Unit 5: Heat Exchangers – Quick Notes

Introduction to Heat Exchangers

• Devices that transfer heat between two fluids at different temperatures; heat transfer involves convection on both sides and conduction through the wall.
• Use an overall heat transfer coefficient U to simplify analysis.
• Governing relation (rate of heat transfer): $Q = U\,A\,\Delta T<em>m$ where $\Delta T</em>m$ is the mean temperature difference between the two fluids.

Classification

• A. By nature of heat transfer process:
  • Direct contact: two immiscible fluids exchange heat (e.g., cooling towers, jet condensers).
  • Recuperators (Transfer type): fluids separated by a solid wall.
  • Regenerators (Storage type): hot and cold fluids alternate on the same surface (matrix stores/releases heat).
• B. By constructional features:
  • Tubular (double-pipe)
  • Shell-and-tube
  • Finned-tube
  • Compact heat exchangers
• C. By flow arrangement:
  • Parallel flow
  • Counter flow
  • Cross flow
• D. By physical state of exchanging fluids:
  • Condensing or evaporating fluids (condenser/boiler/evaporator)

Regenerative heat exchangers (storage type)

• Static-type: porous heat-storage matrix (e.g., ceramic wire mesh).
• Dynamic-type: rotating drum with alternating hot/cold flow regions.

Shell‑and‑tube heat exchangers (key features)

• Fluid on shell side vs tube side; tubes arranged inside a shell.
• Baffles: force shell-side flow across tubes and improve heat transfer and spacing.
• Headers: collect/distribute flow at ends.
• Not typically used in small automotive/aircraft/marine apps due to size/weight.
• Classification by number of shell/tube passes (e.g., 1 shell pass, 2/tube passes; 2 shell passes, 4/tube passes, etc.).

Constructional features (selection)

• Tubular (double-pipe): concentric tubes.
• Shell-and-tube: many tubes in a shell; common in industry.
• Finned-tube: enhanced surface on one side (e.g., radiator).
• Compact: high area density (area/volume) for high heat transfer rates.

Flow arrangement details

• Parallel flow: hot and cold enter same end, flow in same direction.
• Counter flow: enter opposite ends, flow in opposite directions.
• Cross flow: fluids cross; may be unmixed/mixed in respective streams.

Phase-change devices

• Condenser: hot fluid (condensing) temperature is approximately constant.
• Evaporator: cold fluid (evaporating) temperature is approximately constant.
• In phase-change cases, the heat capacity rate effectively becomes infinite, and temperature change tends to zero while heat transfer occurs via latent heat.

Overall heat transfer coefficient (U) and thermal resistance

• Heat transfer resistance network typically includes two convection resistances (hot/cold sides) and wall conduction resistance.
• Total resistance Rtotal combines all resistances; $U = 1/R_total$ and $Q = U A \Delta T</em>m$.
• Note: Ui Ai = Uo Ao only if the respective areas are equal; otherwise U is defined on a chosen reference area.
• Fouling increases thermal resistance, reducing U. Fouling factor Rf adds to total resistance: $R'_{total} = R_{total} + R_f\,,$ hence $U' = 1/R'_{total}.$ The fouling factor depends on temperature, velocity, and service duration.

Analysis assumptions (steady-state approach)

• Long-run steady operation with constant mass flow rates; properties treated as constants over the temperature range.
• Negligible kinetic/potential energy changes; axial conduction along the tube is negligible.
• Energy balance: rate of heat transfer from hot fluid equals rate to cold fluid: $\dot{Q}<em>{hot} = \dot{Q}</em>{cold}$.
• Define heat capacity rate for each fluid: $C = \dot{m} C_p$.

Heat capacity rates and NTU method (ε-NTU)

• Fluid with larger capacity rate experiences smaller temperature change; the fluid with the smaller capacity rate is the minimum fluid.
• Define:
  • $C<em>h = \dot{m}</em>h C<em>{p,h}$, $C</em>c = \dot{m}<em>c C</em>{p,c}$
  • $C<em>{min} = \min(C</em>h, C_c)$
  • $C<em>{max} = \max(C</em>h, C_c)$
  • $\mathcal{C}<em>r = \frac{C</em>{min}}{C_{max}}$
  • $NTU = \frac{U A}{C_{min}}$
• Maximum possible heat transfer rate (assuming perfect exchange within capacity limits): $Q<em>{max} = C</em>{min} (T<em>{h,i} - T</em>{c,i})$
• Actual rate: $Q = \varepsilon Q_{max}$, where $\varepsilon$ is the effectiveness: a property of exchanger geometry/flow.
• For phase-change cases (condenser/boiler/evaporator): the fluid undergoing phase change has effectively infinite C; use the special condensers/boilers/evaporators relations from ε‑NTU tables. In general, for these, the heat transfer rate is set by the latent heat and the other fluid’s capacity.

LMTD method (logarithmic mean temperature difference)

• Used when outlet temperatures are known and UA is prescribed.
• For a given arrangement:
  • Define temperature differences at ends: $\Delta T<em>1 = T</em>{h,i} - T<em>{c,o}$, $\Delta T</em>2 = T<em>{h,o} - T</em>{c,i}$ for counterflow; for parallel flow, use the appropriate end temperatures.
  • Log-mean temperature difference: $\Delta T<em>m = \frac{\Delta T</em>1 - \Delta T<em>2}{\ln\left(\dfrac{\Delta T</em>1}{\Delta T_2}\right)}$
• Correction factor F accounts for deviations from ideal counterflow (multi-pass/cross-flow): $\Delta T<em>m = F\;\Delta T</em>{lm}$ with F in [0,1].
• F = 1 for boiler/condenser/evaporator; charts exist to find F for given geometry and temperatures.

Core equations (summary)

• Heat transfer rate: $Q = U A \Delta T_m$
• Heat capacity rate: $C = \dot{m} C_p$
• Minimum/maximum capacity rates: $C<em>{min}, \; C</em>{max}$; capacity ratio: $C<em>r = \frac{C</em>{min}}{C_{max}}$
• NTU: $NTU = \frac{U A}{C_{min}}$
• Maximum possible rate: $Q<em>{max} = C</em>{min}(T<em>{h,i} - T</em>{c,i})$
• Effectiveness: $\varepsilon = \frac{Q}{Q<em>{max}}$ and $Q = \varepsilon Q</em>{max}$
• Condensers/boilers/evaporators: treat one fluid as having effectively infinite C, and use the specialized NTU relationships from standard tables.

Worked-example themes (high-level)

• Heating water in a counter-flow exchanger with known UA to find length/area via LMTD.
• Cooling a hot oil by water in a multi-pass exchanger using ε‑NTU (since outlet temps are unknown).
• Condensing a chemical in a 2-pass shell-and-tube exchanger using NTU and correction factors.

Quick reference: key concepts for exam prep

• Distinguish direct contact, recuperators, regenerators by heat transfer mechanism.
• Recognize flow arrangements and their impact on ΔTm and ε.
• Use Q = U A ΔTm for overall analysis; use LMTD for design with known in/out temps; use NTU when outlet temps are unknown.
• For phase-change processes, expect infinite C in NTU method and use latent-heat–driven relations.
• Fouling raises R and lowers U; include Rf to modify total resistance.
• For multi-pass/cross-flow, apply correction factor F (F = 1 for condensers/boilers/evaporators) and use P, R parameters from F-charts.

Quick formulas to memorize (LaTeX)

• Heat capacity rate: $C = \dot{m} C_p$
• Minimum/maximum: $C<em>{min} = \min(C</em>h, C<em>c), \; C</em>{max} = \max(C<em>h, C</em>c)$
• Capacity ratio: $C<em>r = \frac{C</em>{min}}{C_{max}}$
• NTU: $NTU = \frac{U A}{C_{min}}$
• Maximum heat transfer: $Q<em>{max} = C</em>{min} (T<em>{h,i} - T</em>{c,i})$
• Actual heat transfer: $Q = \varepsilon Q_{max}$
• Overall heat transfer rate: $Q = U A \Delta T_m$
• Log-mean temperature difference (general): $\Delta T<em>m = \frac{\Delta T</em>1 - \Delta T<em>2}{\ln \left(\dfrac{\Delta T</em>1}{\Delta T_2}\right)}$
• Condensing/boiling: treat their C effectively infinite; Q driven by latent heat or the non-phase-changing fluid.