Abstract
An analysis of temperature-fields and heat transfer in a heat exchanger with codirected cross-flow configuration (see Fig. 1) and tube bundle of arbitrary size has been carried out. This kind of flow arrangement is very suitable for heat transfer between liquid flowing in finned tubes bundle while gas passing across them.
The problem was treated analytically by using the method ofweighted mean value of outside fluid temperature described in [1]. The solution of energy balance equations, valid for this case, is expressed by special polynomials which are appropriate for fast calculation of temperatures. They are analogous to other polynomials found in mathematical physics.
As end result it has been established that for such cross-flow arrangements, with an arbitrary number of tubesn in the bundle, given NTU value and the heat capacity rate ratioR, the relation for thermal effectivenessP has a simple explicit form.
Zusammenfassung
Es wurde eine Berechnung der Temperaturfelder und des Wärmestroms für gleichsinnigen Kreuzgegenstrom (Abb. 1) mit Rohrbündeln beliebiger Größe durchgeführt. Diese Art der Stromführung ist sehr vorteilhaft bei Rippenrohrbündeln, bei denen Flüssigkeit in den Rohren strömt und das Gas außerhalb der Rohre.
Das Problem wurde mit der früher [1] beschriebenen Methode des gewichteten Mittelwertes der äußeren Fluidtemperatur analytisch behandelt. Die Lösung der für diesen Fall gültigen Energiebilanzgleichungen wird durch spezielle Polynome ausgedrückt, die zur schnellen Berechnung der Temperaturen geeignet sind. Die Polynome sind anderen bekannten Polynomen der mathematischen Physik ähnlich.
Als Endergebnis wurde eine einfache explizite Beziehung für die dimensionslose Temperaturänderung (Wirkungsgrad)P als Funktion der Rohrreihenzahln, der Zahl der Übertragungseinheiten NTU und des KapazitätenstromverhältnissesR gefunden.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- A :
-
heat transfer surface area, [m2]
- B :
-
parameter, Eq. (8)
- b 0 :
-
perimeter of tube, [m]
- C :
-
heat capacity rate, [W/K]
- c j :
-
integration constant
- D :
-
determinant, Eq. (26)
- F i (η):
-
auxiliary function, Eq. (12), (13)
- k :
-
overall heat transfer coefficient, [W/(m2K)]
- L i :
-
Laguerre polynomial
- L :
-
Laplace operator
- l 0 :
-
Length of single tube, [m]
- NTU:
-
number of transfer units,kA/C
- ntu:
-
number of transfer units for one tube, NTU/n=kA/(nC)
- ntu⊥ :
-
number of transfer units of outside fluid for one tube, NTU⊥/n=kA/(nC ⊥)
- n :
-
number of tubes in the bundle
- P :
-
effectiveness of the heat exchanger
- p i (σ):
-
polynomial, Eq. (16), (18)
- R :
-
heat capacity rate ratio,C/C ⊥
- T :
-
dimensionless temperature of tube-side, fluid (t−t in)/(t ⊥,in−t in)
- t,t ⊥ :
-
fluids temperature, [K]
- w :
-
auxiliary parmeter, 1/ɛ−w 1
- w j :
-
auxiliary parameter,μ i p i [B(1/μ−1)], Eq. (21)
- y :
-
co-ordinate perpendicular to the tube axis, [m]
- δ i :
-
auxiliary parameter, Eq. (27)
- ɛ :
-
notation of exponent,e −B, Eq. (21)
- Θ :
-
dimensionless mean temperature difference,P/NTU
- ϑ :
-
dimensionless outside fluid temperature, (t ⊥−t in)/(t ⊥,in−t in)
- μ :
-
auxiliary parameter,\(e^{ - ntu_ \bot } \)
- η :
-
dimensionless tube-side flow path,y/l 0
- σ :
-
dimensionless independent variable,ηB(1/μ−1)=η(4/R) sinh (ntu⊥/2)
- ω :
-
weight factor, Eqs. (3), (4)
- i±1/2:
-
on the border betweeni andi+1 ori andi−1 tubes
- (-):
-
transformed polynomial,\(\bar p\) i (s)=L[p i (σ)]
- ⊥:
-
refers to outside fluid
- i,j :
-
current number (0≤i≤n, 0≤j≤n)
- in:
-
inlet
References
Bes, Th.;Roetzel, W.: Verlauf der Fluidtemperaturen im Querstromrohrbündel. Wärme-und Stoffübertragung 28 (1993) 457–463, (written on the basis of prior report for Stifterverband für die Deutsche Wissenschaft 1982/83)
Smith, D.M.: Mean Temperature Difference in Cross-Flow. Engineering, (1934) 479–481 and 606–607
Hausen, H.: Die Wirkung des Austuschers von Rektifikationsböden, Z. ang. Math. Mech. 17 (1937) 25–37
Bowman, R.A.;Mueller, A.C.;Nagle, W.M.: Mean Temperature Difference in Design Transactions of the ASME, 62 (1940) 283–294
Stevens, R.A.;Fernandez, J.;Woolf, J.R.: Mean-Temperature Difference in One, Two and Three-Pass Crossflow Heat Exchangers. Transactions of the ASME 79 (1957) 287–279
Schedwill, H.: Thermische Auslegung von Kreuzstromwärmeaustauschern. Fortschritts-Ber. VDI-Z. Reihe 6 (1968)
Nicole, F.J.L.: Mean Temperature Difference in Cross-Flow Heat Exchange, (applied to multipass air-cooled fin-tube units with a finite number of rows) M.Sc. Thesis, Univ. of Pretoria, CSIR Special Report Chem 223, Nov. 1972
Braun, B.: Wärmeübergang und Temperaturverlauf in Querstrom-Rohrbündeln bei beliebiger Schaltung der Rohrreihen. Forsch. Ing.-Wes. 41 Nr. 6, (1975) 181–191
Gardner, K.;Taborek, J.: Mean Temperature Difference. A Reappraisal AChIE Journal, 23 (1977) 777–786
Shah, R.K.; Mueller, A.C.: Heat Exchangers, (Chapter 4 of 2nd edition in Handbook of Heat Transfer Application, editors: Rohsenow, Hartnett and Ganic) McGraw-Hill Book Company, 1985
Roetzel, W.;Spang, B.: Berechnung von Wärmeübertragern. (Chapt. C of VDI-Wärmeatlas, 6th edition) VDI-Verlag, Düsseldorf 1991
Abramowitz, M.;Stegun, A.: Handbok of Mathematical Functions — (p. 1023) Dover Publications, Inc., New York, 1964
Lebedev, N.N.: Special functions & their applications; Dover Pub. Inc, New York, 1963
Bes, Th.; Roetzel, W.: Thermal Analysis of Codirected CrossFlow Heat Exchangers. Proc. ICHMT Int. Symposium on New Developments in Heat Exchangers, Lisbon, Portugal, paper P4, 1993
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Dr.-Ing. Wilfried Roetzel on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Bes, T. Thermal performances of codirected cross-flow heat exchangers. Heat and Mass Transfer 31, 215–222 (1996). https://doi.org/10.1007/BF02328611
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02328611