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:

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:

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