# Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System

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## Abstract

**:**

## 1. Introduction

## 2. Ejector Description and Modeling

#### 2.1. Mathematical Solution of Ejectors

_{c}) [12]. The geometry of an ejector is shown in Figure 2. The important boundary conditions include throat area of primary nozzle (A

_{t}), area of mixing chamber (A

_{3}), and the inlet pressures of motive fluid (P

_{g}) and secondary fluid (P

_{e}).

_{g}corresponding to saturated vapor temperature from 351 K to 368 K, which lies within the range of ejector refrigeration machinery. The various geometrical specifications of ejector that were used in this study are given in Table 1.

- The working fluid acts as an ideal gas, having constant specific heat (C
_{p}) and specific heat ratio (γ). - Steady, adiabatic, and 1D flow.
- Negligible kinetic energy at secondary flow inlet, primary nozzle inlet, and diffuser exit.
- Use of isentropic relations and constant mixing chamber efficiency.
- The primary and secondary fluid flow mixes at hypothetical throat located within constant area section (Section 2–3).
- Constant pressure mixing (P
_{pH}= P_{sH}). - Choking of entrained flow at hypothetical throat and Mach number of M
_{sH}= 1 is assumed. - Adiabatic ejector walls.

#### 2.2. Mathematical Solution of Ejectors

_{c}without the need of iterations. A regression analysis was conducted to identify the likely relation between ejector performance parameters and ejector operating conditions and geometrical specifications. To correlate the inputs (A

_{3}, A

_{t}, P

_{g}, P

_{e}) with outputs (ω and C

_{P}), regression equations were developed using the least square method. As ω and P

_{c}are two dependent variables, two regression equations were proposed, which required both area ratio (A

_{3}/A

_{t}) and pressure ratio (P

_{g}/P

_{e}) as an input (independent variables). The relationship between dependent variable ω with A

_{3}/A

_{t}and P

_{g}/P

_{e}is shown Figure 3.

_{g}/P

_{e}and A

_{3}/A

_{t}, and there exists a negative nonlinear association between ω and P

_{g}/P

_{e}, that is, decreasing the value of P

_{g}/P

_{e}results in increased entrainment ratio. Similarly, the association between the other dependent variable (P

_{cn}) and independent variables is also shown by Figure 4. Each graphical representation shows that the ejector performance parameters are highly influenced under the variations in inlet flow pressures (P

_{g}, P

_{e}) and ejector geometry (A

_{3}, A

_{t}).

_{cn}/P

_{e}), A

_{3}/A

_{t}, and P

_{g}/P

_{e}. An empirical correlation of second-degree polynomial (that predicts ejector performance in terms of ω as a function of A

_{3}/A

_{t}and P

_{g}/P

_{e}), was developed using polynomial regression techniques because of the existence of the nonlinear behavior among the predictor and response variables, as shown in Figure 3 and Figure 4. Using a forward selection method, a second-degree polynomial, in terms of area ratio and pressure ratio, was selected. The general equation with second-degree polynomial [30] in the present case becomes:

_{0}: intercept; β

_{1}and β

_{2}: linear effect parameters; β

_{11}, β

_{22}: quadratic effect parameters; β

_{12}: interaction parameter; and ε: error. The above equation can be represented in terms of specific independent variables, as given below:

_{0}, β

_{1}, β

_{2}, β

_{11}, β

_{2}, and β

_{12}are coefficient parameters of the equation. These equation parameters were found by applying the least square approach on the matrix form of the above equations, which generated the following matrix relation:

_{3}/A

_{t}and P

_{g}/P

_{e}), and matrix [Y] contained all n values of dependent variable (ω) associated with specific values of independent variables. The above matrix expression was then solved in MATLAB, and the following empirical correlation was obtained:

^{−6}.

#### 2.3. CFD Model Validation

_{t}, A

_{4}, and A

_{3}, are taken from Table 1.

_{g}: 0.4–0.604 MPa, P

_{e}: 0.04–0.047 MPa and P

_{C}), the developed CFD model is employed to validate the results obtained through empirical correlation of entrainment ratio and compression ratio. In addition, these comparisons provide the information about the reliability of the developed correlations.

## 3. Results

#### 3.1. Case Study: Simple Refrigeration Machine

_{Pri}) and evaporator temperature (T

_{Eva}) of 70–100 °C and 10–20 °C, respectively. The ambient sink is considered for condenser, so the condensing temperature (T

_{Cond}) is fixed to 40 °C. The working fluid is taken as R141b, thus the developed correlations are valid. The conditions are summarized in Table 4. For the case study, the refrigeration capacity (Q

_{Cool}) is fixed to 300 W. Using the known Q

_{Cool}and T

_{Eva}, the msec can be calculated. Subsequently, the correlations are applied to calculate m

_{Pri}and P

_{Cond}. The calculation procedure is summarized in Table 5. After solving the system, the coefficient of performance (COP) can be calculated. The pumping work was too small, and is, therefore, neglected in calculating the COP.

#### 3.2. COP Variation with Generation Temperature and the Evaporation Temperature

_{Eva}and T

_{Pri}have a significant impact on the COP of the system, shown in Figure 13. Both can vary, depending on the heat source and the target temperature. For the system shown in Figure 13, the Q

_{Add}can be supplied using any low-grade heat source, such as waste heat or the solar energy [39]. Furthermore, P

_{Eva}is governed by the required target temperature and can also vary, depending on the requirement. Therefore, a parametric study with respect to T

_{Eva}and T

_{Pri}was conducted, and the results are shown in Figure 14. As can be seen from the figures, higher values of T

_{Pri}and T

_{Eva}both favor the performance of the proposed design. The results are rather obvious, since the increase in evaporation or generation temperature both reduce the load on a refrigeration machine. However, the investigation demonstrates the ease of parametric investigation using empirical correlations for the ejector.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

D | Diameter |

A | Area, m^{2} |

C_{p} | Fluid specific heat at constant pressure, KJ kg^{−1} K^{−1} |

C_{v} | Fluid specific heat at constant volume, KJ kg^{−1} K^{−1} |

γ | Ratio of specific heats (Cp/Cv) |

R | Specific gas constant, KJ kg^{−1} K^{−1} |

a | Sonic velocity, ms^{−1} |

V | Fluid velocity, ms^{−1} |

M | Mach number |

m | Mass flowrate, kgs^{−1} |

h | Enthalpy, KJ kg^{−1} |

P_{g} | Fluid pressure at ejector primary nozzle inlet, MPa |

P_{e} | Fluid pressure at ejector suction inlet, MPa |

P_{cn} * | Ejector critical back pressure, MPa |

T | Temperature, K |

T_{g} | Fluid temperature at ejector primary nozzle inlet, K |

T_{e} | Fluid temperature at ejector suction inlet, MPa |

T_{cn} * | Saturated vapor temperature corresponding to P_{cn} *, K |

T_{gs} | Saturated-vapor temperature corresponding to P_{g}, K |

H | Hypothetical throat position |

ƞ | Isentropic efficiency coefficient |

φ | Coefficient representing flow losses |

Superscripts | |

* | Ejector critical operation mode |

Subscripts | |

cn | Condenser, Ejector exit |

e | Entrained flow suction port |

g | Primary nozzle inlet |

M | Mixed flow |

t | Primary nozzle throat |

p4 | Primary fluid at nozzle exit |

sH | Entrained flow at hypothetical throat |

pH | Primary flow at hypothetical throat |

1 | Motive nozzle throat |

2 | Constant area section Entrance |

3 | Constant area section Exit |

4 | Primary Nozzle Exit |

## Appendix A

Geometry | Area Ratio (A_{3}/A_{t}) | Expansion Ratio (P_{g}/P_{e}) | Compression Ratio (P_{c}/P_{e}) | Entrainment Ratio (ω) |

AA | 6.44 | 10 | 2.56 | 0.3257 |

6.44 | 11.62 | 2.84 | 0.288 | |

6.44 | 13.45 | 3.18 | 0.2246 | |

6.44 | 15.1 | 3.54 | 0.1859 | |

6.44 | 9.89 | 2.45 | 0.3398 | |

6.44 | 11.44 | 2.76 | 0.2946 | |

6.44 | 12.85 | 3.05 | 0.235 | |

EG | 6.77 | 15.1 | 3.41 | 0.2043 |

AB | 6.99 | 10 | 2.3 | 0.3922 |

6.99 | 11.62 | 2.66 | 0.3117 | |

6.99 | 13.45 | 3.04 | 0.2718 | |

EC | 7.26 | 15.1 | 3.17 | 0.2273 |

7.26 | 12.85 | 2.74 | 0.304 | |

AG | 7.73 | 10 | 2.26 | 0.4393 |

7.73 | 11.62 | 2.54 | 0.3883 | |

7.73 | 13.45 | 2.95 | 0.304 | |

7.73 | 15.1 | 3.15 | 0.2552 | |

7.73 | 8.51 | 1.93 | 0.6132 | |

7.73 | 9.89 | 2.17 | 0.479 | |

7.73 | 11.44 | 2.45 | 0.4034 | |

7.73 | 12.85 | 2.69 | 0.3503 | |

ED | 8.25 | 15.1 | 3 | 0.2902 |

AC | 8.29 | 10 | 2.09 | 0.4889 |

8.29 | 11.62 | 2.38 | 0.4241 | |

8.29 | 13.45 | 2.67 | 0.3488 | |

8.29 | 15.1 | 2.91 | 0.2814 | |

EE | 9.17 | 15.1 | 2.71 | 0.3505 |

9.17 | 12.85 | 2.31 | 0.4048 | |

AD | 9.41 | 10 | 1.91 | 0.6227 |

9.41 | 11.62 | 2.18 | 0.5387 | |

9.41 | 13.45 | 2.47 | 0.4446 | |

9.41 | 15.1 | 2.66 | 0.3457 | |

9.41 | 8.51 | 1.7 | 0.7412 | |

9.41 | 9.89 | 1.91 | 0.635 | |

9.41 | 11.44 | 2.14 | 0.5422 | |

9.41 | 12.85 | 2.33 | 0.4541 | |

9.83 | 15.1 | 2.6 | 0.3937 | |

9.83 | 12.85 | 2.22 | 0.4989 | |

EH | 10.64 | 15.1 | 2.45 | 0.4377 |

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**Figure 1.**(

**A**): Illustration of schematic of Ejector expansion refrigeration cycle (EERC); (

**B**): Shows corresponding pressure enthalpy (P-h) diagram for EERC cycle.

**Figure 3.**Relation between entrainment ratio (ω) with area (A

_{3}/A

_{t}) and pressure ratio (P

_{g}/P

_{e}).

**Figure 4.**Relation between compression ratio (C

_{p}) with area ratio (A

_{3}/A

_{t}) and pressure ratio (P

_{g}/P

_{e}).

**Figure 5.**Comparative flowchart of ejector performance analysis 2.3. Computational fluid dynamics (CFD) model development.

**Figure 9.**Mach number contours (

**a**) AD Geometry P

_{e}= 0.047 MPa, P

_{g}= 0.538 MPa; (

**b**) AA Geometry P

_{e}= 0.047 MPa, P

_{g}= 0.538 MPa; (

**c**) AD Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa; (

**d**) AC Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa; and (

**e**) AB Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa.

**Figure 10.**Total pressure contours (

**a**) AD Geometry P

_{e}= 0.047 MPa, P

_{g}= 0.538 MPa; (

**b**) AA Geometry P

_{e}= 0.047 MPa, P

_{g}= 0.538 MPa; (

**c**) AD Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa; (

**d**) AC Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa; and (

**e**) AB Geometry P

_{e}= 0.04 MPa, P

_{g}= 0.465 MPa.

**Figure 14.**The figure (

**a**) depicts the influence of T

_{Eva}on system coefficient of performance (COP). The figure (

**b**) displays the influence of T

_{Pri}on system coefficient of performance (COP).

Ejector Geometry | ||||
---|---|---|---|---|

Primary Nozzle | Mixing (Constant Area) Section | |||

Serial No. | Diameter (D_{3}) | |||

Serial No. | Throat Diameter (D_{t}) | Exit Diameter (D_{4}) | A | 6.70 mm |

A | 2.64 mm | 4.50 mm | B | 6.98 mm |

C | 7.60 mm | |||

D | 8.10 mm | |||

E | 2.82 mm | 5.10 mm | E | 8.54 mm |

G | 7.34 mm | |||

H | 9.20 mm |

Step | Inputs | Equations | Output | Comments |
---|---|---|---|---|

1 | ${P}_{g},{T}_{g},$${A}_{t}$ | ${\dot{m}}_{p}=\frac{{P}_{g}{A}_{t}}{\sqrt{{T}_{e}}}\sqrt{\frac{\gamma}{R}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}\sqrt{{\eta}_{p}}$ | ${\dot{m}}_{p}$ | A_{t} choking condition, the mass flow rate through the primary nozzle follows a gas dynamic relation. |

2 | ${A}_{p4}$ | $\frac{{A}_{p4}}{{A}_{t}}=\frac{1}{{M}_{p4}^{2}}{\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}_{p4}^{2}\right)\right]}^{\frac{\gamma +1}{\gamma -1}}$ $\frac{{P}_{g}}{{P}_{p4}}={\left[1+\frac{\gamma -1}{2}{M}_{p4}^{2}\right]}^{\frac{\gamma}{\gamma -1}}$ | ${M}_{p4},$${P}_{p4}$ | The Mach no. M_{4} is calculated by using the Newton–Raphson method. |

3 | ${P}_{e}$ | ${P}_{sH}=\frac{{P}_{e}}{{\left(1+\frac{\gamma -1}{2}{M}_{sH}^{2}\right)}^{\frac{\gamma}{\gamma -1}}}$ | ${P}_{sH}$ | Referring to the assumptions made, the Mach number of secondary flow at hypothetical throat is M_{eH} = 1. |

4 | ${P}_{sH}=$${P}_{pH}$ | $\frac{{P}_{sH}}{{P}_{p4}}=\frac{{\left[1+\frac{\gamma -1}{2}{M}_{p4}^{2}\right]}^{\frac{\gamma}{\gamma -1}}}{{\left[1+\frac{\gamma -1}{2}{M}_{pH}^{2}\right]}^{\frac{\gamma}{\gamma -1}}}$ $\frac{{A}_{pH}}{{A}_{p4}}=\frac{\left(\frac{{\phi}_{p}}{{M}_{pH}}\right){\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}_{pH}^{2}\right)\right]}^{\frac{\gamma +1}{2\left(\gamma -1\right)}}}{\left(\frac{1}{{M}_{p4}}\right){\left[\frac{2}{\gamma +1}\left(1+\frac{\gamma -1}{2}{M}_{p4}^{2}\right)\right]}^{\frac{\gamma +1}{2\left(\gamma -1\right)}}}$ | ${M}_{pH},$${A}_{pH}$ | ${\phi}_{p}$: an isentropic coefficient that represents flow losses as primary fluid flow from section 4-4 to section H-H. |

5 | ${A}_{pH}$ | ${A}_{sH}={A}_{3}+{A}_{pH}$ | ${A}_{sH}$ | If A_{sH} < 0, calculate A_{3} by using A _{3} = A_{pH} + ΔA_{3}, otherwise return to step 4 to recalculate A_{pH}, and again the condition is checked. |

6 | ${A}_{sH},{P}_{e},$ ${T}_{e},\gamma ,$ $R,{\eta}_{s}$ | ${\dot{m}}_{s}=\frac{{P}_{e}{A}_{sH}}{\sqrt{{T}_{e}}}\sqrt{\frac{\gamma}{R}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}\sqrt{{\eta}_{s}}$ | ${\dot{m}}_{s}$ | ${\eta}_{s}$: Isentropic efficiency of entrained flow. |

7 | ${T}_{g},{T}_{e},$ ${M}_{pH},{M}_{sH}$ | $\frac{{T}_{g}}{{T}_{pH}}=1+\frac{\gamma -1}{2}{M}_{pH}^{2}$ $\frac{{T}_{e}}{{T}_{sH}}=1+\frac{\gamma -1}{2}{M}_{sH}^{2}$ | ${T}_{pH,}{T}_{sH}$ | Value of T_{g} and T_{e} can be taken from step 1 and 3, respectively. |

8 | ${T}_{pH,}{T}_{sH},$ $\gamma ,R,$ ${M}_{pH},$ ${M}_{sH},$ ${\dot{m}}_{p},{\dot{m}}_{s}$ | ${\phi}_{m}\left[{\dot{m}}_{p}{V}_{pH}+{\dot{m}}_{s}{V}_{sH}\right]=\left({\dot{m}}_{p}+{\dot{m}}_{s}\right){V}_{m}$ ${V}_{pH}={M}_{pH}{a}_{pH},{V}_{sH}={M}_{sH}{a}_{sH}$ ${a}_{pH}=\sqrt{\gamma R{T}_{pH}},{a}_{sH}=\sqrt{\gamma R{T}_{sH}}$ | ${V}_{M},$ ${V}_{pH},$ ${V}_{sH}$ | ${\phi}_{M}$: mixed flow friction coefficient, P_{M} = P_{pH} = P_{sH}, M_{sH} = 1, and M_{pH} can be taken from step 4. |

9 | ${V}_{M},\gamma ,R,$ ${C}_{p},{T}_{pH},$ ${T}_{sH},{V}_{pH},{V}_{sH},$ ${\dot{m}}_{p},{\dot{m}}_{s}$ | ${\dot{m}}_{p}\left({C}_{p}{T}_{pH}+\frac{{V}_{pH}^{2}}{2}\right)+{\dot{m}}_{s}\left({C}_{p}{T}_{sH}+\frac{{V}_{sH}^{2}}{2}\right)=\left({\dot{m}}_{p}+{\dot{m}}_{s}\right)\left({C}_{p}{T}_{m}+\frac{{V}_{m}^{2}}{2}\right)$ ${a}_{M}=\sqrt{\gamma R{T}_{M}},{M}_{M}=\frac{{V}_{M}}{{a}_{M}}$ | ${T}_{M},{M}_{M}$ | The first equation gives T_{M}, which is then used to find value of α_{M} and M_{M}. |

10 | ${P}_{M},{M}_{M},\gamma $ | ${P}_{3}={P}_{M}\left(1+\frac{2\gamma}{\gamma +1}\left({M}_{M}^{2}-1\right)\right)$ ${M}_{3}=\sqrt{\frac{1+\frac{\gamma -1}{2}{M}_{M}^{2}}{\gamma {M}_{M}^{2}-\frac{\gamma -1}{2}}}$ | ${P}_{3},{M}_{3}$ | Flow is solved after the shock wave, and value of P_{M} can be taken from step 8. |

11 | ${P}_{3},{M}_{3}$ | ${P}_{Cn}={P}_{3}{\left(1+\frac{\gamma -1}{2}{M}_{3}^{2}\right)}^{\frac{\gamma}{\gamma -1}}$ | ${P}_{Cn}$ | Flow pressure at diffuser exit is calculated. |

12 | ${P}_{Cn},{P}_{Cn}^{*}$ | $if{P}_{Cn}\ge {P}_{Cn}^{*}then{A}_{3}={A}_{3}-\Delta {A}_{3}$ $if{P}_{Cn}{P}_{Cn}^{*}then{A}_{3}={A}_{3}+\Delta {A}_{3}$ | ${A}_{3}$ | ${P}_{Cn}^{*}$: critical condenser pressure, and A_{3} must be equal to A_{3} in step 5, otherwise procedure starts again from step 5. |

13 | ${\dot{m}}_{p},{\dot{m}}_{s}$ | $\omega =\frac{{\dot{m}}_{s}}{{\dot{m}}_{p}}$ | $\omega $ | Entrainment ratio is calculated by using ṁ_{s} and ṁ_{p} from step 6 and step 1, respectively. |

Mesh | |

Mesh Type | Structured |

Number of elements | 500,000 |

Element type | Quadratic quadrilateral |

Boundary Conditions | |

Primary flow inlet | Pressure inlet |

Secondary flow inlet | Pressure inlet |

Discharge flow outlet | Pressure outlet |

Numerical Model Setup | |

Solver | Pressure based |

Turbulence model | k-ω-sst |

Method of initialization | Hybrid |

Fluid density | Ideal gas |

Working fluid | R141b |

Discretization scheme | Second order upwind |

Convergence criteria | Residuals <10^{−}^{6} |

Parameters | Values |
---|---|

Refrigeration capacity (Q_{Cool}) | 300 W |

Required condensing saturation temperature (T_{Cond}_{,Req}) | 40 °C |

Generator saturation temperature (T_{Pri}) | 70–100 °C |

Evaporator saturation temperature (T_{Eva}) | 10–20 °C |

Component | Input | Output | Equations | Comment |
---|---|---|---|---|

Evaporator | Q_{Cool}, T_{Eva} | m_{sec} | $m\_\mathrm{sec}={Q}_{Cool}/\Delta {h}_{fg}$ | Since the saturation temperature is known, Δh_{fg} can be calculated. |

Ejector | m_{sec}, A_{3}/A_{T}, P_{Pri}/P_{Eva} | ω, P_{Cond}, m_{Pri} | Using the developed co-relation, the ω and P_{Cond} can be calculated. If ${P}_{Cond}\ne {P}_{Cond,Req}$ then iterate A _{3}/A_{t}. | |

Generator | T_{Pri}, m_{Pri} | Q_{Add} | ${m}_{Pri}={Q}_{Add}/\Delta {h}_{fg}$ | Since the saturation temperature is known, Δh_{fg} can be calculated. |

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## Share and Cite

**MDPI and ACS Style**

Muhammad, H.A.; Abdullah, H.M.; Rehman, Z.; Lee, B.; Baik, Y.-J.; Cho, J.; Imran, M.; Masud, M.; Saleem, M.; Butt, M.S.
Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System. *Energies* **2020**, *13*, 5835.
https://doi.org/10.3390/en13215835

**AMA Style**

Muhammad HA, Abdullah HM, Rehman Z, Lee B, Baik Y-J, Cho J, Imran M, Masud M, Saleem M, Butt MS.
Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System. *Energies*. 2020; 13(21):5835.
https://doi.org/10.3390/en13215835

**Chicago/Turabian Style**

Muhammad, Hafiz Ali, Hafiz Muhammad Abdullah, Zabdur Rehman, Beomjoon Lee, Young-Jin Baik, Jongjae Cho, Muhammad Imran, Manzar Masud, Mohsin Saleem, and Muhammad Shoaib Butt.
2020. "Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System" *Energies* 13, no. 21: 5835.
https://doi.org/10.3390/en13215835