Heat Transfer Chapter 3 STEADY HEAT CONDUCTION Universitry of Technology Materials Engineering Department MaE216: Heat Transfer and Fluid
bjectives Understand the concept of thermal resistance and its limitations, and develop thermal resistance networks for practical heat conduction problems Solve steady conduction problems that involve multilayer rectangular, cylindrical, or spherical geometries Develop an intuitive understanding of thermal contact resistance, and circumstances under which it may be significant Identify applications in which insulation may actually increase heat transfer Analyze finned surfaces, and assess how efficiently and effectively fins enhance heat transfer Solve multidimensional practical heat conduction problems
ADY HEAT CONDUCTION IN PLANE WALLS Heat transfer through the wall of a house can be modeled 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). for steady operation In steady operation, the rate of heat transfer through the wall is constant. Fourier s law of
r 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
mal Resistance Concept uction resistance of the Thermal resistance of the gainst heat conduction. al resistance of a medium ds on the geometry and the al properties of the medium. Analogy between thermal and electrical resistance concepts. rate of heat transfer electric current thermal resistance electrical resistance temperature difference voltage difference
ton s law of cooling vection resistance of the ace: Thermal resistance of the ce against heat convection. Schematic for convection resistance at a surface. en the convection heat transfer coefficient is very large (h ), convection resistance becomes zero and T s T. t is, the surface offers no resistance to convection, and thus it
Radiation resistance of the surface: Thermal resistance of the surface against radiation. tion heat transfer coefficient ined heat transfer icient
Thermal Resistance Network rmal resistance network for heat transfer through a plane wall subjected to tion on both sides, and the electrical analogy.
The temperature drop across a layer is erature drop verall heat sfer coefficient Q is evaluated, the e temperature T 1 can termined from
Multilayer Plane Walls The thermal resistance network for heat transfer through a two-layer plane wall subjected to convection on both sides.
RMAL CONTACT RESISTANCE
hen two such surfaces are ressed against each other, the eaks form good material ontact but the valleys form oids filled with air. hese numerous air gaps of arying sizes act as insulation ecause of the low thermal onductivity of air. hus, an interface offers some esistance to heat transfer, and is resistance per unit interface rea is called the thermal ontact resistance, R c. A typical experimental setup for the
h c thermal contact conductance The value of thermal contact resistance depends on: surface roughness, material properties, temperature and pressure at the interface type of fluid trapped at the interface.
ermal contact resistance can minimized by applying ermal grease such as silicon oil etter conducting gas such as
ERALIZED THERMAL RESISTANCE NETWORKS Thermal
ssumptions in solving complex ltidimensional heat transfer blems by treating them as oneensional using the thermal istance network are y plane wall normal to the x-axis is thermal (i.e., to assume the perature to vary in the x-direction ly) y plane parallel to the x-axis is
T CONDUCTION IN CYLINDERS AND SPHERES is lost from a hot-water pipe to r outside in the radial direction, us heat transfer from a long Heat transfer through the pipe can be modeled as steady and one-dimensional. The temperature of the pipe depends 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 cylindrical pipes and spherical containers.
cylindrical pipe (or spherical with specified inner and outer e temperatures T 1 and T 2.
A spherical shell with specified inner and outer surface temperatures T 1 and T 2.
for a cylindrical layer hermal resistance rk for a cylindrical (or ical) shell subjected for a spherical layer
layered Cylinders and Spheres thermal resistance ork for heat transfer gh a three-layered posite cylinder ected to convection oth sides.
Once heat transfer rate Q has been calculated, the interface temperature T 2 can be determined from any of the following two relations:
ITICAL RADIUS OF INSULATION g more insulation to a wall or attic always decreases heat er since the heat transfer area stant, and adding insulation s increases the thermal nce of the wall without sing the convection nce. cylindrical pipe or a spherical the additional insulation ses the conduction nce of the insulation layer creases the convection nce of the surface because increase in the outer surface or convection. eat transfer from the pipe An insulated cylindrical pipe exposed to convection from the outer surface and the thermal resistance network associated with it.
ritical radius of insulation cylindrical body: ritical radius of insulation spherical shell: largest value of the critical us we are likely to ounter is can insulate hot-water or m pipes freely without The variation of heat transfer
T TRANSFER FROM FINNED SURFACES Newton s law of cooling: The rate of heat transfer from a surface to the surrounding medium T s and T are fixed, two ways to rease the rate of heat transfer are increase the convection heat transfer efficient h. This may require the tallation of a pump or fan, or replacing existing one with a larger one, but this proach may or may not be practical. sides, it may not be adequate. increase the surface area A s by aching to the surface extended surfaces lled fins made of highly conductive terials such as aluminum.
The thin plate fins of a car radiator greatly increase the rate of heat transfer to the air.
Equation Differential
eneral solution of the ential equation dary condition at fin base finitely Long Fin tip = T ) dary condition at fin tip Boundary conditions at the fin base and the fin tip. ariation of temperature along the fin teady rate of heat transfer from the entire fin
Under steady conditions, heat transfer from the exposed surfaces of the fin is equal to heat conduction to 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 and integrating it over the entire surface of the fin:
gligible Heat Loss from the Fin Tip iabatic fin tip, Q fin tip =0) are not likely to be so long that their temperature approaches the unding temperature at the tip. A more realistic assumption is for transfer from the fin tip to be negligible since the surface area of in tip is usually a negligible fraction of the total fin area. dary condition at fin tip ariation of temperature along the fin transfer from the entire fin
ecified Temperature (T fin,tip = T L ) case the temperature at the end of the fin (the fin tip) is t a specified temperature T L. ase could be considered as a generalization of the case of ly Long Fin where the fin tip temperature was fixed at T.
nvection from Fin Tip n tips, in practice, are exposed to the surroundings, and thus the proper ary condition for the fin tip is convection that may also include the effects iation. Consider the case of convection only at the tip. The condition fin tip can be obtained from an energy balance at the fin tip.
ctical way of accounting for the loss from the fin tip is to replace n length L in the relation for the ted tip case by a corrected h defined as hickness of the rectangular fins diameter of the cylindrical fins Corrected fin length L c is defined such that heat transfer from a fin of length L c
Fin Efficiency
Zero thermal resistance or infinite thermal conductivity (T fin = T b )
ncy of straight fins of rectangular, triangular, and parabolic profiles.
ency of annular fins of constant thickness t.
ins with triangular and parabolic profiles contain less material nd are more efficient than the ones with rectangular profiles. he fin efficiency decreases with increasing fin length. Why? ow to choose fin length? Increasing the length of the fin eyond a certain value cannot be justified unless the added enefits outweigh the added cost.
Fin Effectiveness The effectivene ss of a fin e thermal conductivity k of the fin ould be as high as possible. Use uminum, copper, iron. e ratio of the perimeter to the crossctional area of the fin p/a c should be high as possible. Use slender pin fins.
tal rate of heat transfer from a surface ll effectiveness for a finned surface overall fin effectiveness depends e fin density (number of fins per ength) as well as the tiveness of the individual fins. overall effectiveness is a better
r Length of a Fin ml = 5 an infinitely long fin ml = 1 offer a good compromise between heat transfer
mon approximation used in the analysis of fins is to assume the fin erature to vary in one direction only (along the fin length) and the erature variation along other directions is negligible. ps you are wondering if this one-dimensional approximation is a nable 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 rials. es have shown that the error involved in one-dimensional fin analysis ligible (less than about 1 percent) when re is the characteristic thickness of the fin, which is taken to he plate thickness t for rectangular fins and the diameter D for drical ones.
t sinks: Specially igned finned surfaces ch are commonly used in cooling of electronic ipment, and involve one- -kind complex metries. heat transfer formance of heat sinks is ally expressed in terms of ir thermal resistances R. mall value of thermal istance indicates a small perature drop across the t sink, and thus a high fin ciency.
T TRANSFER IN COMMON CONFIGURATIONS we have considered heat transfer in simple geometries such as large plane ong cylinders, and spheres. because heat transfer in such geometries can be approximated as oneional. ny problems encountered in practice are two- or three-dimensional and rather complicated geometries for which no simple solutions are available. ortant class of heat transfer problems for which simple solutions are d encompasses those involving two surfaces maintained at constant atures T 1 and T 2. ady rate of heat transfer between these two surfaces is expressed as nduction shape factor thermal conductivity of the medium between the surfaces onduction shape factor depends on the geometry of the system only. uction shape factors are applicable only when heat transfer between
e the value of the shape factor is known for a specific geometry, the l steady heat transfer rate can be determined from the following ation using the specified two constant temperatures of the two aces and the thermal conductivity of the medium between them.
Summary Steady Heat Conduction in Plane Walls Thermal Resistance Concept Thermal Resistance Network Multilayer Plane Walls Thermal Contact Resistance Generalized Thermal Resistance Networks Heat Conduction in Cylinders and Spheres Multilayered Cylinders and Spheres Critical Radius of Insulation Heat Transfer from Finned Surfaces Fin Equation Fin Efficiency Fin Effectiveness Proper Length of a Fin