Heat Exchangers: Basic Theory and Analysis

Heat Exchangers: Basic Theory and Analysis

Introduction to Heat Exchangers

• Heat exchangers are fundamental in energy conversion and utilization processes.

• They facilitate heat exchange between two fluids.

• A wide array of flow configurations are present in heat exchanger designs.

Heat Exchanger Types

Concentric-Tube Heat Exchangers

• Parallel Flow:

  • Simplest configuration.

  • Fluids flow in the same direction, entering at the same end and exiting at the same end.

  • Temperature profiles show both hot and cold fluid temperatures approaching each other, with the hot fluid always hotter than the cold fluid.

  • $\Delta T<em>1 = T</em>{h,in} - T<em>{c,in}$ and $\Delta T</em>2 = T<em>{h,out} - T</em>{c,out}$.

• Counterflow:

  • Fluids flow in opposite directions, entering at opposite ends.

  • Offers superior performance compared to parallel flow.

  • Temperature profiles show hot fluid entering at one end and cold fluid exiting near that end, and vice-versa. The exit temperature of the cold fluid can exceed the exit temperature of the hot fluid.

  • $\Delta T<em>1 = T</em>{h,in} - T<em>{c,out}$ and $\Delta T</em>2 = T<em>{h,out} - T</em>{c,in}$.

Cross-Flow Heat Exchangers

• In this configuration, fluids move perpendicular to each other, commonly found in compact heat exchangers.

• Finned-Both Fluids Unmixed: Fluid motion and mixing in the transverse direction ($y$) are prevented for finned tubes.

• Unfinned-One Fluid Mixed, the Other Unmixed: Mixing occurs in the transverse direction for unfinned conditions.

• Heat exchanger performance is directly influenced by the extent of fluid mixing.

Shell-and-Tube Heat Exchangers

• The most prevalent type in industrial applications.

• Construction: Consist of numerous tubes (potentially hundreds) enclosed within a shell, with their axes parallel to the shell's axis.

• Heat Transfer Mechanism: One fluid flows inside the tubes, while the other flows outside through the shell.

• Classification by Passes: Further categorized by the number of shell and tube passes:

  • One Shell Pass and One Tube Pass

  • One Shell Pass, Two Tube Passes

  • Two Shell Passes, Four Tube Passes

  • And other multiples (e.g., $n$ shell passes, $2n$ or $4n$ tube passes).

• Baffles: Incorporated within the shell to:

  • Establish a cross-flow pattern for the shell-side fluid.

  • Induce turbulent mixing, significantly enhancing convection heat transfer.

  • Increase cooling rate, promote turbulence, and destroy insulating film layers from laminar flow.

Regenerative Heat Exchangers

• Mechanism: Involves the alternate passage of hot and cold fluid streams through the same flow area.

• Dynamic-Type Regenerator: Features a rotating drum through which hot and cold fluids continuously flow through different sections.

  • Any part of the drum periodically passes through the hot stream, storing heat.

  • It then passes through the cold stream, releasing the stored heat.

Phase-Change Heat Exchangers

• Condenser: One fluid is cooled, causing it to condense as it traverses the heat exchanger.

• Boiler: One fluid absorbs heat, leading to its vaporization.

Compact Heat Exchangers

• Characterized by a high area density ($\beta$), defined as the heat transfer surface area per unit volume.

• Definition: A heat exchanger is classified as compact if its area density is greater than $700 \text{ m}^2/\text{m}^3$.

• Key Features:

  • Achieve large heat transfer rates per unit volume.

  • Especially effective when one or both fluids are gases.

  • Possess large heat transfer surface areas per unit volume.

  • Typically feature small flow passages and laminar flow conditions.

• Examples: Car radiator, human lung.

• Flow Configuration: Usually employ cross-flow, where fluids move perpendicular to each other.

  • Cross-flow can be further classified as unmixed or mixed flow.

• Printed Circuit Heat Exchangers (PCHEs):

  • A prominent type of compact heat exchanger used in industrial applications such as chemical processing, fuel processing, waste heat recovery, and refrigeration.

  • High Surface Density: Typically greater than $2500 \text{ m}^2/\text{m}^3$.

  • Disadvantages: Very high pressure drop and strict requirement for extremely clean fluids to prevent blockages in fine channels (typically $0.5$ to $2 \text{ mm}$ spacing).

  • Materials: Commonly made from stainless steel, titanium, copper, nickel, and nickel alloys.

  • Operating Conditions: Can withstand operating pressures up to $500 \text{ bar}$.

• Illustrations: Examples include fin-tube heat exchangers (flat tubes, continuous plate fins; circular tubes, continuous plate fins; circular tubes, circular fins) and plate-fin exchangers (single pass, multipass).

Plate and Frame Heat Exchangers

• Consist of corrugated flat flow passages.

• Hot and cold fluids flow in alternating passages, ensuring each cold stream is surrounded by two hot streams.

• Results in very effective heat transfer.

• Well-suited for liquid-to-liquid applications.

Heat Exchanger Classification Summary

According to Transfer Process

• Indirect Contact Type:

  • Direct transfer type

  • Storage type

  • Fluidized bed

• Direct Contact Type:

  • Immiscible fluids

  • Gas-Liquid

  • Liquid-Vapor

According to Number of Fluids

• Two-fluid

• Three-fluid

• N-fluid (N > 3)

According to Surface Compactness

• Compact: Surface area density $\ge 700 \text{ m}^2/\text{m}^3$

• Non-compact: Surface area density < 700 \text{ m}^2/\text{m}^3

According to Construction

• Tubular: Double-pipe, Shell-and-tube, Spiral tube, Pipecoils

• Plate-type: Gasketed, Welded, Spiral, Platecoil

• Extended Surface: Plate-fin, Tube-fin

• Regenerative: Rotary, Fixed-matrix

According to Flow Arrangements

• Single-Pass: Counterflow, Parallelflow, Crossflow

• Multipass (for Extended surface, Shell-and-tube, Plate):

  • Cross-counterflow

  • Cross-parallelflow

  • Compound parallel counterflow

  • Split flow

  • Divided flow

According to Heat Transfer Mechanisms

• Single-phase convection on both sides

• Single-phase convection on one side, two-phase convection on other side

• Two-phase convection on both sides

• Combined convection and radiative heat transfer

Overall Heat Transfer Coefficient ($U$)

• Definition: Represents the combined thermal resistance to heat transfer typically involving two flowing fluids separated by a solid wall.

• Heat Transfer Path:

  1. Convection from the hot fluid to the wall.

  2. Conduction through the solid wall.

  3. Convection from the wall to the cold fluid.

• Radiation Effects: Any radiation contributions are generally incorporated into the convection heat transfer coefficients.

• Thermal Resistance Network: For a double-pipe heat exchanger, the overall resistance ($R<em>{total}$) is the sum of convective resistances ($1/(h</em>i A<em>i)$ and $1/(h</em>o A<em>o)$) and conductive resistance ($\ln(D</em>o/D_i)/(2\pi k L)$) for the tube wall.

• Dominance: The overall heat transfer coefficient ($U$) is primarily determined by the smaller convection coefficient.

  • If one convection coefficient is significantly smaller than the other (e.g., hi << ho), then 1/hi >> 1/ho, leading to $U \approx h_i$.

  • This scenario commonly occurs when one fluid is a gas and the other is a liquid.

• Fins: In situations where convection is limited by a gas, fins are frequently employed on the gas side to augment the product $UA$ and thereby enhance heat transfer.

  • Fins are mostly used to enhance heat transfer in a low heat transfer coefficient ($h$) environment and are usually installed on the gas side in gas-liquid heat exchangers.

  • Fin efficiency is for an adiabatic tip.

  • For thin walls, thermal resistance of the wall ($R_W$) is approximately zero.

Fouling in Heat Exchangers

• Definition: The decline in heat exchanger performance over time due to the accumulation of deposits on heat transfer surfaces.

• Impact: The layer of deposits introduces additional thermal resistance to heat transfer.

• Fouling Factor ($R_f$): Represents this additional resistance.

  • For a new heat exchanger, the fouling factor is zero ($R_f = 0$).

  • Increases with operating temperature and service duration.

  • Decreases with the velocity of the fluids.

• Types of Fouling:

  • Precipitation of solid deposits (scaling).

  • Corrosion.

  • Biological fouling (e.g., algae growth in warm fluids).

  • Freezing fouling.

  • Chemical reaction fouling.

• General Expression for Overall Coefficient with Fouling:

  • $\frac{1}{U<em>o A</em>o} = \frac{1}{U<em>i A</em>i} = R<em>{total} = \frac{1}{h</em>i A<em>i} + R</em>{f,i}'' \frac{1}{ A<em>{i}} + \frac{\ln(D</em>o/D<em>i)}{2 \pi k L} + R</em>{f,o}'' \frac{1}{A<em>o} + \frac{1}{h</em>o A_o}$

  • Where $R<em>{f,i}''$ and $R</em>{f,o}''$ are the fouling factors on the inner and outer surfaces, respectively.

• Representative $U$ values: Vary widely based on fluid combinations (e.g., water-to-water: $850-1700 \text{ W/m}^2\text{.K}$), highlighting the diverse applications and operating conditions.

Analysis of Heat Exchangers

Design Calculations Methodologies

• Log Mean Temperature Difference (LMTD) Method: Used to select a heat exchanger that achieves a specified temperature change in a fluid stream with a known mass flow rate. The task is to determine the required heat transfer surface area.

• Effectiveness–NTU Method: Used to predict the outlet temperatures of the hot and cold fluid streams in a specified heat exchanger. The task is to determine the performance of an existing heat exchanger.

Heat Capacity Rate ($C$)

• For a fluid stream, the heat capacity rate is defined as $C = \dot{m} c_p$.

• The rate of heat transfer ($Q$) in a well-insulated heat exchanger is:

  • $Q = C<em>h (T</em>{h,in} - T_{h,out})$

  • $Q = C<em>c (T</em>{c,out} - T_{c,in})$

• Two fluid streams having the same capacity rates ($C<em>h = C</em>c$) experience the same temperature change in a well-insulated heat exchanger.

Log Mean Temperature Difference (LMTD) Method

• Applies Newton's Law of Cooling to heat exchangers using an average temperature difference between the two fluids, specifically the log-mean value.

• Formula: $Q = U A \Delta T_{LMTD}$

• Log Mean Temperature Difference ($\Delta T_{LMTD}$):

  • $\Delta T<em>{LMTD} = \frac{\Delta T</em>1 - \Delta T<em>2}{\ln(\Delta T</em>1 / \Delta T_2)}$

• Counter-Flow Heat Exchanger:

  • $\Delta T<em>1 = T</em>{h,in} - T_{c,out}$

  • $\Delta T<em>2 = T</em>{h,out} - T_{c,in}$

• Parallel-Flow Heat Exchanger:

  • $\Delta T<em>1 = T</em>{h,in} - T_{c,in}$

  • $\Delta T<em>2 = T</em>{h,out} - T_{c,out}$

• Performance Comparison: For equivalent $UA$ and inlet temperatures, a counter-flow heat exchanger requires a smaller surface area to achieve a specified heat transfer rate, indicating superior performance.

  • In parallel flow, $T<em>{c,out}$ cannot exceed $T</em>{h,out}$, but it can in counterflow.

Overall Energy Balance

• Assumptions: Negligible heat transfer between the exchanger and surroundings; negligible potential and kinetic energy changes for each fluid; no liquid/vapor phase change; constant specific heats.

• Application to Hot (h) and Cold (c) Fluids:

  • $Q = \dot{m}<em>h c</em>{p,h} (T<em>{h,in} - T</em>{h,out})$

  • $Q = \dot{m}<em>c c</em>{p,c} (T<em>{c,out} - T</em>{c,in})$

Special Operating Conditions for LMTD

• Case (a): Hot fluid heat capacity rate much greater than cold fluid (Ch >> Cc) or hot fluid is a condensing vapor: Negligible or no change in $T_h$.

• Case (b): Cold fluid heat capacity rate much greater than hot fluid (Cc >> Ch) or cold fluid is an evaporating liquid: Negligible or no change in $T_c$.

• Case (c): Hot and cold fluid heat capacity rates are equal ($C<em>h = C</em>c$).

Condensers and Boilers (LMTD Method)

• Involves one fluid undergoing a phase-change process at a constant temperature.

• The rate of heat transfer is expressed as: $Q = \dot{m} h<em>{fg}$, where $h</em>{fg}$ is the enthalpy of evaporation at the specified pressure and temperature.

Correction Factor ($F$) for LMTD

• Used for shell-and-tube and cross-flow heat exchangers because their temperature profiles deviate from pure counterflow.

• Modified LMTD Equation: $Q = U A F \Delta T_{LMTD,counterflow}$.

• Value of $F$:

  • F < 1 for cross-flow and shell-and-tube heat exchangers.

  • $F = 1$ for pure counterflow heat exchangers.

  • $F = 1$ for boilers or condensers where one fluid undergoes phase change (as $R$ becomes infinite).

• Determination of $F$: Depends on the heat exchanger geometry and the inlet and outlet temperatures of both hot and cold fluid streams.

  • Obtained graphically from charts based on two dimensionless temperature ratios:

    • Temperature Ratio P: $P = \frac{t<em>{out} - t</em>{in}}{T<em>{in} - t</em>{in}}$ ($0 \le P \le 1$)

    • Temperature Ratio R: $R = \frac{T<em>{in} - T</em>{out}}{t<em>{out} - t</em>{in}}$ ($0 \le R \le \infty$)

  • $T$ denotes shell-side temperature and $t$ denotes tube-side temperature.

LMTD Method Procedure for Heat Exchanger Design

1. Select Heat Exchanger Type: Choose a suitable type for the application.

2. Determine Unknown Temperatures and Heat Transfer Rate: Use an energy balance.

3. Calculate Log Mean Temperature Difference ($\Delta T_{lm}$): Apply the appropriate formula for the flow configuration.

4. Calculate Correction Factor ($F$): If necessary, for shell-and-tube or cross-flow exchangers.

5. Obtain Overall Heat Transfer Coefficient ($U$): Either from given data, calculations, or representative tables.

6. Calculate Heat Transfer Surface Area ($A<em>s$): Using the equation $Q = U A F \Delta T</em>{LMTD,counterflow}$.

7. Select Heat Exchanger: Choose an actual heat exchanger with a surface area equal to or larger than the calculated $A_s$.

Effectiveness-NTU Method

• Primarily used when the heat exchanger performance (outlet temperatures) needs to be predicted for a given heat exchanger (known $U$, $A_{s}$).

Key Dimensionless Parameters

• Effectiveness ($\epsilon$):

  • Defined as the ratio of the actual heat transfer rate ($Q<em>{act}$) to the maximum possible heat transfer rate ($Q</em>{max}$).

  • $\epsilon = \frac{Q<em>{act}}{Q</em>{max}}$

  • $Q<em>{act} = \epsilon C</em>{min} (T<em>{h,in} - T</em>{c,in})$

• Maximum Possible Heat Transfer Rate ($Q_{max}$):

  • Achieved if one fluid undergoes an infinite temperature change (i.e., its temperature reaches the inlet temperature of the other fluid).

  • $Q<em>{max} = C</em>{min} (T<em>{h,in} - T</em>{c,in})$

• Heat Capacity Rate Ratio ($C_r$):

  • Ratio of the minimum heat capacity rate to the maximum heat capacity rate.

  • $C<em>r = \frac{C</em>{min}}{C_{max}}$

  • $C<em>{min} = \min(\dot{m}</em>h c<em>{p,h}, \dot{m}</em>c c_{p,c})$

  • $C<em>{max} = \max(\dot{m}</em>h c<em>{p,h}, \dot{m}</em>c c_{p,c})$

• Number of Transfer Units (NTU):

  • A dimensionless group related to the size of the heat exchanger.

  • $NTU = \frac{UA}{C_{min}}$

  • A larger NTU generally indicates a larger heat exchanger surface area ($A_s$).

• Relationship: The effectiveness ($\epsilon$) of a heat exchanger is a function of the NTU and the capacity ratio ($C_r$).

  • $\epsilon = f(NTU, C_r)$

Special Case: Phase Change (Condensers and Boilers)

• When one of the fluids undergoes a phase change, its temperature remains constant, implying an infinitely large heat capacity rate for that fluid. This results in $C_r = 0$.

• For $C_r = 0$, the effectiveness relation for any type of heat exchanger simplifies to:

  • $\epsilon = 1 - \exp(-NTU)$

Effectiveness Relations (Examples)

• Parallel-Flow (Double Pipe): $\epsilon = \frac{1 - \exp[-NTU(1 + C<em>r)]}{1 + C</em>r}$

• Counter-Flow (Double Pipe): $\epsilon = \frac{1 - \exp[-NTU(1 - C<em>r)]}{1 - C</em>r \exp[-NTU(1 - C<em>r)]}$ (for Cr < 1) and $\epsilon = \frac{NTU}{1 + NTU}$ (for $C_r = 1$)

• Shell-and-Tube (One-shell pass, $2,4,…$ tube passes): $\epsilon = 2 \left[ 1+C<em>r+\sqrt{1+C</em>r^2} \frac{1+\exp(-NTU<em>1 \sqrt{1+C</em>r^2})}{1-\exp(-NTU<em>1 \sqrt{1+C</em>r^2})} \right]^{-1}$ (for C_r < 1)

• Cross-Flow (Single-pass, both fluids unmixed): $\epsilon = 1 - \exp \left[ \frac{NTU^{0.22}}{C<em>r} \left( \exp(-C</em>r NTU^{0.78}) - 1 \right) \right]$

• General for $C_r = 0$: $\epsilon = 1 - \exp(-NTU)$

• These relations are also available graphically as charts showing effectiveness vs. NTU for different $C_r$ values, specific to each heat exchanger type.

Examples and Applications

Example: Condenser with Fouling (Illustrates LMTD Method Concept)

• Problem: Calculate $U_o$ without and with fouling for a condenser, then determine an outlet water temperature.

• Given: $D<em>i = 1.5 \text{ cm}$, $h</em>i = 800 \text{ W/m}^2\text{K}$, $R<em>{f,i} = 0.0004 \text{ m}^2\text{K/W}$. $D</em>o = 1.9 \text{ cm}$, $h<em>o = 1200 \text{ W/m}^2\text{K}$, $R</em>{f,o} = 0.0001 \text{ m}^2\text{K/W}$.

• Results:

  • $U_o$ (without fouling): $2255 \text{ W/m}^2\text{K}$.

  • $U_o$ (with fouling): $1800 \text{ W/m}^2\text{K}$.

  • Outlet cooling water temperature (condensing $10 \text{ kg/s}$ steam at $0.0622 \text{ bar}$, inlet water $15^<br>\circ C$): $T_{cw,o} = 29.4^<br>\circ C$.

Example: Shell-and-Tube Heat Exchanger Analysis (Illustrates LMTD Method with Correction Factor)

• Problem: Determine the rate of heat transfer in a two-shell and four-tube pass heat exchanger heating glycerin with hot water.

• Given: $\dot{m}<em>{water}$ and $\dot{m}</em>{glycerin}$ (implicit), $T<em>{c,in} = 20^ \circ C$, $T</em>{c,out} = 50^<br>\circ C$, $T<em>{h,in} = 80^ \circ C$, $T</em>{h,out} = 40^<br>\circ C$, tube diameter $D = 2 \text{ cm}$, total tube length $L = 60 \text{ m}$, $h<em>o = 25 \text{ W/m}^2\text{C}$ (glycerin side), $h</em>i = 160 \text{ W/m}^2\text{C}$ (water side). Fouling factor $R_f = 0.0006 \text{ m}^2\text{C/W}$ on the outer surface.

• Calculations:

  • Heat transfer area: $A = \pi D L = \pi (0.02 \text{ m})(60 \text{ m}) = 3.77 \text{ m}^2$.

  • LMTD for counterflow: $\Delta T_{LMTD,CF} = 24.7^<br>\circ C$.

  • Temperature ratios: $P = 0.67$, $R = 0.75$.

  • Correction factor (from chart for 2-shell, 4-tube passes): $F = 0.92$.

  • (a) No Fouling:

    • Overall heat transfer coefficient: $U = 21.6 \text{ W/(m}^2\text{C)}$.

    • Rate of heat transfer: $Q = U A F \Delta T_{LMTD,CF} = [21.6 \text{ W/(m}^2\text{C)}](3.77 \text{ m}^2)(0.92)(24.7^<br>\circ C) = 1843 \text{ W}$.

  • (b) With Fouling ($R_f = 0.0006 \text{ m}^2\text{C/W})$ on outer surface):

    • Overall heat transfer coefficient: $U = 21.3 \text{ W/(m}^2\text{C)}$.

    • Rate of heat transfer: $Q = U A F \Delta T_{LMTD,CF} = [21.3 \text{ W/(m}^2\text{C)}](3.77 \text{ m}^2)(0.92)(24.7^<br>\circ C) = 1817.4 \text{ W}$.

    • Note: Heat transfer rate decreases with fouling, though not drastically in this case due to a low convection coefficient on the outer surface.

Example: Counterflow Double-Pipe HE (Compares LMTD and Effectiveness-NTU Methods)

• Problem: Determine the length of a counterflow double-pipe HE to heat water from $20^<br>\circ C$ to $80^<br>\circ C$ using geothermal water from $160^<br>\circ C$, given flow rates, overall $U$, and tube diameter.

• Given: Water $\dot{m}<em>c = 1.2 \text{ kg/s}$, $c</em>{p,c} = 4.18 \text{ kJ/(kg}^\circ\text{C})$. Geothermal water $\dot{m}<em>h = 2 \text{ kg/s}$, $c</em>{p,h} = 4.31 \text{ kJ/(kg}^\circ\text{C})$. $U = 640 \text{ W/(m}^2\text{C)}$. Inner tube diameter $D = 1.5 \text{ cm}$.

• Results (LMTD Method):

  • Actual heat transfer rate to water: $Q = 301.0 \text{ kW}$.

  • Geothermal water outlet temperature: $T_{h,out} = 125.1^<br>\circ C$.

  • $\Delta T<em>1 = 105.1^ \circ C$, $\Delta T</em>2 = 80^<br>\circ C$.

  • $\Delta T_{LMTD} = 92.0^<br>\circ C$.

  • Required surface area: $A = 5.11 \text{ m}^2$.

  • Required length: $L = 108.4 \text{ m}$.

• Results (Effectiveness-NTU Method):

  • Heat capacity rates: $C<em>c = 5.02 \text{ kW/}^\circ C$ ($C</em>{min}$), $C<em>h = 8.62 \text{ kW/}^\circ C$ ($C</em>{max}$).

  • Capacity ratio: $C<em>r = C</em>{min}/C_{max} = 0.583$.

  • Maximum heat transfer rate: $Q_{max} = 702.8 \text{ kW}$.

  • Actual heat transfer rate to water: $Q_{act} = 301.1 \text{ kW}$.

  • Effectiveness: $\epsilon = Q<em>{act}/Q</em>{max} = 0.428$.

  • NTU (from effectiveness relation for counter-flow HE): $NTU = 0.651$.

  • Required surface area: $A = NTU \cdot C_{min} / U = 5.11 \text{ m}^2$.

  • Required length: $L = 108.4 \text{ m}$.

• Conclusion: Both the LMTD and Effectiveness-NTU methods yield the same result for the required heat exchanger length.

Other Example Problems Mentioned

• Design of a Shell-and-Tube Heat Exchanger: Calculates the number of tube passes, number of tubes per shell pass, and consistent tube length for heating water with hot water.

• Problem 2.3 (Hodge & Taylor) - Cross-flow Heat Recovery Unit: Involves engine exhaust (mixed flow) and water (unmixed flow). Seeks to find the pressure to prevent boiling, the rate of heat transfer, and the exit temperatures.