Question 4 of 6: B2 — Free, forced, or mixed convection on a vertical plate

Question 4: B2 — Free, forced, or mixed convection on a vertical plate (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Steady laminar flow driven upward past a heated vertical plate ($T_s>T_\infty$), so buoyancy acts in the same direction as the forced flow (an assisting-flow configuration); fluid Prandtl number $Pr=0.2423$.

Find. The dimensionless grouping (and its magnitude ranges) that decides whether forced, free, or mixed convection controls the heat transfer.

Analysis. This is a scaling argument, so the reasoning is presented as flowing derivation rather than numbered steps. Inside the laminar boundary layer the streamwise momentum equation, with the Boussinesq buoyancy term for a heated vertical wall, reads

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\nu\frac{\partial^2 u}{\partial y^2}+g\beta\,(T-T_\infty),$$

where $x$ runs up the plate, $\beta$ is the thermal-expansion coefficient, and the last term is the buoyancy force per unit mass. Non-dimensionalising velocity by the forced-stream value $U$ and lengths by a plate length $L$, the inertia terms scale as $U^2/L$ while the buoyancy term scales as $g\beta\,\Delta T$ (with $\Delta T=T_s-T_\infty$). The ratio of buoyancy to inertia is therefore

$$\frac{\text{buoyancy}}{\text{inertia}}\sim\frac{g\beta\,\Delta T\,L}{U^2}=\frac{g\beta\,\Delta T\,L^3/\nu^2}{(UL/\nu)^2}=\frac{Gr}{Re^2},$$

where $Gr=g\beta\,\Delta T\,L^3/\nu^2$ is the Grashof number and $Re=UL/\nu$ the forced-flow Reynolds number. The single governing group is thus the Richardson number $Ri=Gr/Re^2$, and the regimes follow directly from its size:

$$\boxed{\ \frac{Gr}{Re^2}\ll1:\ \text{forced};\qquad \frac{Gr}{Re^2}\sim O(1):\ \text{mixed};\qquad \frac{Gr}{Re^2}\gg1:\ \text{free (natural)}.\ }$$

In practice the crossovers are placed at $Gr/Re^2\lesssim0.1$ (buoyancy negligible, pure forced convection), $0.1\lesssim Gr/Re^2\lesssim10$ (mixed, both effects retained), and $Gr/Re^2\gtrsim10$ (buoyancy dominant, pure free convection). Because the plate is hot and the flow is upward, buoyancy assists the forced stream, so mixed-convection Nusselt numbers are combined as $Nu^{n}=Nu_{\text{forced}}^{n}+Nu_{\text{free}}^{n}$ (aiding flow; $n\approx3$). The Prandtl number, here $Pr=0.2423<1$, does not change the governing group but enters the individual forced ($Nu_{\text{forced}}\propto Re^{1/2}Pr^{1/3}$) and free ($Nu_{\text{free}}\propto (Gr\,Pr)^{1/4}$) laws; with $Pr<1$ the thermal layer is thicker than the momentum layer, but comparing the two laws, $Nu_{\text{free}}/Nu_{\text{forced}}\sim(Gr/Re^2)^{1/4}Pr^{-1/12}$, and $Pr^{-1/12}=1.13$ for $Pr=0.2423$ (in the $Pr\to0$ limits, $Nu_{\text{free}}\propto(Gr\,Pr^2)^{1/4}$ and $Nu_{\text{forced}}\propto(Re\,Pr)^{1/2}$, and $Pr$ cancels altogether). So this low Prandtl number shifts the regime boundaries only slightly; $Gr/Re^2$ remains the criterion.