Heat Transfer (03)_ Energy balance problems, thermal conductivity, thermal diffusivity

Energy Balance Equations

  • Types of Energy Balance Equations

    • Control volume energy balance

    • Surface energy balance

  • Homework Reminder: Homework due Wednesday, assignments from Chapter 1 include three problems.

Problem Corrections

  • Problem 123:

    • Change power input from 250 watts to 250 horsepower (hp).

  • Problem 130:

    • Correct to: Diameter of the cylinder is 0.5 meters; Length is 1 meter.

    • Include all sides and two ends of the cylinder when solving the problem.

  • Problem 176:

    • Material changed to fireclay brick.

    • K-value can be found in the appendix of the textbook.

Online Resources

  • Lecture Videos: Available on the ME department’s website or YouTube for review.

    • Audio and video quality may vary; retaping for improvement.

    • Useful for studying midterms and finals to refresh on lectures.

Chapter 1 Summary

  • Heating Air in a Duct: Problem involves heating air flowing through a rectangular duct with a heating element.

    • Heat Flux Definitions: q₀'' (heat flux) is important in this context.

    • Insulated Heater: The heater is assumed perfectly insulated to prevent heat loss.

    • Dimensions Given: Duct wall thickness = 10 mm, k-value = 20, h (convection coefficient), Ti (inlet temperature) for air, T₀ (temperature at heater location).

Energy Balance Concepts

  • Surface Energy Balance for Upper Surface:

    • Heat conduction: ( q_{double prime} = \frac{K (T_0 - T_i)}{L} )

    • Heat convection: ( q_{convection} = H (T_{surface} - T_{fluid}) )

    • Energy balance states: ( q_{conduction} = q_{convection} )

  • Surface Energy Balance for Lower Surface:

    • Heat flux from the heater (q₀'') goes up into the wall through conduction.

  • Total Wall Energy Balance:

    • Energy comes in from the heater (q₀'') and goes out by convection.

    • Steady-state condition implies: ( q₀'' = q_{convection} )

Key Equations

  • Surface Energy Balance Equations:

    • Upper surface: ( q_{upper} = \frac{K (T_0 - T_i)}{L} )

    • Lower surface: ( q_{lower} = q_{0}'' = q_{conduction} )

    • Total wall balance: ( q_{0}'' = h(T_0 - T_\infty) )

Chapter 2 Preview: Introduction to Conduction

  • Conductive Heat Transfer: Described using Fourier’s Law.

  • Conduction Directions: Focus on one-dimensional (1D) heat conduction initially, and two-dimensional (2D) later.

  • Thermal Conductivity (K): A material property found in the appendix; examples include:

    • Metals typically have high K values (e.g., Copper = 400, Aluminum = 24).

    • Insulations typically have low K values (e.g., Air = 0.025).

Modes of Energy Transport

  • Solids:

    • Energy transported by lattice vibrations and free electron migration.

  • Liquids and Gases:

    • Thermal energy transport by kinetic energy collisions among molecules, with gases being less efficient.

Insulative Properties of Air

  • Double Pane Windows:

    • Designed to minimize air movement to reduce convection losses.

  • Air acts as an effective insulator when motion is minimized.

Thermal Properties in Insulation

  • Case Studies of Insulation:

    • Importance of trapping air in materials (e.g., Fiberglass, Down Feathers) for insulation.

  • Thermal Diffusivity (( \alpha )): Defines how quickly heat moves through a material:

    • ( \alpha = \frac{K}{\rho C_p} ) where K is thermal conductivity, ( \rho ) is density, and ( C_p ) is specific heat.

  • High thermal diffusivity implies good conduction but poor storage, and vice versa.

Concluding Thoughts

  • Reminder: Bringing concepts of energy balance into practical applications, such as HVAC duct design.

  • Homework is due Wednesday, and students are encouraged to engage in class and utilize resources for their understanding.