# Cooling Ability/Capacity and Exergy Penalty Analysis of Each Heat Sink of Modern Supersonic Aircraft

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## Abstract

**:**

## 1. Introduction

## 2. Heat Sinks of Modern Aircraft

## 3. Estimation Models

#### 3.1. Assumptions

- (1)
- Ignoring the heat leakage, all of the HS can be fully utilized.
- (2)
- The efficiency of engine and lift-drag ratio of aircraft do not change during the whole flight period—this assumption is used to roughly estimate the fuel consumption and output power of engine.
- (3)
- There is always enough heat load to be cooled by various heat sinks, and all the thermal loads are kept at the maximum allowable temperature T
_{loax,}_{max}—this assumption provides an extreme condition for assessing the maximum cooling ability/capacity of various heat sinks, and it also provides the same work condition for different heat sinks for comparing their cooling ability/capacity. - (4)
- The heat transfer efficiency of each HX can be higher than a designed minimum value η
_{HX,}_{min}during the whole flight period—this assumption is used to determine the heat transfer efficiency of HXs under arbitrary conditions, as well as the additional weight caused by these HXs.

#### 3.2. Fuel’s Cooling Ability and Residual Cooling Capacity

#### 3.2.1. Cooling Ability of the Fuel

_{c}, η

_{m}and η

_{t}refer to the combustion efficiency, mechanical efficiency and propulsive efficiency of the engine, respectively. P

_{m,engine}is the total mechanical power the engine supplies, while P

_{t,engine}is engine’s propulsive power, ${\dot{E}}_{k}$ indicates the kinetic energy change rate, ${\dot{E}}_{p}$ is potential energy change rate and ${\dot{E}}_{d}$ is the energy loss resulting from drag. According to the basic flight theory, ${\dot{E}}_{k}$, ${\dot{E}}_{p}$ and ${\dot{E}}_{d}$ all can be determined by the weight, flight velocity and altitude of the aircraft, shown as Equations (3)–(5) respectively. The weight of aircraft m

_{a}will decrease with the consumption of fuel, shown as Equation (6):

#### 3.2.2. Residual Cooling Capacity of Fuel

#### 3.2.3. Exergy Penalty of Fuel

#### 3.3. RA’s Cooling Ability and Exergy Penalty Rate

#### 3.3.1. Cooling Ability of RA

_{RAHX}:

#### 3.3.2. Exergy Penalty Rate of Ra

#### 3.4. EFA’s Cooling Ability and Exergy Penalty Rate

#### 3.4.1. Cooling Ability of EFA

_{ex,EFA}and the heat transfer efficient of FDHX ${\eta}_{FDHX}$:

_{EFA}will be bigger than that of RA, resulting in higher total temperature. Two important performance indexes of turbofan engine are bypass ratio B and power partition coefficient X who will directly influence the Mach number of EFA. The bypass ratio is the ratio between the mass flow rate of the bypass stream and the mass flow rate when entering the core. And the power partition coefficient is the proportion between available work that engine offers ducted fan and all mechanical work produced by engine. The power provided by the engine to per kilogram bypass stream ${w}_{b}$ can be determined by Equation (27) where ${\eta}_{f}$ is the efficiency of the duct fan which indicates the ratio of converting mechanical power into the kinetic energy of bypass stream. The ${P}_{m,engine}$ and ${\dot{m}}_{fuel}$ in these two equations have been obtained by Equation (2) above:

#### 3.4.2. Exergy Penalty Rate of EFA

#### 3.5. SHX’s Cooling Ability and Exergy Penalty Rate

#### 3.5.1. Cooling Ability of SHX

#### 3.5.2. Exergy Penalty Rate of SHX

#### 3.6. EHS’s Cooling Capacity and Exergy Penalty Rate

#### 3.6.1. Cooling Capacity of EHS

#### 3.6.2. Exergy Penalty Rate of EHS

#### 3.7. Cooling Capacity and Exergy Penalty of Heat Sinks

#### 3.8. Calculation Procedure

## 4. Results and Discussion

#### 4.1. Case Design

#### 4.1.1. The Standard “Case 0” for Estimating Study

#### 4.1.2. Flight Profile

#### 4.2. Influence of TMS Parameters on Heat Sinks

#### 4.2.1. Fuel

#### 4.2.2. Ram Air

#### 4.2.3. Engine Fan Air

#### 4.2.4. Skin HXs

#### 4.2.5. Expendable Heat Sinks

#### 4.3. Comparison of Different Heat Sinks

#### 4.3.1. Cooling Ability Comparison of Each Heat Sink

#### 4.3.2. Cooling Capacity and Exergy Penalty Comparison of Each Heat Sink

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbol | Subscript | ||

q | Cooling ability (W) | a | Aircraft |

Q | Cooling capacity (J) | c | Combustion |

$m$ | Mass (kg) | m | Mechanical |

$\dot{m}$ | Mass flow rate (kg/s) | t | Thrust & Propulsive |

$T$ | Temperature (K) | f | Engine duct fan |

E | Energy (J) | k | Kinetic energy |

$\dot{E}$ | Energy change rate (W) | p | Potential energy |

Ex | Exergy penalty (J) | max | Maximum |

$\dot{E}x$ | Exergy penalty rate (W) | min | Minimum |

P | Power (W) | d | Drag |

$\eta $ | efficiency | w | Weight |

$v$ | Velocity (m/s) | $\infty $ | Ambient |

Ma | Mach number of aircraft | 0 | Ambient with sea level |

H | Altitude of aircraft (m) | ex | Total temperature/pressure |

D | Drag force | in | Air inlet |

A | Area (m^{2}) | core | Core of engine |

p | Pressure (Pa) | st | Storage |

t | Time (s) | ||

k | Heat trans coefficient (W/m^{2}K) | Abbreviations | |

$\rho $ | Density (kg/m^{3}) | TMS | Thermal management system |

$\alpha $ | Area density (m^{2}/m^{3}) | HX | Heat exchanger |

$\varphi $ | Combustion value (J/kg) | RA | Ram air |

${c}_{p}$ | Specific heat capacity (J/kg K) | RAHX | Ram air heat exchanger |

$\lambda $ | Thermal conductivity (W/m K) | EFA | Engine fan air |

$\mu $ | Kinetic viscosity (s/m^{2}) | FDHX | Fan duct heat exchanger |

h | Specific enthalpy (J/kg) | SHX | Skin heat exchanger |

$\xi $ | Velocity loss coefficient | EHS | Expendable heat sink |

K | Lift-drag ratio of the aircraft | LA | Liquid ammonia |

B | Bypass ratio of engine | LNG | Liquid nature gas |

X | Power partition coefficient | ECS | Environment control system |

${w}_{b}$ | Power to per kg bypass stream | HTR | Heat transfer rate |

$\theta $ | Air-fuel ratio of engine | PAO | Poly Alpha Olefin |

$\epsilon $ | Surface emissivity | ||

$\chi $ | Cooling-penalty ratio | ||

Constant | |||

R | Gas constant | 8.314 J/mol K | |

g | Acceleration of gravity | 9.807 m/s^{2} | |

$\mathsf{\sigma}$ | Stefan-Boltzmann constant | $5.67\times {10}^{-8}{\mathrm{W}/\mathrm{m}}^{2}{\mathrm{K}}^{4}$ | |

$\mathsf{\gamma}$ | Ambient temperature lapse rate | 0.0065 K/m |

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**Figure 2.**Typical thermal management system (TMS) of modern advantage aircraft. ECS, environment control system; EMA, electro-mechanical actuation; HYS, hydraulic system; RAHX, ram air heat exchanger; FDHX, fan duct heat exchanger; SHX, skin heat exchanger; PAO, Poly Alpha Olefin.

**Figure 3.**(

**a**) Atmosphere temperature with altitude; (

**b**) Total temperature of RA with flight Mach number.

**Figure 7.**The estimation results of fuel in Case 0. (

**a**) cooling ability of consumption fuel flow; (

**b**) cooling capacity of remaining fuel mass in tank.

**Figure 8.**The estimating results of RA in Cases 1~5: (

**a**) The total temperature and (

**b**) The mass flow rate.

**Figure 9.**The estimation results of RA under Cases 1~5. (

**a**) cooling ability; (

**b**) exergy penalty rate; (

**c**) cooling capacity and exergy penalty.

**Figure 11.**The estimation results of EFA under Cases 6~10: (

**a**) cooling ability; (

**b**) exergy penalty rate; (

**c**) cooling capacity and exergy penalty.

**Figure 12.**The estimation results of SHX under Cases 11~15. (

**a**) average skin temperature; (

**b**) cooling ability; (

**c**) exergy penalty rate; (

**d**) cooling capacity and exergy penalty.

**Figure 13.**The cooling capacity and exergy penalty of EHS: (

**a**) liquid natural gas; (

**b**) liquid ammonia.

**Figure 14.**The comparison of cooling ability of various heat sinks at different flight stages. (

**a**) overview; (

**b**) supersonic at Ma 1.8; (

**c**) low supersonic at Ma 1.2; (

**d**) subsonic at Ma 0.9.

**Figure 15.**The comparison of cooling capacity of various heat sinks in different flight stages. (

**a**) stage 1: take-off and climb; (

**b**) stages 2 and 3: supersonic cruise (Ma 1.8) and maneuver; (

**c**) stage 4: low supersonic cruise (Ma1.2); (

**d**) stage 5: subsonic cruise (Ma 0.9); (

**e**) stage 6: descend and land; (

**f**) whole flight mission.

Parameters | Symbol & Value |
---|---|

Temperature | ${T}_{0}=288\text{}\mathrm{K}$ |

Pressure | ${p}_{0}=\mathrm{101,325}\text{\hspace{0.17em}}\mathrm{Pa}$ |

Density | ${\rho}_{0}=1.225\text{\hspace{0.17em}}{\mathrm{kg}/\mathrm{m}}^{3}$ |

**Table 2.**Parameters for the estimating model of “Case 0”. EHS, expendable heat sink; LNG, liquefied natural gas.

Parameters of Heat Sinks | Symbol | Value |
---|---|---|

Take-off weight | ${m}_{a}^{(t=0)}$ | 30,000 |

Engine combustion efficiency | ${\eta}_{c}$ | 0.90 |

Engine mechanical efficiency | ${\eta}_{m}$ | 0.35 |

Engine propulsion efficiency | ${\eta}_{t}$ | 0.80 |

Engine duct fan efficiency | ${\eta}_{f}$ | 0.64 |

Lift-drag ratio of engine | K | 5.00 |

Bypass ratio of engine | B | 0.20 |

Power partition coefficient of engine | X | 0.10 |

Air-fuel ratio of engine | $\theta $ | 18 |

Initial Fuel Mass | ${m}_{fuel}^{(t=0)}$ | 15,000 |

Initial Fuel temperature | ${T}_{fuel}^{(t=0)}$ | 30 °C |

Threshold temperature of Fuel | ${T}_{fuel,threshold}$ | 80 °C |

Special heat of fuel | ${c}_{p,fuel}$ | 2010 |

Combustion value of fuel | ${\varphi}_{fuel}$ | $4.3\times {10}^{7}$ |

Tank Surface Area | ${A}_{tank}$ | 10 |

Heat trans coefficient from tank to skin | ${k}_{tank}$ | 20 |

Emissivity of the skin | ${\epsilon}_{skin}$ | 0.7 |

Initial EHS Mass | ${m}_{EHS}^{(t=0)}$ | 100 |

Substance of EHS | LNG |

**Table 3.**Parameters of RAHX, FDHX, SHX in “Case 0”. HX, heat exchanger; RAHX, ram air heat exchanger; FDHX, fan duct heat exchanger; SHX, skin heat exchanger.

HX Type | Inlet Area/Skin Area | HX Mass Density | Heat Transfer Area Density | Heat Transfer Coefficient | Heat Transfer Efficiency | Velocity Loss Coefficient |
---|---|---|---|---|---|---|

RAHX | ${A}_{in,RA}=0.02$ | ${\rho}_{RAHX}=600$ | ${\alpha}_{RAHX}=1000$ | ${k}_{RAHX}=200$ | ${\eta}_{RAHX}=80\%$ | ${\xi}_{RA}=0.2$ |

FDHX | ${A}_{in,EFA}=0.02$ | ${\rho}_{FDHX}=500$ | ${\alpha}_{FDHX}=800$ | ${k}_{FDHX}=200$ | ${\eta}_{FDHX}=60\%$ | ${\xi}_{EFA}=0.1$ |

SHX | ${A}_{skin}=20$ | ${\rho}_{SHX}=300$ | ${\alpha}_{SHX}=600$ | ${k}_{SHX}=120$ | ${\eta}_{SHX}=80\%$ | --- |

Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|

${A}_{in,RA}=0.01$ | ${A}_{in,RA}=0.02$ | ${A}_{in,RA}=0.04$ | ${A}_{in,RA}=0.07$ | ${A}_{in,RA}=0.10$ |

Case 6 | Case 7 | Case 8 | Case 9 | Case 10 |
---|---|---|---|---|

${A}_{in,EFA}=0.01$ | ${A}_{in,EFA}=0.02$ | ${A}_{in,EFA}=0.04$ | ${A}_{in,EFA}=0.07$ | ${A}_{in,EFA}=0.10$ |

Case 11 | Case 12 | Case 13 | Case 14 | Case 15 |
---|---|---|---|---|

${\eta}_{SHX}=50\%$ | ${\eta}_{SHX}=60\%$ | ${\eta}_{SHX}=70\%$ | ${\eta}_{SHX}=80\%$ | ${\eta}_{SHX}=90\%$ |

Case | Case 16 | Case 17 | Case 18 | Case 19 | Case 20 | Case 21 | Case 22 | Case 23 | Case 24 | Case 25 |
---|---|---|---|---|---|---|---|---|---|---|

Substance | LNG | LNG | LNG | LNG | LNG | LA | LA | LA | LA | LA |

Initial mass | 20 | 50 | 100 | 200 | 300 | 20 | 50 | 100 | 200 | 300 |

Fuel | RA | EFA | SHX | Total | |
---|---|---|---|---|---|

Cruise in Ma 1.8 | 189.1 kW | 0 | 0 | 0 | 189.1 kW |

Cruise in Ma 1.2 | 117.5 kW | 57.4 kW | 19.8 kW | 32.0 kW | 226.7 kW |

Cruise in Ma 0.9 | 105.9 kW | 157.0 kW | 78.1 kW | 53.6 kW | 394.6 kW |

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## Share and Cite

**MDPI and ACS Style**

Mao, Y.-F.; Li, Y.-Z.; Wang, J.-X.; Xiong, K.; Li, J.-X.
Cooling Ability/Capacity and Exergy Penalty Analysis of Each Heat Sink of Modern Supersonic Aircraft. *Entropy* **2019**, *21*, 223.
https://doi.org/10.3390/e21030223

**AMA Style**

Mao Y-F, Li Y-Z, Wang J-X, Xiong K, Li J-X.
Cooling Ability/Capacity and Exergy Penalty Analysis of Each Heat Sink of Modern Supersonic Aircraft. *Entropy*. 2019; 21(3):223.
https://doi.org/10.3390/e21030223

**Chicago/Turabian Style**

Mao, Yu-Feng, Yun-Ze Li, Ji-Xiang Wang, Kai Xiong, and Jia-Xin Li.
2019. "Cooling Ability/Capacity and Exergy Penalty Analysis of Each Heat Sink of Modern Supersonic Aircraft" *Entropy* 21, no. 3: 223.
https://doi.org/10.3390/e21030223