Method for determining the optimum number of circuits for a fin-tube condenser in a heat pump

International Journal of Heat and Mass Transfer 98 (2016) 462–471

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Method for determining the optimum number of circuits for a fin-tube condenser in a heat pump Won-Jong Lee a, Hyun Jung Kim b, Ji Hwan Jeong a,⇑ a b

School of Mechanical Engineering, Pusan National University, Busan 46241, Republic of Korea Department of Mathematics, Hoseo University, Asan 31499, Republic of Korea

a r t i c l e

i n f o

Article history: Received 27 June 2015 Received in revised form 4 February 2016 Accepted 6 February 2016

Keywords: Refrigerant circuitry Number of circuits Fin-tube heat exchanger Condenser

a b s t r a c t Fin-tube heat exchangers are widely used in air-source heat pumps and air-conditioners. Because these heat exchangers are comprised of a number of tubes, there are innumerable ways of organizing the refrigerant circuitry. It is important to determine the number of refrigerant circuits because the circuitry significantly influences the performance of the heat exchangers. A method for determining the optimum number of circuits is proposed in this work. This method is applied to determine the optimum number of circuits for a fin-tube condenser. The performance analyses of fin-tube condensers with various circuit configurations demonstrates that the new method is useful in determining the number of circuits required in a heat exchanger. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The use of air-source heat pumps and air-conditioners has become popular in both developed and developing countries for space heating and cooling. It has been reported that approximately 40% of the energy consumption of buildings is attributed to air-conditioning and 20% of the total electricity is consumed for air-conditioning in developed countries. Due to this large consumption of energy, various efforts have been made to improve the performance and energy efficiency of air-source heat pumps and air-conditioners [1–4]. Because air-source heat pumps and air-conditioners utilize condensers and evaporators to reject and absorb heat to and from the surrounding areas, the performance of condensers and evaporators significantly affects the heat transfer capacity and energy efficiency of the systems. Accordingly, studies on heat exchangers have been, and continue to be, important to increase efficiency and reduce energy consumption. Various studies on heat exchangers have been conducted focusing on the air-side performance [5], refrigerant-side performance [6,7], and circuitry design [8–11] of condensers and evaporators. Fin-tube heat exchangers are widely used in air-source heat pumps and air-conditioners. They consist of a number of heat transfer tubes and fins attached to the outside surface of the tubes. Because fin-tube heat exchangers consist of a large number of heat transfer tubes, there are innumerable tube combinations that can compose ⇑ Corresponding author. Tel.: +82 51 510 3050; fax: +82 51 512 5236. E-mail address: [email protected] (J.H. Jeong). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.02.094 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

refrigerant circuitries. In general, the design of refrigerant circuitries requires a long time and high cost because it is conducted relying on the experience of the designer and repeated tests. Furthermore, a generally accepted analytic method to design refrigerant circuitry does not exist, and the optimal number of refrigerant circuits is determined based on the designers’ experience rather than an analytic method. The optimization of fin-tube heat exchangers’ refrigerant circuitries has been attempted using various methods. Wang et al. [8] experimentally investigated the effect of the refrigerant circuitry on the fin-tube heat exchanger performance in a condenser that uses air as the heat source. They concluded that, although the performance increases when the refrigerant circuitry is countercross arranged, the increase in the performance may be offset by the thermal conduction through the fins. There has been a study that applied the genetic algorithm in order to optimize a refrigerant circuitry [9]. This algorithm can automatically construct every possible refrigerant circuitry in a simulation. This study searched for the refrigerant circuitry with the maximum heat transfer rate among the constructed refrigerant circuitries. This method requires a significant amount of time and it is difficult to confirm whether the circuitry can be practically applied or not. Liang et al. [10] proposed a model that can be used to investigate the heat transfer characteristics and performance evaluation of a refrigerant circuitry through exergy destruction analyses. Although exergy destruction analyses were used to determine which refrigerant circuitry was appropriate for certain operational conditions, it does not provide a direct answer to how many refrigerant

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Nomenclature A D G L Np Pr Q_ R S_ gen Re T U V_ cp h k p pr

area (m2) diameter (m) mass flux (kg m
2 s) length (m) circuit number Prandtl number heat transfer rate (W) thermal resistance (m2 K W
1) entropy generation rate (J K
1 s
1) Reynolds number temperature (K) overall heat transfer coefficient (W m
2 K
1) volume flow rate (m3 s
1) specific heat at constant pressure (J kg
1 K
1) convective heat transfer coefficient (W m
2 K
1) thermal conductivity (W m
1 K
1) pressure (N m
2) actual pressure/critical pressure

Table 1 Correlations used for simulations. Items

Zone

Correlations

Heat transfer in matrix

Air side Single-phase refrigerant Two-phase refrigerant

Wang et al. [12] Gnielinski [13] Shah [14]

Pressure drop in matrix

Air side Single-phase refrigerant Two-phase refrigerant side

Wang et al. [12] Churchill [15] Friedel [16]

Pressure drop in U-bend

Single-phase refrigerant Two-phase refrigerant

Ito [17] Chen et al. [18]

circuits should be used. A model that can be used to minimize the refrigerant-side exergy destruction of a condenser was presented using the entropy generation minimization theory [11]. Although this method is useful, the calculation process is complicated and the refrigerant circuitry cannot be designed intuitively. Studies on heat exchangers have been conducted for numerous years. However, it is not yet clear how to obtain the optimal number of refrigerant circuits. In particular, there is a significant shortage of studies that consider the divergence and mergence of refrigerant circuits. In this paper, the existing evaluation methods for heat exchangers are discussed and a new method is proposed for determining the appropriate number of circuits. This method is applied to a fin-tube heat exchanger and compared with the simulation results in order to prove its validity. This method is also used to review where to diverge or to merge circuits and how many circuits should be diverged and/or merged.

specific volume (m3 kg
1) thermodynamic vapor quality overall surface efficiency viscosity (Pa s)

v x

g l

Subscripts a air c cool side cont contact foul fouling gen generation i inner l liquid o outer r refrigerant w wall

is smooth, (3) the flows of the working fluids are steady, (4) the longitudinal conductive heat transfer through the tubes and fins is negligible, and (5) the curved tube parts (U-bends) do not involve heat transfers but they involve pressure drops. The analysis was conducted such that the refrigerant mass flow rate of each circuit was determined in order to equalize the outlet pressure of each circuit while the total refrigerant flow rate and pressure at the entrance remained constant. Prior to using this numerical simulation model, the performance of some heat exchangers with different circuitries were experimentally measured and compared with the simulation results in order to examine the validity of the simulation model. A representative fin-tube heat exchanger matrix was selected for this study. The selected heat exchanger matrix had a tube configuration of 22 steps and 2 rows. It was used as a heat pump and had a designed capacity of 3.5 kW. The detailed specifications of the heat exchanger matrix are presented in Table 2. Regarding the operation conditions, the median values of the operation parameters required in order to satisfy the load were selected and they are listed in Table 3. Thirty refrigerant circuitries were constructed; these are illustrated in Fig. 1. They are identified using the number of circuits and flow configuration as in Table 3. Seven refrigerant circuitries of 2CUe, 2CNe, 2PUe, 3CUV, 3CUA, and 4Cue were used to verify the simulation model.

Table 2 Specifications of the fin-tube heat exchanger matrix. List

Value

2. Numerical simulation model

Rated capacity Coil type Tube configuration

It is difficult to experimentally measure the performance of a large number of heat exchangers with different refrigerant circuitries due to the limitations of time and cost. In the present work, a numerical simulation model is used to investigate the performance of the heat exchangers. In order to evaluate the performance of these condensers, correlations associated with heat transfers and pressure drops are necessary. The correlations used in the analysis are listed in Table 1. The analysis is conducted with five assumptions: (1) the effect of oil inside the tube is negligible, (2) the internal surface of the tube

Tube length Tube OD Tube thickness Tube horizontal spacing Tube vertical spacing Tube material Fin type Fin thickness Fin spacing Fin height Fin material

3.5 kW Fin-tube HX Staggered 22 steps  2 rows 653.2 mm 7 mm 0.25 mm 12.7 mm 21 mm Copper Louver 0.1 mm 18 fpi 0.7 mm Aluminum

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Table 3 Operating conditions used for the simulations. List

Value

Unit

Refrigerant Pressure at inlet Superheating at inlet Refrigerant mass flow rate Air temperature Air humidity Air velocity

R410A 2840 35 71.8 35 40 1.27

kPa (A) °C kg h
1 °C % m s
1

The performance of the fin-tube condensers with various circuit designs was experimentally measured. The experiments were performed in a psychrometric chamber and a schematic of the experimental apparatus is presented in Fig. 2. The experimental apparatus consisted of an environmental chamber, a cord tester, an air sampler, and a refrigerant supply system. The temperature and humidity of the environmental chamber were controllable in the range of
10 to 50 °C and 30–90%. The heat transfer rate was evaluated using the arithmetic average of the air-side and refrigerant-side heat transfer rates. When the refrigerant subcooling at the heat exchanger exit was not sufficiently large, it was evaluated using the air-side heat transfer rate only. The measurement uncertainties of the temperature, pressure, air flow rate, and refrigerant flow rate were ±0.3 °C, ±1.47 kPa, ±0.25 m3 min
1, and ±1.2 kg h
1, respectively. Test specimens for the fin-tube heat exchangers were fabricated. The specifications of the heat exchanger matrix were the same as those for the simulation and are listed in Table 2; they correspond to 2CUe, 2CNe, 2PUe, 3CUV, 3CUA, and 4Cue in Table 4. The heat exchanger specimens were installed at the inlet of the cord tester and carefully sealed. The experimental conditions were the same as the simulation conditions, which are described in Table 3, except the frontal air velocity. The experiments were performed at frontal air velocities of 0.99, 1.27, and 1.55 m s
1. The measurements were recorded when the experimental apparatus reached a steady state; this typically required approximately two hours to reach a steady state from start up. Figs. 3 and 4 present comparisons of the predicted heat transfer rates and pressure drops against the experimental measurements. The mean absolute errors (MAEs) between the predicted and measured values were 4.3% and 24.4% for heat transfer rate and pressure drop, respectively. These plots demonstrate that the present simulation model predicts the performance of fin-tube condensers well and can be used for further analyses.

3. Existing heat exchanger evaluation methods Various indices have been proposed in order to evaluate and compare the performance of heat transfer enhancement techniques. The indices include the efficiency index, performance evaluation criteria (PEC), area goodness factor, volume goodness factor, entropy generation minimization, and maximum heat transfer rate. The efficiency index, PEC, and goodness factor have been used to evaluate heat transfer augmentation technology; however, these methods are not appropriate for refrigerant circuitry optimization. Previous studies on the refrigerant circuitry have used indices of maximum heat transfer rate and entropy generation minimization. When the maximum heat transfer rate is used as an optimization index, however, the pressure drop limitations should be considered. The entropy generation minimization method might be useful in determining the optimal number of refrigerant circuitries. Assessments using this method were performed using the numerical simulation model for fin-tube condensers. A condenser with

the specifications in Table 2 was simulated with the operating conditions in Table 3. The entropy generation was calculated for the six refrigerant circuitries of 1CUU, 2Cnu, 3Cnu, 4Cnu, 6Cnu, and 11Cnu in order to assess the effects of the different numbers of refrigerant circuits. The number of refrigerant circuits were 1, 2, 3, 4, 6, and 11, respectively. The flow configurations of these constructions were the same as the cross-counter flows; the path length for each circuit in a construction was the same; and the tube connection pattern was similar. The same assumptions and analysis methods were used as described in the previous section. The air-side and refrigerant-side entropy generations were calculated using the following equations.

Tw
Ta v dp _a ¼ dQ_
m ; Ta TaT w

ð1Þ

Tw
Tr v dp _r dS_ gen r ¼ dQ_
m : Tr TrTw

ð2Þ

dS_ gen

a

The terms on the right of the above equations represent the entropy generation due to the heat transfer and pressure loss. Fig. 5 presents the calculated entropy generation from the heat exchangers with different numbers of circuits. From top to bottom in the plot, the areas represent the entropy generation due to the air-side pressure loss, air-side heat transfer, refrigerant-side pressure loss, and refrigerant-side heat transfer. Fig. 5 demonstrates that the entropy generation due to pressure loss was significantly smaller than that by heat transfer. Because the entropy generation was dominated by the heat transfer under the present heat exchanger design and operating conditions, the entropy generation minimization method was also governed by the heat transfer performance. In contrast, the total entropy generated by the construction with 11 circuits appeared to be the smallest among the evaluated constructions. However, this circuit number was not speculated to be the optimum because the heat transfer rate was also the smallest. The exergy destruction ratio defined by Hesselgreaves [19] was calculated using Eq. (3).

Nsl ¼

S_ gen T c;in : Q_

ð3Þ

The calculated exergy destruction ratios are presented in Fig. 6. The exergy destruction ratio of the single circuit design was the smallest in the evaluated circuit number designs because the heat transfer rate was the largest for this circuit design. However, this circuit number was not optimum, but rather it was another extreme case because the pressure drop was very large. Based on the discussions above, it can be understood that both the entropy generation minimization method and the exergy destruction analysis are not useful for determining the optimal number of circuits. This primarily results from the heat transfer dominating the entropy generation of a condenser under the operating conditions of the heat pumps. 4. Proposed method for determining the number of circuits The heat transfer capacity of a heat exchanger is determined by the overall heat transfer coefficient, heat transfer area, and temperature difference between the two working fluids, which can be expressed as follows:

Q_ ¼ UA  ðT r
T a ÞLM :

ð4Þ

Here, (Tr
Ta)LM represents the log mean temperature difference between the refrigerant and the air. If a fin-tube heat exchanger matrix is fixed through determining the size and number of the heat transfer tubes and fins, the total heat transfer area does not change even when the refrigerant circuitry is changed. However, the overall

W.-J. Lee et al. / International Journal of Heat and Mass Transfer 98 (2016) 462–471

465

Fig. 1. Refrigerant circuitries of fin-tube heat exchangers.

heat transfer coefficient and temperature difference between the two working fluids vary if the refrigerant circuitry is changed. The refrigerant can flow through one flow path or be divided into several flow paths. The refrigerant mass flux inside a tube

and the length of each flow path vary as the number of paths changes. Because a change in the mass flux varies the Reynolds number, which affects not only the convective thermal resistance but also the pressure loss of the refrigerant, the effect of the

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Fig. 2. Schematic of the experimental apparatus.

Table 4 Classification of the refrigerant circuitries.

5000 +10%

Cross-counter flow

1

1CN1 1CX3 1CUU

1CX1 1CX4

1CX2 1CX5

2

2Cnu

2CSr 2CNr

2CUe 2CNe

3

3Cnu

3CNe 3CUn 3CNr

3CUV 3CUA

4CNr 11Cnu

4CUe

4 6, 11

4Cnu 6Cnu

Mergence

P4-2

Cross-parallel flow

Cross flow

4500

4000 2PUU 2PUe

2UII 2UZZ

2DII 2DZZ

Simulation (W)

No. of circuits

-10%

3500

Air velocity (m/s)

3000

0.99 1.27 1.55

2500

2000 2000

2500

3000

3500

4000

4500

5000

Experiment (W)

number of circuits on the performance of the heat exchanger is significant. Accordingly, the number of circuits should be selected before designing the refrigerant circuitry. The convective heat transfer can be increased through decreasing the number of circuits. However, a decrease in the number of circuits results in the pumping power being increased. In order to examine the effects of the change in the number of circuits on the heat transfer performance of the refrigerant side and pumping power (V_ DP), the pumping power and heat transfer coefficient were calculated with changes in the mass flux in the heat transfer tube in the range of 0–700 kg/m2 s. Fig. 7 presents the condensation heat transfer coefficient on the internal wall of the heat transfer tube and the pumping power per unit length of the heat transfer tube for a fixed quality of refrigerant flowing inside the heat transfer tube. The conditions of the calculation are described

Fig. 3. Predicted and measured heat transfer rates.

in Table 5. Fig. 7 demonstrates that, although the relationship between the heat transfer coefficient and pumping power is proportional, it is not linear. The pumping power increases as the mass flux increases, and the increase rate of the heat transfer coefficient diminishes as the pumping power increases. However, the pumping power decreases as the mass flux decreases, and the reduction rate of the heat transfer coefficient increases as the pumping power decreases. Therefore, increasing the pumping power without a guide is not an appropriate method to increase the heat transfer rate. In this section, a method to determine the optimal number of circuits for a refrigerant circuitry without divergence/ mergence is proposed.

W.-J. Lee et al. / International Journal of Heat and Mass Transfer 98 (2016) 462–471

transfer performance of a fin-tube heat exchanger, additional cost or work is required such as widening of the area through installing more fins or provision of more flow work through increasing the number of revolutions per minute (rpm) of the fan or compressor. If the work (cost) that can increase gihrAi or gohaAo is the same and the total cost that can be invested in gihrAi or gohaAo is limited, then the relationship can be expressed as follows:

30 +30%

25

Simulation (kPa)

20 -30% 15

gi hr Ai þ go ha Ao ¼ constant: Air velocity (m/s) 0.99 1.27 1.55

10

5

0 0

5

10

15

20

25

30

Experiment (kPa) Fig. 4. Predicted and measured refrigerant pressure drops.

The overall thermal resistance in Eq. (4) can be expressed as in Eq. (5). The terms of the right of the equation represent the thermal resistances related to the convection inside the tube, the conduction of the tube wall, the convection outside the tube, contact, and fouling, respectively.

1 1 lnðDo =Di Þ 1 ¼ þ þ Rcont þ Rfoul : þ UA gi hr Ai 2p  L  k go ha Ao

ð5Þ

Because the conductive, contact, and fouling thermal resistances are regarded to be constant at a certain point in time, the above equation can be expressed as in the following equation:

1 1 1 ¼ þ þ C: UA gi hr Ai go ha Ao

467

ð6Þ

Because convective thermal resistances comprise most of the overall thermal resistance, Eq. (6) is suitable for review of the heat transfer phenomenon. gihrAi and gohaAo represent the heat transfer performances of the refrigerant side and air side, respectively. If the heat exchanger matrix is fixed, the heat transfer performance of the air side depends on the air velocity and is not affected by the refrigerant circuitry construction. In order to enhance the heat

ð7Þ

In Eq. (7), the overall heat transfer coefficient (UA) of Eq. (6) is the largest when gihrAi = gohaAo. That is, regarding the number of circuits, when the number of circuits is selected to make gihrAi and gohaAo equal, the maximized heat transfer performance can be achieved with small work, and this number of circuits is optimal. Now consider the case wherein the shape and number of fins and heat transfer tubes are the same and only the number of refrigerant circuits is changed through connecting the heat transfer tubes. If the heat transfer tubes are connected to make the number of circuits smaller than the optimum number of circuits determined to make gihrAi = gohaAo as described above, the heat transfer performance of the refrigerant side (gihrAi) in each circuit will increase and will result in an increase in the overall heat transfer performance (UA). However, as the heat transfer performance of the air side (gohaAo) is fixed and gihrAi > gohaAo, the heat transfer performance of the air side dominates the overall heat transfer performance. Accordingly, the contribution of the improvement in the heat transfer performance of the refrigerant side to the overall heat transfer performance is reduced. Nevertheless, the increase in the cost (pumping power) increases more than the increase in the benefit (heat transfer performance), because the contribution to the _ increases in proporpumping power of the refrigerant (¼ DP  V)

_ In contrast, if the heat transfer tion to the volume flow rate (V). tubes are connected to make the number of circuits larger than the optimum number of circuits, the cost required for the refrigerant (pumping power) decreases. However, as the heat transfer performance of the air side (gohaAo) is fixed and gihrAi < gohaAo, the heat transfer performance of the refrigerant side dominates the overall heat transfer performance. Accordingly, the contribution of the reduction in the heat transfer performance of the refrigerant side to the reduction in the overall heat transfer performance increases. That is, the reduction in the benefit (heat transfer

Fig. 5. Entropy generation and heat transfer capacity with various numbers of circuits.

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and the conductive, contact, and fouling thermal resistances were negligible, Eq. (5) can be rewritten as in Eq. (8).

5.0

Total exergy destruction ratio Refrigerant side exergy destruction ratio Air side exergy destruction ratio

4.5

Exergy destruction ratio (%)

4.0

1 1 1 þ : ¼ UA hr Ai ha Ao

3.5

ð8Þ

The total area inside the tubes of this heat exchanger was 0.5867 m2 and the total external area of the tubes including the fins was 9.8655 m2. The heat transfer coefficient of the refrigerant side was calculated through obtaining the mean value over the tube length in order to determine the value that could represent the whole heat exchanger. In this study, the condensation heat transfer model of Shah [14] was used, which is as follows:

3.0 2.5 2.0 1.5

"

1.0

hr ¼ hl ð1
xÞ0:8 þ

0.5

# 3:8x0:76 ð1
xÞ0:04 ; x0:38

ð9Þ

0.0 0

2

4

6

8

10

12

Number of circuits Fig. 6. Exergy destruction ratio with various numbers of circuits.

0:4 hl ¼ 0:023 Re0:8 l Prl kl =D;

Rel ¼ Prl ¼

Gr Di

ll

ð9aÞ

;

ð9bÞ

c p ll : kl

ð9cÞ

Assuming negligible change in the transport properties of the liquid phase and pressure along the length of the condenser, the mean heat transfer coefficient over the entire length of the heat transfer tube was obtained as follows:

 ¼ h r

hl ðL2
L1 Þ

Z

L2

"

ð1
xÞ0:8 þ

L1

# 3:8x0:76 ð1
xÞ0:04 dL: pr 0:38

ð10Þ

Assuming that the tube length and vapor quality have a linear relationship, the above equation can be approximated as follows:

r ffi h

"  #x2 hl ð1
xÞ1:8 3:8 x1:76 0:04x2:76 :
þ 0:38
pr ðx2
x1 Þ 1:8 1:76 2:76

ð11Þ

x1

Fig. 7. Relationship between the refrigerant-side heat transfer coefficient and pumping power.

Table 5 Conditions used to estimate the heat transfer coefficient and pumping power. List

Value

Unit

Refrigerant Pressure Inner diameter Roughness Mass flux

R410A 2840 6.5 0 0–700

kPa mm mm kg m
2 s
1

performance) increases more than the reduction in the cost (pumping power). As a result, the heat transfer performance versus the cost invested in the heat exchanger can be maximized through designing the number of circuits to allow gihrAi = gohaAo, and this number of circuits can be regarded as the optimum number of circuits. 5. Optimum number of circuits for the fin-tube condenser The new criterion was applied to the fin-tube heat exchanger that was examined in Section 2 in order to determine the optimum number of circuits. When it is assumed that there was no fin on the refrigerant side, the overall surface efficiency of the air side was 1,

The main area of the condenser where the phase change occurs is where the quality changes from 1 to 0. Substituting x1 = 1 and x2 = 0 in Eq. (11) yields:

r ¼ h ð0:55 þ 2:09=pr 0:38 Þ: h l

ð12Þ

The mean heat transfer coefficient of the refrigerant side can be obtained through applying the operation conditions in Table 3 to Eq. (12), while the heat transfer coefficient of the air side can be obtained using Wang’s model [12]. When all heat transfer tubes are connected in series to form a single refrigerant circuitry, the heat transfer performances of the refrigerant side and air side can be compared, as follows:

hr Ai ð¼ 3750Þ > ha Ao ð¼ 1627Þ:

ð13Þ

The above equation demonstrates that the heat transfer performance of the refrigerant side is superior to that of the air side when the refrigerant flows through one circuit. This result implies that the increase in the overall heat transfer performance is not large in comparison to the cost invested in the refrigerant side because the performance of the refrigerant side is significantly superior to that of the air side. Because the total mass flow rate of the refrigerant is divided into the number of circuits, the refrigerant mass flow rate of each circuit decreases in reverse proportion to the number of circuits. In addition, the length of each flow circuit also decreases in reverse proportion because the entire length of the heat transfer tube is divided into the number of circuits. As described in Eq. (9), the heat transfer performance inside the tube is also affected by the mass flux of the refrigerant. If the mass flux is G in a single circuit

W.-J. Lee et al. / International Journal of Heat and Mass Transfer 98 (2016) 462–471

system, the mass flux of each circuit in a multiple circuit system is G/Np, where Np represents the number of circuits. When this relationship is introduced into Eq. (9) and Eq. (9) into Eq. (8), Eq. (8) can be rewritten as follows:

N0:8 1 1 p ¼ þ : UA hr Ai ha Ao

ð14Þ

Because the number of circuits that comprise the two terms on the right is optimum, the optimum number of circuits can be calculated as follows:

 Np ¼

hr Ai h a Ao

0:81

¼ 2:84:

ð15Þ

The above calculation is made through assuming that the overall surface efficiency of the air side is 1. If the overall surface efficiency of the air side is 0.9, the optimum number of circuits increases to 3.24. Furthermore, it is assumed that the conductive, contact, and fouling thermal resistances are negligible. Accordingly, the value produced by Eq. (15) is an approximate value in order that the optimum number of circuits for the above condenser is determined to be 3. The heat transfer coefficient of the refrigerant side, which has a circuit number of 2.84 obtained from Eq. (15), is 2775 W m
2 K
1. That is, the most effective design is to invest in the cost (pumping power) in order to allow the heat transfer coefficient of the refrigerant side to be 2775 W m
2 K
1 under the given conditions of the air-side heat transfer performance and the internal area of the heat transfer tube. In order to verify the method that determines the optimum number of circuits, the performance of the condenser with the specifications in Table 2 was analyzed using the operation conditions in Table 3. In order to examine the effects of the number of circuits, refrigerant circuitries of 1CUU, 2Cnu, 3Cnu, 4Cnu, 6Cnu, and 11Cnu were used again. The same assumptions and analysis methods were used as described in Section 2 for the present analyses of the heat exchanger performance. The results of the analyses are presented in Fig. 8. The abscissa and ordinate represent the pumping power and heat transfer capacity, respectively. The results exhibit a trend wherein the pumping power decreases and the heat transfer capacity decreases as the number of circuits increases. The pumping power of the condenser with circuit numbers of 1 and 2 appeared to be significantly higher than that of the condenser with a circuit number of 3, while their heat transfer capacities were not significantly higher. In contrast, the pumping powers of the condensers with 6 and 11 circuits were slightly

lower than that of the condenser with 4 circuits, while the heat transfer performance was significantly lower. The configuration with 3 circuits, from which the increase rate of the heat transfer capacity began to sharply decrease in comparison with the large increase in the pumping power, can be regarded as the optimum number of circuits. Three circuits are similar to the 2.84 circuits that were deduced through applying the proposed method to determine the optimum number of circuits. The heat exchanger performance changed with the positions of the inlet and outlet, and the connection method of tubes varied even when the number of circuits remained the same. Additional simulations were performed for various refrigerant circuitries in order to examine the effect of the circuit construction. All refrigerant circuitries in Table 4, except that of the merged refrigerant circuitry, were examined and the simulation results are presented in Fig. 9. This plot demonstrates that the performance of the heat exchanger changed slightly depending on the circuit design. In particular, the pressure drop changes were large compared with the heat transfer capacity. The performance data for the same circuit numbers were similar, but several points such as 2PUU, 2PUe, and 3CUA were located far from their group. This resulted from these circuitries being constructed such that the flow configuration differed from the others and the path length of each circuit varied significantly. The effect of the circuit design diminished as the number of circuits increased. It is noteworthy that the optimal number of circuits remained the same despite the circuitry design. 6. Divergence and mergence of circuits In the previous section, the optimum number of circuits that does not have divergence or mergence was discussed. It is known, however, that the performance of a heat exchanger can be further improved through diverging and/or merging the refrigerant circuits at suitable locations. The condensation heat transfer coefficient changes significantly depending on the quality and the mass flux of the refrigerant flowing inside the heat transfer tubes. Therefore, the performance of the refrigerant side can be changed through the mergence and divergence of the circuits. Fig. 10 presents the condensation heat transfer coefficient and pressure loss inside the heat transfer tube for the same heat exchanger considered in the previous section with 2 circuits. The analysis in Section 5 indicated that the mean refrigerant-side condensation heat transfer coefficient was approximately 2775 W m
2 K
1 when the number of circuits was optimal. Considering this level of heat transfer coefficient, it can be seen

5000 4500

Heat transfer capacity (W)

4000 3500 3000

Refrigerant circuitry 1CUU 2Cnu 3Cnu 4Cnu 6Cnu 11Cnu P4-2

2500 2000 1500 1000 500 0 0

2

4

6

8

10

12

Pumping power (W) Fig. 8. Heat transfer capacity and pumping power at the condenser condition.

469

Fig. 9. Simulation results for various refrigerant circuitries.

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Fig. 10. Heat transfer coefficient and pressure loss of a 2-circuit condenser.

that the heat transfer coefficient is excessively large in the quality range from 0.6 to 0.9, while it is too small in the ranges from 0.0 to 0.2 and from 1.0 to 1.3. Accordingly, it is presumed to be advantageous to reduce the pumping power through increasing the number of circuits in the quality region between 0.6 and 0.9, and it is reasonable to increase the heat transfer coefficient of the refrigerant side through reducing the number of circuits in the region between 0.0 and 0.2. The region of quality between 1.0 and 1.3 is a superheated vapor area, in which the heat transfer coefficient of the refrigerant side is low and the pressure loss is large. Although the heat transfer performance of this part can be improved through reducing the number of circuits, it is not desirable to reduce the number of circuits because the pressure loss may significantly increase and problems of vibration and noise might occur. In order to determine where to diverge or merge circuits, the local heat transfer coefficient should be considered rather than the mean heat transfer coefficient. Table 6 presents the optimal number of circuits calculated at various qualities using Eqs. (9) and (15). It demonstrates that the refrigerant circuitry should be designed such that the number of circuits diminishes as the local quality decreases. That is, the divergence and mergence of circuits should be determined considering the variations of quality and the number of circuits suitable for the conditions should be selected. This approach was applied to the condenser with the specifications in Table 2 under the operating conditions in Table 3. Even though multiple mergences of circuits can be made, a refrigerant circuitry with 4-2 circuits was constructed for simplicity. The 4-2 circuits

Table 6 Optimum number of circuits at various qualities. Quality

Optimum circuit number (Np) Shah

Dobson and Chato

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.36 1.83 2.26 2.64 3.00 3.32 3.61 3.84 3.99

1.28 1.74 2.17 2.57 2.96 3.34 3.70 4.05 4.38

represents the design where there are four parallel tubes for the inlets that are merged into two tubes in the middle in order to create two outlets. The point at which the circuits are merged is selected as the point where the quality is approximately 0.5 referring to Table 6. A schematic for this refrigerant circuitry is depicted as P4-2 in Fig. 1. The analysis results are also plotted in Fig. 8. The heat transfer capacity of P4-2 was evaluated to be slightly lower than that where the number of circuits was two (2Cnu), while the pumping power was significantly smaller. Furthermore, the heat transfer capacity appeared to be larger than that where the number of circuits was four (4Cnu), while the pumping power increased slightly. Therefore, it can be understood that the performance of the P4-2 circuit was improved through merging the circuits. Table 6 presents the optimum circuit numbers evaluated using different condensation heat transfer coefficient models. Evaluations based on the Shah [14] correlation and the Dobson and Chato [20] correlation produced nearly identical results because both correlations predicted similar levels of heat transfer coefficient. A heat transfer coefficient correlation with a higher accuracy over a wide range of qualities should be used because it significantly influences the present analysis for the optimum number of circuits.

7. Conclusion There are innumerable methods of organizing the refrigerant circuitry of fin-tube heat exchangers with large numbers of tubes. Furthermore, the number of refrigerant circuits is an influential design parameter because it changes the pumping power and heat transfer coefficient of the refrigerant side. Analyses have demonstrated that the entropy minimization method is not useful in determining the optimal number of refrigerant circuits because the entropy generation is dominated by heat transfers rather than pressure drops under the operating condition of a heat-pump condenser. In this paper, a new method was proposed for determining the optimal number of circuits for fin-tube condensers. This method is based on the condition of balanced thermal resistance between the refrigerant side and air side. This new method was applied to a fin-tube condenser in order to determine the optimal number of circuits. In order to examine the validity of the new method, the performance of condensers with various circuit constructions were analyzed using a numerical simulation model.

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The results demonstrated that the proposed method is useful in determining the optimal number of circuits. Further analyses also demonstrated that this new method can be used to determine where to merge or diverge refrigerant circuits in order to improve the condenser performance. Acknowledgments This work was supported by the Human Resources Development program (grant no. 20144010200780) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and Basic Science Research Program through the National Research Foundation of Korea (NRF) (NRF-2012R1A1A2007320). References [1] Y.S. Park, J.H. Jeong, B.H. Ahn, Heat pump control method based on direct measurement of evaporation pressure to improve energy efficiency and indoor air temperature stability at a low cooling load condition, Appl. Energy 132 (2014) 99–107. [2] H. Jung, J. Hwang, C. Jeon, An experimental study on performance improvement for an air source heat pump by alternate defrosting of outdoor heat exchanger, Int. J. Air-Condition. Refrig. 22 (3) (2014) 1450017. [3] J.H. Jeong, Y.C. Kweon, K.S. Chang, Variations in the thermal performance of R22 and R410A refrigeration systems depending on operation conditions, Int. J. Air-Condition. Refrig. 12 (1) (2004) 10–20. [4] Y.C. Kwon, J.T. Kwon, J.H. Jeong, S.J. Lee, D.H. Kim, Performance of a 2 room multi-heat pump with a constant speed compressor, Int. J. Air-Condition. Refrig. 12 (4) (2004) 184–191. [5] N.H. Kim, M.H. Kwon, M.G. Go, An experimental investigation on the airside performance of fin-and-tube heat exchangers having nonsymmetrical slit fins, Int. J. Air-Condition. Refrig. 23 (2) (2015) 1550017.

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