Page 455 - Bird R.B. Transport phenomena

§14.3 Heat Transfer Coefficients for Forced Convection in Tubes 435

average bulk temperature and /JL is the viscosity at the arithmetic average wall tempera-
0
1
ture. Then we may write
Nu = Nu(Re, Pr, L/D, fjL /fi ) (14.3-15)
b
0
2
This type of correlation seems to have first been presented by Sieder and Tate. If, in ad-
dition, the density varies significantly, then some free convection may occur. This effect
can be accounted for in correlations by including the Grashof number along with the
other dimensionless groups. This point is pursued further in §14.6.
Let us now pause to reflect on the significance of the above discussion for con-
structing heat transfer correlations. The heat transfer coefficient h depends on eight
physical quantities (D, (v), p, jx , /л , C , k, L). However, Eq. 14.3-15 tells us that this de-
ь
0
p
pendence can be expressed more concisely by giving Nu as a function of only four di-
mensionless groups (Re, Pr, L/D, ^ //x ). Thus, instead of taking data on h for 5 values
/7 0
8
of each of the eight individual physical quantities (5 tests), we can measure h for 5
values of the dimensionless groups (5 4 tests)—a rather dramatic saving of time and
effort.
A good global view of heat transfer in circular tubes with nearly constant wall tem-
2
perature can be obtained from the Sieder and Tate correlation shown in Fig. 14.3-2. This
is of the form of Eq. 14.3-15. It has been found empirically ' 2 3 that transition to turbulence
usually begins at about Re = 2100, even when the viscosity varies appreciably in the ra-
dial direction.
For highly turbulent flow, the curves for L/D > 10 converge to a single curve. For
Re b > 20,000 this curve is described by the equation
( UL \ 0 1 4
Nu l n = 0.026 e Pr 1 / 3 l ^ I (14.3-16)
0 8
R
This equation reproduces available experimental data within about ±20% in the ranges
5
10 4 < Re b e < 10 and 0.6 < Pr < 100.
b
For laminar flow, the descending lines at the left are given by the equation
\ 1/3 /шЛ 01 4
(14.3-17)

1
One can arrive at the viscosity ratio by inserting into the equations of change a temperature-
dependent viscosity, described, for example, by a Taylor expansion about the wall temperature:
= Mo + т ^ (Г - T ) + • • • (14.3-15a)
o
dT
When the series is truncated and the differential quotient is approximated by a difference quotient, we get
t4 Mo ( T _ Го ) (14.3-15Ы
or, with some rearrangement,

Thus, the viscosity ratio appears in the equation of motion and hence in the dimensionless correlation.
2
E. N. Sieder and G. E. Tate, Ind. Eng. Chem., 28,1429-1435 (1936).
3
A. P. Colburn, Trans. AlChE, 29,174-210 (1933). Alan Philip Colburn (1904-1955), provost at the
University of Delaware (1950-1955), made important contributions to the fields of heat and mass transfer,
including the "Chilton-Colburn relations."