# Mathematical Model of a Lithium-Bromide/Water Absorption Refrigeration System Equipped with an Adiabatic Absorber

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## Abstract

**:**

## 1. Introduction

## 2. System Description

- The refrigerant at the exit of the evaporator is saturated vapor (no liquid spillover).
- The solutions which leave the adiabatic absorber and the generator are saturated liquid at the units’ temperatures.
- The adiabatic absorber pressure is equal to the evaporator pressure and the generator pressure is equal to the condenser pressure.
- The solution entering the generator is at the generator pressure.
- The throttling and pumping processes are adiabatic.
- The solution and circulation pumps’ work are negligible.
- The solution leaving the weak solution sub-cooler is at the adiabatic absorber temperature.
- The adiabatic absorber and the accumulator are insulated well.

- A cooling capacity (${\dot{Q}}_{eva}$) at the evaporator of 1 kW.
- The temperature of the condenser (${T}_{con}={T}_{2}$) was set to 35 °C, which is a suitable temperature for the external sink to the condenser.
- The evaporator temperature (${T}_{eva}={T}_{4}$) was set to 10 °C, which is a suitable temperature when using water as refrigerant.
- The range of temperatures of the generator (${T}_{gen}={T}_{1}={T}_{5}$) considered was 65–90 °C.
- The range of temperatures of the absorber (${T}_{abs}={T}_{9}={T}_{11}$) considered was 25–50 °C.
- The ratio (X) of the solution mass flow rate at the circulation pump to that at the solution pump was set to 25.

## 3. Mathematical Model

- Throttling process between stations 2 & 3 and between stations 5 & 6; applying the steady state energy equation across the throttling and expansion valves:$${h}_{3}={h}_{2}\text{and}{h}_{6}={h}_{5}$$
- Pumping process between stations 9 & 10 and between stations 11 & 12; applying the steady state energy equation across the solution and circulation pumps (neglecting pumps’ work):$${h}_{10}={h}_{9}\text{and}{h}_{12}={h}_{11}$$
- Refrigerant circuit between stations 1, 2, 3, & 4; applying the mass conservation equation:$${\dot{m}}_{1}={\dot{m}}_{2}={\dot{m}}_{3}={\dot{m}}_{4}$$
- Evaporation process between stations 3 & 4; applying the steady state energy equation across the evaporator:$${\dot{m}}_{4}=\frac{{\dot{Q}}_{eva}}{({h}_{4}-{h}_{3})}$$
- Condensation process between stations 1 & 2; applying the steady state energy equation across the condenser:$${\dot{Q}}_{con}={\dot{m}}_{1}\times {h}_{1}-{\dot{m}}_{2}\times {h}_{2}$$
- Lithium-bromide mass balance between stations 5, 6, & 7 yields:$${C}_{5}={C}_{6}={C}_{7}$$
- Lithium-bromide mass balance between stations 9 & 10 yields:$${C}_{9}={C}_{10}$$
- Lithium-bromide mass balance between stations 11, 12, & 13 yields:$${C}_{11}={C}_{12}={C}_{13}$$

- Applying the mass and LiBr mass conservation equations and the steady state energy equation across the generator:$${\dot{m}}_{10}-{\dot{m}}_{5}={\dot{m}}_{1}$$$${C}_{10}\times {\dot{m}}_{10}-{C}_{5}\times {\dot{m}}_{5}=0$$$${\dot{m}}_{10}\times {h}_{10}-{\dot{m}}_{5}\times {h}_{5}+{\dot{Q}}_{gen}={\dot{m}}_{1}\times {h}_{1}$$
- Applying the mass conservation equation across the throttling valve:$${\dot{m}}_{5}-{\dot{m}}_{6}=0$$
- Applying the mass conservation equation across the solution pump:$${\dot{m}}_{9}-{\dot{m}}_{10}=0$$
- Applying the steady state energy equation across the weak solution sub-cooler:$${\dot{m}}_{6}\times {h}_{6}-{\dot{m}}_{7}\times {h}_{7}-{\dot{Q}}_{sub1}=0$$
- Applying the mass conservation equation and the steady state energy equation across the control volume shown in Figure 1:$${\dot{m}}_{7}-{\dot{m}}_{9}=-{\dot{m}}_{4}$$$${\dot{m}}_{7}\times {h}_{7}-{\dot{m}}_{9}\times {h}_{9}-{\dot{Q}}_{sub2}=-{\dot{m}}_{4}\times {h}_{4}$$

- From the ratio (X) of the solution mass flow rate at the circulation pump to that at the solution pump:$${\dot{m}}_{11}=\mathrm{X}\times {\dot{m}}_{9}$$
- Applying the mass conservation equation across the circulation pump:$${\dot{m}}_{12}={\dot{m}}_{11}$$
- Applying the mass conservation equation and the steady state energy equation across the strong solution sub-cooler:$${\dot{m}}_{13}={\dot{m}}_{12}$$$${h}_{13}=\frac{{\dot{m}}_{12}\times {h}_{12}-{\dot{Q}}_{sub2}}{{\dot{m}}_{13}}$$
- Applying the mass and LiBr mass conservation equations and the steady state energy equation across the accumulator:$${\dot{m}}_{8}={\dot{m}}_{7}+{\dot{m}}_{13}$$$${C}_{8}=\frac{{C}_{7}\times {\dot{m}}_{7}+{C}_{13}\times {\dot{m}}_{13}}{{\dot{m}}_{8}}$$$${h}_{8}=\frac{{\dot{m}}_{7}\times {h}_{7}+{\dot{m}}_{13}\times {h}_{13}}{{\dot{m}}_{8}}$$
- The coefficient of performance (COP) of a LiBr/water absorption refrigeration cycle is defined as:$$COP=\frac{Coolingcapacityattheevaporator}{Rateofheatinputatthegenerator}=\frac{{\dot{Q}}_{eva}}{{\dot{Q}}_{gen}}$$

## 4. Model Implementation

## 5. Numerical Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

h | Specific enthalpy (kJ/kg) |

$\dot{m}$ | Mass flow rate (kg/s) |

p | Pressure (kPa) |

T | Temperature (°C) |

Q | Heat transfer rate (kW) |

C | LiBr mass concentration (%) |

COP | Coefficient of Performance |

X | Ratio of the solution mass flow rate at the circulation pump to that at the solution pump |

## Subscript

gen | Generator |

con | Condenser |

eva | Evaporator |

abs | Adiabatic absorber |

sub1 | Weak solution sub-cooler |

sub2 | Strong solution sub-cooler |

1, 2, 3… | Represent state point in Figure 1 |

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**Figure 1.**Schematic diagram of a LiBr/water (lithium-bromide/water) absorption refrigeration system equipped with an adiabatic absorber.

**Figure 2.**Program flowchart of an LiBr/water absorption refrigeration system equipped with an adiabatic absorber.

**Figure 3.**Contour plot of the COP (coefficient of performance) of the LiBr/water absorption refrigeration system equipped with an adiabatic absorber $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 4.**Graphs showing the variation of the LiBr mass concentration of: (

**a**) the weak solution with generator temperature $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$; and (

**b**) the strong solution with adiabatic absorber temperature $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 5.**Graph showing the variation of the LiBr mass concentration of the solution entering the adiabatic absorber, with generator and adiabatic absorber temperatures $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 6.**Graph showing the variation of the heat input in the generator, with adiabatic absorber temperature and generator temperature $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 7.**Graph showing the variation of the heat rejection in the weak solution sub-cooler, with generator temperature and adiabatic absorber temperature $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 8.**Graph showing the variation of the heat rejection in the strong solution sub-cooler with generator temperature and adiabatic absorber temperature $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Figure 9.**The influences of: (

**a**) the generator temperature on the absorption refrigeration system performance $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C},{T}_{abs}=30\xb0\mathrm{C})$; and (

**b**) the adiabatic absorber temperature on the absorption refrigeration system performance $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C},{T}_{gen}=80\xb0\mathrm{C})$.

**Figure 10.**Contour plot of the temperature of the LiBr/water solution entering the adiabatic absorber $({T}_{con}=35\xb0\mathrm{C},{T}_{eva}=10\xb0\mathrm{C})$.

**Table 1.**The optimal design parameters of the LiBr/water absorption refrigeration system equipped with an adiabatic absorber.

Point | h (kJ/kg) | T (°C) | P (kPa) | $\dot{\mathbf{m}}$ (g/s) | C (%) | Fluid State |
---|---|---|---|---|---|---|

1 | 2650 | 80 | 5.63 | 0.42 | 0 | Superheated vapor |

2 | 147 | 35 | 5.63 | 0.42 | 0 | Saturated liquid |

3 | 147 | 10 | 1.23 | 0.42 | 0 | Liquid and vapor |

4 | 2519 | 10 | 1.23 | 0.42 | 0 | Saturated vapor |

5 | 194 | 80 | 5.63 | 4.22 | 60.3 | Saturated liquid |

6 | 194 | 80 | 1.23 | 4.22 | 60.3 | liquid |

7 | 117 | 40 | 1.23 | 4.22 | 60.3 | liquid |

8 | 85 | 36 | 1.23 | 120.24 | 55.1 | liquid |

9 | 93 | 40 | 1.23 | 4.64 | 54.8 | Saturated liquid |

10 | 93 | 40 | 5.63 | 4.64 | 54.8 | liquid |

11 | 93 | 40 | 1.23 | 116.02 | 54.8 | Saturated liquid |

12 | 93 | 40 | 1.23 | 116.02 | 54.8 | liquid |

13 | 83 | 35 | 1.23 | 116.02 | 54.8 | liquid |

**Table 2.**The heat transfer in each component and COP of the LiBr/water absorption refrigeration system equipped with an adiabatic absorber.

The System Components | $\dot{\mathit{Q}}$ (W) |
---|---|

The evaporator (cooling capacity) | 1000 |

The generator (heat input from hot resource) | 1504 |

The condenser (heat reject to external sink) | 1055 |

The strong solution sub-cooler (heat reject to external sink) | 1125 |

The weak solution sub-cooler (heat reject to external sink) | 324 |

The coefficient of performance of the system (COP) = 0.66 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Osta-Omar, S.M.; Micallef, C. Mathematical Model of a Lithium-Bromide/Water Absorption Refrigeration System Equipped with an Adiabatic Absorber. *Computation* **2016**, *4*, 44.
https://doi.org/10.3390/computation4040044

**AMA Style**

Osta-Omar SM, Micallef C. Mathematical Model of a Lithium-Bromide/Water Absorption Refrigeration System Equipped with an Adiabatic Absorber. *Computation*. 2016; 4(4):44.
https://doi.org/10.3390/computation4040044

**Chicago/Turabian Style**

Osta-Omar, Salem M., and Christopher Micallef. 2016. "Mathematical Model of a Lithium-Bromide/Water Absorption Refrigeration System Equipped with an Adiabatic Absorber" *Computation* 4, no. 4: 44.
https://doi.org/10.3390/computation4040044