Thermally developing laminar flow of a Bingham plastic in a circular pipe with uniform wall heat flux has been studied analytically. Expressions for the fully developed temperature and Nusselt number are presented in terms of the yield stress, Peclet number, and Brinkman number. The solution to the Graetz problem has been obtained by using the method of separation of variables, where the resulting eigenvalue problem is solved approximately by using the method of weighted residuals. The effects of the yield stress, Peclet, and Brinkman numbers on the Nusselt number are discussed. In particular, it is shown that the heat transfer characteristics in the entrance region are significantly affected by the yield stress with the inclusion of viscous dissipation.

Issue Section:
Forced Convection
Keywords:
Forced Convection, Materials Processing and Manufacturing Process, Non-Newtonian Flows and Systems
Topics:
Convection, Heat flux, Pipes, Yield stress, Eigenvalues, Energy dissipation, Entrance region, Forced convection, Heat transfer, Laminar flow, Manufacturing, Materials processing, Non-Newtonian flow, Separation (Technology), Temperature
1.
Beek
W. J.
, and
Eggink
R.
,
1962
, “
Heat Transfer to Non-Newtonian Fluids in Laminar Flow through Circular Pipes
,”
De Ingenieur
, Vol.
74
, No.
35
, pp.
81
–
89
(in Dutch).
2.
Bird, R. B., Armstrong, R. C., and Hassager, O., 1987, Dynamics of Polymeric Liquids, (Vol. 1, Fluid Mechanics), John Wiley and Sons, New York.
3.
Bird, R. B., Stewart, W. E., and Lightfoot, E. N., 1960, Transport Phenomena, John Wiley and Sons, New York.
4.
Blackwell
B. F.
,
1985
, “
Numerical Solution of the Graetz Problem for a Bingham Plastic in Laminar Tube Flow With Constant Wall Temperature
,”
ASME JOURNAL OF HEAT TRANSFER
, Vol.
107
, pp.
466
–
468
.
5.
Finlayson, B. A., 1972, The Method of Weighted Residuals and Variational Principles, Academic Press, New York.
6.
Hsu
C. J.
,
1967
, “
An Exact Mathematical Solution for Entrance-Region Laminar Heat Transfer with Axial Conduction
,”
Applied Scientific Research
, Vol.
17
, pp.
359
–
376
.
7.
Johnston
P. R.
,
1991
, “
Axial Conduction and the Graetz Problem for a Bingham Plastic in Laminar Tube Flow
,”
International Journal of Heat and Mass Transfer
, Vol.
34
, pp.
1209
–
1217
.
8.
Kakac¸, S., Shah, R. K., and Aung, W., 1987, Handbook of Single-Phase Convective Heat Transfer, John Wiley and Sons, New York.
9.
Kays, W. M., and Crawford, M. E., 1993, Convective Heat and Mass Transfer, 3rd Ed., McGraw-Hill, New York.
10.
LeCroy
R. C.
, and
Eraslan
A. H.
,
1969
, “
The Solution of Temperature Development in the Entrance Region of an MHD Channel by the B. G. Galerkin Method
,”
ASME JOURNAL OF HEAT TRANSFER
, Vol.
91
, pp.
212
–
220
.
11.
Min
T.
,
Yoo
J. Y.
, and
Choi
H.
,
1997
a, “
Laminar Convective Heat Transfer of a Bingham Plastic in a Circular Pipe—I. Analytical Approach—Thermally Fully Developed Flow and Thermally Developing Flow (the Graetz Problem Extended)
,”
International Journal of Heat and Mass Transfer
, Vol.
40
, pp.
3025
–
3037
.
12.
Min
T.
,
Yoo
J. Y.
,
Choi
H. G.
, and
Choi
H.
,
1997
b, “
Laminar Convective Heat Transfer of a Bingham Plastic in a Circular Pipe—II. Numerical Approach—Hydrodynamically Developing Flow and Simultaneously Developing Flow
,”
International Journal of Heat and Mass Transfer
, Vol.
40
, pp.
3689
–
3701
.
13.
Ou
J. W.
, and
Cheng
K. C.
,
1973
, “
Viscous Dissipation Effects on Thermal Entrance Region Heat Transfer in Pipes with Uniform Wall Heat Flux
,”
Applied Scientific Research
, Vol.
28
, pp.
289
–
301
.
14.
Pirkle
J. C.
, and
Sigillito
V. G.
,
1972
, “
Laminar Heat Transfer with Axial Conduction
,”
Applied Scientific Research
, Vol.
26
, pp.
108
–
112
.
15.
Sellars
J. R.
,
Tribus
M.
, and
Klein
J. S.
,
1956
, “
Heat Transfer to Laminar Flow in a Round Tube or Flat Conduit—The Graetz Problem Extended
,”
ASME Transactions
, Vol.
78
, pp.
441
–
448
.
16.
Shah, R. K., 1975, “Thermal Entry Length Solutions for the Circular Tube and Parallel Plates,” Proceedings of National Heat and Mass Transfer Conference, 3rd, Indian Institute of Technology, Bombay, Vol. 1, Paper No. HMT-11-75.
17.
Shah, R. K., and London, A. L., 1978, “Laminar Flow Forced Convection in Ducts,” Advances in Heat Transfer, Supplement 1, Academic Press, New York.
18.
Siegel, R., Sparrow, E. M., and Hallman, T. M., 1958, “Steady Laminar Heat Transfer in a Circular Tube with Prescribed Wall Heat Flux,” Applied Scientific Research, Sect. A7, pp. 386–392.
19.
Vradis
G. C.
,
Dougher
J.
, and
Kumar
S.
,
1993
, “
Entrance Pipe and Heat Transfer for a Bingham Plastic
,”
International Journal of Heat and Mass Transfer
, Vol.
36
, pp.
543
–
552
.
20.
Vradis
G. C.
, and
O¨tu¨gen
M. V.
,
1997
, “
The Axisymmetric Sudden Expansion Flow of a Non-Newtonian Viscoplastic Fluid
,”
ASME Journal of Fluids Engineering
, Vol.
119
, pp.
193
–
200
.
21.
Wissler, E. H., and Schechter, R. S., 1959, “The Graetz-Nusselt Problem (with Extension) for a Bingham Plastic,” Chemical Engineering Progress Symposium Series 29, Vol. 55, pp. 203–208.
This content is only available via PDF.
Copyright © 1999
 by The American Society of Mechanical Engineers
You do not currently have access to this content.