Welcome back to the second part of the second learning module for Fundamentals of Arctic Engineering online. We re going to review in this module the fundamental principles of heat transfer. Exchange of heat energy relates to all aspects of engineering, architecture, construction, and living in the North. Principles of heat transfer apply to heat loss from dwellings or machinery enclosures, to unwanted gain of heat by Arctic permafrost, and to natural adaptations of cold regions plants and animals. Basic knowledge of physical laws associated with heat transfer is essential for good cold regions engineering practice. Page 1 of 32
Transfer of heat energy occurs by three fundamental modes: conduction, convection, and radiation. Conduction is transfer heat energy by direct contact between two substances or through a continuous substance. Convection is a transfer of heat between a solid body and a moving fluid in which it is immersed. Radiation involves emission of electromagnetic waves by all substances whose temperature is above absolute zero. Objects toward which the radiation is directed then absorb this emitted electromagnetic energy. Page 2 of 32
Fourier s Law governs heat transfer by conduction. A temperature gradient in a homogenous body corresponds to an energy transfer from the high temperature region to the low temperature region. The minus sign in Fourier s Law assures that heat flows from high to low temperature. The differential dt/dn is the temperature gradient in the direction of heat flow. Thermal conductivity, k, has units of W/m-K, but can also be expressed in W/m- C, since Celsius and Kelvins have the same value, as a temperature difference. In the British system of units, k has units of Btu/hr-ft-degree Fahrenheit. Since a convenient unit of power, like Watts in SI, doesn t exist in the British system, the time dimension is important in British units for thermal conductivity. Page 3 of 32
The differential dt/dn may be simplified to a ratio of discrete differences, Δ T/Δ x, if the temperature gradient is linear and the heat transfer is steady state. The temperature gradient is expected to be linear in homogenous materials, according to Fourier s Law. The x direction is along the primary direction of heat transfer, from high temperature to low temperature. Page 4 of 32
The table shows representative values of thermal conductivity for some familiar materials. We expect metals to have high thermal conductivity, that is, to be efficient conductors of heat energy. Stone and glass are less efficient conductors of heat energy. Page 5 of 32
Wood has about 5 times the thermal conductivity of glass wool or fiberglass insulation, of interest to those of you thinking about building a log cabin. Water is about 20 times more conductive than air, which relates to various strategies for emergency survival in the cold. It is interesting to note that water vapor is somewhat less conductive than dry air. Page 6 of 32
Considerations of utility pipe material relate to cost per unit weight, to structural strength, and to thermal conductivity. Page 7 of 32
The published thermal conductivities of insulation materials relate in part to the efficiency with which they trap air. In this regard, density of material, as installed, is an indicator of its thermal conductivity. A low thermal conductivity is the mark of a good insulation. Page 8 of 32
Heat transfer is enhanced by fluid motion, such as by air or water flowing past a warmer solid object. Heat is first conducted to the thin, nearly motionless layer of fluid at the solid boundary and then is convected away by adjacent layers of moving fluid. Page 9 of 32
The familiar wind chill system quoted by weather forecasters is developed to relate the accelerated heat transfer in windy conditions to that of still air. Colder still air temperatures are necessary to achieve the same rate of cooling as wind flowing past exposed skin. It is the rate of heat exchange that is affected by fluid motion, not the final equilibrium temperature. Page 10 of 32
Newton s Law of Cooling governs heat transfer by convection. The rate of energy transfer by this mode is also proportional to a temperature difference. The difference between the upstream fluid temperature, Tu, beyond the influence of contact with the solid, and the surface temperature of the solid, Ts, drives the heat exchange between the fluid and the solid. The empirical convective heat transfer coefficient, h, usually must be determined by experiments. The total surface area, A, in contact with fluid is important. Fin-like objects convect heat from both sides. Page 11 of 32
As indicated by these example values for convection coefficients, the size and shape of the object is important, as are the properties of the particular fluid in which the object is immersed. The approximate velocity of the fluid is also important, particularly as it relates to turbulent behavior. The units of h, when multiplied by an area and a temperature difference, are appropriate for a resultant heat transfer rate in Watts or Btu s per hour. Page 12 of 32
The Stefan-Boltzman law is a simple predictor of the rate at which energy is emitted from an ideal black body. The temperature must be in degrees Rankine or Kelvins. Since the temperature is taken to the fourth power, precision of temperature measurements, that is the number of significant figures, is important. Page 13 of 32
Most objects, including those that behave as blackbodies with regard to radiation, do not emit a single wavelength of radiation. Energy is distributed over a range of wavelengths surrounding a wavelength of maximum emissive power. Wien s Displacement Law is another simple relationship that predicts the wavelength of peak spectral power. This wavelength is inversely proportional to the absolute temperature of the emitting body. The higher is the temperature, the shorter will be the wavelength of peak emission. The constant of proportionality, as stated here, has units appropriate to yield a wavelength in micrometers or meters x 10-6. Electromagnetic radiation has a frequency that corresponds to each wavelength. Wien s Law predicts a higher frequency, corresponding to a shorter wavelength, for a higher temperature. Page 14 of 32
The emission spectra of the Sun and the Earth are plotted here on the same wavelength and frequency scales. The Sun, as predicted by Wien s Law, emits maximum radiation intensity in the visible range of the electromagnetic spectrum. This peak wavelength corresponds to the 6,000 degrees Celsius surface temperature of the Sun. The average surface temperature of the Earth is about 18 degrees Celsius. This temperature corresponds to a peak spectral wavelength in the longer-wave infrared range. The bars in the lower half of the figure illustrate the way different substances absorb radiant energy. Gases in the atmosphere are more efficient absorbers at some frequencies than they are at others. The greenhouse gases, including carbon dioxide, don t absorb much visible light, but efficiently absorb longer-wavelength infrared radiation, as emitted by the surface of the Earth. Page 15 of 32
This slide, repeated from the first module, again illustrates the impact of this balance between Earth radiation emission and absorption by greenhouse gases in the atmosphere. If more heat is absorbed in the atmosphere, versus transmitted into space, the Earth will respond by warming. Eventually, a higher Earth temperature will shift the frequency of radiation emissions to a wavelength more readily transmitted through the atmosphere to strike a new balance with incoming solar heat energy. The abundance and distribution of carbon dioxide on Earth are at the crux of global warming studies. Page 16 of 32
A radiant exchange of heat energy between two objects requires radiation by one and absorption by the other. The transfer will be from the object of higher temperature to the object of lower temperature. Both factors of geometry and of efficiency, relative to blackbody behavior, are important in evaluation of an exchange of radiant energy. The efficiency of the exchange is represented by the emissivity parameter. Page 17 of 32
In practice, both the emissivity and geometric considerations must be expressed as functions, rather than constants. As noted, these functions are not necessarily independent. Page 18 of 32
The sketch is intended to show that real situations of heat transfer often involve combined conduction, convection, and radiation. This example implies that the heat lost from the inside of the enclosure through the wall by conduction equals the heat lost by convection and radiation at the outside of the wall. This relation assumes a steady-state condition in which temperatures are not changing and have reached an equilibrium state. Page 19 of 32
Thermal resistance or R factors are useful in problems of combined modes of heat transfer. Since both conduction and convection are proportional to temperature difference, a practical definition of thermal resistance is the temperature difference associated with a given heat transfer rate per unit area. R-factors are functions of both thermal conductivity and thickness of the material along the direction of heat transfer. Page 20 of 32
The application of R-factors is better understood by working an example problem. Let s consider a layer of pipe insulation of inner radius, ri, and outer radius, ro. Fourier s Law for heat radial conduction is shown. The surface area associated with any radius, r, is equal to the circumference, 2 pi, r times the length, L. Page 21 of 32
Fourier s Law can be solved by separation of variables and integration, if the inner and outer temperatures are known and steady state conditions exist. The integration of 1/r with respect to dr between the discrete limits of ri to ro, results in a natural logarithm of the ratio of outer over inner radii. By algebra, the difference of two logs equals the log of their ratio. Page 22 of 32
A general definition of the R-factor for the tubular insulation is shown. Page 23 of 32
The multiple tubular layers shown here might correspond to an inner pipe surrounded by two successive layers of insulation. The radii are numbered 1 to 4 from the inner radius of the pipe to the outer radius of the second layer of insulation. The temperature inside the pipe is T1. The temperature at the outside of the pipe and inside of the first insulation layer is T2, and so on. Page 24 of 32
The solution of Fourier s Law by successive radial integration is shown. The 2 pi L in the numerator, divided by each of the quantities in the denominator, corresponds to a division of the temperature difference by the R-factors for each layer. Fourier s Law can be rearranged to predict heat transfer rate as the inner-to-outer temperature difference divided by the sum of R-factors for each layer, including the pipe itself. This same R-factor form of Fourier s Law can be applied to linear heat transfer, as well. Page 25 of 32
The numerical values of parameters for a two-layer example are given below the sketch. The situation might be that of an industrial process, where the air outside of the pipe cannot be allowed to rise above a certain maximum safe temperature. Page 26 of 32
The numerical solution, using the integrated form, is shown here. You should try on your own to get the same answer for heat transfer per meter length by use of R-factors. Page 27 of 32
This numerical example involves linear heat transfer through a vertical wall composed of inside sheetrock, fiberglass insulation, and outer wood siding. The inner and outer temperatures, inner and outer convection coefficients, layer thicknesses, and layer thermal conductivities are given beside the sketch. This multi-layered example involves both conduction and convection. We are neglecting radiation in this example. Page 28 of 32
The sum of individual thermal resistances, including inner and outer R-factors for convection, is computed here. Convection, in this regard, can be visualized as conduction through a thin film of fluid in contact with the solid surface. The convection R-factors lack the thickness necessary for conduction R-factors. The area is incorporated in each R-factor to predict total heat transfer rate by the modified Fourier s Law. An alternative approach would be to drop the area of the wall to estimate heat transfer per unit area of the wall. Page 29 of 32
The numerical solution of heat transfer rate, as the total temperature difference across the system divided by the sum of R-factors, is presented here. The added simplicity of heat transfer computations using R-factors is apparent in this case. Page 30 of 32
Once the heat transfer rate through the system is known, either form of Fourier s Law can be applied to derive estimates of temperature anywhere along the path of heat transfer. The original form of Fourier s Law with finite differences is advantageous for interpolating temperatures within a conducting layer, such as temperatures within the insulation layer. It is often important, for example, to know if and where the freezing temperature of water occurs within an insulation layer. Page 31 of 32
This concludes the presentation on fundamentals of heat transfer. Two references I can recommend are listed on the slide. Other good references exist as college textbooks by other publishers on the subject of heat transfer. A second set of homework exercises is available now in the Assignments area of the web site. These practice problems will help clarify the principles we have just reviewed. Please post your questions about this presentation or about the heat transfer exercises on the Discussion Board, so others can benefit from the exchange. This is Orson Smith, signing off for now. Page 32 of 32