## ME641A |
## Conduction and Radiation |

## Credits: |
3L-0T-0L-0D (9 Credits) |

##### Course Content:

Conduction: Derivation of heat conduction equation. Summary of basic 1D conduction. Fins with variable cross-section. Multi-dimensional steady and unsteady problems in Cartesian and Cylindrical coordinates. Semi-infinite solids. Duhamel’s Superposition Integral. Solidification and Melting. Inverse heat conduction. Microscale heat transfer. Radiation: Physical mechanism. Laws of thermal radiation. Radiation properties of surfaces. View factors for diffuse radiation. Radiation exchange in black and diffusegray enclosures. Radiation effects in temperature measurement. Enclosure theory for surfaces with wall temperatures that are continuous functions of space. Spectrally diffuse enclosure surfaces. Specularly reflecting surfaces. The equation of radiative properties in participating media. Radiative properties of molecular gases. Approximate solution methods for one-dimensional media: The optically thin and optically thick approximations. Radiation in participating media: Gas radiation. Combined Conduction and Radiation: Example of a spacecraft radiator. Solar radiation. Greenhouse effect.

##### Lecturewise Breakup (based on 75min per lecture)

**I. Conduction: (**12 Lectures**)**

- Derivation of Heat Conduction Equation for Heterogeneous, Isotropic Materials in Cartesian Coordinates. Heat conduction equation for homogeneous, isotropic materials in Cartesian, Cylindrical and Spherical Coordinates. Summary of basic steady 1D heat conduction solutions including concept of resistances.
- Heat transfer from a fin of uniform cross-section. Fin efficiency and fin effectiveness. Fin with variable cross-section.
- Two-dimensional Steady State Heat Conduction: Illustration # 1: A rod with rectangular cross-section with three sides having temperature, To and other side at T = f(x). Solution by Method of Separation of Variables. Isotherms and Heat Flux Lines.
- Illustration #2: 2D Steady State Heat Conduction with Constant Heat Generation in a Long Rod of Rectangular Cross-section with Boundaries at the ambient temperature (large heat transfer coefficient)
- Steady 2D Conduction in Cylindrical Coordinates: Examples of various 2D conduction problems in cylindrical coordinates. Illustration #1: T (r, z), Circular Cylinder of Finite Length (Axi-symmetric Problem with top surface at T = f (r) and other surfaces at T = Tc). Fourier-Bessel Series Solution.
- Illustration #2: Long Cylinder having Circumferential Surface Temperature Variation: T (r, φ) Problem: Periodic boundary conditions in φ-direction. Justification of orthogonality in φ-direction. Solution by Separation of Variables method.
- Treatment of variable conductivity by Kirchhoff transformation.Unsteady State Conduction: Applications. Definition of Lumped and Distributed Systems. Biot Number and its Physical Significance. Characteristic lengths for plane wall, long cylinder and sphere. Lumped System Analysis: Derivation of the governing equation. Solution. T vs. t as a function of hA/ρcV for the cases of heating and cooling. Time Constant and its Physical Significance. Distributed Systems Analysis: Plane Wall: CaseI: Large Heat Transfer Coefficient. Case II: Moderate Heat Transfer Coefficient.
- Long Cylinder: Case I: Large Heat Transfer Coefficient. Case II: Moderate Heat Transfer Coefficient. Introduction to Heisler Charts. Multi-dimensional transient heat conduction: Non-dimensional temperature expressed as a product of 1D transient solution in each direction.
- Semi-Infinite Solid: Definition. 1D Transient Solution by Laplace Transform and Similarity technique (Error function solution) when temperature of the surface at x = 0 is suddenly changed to
*T*∞ (< Ti). Expression of heat flux at x = 0. Other surface boundary conditions: (i) Surface Convection (ii) Constant surface heat flux. Penetration depth. - Time-dependent Boundary Conditions-Duhamel’s Superposition Integral: Principle. Derivation of the integral. Solidification and Melting: Introduction. 1D Solidification Analysis: Stefan (1891) Problem. Melting of a Solid: 1D Analysis.
- Inverse heat conduction: Determination of unknown boundary conditions; Experimental determination of thermal conductivity and heat capacity.
- Microscale heat transfer: hyperbolic heat conduction, speed of propagation of thermal waves, time lag, solution for a thin slab.

**II. Radiation: (**14 Lectures**)**

- Introduction. Physical Mechanism. Laws of Thermal Radiation: Planck’s Law. Wien’s Displacement Law. Stefan-Boltzmann Law. Intensity of Radiation.
- Diffuse and Specular Surfaces. Absorptivity, Reflectivity and Transmissivity. Monochromatic and Total Emissivity. Definition of an ideal gray body. Monochromatic and Total Absorptivity. Kirchhoff’s Law. Restrictions of Kirchhoff’s law. View Factor: Definition. The View Factor Integral. View Factor Relations: Reciprocity relation. Summation Rule for an enclosure.
- View Factor between Any Two Surfaces in a Long Triangular Open-ended Enclosure: Derivation. Hottel’s Crossed-strings Method: Derivation.
- Radiation Exchange in a Black Enclosure: Derivation of the expression for net heat loss from a surface. Radiation Exchange in a Gray Enclosure: Derivation of the expression for net heat loss from a surface. Electric Circuit Analogy: Concept of surface resistance and space resistance. Network for a three-surface enclosure.
- Two-Surface Enclosure: Network, Expression for the net radiation exchange. Special Cases: 1. Large (Infinite) Parallel Planes 2. Long (Infinite) Concentric Cylinders. 3. Concentric Spheres. 4. Small Convex Object in a Large Cavity. Radiation Shields. Radiation Effects in Temperature Measurement (Conduction effects negligible) 1. Expression of error due to radiation 2. Reduction of radiation error with the use of radiation shield.
- Enclosure theory for surfaces with wall temperatures that are continuous functions of space coordinate. Integral equation approach. Method of Solution.
- Spectrally diffuse enclosure surfaces; band approximation; example of energy exchange between parallel walls with spectrally diffuse surface properties.
- Treatment of specularly reflecting surfaces; specular and diffuse reflectivities, modified definition of radiosity, method of images, construction of electrical networks.
- The equation of radiative heat transfer in participating media. Solution methods.
- Radiative properties of molecular gases.
- Approximate solution methods for one-dimensional media: The optically thin approximation. The optically thick approximation (Diffusion Approximation).
- Gas Radiation: Introduction. Beer’s law: Monochromatic intensity variation in a gas layer of thickness x. Monochromatic transmissivity, absorptivity and emissivity of a gas. Mean Beam Length. Gas emissivity charts for CO2 and H2O (vapour) at p = 1 atm. Correction factor charts for p ≠ 1 atm. Heat Exchange between gas volume and black enclosure: Calculation of gas absorptivity using charts. Heat exchange between two black parallel plates at different temperatures T1 and T2 which encloses a gas volume. Heat exchange between surfaces in a black N-sided enclosure containing a gas. Heat exchange between gas volume and gray enclosure: Hottel’s Expression when wall emissivity is greater than 0.8.
- Combined Conduction and Radiation: Example of a Spacecraft Radiator. Solar radiation. Greenhouse effect.

##### References:

- Fundamentals of Heat and Mass Transfer by
*Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine*(*John Wiley and Sons*) - Heat and Mass Transfer by
*Yunus Cengel, Afshin Ghajar(McGraw-Hill)* - Heat Conduction by
*Latif M Jiji (Springer)* - Radiative Heat Transfer by
*Michael F. Modest (Academic Press)*