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A solution to the periodic-flow regenerative heat exchanger problem

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Summary

An analytic solution to the problem of the periodic flow, regenerative heat exchanger is presented. The closed form solution based upon the usual simplifying assumptions permits rapid digital computer calculations of the heat exchanger design characteristics and effectiveness. The regenerator effectiveness calculated from this solution is compared with the result using the finite difference numerical technique.

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Abbreviations

a :

Flow cross section of one tube in matrix, ft2

a n :

Coefficient of the nth term of (43)

a * n :

Coefficient of the nth term of (44)

A :

Metal cross-section of one tube in matrix, ft2

c :

Specific heat of fluids or rotor matrix metal according to subscript, BTU/Lbs, °F

c 1, c 2 :

Constants of integration

C :

Capacity rate (i.e. WC) of fluids or rotor matrix according to subscript BTU/Hr, °F

E :

Regenerator effectiveness

f(ξ) :

Matrix temperature distribution at η=0 when entering the hot fluid

f*(ξ*):

Matrix temperature distribution at η=0 when entering the cold fluid

h :

Coefficient of heat transfer between fluid and matrix, according to subsript, BTU/Hr, ft2, °F

J 0 :

Bessel function of the first kind and zero order

J 1 :

Bessel function of the first kind and first order

K, K*:

Functions defined by (39), (40)

l :

Length of one tube in matrix, ft

− :

Laplace transform operator

n :

Order of the power series represented by (43) and (44)

p :

Perimeter of one tube in matrix, ft

s :

Variable corresponding to ξ in Laplace transform

S :

Heat transfer area on side designated by subscript, ft2

t :

Temperature of matrix in hot or cold fluid depending on subscript, °F

T :

Temperature of fluid according to subscript, °F

T 1 :

Initial hot fluid temperature, °F

T 2 :

Initial cold fluid temperature, °F

W :

Flow rate of fluids or rotor matrix solid phase, Lbs/Hr

x :

Abscissa along matrix positive in the direction of flow, ft

α :

=c r ρ r A/h h P

α*:

=c r ρ r A/h c p

β :

=W h c h /h h p

β*:

=W c c c /h c p

δ :

=aρ h c h /h h p

δ*:

=aρ c c c /h c p

ρ :

Density of fluids or rotor matrix according to subscript, Lbs/ft3

ϑ :

Time, Hr.

ϑ 0 :

Total time between flow reversals for matrix in hot fluid, Hr.

ϑ *0 :

Total time between flow reversals for matrix in cold fluid, Hr.

τ h :

A function defined by (10)

C c /C h :

Capacity rate ratio of the fluids

C r /C c :

Capacity rate ratio of the rotor matrix to the cold stream

(hS) c /(hS) h :

Symmetry factor or the ratio of the thermal resistance of the matrix in cold to hot fluid

NTU 0 :

Overall number of heat transfer units defined by (45)

ξ :

Dimensionless length in hot fluid=X/β

ξ*:

Dimensionless length in cold fluid=X/β*

η :

Dimensionless time in hot fluid = \(\frac{\delta }{{\alpha \beta }}\left( {\frac{\beta }{\delta }\theta - X} \right)\)

η*:

Dimensionless time in cold fluid = \(\frac{{\delta ^* }}{{\alpha ^* \beta ^* }}\left( {\frac{{\beta ^* }}{{\delta ^* }}\theta - X} \right)\)

ξ 0 :

=l/β

ξ *0 :

=l/β*

η 0 :

=ϑ 0/β

η *0 :

=ϑ *0 /β*

h :

refers to the hot fluid

c :

refers to the cold fluid

r :

refers to the rotor matrix

*:

refers to the cold fluid

References

  1. Coppage, J. E. and A. L. London, Trans. Amer. Soc. Mech. Engrs (1953) 779.

  2. Nusselt, W., Z. Ver. dtsch. Ing. 71 (1927) 85.

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  3. Iliffe, C. E., Proc. Instn Mech. Engrs 159 (1948) 363.

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  4. Lambertson, T. J., Trans. Amer. Soc. Mech. Engrs 159 (1948) 586.

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  5. Churchill, Ruel V., Modern Operational Mathematics in Engineering, McGraw-Hill Book Cy., Inc., 1944.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Carnegie Institute of Technology, Pittsburgh, Pa., U.S.A.

    A. N. Nahavandi & A. S. Weinstein

Authors
  1. A. N. Nahavandi
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  2. A. S. Weinstein
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Additional information

This project was done in part by A. N. Nahavandi in partial fulfillment of the requirement for the degree of Master of Science at Carnegie Institute of Technology.

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Nahavandi, A.N., Weinstein, A.S. A solution to the periodic-flow regenerative heat exchanger problem. Appl. sci. Res. 10, 335 (1961). https://doi.org/10.1007/BF00411928

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  • DOI: https://doi.org/10.1007/BF00411928

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