Heat Transfer

Heat transfer is a fundamental aspect of chemical engineering that deals with the movement of thermal energy. Understanding heat transfer principles is essential for designing reactors, heat exchangers, distillation columns, and other process equipment where temperature control is critical.

Modes of Heat Transfer

Conduction

Conduction is heat transfer through a stationary medium due to temperature gradient:

Where:

• $q$ = heat transfer rate (W)
• $k$ = thermal conductivity (W/m·K)
• $A$ = cross-sectional area (m²)
• $\frac{dT}{dx}$ = temperature gradient (K/m)

Convection

Convection is heat transfer between a surface and a moving fluid:

Where:

• $h$ = convective heat transfer coefficient (W/m²·K)
• $T_s$ = surface temperature
• $T_\infty$ = bulk fluid temperature

Radiation

Radiation is heat transfer by electromagnetic waves:

Where:

• $\epsilon$ = emissivity
• $\sigma$ = Stefan-Boltzmann constant ($5.67 \times 10^{-8}$ W/m²·K⁴)

Advanced Heat Transfer Theory

Transient Heat Conduction

For unsteady-state heat transfer, we use the heat conduction equation:

Where $\alpha = \frac{k}{\rho c_p}$ is the thermal diffusivity.

Solutions for Simple Geometries:

• Infinite slab: Error function solution
• Infinite cylinder: Bessel function solution
• Sphere: Similar to cylinder with different constants

Convective Heat Transfer Correlations

Empirical correlations for different flow conditions:

Forced Convection in Tubes:

• Laminar flow ($Re < 2300$): $$Nu = 3.66 \quad \text{(fully developed)}$$
• Turbulent flow ($Re > 10,000$): Dittus-Boelter equation $$Nu = 0.023 Re^{0.8} Pr^n$$ Where $n = 0.4$ for heating, $n = 0.3$ for cooling

Natural Convection: For vertical plates:

Where Grashof number $Gr = \frac{g\beta\Delta T L^3}{\nu^2}$

Radiation Heat Transfer

View Factors: The fraction of radiation leaving surface i that strikes surface j:

Radiation Network Analysis: For multiple surfaces, use resistance network:

• Surface resistance: $R = \frac{1 - \epsilon}{\epsilon A}$
• Space resistance: $R = \frac{1}{A_i F_{ij}}$

Heat Exchanger Design

Types of Heat Exchangers

• Shell and tube: Most common in chemical plants
• Plate: Compact, efficient for low-viscosity fluids
• Double pipe: Simple, for small heat duties
• Air-cooled: When water is scarce
• Spiral: For fouling fluids
• Finned tube: Enhanced surface area

Heat Exchanger Design Methods

LMTD Method: For counter-current flow:

NTU-Effectiveness Method: Useful when outlet temperatures are unknown:

Where $C_{min}$ is the minimum heat capacity rate and NTU = $\frac{UA}{C_{min}}$

Overall Heat Transfer Coefficient

The overall heat transfer coefficient ($U$) accounts for all resistances:

Where:

• $h_h$, $h_c$ = hot and cold side coefficients
• $R_{f,h}$, $R_{f,c}$ = fouling resistances
• $\Delta x$ = wall thickness
• $A_m$ = log mean area for cylindrical geometry

Phase Change Heat Transfer

Boiling Heat Transfer:

• Nucleate boiling: High heat transfer coefficients
• Film boiling: Lower coefficients, risk of burnout
• Critical heat flux: Maximum sustainable heat flux

Condensation Heat Transfer:

• Film condensation: Liquid film forms on surface
• Dropwise condensation: Higher coefficients but unstable

Thermal Insulation

Insulation Materials

• Fiberglass: Common, cost-effective
• Mineral wool: High temperature resistance
• Foam glass: Moisture resistant
• Calcium silicate: High temperature applications

Economic Thickness

Optimum insulation thickness balances capital cost against energy savings.

Real-World Application: Distillation Column Reboiler

A reboiler in a distillation column demonstrates heat transfer principles:

Design Considerations

1. Heat duty: Energy required for vaporization
2. Temperature approach: Minimum temperature difference
3. Fouling factors: Accounting for scale formation
4. Pressure drop: Affecting pump requirements

Example: Steam-Heated Reboiler

Calculate the required heat transfer area:

# Process conditions
vaporization_rate = 5000  # kg/h
latent_heat = 400        # kJ/kg
steam_temperature = 150   # °C
process_temperature = 120 # °C
overall_U = 800          # W/m²·K

# TODO: Calculate heat duty and area
# Steps:
# 1. Convert vaporization rate to kg/s
# 2. Calculate heat duty (Q = m_dot * latent_heat)
# 3. Calculate LMTD
# 4. Calculate required area (A = Q / (U * ΔT_lm))

vaporization_rate_s = 0
heat_duty = 0
delta_T_lm = 0
required_area = 0

print(f"Heat duty: {heat_duty:.0f} kW")
print(f"LMTD: {delta_T_lm:.1f} °C")
print(f"Required area: {required_area:.1f} m²")

Your Challenge: Heat Exchanger Performance Analysis

In this exercise, you'll analyze the performance of a shell-and-tube heat exchanger and determine if it meets process requirements.

Goal: Calculate heat transfer rates and evaluate exchanger performance.

System Description

A shell-and-tube heat exchanger cools a process stream with cooling water:

Hot side (process fluid):

• Inlet temperature: 90°C
• Outlet temperature: 50°C
• Flow rate: 10 kg/s
• Specific heat: 2.5 kJ/kg·K

Cold side (cooling water):

• Inlet temperature: 20°C
• Maximum outlet temperature: 40°C
• Specific heat: 4.18 kJ/kg·K

Exchanger specifications:

• Overall U: 500 W/m²·K
• Area: 100 m²
• Counter-current flow

# Process conditions
m_hot = 10          # kg/s
cp_hot = 2.5        # kJ/kg·K
T_hot_in = 90       # °C
T_hot_out = 50      # °C

T_cold_in = 20      # °C
T_cold_max = 40     # °C
cp_cold = 4.18      # kJ/kg·K

U = 500             # W/m²·K
A = 100             # m²

# TODO: Calculate cooling water flow rate and outlet temperature
# Steps:
# 1. Calculate heat duty from hot side
# 2. Calculate required cold flow rate
# 3. Calculate actual cold outlet temperature
# 4. Calculate LMTD
# 5. Calculate actual heat transfer rate
# 6. Compare with required duty

heat_duty = 0
m_cold_required = 0
T_cold_out = 0
delta_T_lm = 0
actual_heat_transfer = 0

print(f"Heat duty: {heat_duty:.0f} kW")
print(f"Required cooling water: {m_cold_required:.2f} kg/s")
print(f"Cooling water outlet temperature: {T_cold_out:.1f} °C")
print(f"LMTD: {delta_T_lm:.1f} °C")
print(f"Actual heat transfer: {actual_heat_transfer:.0f} kW")

# Check if exchanger is adequate
if actual_heat_transfer >= heat_duty:
    print("Exchanger is adequate for the duty")
else:
    print("Exchanger is undersized")

What would happen if the flow arrangement were parallel instead of counter-current? How could you improve the exchanger performance?