This paper presents the analytical solution for the effectiveness of a counterflow heat exchanger subjected to a uniformly distributed, external heat flux. The solution is verified against conventional $ε$-NTU relations in the limit of zero external heat flux. This situation is of interest in applications such as cryogenic and process engineering, and the analytical solution provides a convenient method for treating differential elements of a heat exchanger in a numerical model.

1.
Incropera
,
F. P.
, and
DeWitt
,
D. P.
, 2002,
Fundamentals of Heat and Mass Transfer
, 5th Edition,
Wiley
, New York.
2.
Chowdhury
,
K.
, and
Sarangi
,
S.
, 1984, “
Performance of Cryogenic Heat Exchangers With Heat Leak From the Surroundings
,”
,
29
, pp.
273
280
.
3.
Barron
,
R. F.
, 1984, “
Effect of Heat Transfer From Ambient on Cryogenic Heat Exchanger Performance
,”
,
29
, pp.
265
272
.
4.
Ameel
,
T. A.
, and
Hewavitharana
,
L.
, 1999, “
Countercurrent Heat Exchangers With Both Fluids Subjected to External Heating
,”
Heat Transfer Eng.
0145-7632,
20
(
3
), pp.
37
44
.
5.
Nellis
,
G. F.
, 2003, “
A Heat Exchanger Model That Includes Axial Conduction, Parasitic Heat Load, and Property Variations
,”
Cryogenics
0011-2275,
43
(
9
), pp.
523
538
.
6.
Nellis
,
G. F.
,
Pfotenhauer
,
J. M.
, and
Klein
,
S. A.
, 2004, “
Actively Cooled Current Leads for Superconducting Electronics Using Mixed-Gas Joule-Thomson Refrigeration
,” 2004 IMECE, ASME Paper No. IMECE2004-60284.
7.
Keppler
,
F.
,
Nellis
,
G.
, and
Klein
,
S. A.
, 2004, “
Optimization of the Composition of a Gas Mixture in a Joule-Thomson Cycle
,”
HVAC&R Res.
1078-9669,
10
(
2
), pp.
213
230
.
8.
Swift
,
W. L.
,
Zagarola
,
M. V.
,
Nellis
,
G. F.
,
McCormick
,
J. A.
,
Sixsmith
,
H.
, and
Gibbon
,
J. A.
, 1999, “
Developments in Turbo Brayton Technology for Low Temperature Applications
,”
Cryogenics
0011-2275,
35
, pp.
989
995
.