# Modelling of Evaporator in Waste Heat Recovery System using Finite Volume Method and Fuzzy Technique

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## Abstract

**:**

## 1. Introduction

## 2. Evaporator in Waste Heat Recovery (WHR) System

**Figure 3.**Plate heat exchanger [27].

Parameter | Quantity | Value |
---|---|---|

A | Heat transfer area of the evaporator | 5.78 m^{2} |

L | Length of each plate of the evaporator | 0.478 m |

W | Width of each plate of the evaporator | 0.124 m |

N_{p} | Number of plates | 100 |

K | Thermal conductivity | 15 W/m K |

## 3. Modelling of Evaporator Using the Finite Volume Method

- There is no pressure loss in either the hot or cold side of the heat exchanger.
- Heat transfer from or to the surrounding environment is negligible.
- Heat exchanger fouling is not included in the model.
- Heat from the hot fluid is completely transferred to the working fluid.

^{2}K) of the hot fluid and refrigerant with the wall. ${A}_{{h}_{j}}$ and ${A}_{{r}_{j}}$ are the heat transfer surface areas, ${T}_{h}$ and ${T}_{r}$ are the hot fluid and refrigerant’s average temperature, within each finite volume, respectively. ${T}_{wall}$ is the wall temperature, which can be obtained from the average of the hot fluid and refrigerant temperature of the cell. These parameters are calculated as follows:

^{3}) of the fluid, $\mu $ is the viscosity (Pa.s) and $V$ is the velocity of the fluids (m/s).

- Step 1:
- All inputs of the model are defined at the beginning of the iteration process. The first segment is then initialized by assigning an initial inlet refrigerant temperature and assuming an initial hot fluid outlet temperature as shown in Figure 6.
- Step 2:
- Set the initial values for the inlet, outlet and wall temperatures of the segment $j=1$ as shown in Figure 7.
- Step 3:
- When all inlet and outlet temperatures of the first segment are known, the wall temperature of the evaporator is iteratively evaluated until the heat transfer rates in Equations (1) and (2) are equal with a selected maximum deviation of ${\epsilon}_{1}=0.1$.
- Step 4:
- The heat transfer rate of the fluids at Step 3 is used to calculate the output variables of each segment by using the energy balance condition of the fluids. The iteration at this step is repeated until the deviations are within the allowable limits of the convergence values as shown in Table 2. The values are a compromise chosen to reduce the computation time while achieving reasonable model accuracy.
- Step 5:
- At this stage, the outlet variables of the first segment are all known. The iteration process continues along the refrigerant flow direction, the output variables of the first segment are used as the input of the second segment as shown in the Figure 4b and the steps 2–4 repeated until the deviations are satisfied. This process is repeated until the ${N}^{th}$ segment as shown in Figure 6.
**Table 2.**Convergence value for iteration loops.Convergence Name Convergence Value ε _{1}0.1 ε _{2}0.1 ε _{3}0.1 ε _{4}0.2 - Step 6:
- At the end of the ${N}^{th}$ segment, the calculated hot fluid temperature at the inlet of the evaporator ${T}_{h,i,cal}$ is obtained. This calculated temperature is then compared with the real hot fluid data. If the error between the calculated and real temperature is less than the deviation shown in Table 2, the iteration process stops. Otherwise, the iteration process is repeated at steps 1–6.

#### Simulation Results of Finite Volume Evaporator Model

Operating System | Windows^{®} 7 Enterprise 64-bits |
---|---|

Processor | Intel^{®} Core^{™} i7-3770 CPU @3.40GHz; 16384MB RAM |

Programming software | MATLAB^{®} R2014a 64 bits |

Input Profile (each profile contains 1470 data sets) | Simulation Time |
---|---|

Ramp profile | 13870 (s) |

Random profile | 14826 (s) |

## 4. Fuzzy Evaporator Model

^{th}fuzzy sets of the input and output variables of the fuzzy system. In this research, trapezoidal functions are used as the membership functions, denoted by μ in Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18. The numbers of rules for the fuzzy model are dependent on the number of input membership functions used to define the system. The rules are determined from intuition and knowledge of characteristics of the evaporator and are shown in Table 5 and in surfaces in Figure 19, Figure 20, Figure 21 and Figure 22.

Rule number | IF ${\dot{m}}_{r}$ is | AND ${\dot{m}}_{h}$ is | AND ${T}_{h}$ is | THEN ${T}_{ev}$ is | AND ${Q}_{ev}$ is |
---|---|---|---|---|---|

1 | L | L | L | LM | VL |

2 | L | L | M | M | L |

3 | L | L | H | VH | L |

4 | M | L | L | VL | L |

5 | M | L | M | LM | LM |

6 | M | L | H | M | M |

7 | H | L | L | VL | LM |

8 | H | L | M | LM | M |

9 | H | L | H | LM | MH |

10 | L | M | L | LM | VL |

11 | L | M | M | MH | L |

12 | L | M | H | VH | L |

13 | M | M | L | L | LM |

14 | M | M | M | M | M |

15 | M | M | H | MH | MH |

16 | H | M | L | L | M |

17 | H | M | M | LM | MH |

18 | H | M | H | MH | VH |

19 | L | H | L | LM | VL |

20 | L | H | M | MH | L |

21 | L | H | H | VH | L |

22 | M | H | L | LM | LM |

23 | M | H | M | M | M |

24 | M | H | H | H | MH |

25 | H | H | L | L | M |

26 | H | H | M | LM | H |

27 | H | H | H | MH | VH |

**Figure 19.**Fuzzy surface for the evaporator outlet temperature with respect to ${\dot{m}}_{r}$ and ${\dot{m}}_{h}$.

**Figure 20.**Fuzzy surface for the evaporator outlet temperature with respect to ${\dot{m}}_{r}$ and ${T}_{h}$.

**Figure 21.**Fuzzy surface for the evaporator power with respect to ${\dot{m}}_{r}$ and ${\dot{m}}_{h}$.

#### Simulation Results of Fuzzy Based Evaporator Model

Performance Indicator | RMSE | Fitness (%) | Simulation Time |
---|---|---|---|

${Q}_{ev}$-Ramp Profile | 0.95 (kW) | 93.68 | Almost instantly (<1 s) |

${T}_{ev}$-Ramp Profile | 1.48 (K) | 89.16 | |

${Q}_{ev}$-Random Profile | 0.92 (kW) | 92.31 | Almost instantly (<1 s) |

${T}_{ev}$-Random Profile | 3.02 (K) | 87.00 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

A | heat transfer area, m^{2} |

C_{p} | specific heat capacity, kJ/kg.K |

D | hydraulic diameter, m |

H | specific enthalpy, kJ/kg |

h | heat transfer coefficient, kW/m^{2}K |

K | thermal conductivity, W/mK |

$\dot{\mathrm{m}}$ | mass flow rate, gm/s |

L | plate length, m |

N | number of segments |

N_{p} | Number of plates |

Nu | Nusselt number, - |

Pr | Prandtl number, - |

Q | heat power, kW |

Re | Reynolds number, - |

T | temperature, K |

V | volume, m^{3} or velocity, m/s |

W | plate width, m |

Y | crisp output |

ε | convergence condition |

ρ | density, kg/m^{3} |

υ | specific volume, m^{3}/kg |

μ | dynamic viscosity, Pa.s |

## Subscripts

b | bulk |

r | refrigerant |

h | hot fluid |

i | in or inlet |

o | out or outlet |

ev | evaporator |

pc | pseudo-critical |

j | segment or cell |

wall | evaporator wall |

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**MDPI and ACS Style**

Chowdhury, J.I.; Nguyen, B.K.; Thornhill, D.
Modelling of Evaporator in Waste Heat Recovery System using Finite Volume Method and Fuzzy Technique. *Energies* **2015**, *8*, 14078-14097.
https://doi.org/10.3390/en81212413

**AMA Style**

Chowdhury JI, Nguyen BK, Thornhill D.
Modelling of Evaporator in Waste Heat Recovery System using Finite Volume Method and Fuzzy Technique. *Energies*. 2015; 8(12):14078-14097.
https://doi.org/10.3390/en81212413

**Chicago/Turabian Style**

Chowdhury, Jahedul Islam, Bao Kha Nguyen, and David Thornhill.
2015. "Modelling of Evaporator in Waste Heat Recovery System using Finite Volume Method and Fuzzy Technique" *Energies* 8, no. 12: 14078-14097.
https://doi.org/10.3390/en81212413