporator capacity corresponded to the highest ®n density (220), the smallest ®n height (1.4 cm) and the thickest ®ns (0.7 mm). In this case, the capacity increased by 14.5% to 178.0 W (compared with 153.5 W of the present design) but this also required 70.5% additional material weight. These results are shown in Fig. 15
Fig. 15. Comparison of evaporator capacity and weight for the present design of egg-crate evaporator (3 tube layers, 8 tube rows) with Cases I (maximum capacity but more weight) and II (optimized capacity per unit weight).
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where the left two adjacent columns represent the capacity and the weight of the present design (both shown as 100%) so that the corresponding improved design performance can be compared with this (as the base case). The middle columns of the ®gure represent the results for this case where the comparative capacity has increased to 114.5% while the evaporator weight has also increased to 170.5%. This obviously does not conform to the interests of the manufacturer where the material savings are as important as the increased evaporator capacity for the process to be economically viable. 5.2. Evaporator with maximum capacity per unit weight (Case II) To ®nd a more economical solution for an improved evaporator design, a new parameter was introduced to yield a design with maximum evaporator capacity per unit weight, called the optimization factor fo . This is de®ned by Eq. (9) as the ratio of the evaporator capacity factor ( fc ) and the evaporator weight factor ( fw ). The evaporator capacity factor fc was de®ned as the ratio of evaporator capacities of the optimized geometry and the present design ( Q100% ). The evaporator weight factor fw was the ratio of the material weights of the optimized design and the present design ( w100% ). fo
fc Q_ opt =Q_ 100% : fw wopt =w100%
9
The evaporator capacity factor should be P 1 (or P 100%), which represents either increased or equal capacity for the geometrically optimized evaporator. The evaporator weight factor should be 6 1 (or 6 100%) to achieve material savings. This means that a geometrically improved evaporator should yield a factor fo P 1. The highest optimization factor fo was found to be 1.5 for an evaporator with 0.2 mm thick ®ns, a density of 160 ®ns/m and ®n height of 1.4 cm. For this, the evaporator capacity increased by 4.7% (to 162.7 W) and the corresponding material weight reduced to 72% of the present evaporator. The capacity and the weight information of this optimized evaporator design is shown in column 3 of Fig. 15. The geometrical dimensions of the optimized and present evaporators are given in Table 2. Fig. 16 reproduces the airside j-Colburn and friction factors as a function of the Reynolds number for the present and the optimized evaporator design. Both the j- and f-factors for the optimized evaporator are higher than the present design. For the same air mass ¯ux rates, the simulated airside heat transfer coecient ho increases by a factor of 2 but the corresponding air
Table 2 Geometrical dimensions for the optimised and the present evaporator Evaporator (3 8 72)
dh (mm)
Amin (m2 )
Ao (m2 )
Finning factor
Optimized Present design
4.6 7.3
0.007571 0.015017
1.76 2.12
5.6 6.7
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767
Fig. 16. Comparison of j- and f-factors for the optimized (Case II) and present evaporators vs. Reynolds number.
pressure drop also increases by a factor of 6 (compared with the present design). This design obviously oers some potential bene®ts to the manufacturers.
6. Conclusions The hydraulic diameter, which is related to the minimum air¯ow area, has a signi®cant in¯uence on the heat transfer performance of the evaporator. For the same air¯ow rates, the smaller the minimum air¯ow area, the higher the airside Reynolds numbers. This results in increased Nusselt numbers due to the turbulent ¯ow conditions and hence better heat transfer performance. Consequently, both the UA and evaporator capacities increase. With increasing ®n heights, the UA decreases (despite increasing outside surface area Ao ) due to the worsening air¯ow conditions (laminar region) with increasing hydraulic diameter and minimum ¯ow area. It shows that an increase in Ao does not necessary lead to an improved heat transfer. The parametric study for an optimum design of the evaporator in terms of maximum capacity per unit weight, revealed two conclusions: (i) thinner (<0.3 mm) and shorter (<2 cm) ®ns yielded maximum heat transfer performance of the evaporator and (ii) higher ®n density increased the evaporator capacity but simultaneously also increased the material weight (i.e. additional cost to the manufacturer). An optimum solution with a combination of ®n density of 160 ®ns/m, ®n height of 1.4 cm and ®n thickness of 0.2 mm, yielded increased evaporator capacity by about 4.7% but with a consequent 72% weight of the material. The j-Colburn and friction factors for the improved new design are
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higher than in the present design. The outside heat transfer coecient is about two times higher than the present design while the UA also improved by 18%. Acknowledgements The authors are thankful to Fisher & Paykel Ltd. for sponsoring the project and providing all necessary support. Appendix A A.1. Fin eciency correlation
The rectangular ®n design of the Ôegg-crateÕ type evaporator (Fig. 17) diers from the common ®n-and-tube heat exchangers and no correlations could be found in the open literature for this type of ®n. Therefore, a correlation proposed by Mills [12] for an annular ®n using Bessel functions of the ®rst and second kind was used here: gf
2r1 =b K1 br1 I1 br2 I1 br1 K1 br2 ; r22 r12 K0 br1 I1 br2 I0 br1 K1 br2
A:1
where K and I are the Bessel functions, r1 is the tube outside radius and r2 is the equal radius of the ®n area (Fig. 17). The ®n parameter b is given as b
ho : kf df
A:2
Fig. 17. Scheme of the ®n dimensions.
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The equal radius is derived from the area of one ®n: r Af : requal p
769
A:3
The surface area of one ®n can be calculated as Af Sl Hf
Wc Hc Wf Sl :
A:4
A.2. Correlation for the hydraulic diameter
The hydraulic diameter for ¯ow inside a non-circular duct is given by dh 4Ac ; L Pw
A:5
where Ac is the air ¯ow cross-sectional area (m2 ), Pw is the wetted perimeter (m) and L is the eective ¯ow length (m). Accounting for the characteristic heat exchanger geometry, the hydraulic diameter may be de®ned as dh 4rL
Afr ; Ao
A:6
where Afr is the frontal area (m2 ) and Ao is the evaporator outside surface area (m2 ). Following Idem et al. (1990) [13], the parameter r is given as ``the ratio of minimum heat exchanger face area per unit length to the frontal area per unit length'' and can be arranged for the Ôegg-crateÕ type evaporator as: r
Hf
do
Hf Wf Hf
do df Sf
:
A:7
The eective ¯ow length of the air through the evaporator and is given as L Np Sl :
A:8
The frontal area is the entrance surface area for the air¯ow and is given as Afr Wl St Nl ;
A:9
where the ®n layer width Wl is Wl
Nf : Sf
A:10
The ®nned area per unit length is given as the product of the surface area of one ®n and the ®n density Sf : Af;u 2Sl Hf
Wc Hc 2Wf Sl Sf ;
A:11
where factor 2 accounts for heat exchange on both sides of the ®n. The total outside area per unit length consists of the ®nned area and the tube outside area minus the tube area, which is bonded with the ®n:
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P.K. Bansal et al. / Applied Thermal Engineering 21 (2001) 751±770
Ao;u Af;u p 1
df Sf do :
A:12
The total evaporator outside surface area is then given as Ao Ao;u Wl Nl Np
Aloss; staging ;
A:13
where Aloss; staging is the reduced area at the leading edge due to staging and is given by For the base evaporator Aloss;
staging
1 Nf Af;u Wl Nl 2Wf Sl Nl : 2 2
For the staged evaporator 3 1 Nf Aloss; staging Af;u Wl Nl 2Wf Sl Nl : 4 2 4
A:14
A:15
The minimum ¯ow area of the Ôegg-crateÕ type evaporator is given by Amin bWf Hf
do
df Wf Hf
do cNl Wl Sf ;
A:16
where all the parameters are given in Table 1.
References [1] P.K. Bansal, T. Wich, J. Chen, M. Browne, A simulation model of new egg-crate type forced ¯ow evaporators in domestic refrigerators and freezers, ASHRAE Transactions, submitted for publication. [2] M. Turanga, N. Tanaka, M. Takeshita, Performance of direct expansion plate ®nned tube coils for air cooling and dehumidi®cation, Int. J. Refrig. 11 (3) (1998) 78±86. [3] Z.Q. Chen, J.X. Ren, Eect of ®n spacing on the heat transfer and pressure drop of a two-row plate ®ne and tube heat exchanger, Int. J. Refrig. 11 (1988) 356±360. [4] Y. Seshimo, M. Fujii, An experimental study on the performance of plate ®n and tube heat exchangers at low Reynolds numbers, ASME/JSME Ther. Engng. Proc. 4 (1991) 449±454. [5] N. Kayansayan, Heat transfer characterization of plate ®n-tube heat exchangers, Int. J. Refrig. 17 (1) (1994) 49±57. [6] I. Horuz, E. Kurem, R. Yamankaradeniz, Experimental and theoretical performance analysis of air-cooled plate®nned-tube evaporators, Int. Comm. Heat and Mass Transfer 25 (6) (1998) 787±798. [7] T. Wich, An NTU-eectiveness model for new forced ¯ow evaporators in domestic refrigerators and parametric study of the evaporator geometry, Diplomarbeit thesis, The University of Auckland, New Zealand, 1999. [8] P.K. Bansal, O. van Neuren, An experimental study on new egg-crate type evaporators in domestic refrigerators and freezers, ASHRAE Trans. 104 (2) (1998). [9] W.M. Kays, A.L. London, Compact Heat Exchangers, third ed., Mc Graw-Hill, New York, 1984. [10] K.M. Nho, M.M. Yovanovich, Eect of oxide layers on measured and theoretical contact conductances in ®nnedtube heat exchangers, in: R.K. Shah, A.D. Kraus, D. Metzger (Eds.), Compact Heat Exchangers, A Festschrift for A.L. London, Hemisphere Publishing Corporation, New York, 1990, pp. 397±420. [11] T.L. Ernest, J.W. Sheeld, H.J. Sauer, Finned tube contact conductance: characterizing the integrity of the mechanical bond, ASHRAE Trans. 91 (2a) (1985) 85±95. [12] A.F. Mills, Heat Transfer, Richard D. Irwin, USA, 1992. [13] S.A. Idem, A.M. Jacobi, V.W. Goldschmidt, Heat transfer characterisation of a ®nned tube heat exchanger (with and without condensation), J. Heat Transfer 112 (1990) 64±79.