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8. APPLICATIONS: ESTIMATION OF TRANSPORT PROPERTIES 339
8.2 ESTIMATION OF THERMAL
CONDUCTIVITY June 22, 2007 14:25 relation for hard-sphere molecules, the following equation is
developed for monoatomic gases.

Thermal conductivity is a molecular property that is required 1 k T
3
B
for calculations related to heat transfer and design and opera- (8.30) k = d 2 π m
3
tion of heat exchangers. It is defined according to the Fourier’s
law: where the parameters are defined in Eq. (8.2). This equation
∂T ∂ (ρC P T) is independent of pressure and is valid up to pressure of 10
q y =−k =−α atm for most gases [1]. The Chapmman–Enskog theory dis-
∂y ∂y
(8.28) cussed in Section 8.1.1 provides a more accurate relation in
k
α = the following form:
ρC P
T
where q y is the heat flux (heat transferred per unit area per (8.31) 1.9 × 10 −4 1/2
M
2
2
unit time, i.e., J/m · s or W/m ) in the y direction, ∂T/∂y k = σ
2
is the temperature gradient, and the negative sign indicates
that heat is being transferred in the direction of decreasing where kis in cal/cm · s · K, σ is in ˚ A, and is a parameter that is
temperature. The proportionality constant is called thermal a weak function of T as given for viscosity or diffusivity. This
conductivity and is shown by k. This equation shows that in function is given later in Section 8.3.1 (Eq. 8.57). From Eq.
the SI unit systems, k has the unit of W/m · K, where K may (8.31) it is seen that thermal conductivity of gases decreases
be replaced by C since it represents a temperature differ- with increase in molecular weight. For polyatomic gases the
◦
ence. In English unit system it is usually expressed in terms Eucken formula for Prandtl number is [1]
of Btu/ft · h · F(= 1.7307 W/m · K). The unit conversions are C P
◦
given in Section 1.7.19. In Eq. (8.28), ρC P T represents heat (8.32) N Pr = C P + 1.25R
per unit volume and coefficient k/ρC P is called thermal diffu-
sivity and is shown by α. A comparison between Eq. (8.28) and where C P is the molar heat capacity in the same unit as for gas
Eq. (8.1) shows that these two equations are very similar in constant R. This relation is derived from theory and errors as
nature as one represents flux of momentum and the other flux high as 20% can be observed.
2
of heat. Coefficients ν and α have the same unit (i.e., cm /s) For pure hydrocarbon gases the following equation is given
and their ratio is a dimensionless number called Prandtl num- in the API-TDB for the estimation of thermal conductivity [5]:
ber N Pr , which is an important number in calculation of heat
transfer by conduction in flow systems. In use of correlations (8.33) k = A + BT + CT 2
for calculation of heat transfer coefficients, N Pr is needed [28].
where k is in W/m · K and T is in kelvin. Coefficients A, B, and
ν μC P
(8.29) N Pr = = C for a number of hydrocarbons with corresponding tem-
α k perature ranges are given in Table 8.3. This equation can be
At 15.5 C (60 F), values of N Pr for n-heptane, n-octane, used for gases at pressures below 3.45 bar (50 psia) and has
◦
◦
benzene, toluene, and water are 6.0, 5.0, 7.3, 6.5, and 7.7, accuracy of ±5%. A generalized correlation for thermal con-
respectively. These values at 100 C (212 F) are 4.2, 3.6, 3.8, ductivity of pure hydrocarbon gases for P < 3.45 bar is given
◦
◦
3.8, and 1.5, respectively [28]. Vapors have lower N Pr num- as follows [5]:
bers, i.e., for water vapor N Pr = 1.06. Thermal conductivity
is a molecular property that varies with both temperature k = 4.911 × 10 −4 T r C P
λ
and pressure. Vapors have k values less than those for (a) only for methane and cyclic compounds at T r < 1
liquids. Thermal conductivity of liquids decreases with an

increase in temperature as the space between molecules k = 11.04 × 10 −5 (14.52T r − 5.14) 2/3 C P
increases, while for vapors thermal conductivity increases λ
with temperature as molecular collision increases. Pressure (b) for all compounds at any T except (a)
increases thermal conductivity of both vapors and liquids. 1/6 1/2
M
However, at low pressures k is independent of pressure. λ = 1.11264 T c 2/3
For some light hydrocarbons thermal conductivities of both P c
gases and liquids versus temperature are shown Fig. 8.3. (8.34)
Methods of prediction of thermal conductivity are very sim-
ilar to those of viscosity. However, thermal conductivity of Equation (8.34) also applies to methane and cyclic com-
gases can generally be estimated more accurately than can pounds at T r > 1, but for other compounds can be used at
liquid viscosity. For dense fluids, residual thermal conductiv- any temperature. The units are as follows: C P in J/mol · K,
ity is usually correlated to the reduced density similar to that T c in K, P c in bar, and k in W/m · K. This equation gives an
of viscosity (i.e., see Eqs. (8.11)–(8.13)). average error of about 5%.
For gas mixtures the following mixing rule similar to
Eq. (8.7) can be used [18]:
8.2.1 Thermal Conductivity of Gases
N
x i k i
Kinetic theory provides the basis of prediction of thermal (8.35) k m = N
conductivity of gases. For example, based on the potential i=1 j=1 x j A ij

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