Handout 4A Heat Transfer

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MM3EM1 - October 21, 2010 EES HT 2010-11.DOCX

1

University of Nottingham

Department of Mechanical, Materials and Manufacturing Engineering

ENERGY MANAGEMENT 1

Handout 4A - Fundamental Concepts - Heat Transfer

4.1 Heat TransferThermodynamics is the study of energy interactions (work & heat). The laws ofthermodynamics govern the quantities and directions of the interactions (eg. howmuch heat can be converted into work). Thermodynamics cannot predict how biga heat exchanger has to be or the rateof heat loss through some insulation.Heat Transfer is the study of the rate of flow of heat and can predict how fast heatenergy transfer takes place. It can be used to design the size of heat transferdevices.

Mechanisms of Heat TransferHeat energy can be transferred by three mechanisms:

Conduction Conduction takes place by interaction of molecules in asubstance. Hotter molecules have a higher kinetic energy andthey transfer this to colder molecules with which they are incontact. In substances with free electrons (eg. metals) heatenergy can also be carried by electron flow, and this is a moreeffective form of heat transfer. Metals generally have muchhigher thermal conductivities.

Convection Convection takes place in fluids. At the boundary of a solidwith a fluid, heat is conducted into the fluid. The fluid may

then move away and convecting heat with it. Convection isthus dominated by fluid mechanics as well asthermodynamics.

Radiation Radiation heat transfer is energy transfer in the form of electro-magnetic radiation. It is unusual in that it does not require thepresence of any intervening matter for transfer to take place.Radiation is the only form of heat transfer that can take placein a vacuum.

4.1.1 Conduction Heat TransferThe rate of heat flow by conduction is described by Fourier's Law:

x

TkAqx

The heat flow in any direction is proportional to the temperature gradient. Thenegative sign indicates that heat flows from a hotter to a colder region.

A - cross-sectional area perpendicular to the direction of heat flow.

k - is a constant known as thermal conductivity. Typical values of


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thermal conductivity are given in the Table 1.

The heat flow is often expressed as a heat flux q"per unit area of cross-section.

xTkqx

Conduction through a Plane WallSteady conduction through a plane wall can be determined by integration ofFourier's Law to give:

21 TT

x

kAq

The term x/kA is known as Thermal Resistance (Rth).Hence:

thR

TTq 21

this is analogous to Ohm's Law for electrical conduction:

R

VI

So thermal conduction can be analysed as a network of resistances.

A composite wallcan be analysed as a number of thermal resistances in series:

R

T-T=q

th

21

k

x+

k

x+

k

x

A

1=R

3

3

2

2

1

1th


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Conduction through Cylinders

Heat flow through a cylinder (eg a pipe) can be analysed by integrating Fourier'sLaw for an elemental cylindrical strip from the inside to the outside of the pipe.Hence:

i

oer

r

TTLkq

log

2 21

This may be expressed as a thermal resistance:

Lk

rr

Ri

e

th2

log 0

When there are multiple layers (eg of insulation) the overall thermal resistance canbe determined by adding the thermal resistances of each layer.

For more complex conduction heat transfer problems, a general differential equation for conduction in 3-dimensions including transient effects can be obtained and solved numerically. See Heat Transfer by ABejan

4.1.2 Convection Heat Transfer

The heat flow by convection from a solid surface into a fluid is described byNewton's Law of Cooling:

fs TTAhq

A = area of surfaceTs = surface temperature


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Tf = fluid temperature (away from surface)h = heat transfer coefficient (W/m2K)

Convection heat transfer from a surface may be expressed in terms of a thermalresistance by:

th

fs

RTTq )(

hARth 1

The heat transfer coefficient h is dependent mainly on the fluid flow conditions overthe surface and the properties of the fluid. Typical values for convective heattransfer coefficients that may be encountered in a number of different heat transfersituations are shown in Figure 1.

Boundary layersWhen a fluid moves over a solid surface, the fluid at the surface is assumed to be

stationary and therefore a layer of slower moving fluid builds up on the surface.This is known as the boundary layer. The edge of the boundary layer is theposition where the fluid is moving at the same velocity (or nearly the same) as it isfar away from the surface.

Thermal boundary layerWhen heat transfer occurs, a thermal boundary layer builds up over a surface asheat is transferred from (or to) the surface. It is a region where the temperaturegradually changes from the surface temperature to the main stream temperature.As the fluid at the solid surface is stationary, heat can only be transferred byconduction into the fluid at the surface. The rate of convection is thus proportionalto the temperature gradient at the surface. The thinner the thermal boundary layerand steeper the temperature gradient, the higher the heat transfer coefficient. Theboundary layer acts as a thermal barrier to heat transfer and should be kept asthin as possible.

Temperature profiles inthe boundary layer.

At the edge of the boundary layer:T - Tw= Ts- Tw= qs

Types of Flow


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At low velocities the flow is laminar. The streamlines in laminar flow are paralleland heat can only pass across the streamlines (ie into the flow) by conduction.Laminar flow heat transfer coefficients are generally low unless the flow is brokenup regularly to keep the boundary layers thin.At higher velocities the flow becomes turbulent and turbulent eddies move fluid

(and heat) across the streamlines. In turbulent boundary layers heat transfercoefficients tend to be much higher.

Types of ConvectionWhen the fluid is forced over a surface driven by an external source (eg pump orfan), it is knows as forced convection. The heat transfer coefficient is largelyindependent of the temperature difference between the fluid and the surface.

When the fluid movement is induced by buoyancy forces caused by densitychanges as heat transfer occurs and temperature changes locally in the fluid, it is

known as free or natural convection. Heat transfer coefficients for free convectiondepend, to a certain extent, upon the magnitude of the temperature differencebetween the surface and the fluid. The heat transfer coefficients vary withorientation as the buoyancy forces are driven by gravity.

Heat Transfer CoefficientsAs a result of the complexity of the fluid flow equations and the difficulty of solvingthem, it is not possible to calculate heat transfer coefficients except for the mostsimple cases (eg. for laminar flow over a flat or in a circular pipe.) Convective heattransfer coefficients are therefore usually derived from experimental data and are

expressed as a correlations between various dimensionless numbers as a means ofgeneralising them. Appendices 1 & 2 (non-examinable) give details of howcorrelations may be used to determine heat transfer coefficients.

4.1.3 Radiation Heat TransferStefan-Boltzmann LawAll substances emit energy as electromagnetic radiation and the maximumradiation that can be emitted by a perfect "black" surface is given by the Stefan-Boltzmann Law.

4

TAq

Where: Ais surface area (m2)

is the Stefan-Boltzmann constant (5.67 x 10-8W/m2K4)Tis the absolute temperature of the surface (K)

(q"bis referred to as the black body emission and sometimes given the symbol Eb)

The emission of radiation varies with wavelength and the peak emission occurs at shorter wavelengths astemperature increases. At temperatures below 500C the wavelengths of emitted radiation are in the infra-red but above 500C an increasing proportion is emitted in the visible spectrum. Hence, hot objects glow

visibly. As radiation emission varies with T4 it is a highly non-linear phenomenon


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and becomes much more important at higher temperatures.Emissivity & Grey SurfacesMost surfaces are not "black" and emit less than the maximum amount ofradiation. The proportion of the "black" body radiation emitted by a surface isknown as the emissivity. So the emission from a real surface is

4TAq

e is the emissivity varying from 0 1.A surface with an emissivity less than 1 (and one on which is assumed not to vary

with wavelength) is known as a greysurface. Typical values of emissivity are shownin Table 2

Radiation Heat Transfer between surfaces

The radiation exchange between two grey surfaces is given by the equation:

4

2

4

121112 TTAq

where 1-2 is the grey body exchange factor and is a function of the geometric

disposition of the surfaces and the emissivities ( 1and 2)

1-1

A

A+

F

1+1-

1

1=

22

1

2-11

2-1

F1-2 is known as the View Factorand is defined as the proportion of the radiationleaving surface 1 that lands on surface 2 it thus depends on the geometricalarrangement of the surfaces. (If the surfaces can see a lot of each other then F1-2isclose to unity. If they cant then the view factor tends to zero.) View factors can beread from diagrams (see Figure 2).

For a small object enclosed by a much larger one, then F1-2= 1 and A1


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where: hrad= 1-2 (T1+ T2)(T12+ T22)

Clearly, hradis also a function of temperature and an iterative solution is often needed.But when T1and T2are fairly close in temperature (eg in heat exchange calculations for

buildings) then T1 T2= T, so hrad= 1-2 4T3

A thermal resistance for radiation can be written in the same way as for convection:

AhR

rad

th

1

Typical values for hrad

If 1-2= 1 then hrad= 4 T3and typical values are:

Temp C hrad [W/m2K]20 5.7100 11.8

250 32.4500 1051000 4681500 1264

Typical convection coefficients for air are:

Natural convection 2 - 20 W/m2KForced convection 10 - 200 W/m2K

Thus radiation is important at temperatures


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Total thermal resistance

R

1+

R

1

1+R+R+R+

R

1+

R

1

1=R

CoRo

321

CiRi

TOT

Heat flow:R

T-T=q

TOT

01

kA

x=RandR,R 321

Ah

1=RandR

c

CiCo

hc= convection heat transfer coefficient

Ah

1=RandR

rad

RiRo

where hrad= 1-2 (T1+ T2)(T12+ T22)

if T1and T2are close together then hrad 1-2 4T3

Overall Heat Transfer CoefficientIn situations where heat is transferred from one fluid through a wall to anotherfluid (eg. in a heat exchanger or across the walls of a building) it is convenient touse an overall heat transfer coefficient:

q = UA(Temp difference between fluids)

where: U = Overall heat transfer coefficient (sometimes known as U-value)

Thus:UA

1=RTOT

The U-value thus contains terms for the convective and radiative heat loss as wellas heat conduction through the wall.

ExampleCalculate overall heat transfer coefficient for the wall of a house using the followingdata:

Convective heat transfer coefficient on the outside = 20 W/m2KConvective heat transfer coefficient on the inside = 3 W/m2KEmissivity on the outside and inside walls is 0.93Wall comprises three layers:

Inner layer of brick 100mm thick k = 0.5 W/mKCavity 50mm wide filled with insulation k = 0.04 W/mKOuter layer of brick 100mm thick k = 0.6 W/mK

Assume that, for the purposes of calculating a radiation heat transfer coefficient,

the wall inner and outer surface temperatures are the same as the inside and


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outside air temperatures of 20C and 2C respectively. Also assume that the greybody exchange factor for the inner and outer walls can be approximated to the

emissivity. (ie = ).

For the purposes of calculating U, assume the area of the wall is 1 m2.

hrad outer = 4(Touter)3 = 4.38 W/m2K RRo= 0.228 Km2/W

hrad inner = 4(Tinner)3 = 5.30 W/m2K RRo= 0.189 Km2/W

hconv outer= 20 W/m2K RCo= 0.05 Km2/Whconv inner= 3 W/m2K RCi= 0.33 Km2/W

R1+

R1

1+R+R+R+

R1+

R1

1=R

CoRo

321

CiRi

TOT

RTOT = 0.12 + 0.2 + 1.25 + 0.167 + 0.041 K/W

RTOT = 1.778 K/W

UA = 1/RTOT so U = 0.56 W/m2K (as area = 1 m2)


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Figure 1

Typical values for Convection Heat Transfer Coefficientshowing effect of fluid type and flow regime


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Figure 2


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Typical Values of Thermal Conductivity Table 1

Substance k [W/mK] at 20oC

Polyurethane foam 0.026

Glass fibre mat 0.033

Cork board 0.043

Ceramic fibre insulation 0.05 (20oC)0.2 (1000oC)

Wood 0.1 to 0.3

Glass 0.7 to 0.9

Concrete 0.8 to 1.7

Stainless steels 14 to 17

Lead 35

Mild steel 50

Zinc 110

Brass (60% Cu, 40% Zn) 120

Aluminium 220

Copper 390

Silver 420

Table 2

Emissivities of Various Surfaces

Surface Temperature oC Emissivity

Aluminium (commercial sheet) 100 0.09

Aluminium (heavily oxidised) 93-505 0.2-0.31

Brass (rolled plate) 22 0.06

Chromium (polished) 100 0.075

Copper (polished) 100 0.052

Steel (polished) 100 0.066

Cast Iron (newly turned) 22 0.44

Sheet steel (shiny oxide layer) 24 0.82

Steel plate (rough) 38-372 0.94-0.97

Stainless steel (polished) 100 0.074

Lead (grey oxidised) 24 0.28

Nichrome wire (bright) 49-1000 0.65-0.79

Platinum filament 27-1230 0.036-0.192

Silver (polished pure) 227-627 0.02-0.032

Tungsten filament 3320 0.39

Zinc (galvanised sheet iron) 28 0.23

Brick (red rough) 21 0.93Fireclay 1000 0.75

Candle soot 97-272 0.952

Enamel (white, fused on iron) 19 0.90

Glass (smooth) 22 0.94

Oak (planed) 21 0.90

Black or white lacquer 38-93 0.80-0.95

Aluminium paints (various) 100 0.27-0.67

Water 0-100 0.95-0.963

Porcelain (glazed) 22 0.92


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APPENDIX 1 - Convective Heat Transfer Correlations (non-examinable)

Heat Transfer CorrelationsConvective heat transfer coefficients are usually derived from experimental dataand are expressed as a correlations between various dimensionless numbers as a

means of generalising them. The most commonly used dimensionless numbersare:

Nusselt number(Ratio of convection heat transfer to conduction in a fluid)

k

hdNud (based on diameter ) or

k

hLNuL based on length

Reynolds number (Ratio of dynamic to viscous forces in a fluid)

dudRe or

uLLRe

Prandtl number (Ratio of momentum diffusivity (viscosity) to thermal diffusivity)

k

cpPr

Grashof number (Ratio of buoyancy to viscous forces in a fluid)

2

23TLg

Gr

(Note: = 1/T for perfect gases)

Typical correlations for free and forced convection are given in Appendix 2.

Use of Correlations

In using correlations for heat transfer coefficients there are several importantpoints to remember:

Only use correlation for application it was intended for. Only use correlation for range of dimensionless numbers it is

applicable for. Evaluate fluid properties at mean fluid temperature for internal flows

and at mean film temperature (Tfreestream+ Twall)/2 for external flows. Be aware of whether correlation is for a local heat transfer coefficient

(at a particular point on a surface) or for an average heat transfercoefficient (over a particular region of surface).

Forced Convectioni) Flow in Pipes

(a) Laminar FlowFully developed thermal and hydraulic boundary layers.

Nud= 4.36 (constant wall heat flux) (circular tube)Nud= 3.66 (constant wall temperature) (circular tube)Nud= 3.65 (constant wall heat flux) (square tube)Nud= 2.95 (constant wall temperature) (square tube)

For non-circular tube use an equivalent hydraulic diameter:

hydraulic diameter = 4 cross section area/perimeter wetted by flow.


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Thermal boundary layer is fully developed if: PrRe0.04>D

x

H

Hydraulic boundary layer is fully developed if: Re0.04>D

x

H

(b) Turbulent Flow in PipesFor fully developed turbulent flow the most commonly usedcorrelation is the Colburn correlation.

PrRe0.023=uN0.330.8

dd

Applicable for Re > 10000 and Pr near 1.Flow is fully developed >60 diameters downsteam of an inlet.Not applicable for temperature differences between wall and fluidgreater than 5oC for liquids and 55oC for gases.

For larger temperature differences a correlation to take into account

variation in viscosity due to temperature variations between the walland bulk fluid can be used:

w

0.14

0.330.8dd PrRe0.027=uN

Where mwis viscosity of the fluid at the wall temperature and m is thefluid viscosity at the bulk fluid temperature.

Correlations are also available for the region near a pipe entry, forrough pipes, curved pipes, rotating pipes, tubes with heating part wayalong......etc.

ii) Flow over Flat Plates(a) Laminar Flow

Nux= 0.332 Re0.5Pr0.33 for constant wall temperatureNux= 0.417 Re0.5Pr0.33 for constant wall heat flux

These formulae are applicable for 0.5 < Pr < 50

(b) Turbulent Flow

Nux= 0.0295 Rex0.8

Prn

n = 0.5 for 0.5 < Pr < 5.0n = 1/3 for Pr > 5.0(Nux= Local Nusselt number)

This correlation applies only to the turbulent boundary layer region.

Transition from laminar to turbulent flow often occurs at Rex= 5 x 105in which case an average Nusselt number for a plate of length L isgiven by:

NuL= Pr0.33(0.037 ReL0.8- 850) for ReL> 5 x 105


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An alternative correlation applicable to liquids where there may be asignificant viscosity variation in the boundary layer is:

w

0.25

0.8L

0.43L 330)-Re(0.036Pr=Nu

iii) Flow Across Cylinders

w

0.25

0.40.66d

0.5dd Pr)Re0.06+Re(0.4=Nu

This is applicable for: 10 < Red< 1050.67 < Pr < 3000.25 < (m/mw) < 5.2

Nudis the average Nusselt number for the whole surface. In reality the heattransfer coefficient varies significantly around the circumference of thecylinder. All fluid properties in this equation should be evaluated at the walltemperature except for m

Natural Convectioniv) Vertical Plates & Vertical Cylinders of Large Diameter

NuL= 0.59(GrLPr)0.25 104< GrLPr < 109NuL= 0.13(GrLPr)0.33 109< GrLPr < 1012

Surfaces must be at constant temperature.

Nusselt number based on height (L) of plate or cylinder.

v) Horizontal Cylinders (at constant temperature)

Nud= 0.525(GrdPr)0.25 104< GrdPr < 109Nud= 0.129(GrdPr)0.33 10 < GrdPr < 1012

Nusselt number based on diameter of cylinder.

vi) Horizontal Flat Surfaces (at constant temperature)For a hot surface facing up or cold surface facing down:

NuL= 0.54(GrLPr)0.25 105< GrLPr < 108NuL= 0.14(GrLPr)0.33 GrLPr > 108

For a hot surface facing down or cold surface facing up:NuL= 0.25(GrLPr)0.25 GrLPr > 105

Nusselt number based on: - side length L for a surface plate.- perimeter/4 for a rectangular plate.- 0.9 x diameter for circular plate.

In the case of natural convection in air at approximately room temperature,the correlations above may be simplified using: GrLPr = 6.4 x 107L3(Tw-

T)


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APPENDIX 2Examples (non-examinable)

Example 1- Natural Convection from a Pipe

Pipe diameter = 150mm

An unlagged steam pipe 150 mm diameter has a surface temperature of 170C.The ambient air is at 10C and there is no significant air movement. Determine theheat losses per metre length of pipe.For natural convection from a horizontal cylinder the correlation for Nusseltnumber depends on the Rayleigh (Gr Pr) number.For: 109< Gr Pr < 1012 Nud= 0.129 (GrdPr)0.333

104< Gr Pr < 109 Nud= 0.525 (GrdPr)0.25

2

23

d

Tdg=Gr

Fluid properties should be evaluated at mean film temperature:

C90=2

170+10

0.003534=363

1=

T

1=

d = 0.15 m

mkg/0.9595=363x287.1

10=RT

p

=3

5

Pas10x2.130=-5

T = 160C

Pr = 0.695

10x2.96=)10x363(2.13

160x)(0.9595x)(0.15x9.81=Gr

7

25-

23

d

Gr Pr = 20.6 106

Use: Nud= 0.525(20.6 106)0.25

k = 0.031 W/mK

10x20.60.525xd

k=h 6

0.25

h = 7.3 W/m2K

Heat loss = hA T

Heat loss per metre length q_' = 7.3 0.15 160 = 551 Watts/m


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Example 2- Forced Convection from a PipeIf the pipe is exposed to an average wind speed of 5 m/s, calculate heat loss.Use:

w

0.25

0.40.66

d

0.5

dd PrRe0.06+Re0.4=uN

ud=Red

w= at 170C = 2.457 x 10-5 Pas

= at 10C = 1.76 x 10-5 Pas

= 1.23 kg/m3at 100C

52400=10x1.76

0.15x5x1.23=Re

5-d

Therefore: h = 25.1 W/m2K

Heat loss per metre length of pipe q_' = 1890 Watts/m

Example 3- Radiation Heat Loss from PipeCalculate heat transfer by radiation from pipe in previous example. Assumeemissivity of surface is 0.9 (oxidised steel).

q_rad= 1-2A1 (T14- T24)

Assume pipe is in large surroundings which have an effective radiation temperatureof 10C.

1-2= e

Therefore: q_rad= A1 (T14- T24) T1= 170C = 443T2= 10C = 293

A1= D per metre length of pipe

= 0.9 0.15 5.67 10-8(4434- 2934)

q_rad= 749 Watts/mRadiation emission depends upon the surface condition of a surface. If the pipe

were painted with aluminium paint then = 0.5.

Therefore: q_rad= 416 Watts/m

2.457

1.76

)(0.7152400x0.06+52400x0.4=k

hd

=

0.25

0.40.660.5

d

140=k

hd


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Example 4- Heat Loss from Insulated Pipe

ri= 0.075ro= 0.125

Steam pipe 150 mm in diameter has a surface temperature of 170C. The pipe islagged with 50 mm of insulation with a thermal conductivity of 0.05 W/mK. Theemissivity of the outside of the lagging is 0.3. The ambient air is at 10C and thereis no bulk air movement. Calculate the losses per metre length of pipe.

The resistance network is:

R1= insulation resistance =kL2

r

r

i

o

elog

R2= convection resistance =Lr2h

1=

Ah

1

occ

R3= radiation resistance =r2_T+TT+T

1=Ah

1

oL2-1

2a

2sasr

r2=

hccan be determined from Nusselt number expression for free convection:

Nud= 0.129(GrdPr)0.333

Method of solution is to guess a value for the surface temperature and iterate asboth hradand hcvary with surface temperature.

Initial guess Tsurf= 40Chc = 4.64 W/m2Khrad = 1.81 W/m2K

Therefore: R1= 1.626R2= 0.274R3= 0.704

R

1+

R

1

1+R=R

32

1TOT


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RTOT= 1.823 mK/W

Therefore heat loss W/m87.8=R

T=q

TOT

Determine new surface temperature from:R

T-T=q

1

surfpipe

Therefore: Tsurf= 27C

So new hc = 3.95 W/m2Khrad = 1.687

So. R1= 1.626R2= 0.323R3= 0.755

RTOT= 1.852 mK/W

Therefore: W/m86.4=R

T=q

TOT

Surface temperature is: 29.5C

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Handout 4A Heat Transfer