In the usual situation all three modes of heat transfer occur simultaneously. In this sec- tion, however, they will be considered separately for clarity and ease of presentation.
Thermal conductionis the mechanism of heat transfer between parts of a contin- uum due to the transfer of energy between particles or groups of particles at the atomic level. The Fourier equation expresses steady-state conduction in one dimension:
(5-1) where:
q=heat transfer rate, Btu/hr or W
k =thermal conductivity, Btu/(hr-ft-F) or W/(m-C) A=area normal to heat flow, ft2or m2
=temperature gradient, F/ft or C/m
Equation 5-1 incorporates a negative sign because qflows in the positive direction of xwhen dxdt is negative.
dt dx
q˙ kA dt
= − dx
119
Consider the flat wall of Fig. 5-1a, where uniform temperatures t1 and t2 are assumed to exist on each surface. If the thermal conductivity, the heat transfer rate, and the area are constant, Eq. 5-1 may be integrated to obtain
(5-2a) A very useful form of Eq. 5-2a is
(5-2b) where R′is the thermal resistance defined by
(5-3a) The thermal resistance for a unit area of material is very commonly used in hand- books and in the HVAC literature. In this book this quantity, sometimes called the “R- factor,” is referred to as the unit thermal resistance, or simply the unit resistance,R.
For a plane wall the unit resistance is
(5-3b) Thermal resistance R′is analogous to electrical resistance, and qand (t2− t1) are anal- ogous to current and potential difference in Ohm’s law. This analogy provides a very convenient method of analyzing a wall or slab made up of two or more layers of dis- similar material. Figure 5-1bshows a wall constructed of three different materials. The heat transferred by conduction is given by Eq. 5-2b, where the resistances are in series (5-4) Although the foregoing discussion is limited to a plane wall where the cross- sectional area is a constant, a similar procedure applies to a curved wall. Consider the long, hollow cylinder shown in cross-section in Fig. 5-2. The surface temperatures ti and to are assumed to be uniform and steady over each surface. The material is assumed to be homogeneous with a constant value of thermal conductivity. Integra- tion of Eq. 5-1 with kand qconstant but A a function of ryields
(5-5) where Lis the length of the cylinder. Here the thermal resistance is
(5-6) Cylinders made up of several layers may be analyzed in a manner similar to the plane wall where resistances in series are summed as shown in Eq. 5-4, except that the indi- vidual resistances are given by Eq. 5-6 with roand ribecoming the outer and the inner radius of each layer.
′ = ( )
R ln
kL
r r o i
2π
˙ ( )
q kL
ln
t t
r r
i o
o i
= 2π( ) −
′ = ′ + ′ + ′ = + +
R R R R x
k A x k A
x
1 2 3 1 k A
1
2 2
3 3
∆ ∆ ∆
R x
= ∆k
′ = −
=
R x x
kA
x kA
2 1 ∆
q˙ t t
= R−
′
2 1
˙ ( )
q kA t t x x
= − −
−
2 1
2 1
Tables 5-1aand 5-1bgive the thermal conductivity kfor a wide variety of build- ing and insulating materials. Other useful data given in Tables 5-1a and 5-1bare the unit thermal conductance C, the density ρ, and the specific heat cp. Note that khas the units of (Btu-in.)/(hr-ft2-F) or W/(m-K). With ∆xgiven in inches or meters, respec- tively, the unit thermal conductance Cis given by
(5-7) Thermal convection is the transport of energy by mixing in addition to conduc- tion. Convection is associated with fluids in motion, generally through a pipe or duct or along a surface. The transfer mechanism is complex and highly dependent on the nature of the flow.
The usual, simplified approach in convection is to express the heat transfer rate as (5-8a) where:
q=heat transfer rate from fluid to wall, Btu/hr or W h=film coefficient, Btu/(hr-ft2-F) or W/(m2-s)
t=bulk temperature of the fluid, F or C tw=wall temperature, F or C
˙ ( )
q =hA t −tw
C R
k
= 1 = x
∆ Btu/(hr-ft -F) or W/(m -K)2 2
5-1 Basic Heat-Transfer Modes 121
Figure 5-1 Nomenclature for conduction in plane walls.
t1 t2
x2 – x1 k
(a) x
t1 t2
x2
∆ x1
∆ ∆x3
k2
(b)
k1 k3
x
Figure 5-2 Radial heat flow in a hollow cylinder.
ti
to k
r
ro
ri
Table 5-1a Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa Conductivity Conductance Specific
Density k, C, Heat,
Thickness, ρ, (Btu-in.)/ Btu/ Btu/
Description in. lbm/ft3 (hr-ft2-F) (hr-ft2-F) (lbm-F)
Building Board
Asbestos–cement board 0.250 120 — 16.500 0.24
Gypsum or plaster board 0.375 50 — 3.100 0.26
Gypsum or plaster board 0.500 50 — 2.220 0.26
Plywood (Douglas fir) — 34 0.80 — —
Plywood (Douglas fir) 0.250 34 — 3.200 —
Plywood (Douglas fir) 0.375 34 — 2.130 —
Plywood (Douglas fir) 0.500 34 — 1.600 —
Plywood or wood panels 0.750 34 — 1.070 0.29
Vegetable fiber board
Sheathing, regular density 0.500 18 — 0.760 0.31
Sheathing intermediate 0.50v 22 — 0.920 0.31
density
Sound deadening board 0.500 15 — 0.740 0.30
Hardboard
Medium density — 50 0.73 — 0.32
Service grade — 55 0.82 —
High-density, standard- — 63 1.00 — —
tempered grade Particle board
Medium density — 50 0.94 — 0.31
Underlayment 0.625 40 — 1.220 0.29
Wood subfloor 0.750 — — 1.060 0.33
Building Membrane
Vapor-permeable felt — — 16.700 —
Vapor-seal, 2 layers of mopped — — — 8.350 —
15 lb felt
Finish Flooring Materials
Carpet and fibrous pad — — 0.480 0.34
Carpet and rubber pad — — — 0.810 0.33
Tile—asphalt, linoleum, vinyl, — — — 20.000 0.30
rubber
Wood, hardwood finish 0.75 — — 1.470 —
Insulating Materials Blanket and Batt
Mineral fiber, fibrous form processed from rock, slag, or glass
approx. 3–4 in. — 0.4–2.0 — 0.091 —
approx. 3.5 in. — 1.2–1.6 — 0.067 —
approx. 5.5–6.5 in. — 0.4–2.0 — 0.053 —
approx. 5.5 in. — 0.6–1.0 — 0.048 —
approx. 6–7.5 in. — 0.4–2.0 — 0.045 —
approx 8.25–10 in. — 0.4–2.0 — 0.033 —
continues
5-1 Basic Heat-Transfer Modes 123
Table 5-1a Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Conductivity Conductance Specific
Density k, C, Heat,
Thickness, ρ, (Btu-in.)/ Btu/ Btu/
Description in. lbm/ft3 (hr-ft2-F) (hr-ft2-F) (lbm-F)
Board and Slabs
Cellular glass — 8.0 0.33 — 0.180
Glass fiber, organic bonded — 4.0–9.0 0.25 — 0.230
Expanded polystyrene, molded — 1.0 0.36 — —
beads.
Mineral fiber with resin binder — 15.0 0.29 — 0.170
Core or roof insulation — 16–17 0.34 — —
Acoustical tile 0.500 — — 0.800 0.310
Acoustical tile 0.750 — — 0.530 —
Loose Fill
Cellulosic insulation (milled — 2.3–32 0.27–0.32 — 0.330
paper or wood pulp)
Perlite, expanded — 2.0–4.1 0.27–0.31 — 0.260
— 4.1–7.4 0.31–0.36 — —
— 7.4–11.0 0.36–0.42 — —
Mineral fiber (rock, slag, or glass)
approx. 3.75–5 in. — 0.6–2.0 — 0.091 0.170
approx. 6.5–8.75 in. — 0.6–2.0 — 0.053 —
approx. 7.5–10 in. — 0.6–2.0 — 0.045 —
approx. 10.25–13.75 in. — 0.6–2.0 — 0.033 —
Mineral fiber (rock, slag, or glass)
approx. 3.5 in. (closed sidewall — 2.0–3.5 — 0.077 —
application)
Vermiculite, exfoliated — 7.0–8.2 0.47 — 0.320
— 4.0–6.0 0.44 — —
Metals
Aluminum (1100) — 171 1536 — 0.214
Steel, mild — 489 314 — 0.120
Steel, stainless — 494 108 — 0.109
Roofing
Asbestos–cement shingles — 120 — 4.760 0.240
Asphalt roll roofing — 70 — 6.500 0.360
Asphalt shingles — 70 — 2.270 0.300
Built-up roofing 0.375 70 — 3.000 0.350
Slate 0.500 — — 20.000 0.300
Wood shingles, plain and — — — 1.060 0.310
plastic-film-faced Plastering Materials
Cement plaster, sand aggregate — 116 5.0 — 0.200
Sand aggregate 0.375 — — 13.300 0.200
Sand aggregate 0.750 — — 6.660 0.200
Gypsum plaster
Lightweight aggregate 0.500 45 — 3.120 —
Lightweight aggregate 0.625 45 — 2.670 —
Lightweight aggregate on 0.750 — — 2.130 —
metal lath
continues
Table 5-1a Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Conductivity Conductance Specific
Density k, C, Heat,
Thickness, ρ, (Btu-in.)/ Btu/ Btu/
Description in. lbm/ft3 (hr-ft2-F) (hr-ft2-F) (lbm-F)
Masonry Materials Masonry Units
Brick, fired clay — 130 6.4–7.8 — —
— 120 5.6–6.8 — 0.19
Clay tile, hollow
1 cell deep 4 — — 0.90 0.21
2 cells deep 6 — — 0.66 —
3 cells deep 8 — — 0.54 —
Concrete blocks
Normal weight aggregate — — — 0.90–1.03 0.22
(sand and gravel), 8 in., 33–36 lb, 126–136 lb/ft3 concrete, 2 or 3 cores
Lightweight aggregate — — — 0.52–0.61 —
(expanded shale, clay, slate or slag, pumice), 6 in., 16–17 lb, 85–87 lb/ft3 concrete, 2 or 3 cores
Same with vermiculite-filled — — — 0.33 —
cores, 8 in., 19–22 lb,
72–86 lb/ft3concrete — — — 0.32–0.54 0.21
Same with vermiculite-filled — — — 0.19–0.26 —
cores Concretes
Sand and gravel or stone — 150 10.0–20.0 — —
aggregate concretes
(concretes with more than 50% — 140 9.0–18.0 — 0.19–0.24
quartz or quartzite sand have — 130 7.0–13.0 — —
conductivities in the higher end of the range)
Limestone concretes — 120 7.9 — —
— 100 5.5 — —
Cement/lime, mortar, and — 100 6.7 — —
stucco — 80 4.5 — —
Lightweight aggregate — 120 6.4–9.1 — —
concretes
Expanded shale, clay, or slate; — 100 4.7–6.2 — 0.20
expanded slags; cinders; — 80 3.3–4.1 — 0.20
pumice (with density up to 100 lb/ft3); and scoria (sanded concretes have conductivities in the higher end of the range)
continues
5-1 Basic Heat-Transfer Modes 125
Table 5-1a Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Conductivity Conductance Specific
Density k, C, Heat,
Thickness, ρ, (Btu-in.)/ Btu/ Btu/
Description in. lbm/ft3 (hr-ft2-F) (hr-ft2-F) (lbm-F)
Siding Materials (on Flat Surface)
Asbestos–cement shingles — 120 — 4.75 —
Wood, drop, 1 ×8 in. — — — 1.27 0.28
Aluminum, steel, or vinyl, over — — — 1.64 0.29
sheathing, hollow-backed
Insulating-board-backed, nominal — — — 0.55 0.32
0.375 in.
Insulating-board-backed, nominal — — — 0.34 —
0.375 in., foil backed
Architectural (soda–lime float) — 158 6.9 — 0.21
glass
Woods (12% Moisture Content) Hardwoods
Oak — 41.2–46.8 1.12–1.25 — 0.39
Softwoods
Hemlock, fir, spruce, pine — 24.5–31.4 0.74–0.90 — 0.39
aValues are for a mean temperature of 75 F and are representative of dry materials for design but may differ depending on installation and workmanship.
Source: Reprinted with permission from ASHRAE Handbook, Fundamentals Volume, 1997.
Table 5-1b Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa Density Conductivity Conductance Specific
Thickness, ρ, k, C, Heat,
Description mm kg/m3 W/(m-C) W/(m2-C) kJ/(kg-C)
Building Board
Asbestos–cement board 6.4 1900 — 93.70 —
Gypsum or plaster board 9.5 800 — 17.60 1.09
Gypsum or plaster board 12.7 800 — 12.60 —
Plywood (Douglas fir) — 540 0.120 — 1.21
Plywood (Douglas fir) 6.4 540 — 18.20 —
Plywood (Douglas fir) 9.5 540 — 12.10 —
Plywood (Douglas fir) 12.7 540 — 9.10 —
Plywood or wood panels 19.0 540 — 6.10 —
Vegetable fiber board — — — — 1.21
Sheathing, regular density 12.7 290 — 4.30 —
Sheathing intermediate density 12.7 350 — 5.20 —
Sound deadening board 12.7 240 — 4.20 1.26
Tile and lay-in panels, plain — 290 0.058 — 0.59
or acoustic Hardboard
Medium density — 800 0.105 9.50 —
High-density, standard- — 1010 0.144 6.93 —
tempered grade
continues
Table 5-1b Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Density Conductivity Conductance Specific
Thickness, ρ, k, C, Heat,
Description mm kg/m3 W/(m-C) W/(m2-C) kJ/(kg-C)
Particleboard
Medium density — 800 0.135 7.35 —
Underlayment 15.9 640 — 6.90 1.210
Wood subfloor 19.0 — — 6.00 1.380
Building Membrane
Vapor-permeable felt — — — 94.90 —
Vapor-seal, 2 layers of mopped — — — 47.40 —
0.73 kg/M2 felt
Finish Flooring Materials
Carpet and fibrous pad — — — 2.73 —
Carpet and rubber pad — — — 4.60 1.380
Tile—asphalt, linoleum, vinyl, — — — 113.60 1.260
rubber
Wood, hardwood finish 19.00 — — 8.35 0.112
Insulating Materials Blanket and Batt
Mineral fiber, fibrous form processed from rock, slag, or glass
approx. 75–100 mm — 6.4–32 — 0.52 —
approx. 90 mm — 19–26 — 0.38 —
approx. 140–165 mm — 6.4–32 — 0.30 —
approx. 140 mm — 10–16 — 0.27 —
approx. 150–190 mm — 6.4–32 — 0.26 —
approx. 210–250 mm — 6.4–32 — 0.19 —
Board and Slabs
Cellular glass — 136 0.050 — —
Glass fiber, organic bonded — 64–140 0.036 — —
Expanded polystrene, molded beads — 16 0.037 — —
Mineral fiber with resin binder — 240 0.042 — —
Core or roof insulation — 260–270 0.049 — —
Acoustical tile 12.70 — — 4.50 —
Acoustical tile 19.00 — — 3.00 —
Loose Fill
Cellulosic insulation (milled — 37–51 0.039–0.046 — 1.398
paper or wood pulp)
Perlite, expanded — 32–66 0.039–0.045 — 1.090
— 66–120 0.045–0.052 — —
— 120–180 0.052–0.060 — —
Mineral fiber (rock, slag, or glass)
approx. 95–130 mm — 9.6–3.2 — 0.52 0.710
approx. 170–220 mm — 9.6–3.2 — 0.31 —
approx. 190–250 mm — 9.6–3.2 — 0.26 —
approx. 260–350 mm — 9.6–3.2 — 0.19 5.280
Mineral fiber (rock, slag or glass)
approx. 90 trim (closed — 32–56 2.1–2.5 — —
sidewall application)
continues
Table 5-1b Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Density Conductivity Conductance Specific
Thickness, ρ, k, C, Heat,
Description mm kg/m3 W/(m-C) W/(m2-C) kJ/(kg-C)
Vermiculite, exfoliated — 110–130 0.068 — 1.34
— 64–96 0.063 15.7 —
Metals
Aluminum (1100) — 2660 221.5 — 0.90
Steel, mild — 7600 45.3 — 0.50
Steel, stainless — 7680 15.6 — 0.46
Roofing
Asbestos–cement shingles — 1900 — 27.0 1.00
Asphalt roll roofing — 1100 — 36.9 1.51
Asphalt shingles — 1100 — 12.9 1.26
Built-up roofing 10 1100 — 17.0 1.46
Slate 13 — — 11.4 1.26
Wood shingles, plain and — — — 6.0 1.30
plastic film faced Plastering Materials
Cement plaster, sand aggregate — 1860 0.72 — 0.84
Sand aggregate 10 — — 75.5 0.84
Sand aggregate 20 — — 37.8 0.84
Gypsum plaster
Lightweight aggregate 13 720 — 17.7 —
Lightweight aggregate 16 720 — 15.2 —
Lightweight aggregate 19 — — 12.1 —
on metal lath Masonry Materials Masonry Units
Brick, fired clay — 2080 0.92–1.12 — —
— 1920 0.81–0.98 — 0.79
Clay tile, hollow
1 cell deep 100 — — 5.11 —
2 cells deep 150 — — 3.75 —
3 cells deep 200 — — 3.07 —
Concrete blocks
Normal mass aggregate (sand — — — 5.1–5.8 0.92
and gravel), 200 mm, 15–16 kg, 2020–2180 kg/m3 concrete, 2 or 3 cores
Low-mass aggregate (expanded — — — 3.0–3.5 —
shale, clay, slate or slag, pumice), 150 mm, 7.3–7.7 kg, 360–1390 kg/m3 concrete, 2 or 3 cores
Same with vermiculite-filled — — — 1.87 —
cores, 200 mm, 8.6–10.0 mm, — — 1.8–3.1 — —
1150–1380 kg/m3concrete Same with vermiculite-filled
cores — — 1.1–1.5 0.93–0.69 —
continues 5-1 Basic Heat-Transfer Modes 127
Table 5-1b Typical Thermal Properties of Common Building and Insulating Materials—Design Valuesa (continued)
Density Conductivity Conductance Specific
Thickness, ρ, k, C, Heat,
Description mm kg/m3 W/(m-C) W/(m2-C) kJ/(kg-C)
Concretes
Sand and gravel or stone — 2400 1.4–2.9 — —
aggregate concretes (concretes with more than 50% quartz or quartzite sand have
conductivities in the — 2240 1.3–2.6 — —
higher end of the range) — 2080 1.0–1.9 — —
Limestone concretes — 1920 1.14 — —
— 1600 0.79 — —
Cement/lime, mortar, and — 1600 0.97 1.04 —
stucco — 1280 0.65 1.54 —
Lightweight aggregate concretes
Expanded shale, clay, or — 1920 0.9–1.3 1.08–0.76 —
slate; expanded slags;
cinders; pumice (with — 1600 0.68–0.89 1.48–1.12 —
density up to 1600 — 1280 0.48–1.19 — 0.84
kg/m3); and scoria (sanded concretes have conductivities in the higher end of the range)
Siding Materials (on Flat Surface)
Asbestos–cement shingles — 1900 — 27.0 —
Wood, drop, 20 ×200 mm — — 7.21 — 1.17
Aluminum, steel, or vinyl, — — 9.13 — 1.22’1
over sheathing, hollow- backed
Insulating-board backed
9.5 mm nominal — — 3.12 — 1.34
9.5 mm nominal, foil- — — 1.93 — —
backed
Architectural (soda–lime — — 56.8 — 0.84
float) glass
Woods (12% Moisture Content) Hardwoods
Oak — 659–749 0.16–0.18 — 1.63
Softwood
Hem–fir, spruce–pine–fir — 392–502 0.107–0.130 — 1.63
aValues are for a mean temperature of 24 C and are representative of dry materials for design but may differ depending on installation and workmanship.
Source: Reprinted with permission from ASHRAE Handbook, Fundamentals Volume, 1997.
The film coefficient h is sometimes called the unit surface conductance or alterna- tively the convective heat transfer coefficient. Equation 5-8a may also be expressed in terms of thermal resistance:
(5-8b) where
(5-9a) so that
(5-9b) The thermal resistance given by Eq. 5-9a may be summed with the thermal resistances arising from pure conduction given by Eqs. 5-3a or 5-6.
The film coefficient happearing in Eqs. 5-8a and 5-9a depends on the fluid, the fluid velocity, the flow channel or wall shape or orientation, and the degree of devel- opment of the flow field (that is, the distance from the entrance or wall edge and from the start of heating). Many correlations exist for predicting the film coefficient under various conditions. Correlations for forced convection are given in Chapter 3 of the ASHRAE Handbook(1) and in textbooks on heat transfer.
In convection the mechanism that is causing the fluid motion to occur is impor- tant. When the bulk of the fluid is moving relative to the heat transfer surface, the mechanism is called forced convection, because such motion is usually caused by a blower, fan, or pump that is forcing the flow. In forced convection buoyancy forces are negligible. In free convection, on the other hand, the motion of the fluid is due entirely to buoyancy forces, usually confined to a layer near the heated or cooled surface. The surrounding bulk of the fluid is stationary and exerts a viscous drag on the layer of moving fluid. As a result inertia forces in free convection are usually small. Free con- vection is often referred to as natural convection.
Natural or free convection is an important part of HVAC applications. However, the predicted film coefficients have a greater uncertainty than those of forced convec- tion. Various empirical relations for natural convection film coefficients can be found in the ASHRAE Handbook(1) and in heat-transfer textbooks.
Most building structures have forced convection due to wind along outer walls or roofs, and natural convection occurs inside narrow air spaces and on the inner walls.
There is considerable variation in surface conditions, and both the direction and magni- tude of the air motion (wind) on outdoor surfaces are very unpredictable. The film coef- ficient for these situations usually ranges from about 1.0 Btu/(hr-ft2-F) [6 W/(m2-C)] for free convection up to about 6 Btu/(hr-ft2-F) [35 W/(m2-C)] for forced convection with an air velocity of about 15 miles per hour (20 ft/sec, 6 m/s). With free convection film coefficients are low, and the amount of heat transferred by thermal radiation may be equal to or larger than that transferred by convection.
Thermal radiation is the transfer of thermal energy by electromagnetic waves, an entirely different phenomenon from conduction and convection. In fact, thermal
R= h1 = C1
* (hr-ft -F Btu or (m -C)/ W2 )/ 2
′ = R hA
1 (hr-ft)/Btu or C/ W q˙ t t
R
= − w
′
5-1 Basic Heat-Transfer Modes 129
*Note that the symbol for conductance is C, in contrast to the symbol for the temperature in Celsius degrees, C.
radiation can occur in a perfect vacuum and is actually impeded by an intervening medium. The direct net transfer of energy by radiation between two surfaces that see only each other and that are separated by a nonabsorbing medium is given by
(5-10) where:
σ=Boltzmann constant, 0.1713 ×10-8 Btu/(hr-ft2-R4) =5.673 ×10-8W/ (m2-K4) T =absolute temperature, R or K
=emittance of surface 1 or surface 2 A=surface area, ft2or m2
F=configuration factor, a function of geometry only (Chapter 6)
In Eq. 5-10 it is assumed that both surfaces are “gray” (where the emittance equals the absorptance α). This assumption often can be justified. The student is referred to textbooks on heat transfer for a more complete discussion of thermal radiation. Fig- ure 5-3 shows situations where radiation is considered to be a significant factor. For the wall
and for the air space
The resistances can be combined to obtain an equivalent overall resistance R′ with which the heat transfer rate can be computed using Eq. 5-2b:
The thermal resistance for radiation is not easily computed, however, because of the fourth power temperature relationship of Eq. 5-10. For this reason and because of the inherent uncertainty in describing the physical situation, theory and experiment have been combined to develop combined or effective unit thermal resistances and unit thermal conductances for many typical surfaces and air spaces. Table 5-2a gives
˙ ( )
q t t
R
o i
= − −
′
˙ ˙ ˙ ˙
qi =qr +qc +qo
˙ ˙ ˙ ˙
qi =qw = qr +qo
˙ ( )
q T T
A A F A
12 14
24
1 1 1 1
1 1 1 12
2 2 2
= −
+ +
− σ −
Figure 5-3 Wall and air space, illustrating thermal radiation effects.
qr to ho
ti
two twi
hi
qo
qo qi
qc
qr
Rr Ro
Rw Ri Ro Ri
Rr
Rc qi
qw
k ko ki
Wall
Air space
5-1 Basic Heat-Transfer Modes 131
Table 5-2aSurface Unit Conductances and Unit Resistances for Aira Direction of Heat Position of SurfaceFlowhr-ft2-Fm2-CBtuWhr-ft2-Fm2-CBtuWhr-ft2-Fm2-CBtuW Still Air HorizontalUpward1.639.260.610.1100.915.21.100.1940.764.31.320.232 Sloping—Upward1.609.090.620.1100.885.01.140.2000.734.11.370.241 45 degrees VerticalHorizontal1.468.290.680.1200.744.21.350.2380.593.41.700.298 Sloping—Downward1.327.500.760.1300.603.41.670.2940.452.62.220.391 45 degrees HorizontalDownward1.086.130.920.1600.372.12.700.4760.221.34.550.800 Moving Air (any position)Any6.0034.00.170.029 Wind is 15 mph or 6.7 m/s (for winter) Wind is 7mph Any4.0022.70.250.044 or 3.4 m/s (for summer) aConductances are for surfaces of the stated emittance facing virtual blackbody surroundings at the same temperature as the ambient air. Values are based on a surface–air temperature difference of 10 F and for a surface temperature of 70 F. Source:Adapted by permission from ASHRAE Handbook,Fundamentals Volume,1989.
1 2
BtuWhr-ft2-Fm2-CBtuWhr-ft2-Fm2-CBtuWhr-ft2-Fm2-C Surface Emittances =0.9=0.05=0.2 hRhRhR
surface film coefficients and unit thermal resistances as a function of wall position, direction of heat flow, air velocity, and surface emittance for exposed surfaces such as outside walls. Table 5-2bgives representative values of emittance for some building and insulating materials. For example, a vertical brick wall in still air has an emittance of about 0.9. In still air the average film coefficient, from Table 5-2a, is about 1.46 Btu/(hr-ft2-F) or 8.29 W/(m2-C), and the unit thermal resistance is 0.68 (hr-ft2-F)/
Btu or 0.12 (m2-C)/W.
If the surface were highly reflective,=0.05, the film coefficient would be 0.59 Btu/(hr-ft2-F) [3.4 W/(m2-C)] and the unit thermal resistance would be 1.7 (hr-ft2-F)/
Btu [0.298 (m2-C)/W]. It is evident that thermal radiation is a large factor when nat- ural convection occurs. If the air velocity were to increase to 15 mph (about 7 m/s), the average film coefficient would increase to about 6 Btu/(hr-ft2-F) [34 W/(m2-C)].
With higher air velocities the relative effect of radiation diminishes. Radiation appears to be very important in the heat gains through ceiling spaces.
Tables 5-3aand 5-3bgive conductances and resistances for air spaces as a func- tion of orientation, direction of heat flow, air temperature, and the effective emittance of the space. The effective emittance Eis given by
(5-11)
1 1 1
1
1 2
E − + −
Table 5-2b Reflectance and Emittance of Various Surfaces and Effective Emittances of Air Spacea
With One
Average Surface Having With Both
Emittance Emittance and Surfaces
Surface Other 0.90 of Emittance
Aluminum foil, 0.05 0.05 0.03
bright
Aluminum foil, with 0.30b 0.29 —
condensate clearly visible (> 0.7 gr/ft2)
Aluminum foil, with 0.7b 0.65 —
condensate clearly visible (> 2.9 gr/ft2)
Regular glass 0.84 0.77 0.72
Aluminum sheet 0.12 0.12 0.06
Aluminum-coated 0.20 0.20 0.11
paper, polished
Steel, galvanized, 0.25 0.24 0.15
bright
Aluminum paint 0.50 0.47 0.35
Building materials— 0.90 0.82 0.82
wood, paper, masonry, nonmetallic paints
aThese values apply in the 4–40 àm range of the electromagnetic spectrum.
bValues are based on data presented by Bassett and Trethowen (1984).
Source: ASHRAE Handbook–Fundamentals.© American Society of Heating, Refrigerating and Air- Conditioning Engineers, Inc., 2001.
Effective Emittance Eof Air Space
where 1 and 2are for each surface of the air space. The effect of radiation is quite apparent in Tables 5-3aand 5-3b, where the thermal resistance may be observed to decrease by a factor of two or three as Evaries from 0.03 to 0.82.
The preceding paragraphs cover thermal resistances arising from conduction, con- vection, and radiation. Equation 5-4 may be generalized to give the equivalent resist- ance of nresistors in series:
(5-12) Figure 5-4 (p. 136) is an example of a wall being heated or cooled by a combination of convection and radiation on each surface and having five different resistances through which the heat must be conducted. The equivalent thermal resistance for the wall is given by Eq. 5-12 as
(5-13)
′ = ′ + ′ + ′ + ′ + ′ Re Ri R1 R2 R3 Ro
′ Re
′ = ′ + ′ + ′ + + ′ Re R1 R2 R3 K Rn
5-1 Basic Heat-Transfer Modes 133
Table 5-3a Thermal Resistances of Plane Air Spacesa
Orientation Direction Mean Temp.
of Air of Heat Temp., Diff., Eb=
Space Flow F F 0.03 0.05 0.2 0.5 0.82 0.03 0.05 0.2 0.5 0.82
Horiz. Up 90 10 2.13 2.03 1.51 0.99 0.73 2.34 2.22 1.61 1.04 0.75 50 30 1.62 1.57 1.29 0.96 0.75 1.71 1.66 1.35 0.99 0.77 50 10 2.13 2.05 1.60 1.11 0.84 2.30 2.21 1.70 1.16 0.87 0 20 1.73 1.70 1.45 1.12 0.91 1.83 1.79 1.52 1.16 0.93 0 10 2.10 2.04 1.70 1.27 1.00 2.23 2.16 1.78 1.31 1.02 –50 20 1.69 1.66 1.49 1.23 1.04 1.77 1.74 1.55 1.27 1.07 –50 10 2.04 2.00 1.75 1.40 1.16 2.16 2.11 1.84 1.46 1.20 45°Slope Up 90 10 2.44 2.31 1.65 1.06 0.76 2.96 2.78 1.88 1.15 0.81 50 30 2.06 1.98 1.56 1.10 0.83 1.99 1.92 1.52 1.08 0.82 50 10 2.55 2.44 1.83 1.22 0.90 2.90 2.75 2.00 1.29 0.94 0 20 2.20 2.14 1.76 1.30 1.02 2.13 2.07 1.72 1.28 1.00 0 10 2.63 2.54 2.03 1.44 1.10 2.72 2.62 2.08 1.47 1.12 –50 20 2.08 2.04 1.78 1.42 1.17 2.05 2.01 1.76 1.41 1.16 –50 10 2.62 2.56 2.17 1.66 1.33 2.53 2.47 2.10 1.62 1.30 Vertical Horiz. 90 10 2.47 2.34 1.67 1.06 0.77 3.50 3.24 2.08 1.22 0.84 50 30 2.57 2.46 1.84 1.23 0.90 2.91 2.77 2.01 1.30 0.94 50 10 2.66 2.54 1.88 1.24 0.91 3.70 3.46 2.35 1.43 1.01 0 20 2.82 2.72 2.14 1.50 1.13 3.14 3.02 2.32 1.58 1.18 0 10 2.93 2.82 2.20 1.53 1.15 3.77 3.59 2.64 1.73 1.26 –50 20 2.90 2.82 2.35 1.76 1.39 2.90 2.83 2.36 1.77 1.39 –50 10 3.20 3.10 2.54 1.87 1.46 3.72 3.60 2.87 2.04 1.56 45°Slope Down 90 10 2.48 2.34 1.67 1.06 0.77 3.53 3.27 2.10 1.22 0.84 50 30 2.64 2.52 1.87 1.24 0.91 3.43 3.23 2.24 1.39 0.99 50 10 2.67 2.55 1.89 1.25 0.92 3.81 3.57 2.40 1.45 1.02 0 20 2.91 2.80 2.19 1.52 1.15 3.75 3.57 2.63 1.72 1.26 0 10 2.94 2.83 2.21 1.53 1.15 4.12 3.91 2.81 1.80 1.30 –50 20 3.16 3.07 2.52 1.86 1.45 3.78 3.65 2.90 2.05 1.57 –50 10 3.26 3.16 2.58 1.89 1.47 4.35 4.18 3.22 2.21 1.66 continues 0.5 in. Air Space 0.75 in. Air Space Air Space Thermal Resistance, (F-ft2-hr)/Btu
Table 5-3a Thermal Resistances of Plane Air Spacesa(continued)
Orientation Direction Mean Temp.
of Air of Heat Temp., Diff., Eb=
Space Flow F F 0.03 0.05 0.2 0.5 0.82 0.03 0.05 0.2 0.5 0.82
Horiz. Down 90 10 2.48 2.34 1.67 1.06 0.77 3.55 3.29 2.10 1.22 0.85 50 30 2.66 2.54 1.88 1.24 0.91 3.77 3.52 2.38 1.44 1.02 50 10 2.67 2.55 1.89 1.25 0.92 3.84 3.59 2.41 1.45 1.02 0 20 2.94 2.83 2.20 1.53 1.15 4.18 3.96 2.83 1.81 1.30 0 10 2.96 2.85 2.22 1.53 1.16 4.25 4.02 2.87 1.82 1.3 –50 20 3.25 115 2.58 1.89 1.47 4.60 4.41 3.36 2.28 1.69 –50 10 3.28 3.18 2.60 1.90 1.47 4.71 4.51 3.42 2.30 1.71
1.5 in. Air Space 3.5 in. Air Space Horiz. Up 90 10 2.55 2.41 1.71 1.08 0.77 2.84 2.66 1.83 1.13 0.80
50 30 1.87 1.81 1.45 1.04 0.80 2.09 2.01 1.58 1.10 0.84 50 10 2.50 2.40 1.81 1.21 0.89 2.80 2.66 1.95 1.28 0.93 0 20 2.01 1.95 1.63 1.23 0.97 2.25 2.18 1.79 1.32 1.03 0 10 2.43 2.35 1.90 1.38 1.06 2.71 2.62 2.07 1.47 1.12 –50 20 1.94 1.91 1.68 1.36 1.13 2.19 2.14 1.86 1.47 1.20 –50 10 2.37 2.31 1.99 1.55 1.26 2.65 2.58 2.18 1.67 1.33 45°Slope Up 90 10 2.92 2.73 1.86 1.14 0.80 3.18 2.96 1.97 1.18 0.82 50 30 2.14 2.06 1.61 1.12 0.84 2.26 2.17 1.67 1.15 0.86 50 10 2.88 2.74 1.99 1.29 0.94 3.12 2.95 2.10 1.34 0.96 0 20 2.30 2.23 1.82 1.34 1.04 2.42 2.35 1.90 1.38 1.06 0 10 2.79 2.69 2.12 1.49 1.13 2.98 2.87 2.23 1.54 1.16 –50 20 2.22 2.17 1.88 1.49 1.21 2.34 2.29 1.97 1.54 1.25 –50 10 2.71 2.64 2.23 1.69 1.35 2.87 2.79 2.33 1.75 1.39 Vertical Horiz. 90 10 3.99 3.66 2.25 1.27 0.87 3.69 3.40 2.15 1.24 0.85 50 30 2.58 2.46 1.84 1.23 0.90 2.67 2.55 1.89 1.25 0.91 50 10 3.79 3.55 2.39 1.45 1.02 3.63 3.40 2.32 1.42 1.01 0 20 2.76 2.66 2.10 1.48 1.12 2.88 2.78 2.17 1.51 1.14 0 10 3.51 3.35 2.51 1.67 1.23 3.49 3.33 2.50 1.67 1.23 –50 20 2.64 2.58 2.18 1.66 1.33 2.82 2.75 2.30 1.73 1.37 –50 10 3.31 3.21 2.62 1.91 1.48 3.40 3.30 2.67 1.94 1.50 45°Slope Down 90 10 5.07 4.55 2.56 1.36 0.91 4.81 4.33 2.49 1.34 0.90 50 30 3.58 3.36 2.31 1.42 1.00 3.51 3.30 2.28 1.40 1.00 50 10 5.10 4.66 2.85 1.60 1.09 4.74 4.36 2.73 1.57 1.08 0 20 3.85 3.66 2.68 1.74 1.27 3.81 3.63 2.66 1.74 1.27 0 10 4.92 4.62 3.16 1.94 1.37 4.59 4.32 3.02 1.88 1.34 –50 20 3.62 3.50 2.80 2.01 1.54 3.77 3.64 2.90 2.05 1.57 –50 10 4.67 4.47 3.40 2.29 1.70 4.50 4.32 3.31 2.25 1.68 Horiz. Down 90 10 6.09 5.35 2.79 1.43 0.94 10.07 8.19 3.41 1.57 1.00 50 30 6.27 5.63 3.18 1.70 1.14 9.60 8.17 3.86 1.88 1.22 50 0 6.61 5.90 3.27 1.73 1.15 11.15 9.27 4.09 1.93 1.24 0 20 7.03 6.43 3.91 2.19 1.49 10.90 9.52 4.87 2.47 1.62 0 10 7.31 6.66 4.00 2.22 1.51 11.97 10.32 5.08 2.52 1.64 –50 20 7.73 7.20 4.77 2.85 1.99 11.64 10.49 6.02 3.25 2.18 –50 10 8.09 7.52 4.91 2.89 2.01 12.98 11.56 6.36 3.34 2.22
aFor multiple air spaces, each air space requires a separate resistance. Resistances of horizontal air spaces with heat flow downward are substantially independent of temperature difference.
bEffective emittance.
Source: Reprinted by permission from ASHRAE Handbook, Fundamentals Volume, 1997.
0.5 in. Air Space 0.75 in. Air Space Air Space Thermal Resistance, (F-ft2-hr)/Btu