Summary of Heat Transfer 18 | Convection Heat Transfer (IV) | Mechanical Engineering | GATE Crash Course

Summary of "Heat Transfer 18 | Convection Heat Transfer (IV) | Mechanical Engineering | GATE Crash Course"

This video lecture focuses on internal forced convection in fluid flow, particularly inside pipes and ducts, as part of a Mechanical Engineering course aimed at GATE exam preparation. The instructor revisits concepts from previous sessions on external forced convection and then elaborates on internal flow characteristics, hydrodynamic and thermal development, velocity and temperature profiles, and related dimensionless numbers.

Main Ideas, Concepts, and Lessons

1. Introduction to Internal Forced Convection

Internal flow refers to fluid flow inside a closed conduit such as pipes or ducts.
Contrast with external flow, where fluid flows over a surface.
Understanding internal flow requires revisiting fluid mechanics fundamentals, especially pipe flow.

2. Internal Flow Characteristics

Flow occurs inside pipes or ducts with a defined radius \( R \).
The coordinate system involves:
- Axial direction (flow direction, \( x \))
- Radial direction (perpendicular to flow, \( r \))
Boundary layer forms near the walls due to viscous effects.
Boundary layer thickness grows along the flow direction until it merges at a certain length.

3. Hydrodynamic Entrance Length and Flow Regions

Entrance Length (\( L_e \)): The length over which the velocity boundary layer develops.
Developing Region: Flow velocity profile changes with both \( r \) and \( x \).
Fully Developed Region: Velocity profile becomes independent of \( x \) and depends only on \( r \).
In fully developed flow:
- Velocity \( u = f(r) \)
- \( \frac{\partial u}{\partial x} = 0 \)
In developing flow:
- Velocity \( u = f(r, x) \)
- \( \frac{\partial u}{\partial x} \neq 0 \)

4. Laminar and Turbulent Flow in Pipes

Flow can be laminar or turbulent depending on Reynolds number \( Re \).
Critical Reynolds number for transition ~ 2300.
Reynolds number definition for internal flow involves Hydraulic diameter \( D_h \) and average velocity \( u_{avg} \).
Hydraulic diameter for non-circular ducts is defined as:
\( D_h = \frac{4 \times \text{Flow Area}}{\text{Wetted Perimeter}} \)
For circular pipes, Hydraulic diameter equals the pipe diameter.

5. Calculating Reynolds number for Internal Flow

Reynolds number:
\( Re = \frac{\rho u_{avg} D_h}{\mu} \)
If mass flow rate \( \dot{m} \) is given instead of velocity:
\( u_{avg} = \frac{4 \dot{m}}{\pi \rho D_h^2} \)
This allows Reynolds number calculation from mass flow rate.

6. Velocity Profiles and Average Velocity

Velocity varies radially due to viscous effects (no-slip condition at wall, max velocity at center).
Average velocity is a hypothetical uniform velocity that would give the same volumetric flow as the actual velocity profile.
Average velocity \( u_{avg} \) is calculated by integrating velocity over the cross-section:
\( u_{avg} = \frac{2}{R^2} \int_0^R u(r) r \, dr \)
For laminar flow, velocity profile is parabolic:
\( u(r) = u_{max} \left(1 - \frac{r^2}{R^2}\right) \)
Maximum velocity is twice the average velocity in laminar flow.

7. Thermal Entrance Length and Thermally Developed Flow

Similar to velocity development, temperature profile develops along the flow.
Thermal entrance length \( L_{t} \) is the length over which the temperature boundary layer develops.
Temperature \( T \) is a function of both \( r \) and \( x \) in the developing region.
Thermally fully developed flow means temperature profile does not change with \( x \), but velocity profile is already fully developed.
Two common thermal boundary conditions:
- Constant wall temperature
- Constant heat flux

8. Mixing Cup (Bulk) Temperature Concept

Mixing Cup Temperature is a hypothetical uniform temperature used for simplifying calculations of enthalpy changes.
It is the temperature that would result if the fluid at varying temperatures mixed uniformly.
Used to relate actual temperature profiles to average thermal properties.

9. Enthalpy Rate and Temperature Profiles

Enthalpy rate is related to mass flow rate and temperature difference.
For small fluid elements, enthalpy rate can be integrated over the cross-section to get