Page 441 - Bird R.B. Transport phenomena

§14.1 Definitions of Heat Transfer Coefficients 423

the Nusselt number ] (a dimensionless wall heat flux or heat transfer coefficient) as a func-
tion of the relevant dimensionless quantities, such as the Reynolds and Prandtl numbers.
This situation is not unlike that in Chapter 6, where we learned how to use dimen-
sionless correlations of the friction factor to solve momentum transfer problems. How-
ever, for nonisothermal problems the number of dimensionless groups is larger, the
types of boundary conditions are more numerous, and the temperature dependence of
the physical properties is often important. In addition, the phenomena of free convec-
tion, condensation, and boiling are encountered in nonisothermal systems.
We have purposely limited ourselves here to a small number of heat transfer formulas
and correlations—just enough to introduce the reader to the subject without attempting to
23 4 5 6
be encyclopedic. Many treatises and handbooks treat the subject in much greater depth. ' ' '

§14.1 DEFINITIONS OF HEAT TRANSFER COEFFICIENTS
Let us consider a flow system with the fluid flowing either in a conduit or around a solid
object. Suppose that the solid surface is warmer than the fluid, so that heat is being trans-
ferred from the solid to the fluid. Then the rate of heat flow across the solid-fluid inter-
face would be expected to depend on the area of the interface and on the temperature
drop between the fluid and the solid. It is customary to define a proportionality factor h
(the heat transfer coefficient) by
Q = hAAT (14.1-1)
in which Q is the heat flow into the fluid (J/hr or Btu/hr), Л is a characteristic area, and AT
is a characteristic temperature difference. Equation 14.1-1 can also be used when the fluid
is cooled. Equation 14.1-1, in slightly different form, has been encountered in Eq. 10.1-2.
Note that h is not defined until the area A and the temperature difference AT have been
specified. We now consider the usual definitions for h for two types of flow geometry.
As an example of flow in conduits, we consider a fluid flowing through a circular tube
of diameter D (see Fig. 14.1-1), in which there is a heated wall section of length L and
varying inside surface temperature T {z), going from T 01 to T . Suppose that the bulk
0
02
temperature T b of the fluid (defined in Eq. 10.8-33 for fluids with constant p and C ) in-
p
creases from T to T b2 in the heated section. Then there are three conventional definitions
M
of heat transfer coefficients for the fluid in the heated section:
Q = /7 (7rDL)(T 01 - T ) = hydrDL)^ (14.1-2)
1
M
2>
й вв( ( Г о 1 " Т м ) + ( T u 2 " * ) = ft 0rDL)AT (14.1-3)
T
Q = я ( 7 Г e fl
Q = MirDL)( J^ 1 1^Z°(T T i i) = MirDL)AT In (14.1-4)
\ln (T - T ) - In (T - T )J
02
b2
1 This dimensionless group is named for Ernst Kraft Wilhelm Nusselt (1882-1857), the German
engineer who was the first major figure in the field of convective heat and mass transfer. See, for
example, W. Nusselt, Zeits. d. Ver. deutsch. Ing., 53,1750-1755 (1909), Forschungsarb. a. d. Geb. d.
Ingenieurwes., No. 80,1-38, Berlin (1910), and Gesundheits-Ing., 38,477-482,490-496 (1915).
M. Jakob, Heat Transfer, Vol. 1 (1949) and Vol. 2 (1957), Wiley, New York.
2
3
W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, 3rd edition, McGraw-Hill,
New York (1993).
H. D. Baehr and K. Stephan, Heat and Mass Transfer, Springer, Berlin (1998).
4
5
W. M. Rohsenow, J. P. Hartnett, and Y. I. Cho (eds.), Handbook of Heat Transfer, McGraw-Hill,
New York (1998).
H. Grober, S. Erk, and U. Grigull, Die Grundgesetze der Warmeiibertragung, Springer, Berlin, 3rd
6
edition (1961).