# Direct Analytical Modeling for Optimal, On-Design Performance of Ejector for Simulating Heat-Driven Systems

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}, η

_{m}and η

_{d}) for the analytical model. The novelty of the presented work is highlighted below:

- A new analytical model is proposed, which is a direct model and does not need iterative processes to get performance prediction;
- This model uses a systematic approach by employing CFD analysis rather than hit-and-trial approach to calculate the ejector efficiencies;
- The proposed model agrees with data published by various researchers for on-design prediction of ejector performance;
- Ejector performance curves produced with the model are presented;
- System simulation and comparison results for ERS and CCP system have been produced;
- The practical applications of the proposed model involve the designing and optimization of thermal systems involving ejectors, for example, ejector refrigeration systems, ejector enhanced ORC systems and other hybrid systems.

## 2. The Analytical Modeling of Ejector

_{g}or P

_{1}, while the secondary fluid inlet pressure is denoted by P

_{e}, where e denotes evaporator from where the secondary fluid enters. At Section 2, the throat of the nozzle, the primary fluid reaches the sonic speed. At Section 3, the two fluids are assumed to have totally mixed to have the same speed. At Section 4, the mixed flow experiences a compression shock. The delivery pressure is denoted by P

_{c}, where c is for the condenser where the mixed fluid is delivered.

_{mc}indicates the pressure of the mixing chamber, whose value is supposed to be constant up to the positioning where shock occurs (Section 4). At Section 2, the primary and secondary fluids are starting to mix at constant pressure. η

_{n}indicates the nozzle isentropic efficiency and accounts for the losses in the convergent-divergent nozzle while the mixing and compression (shock and diffuser) losses are accounted by the efficiencies η

_{m}(mixing) and η

_{d}(diffuser), respectively.

- The model is developed to simulate the on-design, optimum ER values for given conditions. Both motive and suction flows acquire chocked conditions for the critical delivery pressure.
- This model is independent of the size of the ejector, that is, it is non-dimensional or 0-D model and is not able to simulate off-design performance.
- It is assumed that the ejector operates at adiabatic and steady-state conditions.
- Both the inlet velocities are assumed to be negligible, that is, stagnation condition is assumed.
- Both the inlets (motive and suction) are assumed to be at a saturated vapor state.
- The speed at the exit of the ejector is assumed to be negligible.
- The diffuser efficiency accounts for the whole compression (pressure gain) process loss due to shock and diffuser section.
- At Section 2, suction fluid is considered to be chocked, and therefore, it is possible to find the pressure of the constant area mixing section by utilizing the thermodynamic relations.
- For the motive fluid’s expansion calculations, its k-value (exponent for compression and expansion) has been taken as constant.

#### 2.1. Governing Equations

_{g}, P

_{e}, P

_{c}, η

_{n}, η

_{m}and η

_{d}. That is:

_{mc}), the adiabatic equation for suction fluid can be used. That is:

_{mc}is calculated, the values of h

_{6,is}and h

_{1,is}may be calculated. For stagnation inlet conditions, by employing the energy conservation, the velocities V

_{p,2}and V

_{s,2}can be found as:

_{6}and P

_{mc}. Referring to Figure 1, at position 3, both the fluids have the same speeds, and at position 4, shock occurs, the ER is:

_{4}can be obtained.

_{m}), which is the ratio of ideal and actual kinetic energies, becomes:

_{4}because both inlets’ velocities are negligible, hence:

_{4}and h

_{4};

#### 2.2. Computational Procedure

_{mc}is calculated by Equation (3), which helps to determine the values of h

_{6,is}and h

_{1,is}, and then velocities at position 2 are calculated. The three unknown variables at this stage are h

_{4}, V

_{4}and ER, which are obtained by simultaneous solution of Equations (8), (9) and (11), which is a single step.

#### 2.3. Finding the Ejector Efficiencies

#### 2.4. CFD Modelling of Ejectors

_{p}values, which has also been employed by many other researchers [34,49]. While the proposed analytical model of the ejector is using the real gas properties by using the built-in data of the working fluids in the EES software, for the CFD model, as recommended by many researchers, the ideal gas condition is used. While using the ideal gas relations in FLUENT, a good agreement of CFD results with experimental results has been obtained. This is discussed further in the results section. The summary of the settings used in FLUENT is given in Table 1. The CFD results are in good agreement with various published results, and the detailed validation is provided in the results and discussion section.

## 3. Results and Discussion

#### 3.1. Validation of CFD Modelling

#### 3.2. Validation of the Analytical Model

#### 3.3. Ejector Performance Curves

#### 3.4. Thermal Systems Performances

#### 3.4.1. Ejector Refrigeration System (ERS)

#### 3.4.2. Combined Cooling and Power (CCP) System

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Nomenclature

0 D | Zero-dimensional |

1 D | One-dimensional |

2 D | Two-dimensional |

CCP | Combined Cooling and Power |

CFD | Computational fluid dynamics |

COP | Co-efficient of Performance |

D | Diameter, mm |

EES | Engineering Equation Solver |

ER | Entrainment Ratio |

ERS | Ejector Refrigeration System |

EVCC | Enhanced Vapour Compression Cycle |

HVAC | Heating, ventilation and air conditioning |

h | Enthalpy, kJ/kg |

k | Isentropic exponent |

m | Mass flow rate, kg/s |

NXP | Nozzle exit position, mm |

ORC | Organic Rankine Cycle |

P | Pressure, bar |

T | Temperature, °C |

V | Velocity, m/s |

η | Efficiency |

Subscripts | |

1 | Motive (primary) fluid inlet section |

2 | Entrance of the mixing chamber |

3 | Section where the primary and secondary fluids are fully mixed |

4 | Location of section just before the shock wave |

5 | Location of section just after the shock wave |

6 | Secondary (suction) fluid inlet |

7 | Diffuser outlet |

c | Condenser (or delivery) |

d | Diffuser |

e | Evaporator (suction / secondary) |

g | Generator (motive / primary) |

id | Ideal |

is | Isentropic |

m | Mixing-chamber |

mc | mixing chamber (constant-area) |

n | Nozzle (supersonic, converging-diverging) |

p | Primary or motive fluid |

s | Secondary or suction fluid |

t | Throat, primary nozzle |

## Appendix A

#### Appendix A.1. 1-D Model by Huang et al.

_{sy}), and this pressure is also equal to the primary fluid pressure at the same section y-y. This allows the calculation of the primary and secondary fluid areas at section y-y (because the A

_{3}area value is initially assumed); therefore, the secondary mass flow rate can be calculated. At the end of the calculation, the delivery pressure is checked against the required condenser (delivery) pressure, and if the pressure values are not matching, a new value of A

_{3}is assumed, and the calculation is repeated. Therefore, the model is an iterative model, as shown in Figure A2.

**Figure A1.**Notations of ejector used in the 1-D model of Huang et al. [34].

**Figure A2.**Computation procedure of the 1-D model presented by Huang et al. [34].

**Table A1.**Results from the developed EES model based on the 1-D model of Huang et al. [34].

Variable | Value | Units | Variable | Value | Units | Variable | Value | Units |
---|---|---|---|---|---|---|---|---|

A3 | 0.00006642 | m^{2} | Ap1 | 0.0000159 | m^{2} | Apy | 0.00002564 | m^{2} |

Apy_{i} | 0.00002914 | m^{2} | AR | 10.64 | - | Asy | 0.00004078 | m^{2} |

At | 0.000006243 | m^{2} | cp_{g} | 939.3 | J/kg-K | cv_{g} | 807.2 | J/kg-K |

Dp1 | 0.0045 | m | Dt | 0.00282 | m | Eff_{p} | 0.95 | - |

Eff_{s} | 0.85 | - | ER | 0.4682 | - | Fi_{m} | 0.8 | - |

Fi_{p} | 0.88 | - | Kg | 1.164 | - | M3 | 0.6595 | - |

Mm | 1.562 | - | Mp1 | 2.23 | - | Mpy | 2.673 | - |

mp | 0.01069 | kg/s | m_{s} | 0.005006 | kg/s | P3 | 58,291 | Pa |

Pc | 74,748 | Pa | Pe | 40,000 | Pa | Pg | 604,000 | Pa |

Pm | 22,866 | Pa | Pp1 | 53,329 | Pa | Ppy | 22,866 | Pa |

Psy | 22,866 | Pa | Rg | 132.1 | J/kg-K | Te | 281.2 | K |

Tg | 368.1 | K | Tm | 283.7 | K | Tpy | 232.3 | K |

Tsy | 259.9 | K | Vm | 326.2 | m/s | Vpy | 505 | m/s |

Vsy | 199.8 | m/s | - | - | - | - | - | - |

**Table A2.**Validation of the developed EES model based on the 1-D model of Huang et al. [34].

P_{g} (Mpa) | T_{c} (°C) | A_{3}/A_{t} | ω | ||||
---|---|---|---|---|---|---|---|

Theory | Experiment | Difference (%) | Theory | Experiment | Difference (%) | ||

0.604 | 31.3 | 10.87 | 10.64 (EH) | 2.1 | 0.4627 | 0.4377 | 5.7 |

#### Appendix A.2. 0-D Model by Chen

^{’}) it corrects later in a loop against the condenser (delivery) pressure, and then it assumes a value of entrainment ratio (µ

^{’}) that it corrects later with a calculated value (µ), as shown in Figure A3. Because of the double iterative process, the programming is more challenging, and the model is difficult to integrate with other models.

**Figure A3.**Computation procedure used by Chen [40] for their 0-D model.

P_{g}[bar] | T_{g}[°C] | P_{c}[bar] | P_{evaporator}[bar] | ER (Experiment by Huang et al. [34]) | ER (Chen [40]) | ER (Developed EES Model) |
---|---|---|---|---|---|---|

6.05 | 95 | 0.986 | 0.399 | 0.4377 (Model EH) | 0.4387 | 0.4122 |

_{a}(assumed entrainment ratio) value becomes equal to ER

_{cal}(calculated entrainment ratio) as well as when Pc

_{cal}(calculated condenser pressure) becomes equal to the value of P

_{c}(required condenser pressure). The double iterative process makes it difficult for the solution to cover because for every assumed value of one parameter, the other parameter needs to converge hence making it a lengthy process. The successful EES modelling of the 0-D model proposed by Chen [40] enables the calculation of entrainment ratio for various other working conditions for comparison with other models or for using in system analysis and optimization.

**Table A4.**Result from the developed EES model based on 0-D model of Chen [40].

Variable | Value | Units | Variable | Value | Units | Variable | Value | Units |
---|---|---|---|---|---|---|---|---|

AR | 10.45 | - | C4 | 148.6 | m/s | cp | 867.6 | J/kg-K |

cv | 763 | J/kg-K | Eff_{d} | 0.82 | - | Eff_{m} | 0.85 | - |

Eff_{n} | 0.95 | - | ER_{a} | 0.4122 | - | ER_{cal} | 0.4122 | - |

h2 | 274,432 | J/kg | h2_{i} | 271,964 | J/kg | h4 | 285,967 | J/kg |

hc_{ideal} | 317,315 | J/kg | ho | 271,858 | J/kg | hc | 324,196 | J/kg |

h_{eo} | 282,632 | J/kg | h_{go} | 341,329 | J/kg | k | 1.137 | - |

M4 | 1.861 | - | M4c | 1.884 | - | M4st | 1.747 | - |

M5 | 0.5654 | - | Me_{2} | 1.012 | - | Me_{2st} | 1.011 | - |

Mg_{2} | 2.593 | - | Mg_{2st} | 2.218 | - | P2 | 22,750 | Pa |

P4 | 22,750 | Pa | P5 | 82,405 | Pa | Pc | 98,600 | Pa |

Pc_{cal} | 98,639 | Pa | Pe | 39,927 | Pa | Pg | 604,929 | Pa |

P_{ge} | 322,428 | Pa | s4 | 1073 | J/kg-K | sc_{ideal} | 1073 | J/kg-K |

s_{eo} | 1021 | J/kg-K | s_{go} | 1022 | J/kg-K | Te | 281.2 | K |

Tg | 368.2 | K | u2 | 363 | m/s | u4 | 276.5 | m/s |

u4_{i} | 299.9 | m/s | uo | 146.8 | m/s | - | - | - |

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**Figure 1.**Variation of velocity and pressure at different section of ejector [40].

**Figure 2.**Enthalpy–Entropy diagram for ejector’s thermal processes. The blue line presents the secondary fluid; the red line represents the primary fluid; the pink color represents the mixed fluid; the green color represents the constant pressure mixing process.

**Figure 15.**Optimization of a novel CCP using the proposed ejector model and its performance comparison.

Meshing | Structured |

Turbulence | Model: k-ε realizable |

Solver | Axisymmetric, Pressure based |

Energy | Kept ON |

Compressibility | Considered |

Refrigerant | Constant Cp, Ideal gas |

Boundary Conditions | Pressure outlet and inlet |

Initialization | Hybrid |

Discretization | 2nd order scheme |

Residuals | 10^−6 |

**Table 2.**comparison of Results of the proposed model with the experimental results reported by Eames et al. [32].

T_{motive} | T_{suction} | T_{delivery} | Compression Ratio | ER Values, Eames et al. | COP Value of ERS Eames et al. | COP of ERS (Proposed Model) | ER Values (Proposed Model) | Difference in ER Values | Difference in COP Values |
---|---|---|---|---|---|---|---|---|---|

[°C] | [°C] | [°C] | Pdelivery/Psuction | - | - | [%] | [%] | ||

110 | 15 | 33.5 | 2 | 0.94 | 0.67 | 0.6522 | 0.896 | 4.7 | 2.7 |

110 | 12 | 33 | 2.213483146 | 0.76 | 0.54 | 0.56 | 0.778 | 2.4 | 3.7 |

110 | 10 | 32.5 | 2.358536585 | 0.69 | 0.48 | 0.51 | 0.719 | 4.2 | 6.2 |

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## Share and Cite

**MDPI and ACS Style**

Riaz, F.; Yam, F.Z.; Qyyum, M.A.; Shahzad, M.W.; Farooq, M.; Lee, P.S.; Lee, M.
Direct Analytical Modeling for Optimal, On-Design Performance of Ejector for Simulating Heat-Driven Systems. *Energies* **2021**, *14*, 2819.
https://doi.org/10.3390/en14102819

**AMA Style**

Riaz F, Yam FZ, Qyyum MA, Shahzad MW, Farooq M, Lee PS, Lee M.
Direct Analytical Modeling for Optimal, On-Design Performance of Ejector for Simulating Heat-Driven Systems. *Energies*. 2021; 14(10):2819.
https://doi.org/10.3390/en14102819

**Chicago/Turabian Style**

Riaz, Fahid, Fu Zhi Yam, Muhammad Abdul Qyyum, Muhammad Wakil Shahzad, Muhammad Farooq, Poh Seng Lee, and Moonyong Lee.
2021. "Direct Analytical Modeling for Optimal, On-Design Performance of Ejector for Simulating Heat-Driven Systems" *Energies* 14, no. 10: 2819.
https://doi.org/10.3390/en14102819