APPENDIX F
Most Efficient Temperature Difference
in Contraflow
Formal mathematics
F.1 Calculus of variations
A clear exposition of the theory for the calculus of variations is given in Hildebrand
(1976). Other texts are those by Courant & Hilbert (1989), Mathews & Walker
(1970), and Rektorys (1969).
The basic problem concerns a function
and the finding of a maximum or minimum of the integral of this function
where end values xo,xi,y(xo),y(xi) are known. Conditions concerning continuity of
functions and of their derivatives are covered in the reference texts, and the required
solution reduces to solving the Euler equation
Euler equation
Generalization
The problem can be extended to include a constraint in minimization or maximization of the integral
where y(;c) is to satisfy the prescribed end conditions
Advances in Thermal Design of Heat Exchangers: A Numerical Approach: Direct-sizing, step-wise
rating, and transients. Eric M. Smith
Copyright 2005 John Wiley & Sons, Ltd. ISBN: 0-470-01616-7
426
Advances in Thermal Design of Heat Exchangers
as before, but a constraint condition is also imposed in the form
where K is a prescribed constant, then the appropriate Euler equation is found to be
the result of replacing F in equation (F.I) by the auxiliary function
where A is an unknown constant. This constant, which is in the nature of a Lagrange
multiplier, will generally appear in the Euler equation and in its solution, and is to be
determined together with the two constants of integration in such a way that all three
conditions are satisfied.
F.2
Optimum temperature profiles
From definition of LMTD
(Chapter 2, Section 2.4)
From optimum contraflow exchanger
(Chapter 2, Section 2.12)
From general contraflow temperature profiles
[Chapter 3, Section 3.2, equation (3.8)]
Most Efficient Temperature Difference in Contraflow
Hot fluid profile
427
Cold fluid profile
The log mean temperature difference for these profiles depends on choice of the
value for constant a.
References
Courant, R. and Hilbert, D. (1989) Mathematical Methods of Physics, vol. I, John Wiley,
p. 184.
Hildebrand, F.B. (1976) Advanced Calculus for Applications, 2nd edn, Prentice Hall,
New Jersey, p. 360.
Mathews, J. and Walker, R.L. (1970) Mathematical Methods of Physics, 2nd edn,
Addison-Wesley, p. 322 (based on course given by R.P. Feynman at Cornell).
Rektorys, K. (Ed.) (1969) Survey of Applicable Mathematics, MIT Press, Cambridge,
Massachusetts, p. 1020.
