Chapter 3STEADY HEAT CONDUCTION

Heat Transfer

Universitry of Technology Materials Engineering DepartmentMaE216: Heat Transfer and Fluid

ObjectivesUnderstand the concept of thermal resistance and itslimitations, and develop thermal resistance networks forpractical heat conduction problems

Solve steady conduction problems that involve multilayerrectangular, cylindrical, or spherical geometries

Develop an intuitive understanding of thermal contactresistance, and circumstances under which it may besignificant

Identify applications in which insulation may actuallyincrease heat transfer

Analyze finned surfaces, and assess how efficiently andeffectively fins enhance heat transfer

Solve multidimensional practical heat conduction problemsf

EADY HEAT CONDUCTION IN PLANE WALLS

for steady operation

In steady operation, the rate of heat transfer through the wall is constant.

Fourier’s law of heat conduction

Heat transfer through the wall of a house can bemodeled as steady and one-dimensional.The temperature of the wall in this case depends on one direction only (say the x-direction) and can be expressed as T(x).

er steady conditions, the

The rate of heat conduction through a plane wall is proportional to the average thermal conductivity, the wall area, and the temperature difference, but is inversely proportional to the wall thickness.Once the rate of heat conduction is available, the temperature T(x) at any location x can be determined by replacing T by T and L by x

Analogy between thermal and electrical resistance concepts.

rate of heat transfer electric current thermal resistance electrical resistance temperature difference voltage difference

rmal Resistance Concept

duction resistance of the Thermal resistance of the

against heat conduction. mal resistance of a medium nds on the geometry and the

mal properties of the medium.

Electrical resistance

Schematic for convection resistance at a surface.

wton’s law of cooling

nvection resistance of the face: Thermal resistance of the ace against heat convection.

en the convection heat transfer coefficient is very large (h → ), convection resistance becomes zero and Ts T.

at is, the surface offers no resistance to convection, and thus it es not slow down the heat transfer process

Radiation resistance of the surface: Thermal resistance of the surface against radiation.

ation heat transfer coefficient

bined heat transfer ficient

Thermal Resistance Network

ermal resistance network for heat transfer through a plane wall subjected to ction on both sides, and the electrical analogy.

overall heat nsfer coefficient

Q is evaluated, the ce temperature T1 can termined from

perature drop

The temperature drop across a layer is

The thermal resistance network for heat transfer through a two-layer plane wall subjected toconvection on both sides.

Multilayer Plane Walls

ERMAL CONTACT RESISTANCE

A typical experimental setup for the

determination of thermal

When two such surfaces are pressed against each other, the peaks form good material contact but the valleys form voids filled with air. These numerous air gaps of varying sizes act as insulationbecause of the low thermal conductivity of air. Thus, an interface offers some resistance to heat transfer, and his resistance per unit interfacearea is called the thermal contact resistance, Rc.

hc thermal contact conductance

The value of thermal contact resistancedepends on:

• surface roughness,• material properties,• temperature and

pressure at the interface

• type of fluid trappedat the interface.

rmal contact resistance is significant and can even dominate the

hermal contact resistance can minimized by applying

thermal grease such as silicon oilbetter conducting gas such as

ERALIZED THERMAL RESISTANCE NETWORKS

Thermal resistance

assumptions in solving complex ultidimensional heat transfer oblems by treating them as one-mensional using the thermal sistance network are

ny plane wall normal to the x-axis is othermal (i.e., to assume the mperature to vary in the x-direction nly)ny plane parallel to the x-axis is di b ti (i t h t t f

AT CONDUCTION IN CYLINDERS AND SPHERES

is lost from a hot-water pipe toir outside in the radial direction,hus heat transfer from a long

Heat transfer through the pipe can be modeled as steadyand one-dimensional.

The temperature of the pipedepends on one direction only (the radial r-direction) and can be expressed as T = T(r).

The temperature is independent of the azimuthal angle or the axial distance.

This situation is approximated in practice in long cylindricalpipes and spherical containers.

g cylindrical pipe (or sphericalwith specified inner and outer

ce temperatures T1 and T2.

A spherical shell with specified inner and outersurface temperatures T1and T2.

thermal resistance ork for a cylindrical (or rical) shell subjected

for a cylindrical layer

for a spherical layer

ilayered Cylinders and Spheres

thermal resistance work for heat transfer ugh a three-layered

mposite cylinderected to convection

both sides.

Once heat transfer rate Q has been calculated, the interface temperature T2 can be determined from any of the

following two relations:

RITICAL RADIUS OF INSULATIONng more insulation to a wall or attic always decreases heat fer since the heat transfer areanstant, and adding insulation ys increases the thermal ance of the wall without asing the convection ance. cylindrical pipe or a spherical the additional insulation

ases the conduction ance of the insulation layer ecreases the convection ance of the surface because increase in the outer surface for convection. heat transfer from the pipe ncrease or decrease

An insulated cylindrical pipe exposed to convection from the outer surface and the thermal resistance network associated with it.

critical radius of insulation cylindrical body:

critical radius of insulation spherical shell:

The variation of heat transfer rate with the outer radius of the

can insulate hot-water or am pipes freely without

i b t th ibilit f

e largest value of the critical ius we are likely to

counter is

AT TRANSFER FROM FINNED SURFACES

n Ts and T are fixed, two ways to crease the rate of heat transfer are

o increase the convection heat transfer oefficient h. This may require the stallation of a pump or fan, or replacing e existing one with a larger one, but this pproach may or may not be practical.esides, it may not be adequate.o increase the surface area As by taching to the surface extended surfaces

alled fins made of highly conductive aterials such as aluminum.

Newton’s law of cooling: The rate of heat transfer from a surface to the surrounding medium

The thin plate fins of a car radiator greatly increase the

rate of heat transfer to the air.

Equation

me element of a fin at location xDifferentialequation

general solution of the rential equation

ndary condition at fin base

Boundary conditions at the fin base and the fin tip.

nfinitely Long Fin n tip = T)ndary condition at fin tip

variation of temperature along the fin

steady rate of heat transfer from the entire fin

A l i l fi f if

Under steady conditions, heat transfer from the exposed surfaces of the fin is equal to heat conductionto the fin at the base.

The rate of heat transfer from the fin could also be determined by considering heat transfer from a differential volume element of the fin andintegrating it over the entire surface of the fin:

egligible Heat Loss from the Fin Tipiabatic fin tip, Qfin tip = 0)

ndary condition at fin tip

variation of temperature along the fin

transfer from the entire fin

s are not likely to be so long that their temperature approaches the ounding temperature at the tip. A more realistic assumption is for t transfer from the fin tip to be negligible since the surface area of fin tip is usually a negligible fraction of the total fin area.

pecified Temperature (Tfin,tip = TL)

s case the temperature at the end of the fin (the fin tip) is at a specified temperature TL. case could be considered as a generalization of the case of tely Long Fin where the fin tip temperature was fixed at T.

onvection from Fin Tip

n tips, in practice, are exposed to the surroundings, and thus the properdary condition for the fin tip is convection that may also include the effectsdiation. Consider the case of convection only at the tip. The condition fin tip can be obtained from an energy balance at the fin tip.

actical way of accounting for the loss from the fin tip is to replace

fin length L in the relation for the lated tip case by a correctedth defined as

Corrected fin length Lc is defined suchthat heat transfer from a fin of length Lcwith insulated tip is equal to heat transfer

thickness of the rectangular finsdiameter of the cylindrical fins

Fin Efficiency

Zero thermal resistance or infinite thermal conductivity (Tfin = Tb)

ency of straight fins of rectangular, triangular, and parabolic profiles.

iency of annular fins of constant thickness t.

Fins with triangular and parabolic profiles contain less material and are more efficient than the ones with rectangular profiles.The fin efficiency decreases with increasing fin length. Why?How to choose fin length? Increasing the length of the fin beyond a certain value cannot be justified unless the added benefits outweigh the added cost.

Fin Effectiveness

The effectiveness of a fin

he thermal conductivity k of the fin hould be as high as possible. Use luminum, copper, iron.he ratio of the perimeter to the cross-ectional area of the fin p/Ac should be s high as possible. Use slender pin fins.

Low convection heat transfer coefficient

ll effectiveness for a finned surface

overall fin effectiveness dependshe fin density (number of fins per length) as well as the ctiveness of the individual fins. overall effectiveness is a better

f th f f

tal rate of heat transfer from a surface

er Length of a Fin

se of the gradual temperature drop

mL = 5 an infinitely long finmL = 1 offer a good compromise between heat transfer

mmon approximation used in the analysis of fins is to assume the finperature to vary in one direction only (along the fin length) and the perature variation along other directions is negligible. aps you are wondering if this one-dimensional approximation is a onable one. is certainly the case for fins made of thin metal sheets such as the fins car radiator, but we wouldn’t be so sure for fins made of thick

erials. ies have shown that the error involved in one-dimensional fin analysis gligible (less than about 1 percent) when

ere is the characteristic thickness of the fin, which is taken to the plate thickness t for rectangular fins and the diameter D for ndrical ones.

at sinks: Specially signed finned surfacesich are commonly used in

e cooling of electronic uipment, and involve one-a-kind complexometries.e heat transfer rformance of heat sinks is ually expressed in terms of eir thermal resistances R.small value of thermal sistance indicates a small mperature drop across the at sink, and thus a high fin ciency.

AT TRANSFER IN COMMON CONFIGURATIONS we have considered heat transfer in simple geometries such as large plane long cylinders, and spheres.

s because heat transfer in such geometries can be approximated as one-nsional.any problems encountered in practice are two- or three-dimensional and e rather complicated geometries for which no simple solutions are available.portant class of heat transfer problems for which simple solutions are ed encompasses those involving two surfaces maintained at constantratures T1 and T2. eady rate of heat transfer between these two surfaces is expressed as

onduction shape factore thermal conductivity of the medium between the surfacesconduction shape factor depends on the geometry of the system only.duction shape factors are applicable only when heat transfer between wo surfaces is by conduction

ce the value of the shape factor is known for a specific geometry, the al steady heat transfer rate can be determined from the following uation using the specified two constant temperatures of the two faces and the thermal conductivity of the medium between them.

SummarySteady Heat Conduction in Plane Walls Thermal Resistance Concept Thermal Resistance Network Multilayer Plane Walls

Thermal Contact ResistanceGeneralized Thermal Resistance NetworksHeat Conduction in Cylinders and Spheres Multilayered Cylinders and Spheres

Critical Radius of InsulationHeat Transfer from Finned Surfaces Fin Equation Fin Efficiency Fin Effectiveness Proper Length of a Fin

Heat Transfer in Common Configurations