Forced Convection Questions and Answers

Answer: A

The group is the Prandtl number.

\[ Pr = \frac{c_{p}\,\mu}{k} \]

It combines specific heat \(c_{p}\), dynamic viscosity \(\mu\) and thermal conductivity \(k\), and compares how fast momentum diffuses with how fast heat diffuses in the fluid.

The Reynolds number \(\rho V L/\mu\) contains viscosity but no thermal property. The Froude number \(V/\sqrt{gL}\) compares inertia with gravity. The Biot number \(hL/k\) contains the thermal conductivity of a solid and a convective coefficient, but no specific heat or viscosity.

Answer: C

The Nusselt number is a non-dimensional temperature gradient.

At a wall, the convective flux equals the conduction flux through the fluid in contact with it, \(h(T_{s} - T_{\infty}) = -k_{f}\,\partial T/\partial y\) at the wall. Writing \(\theta = (T - T_{s})/(T_{\infty} - T_{s})\) and \(y^{*} = y/L\),

\[ \begin{aligned} Nu &= \frac{hL}{k_{f}} \\ &= \frac{\partial \theta}{\partial y^{*}}\Big|_{y^{*} = 0} \end{aligned} \]

so Nu is the dimensionless temperature gradient at the surface.

The other descriptions are other groups. Internal conduction resistance of a solid over the external convective resistance is the Biot number, \(hL/k_{s}\), which uses the solid's conductivity. Advection over conduction is the Peclet number, \(Re \times Pr\). A non-dimensional velocity gradient at the wall relates to the skin friction coefficient, not to the Nusselt number.

Answer: A

The Nusselt number depends on neither the Reynolds number nor the Prandtl number.

For laminar flow in a circular tube with both the velocity and temperature profiles fully developed, the energy equation gives a constant Nusselt number. For a tube held at constant wall temperature, as here (an isothermal tube), \(Nu = hD/k \approx 3.66\); for constant wall heat flux it is about 4.36. Neither value contains Re or Pr.

Re and Pr enter only in the thermal entry region, through the Graetz number, or in turbulent flow, where correlations such as Dittus-Boelter give \(Nu = 0.023\,Re^{0.8}Pr^{n}\). Those cases are what the options 'both', 'Reynolds only' and 'Prandtl only' describe, and none applies once the flow is laminar and fully developed.

Answer: A

The ratio \(Nu_{2}/Nu_{1}\) is \(\sqrt{2}\).

For laminar flow over a flat plate, both the local and the average Nusselt numbers follow \(Nu \propto Re^{1/2}Pr^{1/3}\); for example the average value is \(\overline{Nu}_{L} = 0.664\,Re_{L}^{1/2}Pr^{1/3}\). With the same fluid and plate, doubling the freestream velocity doubles the Reynolds number:

\[ \begin{aligned} \frac{Nu_{2}}{Nu_{1}} &= \left(\frac{Re_{2}}{Re_{1}}\right)^{1/2} \\ &= \sqrt{2} \end{aligned} \]

2 would require Nu to vary linearly with Re. 1.26 is \(2^{1/3}\), the effect of doubling the Prandtl number rather than the velocity, and 1 would mean the heat transfer did not depend on velocity at all.

Answer: D

The coefficient increases if a denser fluid is used.

For laminar flow over a flat plate the local Nusselt number is \(Nu_{x} = 0.332\,Re_{x}^{1/2}Pr^{1/3}\), so \(h \propto \frac{k}{x}\left(\frac{\rho u x}{\mu}\right)^{1/2}\). A denser fluid raises the Reynolds number, thins the boundary layer and so raises h.

The same relation rules out the others. h rises, not falls, with free-stream velocity. It falls with distance, as \(x^{-1/2}\), because the boundary layer thickens. A more viscous fluid lowers the Reynolds number and so lowers h.

Answer: B

The thermal boundary layer grows in proportion to \(x^{1/2}\).

For laminar flow over a flat plate the velocity boundary layer thickness is \(\delta \approx 5x/\sqrt{Re_{x}}\), and since \(Re_{x} = ux/\nu\),

\[ \delta \propto \frac{x}{\sqrt{x}} = x^{1/2} \]

The thermal boundary layer is \(\delta_{t} \approx \delta\,Pr^{-1/3}\), a constant multiple of \(\delta\), so it too grows as \(x^{1/2}\). Linear growth would apply to neither, and the powers 1/3 and 1/4 do not describe laminar growth on a flat plate.