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Efficient Air Conditioning (A/C) system is the key to reducing energy consumption in building operation. In order to decrease the energy consumption in an A/C system, a method to calculate the optimal tube row number of a direct expansion (DX) cooling coil for minimizing the entropy generation in the DX cooling which functioned as evaporator in the A/C system was developed. The optimal tube row numbers were determined based on the entropy generation minimization (EGM) approach. Parametric studies were conducted to demonstrate the application of the analytical calculation method. Optimal tube row number for different air mass flow rates, inlet air temperatures and sensible cooling loads were investigated. It was found that the optimal tube row number of a DX cooling coil was in the range of 5 - 9 under normal operating conditions. The optimal tube row number was less when the mass flow rate and inlet air temperature were increased. The tube row number increased when the sensible cooling load was increased. The exergy loss when using a non-optimal and optimal tube row numbers was compared to show the advantage of using the optimal tube row number. The decrease of exery loss ranged from around 24% to 70%. Therefore the new analytical method developed in this paper offers a good practice guide for the design of DX cooling coils for energy conservation.

Entropy generation or irreversibility is always present in the heat and mass transfer processes taking place in heat exchangers. It degrades the thermodynamic performance of the heat exchangers, and causes the waste (destruction) of useful energy. For this reason, the minimization of entropy generation in heat exchangers has attracted considerable research interest over the last three decades.

Bejan [

Sahiti et al. [

Although the EGM method had been widely applied in finding the optimal shape or optimal operating conditions of heat exchangers [

The cooling material in the studied heat exchangers is single-phase water, while in DX cooling coil is refrigerant where both two-phase and single-phase exist and the major one is two-phase. This causes that the entropy generation calculation of DX cooling coil is more complicated compared with those of the previous heat exchangers leading to the scarcity of related research.

In an operating DX cooling coil, both sensible heat and latent heat are transferred from air to refrigerant and air pressure drops along the air flow direction. Entropy is generated during air cooling process within the DX coil, degrading the energy quality. Entropy generation in both air side and refrigerant side should be decreased or minimized so as to enhance the thermal performance of DX air cooling coils.

In the present work, optimal tube row number of a DX cooling coil will be found by using an analytical calculation method so as to optimize the shape of the DX cooling coil.

For the purpose, an analysis of entropy generation due to the irreversibility during the air cooling process in a steadily operated DX cooling coil will be firstly carried out. Based on the analysis, a method for analytically calculating optimal tube row number of a DX cooling coil for different steady-state operating conditions will be developed and the numerical values of entropy generation are determined. This will be followed by reporting

parametric studies to illustrate the application of the analytical calculation method developed. The results of the optimal tube row number will be calculated under the variation of three key operating parameters which are inlet air mass flow rate, inlet air temperature and sensible cooling load. Finally, exergy loss when using non-optimal and optimal tube row numbers will be compared to identify the energy saving advantage by using the optimal tube row number.

_{a}_{,in}), a specific static pressure (P_{a}_{,in}) and a specific moisture content (w_{a}_{,in}). After being cooled and dehumidified, the air left the system at a lower dry-bulb temperature, T_{a}_{,out}, a lower static pressure, P_{a}_{,out} and a lower moisture content, w_{a}_{,out}. At the same time, because of dehumidification, the condensate left the system at a temperature of T_{w}. At the refrigerant side, there were two parts along the flow direction of refrigerant. The first part was a two-phase part where liquid refrigerant vaporizes absorbing heat from air; the second part was a superheated region where vapour refrigerant was further heated so that its temperature increased. The liquid phase refrigerant entered the cooling coil at the evaporating temperature, T_{r}_{,e}, and evaporating pressure, P_{r}_{,e}, and left at T_{r}_{,e} + T_{sh} and P_{r}_{,out}, where T_{sh} was the degree of refrigerant superheat.

To simplify the analysis the following two assumptions are made: 1) The entire cooling coil system shown in _{r}_{,e }= P_{r}_{,out}.

The entropy generation rate in the entire control volume is:

At a steady state, the term _{out} − ∑(ms)_{in} is evaluated as follows:

At the air side, air entropy varies because of heat transfer, frictional effect and phase change [

The first term on the right-hand side of Equation (3) represents the irreversibility due to the heat transfer and the frictional effect on moist air stream. The last term in Equation (3) represents the irreversibility due to the condensation of water vapour. The entropy variation at the refrigerant side is

The first and second terms on the right-hand side of Equation (4) denote the irreversibility during the phase change process in the two-phase region and heating process in the superheated region, respectively. Combining Equations (1), (3) and (4), the entropy generation can be determined by:

Based on the fact that in practice w_{a}_{,in }and w_{a}_{,out} are much smaller than 1, and q_{l} = m_{a}(w_{a}_{,in} − w_{a}_{,out})h_{fg}, q_{s} = m_{a}C_{pa}(T_{a}_{,in} − T_{a}_{,out}), Equation (5) can be rewritten as:

The corresponding entropy generation number [

In a steady state, q_{s} and q_{l}, should be equal to the sensible and latent space cooling loads. In addition, the inlet air parameters T_{a}_{,in}, P_{a}_{,in }are constants when indoor air temperature setting is maintained and a supply fan speed remains unchanged. Furthermore, the saturation temperature of moist air, T_{sat}, is equal to the dew point temperature of inlet moist air, T_{dew}, which can be determined by inlet air temperature and moisture content. These parameters are independent of the thermal characteristics of the DX cooling coil. Based on these, Equation (7) is rewritten as:

where N_{c} denotes the entropy generation which is independent of the thermal characteristics of the DX cooling coil, to be calculated by:

Recognizing that in most A/C applications h_{fg}_{,r} ~ 300C_{pr}_{,v}, T_{r}_{,e} < 10˚C and T_{sh}/T_{r}_{,e} < 1 in a DX cooling coil used in a R22 DX A/C system, the second term on the right hand side is much smaller than the first term on the same side of Equation (8); therefore, Equation (8) is simplified to:

Effectiveness-Ntu method is used to evaluate the thermal characteristics of the DX cooling coil. The effectiveness, ε, is

hence,

For a cross-flow heat exchanger, its effectiveness ε is calculated by [

For the DX cooling coil, C_{min} = m_{a}C_{pa}, and C_{max} → ∞, C_{min}/C_{max} → 0. The limit of ε as C_{min}/C_{max} approaches to zero is obtained by performing the Taylor series Expansion to Equation (13), yielding

From the definition of Ntu and Equation (14), ε could be evaluated by:

The air static pressure at the coil outlet, P_{a}_{,out}, is evaluated by P_{a}_{,in} − ∆P, where ∆P is given by [

Combining Equations (8), (12) and (16), and recognizing that ∆P/P_{a}_{,in }is considerably smaller than 1 to obtain:

The overall air side heat transfer area, A_{o}, and air flow length, L, are assumed to be both linearly related to the number of the tube rows, n_{t}. Therefore, Equation (17) becomes:

It can be seen from Equation (18), the first term increases while the second term decreases when n_{t} increases. There certainly exists an optimal value of n_{t} at which the entropy generation number, N_{s}, reaches minimum.

According to the previous studies [_{t} has insignificant effects on both air and refrigerant side heat transfer coefficients and therefore it is assumed to be ignorable. Furthermore, we assume the overall heat transfer coefficient U_{o} is regarded being unrelated to n_{t}, and the friction factor, f, is regard being invariable with n_{t} according to Ref. [

Based on these, the optimal n_{t}, i.e., n_{t}_{,opt}, is the root of the following equation which is obtained by partial differentiating Equation (18) with respect to n_{t}:

The solution of n_{t}_{,opt} obtained from Equation (19) is

where α, β are defined as:

It can be seen from Equations (20)-(23), the solution of n_{t}_{,opt} is determined by three types of parameter: a) DX cooling coil operating parameters such as m_{a}, T_{a}_{,in}, q_{s}; b) thermal properties of air and refrigerant, such as C_{pa} and h_{fg}_{,r}; c) and geometrical parameters such as D, c_{A} and c_{L}. The variations of any these parameters consequently cause the variation of n_{t}_{,opt}.

_{t}_{,opt}. Using these parameters, an optimal tube row number, n_{t}_{,opt}, of 6 was obtained. _{s} − N_{c} at different n_{t} (1 - 50) in. When n_{t} is small, the entropy generation due to air static pressure drop is low according to Equation (18). Hence, the values of (N_{s} − N_{c}) are determined by the irreversibility in heat transfer. When n_{t} is increased, the effectiveness, ε, of the DX coil is enhanced, leading to a decreased entropy generation in heat transfer. At the same time, however, the entropy generation due to the air static pressure drop increases linearly with the increase of n_{t}. At a larger n_{t}, entropy generation caused by air static pressure drop has dominating effect. Hence, (N_{s} − N_{c}) finally increases almost linearly with the increase of n_{t}.

Category | Quantities | Definition | Value (unit) |
---|---|---|---|

Operating conditions | m_{a } | Mass flow rate of moist air | 0.5 (kg/s) |

m_{r } | Mass flow rate of R22 | 0.04 (kg/s) | |

G_{a,dl } | Dimensionless mass velocity | 0.1 | |

P_{ain } | Inlet air pressure | 150 Pa | |

T_{a,in } | Inlet air temperature to cooling coil | 24 (˚C) | |

T_{dew } | Dew point temperature of inlet air | 12.8 (˚C) | |

T_{∞} | Environment temperature | 310 (K) | |

q_{s } | Sensible cooling load | 5.5 (kW) | |

q_{l } | Latent cooling load | 2.5 (kW) | |

U_{o } | Overall heat transfer coefficient | 50 (W/m^{2}×K) | |

f | Friction factor | 3 | |

Thermal properties | h_{fg,r } | Latent heat of vaporization of R22 | 198.4 (kJ/kg) |

C_{pa } | Specific heat of moist air | 1.007 (kJ/kg×˚C) | |

Geometrics | D | Hydraulic diameter | 0.45 m |

c_{A } | Ratio between A_{o} and n_{t } | 3.5 (m^{2}) | |

c_{L } | Ratio between L and n_{t } | 0.02 (m) |

With the method developed, the investigation of the impacts of key operating parameters for a DX cooling coil on the optimal tube row number, n_{t}_{,opt}, can be undertaken. In this section, the optimal tube row number under the variation of three operating parameters, i.e., m_{a}, T_{a}_{,in} and q_{s} for the DX cooling coil of which the other operating and geometric parameters are shown in

The variation ranges for the three parameters were: m_{a}, 0.3 - 0.7 kg/s with interval of 0.05 kg/s; T_{a}_{,in}, 20˚C - 28˚C with interval of 1˚C; q_{s}, 4.5 - 6.5 kW with interval of 0.25 kW, respectively. These variation ranges were selected according to common operating environment for a DX Air Conditioning system. The optimization results based on these operating conditions formed a group of optimal tube row numbers which should be adopted in common operating conditions of a DX A/C system.

As shown in _{a} leads to a decrease in the value of n_{t}_{,opt}. This is because the entropy generated during the heat transfer process decreased with an increased m_{a} and a smaller n_{t}_{,opt} would allow the hydraulic irreversibility as shown as the second term in Equation (18) to dominates. Smaller n_{t}_{,opt} means the lower air flowing resistance and therefore lower hydraulic irreversibility _{a}_{,in} and q_{s} on n_{t}_{,opt}, respectively. It can be seen that when T_{a}_{,in} increases n_{t}_{,opt} decreases and when q_{s} increases, n_{t}_{,opt} increases. These trends indicated the decreasing and increasing effects of T_{a}_{,in} and q_{s} on the entropy generated during heat transfer and pressure drop processes. The explanation related to n_{t}_{,opt} vs T_{a}_{,in} is: higher T_{a}_{,in} increases the temperature difference which drives the heat transfer taking place in the DX cooling coil; smaller n_{t}_{,opt} decreases the heat transfer time period therefore decreases the transfer amount of useful energy in hot air to cold refrigerant, leading to a lower entropy generation. The explanation related to n_{t}_{,opt} vs q_{s} is: larger n_{t}_{,opt} can enlarge the heat transfer area when q_{s} increases so as to decrease the entropy generation. The difference between the variation trends of n_{t}_{,opt} as T_{a}_{,in} and q_{s} increase is for the first one, heat transfer rate can be decreased by decreasing n_{t}_{,opt} even though T_{a}_{,in} increases. For the second one, we can only increase n_{t}_{,opt} for enhancing the heat transfer area because of the increase of q_{s}, the sensible heat transfer rate.

It can be seen that n_{t}_{,opt} is different under different operating conditions from _{t}_{,opt} shown were from 5 - 9. Previous study done by Marathe [

The results of optimal tube row numbers obtained in this paper offered a guidance for the design of a DX cooling coil The design of DX cooling coil is related to the real situations such as limitations of space and cost.

The assumptions adopted in deriving the equations cause the uncertainty of the results as well. However, the results still illustrate how to optimize the tube row number for different operating conditions, so as to enhancing the operating performance of a DX cooling coil.

In this section, the comparison between the exergy losses, E_{xl}, with the optimal tube row number and a non-op- timal one was conducted to show the advantage of the optimal tube row number. In comparison, all the other operating and geometrical conditions of the DX cooling coil were maintained as the same as shown in

As well know, exergy loss, E_{xl}, is proportional to the entropy generation, or

The entropy generation number N_{s} can be determined using Equation (9)-(10) and Equation (18). As shown in Section 3.1, n_{t}_{,opt} under conditions as shown in _{xl} and compared with that with n_{t}_{,opt} of 6. It is found that the exergy loss is around 180 W and 311 W respectively for n_{t}_{,opt} of 6 and 3.

The calculation is also extend to evaluate the exery loss as the operating conditions are varied as shown in Section 3.2. All the corresponding results are shown in Figures 5-7. All of these figures show the exergy loss when using optimal tube row number is less than that when choosing the non-optimal tube row number of 3, suggesting the advantage of optimization of the tube row number of a DX cooling coil.

Saving of E_{xl} if using optimal tube row number is calculated,

In these figures, the relative saving of exergy loss (R_{el}_{,s}) by using optimal tube row number are shown in these figures. We can see that the relative ratio ranges from 24% to 70% identifying a considerable exergy loss decrease and the merit of using optimal tube row in a DX cooling coil.

In this paper, a method for analytically calculating the optimal tube row number, n_{t}_{,opt}, of a DX cooling coil is derived from entropy generation minimization principle. A method for evaluating the entropy generation due to

the irreversibility during the air cooling process in a steadily operated DX cooling coil was firstly developed. Using the method, the variation of entropy generation under the variation of tube row numbers for a DX cooling, in a case study was evaluated and the result indicated that entropy generation decreased firstly and increased secondly when the tube row number increased.

Parametric studies on the impacts of various indoor operating conditions, i.e., mass flow rate of inlet air, indoor air temperature and indoor sensible cooling load, on the optimal tube row number are undertaken. The results suggest that the optimal tube row number lies between 5 and 9 when the indoor operating conditions vary with their suitable ranges and under a set of specified operating conditions for a given DX cooling coil having a fixed geometry.

Result of row number which ranges from 5 to 9 is considered reasonable because it is commonly seen in previous relative literatures. The results also show that increase of m_{a} leads to a decrease of n_{t}_{,opt}; increase of T_{a}_{,in} to an decrease of n_{t}_{,opt} and increase of q_{s} to an increase of n_{t}_{,opt}. The saving of exergy loss due to the application of the optimal tube row number in DX cooling coil has been calculated and reported, illustrating a significant energy saving from 24% - 70% as the operating conditions varies.

The analytical method proposed in this paper can be applied conveniently for calculating optimal tube row number of a DX cooling coil. Even though assumptions are adopted in the development of analytical method, the results from the analytical method are still consistent to ones in previous researches. Energy saving effect of the optimal tube row number reported illustrates its advantage and potential in real application.

We gratefully acknowledge the financial supports from the Small Research Grant (UNNC), Ningbo Natural Science Foundation (2014A610025) and (2014A610172), and Qianjiang Talent Scheme (QJD1402009).

A_{o} overall heat transfer area on the air side of a DX cooling coil, m^{2}

C_{p} specific heat, J/(kg∙K)

c_{A} a linear factor between the overall heat transfer area and the tube row number, m^{2}

c_{L} a linear factor between air flow length and the tube row number, m

D air side hydraulic diameter of a DX cooling coil, m

E_{xl} exergyloss, W

E_{xl}_{,s} saving of exergy loss, W

f friction factor

G mass velocity, kg/(m^{2}/s)

h_{fg} latent heat of vapourization of water, J/kg

h_{fg}_{,r} latent heat of vapourization of liquid refrigerant, J/kg

L air flow length in a DX cooling coil, m

m mass flow rate, kg/s

N_{s} total entropy generation number

N_{c}_{ } entropy generation number defined by Equation (9)

Ntu number of heat transfer unit

n_{t} tube row number

P pressure, Pa

∆P pressure difference, Pa

q_{l} latent cooling load of a DX cooling coil, W

q_{s} sensible cooling load of a DX cooling coil, W

R_{a} ideal gas constant, J/(kg∙K)

S entropy, W/K

S_{gen} entropy generation, W/K

s specific entropy, W/(kgK)

T temperature, ˚C

T_{dew }dew point temperature, ˚C

T_{sat} saturation temperature of moist air, ˚C

T_{sh} degree of superheat, ˚C

T_{w} temperature of condensate, ˚C

T_{∞ }_{ }environment temperature, K

t time, s

U_{o} overall air side heat transfer coefficient of a DX cooling coil, W/(m^{2}∙K)

w moisture content, kg/kg

α parameter defined by Equation (21-1)

β parameter defined by Equation (21-2)

ε effectiveness of a DX cooling coil

ρ density, kg/m^{3}

a air

dl dimensionless

e evapourating

in inlet to the cooling coil

opt optimum

out outlet of the cooling coil

r refrigerant

v vapour

DX direct expansion