Chapter 6: Introduction to Convection

Source files used: Heat Transfer (23), with textbook Chapter 6 as support. Biddle merges Chapter 6 material into the start of Chapter 7. No worked examples are included.

1. Big Picture

Chapter 6 is the bridge between simple Newton’s law of cooling and the detailed convection correlations in Chapters 7–9.

Chapter 1 gave:

$$\dot{q}=hA(T_s-T_\infty)$$

But it did not explain how to find $h$. Biddle’s point is that Chapters 6, 7, 8, and 9 are mainly about finding the convection coefficient.

Chapter 6 introduces the language:

• velocity boundary layer,
• thermal boundary layer,
• local and average convection coefficient,
• Reynolds number,
• Prandtl number,
• Nusselt number,
• empirical correlations.

Biddle does not treat Chapter 6 as a separate long block. He merges it with Chapter 7 when he begins external flow over a flat plate.

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2. Core Ideas

2.1 Convection Needs Fluid Mechanics First

Convection heat transfer depends on fluid motion. Therefore, before solving heat transfer, you must understand the flow situation.

Questions to ask:

• Is the flow external or internal?
• Is it forced or free?
• Is it laminar or turbulent?
• What is the characteristic length?
• What fluid properties are needed?

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2.2 No-Slip Condition

At a solid surface, the fluid velocity equals the surface velocity.

For a stationary wall:

$$u(y=0)=0$$

The fluid far from the wall moves at the free-stream velocity $U_\infty$. Therefore, near the wall, velocity must change from zero to nearly $U_\infty$. This creates a velocity boundary layer.

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2.3 Velocity Boundary Layer

The velocity boundary layer is the region near the surface where the fluid velocity changes from zero at the wall to nearly the free-stream value.

The edge is commonly defined where:

$$\frac{u}{U_\infty}=0.99$$

The boundary layer is important because it controls wall shear stress and strongly affects heat transfer.

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2.4 Thermal Boundary Layer

If the wall temperature differs from the fluid temperature, a thermal boundary layer develops.

At the wall:

$$T=T_s$$

Far from the wall:

$$T=T_\infty$$

The thermal boundary layer is the region where the temperature changes from $T_s$ to nearly $T_\infty$.

The temperature gradient at the wall controls heat flux:

$$q_s''=-k_f\left.\frac{\partial T}{\partial y}\right|_{y=0}$$

This must match Newton’s law:

$$q_s''=h(T_s-T_\infty)$$

So $h$ is connected to the wall temperature gradient in the fluid.

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2.5 Local vs Average $h$

The local convection coefficient $h_x$ applies at one location.

$$q_x''=h_x(T_s-T_\infty)$$

The average convection coefficient applies over a finite surface:

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

For a plate of length $L$:

$$\bar h_L=\frac{1}{L}\int_0^L h_x\,dx$$

Use local $h_x$ for local heat flux. Use average $\bar h$ for total heat transfer over a surface.

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2.6 Dimensionless Groups

Convection correlations are usually written using dimensionless numbers.

Biddle’s practical view: these groups package complicated physics into compact experimental correlations.

The most important ones are:

• Reynolds number $Re$,
• Prandtl number $Pr$,
• Nusselt number $Nu$,
• Grashof number $Gr$,
• Rayleigh number $Ra$.

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3. Main Governing Equations and Formulas

3.1 Newton’s Law of Cooling

$$q''=h(T_s-T_\infty)$$ $$\dot{q}=hA(T_s-T_\infty)$$

Use after $h$ is known.

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3.2 Nusselt Number

$$Nu_L=\frac{hL}{k_f}$$

or

$$Nu_D=\frac{hD}{k_f}$$

where:

• $h$: convection coefficient,
• $L$ or $D$: characteristic length,
• $k_f$: fluid thermal conductivity.

Use to calculate $h$:

$$h=\frac{Nu\,k_f}{L_c}$$

Nusselt number is dimensionless. It represents convection heat transfer relative to pure conduction through the fluid layer.

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3.3 Reynolds Number

$$Re_L=\frac{\rho V L}{\mu}=\frac{VL}{\nu}$$

where:

• $V$: characteristic velocity,
• $L$: characteristic length,
• $\mu$: dynamic viscosity,
• $\nu=\mu/\rho$: kinematic viscosity.

Use to determine laminar/turbulent behavior in forced convection.

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3.4 Prandtl Number

$$Pr=\frac{\nu}{\alpha}=\frac{\mu c_p}{k_f}$$

Prandtl number compares momentum diffusivity to thermal diffusivity.

Physical meaning:

• Small $Pr$: heat diffuses faster than momentum.
• Large $Pr$: momentum diffuses faster than heat.

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3.5 Grashof Number

$$Gr_L=\frac{g\beta(T_s-T_\infty)L^3}{\nu^2}$$

Use in free convection. It plays a role similar to Reynolds number but for buoyancy-driven flow.

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3.6 Rayleigh Number

$$Ra_L=Gr_LPr$$

Use in free convection correlations.

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3.7 Film Temperature

For many external-flow property evaluations:

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

Use properties at $T_f$ unless the problem or correlation specifies otherwise.

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3.8 Boundary Layer Heat Flux Relation

At the wall:

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

This explains what $h$ physically represents: a compact way to express the wall temperature gradient in the fluid.

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4. Problem-Solving Workflow

1. Identify whether the convection is forced or free.
2. Identify whether the flow is external or internal.
3. Pick the correct characteristic length.
4. Evaluate fluid properties at the correct reference temperature.
5. Compute the required dimensionless group: usually $Re$ for forced convection, $Ra$ for free convection.
6. Decide laminar/turbulent or correct geometry category.
7. Select the correct Nusselt correlation.
8. Compute $h=Nu\,k/L_c$.
9. Use Newton’s law of cooling to get heat rate or heat flux.

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5. Decision Rules / Decision Trees

5.1 External, Internal, or Free Convection?

Fluid flows over outside of plate/cylinder/sphere? → external flow, Chapter 7

Fluid flows inside pipe/tube/duct? → internal flow, Chapter 8

Fluid motion caused by buoyancy, no fan/pump? → free/natural convection, Chapter 9

5.2 Local vs Average

Asked for heat flux at a specific x-location? → local $h_x$ and local $Nu_x$

Asked for total heat transfer over a surface? → average $\bar{h}$ and average $\overline{Nu}$

5.3 Property Temperature

External forced flow or free convection with $T_s$ and $T_\infty$? → usually use film temperature $T_f=(T_s+T_\infty)/2$

Internal flow with inlet and outlet bulk temperatures? → usually use mean bulk average temperature

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6. Important Tables / Correlations Needed

Chapter 6 itself mainly defines dimensionless groups. Specific Nusselt correlations are in Chapters 7, 8, and 9.

6.1 Core Dimensionless Groups

Group	Formula	Main Use
$Nu$	$hL_c/k_f$	converts correlation result into $h$
$Re$	$VL_c/\nu$	forced convection regime
$Pr$	$\nu/\alpha$	fluid property ratio
$Gr$	$g\beta\Delta T L^3/\nu^2$	free convection buoyancy/inertia ratio
$Ra$	$GrPr$	free convection correlations

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7. Key Takeaways

• Chapter 6 is the language chapter for convection.
• Biddle merges it into Chapter 7 rather than treating it as a separate long unit.
• Convection problems are mostly about finding $h$.
• $h$ depends on flow physics, not just material properties.
• Boundary layers explain why convection happens.
• Local $h_x$ gives local heat flux; average $\bar h$ gives total heat transfer.
• Nusselt number correlations are the main bridge from fluid mechanics to heat transfer.
• Always compute $h$ before using $\dot{q}=hA\Delta T$.