mse lecture1(heat transfer)

8/6/2019 Mse Lecture1(Heat Transfer)

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Graduate Institute of Ferrous Technology, POSTECH

Rongshan Qin (R. S. Qin)

1. Heat Transfer

1.1 Introduction

Heat is a form of energy created by motion of atoms and molecules. Temperature

measures the average kinetic energy of random motion of atoms and molecules (At

equilibrium, each degree of freedom possesses 0.5kBT kinetic energy, where kB

=1.38x10-23 J/K is Boltzmann constant). Heat transfers from one body/location to

another due to a difference in temperature. Heat transfers from a high temperature area

to a low temperature area spontaneously, but cannot transfer from a low temperature

object to high temperature object without a heat pump.

Heat can be transferred by three modes: Conduction, convection and thermal radiation.

1.2 Heat conduction

Heat conduction is fulfilled by interaction of phonons, free electrons and molecules.

Thermal conductivity is one of parameters describing the heat conduction ability of

materials. Table 1 lists thermal conductivities of a few materials. The heat conduction in

gas is a slow process.

Table 1. Thermal conductivity

Materials Thermal conductivity (in W/ m⋅K)

Diamond 3000

Copper 390

Stainless steel 15

Firebrick 1

Mineral wool 0.048

Water 0.06Air (1 atm) 0.026

Hydrogen (1 atm) 0.18

Bridging two heat tanks with temperatures T1 and T2, respectively, by a material of

thermal conductivity k and length L, as illustrated in figure 1, the rate of heat transfer is





L

T T kAQ 12 −−= (1.1)

Figure 1. Heat conduction.

where Q is the rate of heat transfer by conduction and A is the cross-sectional area ofconducting material. Equation (1.1) can be reorganized into

th R

T T Q 21 −= (1.2)

where ) /(kA L Rth = is called thermal resistance. Equation (1.2) is analogous to Ohm’s

law in electric current passing a conductor. The similarity is useful in the calculation of

heat transfer by composite media, as illustrated in figure 2. The heat resistance of

composite equals a number of heat resistances in serial connection.

Figure 2. Heat conduction by composite media

N

ththth

composite

th R R R R +++= L21 (1.3)

q=Q/A is the rate of heat transfer per unit cross-sectional area, or heat flux. In general, it

is expressed as





T k q ∇−=r

(1.4)

The negative sign represents that heat flows in the direction opposite to temperature

gradient. Equation (1.4) is called Fourier’s first law of heat conduction.

In solidification, solid-liquid interface moves from solid side to liquid side with a velocity

V r

, as demonstrated in figure 3. The generated heat due to heat of fusion (latent heat) in

liquid-solid phase transition is LV ρ , which should be conducted away from interface to

solid and liquid bulk phases.

L LSSS T k T k LV ∇−∇= ρ (1.5)

where L is the heat of fusion. S ρ is the density of solid metal. Sk and Lk are thermal

conductivities of solid and liquid metals, respectively. ST ∇ and LT ∇ are temperature

gradients in solid and liquid at the solid-liquid interface, respectively. Equation (1.5) is

used frequently to the calculation of the migration rate of the solid-liquid interface.

Figure 3. Schematic diagram of solidification

Example 1: In single crystal pulling, solid-liquid interface must be well controlled to

prevent the instability. The instability will happen when LT ∇ <0. Equation (1.5) gives the

maximum growing speed of single crystal ( ) LT k V SSS ρ / max ∇= .

Now we consider an infinite small volume in a thermal conductor, the heat conduction

through the surface of the volume is ( ) T k q ∆=⋅∇− . The rate of heat generation inside

the volume is ε & (The heat generation can be that from phase transition, chemical

reaction, etc.). The heat can be stored in the volume as the change of temperature,

which is t T c ∂∂− / ρ , where ρ and c are the density and specific heat of the material. The

energy conservation defines the Fourier’s second law of heat conduction as





0 / =∂∂−+∆ t T cT k ρ ε & (1.6)

or

t T k T ∂∂=+∆ / / α ε & (1.7)

where α=ρc/k is the thermal diffusivity. Table 2 lists thermal diffusivity of a few materials.

∆ is Laplace vector2

2

2

2

2

2

z y x ∂

∂+

∂

∂+

∂

∂=∆ in Cartesian coordinate. Equation (1.7) is the

governing equation for calculating temperature distribution during heat conduction.

Table 2. Thermal diffusivity

Material α m2 /s×106 at 300K

Cu 114Lead 25

Steel 12

Brick 0.5

Water 0.13

Air 0.3

Example 2: In one dimensional steady state conduction, equation (1.7) reduces into

0 / 22 =dxT d . Its general solution is T=ax+b, where a and b can be determined by the

boundary condition.

Example 3: In one dimensional unsteady heat conduction, equation (1.7) reduces into

T t T ∆=∂∂ / α . Its solution under boundary conditions in figure 4 is

( ) 002

T t

xerfcT T T M +

−=

α (1.8)

Figure 4. Boundary conditions for one dimensional unsteady heat conduction.

Figure 4 is a simplified model of casting in sand mold that illustrated in figure 5. Equation

(1.8) can, therefore, describe the heat conduction in san mold castings.





Figure 5. Heat conduction in sand mold

1.3 Heat transfers by convection

Heat convection is a mixture of conduction and bulk movement of fluid. Heat convection

possesses much larger heat transfer rate than heat conduction. There are two types of

convection: Free convection and forced convection. Free convection is normally due to

the temperature-density relationship. Forced convection is accelerated by external

forces, such as using fan to blow the air in oven. Forced convection produces higher

heat transfer rate than free convection.

Figure 6. Pouring molten metal into mold. (Courtesy to www.casting.org.tw)

In casting processes, as demonstrated in figure 6, liquid metal is poured into a mold for

solidification. Convection can be caused by: a). Residual flow due to filling of the mold.

Liquid

metal Sands

0

T0

TM





b). External forces such as imposed pressure gradients, mechanical stirring or magnetic

forces. c). Density changes due to solidification as well as due to temperature and

chemical compositional changes. d). Surface tension gradients. e). Buoyancy driven

flow. f) Dragging force due to solid motion. Convections in solidifying alloys exist in

multiple scales. For examples, liquid convection exists between arms of dendrites

(mesoscale) as well as in whole liquid space (macroscale). Figure 7 illustrates the

buoyancy driven macroscopic convection.

Figure 7. Rayleigh-Benard instability (Courtesy to online pictures)

Heat transfer from an object to the surround fluid by convection is described by

Newton’s law of cooling.

( )W

T T hAQ −−=∞

(1.9)

where Q is the rate of heat transfer by convection to the surrounding fluid. A is the

object’s exposed area. T∞ and TW are the fluid free-stream temperature and object

temperature, respectively. h is the convection heat transfer coefficient. Liquid flow has

the most important impact on the convection heat transfer coefficient. Equation (1.9) is a

phenomenological format of heat transfer.

Macroscopic flow can be determined by fluid dynamic theory. In mesoscale, liquid can

flow through dendritic arms before trapped. The driving force for convection is the

volume shrink or density inhomogeneous. Liquid convection among dendritic arms can

seriously affect segregation and void formation. Because of the nature of fragmentation,

liquid flow in dendritic arms is usually described by the Darcy law, which is used for flow

in porous media like rock or soil.

( )gP f

k v L

L

ρ η

+∇−= (1.10)





where v is the flow velocity, k is the permeability which depends on distance between

dendritic arms and volume fraction, η here is the kinematic viscosity, P is pressure, ρL is

the liquid density and g is the acceleration due to gravity.

1.4 Thermal radiation

The thermal radiation is by electromagnetic wave propagation. There is no medium

needed for thermal radiation, although the electromagnetic waves can be transferred

through gases. The reason for thermal radiation is its temperature. The rate of energy

emission depends on the surface temperature and surface condition. The thermal

radiation power of black body is

4T Aeb ⋅⋅=σ (1.11)

where σ is the Stefan-Boltzmann Constant and has 4281067.5

−−− ⋅⋅×= K mW σ . For

surface which is not black body, the thermal radiation power is

bee ε = (1.12)

where ε is emissivity of the material. A laboratory black body is demonstrated in figure 8.

Figure 8. Laboratory black bodies.

Wall of cavity,

roughened

Thermal insulation

Heater

Metal enclosure

Aperture





REFERENCES

1. R.V. Kumar, Heat and mass transfer, Cambridge.

2. C. Beckermann and R. Viskanta, Appl. Mech. Rev., 46 (1993) 1-27.

3. M.C. Flemings, Solidification processing, 1974.

mse lecture1(heat transfer)

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