Chapter 7: External Flow

Source files used: Heat Transfer (23), (24), (25), (26), and combined-mode use in (35); textbook Chapter 7 as support. No worked examples are included.

1. Big Picture

Chapter 7 is about forced convection over external surfaces. The fluid flows over the outside of an object.

Main geometries in Biddle’s lectures:

• flat plates,
• circular cylinders,
• non-circular cylinders.

The goal is still the same:

$$\boxed{\text{Find }h\text{, then use }\dot{q}=hA\Delta T.}$$

The most important first step is usually the Reynolds number. For flat plates, Reynolds number tells where the boundary layer is laminar, turbulent, or mixed. For cylinders, Reynolds number determines which empirical constants are used.

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2. Core Ideas

2.1 External Flow over a Flat Plate

A fluid flows over a plate with free-stream velocity $U_\infty$. Because of the no-slip condition, the fluid at the wall has zero velocity, and a boundary layer grows along the plate.

The flow usually starts laminar at the leading edge. If the plate is long enough or velocity is high enough, the boundary layer transitions to turbulent.

Biddle’s default rule:

Unless the problem says turbulent from the leading edge, assume the flow starts laminar.

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2.2 Local vs Average Heat Transfer

For a flat plate:

• local $h_x$: heat transfer coefficient at a specific location $x$,
• average $\bar h_L$: average over length $L$.

Use local equations for local heat flux. Use average equations for total heat transfer over the plate.

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2.3 Critical Reynolds Number

For a flat plate, transition is commonly taken at:

$$Re_{x,c}=5\times10^5$$

If $Re_L<5\times10^5$, flow is laminar over the whole plate.

If $Re_L>5\times10^5$, flow is mixed: laminar from the leading edge to $x_c$, turbulent after $x_c$.

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2.4 Flow over Cylinders

For cylinders, the boundary layer wraps around the object and usually separates, forming a wake. The physics is difficult, so Biddle uses empirical correlations.

The main cylinder correlation in his lectures is the Hilpert equation:

$$\overline{Nu}_D=CRe_D^mPr^{1/3}$$

where $C$ and $m$ come from a table based on Reynolds number range.

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2.5 Property Evaluation

For external forced convection, properties are commonly evaluated at the film temperature:

$$T_f=\frac{T_s+T_\infty}{2}$$

Use fluid properties at $T_f$, unless the problem states otherwise.

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3. Main Governing Equations and Formulas

3.1 Reynolds Number for Flat Plate

$$Re_x=\frac{U_\infty x}{\nu}$$

At the end of the plate:

$$Re_L=\frac{U_\infty L}{\nu}$$

Use $Re_L$ to decide whether the whole plate is laminar or whether transition occurs.

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3.2 Critical Transition Location

If transition occurs at $Re_{x,c}=5\times10^5$:

$$x_c=\frac{Re_{x,c}\nu}{U_\infty}$$

Use when $Re_L>Re_{x,c}$.

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3.3 Laminar Flat Plate: Local Nusselt Number

$$Nu_x=0.332Re_x^{1/2}Pr^{1/3}$$ $$h_x=\frac{Nu_x k_f}{x}$$

Use for local heat transfer at location $x$ in laminar flow.

Typical restrictions: laminar boundary layer, moderate Prandtl number, constant surface temperature.

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3.4 Laminar Flat Plate: Average Nusselt Number

$$\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}$$ $$\bar h_L=\frac{\overline{Nu}_L k_f}{L}$$

Use for total heat transfer over a laminar plate.

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3.5 Turbulent Flat Plate from Leading Edge: Local Nusselt Number

$$Nu_x=0.0296Re_x^{4/5}Pr^{1/3}$$

Use when the boundary layer is turbulent from the leading edge or when the problem states that it is turbulent from the start.

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3.6 Turbulent Flat Plate from Leading Edge: Average Nusselt Number

$$\overline{Nu}_L=0.037Re_L^{4/5}Pr^{1/3}$$

Use for average heat transfer if the whole plate is turbulent from the leading edge.

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3.7 Mixed Laminar-Turbulent Flat Plate

For transition from laminar to turbulent:

$$\overline{Nu}_L=\left(0.037Re_L^{4/5}-871\right)Pr^{1/3}$$

Use when the boundary layer starts laminar and transitions at $Re_{x,c}=5\times10^5$.

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3.8 Heat Transfer from Plate

For one side of a plate:

$$\dot{q}=\bar h_L A(T_s-T_\infty)$$

where:

$$A=WL$$

If both sides are exposed, include both sides if appropriate.

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3.9 Reynolds Number for Cylinder

$$Re_D=\frac{VD}{\nu}$$

where $D$ is cylinder diameter.

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3.10 Hilpert Correlation for Circular Cylinder

$$\overline{Nu}_D=CRe_D^mPr^{1/3}$$ $$\bar h=\frac{\overline{Nu}_D k_f}{D}$$

Use for forced convection over a cylinder in crossflow.

Typical constants:

$Re_D$ range	$C$	$m$
0.4–4	0.989	0.330
4–40	0.911	0.385
40–4000	0.683	0.466
4000–40,000	0.193	0.618
40,000–400,000	0.027	0.805

Use the row corresponding to your calculated Reynolds number.

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3.11 Heat Transfer from Cylinder

For total heat transfer from a cylinder:

$$\dot{q}=\bar h A_s(T_s-T_\infty)$$

where side area is:

$$A_s=\pi DL$$

For heat transfer per unit length:

$$q'=\bar h\pi D(T_s-T_\infty)$$

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3.12 Non-Circular Cylinders

Biddle notes that non-circular cylinders use a correlation of similar form:

$$\overline{Nu}_D=CRe_D^mPr^{1/3}$$

but constants depend on geometry, such as square or hexagonal cross-sections. Use the textbook table for $C$ and $m$.

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3.13 Sphere Correlation, Textbook Support

If a sphere appears and the course allows textbook correlations, use a sphere-specific correlation such as:

$$\overline{Nu}_D=2+\left(0.4Re_D^{1/2}+0.06Re_D^{2/3}\right)Pr^{0.4}\left(\frac{\mu}{\mu_s}\right)^{1/4}$$

Use only when the problem is explicitly a sphere problem and the correlation validity range is satisfied.

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4. Problem-Solving Workflow

4.1 Flat Plate Workflow

1. Evaluate properties at film temperature.
2. Compute $Re_L=U_\infty L/\nu$.
3. Compare with $5\times10^5$.
4. If $Re_L<5\times10^5$, use laminar average correlation.
5. If $Re_L>5\times10^5$, assume mixed flow unless stated turbulent from leading edge.
6. If mixed, compute $x_c$ if needed.
7. Compute $\overline{Nu}_L$.
8. Compute $\bar h=\overline{Nu}_L k/L$.
9. Compute heat rate $\dot{q}=\bar hA\Delta T$.

4.2 Cylinder Workflow

1. Evaluate properties at film temperature.
2. Compute $Re_D=VD/\nu$.
3. Use Reynolds number range to get $C$ and $m$.
4. Compute $\overline{Nu}_D=CRe_D^mPr^{1/3}$.
5. Compute $\bar h=\overline{Nu}_Dk/D$.
6. Compute $\dot{q}$ or $q'$.

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5. Decision Rules / Decision Trees

5.1 Flat Plate Regime

Compute $Re_L=U_\infty L/\nu$.

$Re_L<5\times10^5$? → laminar over entire plate

$Re_L>5\times10^5$ and problem does not say turbulent from leading edge? → mixed laminar + turbulent

Problem says turbulent from leading edge? → turbulent-from-leading-edge correlation

5.2 Local vs Average for Plate

Asked $q''$ at $x$? → use local $Nu_x$ and $h_x$

Asked total $\dot{q}$ over plate? → use average $\overline{Nu}_L$ and $\bar{h}_L$

5.3 Cylinder Correlation

Flow over circular cylinder? → Hilpert correlation with $C,m$ from $Re_D$ table

Flow over square/hexagonal/noncircular cylinder? → same form, geometry-specific table

Flow over sphere? → use sphere-specific textbook correlation, not cylinder Hilpert table

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6. Important Tables / Correlations Needed

6.1 Flat Plate Summary

Case	Correlation
Laminar local	$Nu_x=0.332Re_x^{1/2}Pr^{1/3}$
Laminar average	$\overline{Nu}_L=0.664Re_L^{1/2}Pr^{1/3}$
Turbulent local	$Nu_x=0.0296Re_x^{4/5}Pr^{1/3}$
Turbulent average	$\overline{Nu}_L=0.037Re_L^{4/5}Pr^{1/3}$
Mixed average	$\overline{Nu}_L=(0.037Re_L^{4/5}-871)Pr^{1/3}$

6.2 Cylinder Hilpert Table

$Re_D$ range	$C$	$m$
0.4–4	0.989	0.330
4–40	0.911	0.385
40–4000	0.683	0.466
4000–40,000	0.193	0.618
40,000–400,000	0.027	0.805

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7. Key Takeaways

• Chapter 7 is forced convection over external surfaces.
• Find $h$ first, then use Newton’s law.
• Flat plate flow usually starts laminar unless stated otherwise.
• Critical Reynolds number for flat plates is usually $5\times10^5$.
• Use local correlations for local heat flux and average correlations for total heat rate.
• Cylinder flow uses empirical correlations because separation and wakes make analysis difficult.
• Use film temperature for property evaluation unless told otherwise.
• Do not use cylinder correlations for spheres or non-circular shapes unless the table/correlation is for that geometry.

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Derivation: From Continuity to h_x (Laminar Flat Plate)

Sources: Lecture 23 (t ≈ 05–37 min), Lecture 24, Chapter 7 Section 7.2.1, Eq. 7.23

Step 1 — Continuity

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$

Conservation of mass in the boundary layer. u and v are coupled — they cannot vary independently.

Step 2 — x-Momentum PDE

$u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = \nu \frac{\partial^2 u}{\partial y^2}$

Left side: momentum transport by the flow. Right side: viscous diffusion (shear stress term). This is the Navier-Stokes x-momentum equation simplified under boundary layer assumptions — zero pressure gradient, flat plate, so the ∂²u/∂x² and pressure terms are dropped.

Step 3 — Blasius Similarity Transformation

Introduce the similarity variable:

$\eta = y\sqrt{\frac{u_\infty}{\nu x}}$

Substituting into the momentum PDE collapses it from a PDE in (x, y) to a single nonlinear ODE in η (Ch. 7, Eq. 7.17):

$2\frac{d^3f}{d\eta^3} + f\frac{d^2f}{d\eta^2} = 0$

Solved numerically. Key result at the wall (η = 0):

$\left.\frac{d^2f}{d\eta^2}\right|_{\eta=0} = 0.332$

This value — 0.332 — is a numerical constant, not derived analytically.

Step 4 — Energy PDE

$u \frac{\partial T}{\partial x} + v \frac{\partial T}{\partial y} = \alpha \frac{\partial^2 T}{\partial y^2}$

Same mathematical form as the momentum equation, with T replacing u and α replacing ν. Using the same similarity variable, this reduces to an ODE. For Pr ≥ 0.6, the dimensionless wall temperature gradient is (Ch. 7, Eq. 7.23):

$\left.\frac{dT^*}{d\eta}\right|_{\eta=0} = 0.332 \, \Pr^{1/3}$

The Pr^(1/3) factor comes from the ratio of boundary layer thicknesses: δ/δ_t = Pr^(1/3).

Step 5 — Connecting Wall Gradient to h_x

Apply Fourier’s law at the wall and equate to Newton’s law of cooling:

$q_s'' = -k\left.\frac{\partial T}{\partial y}\right|_{y=0} = h_x(T_s - T_\infty)$

Substituting the result from Step 4:

$\boxed{h_x = 0.332 \, k \left(\frac{u_\infty}{\nu x}\right)^{1/2} \Pr^{1/3}}$

In dimensionless form (Ch. 7, Eq. 7.23):

$\text{Nu}_x = \frac{h_x x}{k} = 0.332 \, \text{Re}_x^{1/2} \, \Pr^{1/3} \quad (\Pr \geq 0.6, \; \text{Re}_x < 5 \times 10^5)$

Note: As x → 0, h_x → ∞ (boundary layer is thinnest at the leading edge, so most heat transfer occurs there). As x increases, the boundary layer thickens and h_x decreases. (Lecture 23, t ≈ 33 min)