Page 331 - Bird R.B. Transport phenomena

§10.8 Forced Convection 315

Uniform Plane at arbitrary
temperature 7^ downstream position z

t /
No energy enters here, Heat in by Energy leaving here is
since datum temperature heating coil rR
was chosen to be Tj is 2irRzq 0 2тг рС р (Т -T{)v z r'dr
Jo

Fig. 10.8-4. Energy balance used for boundary condition 4
given in Eq. 10.8-24.

This equation may be integrated twice with respect to £ and the result substituted into
Eq. 10.8-23 to give

•, 0 = Crf + C o l j - ^ I + Q In f + C 2 (10.8-27)

The three constants are determined from the conditions 1,2, and 4 above:
B.C. 1: Q = 0 (10.8-28)
B.C. 2: C o = 4 (10.8-29)
Condition 4: C 2 = - £ (10.8-30)
Substitution of these values into Eq. 10.8-27 gives finally
(10.8-31)

This result gives the dimensionless temperature as a function of the dimensionless radial
and axial coordinates. It is exact in the limit as £ -» °o; for £ > 0.1, it predicts the local
value of © to within about 2%.
Once the temperature distribution is known, one can get various derived quantities.
There are two kinds of average temperatures commonly used in connection with the
flow of fluids with constant p and C p:
Г27Г CR Т(Г
J J ' drdO n l ?
z)r
«'О •'O
rdrde
/ г
Г2тг ГК
v {r)T{r,z)rdrd0
z
J о J о

Both averages are functions of z. The quantity (Г) is the arithmetic average of the temper-
atures over the cross section at z. The "bulk temperature" T is the temperature one
b
would obtain if the tube were chopped off at z and if the fluid issuing forth were col-
lected in a container and thoroughly mixed. This average temperature is sometimes re-
ferred to as the "cup-mixing temperature" or the "flow-average temperature."