Heat Transfer Processes Exhaustive Study Guide

Fundamental Concepts in Transport Phenomena

• Transport phenomena involve the movement of specific properties through a system via molecular movement. The three primary properties are mass, momentum, and thermal energy (heat).
• The discipline utilizes two main types of equations:     * Balance or Conservation Equations: These include the equation of continuity (mass), equation of motion (momentum), and equations for energy and chemical components. They are referred to as the equations of change as they describe variations in density ($\rho$), velocity ($v$), temperature ($T$), and concentration ($c_i$) across time and position.     * Phenomenological (Rate) Equations: These describe natural processes by treating substances as a continuous medium (continuum), ignoring microscopic structures. They relate the rate of a process to its driving force using the relationship: $\text{rate of transport process} = \frac{\text{driving force}}{\text{resistance}}$ or $\text{rate} = \text{rate coefficient} \times \text{driving force}$.
• Heat transfer is specifically defined as energy in transit resulting from a temperature gradient or difference, which acts as the driving force.
• System Classifications:     * Isolated System: No exchange of energy or matter with surroundings.     * Closed System (Non-flow): No matter exchange; only heat and work cross the boundary.     * Open System (Flow): Both matter and energy (heat/work) are exchanged with surroundings.

Thermodynamics and the Conservation of Energy

• The First Law of Thermodynamics: This is an axiom stating that energy is conserved in an isolated system. For a closed system, the change in internal energy ($U$) equals the difference between heat supplied ($Q$) and work done ($W$):     * $Q - W = U_2 - U_1$     * Internal energy ($U$) comprises molecular motion (translational, rotational, vibrational), chemical bonding energy, and subatomic particle energy. Macroscopically, an increase in $U$ manifests as a temperature rise.     * In an isolated system, $Q = 0$ and $W = 0$, thus $\Delta U = 0$.     * For reversible, constant pressure processes in a closed system: $Q = \Delta H$, where $H$ is enthalpy ($H = U + pV$).
• Steady-Flow Energy Equation (Open Systems):     * $Q - W = (H_2 - H_1) + \frac{1}{2} m (v_2^2 - v_1^2) + mg (z_2 - z_1)$
• The Second Law of Thermodynamics: This axiom distinguishes heat from work, stating that net work can never exceed supplied heat. It establishes that work can be completely converted to heat, but heat cannot be completely and continuously converted to work.     * It defines entropy ($S$): For a reversible process, $\Delta S = \frac{\Delta Q}{T}$.     * Entropy of an isolated system never decreases: $S_2 \ge S_1$.     * Heat transfer is always accompanied by entropy transfer, whereas work transfer carries zero entropy change.

Temperature Definitions and Absolute Scales

• Temperature is a macroscopic measure of molecular activity (kinetic energy) and determines if systems are in thermal equilibrium.
• Standard Scales:     * Celsius: Reference points are $0^{\circ}\text{C}$ (ice-point) and $100^{\circ}\text{C}$ (boiling point).     * Fahrenheit: Reference points are $32^{\circ}\text{F}$ (ice-point) and $212^{\circ}\text{F}$ (boiling point). Conversion: $T(^{\circ}\text{C}) = \frac{5}{9} [T(^{\circ}\text{F}) - 32]$.     * Kelvin (Absolute Scale): Defined such that a change of $1\,K$ equals $1^{\circ}\text{C}$. Absolute zero is $0\,K$. $T(K) = T(^{\circ}\text{C}) + 273.15$.
• Modern definition of Kelvin (since 2019): Based on fixed Boltzmann constant $k = 1.380649 imes 10^{-23}\,J/K$. Temperature is linked to energy ($E \sim kT$). For an ideal gas, average translational kinetic energy per molecule is $E_k = 1.5 kT$.
• Temperature Examples in the Universe:     * Space: $2.7\,K$.     * Supermassive black hole: $1.4 imes 10^{-14}\,K$.     * Boomerang Nebula: $1\,K$.     * Lowest lab temp on Earth (Germany): $38 imes 10^{-12}\,K$.

Modes of Heat Transfer: Conduction, Convection, and Radiation

• Conduction: Energy transfer through a body or medium via molecular interaction with no macroscopic movement.     * In fluids, energy is transferred via molecular collisions.     * In solids, conduction occurs through free electron motion (metals) or lattice vibrations (nonmetals).     * Governed by Fourier's Law.
• Convection: Energy transfer between a solid surface and an adjacent moving fluid, combining conduction and bulk fluid motion.     * Natural (Free) Convection: Motion due to density differences from temperature gradients.     * Forced Convection: Motion driven by external means (pumps/fans).     * Sensible heat calculation: $Q = m c_p (T_{fl,in} - T_{fl,out})$.     * Governed by Newton's Law of Cooling.
• Radiation: Energy transport by electromagnetic waves ($0.1$ to $100\,\mu m$) resulting from electronic configuration changes.     * No medium required.     * Dominant above $500^{\circ}\text{C}$.     * Governed by the Stefan-Boltzmann Law.

Fourier’s Law and Thermal Conductivity

• Fourier's Law (1822) for one-dimensional steady heat flow:     * $q_x = -k A \frac{dT}{dx}$     * $q_x$ is heat flow rate ($W$), $A$ is normal area ($m^2$), and $k$ is thermal conductivity ($W/mK$). The negative sign indicates heat flows in the direction of decreasing temperature.
• Thermal Conductivity ($k$):     * Indicates how fast heat is conducted. For small ranges, $k$ is constant. For large ranges: $k = k_o + bT$.     * Values: Metals ($50$-$400\,W/mK$), Alloys ($10$-$120\,W/mK$), Water ($0.598\,W/mK$ at $20^{\circ}\text{C}$), Air ($0.0251\,W/mK$ at $20^{\circ}\text{C}$), Insulators ($0.04$-$0.2\,W/mK$).
• Specific heat ($c_p$) and density ($\rho$) determine the resulting temperature rise from a given heat input.

Newton’s Law of Cooling and Interphase Transport

• Newton's Law (1701): $q = h A (T_W - T_{fl})$     * $h$ is the convective heat transfer coefficient ($W/m^2K$). It is an empirical parameter, not a material property.     * It depends on fluid properties ($\mu$, $c_p$, $\rho$, $k$), system geometry, flow velocity, and temperature distribution.
• Boundary Layers:     * Hydrodynamic Boundary Layer: Region where velocity changes from zero at the wall to $v_{\infty}$.     * Thermal Boundary Layer ($\delta_T$): Region where temperature changes from $T_W$ to $T_{\infty}$. Defined where $T_W - T = 0.99 (T_W - T_{\infty})$.
• Typical $h$ values:     * Free convection (Air): $5$-$30\,W/m^2K$.     * Forced convection (Water): $300$-$18,000\,W/m^2K$.     * Boiling Water: $3,000$-$60,000\,W/m^2K$.     * Condensation of Steam: $6,000$-$120,000\,W/m^2K$.

Thermal Radiation and Surface Interactions

• Absorption, Reflection, and Transmission:     * $\alpha + \rho + \tau = 1$, where $\alpha$ is absorptivity, $\rho$ is reflectivity, and $\tau$ is transmissivity.     * For most solids, $\tau \approx 0$, so $\alpha + \rho = 1$.     * Earth: Absorbs $\%70$ solar radiation ($20\%$ atmosphere, $50\%$ surface) and reflects $\%30$.
• Blackbody: An ideal surface that absorbs all incident radiation ($\alpha = 1.0$) and has maximum emissivity ($\epsilon = 1.0$).
• Stefan-Boltzmann Equation: $q_r = A \epsilon \sigma T^4$     * $\sigma = 5.676 \times 10^{-8}\,W/m^2K^4$.     * For a gray body, $\epsilon < 1.0$ and is assumed constant over all wavelengths.
• Kirchhoff's Law: At a given temperature, $\alpha = \epsilon$.
• Radiation Shields: Used to reduce heat transfer between surfaces by adding resistance. For parallel planes with equal emissivity surfaces, a single shield reduces heat flow to exactly one-half.
• Greenhouse Effect: Water vapor, $CO_2$, and methane absorb Earth's longwave infrared radiation, warming the lower atmosphere.

The General Heat Conduction Equation

• Conservation of Energy Statement: Rate of Accumulation = Rate In - Rate Out + Rate of Production.
• General 3D Unsteady-State Equation (Rectangular Coordinates):     * $\rho c_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}(k \frac{\partial T}{\partial x}) + \frac{\partial}{\partial y}(k \frac{\partial T}{\partial y}) + \frac{\partial}{\partial z}(k \frac{\partial T}{\partial z}) + q'''$
• Vector Notation: $\rho c_p \frac{\partial T}{\partial t} = k \nabla^2 T + q'''$
• Thermal Diffusivity: $\alpha = \frac{k}{\rho c_p}$, with units $m^2/s$. It measures the ability of a material to conduct thermal energy relative to its ability to store it.
• Simplified Forms:     * Fourier Equation (no source): $\frac{\partial T}{\partial t} = \alpha \nabla^2 T$     * Poisson Equation (steady state with source): $\nabla^2 T + \frac{q'''}{k} = 0$     * Laplace Equation (steady state, no source): $\nabla^2 T = 0$

Steady-State One-Dimensional Conduction

• Plane Wall: $\frac{d^2 T}{dx^2} = 0$. Temperature profile is linear: $T(x) = C_1 x + C_2$. Heat flow: $q_x = \frac{T_{1} - T_{2}}{\delta / kA}$.
• Thermal Resistance ($R$):     * Plane wall: $R = \frac{\delta}{kA}$.     * Cylindrical wall: $R = \frac{\ln(R_2/R_1)}{2 \pi k L}$.     * Convection: $R = \frac{1}{hA}$.
• Composite Walls (Series): Total resistance is the sum of individual resistances: $q = \frac{\Delta T_{\text{overall}}}{\sum R}$.
• Overall Heat Transfer Coefficient ($U$): Defined such that $q = U A \Delta T_{\text{overall}}$. For a plane wall: $\frac{1}{U} = \frac{1}{h_{\text{hot}}} + \frac{\delta}{k} + \frac{1}{h_{\text{cold}}}$.
• Hollow Cylinder: Temperature profile is logarithmic: $T = C_1 \ln(r) + C_2$. Heat flow: $q = \frac{2 \pi k L (T_1 - T_2)}{\ln(R_2 / R_1)}$.

Critical Thickness of Insulation and Internal Heat Generation

• Critical Radius ($R_{\text{ins crit}}$): Adding insulation to a pipe increases conduction resistance but decreases convection resistance due to increased surface area.     * For a cylinder: $R_{\text{ins crit}} = \frac{k}{h_o}$.     * For a sphere: $R_{\text{ins crit}} = \frac{2k}{h_o}$.     * If the outer radius $R_o < R_{\text{ins crit}}$, adding insulation actually increases heat loss until $R_{\text{ins crit}}$ is reached.
• Internal Heat Generation ($q'''$): In a slab of thickness $2\delta$ with constant $q'''$ and both surfaces at $T_W$, the temperature profile is parabolic:     * $T(x) = \frac{q'''}{2k} (\delta^2 - x^2) + T_W$     * Maximum temperature occurs at the center ($x = 0$): $T_{\max} = \frac{q''' \delta^2}{2k} + T_W$.

Unsteady-State Heat Conduction Models

• Lumped Capacitance Model: Assumes internal resistance is negligible compared to surface resistance ($Bi < 0.1$). Temperature is uniform throughout the body and varies only with time.     * Equation: $\frac{T - T_{\infty}}{T_i - T_{\infty}} = \exp(-\frac{hAt}{\rho c_p V})$     * Biot Number ($Bi$): $Bi = \frac{h L_{ch}}{k}$. Ratio of internal to surface resistance.     * Fourier Number ($Fo$): $Fo = \frac{\alpha t}{L_{ch}^2}$. Dimensionless time.
• Systems with Finite Resistance ($0.1 < Bi < 1$): Temperature varies with both position and time. Solved using Heisler and Gr\u00f6ber charts, often truncated to the first term of the Fourier series if $Fo > 0.2$.
• Semi-Infinite Solid: Body with one surface extending to infinity. Used if thermal penetration thickness $\delta_T \approx 4 \sqrt{\alpha t}$ is small compared to total thickness.     * Solution for constant surface temperature $T_s$: $\frac{T - T_s}{T_i - T_s} = \text{erf}(\frac{x}{2 \sqrt{\alpha t}})$.

Convective Heat Transfer Equations of Change

• Equation of Continuity (Conservation of Mass): $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$.
• Equation of Motion (Conservation of Momentum/Navier-Stokes): $\rho (\frac{\partial v}{\partial t} + v \cdot \nabla v) = -\nabla p + \mu \nabla^2 v + \rho g$.
• Equation of Thermal Energy: $\rho c_p (\frac{\partial T}{\partial t} + v \cdot \nabla T) = k \nabla^2 T + \Phi_v$, where $\Phi_v$ is the viscous dissipation function.
• Laminar Falling Film Example: Steady flow over an inclined plane produces a parabolic velocity profile and, neglecting viscous dissipation, a linear temperature profile between the film surface and the wall.
• Bulk (Cup-Mixing) Temperature: Weighted average of temperature based on velocity: $T_{\text{bulk}} = \frac{\iint v_z T dA}{\iint v_z dA}$.

Boundary Layer Theory and Correlations

• Prandtl's Boundary Layer Theory (1904): Effects of viscosity are confined to a thin layer near the wall.     * Hydrodynamic Boundary Layer ($\delta$): Defined where $v_x \approx 0.99 v_{\infty}$. For laminar flow over a flat plate: $\delta \approx \frac{4.96 x}{\sqrt{Re_x}}$.     * Thermal Boundary Layer ($\delta_T$): Region of temperature gradient.
• Dimensionless Numbers:     * Reynolds Number ($Re$): $\frac{\rho v L}{\mu}$. Ratio of inertia to viscous forces. Transition in plates occurs at $Re \approx 5 \times 10^5$.     * Prandtl Number ($Pr$): $\frac{\nu}{\alpha}$. Ratio of momentum diffusivity to thermal diffusivity. If $Pr > 1$, $\delta > \delta_T$.     * Nusselt Number ($Nu$): $\frac{hL}{k}$. Ratio of convective to conductive heat transfer.
• Correlations for Flat Plate (Laminar, $Pr \ge 0.6$):     * Local: $Nu_x = 0.332 Pr^{1/3} Re_x^{1/2}$.     * Average: $Nu_L = 0.664 Pr^{1/3} Re_L^{1/2}$.
• Correlations for Flat Plate (Turbulent, $5 \times 10^5 < Re < 10^7$):     * Average: $Nu = 0.037 Re^{4/5} Pr^{1/3}$.
• Flow Around Immersed Bodies:     * Zukauskas correlation for cylinders: $Nu = C Re_D^m Pr^n (\mu / \mu_W)^{1/4}$.     * Whitaker correlation for spheres: $Nu = 2 + (0.4 Re_D^{1/2} + 0.06 Re_D^{2/3}) Pr^{0.4} (\mu / \mu_W)^{1/4}$.