# An Approach to the Integrated Design of PCM-Air Heat Exchangers Based on Numerical Simulation: A Solar Cooling Case Study

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. DOE Approach

#### 1.2. Numerical Simulations Based On DOE

- The overall frame is the DOE approach which includes the response surface technique for multi-objective optimization purposes. One important point of the response surface is that once an optimal solution is calculated, the optimization plot is obtained. The optimal solution serves as the starting point for this plot and then the settings can be modified interactively to see how different settings affect responses. No additional simulation runs are required to achieve any other design allowing to modify factors and getting new responses.
- This approach is easier and more intuitive to use for “non-expert” designers and decision-makers. The main idea is to provide a more attractive, though rigorous, tool for designers. Traditional optimization techniques usually require a medium-high level of mathematics. Although not trivial, it is much easier to understand and interpret design criteria, as well as the main parameters that define the analysis (such as the weight and the importance).
- Numerical simulations are not supposed to suffer from variability; however, it is well known that uncertainties in the model input variables/parameters can lead to completely different results in the simulations. In fact, in the type of systems studied here, this occurs with the uncertainty in the inlet temperature (among others). Although not considered in this article, the DOE approach can deal with such uncertainties.

#### 1.3. Design and Optimization of TES Systems for Solar Cooling

_{p}(T), h(T), λ(T), ρ(T), rheological properties, thermal cycling, etc.).

## 2. Methodology

#### 2.1. Similarity Analysis: Scaling the Model

#### 2.2. Validity Range

- Re is the Reynolds number defined as Re = ρ·v·D
_{h}/µ, where D_{h}= 4·A/P; - NTU is the number of transfer units defined as NTU = (h·Δx·w)/C
_{air} - Bi is the Biot number defined as Bi = h·e/λ
_{encapsulation} - λ
_{eff}/λ is the thermal conductivities ratio that quantifies the effect of natural convection within the PCM inside the plate. As this relation goes beyond one, the effect of natural convection is more substantial.

Re | NTU | Bi | λ_{eff}/λ | Ramp (°C/min) | |||||
---|---|---|---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | Min | Max | Min | Max |

917 | 2577 | 0.013 | 0.039 | 0.088 | 0.875 | 1.00 | 1.74 | 0.05 | 0.25 |

_{1}is the air temperature in the TES unit (average between the inlet and the outlet); T

_{2}corresponds to T

_{melt}; the volumetric expansion coefficient, β, is given by the PCM manufacturer (0.001 K

^{−}

^{1}); the dynamic viscosity value, µ (3.38 × 10

^{−}

^{3}Pa·s at 40 °C) is provided by the manufacturer and is used as well as those measured in the Malvern laboratory with a Gemini II rheometer; and Nu, Ra, and Pr correspond to the Nusselt, Rayleigh, and Prandtl numbers:

## 3. Calculation

#### 3.1. Potential Application: Case Study of Absorption Cycles for Solar Cooling

#### 3.2. A Real Case of an Absorption Cycle and a Proposed Combi-System with a PCM Storage Unit

_{2}O pair, single effect) are available. The system is installed and is currently working in the sports hall at the University of Zaragoza (Spain). The machine ejects heat from the system through a finned cooling coil and an axial fan that allows blowing outside air to the coil. Hot water is cooled as it transfers heat from the tube walls of the coil to the ambient air. Thus, in the hottest days of the year, if the air temperature is above 35 °C the absorption machine will not work properly [27].

^{3}/h. Given these conditions, it is proposed to place the TES unit just before the entrance to the air-condensing unit, which will soften the increase in the ambient air temperature at the condenser inlet. Thus, as the ambient air temperature increases and the TES unit cushions its effect, both the COP and the cooling power of the system will improve compared to the same system without the TES unit.

_{2}O solution in the generator):

_{g,i}and T

_{g,o}are measured at the inlet and outlet entries of the generator, respectively. ${\dot{W}}_{ch}$ is the chilling power (heat exchanged from the chilling circuit to the evaporator):

_{ev,i}and T

_{ev,o}are measured at the inlet and outlet entries of the evaporator, respectively.

#### 3.3. Design

- Reducing the maximum temperature of air flowing through the condensing unit.
- Smoothing the air temperature curve at the outlet of the condenser.
- Reducing the pressure drop in order to achieve a low electrical consumption of the fan.
- Improving the melting degree (i.e., efficiency: Ratio between the energy exchanged and the theoretical energy available of the PCM).

_{PCM}); length of the PCM plate system (L

_{sist}); thickness of the PCM plate (e

_{plate}); and width of the air gap between PCM plates (e

_{air}).

- Volumetric airflow is determined by the condensing unit and is set at 5500 m
^{3}/h. - Number of plate walls is set at 18 (the same number as in the experimental prototype).
- The only PCM considered was the commercial one used in the experimental prototype: RT27 (Rubitherm [29]).
- Finishing of the simulated plates is the same as the plates in the prototype (round bulges).
- Heat generation of the fan is considered constant and equal to 300 W.

- Air maximum temperature, T
_{max}. - Pressure drop (necessary to determine which fan to use, electrical consumption, and noise), Δp.
- PCM melting degree for design conditions, ° Melt.

- Width of the TES unit (along with the length and depth allowing the floor space and the volume occupied by the unit to be calculated).
- Maximum heat transfer rate supplied by the TES unit,${\dot{Q}}_{max}$.

- The amount of PCM has to be theoretically sufficient for the stored energy to cover the needs of the average heat rate (13 kW);
- Demand: 13 kW during eight hours 374,400 kJ;
- Mass of RT27 required to cover the demand: h
_{sl}= 179 kJ/kg (which corresponds to the PCM latent heat in the thermal window of 20 to 32 °C) 2092 kg.

Factors | Domain | |
---|---|---|

Level (−) | Level (+) | |

PCM mass (kg) | 1000 | 3000 |

Length of the plate system (m) | 1 | 5 |

Thickness of the PCM plate (mm) | 5 | 15 |

Thickness of air duct gap (mm) | 5 | 55 |

- The length of the plate system is one of the two dimensions that will determine the unit’s floor space. Depending on the location of the unit (on the roof or inside ducts), this value will depend on the available space. In this case, for the factor length, we have selected a lower level of 1 m and a higher level of 5 m (that allows the placement on the roof with sufficient room).
- The plate thickness is set at a lower level of 5 mm and a higher level of 15 mm. This value of maximum thickness should always obey the relation L/e > 10 in order to assume one-dimensional conduction (in this specific case 1000/15 = 66.6 >> 10).
- The air gap has a lower level of 5 mm and a higher level of 55 mm. The idea behind this is to leave ample room to find a trade-off between compactness on the one hand and a small pressure drop on the other.

## 4. Results and Discussion

#### 4.1. Response Optimization

- Minimizing the maximum temperature reached by the air (32 °C is the objective, the upper limit being 33.5 °C, which is 3 °C below the maximum outdoors temperature).
- Maximizing the melting degree (i.e., efficiency) with the objective of minimizing the amount of PCM, but accepting as a minimum an efficiency of 50%.
- Minimizing the pressure drop reducing the electrical consumption, accepting values close to 25 Pa, but not beyond 50 Pa.

Response | Goal | Lower | Target | Upper | Weight | Importance |
---|---|---|---|---|---|---|

T_{max} (°C) | Maximum | 20 | 32 | 33.5 | 1 | 5 |

°Melt (%) | Maximum | 50 | 100 | 100 | 1 | 10 |

Δp (Pa) | Minimum | 1 | 25 | 50 | 1 | 1 |

- minimize the response,
- target the response,
- maximize the response.

**Figure 3.**Default desirability function for a minimization goal (adapted from [31]).

- less than 1 (minimum is 0.1) places less emphasis on the target,
- equal to 1 places equal importance on the target and the bounds,
- greater than 1 (maximum is 10) places more emphasis on the target.

**Figure 4.**Shape of the desirability function changes depending on the weight (adapted from [31]).

**Figure 5.**Summary of desirability functions shapes (adapted from [31]).

#### 4.2. Suggested TES Unit

^{3}/h, with 2200 kg of RT27 arranged in plates of 20 mm thickness and air channels of 40 mm. The plate system length is 4.9 m.

^{2}and its volume is 8.4 m

^{3}. The results obtained from the simulation fit quite well with those predicted by the optimization in terms of T

_{max}, although they are somewhat better for the efficiency and show higher values for the pressure drop. However, it is also shown that the simulation follows the trends predicted by the optimization as, in the simulation, a smaller amount of PCM (increased efficiency) and a narrower air gap (increased Δp) have been considered compared to the optimal values.

M_{PCM} (kg) | L_{sist} (m) | e_{plate} (mm) | e_{air} (mm) | ${\dot{Q}}_{max}$ (W) | T_{max} (°C) | v_{air} (m/s) | w (m) | °Melt. (%) | Δp (Pa) |
---|---|---|---|---|---|---|---|---|---|

2200 | 4.9 | 20 | 40 | 10257 | 32.9 | 1.43 | 1.48 | 66.07 | 63 |

#### 4.3. Dimensional Analysis and Similarity

_{eff}/λ ratio). The range of experimental validity is shown in Table 1. Figure 7 shows the results of the calculation of every parameter for each simulation. In these plots, what is reported is whether or not each of the units simulated according to the DOE plan (diamonds) matches the validity range (grey zone), and in particular if the suggested design (circle) does so.

_{eff}/λ ratio obtained is high, mainly due to the thickness of the selected PCM (20 mm). Re is also outside the experimental validity range. As the airflow is set, reconciling the application requirements with the adjustment of the Re to the experimental validity range is only achievable through the hydraulic diameter. Since the theoretical model itself takes into account both the effective thermal conductivity as well as the different flow regime, it seems reasonable to extrapolate by means of the simulations outside the experimental validity range. However, to be fully rigorous, the suggested unit proposed here would require additional experimentation for empirical validation, although the Bi and the NTU are within the experimental validity range. Therefore, it is concluded that the analysis performed for this solar cooling application is useful for a pre-design stage and, once validated with additional experimentation, would also be valid in a design stage.

# | Simulation plan | Responses | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

M_{PCM} (kg) | L_{sist} (m) | e_{plate} (mm) | e_{aire} (mm) | ${\dot{Q}}_{max}$ (W) | T_{max} (°C) | v_{air} (m/s) | Width (m) | °Melt (%) | Δp (Pa) | |

1 | 1000 | 1 | 5 | 5 | 8227 | 36.2 | 1.33 | 12.77 | 74.09 | 117 |

2 | 3000 | 1 | 5 | 5 | 15,500 | 34.5 | 0.44 | 38.31 | 65.34 | 56 |

3 | 1000 | 5 | 5 | 5 | 11,935 | 36.3 | 6.65 | 2.55 | 100 | 170 |

4 | 3000 | 5 | 5 | 5 | 15,501 | 34.5 | 2.22 | 7.66 | 65.34 | 150 |

5 | 1000 | 5 | 15 | 55 | 7385 | 36.3 | 3.99 | 4.26 | 71.75 | 120 |

6 | 3000 | 5 | 15 | 55 | 12,673 | 31.8 | 1.33 | 12.77 | 51.83 | 115 |

7 | 100 | 3 | 10 | 30 | 7621 | 36.3 | 19.94 | 0.85 | 73.02 | 250 |

8 | 4000 | 3 | 10 | 30 | 16,085 | 32.3 | 6.65 | 2.55 | 67.65 | 180 |

9 | 2000 | 0.25 | 10 | 30 | 3912 | 34.2 | 0.12 | 12.77 | 47.00 | 25 |

10 | 2000 | 7 | 10 | 30 | 9310 | 31.2 | 0.04 | 38.31 | 38.63 | 25 |

11 | 2000 | 3 | 1 | 30 | 7543 | 36.2 | 0.60 | 2.55 | 72.91 | 27 |

12 | 2000 | 3 | 20 | 30 | 9334 | 31.1 | 0.20 | 7.66 | 38.72 | 26 |

13 | 2000 | 3 | 10 | 3 | 4284 | 34.2 | 0.36 | 4.26 | 50.29 | 25 |

14 | 2000 | 3 | 10 | 80 | 4830 | 33.5 | 0.12 | 12.77 | 20.13 | 25 |

15 | 2000 | 3 | 10 | 30 | 4844 | 34.2 | 1.81 | 0.85 | 55.99 | 53 |

16 | 2000 | 3 | 10 | 30 | 11,543 | 31.2 | 0.60 | 2.55 | 47.20 | 27 |

17 | 2000 | 3 | 10 | 30 | 1077 | 36.3 | 13.29 | 0.21 | 61.59 | 185 |

18 | 2000 | 3 | 10 | 30 | 10,608 | 30.4 | 0.33 | 8.52 | 33.16 | 27 |

19 | 2000 | 3 | 10 | 30 | 6400 | 32.7 | 0.06 | 51.08 | 39.24 | 30 |

20 | 2000 | 3 | 10 | 30 | 12,222 | 35.4 | 1.55 | 1.82 | 70.69 | 76 |

21 | 2000 | 3 | 10 | 30 | 13,967 | 35.1 | 0.07 | 42.57 | 87.99 | 31 |

22 | 1000 | 5 | 15 | 55 | 9553 | 33.1 | 1.33 | 2.13 | 56.54 | 37 |

23 | 3000 | 5 | 15 | 55 | 14,791 | 35.5 | 6.65 | 4.26 | 90.19 | 230 |

24 | 100 | 3 | 10 | 30 | 7865 | 32.3 | 0.25 | 4.26 | 47.49 | 25 |

25 | 4000 | 3 | 10 | 30 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

26 | 2000 | 0.25 | 10 | 30 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

27 | 2000 | 7 | 10 | 30 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

28 | 2000 | 3 | 1 | 30 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

29 | 2000 | 3 | 20 | 30 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

30 | 2000 | 3 | 10 | 3 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

31 | 2000 | 3 | 10 | 80 | 10,592 | 34.2 | 0.66 | 4.26 | 62.61 | 28 |

**Figure 7.**Re numbers (

**a**); Bi numbers (

**b**); NTU numbers (

**c**); and λ

_{eff}/λ ratio (

**d**) for each simulation and experimental validity.

#### 4.4. Evaluation of the System Performance During the Operation Period (Cooling)

System setup | Average COP | Hours working with T > 27 °C |
---|---|---|

without TES | 0.51 | 456 |

with TES | 0.583 | 399.3 |

#### 4.5. Initial Investment Evaluation

**Table 7.**Order of magnitude of the TES unit casing cost as a function of the PCM volume inside the unit.

Volume (m^{3}) | Cost (€) |
---|---|

0.5 | 174 |

1.3 | 376 |

2.0 | 658 |

Flow (m^{3}/h) | Type | Consumption (W) | Cost (€) |
---|---|---|---|

2000 | RER10 11-315 | 290 | 547 |

5000 | RER10 11-384 | 450 | 591 |

8000 | RER10 11-577 | 720 | 837 |

^{3}/h is estimated at 47,460 €. This value is highly restrictive so that in these conditions it will not be economically feasible. To make the TES unit competitive against conventional equipment in terms of initial investment, the price of the macroencapsulated PCM would have to be reduced down to about 5 €/kg. This can be deduced from Equation (9) that shows the linear relation between the unit initial investment costs (y) depending on the price of the macroencapsulated PCM in aluminum plate (x). Only a reduction in the price of the macroencapsulated PCM to the lowest values (5 €/kg

_{PCMencapsulated}) would establish a scenario in which the investment cost of the TES unit would be economically competitive compared to conventional cooling equipment.

## 5. Conclusions

- Reducing the number of simulation runs and the time invested. In the case study described in this paper, each numerical simulation corresponds to one single cycle at most (with a duration of two to six minutes per run on an average desktop computer), and only the behavior of the TES unit is simulated. Simulations of buildings are usually for one-year periods and include a complete system: an annual simulation with a simple system, a building and incorporating the TES unit can last 12 h with the numerical model.
- Provided you have a PCM-TES numerical model empirically validated, the methodology can:
- -
- Adapt the design of the TES unit depending on the final requirements (responses) and in accordance to a series of variables or parameters (factors) influencing the system.
- -
- Find optimal points of operation based on multi-response functions criteria. These functions can be shape-defined by the designer depending on their goal (maximization of the function, minimization, or target objective), their weight, and their importance. This is done through desirability functions. In the approach to optimization, each of the response values are transformed using a specific desirability function. The weight determines how the desirability is distributed on the interval between the lower (or upper) bound and the target. It determines the shape of the desirability function that is used to translate the response scale to the zero-to-one desirability scale to determine the individual desirability of a response. In addition, the individual desirabilities are weighted according to the importance that we assign each response.
- -
- A key feature is that without having to run more numerical simulations, a designer can identify the factor settings that enable any other requirements to be fulfilled.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature, Acronyms and Definitions

A [m^{2}] | area |

c_{p} [J/(kg·K)] | specific heat capacity at constant pressure |

d | desirability parameter (ranges from 0 to 1) |

D_{h} [m] | hydraulic diameter |

e_{air} [m] | thickness of the air gap between two PCM plates |

e_{enc} [m] | thickness of the plate encapsulation material |

e_{plate} [m] | thickness of the PCM plate |

h [W/(m^{2}·K)] | air convection coefficient |

L_{sist} [m] | Length of the PCM system in the TES unit |

$\dot{m}$ [kg/s] | mass flow |

M_{PCM} [kg] | PCM mass |

N | number of PCM plate walls (inside the TES unit) |

P [m] | wet perimeter |

$\dot{Q}$ [W] | heat transfer rate |

T [°C] | temperature |

T_{amb} [°C] | outdoors air temperature |

v [m/s] | velocity |

V [m^{3}] | volume |

w [m] | width |

°Melt | ratio of PCM melted, percentage |

Δp [Pa] | pressure difference |

λ [W/(m·K)] | thermal conductivity |

λ_{eff} [W/(m·K)] | effective thermal conductivity |

ρ [kg/m^{3}] | density |

µ [Pa·s] | dynamic viscosity |

Bi = h·e/λ_{enc} | Biot number |

${C}_{air}={\rho}_{air}\xb7\frac{\dot{V}}{2\xb7N}\xb7{c}_{pair}$ | heat capacity [J/(s·K)] |

Re, = ρ·v·D_{h}/μ | Reynolds number |

NTU = (h·Δx·w)/C_{air} | number of transfer units |

COP | Coefficient of Performance |

CSM | Compact Storage Module |

DOE | Design of Experiments |

PCM | Phase Change Material |

TES | Thermal Energy Storage |

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## Share and Cite

**MDPI and ACS Style**

Dolado, P.; Lazaro, A.; Delgado, M.; Peñalosa, C.; Mazo, J.; Marin, J.M.; Zalba, B.
An Approach to the Integrated Design of PCM-Air Heat Exchangers Based on Numerical Simulation: A Solar Cooling Case Study. *Resources* **2015**, *4*, 796-818.
https://doi.org/10.3390/resources4040796

**AMA Style**

Dolado P, Lazaro A, Delgado M, Peñalosa C, Mazo J, Marin JM, Zalba B.
An Approach to the Integrated Design of PCM-Air Heat Exchangers Based on Numerical Simulation: A Solar Cooling Case Study. *Resources*. 2015; 4(4):796-818.
https://doi.org/10.3390/resources4040796

**Chicago/Turabian Style**

Dolado, Pablo, Ana Lazaro, Monica Delgado, Conchita Peñalosa, Javier Mazo, Jose M. Marin, and Belen Zalba.
2015. "An Approach to the Integrated Design of PCM-Air Heat Exchangers Based on Numerical Simulation: A Solar Cooling Case Study" *Resources* 4, no. 4: 796-818.
https://doi.org/10.3390/resources4040796