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Explicit Solution to Predict the Temperature Distribution and Exit Temperatures in a Heat Exchanger Using Differential Transform Method

  • Research Article - Mechanical Engineering
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Abstract

In this paper, a simple mathematical model based on energy transfer between the hot and cold fluids in a single-pass cross-flow plate-type heat exchanger is considered and the governing equations are solved using the differential transform method (DTM). DTM transforms the governing equations and its boundary conditions into a set of recurrence relations with some algebraic equations. These relations are further reduced, and then from the inverse transform, the dependent variable (temperature distribution) is obtained as a series solution in explicit form. The temperature distribution and the exit temperatures inside a single-pass cross-flow heat exchanger under steady state are obtained successfully for a single channel. The results obtained are then compared with those derived from Laplace Adomian decomposition method, finite difference method and the effectiveness-NTU approach. The comparison of the solution by DTM with other widely used industrial solutions is also shown, and the merits of the proposed method are presented. The predicted temperature distribution by the DTM is accurate, and it has excellent agreement with the effectiveness-NTU results and other well-known published results.

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Abbreviations

T :

Temperature of cold fluid (K)

\({\widetilde{T}}\) :

Temperature of hot fluid (K)

Q :

Heat transfer rate (kW)

U :

Overall heat transfer coefficient \({\left({\rm W}/{\rm m}^{2}{\rm K}\right)}\)

\({\Delta T_{\rm ln}}\) :

Log mean temperature difference \({\left({\rm K}\right)}\)

\({A_{\rm tot}}\) :

Total effective heat surface area \({\left({\rm m}^{2}\right)}\)

\({A_{\rm cr}}\) :

Cross-sectional area \({\left({\rm m}^{2}\right)}\)

m :

mass \({({\rm kg})}\)

D :

Diameter \({({\rm m}^{2})}\)

\({q^{\prime\prime\prime}}\) :

Heat generation term \({({\rm W}/{\rm m}^3)}\)

h :

Convection coefficient between fluid flow and heat surface \({\left({\rm W}/{\rm m}^{2}{\rm K}\right)}\)

\({\dot {\rm m}}\) :

Mass flow rate \({\left({\rm kg}/{\rm s}\right)}\)

u, v :

velocities along X and Y directions \({\left({\rm m}/{\rm s}\right)}\)

\({c_{p}}\) :

Specific heat of the fluid \({({\rm J}/{\rm kg}\,{\rm K})}\)

\({\rho}\) :

Density of the fluid \({({\rm kg}/{\rm m}^{3})}\)

C :

Heat capacity rate =\(\frac{\mbox{mass flow rate} \times \mbox{specific heat of the fluid}}{\mbox{number of plates} \times \mbox{perpendicular length}}\) \({\left({\rm W}/{\rm mK}\right)}\)

\({{\bf k}, \widetilde{\bf k}}\) :

Thermal conductivity \({({\rm W}/{\rm mK})}\)

s :

Thickness of the heat exchanger wall \({\left({\rm m}\right)}\)

k :

Thermal conductivity of the heat exchanger wall \({\left({\rm W}/{\rm mK}\right)}\)

F :

Correction factor to fix the error got from the assumption for the log mean temperature

Re :

Reynolds number

f :

Friction factor

Pr :

Prandtl number

Pe :

Peclet number

Nu :

Nusselt number

X :

X-coordinate axis

Y :

Y-coordinate axis

l :

Length of the axis

\({\epsilon}\) :

Effectiveness

\({\beta, \gamma}\) :

Constants

\({\delta}\) :

Kronecker delta function

W :

Differential transform of \({T(x, y)}\)

V :

Differential transform of \({\widetilde{T}(x, y)}\)

P :

Differential transform of p(x, y)

cold:

Cold fluid flow

hot:

Hot fluid flow

in:

Fluid inlet

out:

Fluid outlet

tot:

Total

DTM:

Differential transform method

LADM:

Laplace Adomian decomposition method

FDM:

Finite difference method

NTU:

Number of transfer units

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Authors and Affiliations

  1. Department of Mathematics, PSG College of Technology, Coimbatore, 641004, India

    Bagyalakshmi Morachan

  2. Department of Mechanical Engineering, PSG College of Technology, Coimbatore, 641004, India

    Madhu Ganesh

  3. Department of Applied Mathematics and Computational Sciences, PSG College of Technology, Coimbatore, 641004, India

    SaiSundarakrishnan Gangadharan

Authors
  1. Bagyalakshmi Morachan
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  2. Madhu Ganesh
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  3. SaiSundarakrishnan Gangadharan
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Correspondence to Bagyalakshmi Morachan.

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Morachan, B., Ganesh, M. & Gangadharan, S. Explicit Solution to Predict the Temperature Distribution and Exit Temperatures in a Heat Exchanger Using Differential Transform Method. Arab J Sci Eng 41, 1825–1834 (2016). https://doi.org/10.1007/s13369-015-1978-1

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  • DOI: https://doi.org/10.1007/s13369-015-1978-1

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