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Laminar heat transfer in parallel plate flow

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Abstract

An exact solution for the fluid temperature due to laminar heat transfer in parallel plate flow is found. The formulas obtained are valid for an arbitrary velocity profile. The basic problem encountered involves finding certain expansion coefficients in a series of nonorthogonal eigenfunctions. This problem is solved by passing to a vector system of equations having orthogonal eigenvectors. The method is applicable to more general problems.

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Abbreviations

A :

matrix defined in (11)

B :

matrix defined in (11)

C + n :

expansion coefficients of T *1

C − n :

expansion coefficients of T *2

C :

specific heat

f(y):

\( - \mu \left( {\frac{{\partial \upsilon }}{{\partial y}}} \right)^2 kpc\)

k :

thermometric conductivity

N ± n :

normalization factors for \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Z} _n^ \pm\) respectively

T :

temperature

T ∞ :

particular solution of (1) for x>0

T -∞ :

particular solution of (1) for x<0

T *1 :

homogenous solution of (1) for x<0 defined by (6) and (7)

T *2 :

same as T *1 but for x>0

V :

velocity profile of fluid

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{Z} _n^ \pm\) :

(ψ ± n , ψ /± n ′/λ /± n )

γ /+ n :

C /+ n /λ /+ n

γ /− n :

C /− n /λ /− n

\(\hat \theta\) :

(O, ΔT(δ(y+1)−δ(y−1))

λ :

eigenvalue parameter for (8)

λ /+ n :

positive eigenvalues of (8)

λ /− n :

negative eigenvalues of (8)

ρ :

density of fluid

ψ /+ n :

eigenvectors of (8) corresponding to λ /+ n

ψ /− n :

eigenvectors of (8) corresponding to λ /− n

References

  1. Agrawal, H. C., Appl. Sci. Res. Sect. A, 9 (1959/60) 177.

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  2. Langer, R. E., Trans. American Math. Society, Oct. (1929) 868.

  3. Friedman, B., Principles and Techniques of Applied Mathematics, John Wiley and Sons, New York, 1956, 154.

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  4. Deavours, C. A., J. on Math. Analysis (SIAM), 1 ≅ 2 (1971) 168.

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  5. Deavours, C. A., J. Appl. Mech. 38 (1971) 708.

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Authors and Affiliations

  1. Dept. of Mathematics, Kean College of New Jersey Union, 07083, New Jersey, USA

    C. A. Deavours

Authors
  1. C. A. Deavours
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Deavours, C.A. Laminar heat transfer in parallel plate flow. Appl. Sci. Res. 29, 69–76 (1974). https://doi.org/10.1007/BF00384132

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  • DOI: https://doi.org/10.1007/BF00384132

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