Thermal conductivity, k, is a material property that is expressed in Btu/h ft F (W/m K) and is dependent on the chemical composition and physical characteristics of the substance. The relative order of magnitude of val- ues for various substances is shown in Table 7. Thermal conductivities are generally highest for solids, lower for liquids and lower yet for gases. Insulating materials have the lowest conductivities of solid materials.
Thermal conductivities of pure metals generally decrease with an increase in temperature, while al- loy conductivities may either increase or decrease. (See Fig. 7.) Conductivities of several steels and alloys are shown in Table 7. Thermal conductivities of various
refractory materials are shown in Chapter 23, Fig. 10. For many heat transfer calculations it is sufficiently ac- curate to assume a constant thermal conductivity that corresponds to the average temperature of the material. The effective thermal conductivity of ash deposits on water wall heating surfaces varies widely depend- ing on temperature, composition, heating cycle and physical characteristics of the deposits. The lower limit is close to the thermal conductivity of air or lower (0.03 Btu/h ft F or 0.05 W/m K), and the upper limit does not exceed values for refractory materials (1.4 Btu/h ft F or 2.4 W/m K). The effective thermal conductiv- ity of a friable particulate layer is near the lower limit and is fairly independent of temperature below 1650 to 2200F (899 to 1204C) at which sintering usually occurs. Above this temperature, particles fuse together and thermal contact between particles increases, result- ing in a sharp increase in thermal conductivity. The high- est conductivity is achieved with complete melting. The physical changes caused by fusion and melting are ir- reversible upon cooling, and thermal conductivity of fused deposits decreases with decreasing temperature.
Table 7
Properties of Various Substances at Room Temperature (see Note 1)
ρ cp k lb Btu Btu ft3 lb F h ft F METALS Copper 559 0.09 223 Aluminum 169 0.21 132 Nickel 556 0.12 52 Iron 493 0.11 42 Carbon steel 487 0.11 25
Alloy steel 18Cr 8Ni 488 0.11 9.4
NONMETAL SOLIDS Limestone 105 ~0.2 0.87 Pyrexglass 170 ~0.2 0.58 Brick K-28 27 ~0.2 0.14 Plaster 140 ~0.2 0.075 Kaowool 8 ~0.2 0.016 GASES Hydrogen 0.006 3.3 0.099 Oxygen 0.09 0.22 0.014 Air 0.08 0.24 0.014 Nitrogen 0.08 0.25 0.014
Steam (see Note 2) 0.04 0.45 0.015
LIQUIDS
Water 62.4 1.0 0.32
Sulfur dioxide (liquid) 89.8 0.33 0.12
Notes:
1. SI conversions: ρ, 1 lb/ft3
= 16.018 kg/m3
; cp, 1 Btu/lb F =
4.1869 kJ/kg K; k, 1 Btu/h ft F = 1.7307 W/m K. 2. Reference temperature equals 32F (0C) except for steam
Thermal conductance of ash deposits (k/x) is less sen- sitive to changing conditions than thermal conductivity. As the deposit grows in thickness (x), thermal conduc- tivity (k) also increases due to fusion and slagging. The net effect is that unit thermal conductance may only vary by a factor of four, 25 to 100 Btu/h ft2 F (142 to 568 W/
m2 C), while variations in thermal conductivity are an
order of magnitude larger. The thermal effects of coal- ash deposits are further described by Wall et al.5
The thermal conductivity of water ranges from 0.33 Btu/h ft F (0.57 W/m K) at room temperature to 0.16 Btu/h ft F (0.28 W/m K) near the critical point. Water properties are relatively insensitive to pressure, par- ticularly at pressures far from the critical point. Most other nonmetallic liquid thermal conductivities range from 0.05 to 0.15 Btu/h ft F (0.09 to 0.26 W/m K). In addition, thermal conductivities of most liquids de- crease with temperature.
The thermal conductivities of gases increase with temperature and are independent of pressure at nor- mal boiler conditions. These conductivities generally decrease with increasing molecular weight. The rela- tively high conductivity of hydrogen (a low molecu- lar weight gas) makes it a good cooling medium for electric generators. The relatively low conductivity of argon (a high molecular weight gas) makes a good insulating medium for thermal pane windows.
When calculating the conductivity of nonhomo- geneous materials, the designer must use an apparent thermal conductivity to account for the porous or lay- ered construction materials. In boilers and furnaces with refractory walls, thermal conductivity may vary from site to site due to variations in structure, compo- sition, density, or porosity when the materials were in- stalled. The thermal conductivities of these materials are strongly dependent on their apparent bulk den- sity (mass per unit volume). For higher temperature
insulations, the apparent thermal conductivity of fi- brous insulations and insulating firebrick decreases as bulk density increases, because the denser mate- rial attenuates the radiation. However, an inflection occurs at some point at which a further increase in den- sity increases the thermal conductivity due to conduc- tion in the solid material.
Theory shows that specific heats of solids and liq- uids are generally independent of pressure. Table 7 lists specific heats of various metals, alloys and nonhomogeneous materials at 68F (20C). These val- ues may be used at other temperatures without sig- nificant error.
The temperature dependence of the specific heat for gases is more pronounced than for solids and liquids. In boiler applications, pressure dependence may gen- erally be neglected. Tables 8a and 8b give specific heat data for air and other gases.
In the case of steam and water, property variations (specific heat and thermal conductivity) can be signifi- cant over the ranges of temperature and pressure found in boilers. It is therefore recommended that the properties as compiled in the American Society of Mechanical Engineers (ASME) Steam Tables6 be used.
Radiation properties
Bodies that are good radiation absorbers are equally good emitters and Kirchhoff ’s law states that, for gray surfaces at thermal equilibrium, their emissivities are equal to their absorptivities. A blackbody is one which absorbs all incident radiant energy while reflecting or transmitting none of it. The absorptivity and emissiv- ity of a blackbody are, by definition, each equal to one. This terminology does not necessarily mean that the body appears to be black. Snow, for instance, absorbs only a small portion of the incident visible light, but to the longer wavelengths (the bulk of thermal radia- tion), snow is almost a blackbody. At a temperature of 2000F (1093C) a blackbody glows brightly, because a non-negligible part of its radiation is in the visible range. Bodies are never completely black, but a hole through the wall of a large enclosure can be used to approximate blackbody conditions, because radiation entering the hole undergoes multiple reflections and absorptions. As a result, most of the radiation is retained in the enclosure, and surfaces are treated as gray.
Fortunately, a number of commercial surfaces, par- ticularly at high temperatures, have emissivities of 0.80 to 0.95 and behave much like blackbodies. Typi- cal average emissivity values are noted in Table 9. Although emissivity depends on the surface composi- tion and roughness and wavelength of radiation, the wavelength dependence is often neglected in practi- cal boiler calculations and surfaces are treated as gray.
Ash deposits The emittance and thermal proper-
ties of furnace ash deposits have a large effect on boiler heat transfer. The emittance depends on the tempera- ture, chemical composition, structure and porosity of the particulate layer, and whether deposits are par- tially fused or molten. The same ash at different loca- tions within the same boiler (or the same location in different boilers) may have significantly different values of surface emittance. Reported values in the
Thermal Conductivity, Btu/h ft F
(W/m K) 0 40 (69) 30 (52) 20 (35) 10 (17) 100 (38) (149)300 (260)500 700 (371) (482)900 (593)1100 1500 (816) 1300 (704) Temperature, F (C) Alloy 600 Carbon Steel, SA210A1, SA106A,B,C Alloy 625 Alloy 800 Alloy 825
Low Alloy (1-1/4 Cr-1/2 Mo-Si) SA213T2,T12,T11
Stainless Steel SA213TP304 Low Alloy (2-1/4 Cr-1Mo)
SA213T22
Medium Alloy (9Cr-1Mo-V) SA213T9, T91
Stainless Steel SA213TP309, TP310, TP316, TP317, TP321, TP347
Fig. 7 Thermal conductivity, k, of some commonly used steels and
literature claim emittances between 0.5 and 0.9 for most ash and slag deposits.
The effect of coal ash composition, structure, and temperature on deposit emittance5,7 is shown in Fig.
8. A friable particulate material has low emittance be- cause radiation is scattered (and reflected) from indi- vidual particles and does not penetrate beyond a thin layer (~1 mm) near the surface. Emittance of friable ash deposits decreases with increasing surface tem- perature, until sintering and fusion changes the struc- ture of the deposit. A sharp increase in emittance is associated with ash fusion as particles grow together (pores close) and there are fewer internal surfaces to scatter radiation. Completely molten ash or slag is partially transparent to radiation, and emittance may depend upon substrate conditions. The emittance of completely fused deposits (molten or frozen slag) on oxidized carbon steel is about 0.9. Emittance increases Table 8a
Properties of Selected Gases at 14.696 psi (101.33 kPa) (see Note 1)
cp k µ T ρ Btu/ Btu/ lbm/ F lb/ft3 lb F h ft F ft h Pr Air 0 0.0860 0.239 0.0133 0.0400 0.719 100 0.0709 0.240 0.0154 0.0463 0.721 300 0.0522 0.243 0.0193 0.0580 0.730 500 0.0413 0.247 0.0231 0.0680 0.728 1000 0.0272 0.262 0.0319 0.0889 0.730 1500 0.0202 0.276 0.0400 0.1080 0.745 2000 0.0161 0.286 0.0471 0.1242 0.754 2500 0.0134 0.292 0.0510 0.1328 0.760 3000 0.0115 0.297 0.0540 0.1390 0.765
Carbon Dioxide (CO2)
0 0.1311 0.184 0.0076 0.0317 0.767 100 0.1077 0.203 0.0100 0.0378 0.767 300 0.0793 0.226 0.0149 0.0493 0.748 500 0.0628 0.247 0.0198 0.0601 0.750 1000 0.0413 0.280 0.0318 0.0828 0.729 1500 0.0308 0.298 0.0420 0.1030 0.731 2000 0.0245 0.309 0.0500 0.1188 0.734 2500 0.0204 0.316 0.0555 0.1300 0.739 3000 0.0174 0.322 0.0610 0.1411 0.745 Water Vapor (H2O) 212 0.0372 0.451 0.0145 0.0313 0.974 300 0.0328 0.456 0.0171 0.0360 0.960 500 0.0258 0.470 0.0228 0.0455 0.938 1000 0.0169 0.510 0.0388 0.0691 0.908 1500 0.0127 0.555 0.0570 0.0889 0.866 2000 0.0100 0.600 0.0760 0.1091 0.861 2500 0.0083 0.640 0.0960 0.1289 0.859 3000 0.0071 0.670 0.1140 0.1440 0.846 Oxygen (O2) 0 0.0953 0.219 0.0131 0.0437 0.730 100 0.0783 0.220 0.0159 0.0511 0.707 300 0.0577 0.227 0.0204 0.0642 0.715 500 0.0457 0.235 0.0253 0.0759 0.705 1000 0.0300 0.253 0.0366 0.1001 0.691 1500 0.0224 0.264 0.0465 0.1195 0.677 2000 0.0178 0.269 0.0542 0.1414 0.701 2500 0.0148 0.275 0.0624 0.1594 0.703 3000 0.0127 0.281 0.0703 0.1764 0.703 Nitrogen (N2) 0 0.0835 0.248 0.0132 0.0380 0.713 100 0.0686 0.248 0.0154 0.0440 0.710 300 0.0505 0.250 0.0193 0.0547 0.710 500 0.0400 0.254 0.0232 0.0644 0.704 1000 0.0263 0.269 0.0330 0.0848 0.691 1500 0.0196 0.284 0.0423 0.1008 0.676 2000 0.0156 0.292 0.0489 0.1170 0.699 2500 0.0130 0.300 0.0565 0.1319 0.700 3000 0.0111 0.305 0.0636 0.1460 0.701 Note: 1. SI conversions: T(C) = [T(F) − 32]/1.8; ρ, 1 lb/ft3 = 16.018 kg/m3; c p, 1 Btu/lb F = 4.1869 kJ/kg K; k, 1 Btu/h ft F = 1.7307 W/m K; µ, 1 lbm/ft h = 0.0004134 kg/m s. Table 8b
Properties of Selected Gases at 14.696 psi (101.33 kPa) (see Note 1)
cp k µ
T ρ Btu/ Btu/ lbm/
F lb/ft3 lb F h ft F ft h Pr
Flue gas − natural gas (see Note 2)
300 0.0498 0.271 0.0194 0.0498 0.694 500 0.0394 0.278 0.0237 0.0593 0.694 1000 0.0259 0.298 0.0345 0.0803 0.694 1500 0.0193 0.317 0.0452 0.0989 0.693 2000 0.0154 0.331 0.0555 0.1160 0.692 2500 0.0128 0.342 0.0651 0.1313 0.691 3000 0.0109 0.351 0.0742 0.1456 0.689
Flue gas − fuel oil (see Note 3)
300 0.0524 0.259 0.0192 0.0513 0.692 500 0.0415 0.266 0.0233 0.0608 0.694 1000 0.0273 0.287 0.0336 0.0817 0.696 1500 0.0203 0.304 0.0436 0.1001 0.697 2000 0.0162 0.316 0.0531 0.1169 0.697 2500 0.0134 0.326 0.0618 0.1318 0.696 3000 0.0115 0.334 0.0700 0.1459 0.695
Flue gas − coal (see Note 4)
300 0.0537 0.254 0.0191 0.0519 0.691 500 0.0425 0.261 0.0232 0.0615 0.693 1000 0.0279 0.282 0.0333 0.0824 0.697 1500 0.0208 0.299 0.0430 0.1007 0.699 2000 0.0166 0.311 0.0521 0.1173 0.700 2500 0.0138 0.320 0.0605 0.1322 0.701 3000 0.0118 0.328 0.0684 0.1462 0.701 Notes: 1. SI conversions: T(C) = [T(F) − 32]/1.8; ρ, 1 lb/ft3 = 16.018 kg/m3 ; cp, 1 Btu/lb F = 4.1869 kJ/kg K; k, 1 Btu/h ft F = 1.7307 W/m K; µ, 1 lbm/ft h = 0.0004134 kg/m s. 2. Flue gas composition by volume (natural gas, 15% excess air):
71.44% N2, 2.44% O2, 8.22% CO2, 17.9% H2O.
3. Flue gas composition by volume (fuel oil, 15% excess air): 74.15% N2, 2.54% O2, 12.53% CO2, 0.06% SO2, 10.72% H2O.
4. Flue gas composition by volume (coal, 20% excess air): 74.86% N2, 3.28% O2, 13.97% CO2, 0.08% SO2, 7.81% H2O.
with increasing particle size of friable particulate de- posits (Fig. 8a), because larger particles have less ca- pacity to back-scatter incident radiation. Emittance increases with increasing iron oxide (Fe2O3) and un-
burned carbon content of the ash (Fig. 8b) because these components have a greater capacity to absorb radiation. Low emittance of some lignitic ash depos- its, known as reflective ash, may be attributed to low Fe2O3 content, although this alone is not a reliable
indicator of a reflective ash. Emittance is also indi- rectly dependent upon oxidizing and reducing envi- ronment of the flue gas, due to the effect on the melt- ing characteristics and unburned carbon content in the ash. The thermal and radiative effects of coal-ash deposits are further described by Wall et al.5
Combustion gases Although many gases, such as
oxygen and nitrogen, absorb or emit only insignificant amounts of radiation, others, such as water vapor, carbon dioxide, sulfur dioxide and carbon monoxide, substantially absorb and emit. Water vapor and car- bon dioxide are important in boiler calculations be- cause of their presence in the combustion products of hydrocarbon fuels. These gases are selective radiators. They emit and absorb radiation only in certain bands (wavelengths) of the spectrum that lie outside of the visible range and are consequently identified as nonluminous radiators. Whereas the radiation from a furnace wall is a surface phenomenon, a gas radi- ates and absorbs (within its absorption bands) at ev- ery point throughout the furnace. Furthermore, the emissivity of a gas changes with temperature, and the presence of one radiating gas may have characteris-
tics that overlap with the radiating characteristics of another gas when they are mixed. The energy emit- ted by a radiating gaseous mixture depends on gas temperature, the partial pressures, p, of the constitu- ents and a beam length, L, that depends on the shape and dimensions of the gas volume. An estimate of the mean beam length is L = 3.6 V/A for radiative trans- fer from the gas to the surface of the enclosure, where V is the enclosure volume and A is the enclosure sur- face area. The factor 3.6 is approximate, and values between 3.4 to 3.8 have been recommended depend- ing on the actual geometry.4
Figs. 9 and 10 show the emissivity for water vapor and carbon dioxide.8 The accuracy of these charts has
gained greater acceptance than the more widely known charts of Hottel,4 particularly at high temperatures and
short path lengths. The effective emissivity of a water vapor-carbon dioxide mixture is calculated as follows:
ε = εH O2 +εCO2 −∆ε (41)
where ∆ε is a correction factor that accounts for the effect of overlapping spectral bands. This equation neglects pressure corrections and considers boilers op- erating at approximately 1 atm. The factors shown in Fig. 11 depend on temperature, the partial pressures, p, of the constituents and the beam length, L. The pres- ence of carbon monoxide and sulfur dioxide can typi- cally be neglected in combustion products, because CO and SO2 are weakly participating and overlap with the
infrared spectrum of H2O and CO2.
When using Figs. 9 to 11 to evaluate absorptivity,
α, of a gas, Hottel4 recommends modification of the pL
product by a surface to gas temperature ratio. This is illustrated in Example 6 at the end of this chapter. Table 9
Normal Emissivities,
ε
, for Various Surfaces13 (see Note 1)Material Emissivity, ε Temp., F Description
Aluminum 0.09 212 Commercial sheet
Aluminum
oxide 0.63 to 0.42 530 to 930
Aluminum Varying age and Al
paint 0.27 to 0.67 212 content
Brass 0.22 120 to 660 Dull plate
Copper 0.16 to 0.13 1970 to 2330 Molten
Copper 0.023 242 Polished
Cuprous
oxide 0.66 to 0.54 1470 to 2012
Iron 0.21 392 Polished, cast
Iron 0.55 to 0.60 1650 to 1900 Smooth sheet
Iron 0.24 68 Fresh emeried
Iron oxide 0.85 to 0.89 930 to 2190
Steel 0.79 390 to 1110 Oxidized at 1100F
Steel 0.66 70 Rolled sheet
Steel 0.28 2910 to 3270 Molten
Steel (Cr-Ni) 0.44 to 0.36 420 to 914 18-8 rough, after
heating
Steel (Cr-Ni) 0.90 to 0.97 420 to 980 25-20 oxidized in
service
Brick, red 0.93 70 Rough
Brick, fireclay 0.75 1832 Carbon, lamp-
black 0.945 100 to 700 0.003 in. or thicker
Water 0.95 to 0.963 32 to 212 Note:
1. SI conversion: T(C) = [T(F) − 32]/1.8; 1 in. = 25.4 mm.
Fig. 8 Effect of coal ash composition, structure and temperature on
deposit emittance.5,7 0.9 1100 900 Increasing Absorption 700 500 300 100 0.4 0.5 0.6 0.7 0.8 0.9 (b) Surface Temperature, T , C With Carbon With Fe O Colorless Increasing Particle Size 1.0 0.8 0.7 0.6 0.5 0.4 (a) 211-422 µm 211 µm 104- <44 µm 53-104 µm Cooling Particulates Heating Fused Sintering Fusion 1.0
Radiation properties of gases can be calculated more accurately based on fundamental models for spectral gas radiation. The exponential wide band model9 pre-
dicts spectral absorption and emission properties of single and multi-component gases including H2O,
CO2, CO, CH4, NO, and SO2 as a function of tempera-
ture and pressure. Diatomic gases N2, O2 and H2 may
contribute to the total gas volume and pressure of the mixture, but are considered transparent to infrared radiation. Radiation properties are conveniently ex- pressed as emission and absorption coefficients that depend on local variations in gas composition, tempera- ture, and pressure. This approach is suitable for nu- merical modeling of radiation with participating media, which requires frequent evaluation of gas properties at a large number of control volumes.
Entrained particles Combustion usually involves some form of particulate that is entrained in combus- tion gases. Particles are introduced as the fuel which undergo transformations of combustion and/or are formed by the processes of condensation and agglom- eration of aerosol particles. Entrained particles have a significant role in radiation heat transfer because they absorb, emit, and scatter radiation. Scattering effec- tively extends the beam length of radiation in an en- closure, because the beam changes direction many times before it reaches a wall. Radiation from entrained par- ticles depends on the particle shape, size distribution, chemical composition, concentration, temperature, and the wavelength of incident radiation.
Particulates in boilers are comprised of unreacted fuel (coal, oil, black liquor), char, ash, soot, and other aerosols. Soot is an example of an aerosol that con-
0.05 1.0 0.06 0.07 0.04 0.03 0.02 0.01 0.00 0.0 0.2 0.4 0.6 0.8 90 bar cm 30 bar cm 60 bar cm 90 bar cm 1700F (925C) and Above p L + p L = 120 bar cm p p + p
Fig. 11 Radiation heat transfer correction factor associated with
mixtures of water vapor and carbon dioxide.8 (1 bar-cm = 0.0324 ft-atm) Fig.10 Emissivity of carbon dioxide at one atmosphere total pressure:
pcL= partial pressure in atmospheres x mean beam length in feet.8
(1 bar-cm = 0.0324 ft-atm; T(F) = [T(C) x 1.8] + 32)
Emissivity of Carbon Dioxide
0.005 0.05 0.10 0.20 0.30 0.01 2200 2000 1600 1200 800 200 400 600 1000 1400 1800 Carbon Dioxide
Total Pressure 1 bar Partial Pressure 0 bar
0.3 bar cm 0.15 bar cm 2 bar cm 4 bar cm 15 bar cm 40 bar cm 8 bar cm 1 bar cm 0.6 bar cm 100 bar cm Temperature, C (T)F = (T(C) x 1.8) + 32 0.04 0.03 0.02 p cL = 0.5 bar cm
Fig. 9 Emissivity of water vapor at one atmosphere total pressure:
pwL= partial pressure in atmospheres x mean beam length in feet.8
(1 bar-cm = 0.0324 ft-atm; T(F) = [T(C) x 1.8] + 32)
Emissivity of Water Vapor
0.70 0.50 0.10 0.05 0.01 2200 2000 1600 1200 800 200 Temperature, C T(F) = (T(C) x 1.8) + 32 1000 1400 1800 400 600 Water Vapor Total Pressure 1 bar Partial Pressure 0 bar
40 bar cm 80 bar cm 150 bar cm 400 bar cm 10 bar cm 20 bar cm 3 bar cm 0.5 bar cm 1.5 bar cm 6 bar cm 0.08 0.04 0.03 0.02 0.20 pw L = 0.2 bar cm
tributes to radiation from gas flames in boilers. Ne- glecting the effect of soot on radiation heat transfer in the flame could lead to significant errors in the cal- culated flame temperature, and radiation heat trans- fer to the furnace walls in the flame zone. Ash is an example of particulate that contributes to radiation in coal-fired boilers. Scattering by ash particles effectively redistributes radiation in the furnace, and smooths out variations in radiation heat flux, analogous to the way a cloud distributes solar radiation on the earth. The absorption and emission characteristics of flyash par- ticles increase, and scattering decreases with the rela- tive amount of iron oxide or residual carbon, which acts as a coloring agent in the ash.
Analytical methods such as Equation 39 that de- pend upon emissivity and absorptivity of the partici- pating media are inaccurate when particles other than soot are involved, because the effects of scatter- ing are neglected. Numerical methods which solve the general form of the radiative transport equation in- clude the effects of scattering (see Numerical methods). Mie Theory10 is a general method for calculating the
radiation properties of spherical particles as a func- tion of particle composition, concentration, diameter and wavelength. Rigorous calculations by this method can only be performed with the aid of a computer and require that optical properties (complex refractive in- dex as a function of wavelength) of the particle mate- rials are known. The complex refractive index of lig- nite, bituminous, and anthracite coals, and corre- sponding properties of char and ash have been mea- sured, as well as other materials that are typically en- countered in combustion systems. Radiation properties of particles are conveniently expressed as total emission, absorption, and scattering efficiencies that depend on particle composition, diameter and temperature. Particle properties must be combined with gas properties in an analysis of radiation with participating media.