Heat transfer and thermal properties of materials Background Effects of temperature on the properties of materials Heat transfer mechanisms Conduction Radiation Convection
Background: Thermal management Increasing demands for functionality results into increase in the level of integration. Reliability requirements are also increasing. All this induces stringent requirements for thermal management. Production of heat Heat Transfer and Their effect on functionality and reliability of the device
Essential is to be able to predict the temperature, which the device will experience during operation. Waste heat (intrinsic) Temperature of the environment Thermal management of the device is influenced by: Components Layout Materials etc. Thermal management Typically each material parameter dependes on temperature (linearly/non-linearly) and it is important to know their cumulative effects For example the temperature dependence of thin film resistors may be several hundreds of ppm/ C Speed of chemical reactions doubles with every 10 C increase in temperature.
Thermal management of electrical devices tries to answer the following questions: 1) How, how much and where heat is generated? Active/passive components Location of heat sources and their power 2) How to determine temperature at certain location? Thermal simulation is not an exact science Heat conduction is well established but convection and radiation still rely on empirical correlations especially with more complex geometries. In addition large amounts of computer resources are required for complex thermal problems. 3) How the excess heat can be transfered? Thermal management One must provide an effective path for the heat from where it is generated to outside world.
Thermal management Effects of temperature can be divided to three categories
Temperature and temperature change Materials absorb and emit heat as phonons Heat Capacity is the amount of energy needed to increase temperature 2 by one degree for one mole of substance α C p CV = TV, C p heat capacity under constant pressure κ C v heat capacity under constant volume κ is isothermal compressibility Specific heat is the energy required to increase the temperature of a certain mass (g) of substance by one degree Specifiche at = c = Heatcapacity Atomicweight Note that the value of heat capacity (specific heat) does not depend on the defect structure of the material, but is heavily dependent on the crystal structure
Temperature and temperature change Specific heats of materials 27 C Specific heat of iron as a function of temperature
Temperature and temperature change When temperature increases, distances between atoms change and this results into dimensional changes Thermal expansion This property is described by linear coefficient of thermal expansion, (CTE, α) CTE = α = l lo T CTE- values for different materials
Temperature and temperature change Volume coefficient of thermal expansion (α V ) for isotropic material is α = 3 α v Value of the coefficient of thermal expansion reflects the stregth of the atomic bonds (as does also Young s modulus E) Strong materials have typically low CTE High melting temperature materials have low CTE (Figure 2.)
Thermal stresses are formed when dimensional changes of materials are inhibited σ Temperature and temperature change thermal = αe T
Stresses T < T' IC:n paketointi IC Juotokset In thermal cycling different materials respond differently to temperature variation γ = (α subs. - α kot. )(T 2 - T 1 ) (L/2h) T > T' IC:n paketointi IC 2L α(fr4) = n. 16 10-6 1/K, α(si) = 2.7 10-6 1/K (av.) α(epoxies) > 50 10-6 1/K, α(snpb) = n. 24 10-6 1/K, α(al 2 O 3 ) = 6 10-6 1/K 2h Under deflection the surface of the component board bends -> shear deformation to interconnections shear γ = (hxl)/r, R = radius of curvature vertical displacement δ = [1- cos(l/r)]r
Heat transfer mechanisms Heat is materials internal energy and heat transfers is energy transfer caused by temperature difference. Three mechanisms: conduction, convection and radiation
Heat transfer mechanisms
Conduction At macroscopic level conduction means transfer of heat from a warmer part of a material (solid, liquid or gas) to the colder part or from one body to another (to colder one) by assuming there is physical contact between the two bodies. At microscopic level conduction is transfer of energy from more energetic particles to the less energetic ones as particles collide. There are two main types of conduction: In amorphous solids, liquids and gases heat conduction is transfer of movement from one species to another. This is called diffusion of heat. In crystalline material, the whole lattice starts to vibrate as a consequence of the atom movements (+ heat transferred by free electrons).
Conduction Heat conduction can be demonstrated as follows: T=T 1 -T 2 q A 1, T 1 x A 2, T 2 A rod made out of known material is insulated except for its ends. Ends of the rod are at different temperatures T 1 > T 2. Temperature difference induces a heat flux to the direction of positive x. Heat flux Q can be measured and it is a function of temperature difference, length of the rod and cross-sectional area (dt, dx, A).
Conduction Rate of heat conduction (amount of energy/unit time) for a one dimensional situation can be calculated from Fourier s law dq dt dt = λa( ) dx Unit J/s = W Where A is area, λ is heat conductivity (W/mK) and (dt/dx) is temperature gradient. Q=qA, where q is heat flux (W), which is perpendicular to the cross-sectional area A Minus sign indicates that heat flows from higher T to lower T (heat flux is positive as temperature decreases) In 3D (isotropic material) Fourier s law can be written as: q = λ T T = λ i x + T j y + k T z
Fourier s law can be expressed as: Q dt k = dx λa by integrating and assuming that temperature gradient is linear one gets a temperature drop within the length L as: T = Q L k λa Thermal resistance θ is given as: L θ = = λa T Q k unit C/W Conduction When heat source produces Q k Watts and temperature of the heat sink is kept constant T heat source = Theat sink + θ Qk
Conduction Analogies between heat and electrical conduction. Temperature drop T Voltage drop V Heat flux Q Current I Heat resistance θ Electrical resistance R Heat conduction λ Electrical conduction σ When multiple materials are joined together the heat resistance is given by: θ equiv = θ 1 + θ 2 + θ 3 + θ 4 + + θ N
Conduction Temperature at certain interface is given as: T j, j 1 = Theat sink + Qk θ j hs where T j,j-1 is T at the interface between layers j and j-1 and Σθ j,j-1:hs is the sum of the thermal resistances from the heat sink to the interface in question.
Conduction If there are several paths for the heat to propagate the thermal resistances are connected in parallel: θ 1 equiv = 1 θ 1 1 + θ 2 1 + θ 3 1 + + θ N
Conduction
Heat conduction in solids There are two mechanims: Transfer by free electrons λ e and Lattice vibrations (phonons) λ h. Conduction λ = λ + λ e h where λ e is directly proportional to electrical conductivity. With metals λ e dominates and λ h has relatively small effect
Conduction With pure metals there is a relation between λ and σ given by: λ = σ LT where T is temperature (K) and L (2.45 10 8 W Ω K 2 ) is Lorenz s number. As electrical conductivity decreases the importance of λ h increases and becomes dominant with non-metallic elements Symmetry of the lattice improves the conduction by lattice vibrations
Conduction Structure of liquids is less dense than that of solids. In liquids there is a third component contributing to heat conduction, which is photons λ f. λ = λ e + λ h + λ f Liquid metals conduct heat much better that other liquids because of the free electrons. Presence of the lattice component is based on the fact that many metallic melts have loosely cubic structures The heat conduction by photons comes from the fact that many polymer melts are infrared transparent.
Conduction Gases are least dense=> interactions between molecules small -> heat conduction small. In gases heat is conducted as molecules collide λ = n c v, m u k Cv k 3 = u 3 _ u where n number of particles in unit volume, is average molecular speed, k is the mean free path, c v,m is the molar specific heat and C v is heat capacity under constant volume
Transient heat flux Transient heat flux: Transient heat flux is given as: where the last term brings in the time dependence. From the equation one can get heat capacitance C Θ where ρ is density, C T is specific heat, V is volume, l is length and A crosssectional area Steady state t T C Q z T z y T y x T x v z y x + ρ = λ + λ + λ Θ ρ = ρ = = Θ L 0 T T T V C Adl C M C C v z y x Q z T z y T y x T x = λ + λ + λ
Heat capacitance Kovar sheet (Fe54Ni29Co17 w-%) C Θ = CT ρv = 0.023Ws/ 0 C Electrical analogue (RC-circuit) V C V 1 e ( t ) = RC 0 RC = time constant = τ
Heat capacitance For temperature and temperature response Temperature difference over the material is T T = T E 1 e where T E is the equilibrium temperature t θc θ thermal time constant τ = θc θ
Heat capacitance For Kovar-sheet, when heat flow is from top to bottom, heat resistance is θ = And time constant t 0 λa = 1.22 0 C W τ = θc 1.22 C 0.023 W Ws θ = 0 C = 0.921s Thus time to achive 99.9 % out of the final equilibrium temperature is given as: T T t E = = 1 6.36s e ( t ) τ = 0.999
Thermal spreading In microelectronics the heat source is often small and the heat is spread purposely to a much larger area By implementing a good thermal conductor between chip and the heat sink the heat flux to x, y- directions becomes larger than that to z-direction Heat spreads because the effective cross-sectional area has been increased Total heat resistance becomes smaller
Contact resistance Interfaces typically contribute greatly to heat conduction as at the microscopic level there are Point contacts Air pockets Elimination of air pockets at the Si heat sink interface is very important This can be achieved by soldering, however the heat involved may become a problem as many cooling device do not endure temperatures higher than 150 0 C. Also flux residues cause voids at the interface=> increase in heat resistance. In some cases CTE-differences may become important
Thus, typically one uses Thermal greases Elastomers Conductive adhesives Thermal greases are typically silicone based materials which are filled with heat conducting particles (aluminiumor zinc oxide) Adhesion of thermal greases is often a problem Elastomers are typically silicon pads (problem with bonding force) R '' t, c Contact resistance T A T q Conductive adhesives are typically filled with Ag flakes or ceramic particles. Adhesives can be obtained in liquid and solid (tape) form. B
All material bodies above absolute zero 0 K emit electromagnetic radiation Radiation 0.1-100 µm
Radiation Radiation: transfer of electromagnetic energy from one surface to another across a medium. Radiation turns into heat only when it is absorbed on a surface. In coming radiation can be reflected, absorbed or permeate the substance. α + ν + τ =1 Where α is absorptivity, υ is reflectivity and τ is transmissivity
Radiation The amount of energy transferred by radiation does not depend on temperature differences. Heat can be transferred between two bodies separated by a medium which is colder than either one of the bodies With respect to heat transfer the most important range of electromagnetic radiation is that of infra red (0,76-100 µm). As the temperature of the emitter increases the wave length of the emitted radiation decreases (energy increases). Above 500 ºC visible spectrum becomes important colours at normal temperatures do not appear because of emission but owing to absorption and reflection.
Radiation Transfer of energy by radiation does not require medium (any substance) Thus radiation energy is best transferred in vacuum Radiation is caused by changes in the electronic configuration of atoms and molecules (+vibration and rotation). Radiation energy is given by Stefan-Boltzmann law Q Aσ T 4 s = (5.670 10-8 W/m 2. K) where T s is the surface temperature (in K), σ is Stefan-Boltzmann constant and A is the surface area. Surface that strictly obeys the S-B law is called ideal emitter or black body (ε=1). (absorbs all radiation) The amount of emitted radiation is a function of material, its surface properties and temperature black body emits maximum amount of radiation
Radiation Real bodies are not like black bodies. Absorbtivity (0 α 1) and emissivity (0 ε 1) give the properties of normal substances with regard to the black body (these are not constant, but (T, λ, and direction) For the radiation emitted by a substance 4 4 Q = A ε σ (Ts Tsur For radiation absorbed by a substance Q = A α σ (T 4 s T 4 sur If the absorbing surface is partly shielded from the emitting surface S = shielding factor, view factor, is used which has values between 0 < S < 1 Q = S A ε σ (T 4 1 ) ) T 4 2 )
Example: Temperature of a heat sink is 150 C and that of the surrounding air 25 C. Heat sink is Ni-plated its area is 4 in 2 and shielding factor is 1.0. Heat power by radiation is Q P = P [(150 + = S A ε σ (T P = 1 4 0.11 3.65 10 = 273) 0.388W 4 (25 + Radiation 4 1 11 273) T 4 4 2 ] )
Radiation One can show that also radiation has thermal resistance Thermal resistance for radiation has signifigance when a warm surface is surrounded by a large volume of gas. Thermal resistance is given by R t, rad = T s q T rad sur Convection thermal resistance and radiation thermal resistance are connected in parallel.
Why car acts as a heat trap? All material bodies above absolute zero 0 K emit electromagnetic radiation Spectral transmissivity of low-iron glass at room temperature for different thicknesses
Convection Convection is heat transfer with the aid of moving fluid (liquid or gas) from warmer parts of the system to colder parts. Convection is the only heat transfer mechanism where macroscopic movement is required. Heat transfer from the body to fluid and vice versa takes place by conduction or radiation. Convective movement is formed when a surface and the fluid in contact with it are at different temperatures and it is composed of Random molecular movement Macroscopic flow of the fluid If flow is caused by density differences the phenomenon is called natural convection If flow is caused by pumps etc. it is called forced convection
Convection Heat transfer rate for convection is given by Newton s cooling law: Q c = h c A s ( T T ) = h A T s A c s where Q c is heat power from surface to environment [W], A is area and T is temperature difference between surface and fluid h c is the heat transfer coefficient for convection and depends on Shape and size of the body, Temperature of the fluid and velocity of the flow Physical properties of the fluid. Heat absorption Q can be influenced by Increasing/decreasing convection Changing the temperature difference between surface and the fluid T= t s - t f.
Convection Newton s cooling equation can be written as: 1 T = h A c s Q c From where the thermal resistance for convection can be obtained θ s = 1 h A c s Convection and conduction resistances are in series Natural convection is caused by the decrease in density of the fluid near the interface due to the thermal energy transferred from surface to fluid Buoyancy forces drive the energetic particles away from the surface into the fluid. Often results into a warmer boundary layer next to the surface
Convection For natural convection h = B T L 0.25 0.25 where B = DE is a constant, where D describes flow properties of the fluid, E shape of the surface and L is the characteristic length of the surface along the flow direction. For forced convection V h = B L 0.75 0.25 where B is constant, which depends on the structure of the surface and fluid properties and V is the velocity of fluid flow. As can be seen in the case of forced convection V is the decisive factor
Example natural convection Convection Sheet-like (2 2 ) heat sink (L=2, E=1.9 10-4 ) lower surface is at 125 0 C and surrounding air at 25 0 C. Heat transfer coefficient for convention h = T DE L 0.25 0.25 Thermal resistance θ = 0.26 1.9 10 1 1 0 s = = 1.9 2 hcas 1.31 10 4 = 4 C W (125 2.0 25) 0.25 0.25 = 1.31 10 2
For example, forced convection Convection Air with velocity 500 ft/min is blown over a plate (L=2in, area 4in 2 and B =1.0 10-3 ) Heat transfer coefficient for convection 0.75 V 3 (500 12) h = B = 1.0 10 = 0.482 W 0.25 0.25 2 0 L 2 in C Thermal resistance θ 1 1 0 s = = = 0.518 has 0.482 4 C W 0.75
When to add milk to coffee? Q = c h = c A h s c A s T ( T T ) s A
Forced convection can be divided into Laminar flow (V air <~55 m/min) Turbulent flow (V air >~55 m/min) Convection Transfer of heat and momentum is more efficient here Boundary layer thickness f(v) u = 0.99u Turbulent layer buffer layer Laminar layer
Dimensionless analysis of convective heat transfer Virtually all convection heat transfer data are correlated to dimensionless parameters due to the complex nature of this heat transfer mode. They are convenient in terms of data presentation Often results deeper understanding of the underlying physical processes and mechanisms The primary goal of dimensionless analysis is to predict the convective heat transfer coefficient utilizing the Nusselt (Nu) number Empirical Nu correlations for many flow regimes, situations and geometry s have been generated Nu Nu hl λ m n = = C (Re) (Pr) Forced convection regimes f hl = = C (Gr Pr) λ f n = C (Ra) n Natural convection regimes
Convection To discriminate between turbulent and laminar flow one can use dimensionless numbers Reynolds number Re describes the ratio of inertia and viscous forces Re x ρu x µ Where ρ is density of the fluid, u is speed of the fluid, x is critical length (dimension of the body) and µ is viscosity Re is a function of T (as density and viscosity are f(t)) For free flow Laminar flow Re < 5 10 5 (viscous forces are large enough to keep flow ordered ) Turbulent flow Re > 5 10 5 (inertia forces are large -> breaking down of the lamellar structure of the fluid)
Convection Prandtl number Pr is the ratio between moment and heat conduction Pr c pµ λ It describes the ratio of fluid speed and the thickness of thermal boundary layer. For air it is 0.7 for flat plate surface (metallic melts Pr <<1 and for oils Pr >> 1). Pr is not a strong function of temperature
Convection Nusselt number Nu is defined as (x is the characteristic length) Nu = hx λ Grashof number Gr is the ratio between Bouyancy and viscous forces Gr = ρ 2 gαx ( T T ) Rayleigh number Ra is defined as Gr Pr 3 µ 2 s Relations between the numbers for natural convection F Effectiveness of convection Natural convection ( Gr Pr) n c( Ra) n Nu = c = And for forced convection n Pr m Nu = cre c,n,m constants
Natural convection Ra x The Rayleigh number, which is simply the product of the Grashof and Prandtl numbers, can be used to determine the occurrence of turbulence in free convection. 3 gβ ( Ts T ) x 9, c Grx c Pr = 10, γα
Dimensionless analysis of convective heat transfer
Summary How excess heat influences electrical devices Heat sources in electronics Heat conduction in different material types Transient heat flux Contact resistance Heat transfer by radiation Heat transfer by convection Forced and natural convection Dimensionless analysis of convection