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CHAPTER 3
HEAT TRANSFER
Heat Transfer Processes ........................................................... 3.1 Overall Heat Transfer Coefficient............................................ 3.18
Thermal Conduction ................................................................. 3.2 Heat Transfer Augmentation ................................................... 3.19
Thermal Radiation .................................................................... 3.8 Heat Exchangers ..................................................................... 3.28
Thermal Convection ................................................................ 3.13 Symbols ................................................................................... 3.34
EAT transfer is energy in transit because of a temperature dif- Fourier’s law for conduction is
Hference. The thermal energy is transferred from one region to
another by three modes of heat transfer: conduction, radiation, and ∂t
convection. Heat transfer is among a group of energy transport phe- q″ = –k----- (1a)
nomena that includes mass transfer (see Chapter 5), momentum ∂x
transfer (see Chapter 2), and electrical conduction. Transport phe- where
nomena have similar rate equations, in which flux is proportional to 2
a potential difference. In heat transfer by conduction and convec- q″ =heat flux (heat transfer rate per unit area A), Btu/h·ft
tion, the potential difference is the temperature difference. Heat, ∂t/∂x = temperature gradient, °F/ft
mass, and momentum transfer are often considered together k=thermal conductivity, Btu/h·ft2·°F
because of their similarities and interrelationship in many common The magnitude of heat flux q″ in the x direction is directly pro-
physical processes. portional to the temperature gradient ∂t/∂x. The proportionality fac-
This chapter presents the elementary principles of single-phase tor is thermal conductivity k. The minus sign indicates that heat
heat transfer with emphasis on HVAC applications. Boiling and flows in the direction of decreasing temperature. If temperature is
condensation are discussed in Chapter 4. More specific information steady, one-dimensional, and uniform over the surface, integrating
on heat transfer to or from buildings or refrigerated spaces can be Equation (1a) over area A yields
found in Chapters 25 through 32 of this volume and in Chapter 12
of the 2002 ASHRAE Handbook—Refrigeration. Physical proper- ∂t
= -----
ties of substances can be found in Chapters 18, 22, 24, and 39 of this qk– A (1b)
volume and in Chapter 8 of the 2002 ASHRAE Handbook—Refrig- ∂x
eration. Heat transfer equipment, including evaporators, condens- Equation (1b) applies where temperature is a function of x only. In
ers, heating and cooling coils, furnaces, and radiators, is covered in the equation, q is the total heat transfer rate across the area of cross
the 2004 ASHRAE Handbook—HVAC Systems and Equipment. For section A perpendicular to the x direction. Thermal conductivity val-
further information on heat transfer, see the Bibliography. ues are sometimes given in other units, but consistent units must be
HEAT TRANSFER PROCESSES used in Equation (1b).
If A (e.g., a slab wall) and k are constant, Equation (1b) can be
In the applications considered here, the materials are assumed to integrated to yield
behave as a continuum: that is, the smallest volume considered
kA t t
contains enough molecules so that thermodynamic properties (e.g., ()–t –t
1 2 1 2
-------------------------- ------------------
density) are valid. The smallest length dimension in most engineer- q == (2)
–1 L Lk⁄ ()A
ing applications is about 100 µm (10 mm). At a standard pressure
10
of 14.696 psi at 32°F, there are about 3 × 10 molecules of air in a where L is wall thickness, t is the temperature at x = 0, and t is the
–3 3 1 2
volume of 10 mm ; even at a pressure of 0.0001 psi, there are 3 × temperature at x = L.
7 molecules. With such a large number of molecules in a very
10 Thermal Radiation. In conduction and convection, heat transfer
small volume, the variation of the mass of gas in the volume result- takes place through matter. In thermal radiation, energy is emitted
ing from variation in the number of molecules is extremely small, from a surface and transmitted as electromagnetic waves, and then
and the mass per unit volume at that location can be used as the den- absorbed by a receiving surface. Whereas conduction and convec-
sity of the material at that point. Other thermodynamic properties tion heat transfer rates are driven primarily by temperature gradients
(e.g., temperature) can also serve as point functions. and somewhat by temperature because of temperature-dependent
Thermal Conduction. This heat transfer mechanism transports properties, radiative heat transfer rates are driven by the fourth
energy between parts of a continuum by transfer of kinetic energy absolute temperature and increase rapidly with tem-
between particles or groups of particles at the atomic level. In power of the
gases, conduction is caused by elastic collision of molecules; in liq- peratures. Unlike conduction and convection, no medium is
uids and electrically nonconducting solids, it is believed to be required to transmit electromagnetic energy.
caused by longitudinal oscillations of the lattice structure. Thermal Every surface emits energy. The rate of emitted energy per unit
conduction in metals occurs, like electrical conduction, through the area is the emissive power of the surface. At any given temperature,
motion of free electrons. Thermal energy transfer occurs in the the emissive power depends on the surface characteristics. At a
direction of decreasing temperature. In opaque solid bodies, ther- defined surface temperature, an ideal surface (perfect emitter) emits
mal conduction is the significant heat transfer mechanism because the highest amount of energy. Such a surface is also a perfect
no net material flows in the process and radiation is not a factor. absorber (i.e., it absorbs all incident radiant energy) and is called a
blackbody. The blackbody emissive power W is given by the
Stefan-Boltzmann relation b
The preparation of this chapter is assigned to TC 1.3, Heat Transfer and = σT 4
Fluid Flow. W
b
3.1
3.2 2005 ASHRAE Handbook—Fundamentals
–8 2 4
where σ = 0.1712 × 10 Btu/h·ft ·°R is the Stefan-Boltzmann Fig. 1 Thermal Circuit
2
constant. W is in Btu/ft and T is in °F.
b
The ratio of the emissive power of a nonblack surface to the
blackbody emissive power at the same temperature is the surface
emissivity ε, which varies with wavelength for some surfaces (see
discussion in the section on Actual Radiation). Gray surfaces are
those for which radiative properties are wavelength-independent.
When a gray surface (area A , temperature T ) is completely
1 1
enclosed by another gray surface (area A2, temperature T2) and sep-
arated by a transparent gas, the net radiative heat transfer rate from
surface 1 is Fig. 1 Thermal Circuit
A ()W –W
1 b1 b2
q1 = ---------------------------------------------------------------
1 ⁄ ε +A ⁄εA ()1 – ε ⁄ Thermal Resistance R. In Equation (2) for conduction in a
1 1 2 2 2
where W and ε are the blackbody emissive power and emissivity of slab, Equation (3a) for radiative heat transfer rate between two sur-
b is defined faces, and Equation (4) for convective heat transfer rate from a sur-
the surfaces. The radiative heat transfer coefficient h face, the heat transfer rate can be expressed as a temperature
as r difference divided by a thermal resistance R. Thermal resistance is
2 2 analogous to electrical resistance, with temperature difference and
σ T heat transfer rate instead of potential difference and current,
q ⁄ A ()+T ()T +T
1 1 1 2 1 2
----------------- ---------------------------------------------------------------
hr == (3a) respectively. All the tools available for solving series electrical
T –T 1 ⁄ ε +A ⁄εA ()1 – ε ⁄
1 2 1 1 2 2 2 resistance circuits can also be applied to series heat transfer cir-
For two common cases, Equation (3a) simplifies to cuits. For example, consider the heat transfer rate from a liquid to
the surrounding gas separated by a constant cross-sectional area
2 2 solid, as shown in Figure 1. The heat transfer rate from the fluid to
σ()T +T ()T +T
1 2 1 2 the adjacent surface is by convection, then across the solid body by
A1 = A2: hr = -------------------------------------------------- (3b)
1 ⁄εε +1⁄ –1 conduction, and finally from the solid surface to the surroundings
1 2 by convection and radiation, as shown in the figure. A series circuit
A 2 2 using the equations for the heat transfer rates for each mode is also
>> A : h = σε ()T +T ()T +T (3c)
2 1 r 1 1 2 1 2 shown.
Note that h is a function of the surface temperatures, one of which From the circuit, the heat transfer rate is computed as
r
is often unknown. tf 1 – tf 2
Thermal Convection. When fluid flows are produced by exter- q = ---------------------------------------------------------------------
1 h
nal sources such as blowers and pumps, the solid-to-fluid heat trans- 1 L ()⁄ A ()1 ⁄ h A
c r
------- ------ ------------------------------------------
fer is called forced convection. If fluid flow is generated by density ++
hA kA 1⁄hcA+1⁄hrA
differences caused solely by temperature variation, the heat transfer
is called natural convection. Free convection is sometimes used to For steady-state problems, thermal resistance can be used
denote natural convection in a semi-infinite fluid. • With several layers of materials having different thermal conduc-
For convective heat transfer from a solid surface to an adjacent tivities, if temperature distribution is one-dimensional
fluid, the convective heat transfer coefficient h is defined by • With complex shapes for which exact analytical solutions are not
q″ ht available, if conduction shape factors are available
= ()–t
s ref • In many problems involving combined conduction, convection,
where and radiation
q″ = heat flux from solid surface to fluid Although the solutions are exact in series circuits, the solutions
ts = solid surface temperature, °F with parallel circuits are approximate because a one-dimensional
tref = fluid reference temperature for defining convective heat transfer temperature distribution is assumed.
coefficient Further use of the resistance concept is discussed in the sections
If the convective heat transfer coefficient, surface temperature, on Thermal Conduction and Overall Heat Transfer Coefficient.
and fluid temperature are uniform, integrating this equation over
surface area A gives the total convective heat transfer rate from the THERMAL CONDUCTION
surface: s
Steady-State Conduction
ts – tref One-Dimensional Conduction. Solutions to steady-state heat
-----------------
q ==hA ()t – t (4)
c s s ref 1 ⁄ hA transfer rates in (1) a slab of constant cross-sectional area with
s parallel surfaces maintained at uniform but different temperatures,
Combined Heat Transfer Coefficient. For natural convection (2) a hollow cylinder with heat transfer across cylindrical surfaces
from a surface to a surrounding gas, radiant and convective heat only, and (3) a hollow sphere are given in Table 1.
transfer rates are usually of comparable magnitudes. In such a case, Mathematical solutions to a number of more complex heat con-
the combined heat transfer coefficient is the sum of the two heat duction problems are available in Carslaw and Jaeger (1959). Com-
transfer coefficients. Thus, plex problems can also often be solved by graphical or numerical
methods such as described by Adams and Rogers (1973), Croft and
h = h + h Lilley (1977), and Patankar (1980).
c r Two- and Three-Dimensional Conduction: Shape Factors.
where h, h , and h are the combined, natural convection, and radi- There are many steady cases with two- and three-dimensional tem-
c r
ation heat transfer coefficients, respectively. perature distribution where a quick estimate of the heat transfer rate
Heat Transfer 3.3
Table 1 Heat Transfer Rate and Thermal Resistance for hV()⁄ A
s
Sample Configurations Bi = --------------------- ≤0.1
k
Heat Transfer Thermal where
Configuration Rate Resistance Bi = Biot number [ratio of (1) internal temperature difference
Constant t1 – t2 L required to move energy within the solid of liquid to (2)
cross- q = kA -------------- --------- temperature difference required to add or remove the same
x x L kA
sectional x energy at the surface]
area slab h = surface heat transfer coefficient
V = material’s volume
As = surface area exposed to convective heat transfer
k = material’s thermal conductivity
The temperature is given by
dt
Mc ----- = q +q (7)
Hollow pdτ net gen
2πkL()t – t ()⁄r
i o ln ro i
cylinder qr = -------------------------------- R = ---------------------- where
ro 2πkL
with ⎛⎞
ln ---- M=body mass
⎝⎠
negligible ri c = specific heat at constant pressure
heat transfer q p = internal heat generation
from end gen
q = net heat transfer rate to substance (into substance is positive, and
surfaces net out of substance is negative)
Equation (7) is applicable when pressure around the substance is
constant; if the volume is constant, replace c with the constant-
Hollow p
4πkt()– t 1 ⁄ r – 1 ⁄ r volume specific heat c . Note that with the density of solids and liq-
i o i o v
sphere qr = ---------------------------- R = ---------------------------- uids being substantially constant, the two specific heats are almost
1 1 4πk
--- +---- equal. The term q may include heat transfer by conduction, con-
ri ro net
vection, or radiation and is the difference between the heat transfer
rates into and out of the body. The term qgen may include a chemical
reaction (e.g., curing concrete) or heat generation from a current
passing through a metal.
A common case for lumped analysis is a solid body exposed to a
fluid at a different temperature. The time taken for the solid temper-
ature to change to t is given by
is desired. Conduction shape factors provide a method for getting f
such estimates. Heat transfer rates obtained by using conduction tf – t∞ hAτ
shape factors are approximate because one-dimensional tempera- ln --------------- = –---------- (8)
to – t∞ Mc
ture distribution cannot be assumed in those cases. Using the con- p
duction shape factor S, the heat transfer rate is expressed as where
M=mass of solid
q = Sk(t – t )(5)
1 2 c = specific heat of solid
p
where k is the material’s thermal conductivity, and t and t are the A = surface area of solid
1 2 h = surface heat transfer coefficient
temperatures of two surfaces. Conduction shape factors for some τ = time required for temperature change
common configurations are given in Table 2. When using a conduc- t = final solid temperature
tion shape factor, the thermal resistance is t f = initial uniform solid temperature
t o = surrounding fluid temperature
R = 1/Sk (6) ∞
Example 1. A 0.0394 in. diameter copper sphere is to be used as a sensing
Transient Conduction element for a thermostat. It is initially at a uniform temperature of
69.8°F. It is then exposed to the surrounding air at 68°F. The combined
Often, heat transfer and temperature distribution under tran- 2·°F. Determine the time
sient (varying with time) conditions must be known. Examples are heat transfer coefficient is h = 10.63 Btu/h·ft
taken for the temperature of the sphere to reach 69.6°F. The properties
(1) cold-storage temperature variations on starting or stopping a of copper are
refrigeration unit, (2) variation of external air temperature and ρ = 557.7 lb /ft3 c = 0.0920 Btu/lb ·°F k = 232 Btu/h·ft·°F
solar irradiation affecting the heat load of a cold-storage room or m p m
–5
wall temperatures, (3) the time required to freeze a given material Bi = hR/k = 10.63(0.0394/12/2)/232 = 7 × 10 , which is much less
under certain conditions in a storage room, (4) quick-freezing than 1. Therefore, lumped analysis is valid.
3 –6
M = ρ(4πR /3) = 10.31 × 10 lb
objects by direct immersion in brines, and (5) sudden heating or m
cooling of fluids and solids from one temperature to another. Using Equation (8), τ = 2.778 × 10–4 h = 1 s.
For slabs of constant cross-sectional areas, cylinders, and Nonlumped Analysis. In cases where the Biot number is greater
spheres, analytical solutions in the form of infinite series are avail- than 0.1, the variation of temperature with location within the mass
able. For solids with irregular boundaries, use numerical methods. must be accounted for. This requires solving multidimensional par-
Lumped Mass Analysis. One elementary transient heat transfer tial differential equations. Many common cases have been solved
model predicts the rate of temperature change of a body or material and presented in graphical forms (Jakob 1957; Myers 1971; Sch-
with uniform temperature, such as a well-stirred reservoir of fluid neider 1964). In other cases, it is simpler to use numerical methods
whose temperature is a function of time only and spatially uniform (Croft and Lilley 1977; Patankar 1980). When convective boundary
at all instants. Such an approximation is valid if conditions are required in the solution, values of h based on steady-
3.4 2005 ASHRAE Handbook—Fundamentals
Table 2 Conduction Shape Factors
Configuration Shape Factor S, ft Restriction
Edge of two adjoining walls 0.54WW > L/5
Corner of three adjoining walls (inner surface at T and 0.15LL << length
outer surface at T ) 1 and width of
2 wall
Isothermal rectangular block embedded in semi- 2.756L H 0.078 L > W
⎛⎞
infinite body with one face of block parallel to surface --------------------------------------- -----
0.59⎝⎠
d d L >> d, W, H
of body ⎛⎞
ln 1 + -----
⎝⎠
W
Thin isothermal rectangular plate buried in semi- πW
infinite medium ------------------------- d = 0, W > L
ln()4WL⁄
2πW d >> W
-------------------------
ln()4WL⁄ W > L
2πW d > 2W
---------------------------
ln()2πdL⁄ W >> L
Cylinder centered inside square of length L 2πL L >> W
--------------------------------- W > 2R
ln()0.54WR⁄
Isothermal cylinder buried in semi-infinite medium 2πL
------------------------------- L >> R
cosh–1()dR⁄
2πL L >> R
-------------------------
ln()2dR⁄ d > 3R
2πL
------------------------------------------------ d >> R
L ln()L ⁄ 2d L >> d
ln--- 1 – -----------------------
R ln()LR⁄
Horizontal cylinder of length L midway between two 2πL
infinite, parallel, isothermal surfaces ------------------ L >> d
4d
⎛⎞
ln ------
⎝⎠
R
Isothermal sphere in semi-infinite medium 4πR
----------------------------
1 – ()R ⁄ 2d
Isothermal sphere in infinite medium 4πR
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