Comprehensive Notes on Heat Transfer: Heat Exchangers

Components and Materials of Heat Exchangers

• Tubes: These are primary heat transfer components available in various materials including Copper, 90/10 CuNi (Copper-Nickel), 316 Stainless Steel, Admiralty, or Carbon Steel. A key construction feature is that tubes are roller expanded.
• Shells: Constructed for ruggedness, shells are typically available in Steel and 316 Stainless Steel. Design features include minimum clearances between the shell and baffles to reduce by-pass and maximize heat transfer efficiency.
• Supports: Fabricated Carbon Steel supports are moveable and available for all exchanger sizes.
• Heads: Available materials include Cast Iron, Brass, 316 Stainless Steel, or Fabricated Carbon Steel.
• Tubesheets: These are thick components made from Carbon Steel, 316 Stainless Steel, or 90/10 CuNi.
• Baffles: These are precision punched to assure effective circulation. They provide minimum clearances between the tubes and tube holes. Baffle cuts and spacing are tailored for each diameter according to best practices. Standard materials include Carbon Steel, Brass, and 316 Stainless Steel.

Introduction to Heat Exchangers

• Definition: Heat exchangers (HE) are devices that facilitate the exchange of heat between two fluids at different temperatures while preventing them from mixing.
• Distinction from Mixing Chambers: Unlike mixing chambers, heat exchangers do not allow the two fluids to mix.
• Practical Example (Car Radiator): Heat is transferred from hot water flowing through radiator tubes to cold air flowing through closely spaced thin plates (fins) attached to the outside of the tubes.
• Mechanism of Heat Transfer: Heat transfer in an HE involves convection in each fluid and conduction through the wall separating them.

Classifications and Types of Heat Exchangers

• General Types:     - Double-pipe     - Parallel flow     - Counter flow     - Cross-flow     - Compact     - Shell-and-Tube (One-shell pass, two or more tube passes)     - Plate and frame
• Double-Pipe HE: The simplest type, consisting of two concentric pipes of different diameters. One fluid flows in the smaller pipe while the other flows in the annular space.     - Parallel Flow: Both fluids enter at the same end and move in the same direction.     - Counter Flow: Fluids enter at opposite ends and flow in opposite directions.
• Compact HE: Specifically designed for large surface area per unit volume.     - Area Density ($\beta$): Defined as the ratio of the heat transfer surface area to its volume ($A_s / V$).     - An HE is classified as compact if $\beta \ge 700\,m^2/m^3$ (or $200\,ft^2/ft^3$).     - Examples of Area Density:         - Car Radiators: $\beta = 1000\,m^2/m^3$         - Glass ceramic gas turbine HE: $\beta = 6000\,m^2/m^3$         - Regenerator of a Stirling engine: $\beta = 15,000\,m^2/m^3$         - Human lung: $\beta = 20,000\,m^2/m^3$
• Cross-Flow HE: Often used where one fluid is a gas.     - Mixed flow: Fluid is free to move laterally in the cross-flow direction.     - Unmixed flow: Fluid is confined in channels (e.g., tubes or fins), preventing lateral movement.
• Shell-and-Tube HE: The most common type, though not suitable for automotive or aircraft use due to large size and weight.     - Tubes: Large numbers (sometimes hundreds) packed in a shell with parallel axes.     - Baffles: Placed in the shell to force the shell-side fluid to flow across the tubes, enhancing heat transfer and maintaining tube spacing.     - Headers: Large flow areas at ends of the shell where tube-side fluid accumulates before and after passing through tubes, acting as a flow stabilizer.
• Plate-and-Frame HE: Composed of very thin grooved stainless steel plates clamped in a frame. Hot and cold fluids flow in alternate passages, making it very effective.     - Capacity can increase by adding more plates.     - Rubber gaskets between plates prevent mixing and leaking.     - Best suited for liquid-to-liquid applications (without pulp) at similar pressures.
• Specific Function-Based Names:     - Condenser: One fluid is cooled and condenses.     - Boiler: One fluid absorbs heat and vaporizes.     - Space radiator/heater: Transfers heat to surrounding space via radiation.

The Overall Heat Transfer Coefficient ($U$)

• Thermal Resistance Network: Heat transfer involves three resistances in series: convection from hot fluid to wall, conduction through the wall, and convection from the wall to the cold fluid.
• General Equation:   $Q = U A_s \Delta T_m$   where $U$ is the overall heat transfer coefficient ($W/m^2\cdot^\circ C$).
• Resistance Summation:   $\frac{1}{U A_s} = R_{total} = R_{conv,i} + R_{wall} + R_{conv,o}$$\frac{1}{U A_s} = \frac{1}{h_i A_i} + \frac{\ln(D_o/D_i)}{2\pi k L} + \frac{1}{h_o A_o}$
• Simplified Case (Thin Tube): If wall thickness is small and thermal conductivity is high:   $R_{wall} \simeq 0$$A_i \simeq A_o \simeq A_s$$\frac{1}{U} \simeq \frac{1}{h_i} + \frac{1}{h_o}$
• Representative Values of $U$ ($W/m^2\cdot^\circ C$):     - Water-to-water: $850 - 1700$     - Water-to-oil: $110 - 350$     - Water-to-gasoline: $300 - 600$     - Feedwater heater: $1100 - 8500$     - Steam-to-light fuel oil: $190 - 340$     - Steam-to-heavy fuel oil: $60 - 130$     - Steam condenser: $1000 - 6000$     - Freon condenser (water-cooled): $300 - 1000$     - Ammonia condenser (water-cooled): $800 - 1400$     - Alcohol condenser (water-cooled): $250 - 700$     - Gas-to-gas: $10 - 40$     - Water-to-air in finned tubes: $400 - 850$ (water inside tubes), $20 - 50$ (air outside tubes).

Fouling Factor ($R_f$)

• Definition: An additional thermal resistance introduced by the accumulation of deposits on heat transfer surfaces over time, causing a decrease in heat transfer rate.
• Types of Fouling:     - Precipitation (Physical): Most common, e.g., calcium-based scale. Controlled by water treatment.     - Corrosion (Chemical): Accumulation of reaction products on surfaces. Avoided by using plastic or glass-coated pipes.     - Biological (Algae growth): Common in warm fluids. Prevented by chemical treatment.
• Accounting for Fouling in $U$:   $\frac{1}{U A_s} = \frac{1}{h_i A_i} + \frac{R_{f,i}}{A_i} + R_{wall} + \frac{R_{f,o}}{A_o} + \frac{1}{h_o A_o}$
• Numerical Values for $R_f$ ($m^2 \cdot ^\circ C/W$):     - Distilled/Sea/River water ($< 50^\circ C$): $0.0001$     - Distilled/Sea/River water ($> 50^\circ C$): $0.0002$     - Fuel oil: $0.0009$     - Steam (oil-free): $0.0001$     - Refrigerants (liquid): $0.0002$     - Refrigerants (vapor): $0.0004$     - Alcohol vapors: $0.0001$     - Air: $0.0004$
• Default Assumption: In the absence of specific data, assume a $0.2\,mm$ thick limestone layer ($k = 2.9\,W/m \cdot ^\circ C$) per unit surface area.

Nusselt Number ($Nu$) Correlations for Tubes and Annulus

• Laminar Flow (Fully Developed): $Nu = 3.66$
• Developing Laminar Flow:   $Nu = 3.66 + \frac{0.065(D/L)Re Pr}{1 + 0.04[(D/L)Re Pr]^{2/3}}$
• Hydraulic Diameter ($D_h$): For plate spacing, $D_h$ is twice the spacing. For circular annulus:   $D_h = D_o - D_i$
• Turbulent Flow:     - Petukhov Equation: $f = (0.790 \ln Re - 1.64)^{-2}$ for 10^4 < Re < 10^6     - Colburn Equation: $Nu = 0.023 Re^{0.8} Pr^{1/3}$     - Dittus-Boelter Equation: $Nu = 0.023 Re^{0.8} Pr^n$ where $n = 0.4$ for heating and $0.3$ for cooling.     - Gnielinski Equation:       $Nu = \frac{(f/8)(Re - 1000)Pr}{1 + 12.7(f/8)^{0.5}(Pr^{2/3} - 1)}$       Valid for 3 \times 10^3 < Re < 5 \times 10^6 and $0.5 \le Pr \le 2000$

Thermodynamics and Heat Balance

• Assumptions:     - Steady-flow operation.     - Constant mass flow rates and fluid properties.     - Negligible kinetic/potential energy changes.     - Constant specific heat ($C_p$) at average temperature.     - Insignificant axial heat conduction.     - Outer surface is perfectly insulated.
• Energy Balance Equation:   $Q = \dot{m}<em>c C</em>{pc} (T_{c,out} - T_{c,in}) = \dot{m}<em>h C</em>{ph} (T_{h,in} - T_{h,out})$
• Heat Capacity Rate ($C$):   $C_c = \dot{m}<em>c C</em>{pc}$$C_h = \dot{m}<em>h C</em>{ph}$   - A fluid with a large heat capacity rate experiences a small temperature change.
• Phase Change (Condensers/Boilers):   $Q = \dot{m} h_{fg}$   - During phase change, the heat capacity rate ($C$) approaches infinity ($C \rightarrow \infty$) because $\Delta T \rightarrow 0$.

Analysis: The Log Mean Temperature Difference (LMTD) Method

• General Formula:   $Q = U A_s \Delta T_{lm}$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$
• Temperature Gradients:     - Parallel Flow: $\Delta T_1 = T_{h,in} - T_{c,in}$; $\Delta T_2 = T_{h,out} - T_{c,out}$. $\Delta T$ decreases exponentially toward the outlet.     - Counter Flow: $\Delta T_1 = T_{h,in} - T_{c,out}$; $\Delta T_2 = T_{h,out} - T_{c,in}$. $\Delta T_{lm,CF}$ is always greater than $\Delta T_{lm,PF}$, requiring a smaller surface area.
• Multi-Pass and Cross-Flow:   Use a correction factor periodically:   $Q = U A_s F \Delta T_{lm,CF}$   - Correction Factor ($F$): Measured deviation from counter-flow. $F \le 1$. Determined via charts using ratios:     - $P = \frac{t_2 - t_1}{T_1 - t_1}$     - $R = \frac{T_1 - T_2}{t_2 - t_1}$     (where $T$ is shell-side and $t$ is tube-side).

Analysis: The Effectiveness-NTU Method

• Utility: Practical for predicting outlet temperatures when the heat exchanger size/type is known.
• Effectiveness ($\epsilon$):   $\epsilon = \frac{Q}{Q_{max}}$
• Maximum Possible Heat Transfer ($Q_{max}$):   $Q_{max} = C_{min} (T_{h,in} - T_{c,in})$   - $C_{min}$ is the smaller of $C_c$ and $C_h$.
• Number of Transfer Units (NTU):   $NTU = \frac{U A_s}{C_{min}}$
• Capacity Ratio ($c$):   $c = \frac{C_{min}}{C_{max}}$
• Effectiveness Relations:     - All HE with $c = 0$ (phase change): $\epsilon = 1 - e^{-NTU}$     - Parallel-flow: $\epsilon = \frac{1 - e^{-NTU(1+c)}}{1+c}$     - Counter-flow: $\epsilon = \frac{1 - e^{-NTU(1-c)}}{1 - c e^{-NTU(1-c)}}$
• Economic Justification: Increasing NTU beyond $1.5$ provides diminishing returns in effectiveness. Values of NTU significantly larger than $3$ are usually not economically justified.

Numerical Examples

• Example 1 ($U$ determination):     - Inner tube: Copper, $D_i = 2\,cm$, negligible thickness.     - Annulus: $D_{outer, inner} = 3\,cm$.     - Water (Tube): $0.5\,kg/s$, $45^\circ C$. Properties: $Pr = 3.91$, $k=0.637$, $\nu=0.602 \times 10^{-6}$.     - Oil (Shell): $0.8\,kg/s$, $80^\circ C$. Properties: $Pr = 490$, $k=0.138$, $\nu=37.5 \times 10^{-6}$.     - Water flow is Turbulent ($Re=53,490$); $h_i=7663\,W/m^2\cdot^circ C$.     - Oil flow is Laminar ($Re=637$); $h_o=75.2\,W/m^2\cdot^circ C$.     - Result: $U = 74.5\,W/m^2 \cdot ^\circ C$.
• Example 2 (Steam Condenser):     - Steam at $30^\circ C$, Water enters at $14^\circ C$ and leaves at $22^\circ C$.     - $A_s = 45\,m^2$, $U = 2100\,W/m^2 \cdot ^\circ C$.     - $\Delta T_{lm} = 11.5^\circ C$. $Q = 1087\,kW$.     - Cooling water rate: $32.5\,kg/s$. Condensation rate: $0.45\,kg/s$.
• Example 3 (Double-Pipe HE):     - Water: $1.2\,kg/s$, heated $20^\circ C \rightarrow 80^\circ C$.     - Geothermal: $2\,kg/s$ enters at $160^\circ C$.     - $U = 640\,W/m^2 \cdot ^\circ C$. Result: $L = 108\,m$.
• Example 4 (Fouling Impact):     - Glycerin heated $20^\circ C \rightarrow 50^\circ C$. Water: $80^\circ C \rightarrow 40^\circ C$.     - $U_{no-fouling} = 21.6\,W/m^2 \cdot ^\circ C$, $Q = 1830\,W$.     - With fouling $R_f = 0.0006$, $U = 21.3\,W/m^2 \cdot ^\circ C$, $Q = 1805\,W$.
• Example 5 ($Q_{max}$):     - Counter-flow HE. Water 1: $8\,kg/s$, $10^\circ C$. Water 2: $2\,kg/s$, $70^\circ C$.     - $C_{min} = 8.36\,kW/^\circ C$. $Q_{max} = 502\,kW$.     - Limiting outlet temps: $T_{c,out} = 25^\circ C$, $T_{h,out} = 10^\circ C$.
• Example 7 (Multi-Pass HE):     - Oil ($0.3\,kg/s$) cooled by Water ($0.2\,kg/s$).     - Oil in at $150^\circ C$, Water in at $20^\circ C$.     - $A_s = 1.76\,m^2$, $U = 310$. Result: $Q = 39.1\,kW$.

Practice Exercises

• Test Yourself 1: Counter-flow double-pipe. Cooling water ($0.2\,kg/s$, $30^\circ C$), Lubricating oil ($0.1\,kg/s$, $100^\circ C \rightarrow 60^\circ C$). Result: $L = 65.9\,m$.
• Test Yourself 2: Cross-flow HE. Pressurized water ($1\,kg/s$, $35^\circ C \rightarrow 125^\circ C$), Exhaust gases ($300^\circ C \rightarrow 100^\circ C$). $U_h = 100$. Result: $A_h = 37.8\,m^2$.
• Test Yourself 3: Cross-flow HE. Water $1\,kg/s$, $35^\circ C$. Exhaust gas $1.5\,kg/s$, $250^\circ C$. $U = 100$, $A = 40\,m^2$. Results: $Q = 2.65 \times 10^5\,W$, $T_{oil,o} = 73.3^\circ C$, $T_{w,o} = 98.1^\circ C$.
• Test Yourself 4: Shell-and-tube condenser. 30,000 tubes, 2 passes. Steam at $50^\circ C$, Water at $20^\circ C$. $Q = 2 \times 10^9\,W$. Results: Water outlet $T = 36.0^\circ C$, Required length per pass $L = 4.51\,m$.