Classification for thermal development of fluid flow

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Classification for thermal development of fluid
flow

• Prof. S.P. Mahulikar
• Department of Aerospace Engineering
• IIT Bombay

Re-examination of general criterion for thermal
development of flow. Thermal development ratio
various flow condition Thermal undevelopment of
flow
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Background

• Poiseuille (1840)
• solved developing temperature profile for fluid
  under forced convection inside circular pipe.
• Recherches experimentelles sur le mouvement
  des liquides dans lestubes de tres petits
  diametres
• Comptes Rendus 11
• Sellars (1956)
• Accurate results for developing temperature.
• Heat transfer to laminar flow in a round tube or
  flat conduit the Graetz problem extended
• Trans. ASME J. Heat Transfer 78
• Seban Shimazaki (1951)
• Reported region of thermal development of flow
• Heat transfer to a fluid flowing turbulently in
  a smooth pipe with walls at constant temperature
  
• Trans. ASME J. Heat Transfer 73

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Introduction

• RTD 0
• is criterion for thermally developed flow,
  applied to
• const. wall heat flux boundary condition (H1),
• const. wall temp. boundary condition (T1),
• hydrodynamically developed laminar flow.
• where is dimensionless temp.
  profile

• Case of RTD 0
• Variation of fluid temp. linearly proportional to
  variation of wall to bulk mean fluid temp.
  difference
• Fluid temp. profile follows variation of mean of
  profile along flow

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Illustration of flow condition segregation based
on RTD.

• (b) Case of RTD gt 0
• Thermally developing flow.

• (c) Case of RTD lt 0
• Ratio of fluid temp. profile to mean of fluid
  temp. profile decreases along flow.
• This occurs especially when boundary condition
  varies drastically.
• This flow condition is referred as thermally
  undeveloping flow

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Derivation of alternative criterion
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New criterion for thermal development

• Thermal development ratio defined-

• The criterion for thermally developed flow
  represented by-

• In developing flow region passive term dominate
  over active

• When active term dominates over passive term

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• In thermal undevelopment of flow, lag between
  active variation of qwz passive variation of
  fluid temp. profile increases.
• In thermal development, this lag decreases.
• In thermally developed flow there is no lag.
• temperature profile Tw(z) - Tf(r, z)
• follows active variation of qwz.
• In thermally undeveloped flow lag remains
  unchanged
• variation of fluid temperature profile follows BC
  variation.

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List of possible flow condition changes
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Flow continuously undevelops
diverging thermally undeveloping flow
converging thermally undeveloping flow
converging to thermally undeveloped flow with
fixed lag

• For flow to be thermally developed
• RTD 0 must be satisfied.
• Flow must be thermally developing

• For thermally-developed flow, RTD gt 1 must be
  satisfied before RTD 1 is satisfied.
• For flow to be thermally undeveloped, RTD lt 1
  must be satisfied before RTD 1 is satisfied.

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• Step change from one H1 to another H1 boundary
  condition
• For thermally developed flow to develop again, it
  must first undevelop.

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HG type boundary condition general thermal
entry length

• For fluctuating BC, flow may never get
  opportunity to develop
• Thermal entry length may be undefined for
  certain HG type BCs

Hydrodynamic undevelopment of internal flow

• Undevelopment applicable when there is sudden
  change in slip at boundary.
• Hydrodynamic undevelopment can occur for no-slip
  BC at wall Couette flow problem.

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Conclusions

• For thermally developing flow, RTDgt1.
• When RTDlt1,thermally undeveloping flow occurs.
• For flow to be thermally developed, RTD gt 1
  must be satisfied before satisfying RTD 1.
• For flow to be thermally undeveloped, RTD lt 1
  must be satisfied before satisfying RTD 1.
• Unusual observations in convection due to certain
  varying BCs can be explained by thermal
  undevelopment of flow
• Lth exists only for certain standard thermal BCs.
• length of flow from thermally undeveloped
  thermally developed