Heat transfer — For free convection over a vertical flat plate, the Nusselt number Nu is related to Grashof number Gr. The dependence in turbulent and laminar flow, respectively, is:
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AGr^0.25, Gr
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BGr^0.25, Gr^0.33
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CGr, Gr^0.25
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DGr^0.33, Gr^0.25
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EGr^0.5, Gr^0.25
Answer
Correct Answer: Gr^0.33, Gr^0.25
Explanation
Introduction:In natural (free) convection, buoyancy drives flow, and the Nusselt number correlates heat-transfer coefficient to fluid properties and geometry via dimensionless groups. For a vertical plate, different exponents apply in laminar and turbulent regimes, reflecting changes in boundary-layer behavior.
Given Data / Assumptions:
- Vertical flat plate, uniform surface temperature.
- Property variations are modest; Prandtl number Pr is accounted for but here the focus is on Gr exponents.
- Correlation form Nu ∝ (Gr * Pr)^n with n depending on regime.
Concept / Approach:
Classical correlations give Nu_L ≈ C * (Gr_L * Pr)^1/4 for laminar and Nu_L ≈ C * (Gr_L * Pr)^1/3 for turbulent flow over a vertical surface. When emphasizing Gr only, the exponents are 0.25 (laminar) and 0.33 (turbulent). The question asks for 'turbulent & laminar flow respectively', hence the ordered pair is Gr^0.33 for turbulent and Gr^0.25 for laminar.
Step-by-Step Solution:
Recall laminar natural convection: Nu ∝ (Gr * Pr)^0.25.Recall turbulent natural convection: Nu ∝ (Gr * Pr)^0.33.Match order 'turbulent, then laminar': (0.33, 0.25).Select option with Gr^0.33, Gr^0.25.Verification / Alternative check:
Heat-transfer handbooks list Churchill–Chu or similar correlations that reduce to these exponents in limiting regimes, supporting the stated powers.
Why Other Options Are Wrong:
A–C/E present incorrect order or exponents that do not match standard natural convection theory for vertical plates.
Common Pitfalls:
Mixing forced convection (Nu ∝ Re^m Pr^n) with natural convection correlations or confusing the order in which regimes are listed.
Final Answer:
Gr^0.33, Gr^0.25