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Heat transfer
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Simulation of thermal convection in the Earth's mantle. Colors span from red and green to blue with decreasing temperatures. A hot, less-dense lower boundary layer sends plumes of hot material upwards, and cold material from the top moves downwards.
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes. Engineers also consider the transfer of mass of differing chemical species, either cold or hot, to achieve heat transfer. While these mechanisms have distinct characteristics, they often occur simultaneously in the same system.
Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium. Such spontaneous heat transfer always occurs from a region of high temperature to another region of lower temperature, as described in the second law of thermodynamics.
Heat convection occurs when bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid. The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. The latter process is often called "natural convection". All convective processes also move heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is forced to flow by use of a pump, fan or other mechanical means.
Thermal radiation occurs through a vacuum or any transparent medium (solid or fluid). It is the transfer of energy by means of photons in electromagnetic waves governed by the same laws.[1]
Contents
[hide]
1 Overview
2 Mechanisms
2.1 Advection
2.2 Conduction
2.3 Convection
2.3.1 Convection-cooling
2.4 Convection vs. conduction
2.5 Radiation
3 Phase transition
3.1 Boiling
3.2 Condensation
3.3 Melting
4 Modeling approaches
4.1 Climate models
4.2 Heat equation
4.3 Lumped system analysis
5 Engineering
5.1 Insulation, radiance and resistance
5.2 Devices
5.2.1 Heat exchangers
6 Applications
6.1 Architecture
6.2 Climate engineering
6.3 Greenhouse effect
6.4 Heat transfer in the human body
6.5 Cooling techniques
6.5.1 Evaporative cooling
6.5.2 Laser cooling
6.5.3 Magnetic cooling
6.5.4 Radiative cooling
6.6 Thermal energy storage
7 See also
8 References
9 External links
Overview[edit]
See also: Heat transfer physics
Earth's long wave thermal radiation intensity, from clouds, atmosphere and surface.
Heat is defined in physics as the transfer of thermal energy across a well-defined boundary around a thermodynamic system. The thermodynamic free energy is the amount of work that a thermodynamic system can perform. Enthalpy is a thermodynamic potential, designated by the letter "H", that is the sum of the internal energy of the system (U) plus the product of pressure (P) and volume (V). Joule is a unit to quantify energy, work, or the amount of heat.
Heat transfer is a process function (or path function), as opposed to functions of state; therefore, the amount of heat transferred in a thermodynamic process that changes the state of a system depends on how that process occurs, not only the net difference between the initial and final states of the process.
Thermodynamic and mechanical heat transfer is calculated with the heat transfer coefficient, the proportionality between the heat flux and the thermodynamic driving force for the flow of heat. Heat flux is a quantitative, vectorial representation of heat-flow through a surface.[2]
In engineering contexts, the term heat is taken as synonymous to thermal energy. This usage has its origin in the historical interpretation of heat as a fluid (Caloric) that can be transferred by various causes,[3] and that is also common in the language of laymen and everyday life.
The transport equations for thermal energy (Fourier's law), mechanical momentum (Newton's law for fluids), and mass transfer (Fick's laws of diffusion) are similar,[4][5] and analogies among these three transport processes have been developed to facilitate prediction of conversion from any one to the others.[5]
Thermal engineering concerns the generation, use, conversion, and exchange of heat transfer. As such, heat transfer is involved in almost every sector of the economy.[6] Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation, and transfer of energy by phase changes.
Mechanisms[edit]
The fundamental modes of heat transfer are:
Advection
Advection is the transport mechanism of a fluid from one location to another, and is dependent on motion and momentum of that fluid.
Conduction or diffusion
The transfer of energy between objects that are in physical contact. Thermal conductivity is the property of a material to conduct heat and evaluated primarily in terms of Fourier's Law for heat conduction.
Convection
The transfer of energy between an object and its environment, due to fluid motion. The average temperature is a reference for evaluating properties related to convective heat transfer.
Radiation
The transfer of energy by the emission of electromagnetic radiation.
Advection[edit]
By transferring matter, energy—including thermal energy—is moved by the physical transfer of a hot or cold object from one place to another.[7] This can be as simple as placing hot water in a bottle and heating a bed, or the movement of an iceberg in changing ocean currents. A practical example is thermal hydraulics.[citation needed] This can be described by the formula:
Q = v ρ c p Δ T {\displaystyle Q=v\rho c_{p}\Delta T}
where Q is heat flux (W/m²), ρ is density (kg/m³), c p {\displaystyle c_{p}} is heat capacity at constant pressure (J/kg·K), ΔT is the change in temperature (K), v {\displaystyle v} is velocity (m/s).
Conduction[edit]
Main article: Thermal conduction
On a microscopic scale, heat conduction occurs as hot, rapidly moving or vibrating atoms and molecules interact with neighboring atoms and molecules, transferring some of their energy (heat) to these neighboring particles. In other words, heat is transferred by conduction when adjacent atoms vibrate against one another, or as electrons move from one atom to another. Conduction is the most significant means of heat transfer within a solid or between solid objects in thermal contact. Fluids—especially gases—are less conductive. Thermal contact conductance is the study of heat conduction between solid bodies in contact.[8] The process of heat transfer from one place to another place without the movement of particles is called conduction. Example: Heat transfer through Metal rods.[clarification needed] Steady state conduction (see Fourier's law) is a form of conduction that happens when the temperature difference driving the conduction is constant, so that after an equilibration time, the spatial distribution of temperatures in the conducting object does not change any further.[9] In steady state conduction, the amount of heat entering a section is equal to amount of heat coming out.[8]
Transient conduction (see Heat equation) occurs when the temperature within an object changes as a function of time. Analysis of transient systems is more complex and often calls for the application of approximation theories or numerical analysis by computer.[8]
Convection[edit]
Main article: Convection
The flow of fluid may be forced by external processes, or sometimes (in gravitational fields) by buoyancy forces caused when thermal energy expands the fluid (for example in a fire plume), thus influencing its own transfer. The latter process is often called "natural convection". All convective processes also move heat partly by diffusion, as well. Another form of convection is forced convection. In this case the fluid is forced to flow by using a pump, fan or other mechanical means.
Convective heat transfer, or convection, is the transfer of heat from one place to another by the movement of fluids, a process that is essentially the transfer of heat via mass transfer. Bulk motion of fluid enhances heat transfer in many physical situations, such as (for example) between a solid surface and the fluid.[10] Convection is usually the dominant form of heat transfer in liquids and gases. Although sometimes discussed as a third method of heat transfer, convection is usually used to describe the combined effects of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming.[11] The process of transport by fluid streaming is known as advection, but pure advection is a term that is generally associated only with mass transport in fluids, such as advection of pebbles in a river. In the case of heat transfer in fluids, where transport by advection in a fluid is always also accompanied by transport via heat diffusion (also known as heat conduction) the process of heat convection is understood to refer to the sum of heat transport by advection and diffusion/conduction.
Free, or natural, convection occurs when bulk fluid motions (streams and currents) are caused by buoyancy forces that result from density variations due to variations of temperature in the fluid. Forced convection is a term used when the streams and currents in the fluid are induced by external means—such as fans, stirrers, and pumps—creating an artificially induced convection current.[12]
Convection-cooling[edit]
See also: Nusselt number
Convective cooling is sometimes described as Newton's law of cooling:
The rate of heat loss of a body is proportional to the temperature difference between the body and its surroundings.
However, by definition, the validity of Newton's law of Cooling requires that the rate of heat loss from convection be a linear function of ("proportional to") the temperature difference that drives heat transfer, and in convective cooling this is sometimes not the case. In general, convection is not linearly dependent on temperature gradients, and in some cases is strongly nonlinear. In these cases, Newton's law does not apply.
Convection vs. conduction[edit]
In a body of fluid that is heated from underneath its container, conduction and convection can be considered to compete for dominance. If heat conduction is too great, fluid moving down by convection is heated by conduction so fast that its downward movement will be stopped due to its buoyancy, while fluid moving up by convection is cooled by conduction so fast that its driving buoyancy will diminish. On the other hand, if heat conduction is very low, a large temperature gradient may be formed and convection might be very strong.
The Rayleigh number ( R a {\displaystyle Ra} ) is the product of the Grashof and Prandtl numbers. It is a measure which determines the relative strength of conduction and convection.[13]
R a = G r P r = g Δ ρ L 3 μ α = g β Δ T L 3 ν α {\displaystyle Ra=GrPr={\frac {g\Delta \rho L^{3}}{\mu \alpha }}={\frac {g\beta \Delta TL^{3}}{\nu \alpha }}}
where
g is acceleration due to gravity,
ρ is the density with Δ ρ {\displaystyle \Delta \rho } being the density difference between the lower and upper ends,
μ is the dynamic viscosity,
α is the Thermal diffusivity,
β is the volume thermal expansivity (sometimes denoted α elsewhere),
T is the temperature,
ν is the kinematic viscosity, and
L is characteristic length.
The Rayleigh number can be understood as the ratio between the rate of heat transfer by convection to the rate of heat transfer by conduction; or, equivalently, the ratio between the corresponding timescales (i.e. conduction timescale divided by convection timescale), up to a numerical factor. This can be seen as follows, where all calculations are up to numerical factors depending on the geometry of the system.
The buoyancy force driving the convection is roughly g Δ ρ L 3 {\displaystyle g\Delta \rho L^{3}} , so the corresponding pressure is roughly g Δ ρ L {\displaystyle g\Delta \rho L} . In steady state, this is canceled by the shear stress due to viscosity, and therefore roughly equals μ V / L = μ / T c o n v {\displaystyle \mu V/L=\mu /T_{conv}} , where V is the typical fluid velocity due to convection and T c o n v {\displaystyle T_{conv}} the order of its timescale.[citation needed] The conduction timescale, on the other hand, is of the order of T c o n d = L 2 / α {\displaystyle T_{cond}=L^{2}/\alpha } .
Convection occurs when the Rayleigh number is above 1,000–2,000.
Radiation[edit]
Red-hot iron object, transferring heat to the surrounding environment through thermal radiation
Thermal radiation occurs through a vacuum or any transparent medium (solid or fluid). It is the transfer of energy by means of photons in electromagnetic waves governed by the same laws.[1]
Thermal radiation is energy emitted by matter as electromagnetic waves, due to the pool of thermal energy in all matter with a temperature above absolute zero. Thermal radiation propagates without the presence of matter through the vacuum of space.[14]
Thermal radiation is a direct result of the random movements of atoms and molecules in matter. Since these atoms and molecules are composed of charged particles (protons and electrons), their movement results in the emission of electromagnetic radiation, which carries energy away from the surface.
The Stefan-Boltzmann equation, which describes the rate of transfer of radiant energy, is as follows for an object in a vacuum :
Q = ϵ σ T 4 . {\displaystyle Q=\epsilon \sigma T^{4}.}
For radiative transfer between two objects, the equation is as follows:
Q = ϵ σ ( T a 4 − T b 4 ) , {\displaystyle Q=\epsilon \sigma (T_{a}^{4}-T_{b}^{4}),}
where Q is the heat flux, ε is the emissivity (unity for a black body), σ is the Stefan–Boltzmann constant, and T is the absolute temperature (in kelvins or degrees Rankine). Radiation is typically only important for very hot objects, or for objects with a large temperature difference.
Radiation from the sun, or solar radiation, can be harvested for heat and power.[15] Unlike conductive and convective forms of heat transfer, thermal radiation can be concentrated in a small spot by using reflecting mirrors, which is exploited in concentrating solar power generation.[16] For example, the sunlight reflected from mirrors heats the PS10 solar power tower and during the day it can heat water to 285 °C (545 °F).[citation needed]
Phase transition[edit]
Lightning is a highly visible form of energy transfer and is an example of plasma present at Earth's surface. Typically, lightning discharges 30,000 amperes at up to 100 million volts, and emits light, radio waves, X-rays and even gamma rays.[17] Plasma temperatures in lightning can approach 28,000 kelvins (27,726.85 °C) (49,940.33 °F) and electron densities may exceed 1024 m−3.
Phase transition or phase change, takes place in a thermodynamic system from one phase or state of matter to another one by heat transfer. Phase change examples are the melting of ice or the boiling of water. The Mason equation explains the growth of a water droplet based on the effects of heat transport on evaporation and condensation.
Phase transitions involve the four fundamental states of matter:
Solid – Deposition, freezing and solid to solid transformation.
Gas – Boiling / evaporation, recombination / deionization, and sublimation.
Liquid – Condensation and melting / fusion.
Plasma – Ionization.
Boiling[edit]
Nucleate boiling of water.
The boiling point of a substance is the temperature at which the vapor pressure of the liquid equals the pressure surrounding the liquid[18][19] and the liquid evaporates resulting in an abrupt change in vapor volume.
Saturation temperature means boiling point. The saturation temperature is the temperature for a corresponding saturation pressure at which a liquid boils into its vapor phase. The liquid can be said to be saturated with thermal energy. Any addition of thermal energy results in a phase transition.
At standard atmospheric pressure and low temperatures, no boiling occurs and the heat transfer rate is controlled by the usual single-phase mechanisms. As the surface temperature is increased, local boiling occurs and vapor bubbles nucleate, grow into the surrounding cooler fluid, and collapse. This is sub-cooled nucleate boiling, and is a very efficient heat transfer mechanism. At high bubble generation rates, the bubbles begin to interfere and the heat flux no longer increases rapidly with surface temperature (this is the departure from nucleate boiling, or DNB).
At similar standard atmospheric pressure and high temperatures, the hydrodynamically-quieter regime of film boiling is reached. Heat fluxes across the stable vapor layers are low, but rise slowly with temperature. Any contact between fluid and the surface that may be seen probably leads to the extremely rapid nucleation of a fresh vapor layer ("spontaneous nucleation"). At higher temperatures still, a maximum in the heat flux is reached (the critical heat flux, or CHF).
The Leidenfrost Effect demonstrates how nucleate boiling slows heat transfer due to gas bubbles on the heater's surface. As mentioned, gas-phase thermal conductivity is much lower than liquid-phase thermal conductivity, so the outcome is a kind of "gas thermal barrier".
Condensation[edit]
Condensation occurs when a vapor is cooled and changes its phase to a liquid. During condensation, the latent heat of vaporization must be released. The amount of the heat is the same as that absorbed during vaporization at the same fluid pressure.[20]
There are several types of condensation:
Homogeneous condensation, as during a formation of fog.
Condensation in direct contact with subcooled liquid.
Condensation on direct contact with a cooling wall of a heat exchanger: This is the most common mode used in industry:
Filmwise condensation is when a liquid film is formed on the subcooled surface, and usually occurs when the liquid wets the surface.
Dropwise condensation is when liquid drops are formed on the subcooled surface, and usually occurs when the liquid does not wet the surface.
Dropwise condensation is difficult to sustain reliably; therefore, industrial equipment is normally designed to operate in filmwise condensation mode.
Melting[edit]
Ice melting
Melting is a thermal process that results in the phase transition of a substance from a solid to a liquid. The internal energy of a substance is increased, typically with heat or pressure, resulting in a rise of its temperature to the melting point, at which the ordering of ionic or molecular entities in the solid breaks down to a less ordered state and the solid liquefies. Molten substances generally have reduced viscosity with elevated temperature; an exception to this maxim is the element sulfur, whose viscosity increases to a point due to polymerization and then decreases with higher temperatures in its molten state.[21]
Modeling approaches[edit]
Heat transfer can be modeled in the following ways.
Climate models[edit]
Climate models study the radiant heat transfer by using quantitative methods to simulate the interactions of the atmosphere, oceans, land surface, and ice.
Heat equation[edit]
The heat equation is an important partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. In some cases, exact solutions of the equation are available;[22] in other cases the equation must be solved numerically using computational methods.
Lumped system analysis[edit]
Lumped system analysis often reduces the complexity of the equations to one first-order linear differential equation, in which case heating and cooling are described by a simple exponential solution, often referred to as Newton's law of cooling.
System analysis by the lumped capacitance model is a common approximation in transient conduction that may be used whenever heat conduction within an object is much faster than heat conduction across the boundary of the object. This is a method of approximation that reduces one aspect of the transient conduction system—that within the object—to an equivalent steady state system. That is, the method assumes that the temperature within the object is completely uniform, although its value may be changing in time.
In this method, the ratio of the conductive heat resistance within the object to the convective heat transfer resistance across the object's boundary, known as the Biot number, is calculated. For small Biot numbers, the approximation of spatially uniform temperature within the object can be used: it can be presumed that heat transferred into the object has time to uniformly distribute itself, due to the lower resistance to doing so, as compared with the resistance to heat entering the object.[23]
Engineering[edit]
Heat exposure as part of a fire test for firestop products
Heat transfer has broad application to the functioning of numerous devices and systems. Heat-transfer principles may be used to preserve, increase, or decrease temperature in a wide variety of circumstances.[citation needed] Heat transfer methods are used in numerous disciplines, such as automotive engineering, thermal management of electronic devices and systems, climate control, insulation, materials processing, and power station engineering.
Insulation, radiance and resistance[edit]
Thermal insulators are materials specifically designed to reduce the flow of heat by limiting conduction, convection, or both. Thermal resistance is a heat property and the measurement by which an object or material resists to heat flow (heat per time unit or thermal resistance) to temperature difference.
Radiance or spectral radiance are measures of the quantity of radiation that passes through or is emitted. Radiant barriers are materials that reflect radiation, and therefore reduce the flow of heat from radiation sources. Good insulators are not necessarily good radiant barriers, and vice versa. Metal, for instance, is an excellent reflector and a poor insulator.
The effectiveness of a radiant barrier is indicated by its reflectivity, which is the fraction of radiation reflected. A material with a high reflectivity (at a given wavelength) has a low emissivity (at that same wavelength), and vice versa. At any specific wavelength, reflectivity=1 - emissivity. An ideal radiant barrier would have a reflectivity of 1, and would therefore reflect 100 percent of incoming radiation. Vacuum flasks, or Dewars, are silvered to approach this ideal. In the vacuum of space, satellites use multi-layer insulation, which consists of many layers of aluminized (shiny) Mylar to greatly reduce radiation heat transfer and control satellite temperature.[citation needed]
Devices[edit]
Schematic flow of energy in a heat engine.
A heat engine is a system that performs the conversion of a flow of thermal energy (heat) to mechanical energy to perform mechanical work.[24][25]
A thermocouple is a temperature-measuring device and widely used type of temperature sensor for measurement and control, and can also be used to convert heat into electric power.
A thermoelectric cooler is a solid state electronic device that pumps (transfers) heat from one side of the device to the other when electric current is passed through it. It is based on the Peltier effect.
A thermal diode or thermal rectifier is a device that causes heat to flow preferentially in one direction.
Heat exchangers[edit]
A heat exchanger is used for more efficient heat transfer or to dissipate heat. Heat exchangers are widely used in refrigeration, air conditioning, space heating, power generation, and chemical processing. One common example of a heat exchanger is a car's radiator, in which the hot coolant fluid is cooled by the flow of air over the radiator's surface.[citation needed]
Common types of heat exchanger flows include parallel flow, counter flow, and cross flow. In parallel flow, both fluids move in the same direction while transferring heat; in counter flow, the fluids move in opposite directions; and in cross flow, the fluids move at right angles to each other. Common constructions for heat exchanger include shell and tube, double pipe, extruded finned pipe, spiral fin pipe, u-tube, and stacked plate.[further explanation needed]
A heat sink is a component that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems or the radiator in a car. A heat pipe is another heat-transfer device that combines thermal conductivity and phase transition to efficiently transfer heat between two solid interfaces.
Applications[edit]
Architecture[edit]
Efficient energy use is the goal to reduce the amount of energy required in heating or cooling. In architecture, condensation and air currents can cause cosmetic or structural damage. An energy audit can help to assess the implementation of recommended corrective procedures. For instance, insulation improvements, air sealing of structural leaks or the addition of energy-efficient windows and doors.[26]
Smart meter is a device that records electric energy consumption in intervals.
Thermal transmittance is the rate of transfer of heat through a structure divided by the difference in temperature across the structure. It is expressed in watts per square meter per kelvin, or W/(m²K). Well-insulated parts of a building have a low thermal transmittance, whereas poorly-insulated parts of a building have a high thermal transmittance.
Thermostat is a device to monitor and control temperature.
Climate engineering[edit]
See also: Anthropogenic heat
An example application in climate engineering includes the creation of Biochar through the pyrolysis process. Thus, storing greenhouse gases in carbon reduces the radiative forcing capacity in the atmosphere, causing more long-wave (infrared) radiation out to Space.
Climate engineering consists of carbon dioxide removal and solar radiation management. Since the amount of carbon dioxide determines the radiative balance of Earth atmosphere, carbon dioxide removal techniques can be applied to reduce the radiative forcing. Solar radiation management is the attempt to absorb less solar radiation to offset the effects of greenhouse gases.
Greenhouse effect[edit]
A representation of the exchanges of energy between the source (the Sun), the Earth's surface, the Earth's atmosphere, and the ultimate sink outer space. The ability of the atmosphere to capture and recycle energy emitted by the Earth surface is the defining characteristic of the greenhouse effect.
The greenhouse effect is a process by which thermal radiation from a planetary surface is absorbed by atmospheric greenhouse gases, and is re-radiated in all directions. Since part of this re-radiation is back towards the surface and the lower atmosphere, it results in an elevation of the average surface temperature above what it would be in the absence of the gases.
Heat transfer in the human body[edit]
See also: Wet-bulb temperature
The principles of heat transfer in engineering systems can be applied to the human body in order to determine how the body transfers heat. Heat is produced in the body by the continuous metabolism of nutrients which provides energy for the systems of the body.[27] The human body must maintain a consistent internal temperature in order to maintain healthy bodily functions. Therefore, excess heat must be dissipated from the body to keep it from overheating. When a person engages in elevated levels of physical activity, the body requires additional fuel which increases the metabolic rate and the rate of heat production. The body must then use additional methods to remove the additional heat produced in order to keep the internal temperature at a healthy level.
Heat transfer by convection is driven by the movement of fluids over the surface of the body. This convective fluid can be either a liquid or a gas. For heat transfer from the outer surface of the body, the convection mechanism is dependent on the surface area of the body, the velocity of the air, and the temperature gradient between the surface of the skin and the ambient air.[28] The normal temperature of the body is approximately 37 °C. Heat transfer occurs more readily when the temperature of the surroundings is significantly less than the normal body temperature. This concept explains why a person feels “cold” when not enough covering is worn when exposed to a cold environment. Clothing can be considered an insulator which provides thermal resistance to heat flow over the covered portion of the body.[29] This thermal resistance causes the temperature on the surface of the clothing to be less than the temperature on the surface of the skin. This smaller temperature gradient between the surface temperature and the ambient temperature will cause a lower rate of heat transfer than if the skin were not covered.
In order to ensure that one portion of the body is not significantly hotter than another portion, heat must be distributed evenly through the bodily tissues. Blood flowing through blood vessels acts as a convective fluid and helps to prevent any buildup of excess heat inside the tissues of the body. This flow of blood through the vessels can be modeled as pipe flow in an engineering system. The heat carried by the blood is determined by the temperature of the surrounding tissue, the diameter of the blood vessel, the thickness of the fluid, velocity of the flow, and the heat transfer coefficient of the blood. The velocity, blood vessel diameter, and the fluid thickness can all be related with the Reynolds Number, a dimensionless number used in fluid mechanics to characterize the flow of fluids.
Latent heat loss, also known as evaporative heat loss, accounts for a large fraction of heat loss from the body. When the core temperature of the body increases, the body triggers sweat glands in the skin to bring additional moisture to the surface of the skin. The liquid is then transformed into vapor which removes heat from the surface of the body.[30] The rate of evaporation heat loss is directly related to the vapor pressure at the skin surface and the amount of moisture present on the skin.[28] Therefore, the maximum of heat transfer will occur when the skin is completely wet. The body continuously loses water by evaporation but the most significant amount of heat loss occurs during periods of increased physical activity.
Cooling techniques[edit]
Evaporative cooling[edit]
A traditional air cooler in Mirzapur, Uttar Pradesh, India
Evaporative cooling happens when water vapor is added to the surrounding air. The energy needed to evaporate the water is taken from the air in the form of sensible heat and converted into latent heat, while the air remains at a constant enthalpy. Latent heat describes the amount of heat that is needed to evaporate the liquid; this heat comes from the liquid itself and the surrounding gas and surfaces. The greater the difference between the two temperatures, the greater the evaporative cooling effect. When the temperatures are the same, no net evaporation of water in air occurs; thus, there is no cooling effect.
Laser cooling[edit]
In quantum physics, laser cooling is used to achieve temperatures of near absolute zero (−273.15 °C, −459.67 °F) of atomic and molecular samples to observe unique quantum effects that can only occur at this heat level.
Doppler cooling is the most common method of laser cooling.
Sympathetic cooling is a process in which particles of one type cool particles of another type. Typically, atomic ions that can be directly laser-cooled are used to cool nearby ions or atoms. This technique allows cooling of ions and atoms that cannot be laser cooled directly.[citation needed]
Magnetic cooling[edit]
Main articles: Magnetic refrigeration and Magnetic evaporative cooling
Magnetic evaporative cooling is a process for lowering the temperature of a group of atoms, after pre-cooled by methods such as laser cooling. Magnetic refrigeration cools below 0.3K, by making use of the magnetocaloric effect.
Radiative cooling[edit]
Radiative cooling is the process by which a body loses heat by radiation. Outgoing energy is an important effect in the Earth's energy budget. In the case of the Earth-atmosphere system, it refers to the process by which long-wave (infrared) radiation is emitted to balance the absorption of short-wave (visible) energy from the Sun. Convective transport of heat and evaporative transport of latent heat both remove heat from the surface and redistribute it in the atmosphere.
Thermal energy storage[edit]
Thermal energy storage includes technologies for collecting and storing energy for later use. It may be employed to balance energy demand between day and nighttime. The thermal reservoir may be maintained at a temperature above or below that of the ambient environment. Applications include space heating, domestic or process hot water systems, or generating electricity.
See also[edit]
Combined forced and natural convection
Heat capacity
Heat transfer physics
Stefan–Boltzmann law
Thermal contact conductance
Thermal physics
Thermal resistance in electronics
Thermal science
Heat transfer enhancement
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Jump up ^ David.E. Goldberg (1988). 3,000 Solved Problems in Chemistry (1st ed.). McGraw-Hill. Section 17.43, page 321. ISBN 0-07-023684-4.
Jump up ^ Louis Theodore, R. Ryan Dupont and Kumar Ganesan (Editors) (1999). Pollution Prevention: The Waste Management Approach to the 21st Century. CRC Press. Section 27, page 15. ISBN 1-56670-495-2. CS1 maint: Extra text: authors list (link)
Jump up ^ Tro, Nivaldo (2008). Chemistry: A Molecular Approach. Upper Saddle River, New Jersey: Prentice Hall. p. 479. When a substance condenses from a gas to a liquid, the same amount of heat is involved, but the heat is emitted rather than absorbed.
Jump up ^ C. Michael Hogan (2011) Sulfur, Encyclopedia of Earth, eds. A. Jorgensen and C. J. Cleveland, National Council for Science and the environment, Washington DC
Jump up ^ Wendl, M. C. (2012). Theoretical Foundations of Conduction and Convection Heat Transfer. Wendl Foundation.
Jump up ^ "How to simplify for small Biot numbers". Retrieved December 21, 2016.
Jump up ^ Fundamentals of Classical Thermodynamics, 3rd ed. p. 159, (1985) by G. J. Van Wylen and R. E. Sonntag: "A heat engine may be defined as a device that operates in a thermodynamic cycle and does a certain amount of net positive work as a result of heat transfer from a high-temperature body and to a low-temperature body. Often the term heat engine is used in a broader sense to include all devices that produce work, either through heat transfer or combustion, even though the device does not operate in a thermodynamic cycle. The internal-combustion engine and the gas turbine are examples of such devices, and calling these heat engines is an acceptable use of the term."
Jump up ^ Mechanical efficiency of heat engines, p. 1 (2007) by James R. Senf: "Heat engines are made to provide mechanical energy from thermal energy."
Jump up ^ "EnergySavers: Tips on Saving Money & Energy at Home" (PDF). U.S. Department of Energy. Retrieved March 2, 2012.
Jump up ^ Hartman, Carl; Bibb, Lewis. (1913). "The Human Body and Its Enemies". World Book Co., p. 232.
^ Jump up to: a b Cengel, Yunus A. and Ghajar, Afshin J. "Heat and Mass Transfer: Fundamentals and Applications", McGraw-Hill, 4th Edition, 2010.
Jump up ^ Tao, Xiaoming. "Smart fibres, fabrics, and clothing", Woodhead Publishing, 2001
Jump up ^ Wilmore, Jack H.; Costill, David L.; Kenney, Larry. "Physiology of sport and exercise". Human Kinetics. 2008: 256.
External links[edit]
A Heat Transfer Textbook - (free download).
Thermal-FluidsPedia - An online thermal fluids encyclopedia.
Hyperphysics Article on Heat Transfer - Overview
Interseasonal Heat Transfer - a practical example of how heat transfer is used to heat buildings without burning fossil fuels.
Aspects of Heat Transfer, Cambridge University
Thermal-Fluids Central
Energy2D: Interactive Heat Transfer Simulations for Everyone
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text:
2017–18 UEFA Champions League knockout phase
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
The 2017–18 UEFA Champions League knockout phase began on 13 February and ended on 26 May 2018 with the final at the NSC Olimpiyskiy Stadium in Kiev, Ukraine, to decide the champions of the 2017–18 UEFA Champions League.[1] A total of 16 teams competed in the knockout phase.[2]
Times up to 24 March 2018 (round of 16) are CET (UTC+1), thereafter (quarter-finals and beyond) times are CEST (UTC+2).
Contents
[hide]
1 Round and draw dates
2 Format
3 Qualified teams
4 Bracket
5 Round of 16
5.1 Summary
5.2 Matches
6 Quarter-finals
6.1 Summary
6.2 Matches
7 Semi-finals
7.1 Summary
7.2 Matches
8 Final
9 Notes
10 References
11 External links
Round and draw dates[edit]
The schedule of the knockout phase was as follows (all draws are held at the UEFA headquarters in Nyon, Switzerland).[1]
Round
Draw
First leg
Second leg
Round of 16 11 December 2017 13–14 & 20–21 February 2018 6–7 & 13–14 March 2018
Quarter-finals 16 March 2018 3–4 April 2018 10–11 April 2018
Semi-finals 13 April 2018 24–25 April 2018 1–2 May 2018
Final 26 May 2018 at NSC Olimpiyskiy Stadium, Kiev
Format[edit]
The knockout phase involves the 16 teams which qualify as winners and runners-up of each of the eight groups in the group stage.
Each tie in the knockout phase, apart from the final, is played over two legs, with each team playing one leg at home. The team that scores more goals on aggregate over the two legs advance to the next round. If the aggregate score is level, the away goals rule is applied, i.e. the team that scored more goals away from home over the two legs advances. If away goals are also equal, then thirty minutes of extra time is played. The away goals rule is again applied after extra time, i.e. if there are goals scored during extra time and the aggregate score is still level, the visiting team advances by virtue of more away goals scored. If no goals are scored during extra time, the tie is decided by penalty shoot-out. In the final, which is played as a single match, if scores are level at the end of normal time, extra time is played, followed by penalty shoot-out if scores remain tied.[2]
The mechanism of the draws for each round is as follows:
In the draw for the round of 16, the eight group winners are seeded, and the eight group runners-up are unseeded. The seeded teams are drawn against the unseeded teams, with the seeded teams hosting the second leg. Teams from the same group or the same association cannot be drawn against each other.
In the draws for the quarter-finals onwards, there are no seedings, and teams from the same group or the same association can be drawn against each other.
Qualified teams[edit]
Group
Winners
(seeded in round of 16 draw)
Runners-up
(unseeded in round of 16 draw)
A
Manchester United Basel
B
Paris Saint-Germain Bayern Munich
C
Roma Chelsea
D
Barcelona Juventus
E
Liverpool Sevilla
F
Manchester City Shakhtar Donetsk
G
Beşiktaş Porto
H
Tottenham Hotspur Real Madrid
Bracket[edit]
Round of 16 Quarter-finals Semi-finals Final (26 May – Kiev)
Sevilla 0 2
2
Manchester United 0 1
1
Sevilla 1 0
1
Bayern Munich 2 0
2
Bayern Munich 5 3
8
Beşiktaş 0 1
1
Bayern Munich 1 2
3
Real Madrid 2 2
4
Juventus 2 2
4
Tottenham Hotspur 2 1
3
Juventus 0 3
3
Real Madrid 3 1
4
Real Madrid 3 2
5
Paris Saint-Germain 1 1
2
Real Madrid
3
Liverpool
1
Porto 0 0
0
Liverpool 5 0
5
Liverpool 3 2
5
Manchester City 0 1
1
Basel 0 2
2
Manchester City 4 1
5
Liverpool 5 2
7
Roma 2 4
6
Chelsea 1 0
1
Barcelona 1 3
4
Barcelona 4 0
4
Roma (a) 1 3
4
Shakhtar Donetsk 2 0
2
Roma (a) 1 1
2
Round of 16[edit]
The draw for the round of 16 was held on 11 December 2017, 12:00 CET.[3]
With five English teams in the round of 16, this was the first time five teams from one association qualified for the Champions League knockout phase.[4]
Summary[edit]
The first legs were played on 13, 14, 20 and 21 February, and the second legs were played on 6, 7, 13 and 14 March 2018.
Team 1
Agg.
Team 2
1st leg
2nd leg
Juventus 4–3 Tottenham Hotspur 2–2 2–1
Basel 2–5 Manchester City 0–4 2–1
Porto 0–5 Liverpool 0–5 0–0
Sevilla 2–1 Manchester United 0–0 2–1
Real Madrid 5–2 Paris Saint-Germain 3–1 2–1
Shakhtar Donetsk 2–2 (a) Roma 2–1 0–1
Chelsea 1–4 Barcelona 1–1 0–3
Bayern Munich 8–1 Beşiktaş 5–0 3–1
Matches[edit]
13 February 2018 (2018-02-13)20:45
Juventus
2–2
Tottenham Hotspur
Higuaín 2', 9' (pen.)
Report
Kane 35'
Eriksen 71'
Juventus Stadium, Turin
Attendance: 41,232[5]
Referee: Felix Brych (Germany)
7 March 2018 (2018-03-07)20:45
Tottenham Hotspur
1–2
Juventus
Son Heung-min 39'
Report
Higuaín 64'
Dybala 67'
Wembley Stadium, London
Attendance: 84,010[6]
Referee: Szymon Marciniak (Poland)
Juventus won 4–3 on aggregate.
13 February 2018 (2018-02-13)20:45
Basel
0–4
Manchester City
Report
Gündoğan 14', 53'
B. Silva 18'
Agüero 23'
St. Jakob-Park, Basel
Attendance: 36,000[7]
Referee: Jonas Eriksson (Sweden)
7 March 2018 (2018-03-07)20:45
Manchester City
1–2
Basel
Gabriel Jesus 8'
Report
Elyounoussi 17'
Lang 71'
City of Manchester Stadium, Manchester
Attendance: 49,411[8]
Referee: Pavel Královec (Czech Republic)
Manchester City won 5–2 on aggregate.
14 February 2018 (2018-02-14)20:45
Porto
0–5
Liverpool
Report
Mané 25', 53', 85'
Salah 29'
Firmino 69'
Estádio do Dragão, Porto
Attendance: 47,718[9]
Referee: Daniele Orsato (Italy)
6 March 2018 (2018-03-06)20:45
Liverpool
0–0
Porto
Report
Anfield, Liverpool
Attendance: 48,768[10]
Referee: Felix Zwayer (Germany)
Liverpool won 5–0 on aggregate.
21 February 2018 (2018-02-21)20:45
Sevilla
0–0
Manchester United
Report
Ramón Sánchez Pizjuán, Seville
Attendance: 39,725[11]
Referee: Clément Turpin (France)
13 March 2018 (2018-03-13)20:45
Manchester United
1–2
Sevilla
Lukaku 84'
Report
Ben Yedder 74', 78'
Old Trafford, Manchester
Attendance: 74,909[12]
Referee: Danny Makkelie (Netherlands)
Sevilla won 2–1 on aggregate.
14 February 2018 (2018-02-14)20:45
Real Madrid
3–1
Paris Saint-Germain
Ronaldo 45' (pen.), 83'
Marcelo 86'
Report
Rabiot 33'
Santiago Bernabéu, Madrid
Attendance: 78,158[13]
Referee: Gianluca Rocchi (Italy)
6 March 2018 (2018-03-06)20:45
Paris Saint-Germain
1–2
Real Madrid
Cavani 71'
Report
Ronaldo 51'
Casemiro 80'
Parc des Princes, Paris
Attendance: 46,585[14]
Referee: Felix Brych (Germany)
Real Madrid won 5–2 on aggregate.
21 February 2018 (2018-02-21)20:45
Shakhtar Donetsk
2–1
Roma
Ferreyra 52'
Fred 71'
Report
Ünder 41'
Metalist Stadium, Kharkiv[note 1]
Attendance: 35,124[15]
Referee: Willie Collum (Scotland)
13 March 2018 (2018-03-13)20:45
Roma
1–0
Shakhtar Donetsk
Džeko 52'
Report
Stadio Olimpico, Rome
Attendance: 47,693[16]
Referee: Alberto Undiano Mallenco (Spain)
2–2 on aggregate. Roma won on away goals.
20 February 2018 (2018-02-20)20:45
Chelsea
1–1
Barcelona
Willian 62'
Report
Messi 75'
Stamford Bridge, London
Attendance: 37,741[17]
Referee: Cüneyt Çakır (Turkey)
14 March 2018 (2018-03-14)20:45
Barcelona
3–0
Chelsea
Messi 3', 63'
Dembélé 20'
Report
Camp Nou, Barcelona
Attendance: 97,183[18]
Referee: Damir Skomina (Slovenia)
Barcelona won 4–1 on aggregate.
20 February 2018 (2018-02-20)20:45
Bayern Munich
5–0
Beşiktaş
Müller 43', 66'
Coman 53'
Lewandowski 79', 88'
Report
Allianz Arena, Munich
Attendance: 70,000[19]
Referee: Ovidiu Hațegan (Romania)
14 March 2018 (2018-03-14)18:00
Beşiktaş
1–3
Bayern Munich
Vágner Love 59'
Report
Thiago 18'
Gönül 46' (o.g.)
Wagner 84'
Vodafone Park, Istanbul
Attendance: 36,885[20]
Referee: Michael Oliver (England)
Bayern Munich won 8–1 on aggregate.
Quarter-finals[edit]
The draw for the quarter-finals was held on 16 March 2018, 12:00 CET.[21][22]
Summary[edit]
The first legs were played on 3 and 4 April, and the second legs were played on 10 and 11 April 2018.
Team 1
Agg.
Team 2
1st leg
2nd leg
Barcelona 4–4 (a) Roma 4–1 0–3
Sevilla 1–2 Bayern Munich 1–2 0–0
Juventus 3–4 Real Madrid 0–3 3–1
Liverpool 5–1 Manchester City 3–0 2–1
Matches[edit]
4 April 2018 (2018-04-04)20:45
Barcelona
4–1
Roma
De Rossi 38' (o.g.)
Manolas 55' (o.g.)
Piqué 59'
L. Suárez 87'
Report
Džeko 80'
Camp Nou, Barcelona
Attendance: 90,106[23]
Referee: Danny Makkelie (Netherlands)
10 April 2018 (2018-04-10)20:45
Roma
3–0
Barcelona
Džeko 6'
De Rossi 58' (pen.)
Manolas 82'
Report
Stadio Olimpico, Rome
Attendance: 56,580[24]
Referee: Clément Turpin (France)
4–4 on aggregate. Roma won on away goals.
3 April 2018 (2018-04-03)20:45
Sevilla
1–2
Bayern Munich
Sarabia 31'
Report
Navas 37' (o.g.)
Thiago 68'
Ramón Sánchez Pizjuán, Seville
Attendance: 40,635[25]
Referee: Daniele Orsato (Italy)
11 April 2018 (2018-04-11)20:45
Bayern Munich
0–0
Sevilla
Report
Allianz Arena, Munich
Attendance: 70,000[26]
Referee: Willie Collum (Scotland)
Bayern Munich won 2–1 on aggregate.
3 April 2018 (2018-04-03)20:45
Juventus
0–3
Real Madrid
Report
Ronaldo 3', 64'
Marcelo 72'
Juventus Stadium, Turin
Attendance: 40,849[27]
Referee: Cüneyt Çakır (Turkey)
11 April 2018 (2018-04-11)20:45
Real Madrid
1–3
Juventus
Ronaldo 90+8' (pen.)
Report
Mandžukić 2', 37'
Matuidi 61'
Santiago Bernabéu, Madrid
Attendance: 75,796[28]
Referee: Michael Oliver (England)
Real Madrid won 4–3 on aggregate.
4 April 2018 (2018-04-04)20:45
Liverpool
3–0
Manchester City
Salah 12'
Oxlade-Chamberlain 21'
Mané 31'
Report
Anfield, Liverpool
Attendance: 50,685[29]
Referee: Felix Brych (Germany)
10 April 2018 (2018-04-10)20:45
Manchester City
1–2
Liverpool
Gabriel Jesus 2'
Report
Salah 56'
Firmino 77'
City of Manchester Stadium, Manchester
Attendance: 53,461[30]
Referee: Antonio Mateu Lahoz (Spain)
Liverpool won 5–1 on aggregate.
Semi-finals[edit]
The draw for the semi-finals was held on 13 April 2018, 13:00 CEST.[31]
For the first time since the 2009–10 UEFA Champions League, all the four teams at this stage represent different national associations.[32]
Summary[edit]
The first legs were played on 24 and 25 April, and the second legs were played on 1 and 2 May 2018.
Team 1
Agg.
Team 2
1st leg
2nd leg
Bayern Munich 3–4 Real Madrid 1–2 2–2
Liverpool 7–6 Roma 5–2 2–4
Matches[edit]
25 April 2018 (2018-04-25)20:45
Bayern Munich
1–2
Real Madrid
Kimmich 28'
Report
Marcelo 44'
Asensio 57'
Allianz Arena, Munich
Attendance: 70,000[33]
Referee: Björn Kuipers (Netherlands)
1 May 2018 (2018-05-01)20:45
Real Madrid
2–2
Bayern Munich
Benzema 11', 46'
Report
Kimmich 3'
Rodríguez 63'
Santiago Bernabéu, Madrid
Attendance: 77,459[34]
Referee: Cüneyt Çakır (Turkey)
Real Madrid won 4–3 on aggregate.
24 April 2018 (2018-04-24)20:45
Liverpool
5–2
Roma
Salah 36', 45+1'
Mané 56'
Firmino 61', 69'
Report
Džeko 81'
Perotti 85' (pen.)
Anfield, Liverpool
Attendance: 51,236[35]
Referee: Felix Brych (Germany)
2 May 2018 (2018-05-02)20:45
Roma
4–2
Liverpool
Milner 15' (o.g.)
Džeko 52'
Nainggolan 86', 90+4' (pen.)
Report
Mané 9'
Wijnaldum 25'
Stadio Olimpico, Rome
Attendance: 61,889[36]
Referee: Damir Skomina (Slovenia)
Liverpool won 7–6 on aggregate.
Final[edit]
Main article: 2018 UEFA Champions League Final
The final was played at the NSC Olimpiyskiy Stadium in Kiev on 26 May 2018. The "home" team (for administrative purposes) was determined by an additional draw held after the semi-final draw.[31]
26 May 2018 (2018-05-26)20:45
Real Madrid
3–1
Liverpool
Benzema 51'
Bale 64', 83'
Report
Mané 55'
NSC Olimpiyskiy Stadium, Kiev
Attendance: 61,561[37]
Referee: Milorad Mažić (Serbia)
Notes[edit]
Jump up ^ Shakhtar Donetsk played their home match at Metalist Stadium, Kharkiv, instead of their regular stadium Donbass Arena, Donetsk, due to the war conditions in Eastern Ukraine.
References[edit]
^ Jump up to: a b "2017/18 Champions League match and draw calendar". UEFA.com. 11 January 2017.
^ Jump up to: a b "Regulations of the UEFA Champions League 2017/18 Season" (PDF). UEFA.com. 4 April 2017.
Jump up ^ "Round of 16 draw". UEFA.com.
Jump up ^ "Knockout stage first: five English clubs through". UEFA.com. 6 December 2017.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Juventus v Tottenham Hotspur" (PDF). UEFA.com. Union of European Football Associations. 13 February 2018. Retrieved 13 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Tottenham Hotspur v Juventus" (PDF). UEFA.com. Union of European Football Associations. 7 March 2018. Retrieved 7 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Basel v Manchester City" (PDF). UEFA.com. Union of European Football Associations. 13 February 2018. Retrieved 13 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Manchester City v Basel" (PDF). UEFA.com. Union of European Football Associations. 7 March 2018. Retrieved 7 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Porto v Liverpool" (PDF). UEFA.com. Union of European Football Associations. 14 February 2018. Retrieved 14 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Liverpool v Porto" (PDF). UEFA.com. Union of European Football Associations. 6 March 2018. Retrieved 6 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Sevilla v Manchester United" (PDF). UEFA.com. Union of European Football Associations. 21 February 2018. Retrieved 21 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Manchester United v Sevilla" (PDF). UEFA.com. Union of European Football Associations. 13 March 2018. Retrieved 13 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Real Madrid v Paris Saint-Germain" (PDF). UEFA.com. Union of European Football Associations. 14 February 2018. Retrieved 14 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Paris Saint-Germain v Real Madrid" (PDF). UEFA.com. Union of European Football Associations. 6 March 2018. Retrieved 6 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Shakhtar Donetsk v Roma" (PDF). UEFA.com. Union of European Football Associations. 21 February 2018. Retrieved 21 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Roma v Shakhtar Donetsk" (PDF). UEFA.com. Union of European Football Associations. 13 March 2018. Retrieved 13 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Chelsea v Barcelona" (PDF). UEFA.com. Union of European Football Associations. 20 February 2018. Retrieved 20 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Barcelona v Chelsea" (PDF). UEFA.com. Union of European Football Associations. 14 March 2018. Retrieved 14 March 2018.
Jump up ^ "Full Time Summary Round of 16 1st Leg – Bayern Munich v Beşiktaş" (PDF). UEFA.com. Union of European Football Associations. 20 February 2018. Retrieved 20 February 2018.
Jump up ^ "Full Time Summary Round of 16 2nd Leg – Beşiktaş v Bayern Munich" (PDF). UEFA.com. Union of European Football Associations. 14 March 2018. Retrieved 14 March 2018.
Jump up ^ "Quarter-final draw". UEFA.com.
Jump up ^ "UEFA Champions League quarter-final draw". UEFA.com.
Jump up ^ "Full Time Summary Quarter-finals 1st Leg – Barcelona v Roma" (PDF). UEFA.com. Union of European Football Associations. 4 April 2018. Retrieved 4 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 2nd Leg – Roma v Barcelona" (PDF). UEFA.com. Union of European Football Associations. 10 April 2018. Retrieved 10 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 1st Leg – Sevilla v Bayern Munich" (PDF). UEFA.com. Union of European Football Associations. 3 April 2018. Retrieved 3 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 2nd Leg – Bayern Munich v Sevilla" (PDF). UEFA.com. Union of European Football Associations. 11 April 2018. Retrieved 11 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 1st Leg – Juventus v Real Madrid" (PDF). UEFA.com. Union of European Football Associations. 3 April 2018. Retrieved 3 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 2nd Leg – Real Madrid v Juventus" (PDF). UEFA.com. Union of European Football Associations. 11 April 2018. Retrieved 11 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 1st Leg – Liverpool v Manchester City" (PDF). UEFA.com. Union of European Football Associations. 4 April 2018. Retrieved 4 April 2018.
Jump up ^ "Full Time Summary Quarter-finals 2nd Leg – Manchester City v Liverpool" (PDF). UEFA.com. Union of European Football Associations. 10 April 2018. Retrieved 10 April 2018.
^ Jump up to: a b "Semi-final and final draws". UEFA.com.
Jump up ^ "Champions League last four: which nations dominate?". UEFA.com. 12 April 2018.
Jump up ^ "Full Time Summary Semi-finals 1st Leg – Bayern Munich v Real Madrid" (PDF). UEFA.com. Union of European Football Associations. 25 April 2018. Retrieved 25 April 2018.
Jump up ^ "Full Time Summary Semi-finals 2nd Leg – Real Madrid v Bayern Munich" (PDF). UEFA.com. Union of European Football Associations. 1 May 2018. Retrieved 1 May 2018.
Jump up ^ "Full Time Summary Semi-finals 1st Leg – Liverpool v Roma" (PDF). UEFA.com. Union of European Football Associations. 24 April 2018. Retrieved 24 April 2018.
Jump up ^ "Full Time Summary Semi-finals 2nd Leg – Roma v Liverpool" (PDF). UEFA.com. Union of European Football Associations. 2 May 2018. Retrieved 2 May 2018.
Jump up ^ "Full Time Report Final – Real Madrid v Liverpool" (PDF). UEFA.com. Union of European Football Associations. 26 May 2018. Retrieved 26 May 2018.
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