'I
LEONARD TOPPER1 Johns Hopkins University, Baltimore, Md.
Forced Heat Convection in Pipes A steady-state sotution to the forced convection heat equation, applicable within entrance region and downstream, when wall heat flux remains constant
T H E operation of nuclear reactors and also of some other kinds of heat sources is essentially to warm a fluid that flows through a tube, the wall heat flux remaining constant. The prediction of the temperature distribution in the fluid was studied by Jakob and Rees (7) and more recently by Poppendiek (2). These workers investigated the temperature distribution downstream of the entrance region : they assumed that "the axial temperature gradient bT/bX was uniform with respect to the radius and equal to the mixed-mean axial temperature gradient, b T,/bX" (2). T h e mixed-mean fluid temperature at any axial position is
The results may also be of interest in the kinetics of zero-order heterogeneous
depletion or production chqmical processes in a tubular reactor, since the chemical problem and the thermal problem studied here are similar boundary-value problems. T h e temperature of the fluid satisfies the differential equation:
T,,, = V (2irr)dr
=
&
T V rdr
(3)
and the boundary condition
\
(1)
Present address, Engineering Research Center, Columbia University, New York 17, N. Y.
pansion of
No
+
must be a Dini ex-
NnJo(Anw)
n=l
that
W2
- -.
The calculation of 2 the coefficients is discussed in Watson ( 3 ) . The result is:
4
dT Atr = s,K= F br
This paper presents a steady-state solution to the forced convection heat equation that is applicable within the entrance region and also downstream. The solution is for plug flow (uniform velocity), and uses the assumption of temperature-independent fluid properties. T h e plug flow solution, presented here, may also be used in turbulent flow problems, provided that the turbulent eddy thermal diffusivity is used in place of the molecular thermal diffusivity.
2
states
r br
At X = 0 , T = To
TV(2rr)dk
6
Equation
T h e initial condition
PS
Jo
T h e N , of Equation 5 are determined by the initial condition.
(4)
so that the solution to the plug flow problem is: K - ( T - To) Fs
2ax w 2 - fvs s 2
We can show that
where W =
-
E], a
z,
(P,,
S
=
[d(s>l+
1
2
(2Xn)2
and X, is the nth zero of the
Bessel function of the first kind and first order [Jl(X,J= 01.
For numerical purposes, we tabulate the The values first three A, and Ja (A,). for n greater than three are sometimes needed in computations and are available in (3) and in the Jahnke-Emde tables. n 1 2 3
Xn
JdL)
3.83 7.02 10.17
-0.403 0.300 -0.250
The range of validity of the JakobPoppendiek assumption stated in the VOL. 48, NO. 8
AUGUST 1956
1379
Es such that
The vertical braces indicate that the inequality is applied to the absolute value of the sum on the right-hand side. The dimensionless distance downstream,
X -, a t which this sum (denoted by 01
amounts to only 1% of computed and
-
vs
tabulated
8)
has been
below,
for
CY -.vs
and
various values of the parameter
for W = 0 and W = 1 The parameter a - is the same as the inverse of the prodvs uct of Reynolds number and Prandtl number. Dzstance Dorun,truam
W
Figure 1. L5=10-2
SV
a T m - 2aF
dX
KVs
(*)
Differentiation of Equation 7 with respect to A’, on the other hand, yields
!
10-2
10-1
1
1 2
31
3 5
033
lo-. =
mJ
38
(9)
Comparing Equations 8 and 9, it appears that thc assumption made by the earlier workers is valid only when the second term on the right-hand side of Equation 9 is negligibly small; that is, the condition is satisfied only for values of W and
4 2
This analysis demonstrates that for small values of the parameter a ; 1 7 ~ , the Jakob-Poppendiek assumption mav not be valid for an appreciable distanw downstream
Nomenclature F
1.4 I
1x1 =
W = l
W=O
S
opening paragraph can be computed from the exact solution (Equation 7). Inserting Equation 7 in Equation 1 and carrying through the indicated integrations, it is found that:
Whev
0.09 5 vs
Radial temperature profiles X 1-=20 2 -X= I O 3 X - = 4 s
):(
=
heat flux a t wall
Jo,J1 = Bessel functions of first kind, and zero and first order
K = thermal conductivity of Auid Ai, = the nth coefficient in Equation 5 T5 T,,, To = temperature: local, mean a t distance X,initial V, V,, = velocity, mean velocity n = index of summation r = radial coordinate s = tube radius W = r/s X = axial coordinate a = thermal diffusivity of fluid A,, = the nth zero of J I
Acknowledgment Leo F. Epsteiri of Knolls Atomic Po\$er Laboratorv gave important assistance in his revieLv.
literature Cited
ocx sv s Figure 2. a = 1 1SV
1380
Temperature along tube axis L2 2 = 10-1 sv
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
3
2. sv
10-2
( 1 ) Jakob, 14., Rees, K. A , , Trans. Ani. Inst. Chem. Engrs. 37, 619 (1941). ( 2 ) Poppendiek, H. F., Chem. Eng. Pmgr. Symp. Ser. 50, No. 1 1 , 93 (1954). ( 3 ) Watson, G. N., “Treatise on the Theory of Bessel Functions,” 2nd ed., Cambridge University Press, Cambridge, 1944.
RECEIVEDfor review February 8, 1955 ACCEPTEDMarch 29,1956
