Heat Transfer Lectures2

Heat TransferHeat Transfer

What is Heat Transfer?

Heat transfer is energy in transit due to a temperature difference. It occurs through various modes:

• Conduction – medium, temperature gradient exists

• Convection – between surface and moving fluid

• Radiation – via emitted electromagnetic waves

1. Introduction1. Introduction

Figure 1 – Various heat transfer modes

1.1 Conduction

• Transfer of energy from more energetic particles (atomic or molecular) to less energetic particles through a medium.

• Due to temperature gradient

• No bulk motion.

• Can occur in any medium (solid > liquid > gas)

Figure 2 – Conduction heat transfer due to molecular activity


Fourier’s Law

Rate equation computing amount of energy being transferred per unit time. q = -kA dT/dx [W]

qx = -k (dT/dx) [W/m2 ]


qx = heat flux (heat transfer rate) in x direction (W/m2)

dT/dx = temperature gradientK = thermal conductivity (W/mK)

dT = T2 – T1

dx L

Under steady state conditions:

qx = k T

L

1.2 Convection

• Heat transfer between a fluid in motion and bounding surface due to a temperature gradient.

• Convection heat transfer mode comprises two mechanisms:

• Energy transfer due to random molecular motion (diffusion)

• Bulk, or macroscopic motion of the fluid

• If only by bulk fluid motion - Advection


Boundary Layer Concept

• Fluid-surface interaction – region in the fluid where velocity varies from zero (at surface) to a finite value (u) associated with

the flow.

• Known as hydrodynamic or velocity boundary layer

• If the surface (Ts) & fluid temperature (T) is different,

temperature boundary layer will also exist.


At surface, fluid velocity is zero, transfer mechanism is by diffusion (random molecular motion) only

Types of Convection Heat Transfer

• Forced convection

• Natural convection – buoyancy forces via density difference

• Boiling – latent heat exchange (change of state)

• Condensation – latent heat exchange (change of state)


Newton’s Law of Cooling

Rate equation computing amount of energy being transferred per unit time.

q= h (Ts - T)

q = heat flux (heat transfer rate) (W/m2)

h = convection heat transfer coefficient (W/m2K)K = thermal conductivity (W/mK)

h depends on conditions in boundary layer, which are influenced by surface geometry, nature of fluid motion, fluid thermodynamic and transport properties.


1.3 Radiation

• Energy of the radiation field is transported by

electromagnetic waves (photons).

• No medium required (most efficient in vacuum)

• Can take place in solids, liquids and gases

• Focus will be on solids


Stefan-Boltzmann Law

Surface emissive power, E is the rate at which energy is released per unit area (W/m2)

Eb = Ts4 (Ideal radiator, Blackbody)

For a real surface, heat flux emitted is:

E = Ts4

E = surface emissive power (W/m2)

= Stefan-Boltzmann constant (5.67x10-8 W/m2K4) = emissivity, radiative property of the material (0 to 1)


• Irradiation, G, is the radiation incident on a unit area of surface

• Absorptivity, , is the rate at which radiant energy is absorbed per unit surface area (0 to 1).

• Any radiation not absorbed is either reflected (opaque surface) or transmitted (semi-transparent).


1.4 Analysis of Heat Transfer Problems

Methodology

• Known – state briefly and concisely what is known

• Find – state briefly and concisely what must be found

• Schematic – draw schematic of the physical system

• Assumptions – list all simplifying assumptions

• Properties – compile property values needed and identify source

• Analysis – apply appropriate conservation laws and rate equations.

• Comments – discuss results & perform “what-if” and parameter sensitive calculations.


2.1 Conduction Rate Equation

Most calculations are performed on a one-dimensional system, although in reality, heat transfer occurs in all 3 dimensions.

T + T + T x y z( )-kq = i j k

2. Introduction to Conduction2. Introduction to Conduction

2.2 Solid State Conduction

• Solid comprises of free electrons and atoms bound in a

periodic arrangement called lattice

• Transport of thermal energy due to 2 effects: migration of

free electrons + lattice vibrational waves.

• Fluid (liquid & gas) state conduction is less effective due to

larger intermolecular spacing. Gas conductivity affected by

temperature, pressure and chemistry.


2.3 Thermal Conductivity

Provides an indication of the rate at which energy is transferred by the diffusion process.


Insulation Systems

• Most insulation systems make use of air as the insulator; air has a very low thermal conductivity.

• Fiber, powder or flake type insulation

• Foam insulation system

•The effectiveness of insulation materials is expressed in terms of their R-value: R = L/k {thickness/ thermal conductivity}


2.4 Other Relevant Properties

• Density ()

• Specific Heat (cp)

• Volumetric Heat Capacity = x cp

• Thermal Diffusivity = k x cp

Ratio of thermal conductivity to heat

capacity


Boundary condition

Solution depends on the

boundaries of the medium

Initial condition

Solution exists at specific

point in time

2.5 Boundary & Initial Conditions


3.1 The Plane Wall

Temperature Distribution

• Temperature function of x coordinate only

• Consider all conditions within the wall

• No heat generation, heat flux is constant

3. One-Dimensional Steady State Conduction3. One-Dimensional Steady State Conduction


(Solve heat equation to get general solution:

d dTdx dx= 0k )

T(x) = C1x + C2

Integrate twice:

T(0) =Ts,1 and T(0) = Ts,2

Boundary conditions (x=0 and x=L):

C2 = Ts,1

Ts,2 = C1L + C2 = C1L + Ts,1

C1 = (Ts,2 - Ts,1)/L


General Equation:

T(x) = (Ts,2 - Ts,1)x/L + Ts,1

Conduction Heat Transfer Rate:

qx = - k A dT/dx

= kA/L (Ts,1 – Ts,2)

Plane Wall Conduction Heat Transfer Rate:

qx = k/L (Ts,1 – Ts,2)

Thermal Resistance

• Analogy between diffusion and electrical charge


Rt,cond = (Ts,1 - Ts,2) / qx = L / KA

Rt,conv = (Ts - T) / q = 1 / hA

Rt,rad = (Ts - Tsur) / qrad = 1 / hrA

Simple Composite Wall

• Equivalent to series electrical resistance


Rtotal = Rt = T/q = 1 / UA

qx = UA T

Complex Composite Wall

• Series – Parallel Composite Wall


Contact Resistance

• Thermal contact resistance, Rt,c = (TA – TB )qx

• Due to surface roughness effects

• Gaps are usually filled with air (low thermal conductivity)

• Lower contact resistance by pressure, reduced surface roughness, fluid of high thermal conductivity.

• Thermal grease used to lower contact resistance; grease filled with indium, lead, silver etc.


3.2 Radial Systems

The Cylinder

• Hollow cylinder with inner and outer surfaces exposed to fluids of different temp



qr = 2Lk (Ts,1 - Ts,2)

ln (r2/r1)

Rt,cond = ln (r2/r1) 2Lk

The Sphere

• Hollow sphere


qr = 4k (Ts,1 - Ts,2)

(1/r1) – (1/ r2)

Rt,cond = 1 1 14k r1 r2

( )

3.3 Extended Surfaces

• Cooling fins

• Combined conduction and convection effects

• Used to enhance heat transfer rate by increasing surface area


Fin Performance

• Ratio of fin heat transfer rate to the heat transfer rate that would exist without the fin.


Non-Uniform Cross Section

• Ratio of fin heat transfer rate to the heat transfer rate that would exist without the fin.


Fin effectiveness & Fin efficiency

• The performance of the fin is expressed in terms of fin effectiveness: €fin= Qfin/Qnofin €fin= Qfin/hAb(Tb- T∞)

• Fin efficiency: µfin = Qfin/Qfin,max

• µfin = actual heat transfer rate from the fin/ideal heat transfer rate from the fin or Qfin= µfin Qfin,max= µfinh Afin(Tb- T∞)

4.1 Velocity Boundary Layer

• Consider flow over a flat plate

• Fluid particles in contact with surface has zero velocity

• These particles retard particles next to them

• At y = (boundary layer thickness), effect is negligible, so velocity u = u

4. Introduction to Convection4. Introduction to Convection

4.2 Thermal Boundary Layer

• Consider flow over a flat plate hotter than ambient

• Energy transfer from plate to fluid

• These particles transfer to particles next to them

• Boundary layer thickness (t) defined when (Ts-T)/(Ts-T,)=0.99


4.3 Concentration Boundary Layer

• Mixture of fluid A & B flows over surface

• If conc. of A at surface CA,s differs from free stream CA,, conc. boundary layer develops

• Concentration boundary layer thickness (c)

• Defined when (CA,s- CA)/(CA,s- CA,,) = 0.99


4.4 Laminar & Turbulent Flow

• Surface friction and convection transfer rate depend strongly on type of flow

• Laminar Flow

• Fluid motion highly ordered

• Streamlines can be easily identified, characterized by velocity components

• Turbulent Flow

• Fluid motion highly irregular

• Characterized by velocity fluctuations

• Fluctuations enhance transfer of momentum, energy & species

• Also increase surface friction & convection transfer rates



• Boundary layer initially laminar

• As distance increase, small disturbances amplified and transition to turbulent flow occurs

• Fluid fluctuations develop in transition region

• Finally becomes completely turbulent



• Reasonable to use Reynold’s number to determine flow

Rex = (ux) /

• x = length from leading edge

• = fluid density

• u = fluid free stream velocity

• = fluid viscosity

• Transition occurs at 105 < Rex , 3 x 106

• Prandtl Number: Pr = v / (v is kinematic viscosity, is thermal diffusivity)

• Schmidt Number: Sc = v / DAB (DAB is binary mass diffusion coefficient)


4.5 Evaporative Cooling

• Occurs when gas flows over a liquid

• Energy associated with the phase change is the latent heat of vaporization of the liquid

• Liquid molecules near surface experience collisions that increase their energy above that needed to overcome surface binding energy

• Energy to sustain evaporation comes from liquid

• In order to maintain equilibrium, energy lost from liquid must be transferred from surrounding


4.5 Evaporative Cooling

qevap = qadd + qconv

qevap = nA hfg

• nA = evaporative mass flux

• hfg = latent heat of vaporization

• Magnitude of cooling effect can be expressed as:

h (T-Ts) = hfg (hm/h) [(A,sat (Ts) - A, ]


4.6 Methodology for Convection Calculations

• Identify flow geometry – flat plate, sphere, cylinder?

• Specify appropriate reference temperature and evaluate fluid properties at that temperature

• Calculate Reynolds number – determines boundary layer conditions )laminar / turbulent)

• Decide whether local or surface average coefficient is required – for constant surface temperature or vapor density, local coefficient is sued to determine the flux at a particular point on the surface, whereas the average coefficient determines the transfer rate for the entire surface.

• Select the appropriate correlation


5. External Convection5. External Convection

5.1 Flat Plate in Parallel Flow

Laminar Flow

• Despite its simplicity, this occurs in numerous engineering applications

• Leading edge – laminar layer development

• Transition to turbulence occurs downstream, where critical Re is reached

• Velocity boundary layer u/u = 0.99, boundary layer thickness is

= 5.0 = 5xu/ vx Rex

• Local friction coefficient is,

C f,x = 0.664Rex-1/2

• Taking into consideration other boundary layers (concentration & thermal), the local friction coefficient is

C f,x = 1.328Rex-1/2

• Average heat transfer coefficient for laminar flow is:

Nux = hxx / k = 0.664Rex1/2Pr1/3 (Pr 0.6)

Shx = hm,xx / DAB = 0.664Rex1/2Sc1/3 (Sc 0.6)

Note : hx = 2hx

• If flow is laminar over entire surface, then subscript x may be replaced by L, and the above equations used to predict average conditions for the entire surface.


Turbulent Flow

• Local friction coefficient is well correlated by:

C f,x = 0.0592Rex-1/5 (Rex 107)

• Velocity boundary layer may be expressed by:

= 0.37xRex-1/5 (Rex 107)

• Local numbers are:

Nux = St Rex Pr = 0.0296 Rex4/5 Pr1/3 (0.6 < Pr < 60)

Shx = Stm Rex Sc = 0.0296 Rex4/5Sc1/3 (0.6 < Sc < 3000)


Mixed Flow

• Occurs when transition occurs sufficiently upstream of the trailing edge

• xc / L < 0.95


5.2 Cylinder in Cross Flow

• Fluid motion normal to axis of circular cylinder

• Free stream fluid brought to rest at forward stagnation point, with accompanying rise in pressure

• From this point, pressure decreases with increases x, the streamline coordinate and boundary layer develops under the influence of a favourable pressure gradient (dp/dx<0)

• Pressure eventually reaches a minimum and further boundary layer development occurs at rear of cylinder in the presence of an adverse pressure gradient (dp/dx>0).


• At separation point, fluid near the surface lacks sufficient momentum to overcome pressure gradient and continued downstream movement is impossible – boundary layer separation occurs

• Wake occurs in downstream region.

• Since momentum of fluid in a turbulent boundary layer is larger than in the laminar boundary layer, transition delays the occurrence of separation.

• If Re < 2 x 105 boundary layer remains laminar and separation occurs at 80. If Re > 2 x 105, boundary layer transition occurs and separation is delayed to 80.


• The drag force, Fd acting on the cylinder is influenced by:

• Boundary layer surface shear stress (friction drag)

• Pressure differential in the flow direction resulting from formation of the wake (form, or pressure drag)

•Dimensionless drag coefficient, CD may be defined:

CD = FD

Af (V2/2)

where Af is the cylinder frontal area.


5.3 Spheres / Row of Tubes / Annular Tubes

Spheres

• Boundary layer effects associated with flow over a sphere are much like those for the circular cylinder with transition and separation playing prominent roles.

• Numerous heat transfer correlations have been proposed.


Rows of Tubes

• Relevant to numerous industrial applications such as steam generation in a boiler or air cooling in the coil of an air conditioner.

• One fluid moves over the tubes while a second fluid at a different temperature passes through the tubes

• Coefficient of 1st tube similar to single tube coefficient, while larger heat transfer coefficients are associated with tubes of the inner rows, as they act as a turbulence grid.

• Stabilization occurs at about 4th or 5th tube.


• Interest is in the average heat transfer coefficient for the entire tube bundle


6. Internal Convection6. Internal Convection6.1 Hydrodynamic Considerations

• For external flow, only considerations are laminar or turbulent flow.

• For internal flow, other considerations include the existence of entrance and fully developed regions.

• Onset of turbulence occurs at Re 2300. Full turbulence Re 10,000

• Generally, fully turbulent flow occurs when (x/D) > 10

Fluid makes contact with surfaceViscous effects become importantBoundary layer develops with increasing xShrinking inviscid flow regionBoundary layer merge at centerline

Viscous effects extend over cross sectionVelocity profile no longer changes with increasing x

Pressure Gradient & Friction Factor

Pressure drop can be estimated by using the Moody (or Darcy) friction factor

• For fully developed laminar flow,

f = 64 / ReD

• For fully developed turbulent flow,

f = 0.316ReD-1/4 ReD < 2x104

f = 0.184ReD-1/5 ReD > 2x104

6. Internal Convection6. Internal Convection

6.2 Thermal Considerations

• For laminar flow, the thermal entry length is:

(x/D) 0.05ReDPr

• If Pr > 1, hydrodynamic boundary layer develops more rapidly than thermal boundary layer, inverse is true for PR < 1.

• Usually use a mean temperature, Tm as a reference temperature for internal flow.

• Newton’s law of cooling is : qs = h (Ts-Tm)


6.4 Turbulent Flow in Circular Tubes

6.5 Noncircular Tubes

• Concept of hydraulic diameter, Dh = 4Ac/P where Ac/ and P are the flow cross

sectional area and wetted perimeter respectively. This diameter is used to calculate Re and Nu.

• Turbulent Flow – approximation is reasonably accurate if Re > 2300 and correlations from the previous section are valid (Pr > 0.7).

• Laminar Flow – less accurate, especially when cross sections are characterized by sharp corners. For such cases, Nu corresponding to fully developed conditions can be obtained from the table in the next page.



6.6 Concentric Tube Annulus

• Very common configuration in industrial applications

• Heat flux from each surface may be computed with:

• qi = hi (Ts,i – Tm)

• qo = ho (Ts,o – Tm)

• Convection coefficients are separately calculated too:

• Nui = hiDh / k

• Nuo = hoDh / k

• Note that hydraulic diameter is used, Dh

• Dh = 4(/4)(D2o – D2

i)

Do + Di


6.7 Heat Transfer Enhancements

• Increase convection coefficient (Eg. Surface roughness to enhance turbulence)

• Increase convection surface area


6.8 Heat Exchangers

• Classified according to flow arrangement and type of construction

• Parallel Flow

• Counter Flow

• Cross Flow

• Shell & Tube – baffles used to increase convection coefficient by inducing turbulence

• Compact Heat Exchangers – Very large heat transfer surface area per unit volume


Fouling Factor, Rf

• Deposition of film or scale on surface greatly increases resistance to heat transfer

7. Radiation7. Radiation


7.1 Radiation Intensity

7. Radiation7. Radiation7.2 Blackbody Radiation

• Ideal surface with the following properties

• Absorbs all incident radiation, regardless of wavelength and direction

• For a prescribed temperature and wavelength, no surface can emit more energy than a blackbody

• Although radiation emitted by a blackbody is a function of wavelength and temperature, it is independent of direction.

• Total emissive power of a blackbody may be expressed by:

• Eb = T4 ( is the Stefan-Boltzmann constant, 5.67x10-8 W/m2K4)


7.3 Surface Emission

• Ratio of radiation emitted by the surface to the radiation emitted by a blackbody at the same temperature


7.4 Surface Absorption, Reflection and Transmission

• Spectral irradiation, G (W/m2.m) is the rate at which radiation of wavelength, is incident on a surface per unit area of the surface and per unit wavelength interval d about .

• Total irradiation encompasses all spectral contributions of absorption, reflection and transmission:

G = G,ref + G,abs + Gtr


• In most applications, the surface is opaque to the incident radiation, and transmitted radiation is zero. The remaining absorption and reflection processes may be treated as surface phenomena.

• Relevant terms

• Absorptivity

• Reflectivity

• Transmissivity


7.5 Radiation Exchange between Surfaces

View Factor Integral

• Fraction of radiation leaving surface i that is intercepted by surface j.

Example 7.1

A furnace cavity, which is in the form of a cylinder

of 75mm diameter and 150mm length is open at

one end to large surroundings that are at 27degC.

The sides and bottom may be approximated as

blackbodies, are heated electrically, are well

insulated and are maintained at temperatures of

1350degC and 1650degC, respectively. How

much power is required to maintain the furnace

conditions?

8. ASSIGNMENT8. ASSIGNMENTA semiconductor packaging test company has just received a contract to design and

build a reliability testing station for new ball grid array packages. The test station is to

be designed to test 800 packages at one go. There will be 10 shelves within the test

cabinet, with each shelf holding 80 packages.

Each package is 30mm by 30mm and is made from 96% Aluminum Oxide ceramic.

When powered up, each package is estimated to give off about 100W of heat. This

100W of heat is to be removed via a cooling fin attached to each package, and the

temperature of each package should never exceed 130C.

The cooling fins should not exceed a volume of 30mm x 30mm x 60mm. Two potential

materials are being considered for the fins: aluminum and carbon fibres (anisotropic

heat transfer). You are to recommend which material is better based on heat transfer

calculations and design a fin based on the heat conducted to the fin by the package.

8. ASSIGNMENT8. ASSIGNMENTThe heat is to be removed from each cooling fin by forced convection of air at an

ambient temperature of 25C. Design the size of the test cabinet, bearing in mind that

a larger footprint will cost more in terms of real estate, but you are limited by a height

of 2m for practical reasons. For this cabinet size, what airflows are required to ensure

a maximum allowable temperature of 130C?

Considerations:

• Direction of airflow within the cabinet and within each shelf.

• Is ambient air sufficient or is chilled air required?

• There is a maximum allowable volumetric flow rate of air for safety reasons.

•

8. ASSIGNMENT8. ASSIGNMENT

Testing Cabinet Air Flow

Each tray holding 40 packages

8. ASSIGNMENT8. ASSIGNMENT

Shelf Air Flow

Cooling Fin

Test Package

Air Flow

Heat Transfer Lectures2

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