🦫Intro to Chemical Engineering Unit 6 – Heat Transfer

Key Concepts and Definitions

• Heat transfer involves the exchange of thermal energy between systems or within a system due to temperature differences
• Temperature measures the average kinetic energy of molecules in a substance
• Thermal conductivity ($k$) quantifies a material's ability to conduct heat
  • Materials with high $k$ values (metals) are good thermal conductors
  • Materials with low $k$ values (insulators) are poor thermal conductors
• Thermal resistance ($R$) measures a material's resistance to heat flow
  • Thermal resistance is the reciprocal of thermal conductivity ($R = 1/k$)
• Steady-state heat transfer occurs when the temperature distribution within a system does not change with time
• Transient heat transfer involves changes in temperature distribution over time
• Thermal boundary layer is a region near a surface where the temperature gradient is steep due to heat transfer between the surface and the fluid

Modes of Heat Transfer

• Conduction is the transfer of heat through a solid material by molecular vibrations and collisions
  • Occurs in the direction of decreasing temperature
  • Governed by Fourier's law of heat conduction
• Convection is the transfer of heat between a surface and a moving fluid
  • Involves the combined effects of conduction and fluid motion
  • Can be natural (buoyancy-driven) or forced (externally-induced flow)
• Radiation is the transfer of heat through electromagnetic waves
  • Does not require a medium for transmission
  • Governed by the Stefan-Boltzmann law
• Phase change heat transfer involves the transfer of heat during a change in the physical state of a substance (melting, solidification, evaporation, condensation)
• Combined heat transfer modes often occur simultaneously in real-world applications (convection and radiation in a heat exchanger)

Heat Transfer Equations

• Fourier's law of heat conduction: $q = -kA\frac{dT}{dx}$
  • $q$ is the heat transfer rate (W)
  • $k$ is the thermal conductivity (W/m·K)
  • $A$ is the cross-sectional area (m²)
  • $\frac{dT}{dx}$ is the temperature gradient (K/m)
• Newton's law of cooling: $q = hA(T_s - T_∞)$
  • $h$ is the convective heat transfer coefficient (W/m²·K)
  • $T_s$ is the surface temperature (K)
  • $T_∞$ is the fluid temperature (K)
• Stefan-Boltzmann law: $q = \epsilon\sigma A(T_s^4 - T_∞^4)$
  • $\epsilon$ is the surface emissivity (dimensionless)
  • $\sigma$ is the Stefan-Boltzmann constant (5.67 × 10⁻⁸ W/m²·K⁴)
• Overall heat transfer coefficient ($U$) relates the total heat transfer rate to the temperature difference and surface area in heat exchangers: $q = UA\Delta T_{LMTD}$
  • $\Delta T_{LMTD}$ is the log-mean temperature difference (K)

Heat Exchangers and Equipment

• Heat exchangers are devices that facilitate the transfer of heat between two or more fluids
• Common types of heat exchangers include shell-and-tube, plate, and finned tube
  • Shell-and-tube heat exchangers consist of a bundle of tubes enclosed within a shell
  • Plate heat exchangers use a series of parallel plates to create flow channels for fluids
• Evaporators are used to vaporize a liquid by transferring heat from a heating medium
• Condensers are used to condense a vapor into a liquid by removing heat using a cooling medium
• Furnaces and boilers are used to generate heat for process heating or power generation
• Insulation materials (fiberglass, mineral wool) are used to reduce heat losses in equipment and piping

Applications in Chemical Engineering

• Heat integration in process design to minimize energy consumption and costs
  • Pinch analysis is a technique used to optimize heat recovery and utility usage
• Reactor design and temperature control to maintain optimal reaction conditions
  • Exothermic reactions require heat removal to prevent runaway reactions
  • Endothermic reactions require heat input to maintain the desired reaction rate
• Distillation columns for separating liquid mixtures based on differences in boiling points
  • Heat is supplied to the reboiler to vaporize the liquid
  • Cooling is provided in the condenser to condense the vapor
• Heat transfer in heat exchangers for process heating, cooling, and energy recovery
• Thermal insulation of process equipment and piping to minimize heat losses and maintain process temperatures

Problem-Solving Techniques

• Identify the mode(s) of heat transfer involved in the problem (conduction, convection, radiation)
• Determine the appropriate heat transfer equation based on the mode and geometry
• Identify the given information and the unknown variable to be solved
• Apply the relevant boundary conditions and initial conditions
• Solve the equation analytically or numerically to obtain the desired quantity
  • Analytical solutions involve solving the equation directly using mathematical techniques
  • Numerical solutions involve discretizing the problem and solving it iteratively using computational methods
• Verify the solution by checking the units, magnitude, and physical reasonableness of the result

Real-World Examples

• Heat transfer in a cooking pan
  • Conduction through the pan material
  • Convection from the hot surface to the food
  • Radiation from the heating element to the pan
• Insulation in buildings
  • Reduces heat loss in winter and heat gain in summer
  • Commonly used insulation materials include fiberglass, cellulose, and foam
• Cooling of electronic devices
  • Heat generated by electronic components must be dissipated to prevent overheating
  • Techniques include heat sinks, fans, and liquid cooling systems
• Solar water heaters
  • Use solar radiation to heat water for domestic or industrial use
  • Employ a combination of radiation absorption and convective heat transfer
• Heat exchangers in power plants
  • Used to transfer heat from the high-temperature combustion gases to the working fluid (water/steam)
  • Efficiency of heat transfer directly impacts the overall power plant efficiency

Review and Practice Problems

1. A steel pipe (k = 50 W/m·K) with an inner diameter of 0.1 m and an outer diameter of 0.12 m is used to transport steam. The inner surface temperature is 200°C, and the outer surface temperature is 150°C. Calculate the heat transfer rate through the pipe per unit length.

2. A flat plate with a surface area of 2 m² is exposed to air at 25°C. The plate is maintained at a constant temperature of 80°C. If the convective heat transfer coefficient is 10 W/m²·K, determine the rate of heat transfer from the plate to the air.

3. A black body (ε = 1) with a surface area of 0.5 m² is at a temperature of 500 K. Calculate the rate of heat transfer by radiation if the surrounding temperature is 300 K.

4. A counterflow heat exchanger is used to cool oil (Cp = 2 kJ/kg·K) from 100°C to 60°C using water (Cp = 4.18 kJ/kg·K) entering at 20°C and leaving at 40°C. If the mass flow rate of oil is 2 kg/s, determine the required mass flow rate of water.

5. A reactor wall made of concrete (k = 1 W/m·K) is 0.2 m thick. The inner surface temperature is 50°C, and the outer surface temperature is 30°C. If the reactor has a height of 5 m and an inner diameter of 2 m, calculate the rate of heat loss through the reactor wall.