Chapter 8: Internal Flow

Source files used: Heat Transfer (27), (28), (29), (30), and enhancement/hardware comments in (36); textbook Chapter 8 as support. No worked examples are included.

1. Big Picture

Chapter 8 is about forced convection inside tubes, pipes, and ducts.

Chapter 7 asked: what happens when fluid flows over the outside of a surface? Chapter 8 asks: what happens when fluid flows inside a tube?

The engineering goal is still:

$$\boxed{\text{Find }h\text{, then find }\dot{q}\text{ and/or outlet temperature.}}$$

But internal flow adds new ideas:

• mean velocity,
• mean/bulk temperature,
• hydrodynamic entrance length,
• thermal entrance length,
• constant surface temperature vs constant heat flux,
• energy balance along the tube.

Biddle gives a decision tree for Chapter 8. The first branch is laminar vs turbulent using Reynolds number.

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2. Core Ideas

2.1 Mean Velocity

In a tube, velocity varies over the cross-section. Instead of using a single free-stream velocity, internal flow uses mean velocity $u_m$.

$$u_m=\frac{\dot{m}}{\rho A_c}$$

where $A_c$ is tube cross-sectional area.

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2.2 Mean Fluid Temperature

Internal flow also uses mean or bulk temperature $T_m$. This represents the energy-average temperature of the fluid across the cross-section.

Unlike external flow, the fluid temperature changes along the tube:

$$T_m=T_m(x)$$

The wall may have:

• constant surface temperature, or
• constant surface heat flux.

The solution form depends strongly on which condition is given.

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2.3 Hydrodynamic Entrance Region

At the tube inlet, the velocity profile may be nearly uniform. Boundary layers grow from the wall until they meet at the centerline. After that, the velocity profile is fully developed.

The distance required is the hydrodynamic entrance length.

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2.4 Thermal Entrance Region

If the wall temperature differs from the inlet fluid temperature, a thermal boundary layer also develops. The distance required for the temperature profile shape to become fully developed is the thermal entrance length.

Hydrodynamic and thermal fully developed regions are not necessarily reached at the same axial location.

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2.5 Laminar vs Turbulent Internal Flow

Biddle’s internal-flow transition criterion:

$$Re_D<2300 \quad \Rightarrow \quad \text{laminar}$$ $$Re_D>2300 \quad \Rightarrow \quad \text{turbulent, for this course decision tree}$$

Real transition can be more gradual, but Biddle uses 2300 as the practical dividing value.

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2.6 Constant Surface Temperature vs Constant Heat Flux

Internal laminar correlations depend on wall boundary condition:

• constant surface temperature: $T_s=\text{constant}$,
• constant surface heat flux: $q_s''=\text{constant}$.

For turbulent flow, Biddle simplifies the decision tree: use the turbulent correlation regardless of constant temperature or constant heat flux for the problems considered.

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3. Main Governing Equations and Formulas

3.1 Tube Cross-Section and Surface Area

For a circular tube:

$$A_c=\frac{\pi D^2}{4}$$ $$A_s=\pi DL$$

Perimeter:

$$P=\pi D$$

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3.2 Mean Velocity

$$u_m=\frac{\dot{m}}{\rho A_c}$$

For circular tubes:

$$u_m=\frac{4\dot{m}}{\rho\pi D^2}$$

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3.3 Reynolds Number in a Tube

$$Re_D=\frac{\rho u_mD}{\mu}=\frac{u_mD}{\nu}$$

Using mass flow rate:

$$Re_D=\frac{4\dot{m}}{\pi D\mu}$$

Use to decide laminar or turbulent.

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3.4 Hydraulic Diameter

For non-circular ducts:

$$D_h=\frac{4A_c}{P}$$

Use $D_h$ in Reynolds and Nusselt numbers when the duct is not circular.

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3.5 Hydrodynamic Entrance Length

Laminar approximation:

$$\frac{x_{fd,h}}{D}\approx0.05Re_D$$

Turbulent estimate:

$$\frac{x_{fd,h}}{D}\approx10$$

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3.6 Thermal Entrance Length

Laminar approximation:

$$\frac{x_{fd,t}}{D}\approx0.05Re_DPr$$

Turbulent estimate:

$$\frac{x_{fd,t}}{D}\approx10$$

Use entrance length estimates to decide whether fully developed correlations are appropriate.

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3.7 Energy Balance for Internal Flow

$$\dot{q}=\dot{m} c_p(T_{m,o}-T_{m,i})$$

Use for heating of the fluid.

For cooling, sign may reverse, but magnitude is:

$$|\dot{q}|=\dot{m} c_p|T_{m,o}-T_{m,i}|$$

This is often the cleanest equation in tube problems.

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3.8 Constant Surface Heat Flux

If $q_s''$ is constant:

$$\dot{q}=q_s''A_s=q_s''\pi DL$$

Mean temperature variation:

$$T_m(x)=T_{m,i}+\frac{q_s''Px}{\dot{m} c_p}$$

For a circular tube:

$$T_m(x)=T_{m,i}+\frac{q_s''\pi D x}{\dot{m} c_p}$$

Once $h$ is known:

$$T_s-T_m=\frac{q_s''}{h}$$

for fully developed conditions with constant $h$.

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3.9 Constant Surface Temperature

If $T_s$ is constant:

$$\frac{T_s-T_{m,o}}{T_s-T_{m,i}}=\exp\left(-\frac{\bar h A_s}{\dot{m} c_p}\right)$$

where:

$$A_s=\pi DL$$

Heat rate:

$$\dot{q}=\dot{m} c_p(T_{m,o}-T_{m,i})$$

or

$$\dot{q}=\bar hA_s\Delta T_{\text{lm}}$$

where:

$$\Delta T_{\text{lm}}=\frac{(T_s-T_{m,i})-(T_s-T_{m,o})}{\ln\left[(T_s-T_{m,i})/(T_s-T_{m,o})\right]}$$

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3.10 Laminar, Fully Developed, Constant Surface Temperature

$$Nu_D=3.66$$

Use for laminar, fully developed internal flow in a circular tube with constant wall temperature.

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3.11 Laminar, Fully Developed, Constant Heat Flux

$$Nu_D=4.36$$

Use for laminar, fully developed internal flow in a circular tube with constant wall heat flux.

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3.12 Laminar Developing Flow, Constant Surface Temperature

A common textbook correlation is:

$$\overline{Nu}_D=3.66+\frac{0.0668(D/L)Re_DPr}{1+0.04[(D/L)Re_DPr]^{2/3}}$$

Use when laminar flow is thermally developing and the wall temperature is constant.

Another common entrance-region form is:

$$\overline{Nu}_D=1.86\left(\frac{Re_DPrD}{L}\right)^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}$$

Use only if the problem/correlation conditions match.

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3.13 Turbulent Internal Flow — Dittus–Boelter

$$Nu_D=0.023Re_D^{0.8}Pr^n$$

where:

$$n=0.4 \quad \text{for heating the fluid}$$ $$n=0.3 \quad \text{for cooling the fluid}$$

Use for turbulent internal flow in smooth circular tubes within the usual validity range.

Then:

$$h=\frac{Nu_Dk_f}{D}$$

Biddle’s simplified decision tree for turbulent internal flow: use the turbulent correlation regardless of whether the wall is constant heat flux or constant temperature for the problems considered.

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3.14 Overall Heat Transfer Coefficient for Tubes

For combined inside convection, tube-wall conduction, and outside convection:

$$\frac{1}{UA}=R_{\text{total}}$$

For a tube:

$$\frac{1}{U_iA_i}=\frac{1}{h_iA_i}+\frac{\ln(r_o/r_i)}{2\pi kL}+\frac{1}{h_oA_o}$$

where:

$$A_i=2\pi r_iL,\qquad A_o=2\pi r_oL$$

Use when heat must pass from one fluid through a tube wall to another fluid.

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4. Problem-Solving Workflow

4.1 Finding $h$ in Tube Flow

1. Identify the fluid and tube geometry.
2. Determine property temperature.
   • If inlet and outlet mean temperatures are known, use their average.
   • If outlet is unknown, estimate or iterate if needed.
3. Compute $Re_D$.
4. Decide laminar or turbulent.
5. If laminar, decide fully developed or developing and constant $T_s$ or constant $q_s''$.
6. If turbulent, use Dittus–Boelter or specified turbulent correlation.
7. Compute $Nu_D$.
8. Compute $h=Nu_Dk/D$.

4.2 Solving Tube Heating/Cooling Problems

1. Find or compute $h$.
2. Identify wall condition:
   • constant heat flux,
   • constant surface temperature,
   • heat transfer through tube wall with external resistance.
3. Use energy balance:

$$\dot{q}=\dot{m} c_p(T_{o}-T_i)$$

1. Use the appropriate heat-transfer relation:
   • constant heat flux: $\dot{q}=q_s''\pi DL$,
   • constant surface temperature: exponential/LMTD relation,
   • tube wall: $\dot{q}=UA\Delta T$ or exchanger relation.
2. Solve for requested unknown: outlet temperature, length, heat rate, or surface temperature.

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5. Decision Rules / Decision Trees

5.1 Main Internal Flow Decision

Compute $Re_D$.

$Re_D<2300$? → laminar

$Re_D>2300$? → turbulent for Biddle’s course decision tree

5.2 Laminar Decision

Laminar flow:

Hydrodynamically and thermally fully developed? → use $Nu=3.66$ for constant $T_s$ → use $Nu=4.36$ for constant $q_s''$

Developing? → use entrance-region correlation

5.3 Turbulent Decision

Turbulent flow: → use Dittus–Boelter / assigned turbulent correlation → do not separately branch on constant $T_s$ vs constant $q_s''$ for Biddle’s simplified problems

5.4 Constant Heat Flux vs Constant Surface Temperature

Problem gives $q_s''$? → constant heat flux route → $\dot{q}=q_s''\pi DL$ → $T_m$ changes linearly with $x$

Problem gives $T_s$ constant? → constant surface temperature route → exponential temperature relation or LMTD form

5.5 Property Evaluation

Internal water table has saturated-liquid and saturated-vapor columns? → use saturated-liquid properties for liquid water problems

Inlet and outlet temperatures known? → properties at average bulk temperature

Outlet unknown and property variation important? → estimate, solve, update properties if needed

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6. Important Tables / Correlations Needed

6.1 Internal Flow Nusselt Summary

Case	Correlation
Laminar, fully developed, constant $T_s$	$Nu_D=3.66$
Laminar, fully developed, constant $q_s''$	$Nu_D=4.36$
Laminar developing, constant $T_s$	$\overline{Nu}_D=3.66+\frac{0.0668(D/L)Re_DPr}{1+0.04[(D/L)Re_DPr]^{2/3}}$
Turbulent, smooth tube	$Nu_D=0.023Re_D^{0.8}Pr^n$

6.2 Entrance Lengths

Region	Laminar	Turbulent
Hydrodynamic	$x_{fd,h}/D\approx0.05Re_D$	$x_{fd,h}/D\approx10$
Thermal	$x_{fd,t}/D\approx0.05Re_DPr$	$x_{fd,t}/D\approx10$

6.3 Heat Transfer Enhancement

Biddle briefly mentions heat transfer enhancement in tubes:

• roughen the inside surface,
• insert coil/spring wire,
• increase turbulence,
• increase surface area.

These increase heat transfer, but often also increase pressure drop. The engineering tradeoff is better heat transfer versus pumping cost.

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7. Key Takeaways

• Chapter 8 is convection inside tubes and pipes.
• Start with Reynolds number.
• Use $Re_D=2300$ as the practical laminar/turbulent cutoff.
• Laminar internal flow has more decision branches than turbulent flow.
• $Nu=3.66$ is for laminar, fully developed, constant wall temperature.
• $Nu=4.36$ is for laminar, fully developed, constant wall heat flux.
• Dittus–Boelter is the main turbulent tube correlation.
• Constant heat flux gives linear mean temperature change.
• Constant wall temperature gives exponential mean temperature change.
• Energy balance $\dot{q}=\dot{m} c_p\Delta T_m$ is central.