Shell-and-Tube Heat Exchanger Rating: Bell-Delaware Shell-Side Hydraulics & LMTD

Executive Summary & Engineering Relevance

Shell-and-tube heat exchangers (STHE) are the workhorses of chemical process thermal integration, accounting for over $65\%$ of all thermal transfer equipment installed across petrochemical plants, petroleum refineries, and power generation stations.

Thermal design of an industrial exchanger involves two coupled disciplines: 1. Thermodynamic Rating: Establishing the effective temperature driving force (Logarithmic Mean Temperature Difference, LMTD) and determining the Multipass Correction Factor ($F$) to account for non-ideal co-current/counter-current flow passes within standard TEMA configurations. 2. Shell-Side Hydrodynamics: Predicting heat transfer coefficients and pressure drops across the tube bundle using the rigorous Bell-Delaware Method (Tinker/Delaware model), which replaces simplistic Kern equations by quantifying the five internal leakage, bypass, and crossflow streams.

A critical design pitfall in process revamps is operating with an $F$-factor below $0.80$ or failing to anticipate a temperature cross—where the cold stream outlet temperature exceeds the hot stream outlet temperature ($T_{c,out} > T_{h,out}$). In a 1-2 multipass exchanger, this causes severe thermal inefficiency, requiring multiple shell passes in series or pure counter-current configurations.

This engineering guide provides the mathematical formulations for counter-current LMTD, analytical $F$-factor derivation for 1-2 and 2-4 TEMA exchangers, the five Bell-Delaware flow streams, shell-side pressure drop evaluation, and a step-by-step worked industrial example for a kerosene-to-crude preheat exchanger.

1. Thermodynamic Driving Force: Counter-Current LMTD

For steady-state heat exchange without phase change where overall heat transfer coefficient ($U$) and fluid heat capacities ($C_p$) are constant along the exchanger length, the rate equation is:

$$Q = U \cdot A \cdot \Delta T_{eff} = U \cdot A \cdot (F \cdot \Delta T_{lm})$$

Temperature Profiles in Pure Counter-Current Flow:
Temperature T [°C]
Th,in +-------------------------\
      |                          \  Hot Stream (Th,in -> Th,out)
      |                           \
      |                            + Th,out
      |
Tc,out+                            + Tc,in
       \                          /
        \                        /  Cold Stream (Tc,in -> Tc,out)
         \----------------------/
  Position: x = 0 (Left End)        Position: x = L (Right End)
  Delta T1 = Th,in - Tc,out        Delta T2 = Th,out - Tc,in

1.1 Pure Counter-Current Logarithmic Mean ($\Delta T_{lm}$)

$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln\left(\frac{\Delta T_1}{\Delta T_2} \right)}$$

Where: * $\Delta T_1 = T_{h,in} - T_{c,out}$: Temperature approach at the hot end ($K$ or $^\circ C$). * $\Delta T_2 = T_{h,out} - T_{c,in}$: Temperature approach at the cold end ($K$ or $^\circ C$). * (Special Case: If $\Delta T_1 = \Delta T_2$, then $\Delta T_{lm} = \Delta T_1 = \Delta T_2$).

2. Multipass Correction Factor ($F$) & Temperature Cross

In most industrial exchangers (e.g., TEMA E shells with 2, 4, or 6 tube passes), fluid in the tubes flows co-currently with the shell fluid during half the passes and counter-currently during the other half. The true mean temperature difference is smaller than pure counter-current LMTD:

$$\Delta T_{eff} = F \cdot \Delta T_{lm} \quad (F \le 1.0)$$

2.1 Thermal Dimensionless Parameters ($P$ and $R$)

The correction factor is parameterized by two dimensionless numbers:

1. Thermal Effectiveness / Temperature Efficiency ($P$): $$P = \frac{T_{c,out} - T_{c,in}}{T_{h,in} - T_{c,in}} \quad (0 < P < 1.0)$$ Measures the heating of the cold stream relative to the maximum theoretical temperature difference.

2. Heat Capacity Rate Ratio ($R$): $$R = \frac{T_{h,in} - T_{h,out}}{T_{c,out} - T_{c,in}} = \frac{(\dot{m} C_p)_c}{(\dot{m} C_p)_h}$$ Ratio of cold fluid capacity rate to hot fluid capacity rate.

TEMA 1-2 Exchanger Multipass Configuration:
               +-------------------------------------------+
Hot Shell In ->| ===== Shell Flow (Crosses Baffles) ======>| -> Hot Shell Out
               |  +-------------------------------------+  |
               |  | === Tube Pass 1 (Co-current) ===>   |  |
Cold Tube In ->|==|                                     |==| (Floating Head / U-Tube)
               |  | <=== Tube Pass 2 (Counter-current)==|  |
Cold Tube Out<-|==+-------------------------------------+  |
               +-------------------------------------------+

2.2 Analytical Formula for 1 Shell Pass, 2 or more Tube Passes (1-2 TEMA E)

$$F = \frac{\sqrt{R^2 + 1}}{R - 1} \cdot \frac{\ln\left(\frac{1 - P}{1 - P \cdot R} \right)}{\ln\left[ \frac{2 - P \left(R + 1 - \sqrt{R^2 + 1} \right)}{2 - P \left(R + 1 + \sqrt{R^2 + 1} \right)} \right]}$$

2.3 The $F \ge 0.80$ Rule & Temperature Cross

• When a temperature cross occurs ($T_{c,out} > T_{h,out}$), the denominator argument inside the logarithm approaches zero, driving $F$ toward zero vertically.
• TEMA Best Practice: Never design a heat exchanger with $F < 0.80$. Below $0.80$, the slope $dF/dP$ is extremely steep; any small operational fluctuation in flow or fouling will cause the exchanger to suffer an immediate loss of heat duty.
• Solution for $F < 0.80$: Split the duty into two shell passes in series (2-4 exchanger, TEMA F) or multiple TEMA E shells in series.

3. Shell-Side Hydraulics: The Bell-Delaware Method

Simplistic sizing methods (like Donald Q. Kern's 1950 correlation) assume that 100% of the shell fluid traverses the tube bundle in pure crossflow. In reality, manufacturing tolerances between tubes, baffles, and shell produce internal bypassing and leakage.

The Bell-Delaware Method (Delaware Project, 1963) divides the shell-side flow into five distinct streams:

Bell-Delaware Five Shell-Side Flow Streams:
+-----------------------------------------------------------------------------+
| Stream B (Main Crossflow Stream):                                           |
|   - The ONLY effective heat transfer stream (typically 40% - 65% of total). |
|   - Traverses the tube bundle in crossflow across each baffle bay.          |
+-----------------------------------------------------------------------------+
| Stream A (Tube-to-Baffle Hole Leakage):                                     |
|   - Leaks through annular gaps between tube OD and drilled baffle holes.    |
|   - Reduces pressure drop, but moderately effective for heat transfer.      |
+-----------------------------------------------------------------------------+
| Stream E (Baffle-to-Shell Diametral Leakage):                               |
|   - Bypasses through the clearance between baffle outer edge and shell ID.  |
|   - HIGHLY INEFFECTIVE: Forms a stagnant thermal boundary near shell wall.  |
+-----------------------------------------------------------------------------+
| Stream C (Bundle Bypass Stream):                                            |
|   - Flows around the outer tube limit (OTL) clearance to shell.             |
|   - Controlled by adding sealing strips and dummy tie-rods.                 |
+-----------------------------------------------------------------------------+
| Stream F (Pass-Partition Lane Bypass):                                      |
|   - Flows through tube pass partition lanes in multipass tube layouts.      |
+-----------------------------------------------------------------------------+

Bell-Delaware Correction Factors

The true shell-side heat transfer coefficient ($h_s$) is calculated by derating an idealized crossflow bank coefficient ($h_{ideal}$):

$$h_s = h_{ideal} \cdot J_c \cdot J_l \cdot J_b \cdot J_s \cdot J_r$$

Where: * $J_c$: Baffle cut and window correction factor (typically $0.85 - 1.15$). * $J_l$: Baffle leakage correction for Streams A and E (typically $0.65 - 0.85$). * $J_b$: Bundle bypass correction for Stream C ($0.70 - 0.95$, improved by sealing strips). * $J_s$: Unequal baffle spacing correction at inlet/outlet nozzles ($0.85 - 1.0$). * $J_r$: Laminar flow adverse temperature gradient correction.

Ideal Shell-Side Crossflow Area ($S_m$)

The minimum flow area between tubes at the shell centerline:

$$S_m = B \cdot \left[ (D_s - D_{otl}) + \frac{D_{otl} - d_o}{p_t} \cdot (p_t - d_o) \right]$$

Where: * $B$: Central baffle spacing ($m$, typically $0.3 - 0.5 D_s$). * $D_s$: Shell inside diameter ($m$). * $D_{otl}$: Outer tube limit diameter ($m$). * $d_o$: Tube outside diameter ($m$). * $p_t$: Tube pitch ($m$, standard triangular $30^\circ$ or square $90^\circ$, pitch ratio $p_t / d_o \ge 1.25$).

4. Shell-Side Pressure Drop Formulation

Total shell-side pressure drop ($\Delta P_s$) combines crossflow zone losses, window zone losses, and end baffle spaces:

$$\Delta P_s = \Delta P_{bi} \cdot (N_b - 1) \cdot R_l \cdot R_b + N_b \cdot \Delta P_w \cdot R_l + 2 \Delta P_{bi} \cdot \left(1 + \frac{N_{cw}}{N_c} \right) \cdot R_b \cdot R_s$$

Where: * $N_b$: Number of baffles ($N_b = L / B - 1$). * $\Delta P_{bi}$: Ideal crossflow pressure drop between baffle tips. * $\Delta P_w$: Ideal window pressure drop. * $R_l, R_b, R_s$: Bell-Delaware pressure drop correction factors for leakage, bypass, and nozzle spacing.

5. Comprehensive Step-by-Step Industrial Worked Example

Design Problem Statement:

An oil refinery crude distillation unit (CDU) uses hot kerosene product to preheat cold desalted crude oil: * Thermal Duty: $Q = 2,400\text{ kW}$ ($2.40 \times 10^6\text{ W}$). * Hot Stream (Kerosene, Shell Side): * Inlet Temp: $T_{h,in} = 175.0^\circ\text{C}$ * Outlet Temp: $T_{h,out} = 95.0^\circ\text{C}$ * Cold Stream (Crude Oil, Tube Side): * Inlet Temp: $T_{c,in} = 40.0^\circ\text{C}$ * Outlet Temp: $T_{c,out} = 85.0^\circ\text{C}$ * Exchanger Configuration: TEMA E Shell with 1 Shell Pass and 2 Tube Passes (1-2 STHE). * Assumed Overall Heat Transfer Coefficient: $U = 380\text{ W/m}^2\cdot\text{K}$ ($66.9\text{ Btu/h}\cdot\text{ft}^2\cdot^\circ\text{F}$). * Mechanical Geometry: Shell ID $D_s = 650\text{ mm}$ ($0.65\text{ m}$), Baffle Spacing $B = 300\text{ mm}$, $25\%$ segmental baffle cut.

Calculate: 1. Counter-current LMTD ($\Delta T_{lm}$). 2. Dimensionless parameters $P$ and $R$. 3. Multipass correction factor ($F$) and verify against the $0.80$ TEMA threshold. 4. Effective temperature difference ($\Delta T_{eff}$). 5. Required total heat transfer surface area ($A_{req}$). 6. Shell-side crossflow area ($S_m$).

Step 1: Counter-Current LMTD ($\Delta T_{lm}$)

Temperature approaches: $$\Delta T_1 = T_{h,in} - T_{c,out} = 175.0 - 85.0 = \mathbf{90.0^\circ\text{C}}$$ $$\Delta T_2 = T_{h,out} - T_{c,in} = 95.0 - 40.0 = \mathbf{55.0^\circ\text{C}}$$

Logarithmic mean: $$\Delta T_{lm} = \frac{90.0 - 55.0}{\ln(90.0 / 55.0)} = \frac{35.0}{\ln(1.6364)} = \frac{35.0}{0.4925} = \mathbf{71.07^\circ\text{C}}$$

Step 2: Dimensionless Parameters ($P$ and $R$)

$$P = \frac{T_{c,out} - T_{c,in}}{T_{h,in} - T_{c,in}} = \frac{85.0 - 40.0}{175.0 - 40.0} = \frac{45.0}{135.0} = \mathbf{0.3333}$$

$$R = \frac{T_{h,in} - T_{h,out}}{T_{c,out} - T_{c,in}} = \frac{175.0 - 95.0}{85.0 - 40.0} = \frac{80.0}{45.0} = \mathbf{1.7778}$$

Check product $P \cdot R$: $$P \cdot R = 0.3333 \times 1.7778 = 0.5926 < 1.0 \implies \text{No temperature cross!}$$

Step 3: Multipass Correction Factor ($F$)

Radical term: $$\sqrt{R^2 + 1} = \sqrt{(1.7778)^2 + 1} = \sqrt{3.1606 + 1} = \sqrt{4.1606} = 2.0397$$

Numerator argument: $$\frac{1 - P}{1 - P \cdot R} = \frac{1 - 0.3333}{1 - 0.5926} = \frac{0.6667}{0.4074} = 1.6365$$ $$\ln(1.6365) = 0.4925$$

Numerator value: $$\text{Num} = \left(\frac{2.0397}{1.7778 - 1} \right) \times 0.4925 = \left(\frac{2.0397}{0.7778} \right) \times 0.4925 = 2.6224 \times 0.4925 = \mathbf{1.2915}$$

Denominator argument: $$R + 1 = 1.7778 + 1 = 2.7778$$ $$2.7778 - 2.0397 = 0.7381$$ $$2.7778 + 2.0397 = 4.8175$$

$$2 - P \cdot (0.7381) = 2 - (0.3333 \times 0.7381) = 2 - 0.2460 = 1.7540$$ $$2 - P \cdot (4.8175) = 2 - (0.3333 \times 4.8175) = 2 - 1.6057 = 0.3943$$

$$\frac{1.7540}{0.3943} = 4.4484$$ $$\text{Denom} = \ln(4.4484) = \mathbf{1.4926}$$

Calculating $F$: $$F = \frac{1.2915}{1.4926} = \mathbf{0.8653}$$

Assessment against TEMA Standards: $$F = 0.865 > 0.800$$ The 1-2 configuration is highly thermally efficient and stable. A single TEMA E shell pass is fully validated.

Step 4: Effective Temperature Difference & Required Area

$$\Delta T_{eff} = F \cdot \Delta T_{lm} = 0.8653 \times 71.07^\circ\text{C} = \mathbf{61.50^\circ\text{C}}$$

Required heat transfer area: $$A_{req} = \frac{Q}{U \cdot \Delta T_{eff}} = \frac{2,400,000\text{ W}}{380\text{ W/m}^2\cdot\text{K} \times 61.50\text{ K}} = \frac{2,400,000}{23,370} = \mathbf{102.7\text{ m}^2} \quad (1,105.4\text{ ft}^2)$$

Step 5: Shell-Side Crossflow Area ($S_m$)

With $D_s = 0.65\text{ m}$, $B = 0.30\text{ m}$, and standard clearances: $$S_m \approx B \cdot D_s \cdot 0.22 = 0.30 \times 0.65 \times 0.22 = \mathbf{0.0429\text{ m}^2} \quad (429\text{ cm}^2)$$

If kerosene mass flow rate is $22.5\text{ kg/s}$ (density $\rho = 780\text{ kg/m}^3$): $$v_{cross} = \frac{\dot{m}}{\rho \cdot S_m} = \frac{22.5}{780 \times 0.0429} = \mathbf{0.672\text{ m/s}}$$

Hydraulic Check: A crossflow velocity of $0.67\text{ m/s}$ is well within the recommended non-erosive, non-vibrating liquid window of $0.5 - 1.2\text{ m/s}$, ensuring low shell-side pressure drop ($< 0.4\text{ bar}$) and preventing tube acoustic resonance.

6. Common Design Traps & Best Practice Checklist

1. Beware of Low $F$-Factor ($F < 0.80$): If $F < 0.80$, never attempt to compensate by simply adding more tube surface area. The exchanger will suffer operational instability during plant turn-down. Reconfigure to two shells in series ($N_s = 2$).
2. Optimize Baffle Cut (20% to 30%): Baffle cuts smaller than $15\%$ cause extreme window pressure drops and stagnant zones; cuts larger than $35\%$ cause longitudinal flow rather than crossflow, destroying heat transfer efficiency.
3. Control Stream C with Sealing Strips: For tube bundles with large bypass lanes (e.g., pull-through floating heads TEMA T), always install sealing strips every 5 to 7 tube rows to force bypass fluid back into crossflow Stream B.
4. Vibration Screening (TEMA Section 5): Always calculate the vortex shedding frequency ($f_v = S_t v / d_o$) and fluid elastic instability constant ($V_{crit}$) to ensure flow velocities do not induce destructive acoustic or mechanical tube vibration.

References & Industry Standards

1. Tubular Exchanger Manufacturers Association. (2019). Standards of the Tubular Exchanger Manufacturers Association (10th Edition). TEMA, Tarrytown, NY.
2. Bell, K. J. (1981). Delaware Method for Shell-Side Design. In Heat Exchangers: Thermal-Hydraulic Fundamentals and Design, Hemisphere Publishing, pp. 581–618.
3. Kern, D. Q. (1950). Process Heat Transfer. McGraw-Hill Book Company, New York.
4. Serth, R. W., & Lestina, T. G. (2014). Process Heat Transfer: Principles, Applications and Rules of Thumb (2nd Edition). Academic Press.
5. Bowman, R. A., Mueller, A. C., & Nagle, W. M. (1940). Mean Temperature Difference in Design. Transactions of ASME, 62, 283–294.