Exhaustive University Study Notes: Engineering Heat Transfer

Unit 1.0: Conduction and Basic Heat Transfer Modes

• 1.1 Fundamentals of Heat Transfer Modes

  • Heat transfer is the transition of thermal energy from a higher-temperature object to a lower-temperature object. It occurs via three primary mechanisms:
  • Conduction: The process by which heat is transferred through a substance without movement of the substance as a whole. In solids, this is achieved through the vibration of molecules and the movement of free electrons. In fluids (liquids and gases), it occurs through molecular collisions.
  • Convection: The transfer of heat between a solid surface and a fluid (liquid or gas) that is in motion. It involves the combined effects of conduction and advection (bulk fluid movement).
  • Radiation: The transfer of energy through space by means of electromagnetic waves. Unlike conduction and convection, radiation does not require a physical medium and can occur in a vacuum.

• 1.2 Real-World Examples of Heat Transfer

  • Conduction Examples: A metal spoon becoming hot while sitting in a bowl of hot soup; the handle of a cast-iron skillet heating up on a stove.
  • Convection Examples: The cooling of a hot cup of coffee by blowing air over it; the circulation of air in a room caused by a radiator (natural convection); water circulating in a car engine (forced convection).
  • Radiation Examples: Feeling the warmth of the sun on your skin; the heat emitted by a glowing heating element in a toaster or an electric heater.

• 1.3 Fourier's Law of Heat Conduction

  • Statement: The rate of heat flow through a uniform material is directly proportional to the area of the section through which the heat flows (measured perpendicular to the direction of flow) and the temperature gradient in that direction.
  • Mathematical Expression:     $q = -k A \frac{dT}{dx}$
  • Where:
  • $q$ is the rate of heat flow in Watts ($W$).
  • $k$ is the thermal conductivity of the material ($W\,m^{-1}\,K^{-1}$).
  • $A$ is the surface area through which heat is transferred ($m^2$).
  • $\frac{dT}{dx}$ is the temperature gradient ($K\,m^{-1}$).
  • The negative sign indicates that heat flows in the direction of decreasing temperature.

• 1.4 Thermal Conductivity ($k$)

  • Definition: It is a physical property of a material that represents its ability to conduct heat. It is defined as the amount of heat conducted per unit time through a unit area of a material of unit thickness when a unit temperature difference is maintained across its faces.
  • Units: In SI units, thermal conductivity is expressed as $W\,m^{-1}\,K^{-1}$ or $J\,s^{-1}\,m^{-1}\,K^{-1}$.

• 1.5 Conductors and Insulators

  • Conductors: Materials with high thermal conductivity that allow heat to pass through them quickly.
  • Example: Metals such as Copper ($k \approx 400\,W\,m^{-1}\,K^{-1}$) and Aluminum ($k \approx 235\,W\,m^{-1}\,K^{-1}$).
  • Insulators: Materials with low thermal conductivity that resist the flow of heat.
  • Example: Asbestos, cork, glass wool, and polyurethane foam ($k$ typically less than $0.1\,W\,m^{-1}\,K^{-1}$).

• 1.6 Steady State Heat Transfer through Composite Structures

  • Composite Slabs: Heat flows through multiple layers of different materials in series. The total rate of heat transfer is given by:     $q = \frac{T_{begin} - T_{end}}{\sum \frac{L_i}{k_i A}}$     Where $L_i$ is the thickness of the $i$-th layer and $k_i$ is its thermal conductivity.
  • Composite Cylinders: Heat flows radially through concentric layers (e.g., an insulated pipe). The rate of heat flow is:     $q = \frac{2 \pi L (T_{inner} - T_{outer})}{\sum \frac{\ln(r_{i+1}/r_i)}{k_i}}$

• 1.7 Optimum (Critical) Thickness of Insulation

  • Adding insulation to a flat surface always decreases heat loss. However, for curved surfaces (cylinders or spheres), adding insulation increases the outer surface area, which can increase heat loss by convection.
  • The Critical Radius ($r_c$) is the radius at which heat transfer is maximized. Beyond this radius, adding more insulation decreases heat transfer.
  • For a cylinder: $r_c = \frac{k}{h}$
  • For a sphere: $r_c = \frac{2k}{h}$
  • Where $k$ is the thermal conductivity of the insulation and $h$ is the external convective heat transfer coefficient.

Unit 2.0: Natural and Forced Convection

• 2.1 Mechanisms of Convection

  • Natural Convection (Free Convection): Fluid motion is caused by buoyancy forces resulting from density gradients near the heated or cooled surface. (e.g., hot air rising from a chimney).
  • Forced Convection: Fluid motion is induced by an external source such as a pump, fan, or atmospheric winds.

• 2.2 Individual and Overall Heat Transfer Coefficients

  • Individual Coefficient ($h$): Defined by Newton's Law of Cooling: $q = h A \Delta T$.
  • Overall Heat Transfer Coefficient ($U$): Represents the total resistance to heat transfer from one fluid to another through a separating wall. It accounts for conduction through the wall and convection on both sides.     $\frac{1}{U} = \frac{1}{h_{inner}} + \frac{L}{k} + \frac{1}{h_{outer}}$

• 2.3 Effect of Fouling

  • Over time, surfaces in contact with fluids accumulate deposits (scale, rust, algae), creating additional thermal resistance known as Fouling Resistance ($R_f$). This reduces the effective overall heat transfer coefficient:     $\frac{1}{U_{dirty}} = \frac{1}{U_{clean}} + R_{f, inner} + R_{f, outer}$

• 2.4 Heat Transfer in Laminar and Turbulent Flow

  • Heat transfer is significantly higher in Turbulent Flow due to the vigorous mixing of fluid particles across the cross-section compared to the orderly streamlines of Laminar Flow.

• 2.5 Buckingham Pi Theorem

  • Used for dimensional analysis to correlate different dimensionless groups. It states that if there are $n$ variables and $m$ fundamental dimensions, there will be $(n - m)$ independent dimensionless ($\pi$) groups.

• 2.6 Sider-Tate and Dittus-Boelter Equations

  • Sider-Tate Equation (Laminar Flow): Account for variations in viscosity near the wall.
  • Dittus-Boelter Equation (Turbulent Flow):     $Nu = 0.023 Re^{0.8} Pr^n$
  • $n = 0.4$ for heating of the fluid.
  • $n = 0.3$ for cooling of the fluid.

• 2.7 Important Dimensionless Numbers

  • Reynolds Number ($Re$): Ratio of inertial forces to viscous forces. $Re = \frac{\rho v D}{\mu}$.
  • Nusselt Number ($Nu$): Ratio of convective to conductive heat transfer. $Nu = \frac{h D}{k}$.
  • Prandtl Number ($Pr$): Ratio of momentum diffusivity to thermal diffusivity. $Pr = \frac{C_p \mu}{k}$.
  • Grashof Number ($Gr$): Ratio of buoyancy to viscous forces (crucial for natural convection).
  • Stanton Number ($St$): Ratio of heat transferred into a fluid to the thermal capacity of the fluid. $St = \frac{Nu}{Re \times Pr}$.

• 2.8 Log Mean Temperature Difference (LMTD)

  • Used to find the temperature driving force for heat transfer in flow systems.     $\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\frac{\Delta T_1}{\Delta T_2})}$
  • Co-current (Parallel) Flow: Hot and cold fluids enter at the same end.
  • Counter-current Flow: Fluids enter at opposite ends. This is generally more efficient as it provides a more uniform temperature difference and allows the cold exit to be hotter than the hot exit.

• 2.9 Heat Transfer in Boiling Liquids

  • Involves phase change. Includes regimes such as pool boiling, nucleate boiling (most efficient), transition boiling, and film boiling (Leidenfrost effect).

• 2.10 Condensation

  • Types:
  • Drop-wise Condensation: Liquid forms droplets that fall away. It is highly efficient due to the exposure of the surface.
  • Film-wise Condensation: Liquid forms a continuous film. The film acts as a thermal barrier, reducing heat transfer efficiency.

Unit 3.0: Heat Transfer by Radiation

• 3.1 Concept of Radiation

  • Energy is emitted by all matter above absolute zero temperature ($0\,K$) in the form of electromagnetic waves (photons). It requires no medium.

• 3.2 Radiation Properties

  • Reflectivity ($\rho$): Fraction of incident radiation reflected by a surface.
  • Absorptivity ($\alpha$): Fraction of incident radiation absorbed by a surface.
  • Transmissivity ($\tau$): Fraction of incident radiation transmitted through a body.
  • Relationship: $\alpha + \rho + \tau = 1$.
  • Emissivity ($\epsilon$): Ratio of energy emitted by a surface to energy emitted by a black body at the same temperature.
  • Emissive Power ($E$): Total amount of radiation energy emitted per unit area per unit time ($W\,m^{-2}$).

• 3.3 Black Body and Grey Body

  • Black Body: An idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence ($\alpha = 1, \rho = 0, \tau = 0$).
  • Grey Body: A body whose absorptivity and emissivity are constant for all wavelengths and temperatures, but less than unity ($0 < \alpha < 1$).

• 3.4 Fundamental Laws of Radiation

  • Stefan-Boltzmann Law: The total emissive power of a black body is proportional to the fourth power of its absolute temperature.     $E_b = \sigma T^4$
  • Where $\sigma = 5.67 \times 10^{-8}\,W\,m^{-2}\,K^{-4}$.
  • Kirchhoff's Law: For an arbitrary body in thermal equilibrium, the emissivity is equal to its absorptivity ($\epsilon = \alpha$).
  • Planck's Law: Describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature $T$.

• 3.6 Concept of Radiation Shields

  • Thin layers of materials with low emissivity (high reflectivity) placed between two surfaces to reduce the net rate of radiation heat transfer between them.

Unit 4.0: Heat Exchange Equipment

• 4.1 - 4.2 Definition and Classification

  • Heat Exchanger: A device designed to transfer heat between two or more fluids at different temperatures.
  • Classification: Based on process (recuperators, regenerators), compactness, or construction type.

• 4.3 Flow Patterns

  • Parallel Flow: Fluids move in the same direction.
  • Counter Flow: Fluids move in opposite directions (maximum efficiency).
  • Cross Flow: Fluids move perpendicular to each other (common in air-coolers).

• 4.4-4.5 Shell and Tube Heat Exchangers

  • Consist of a bundle of tubes inside a large cylindrical shell.
  • Passes: Refers to how many times the fluid travels the length of the exchanger.
  • 1-2 Pass: Shell fluid passes once, tube fluid passes twice.
  • 2-4 Pass: Shell fluid passes twice, tube fluid passes four times.

• 4.6 Other Types of Heat Exchangers

  • Double Pipe: Consists of one pipe inside another; simplest type, used for small flow rates.
  • Finned Tube: Tubes with extended surfaces (fins) to increase heat transfer area, often used when one fluid is a gas (low heat transfer coefficient).
  • Plate Type: Uses a series of thin plates to transfer heat; very compact and efficient.

• 4.7 - 4.9 Design and Performance Calculations

  • Rate of Heat Transfer: $q = U A \Delta T_{lm}$.
  • Comparison of Area vs. Rate: Calculations can determine which exchanger type provides the highest rate of transfer for a fixed area ($A$), or the minimum area required for a target heat duty ($q$).

Unit 5.0: Evaporators

• 5.1 Types of Evaporators

  • Examples include: Open pan, Horizontal Tube, Vertical Tube (Short tube/Long tube), Forced Circulation, and Falling Film evaporators.

• 5.2 Performance Metrics

  • Steam Capacity: The mass of water evaporated per unit time.
  • Steam Economy: The mass of vapor produced per unit mass of steam used. (In a single effect, typically $< 1$; in multiple effects, this increases).
  • Boiling Point Elevation (BPE): The increase in the boiling point of a solution compared to the pure solvent, caused by the presence of dissolved solids.

• 5.3 Single Effect Evaporator Balances

  • Mass Balance: $F = L + V$, where $F$ is feed, $L$ is liquid concentrate, and $V$ is vapor.
  • Enthalpy (Energy) Balance: $F h_f + S \lambda_s = L h_l + V H_v + S h_{sc}$
  • Where $S$ is steam flow rate and $\lambda_s$ is the latent heat of steam.

• 5.4 Multiple Effect Evaporator

  • A series of evaporators where the vapor from one effect is used as the heating medium for the next effect. This significantly increases steam economy.

• 5.5 Methods of Feeding

  • Forward Feed: Feed and steam enter the first effect and travel together. Best for high-temperature sensitive materials.
  • Backward Feed: Feed enters the last (lowest pressure) effect and moves toward the first. Requires pumps between effects but is efficient for viscous products.
  • Parallel Feed: Feed is split and enters each effect simultaneously.
  • Mixed Feed: A combination of forward and backward feeding patterns.