Heat Exchanger Rating

1. Principles of Heat Exchanger Thermal Rating

Heat exchangers are fundamental process units designed to transfer thermal energy between two fluid streams without intermixing. The rigorous thermal rating of a shell-and-tube or plate heat exchanger requires solving coupled energy and mass balances to determine the required heat transfer area, leaving temperatures, and hydraulic pressure drops.

The Heat Exchanger Rating tool automates these intensive calculations using the foundational Logarithmic Mean Temperature Difference (LMTD) method and the rigorous $\varepsilon$-NTU (Effectiveness-Number of Transfer Units) method.

2. The Fundamental Heat Transfer Equation

The total heat duty ($Q$) transferred across the exchanger surface is governed by the global rate equation:

$$Q = U \cdot A \cdot \Delta T_{LM} \cdot F$$

Where:

• $Q$ = Total heat duty (W or BTU/hr)
• $U$ = Overall heat transfer coefficient ($\text{W}/m^2\cdot\text{K}$)
• $A$ = Effective heat transfer surface area ($m^2$)
• $\Delta T_{LM}$ = Log Mean Temperature Difference (K or $^\circ$C)
• $F$ = LMTD Correction factor (dimensionless)

The overall coefficient ($U$) is the reciprocal sum of thermal resistances, including convective films, wall conduction, and critical fouling factors (dirt films):

$$\frac{1}{U} = \frac{1}{h_{hot}} + R_{f,hot} + \frac{t_{wall}}{k_{wall}} + R_{f,cold} + \frac{1}{h_{cold}}$$

3. The LMTD and F-Factor Correction

For pure counter-current flow, the driving temperature force is the LMTD:

$$\Delta T_{LM} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}$$

However, true counter-current flow is rarely achieved in multipass shell-and-tube heat exchangers (e.g., 1-2, 2-4 TEMA configurations). Due to cross-flow and co-current passes, the effective temperature difference is reduced. The correction factor $F$ accounts for this geometric deviation.

If $F < 0.80$, a single shell pass is typically thermodynamically inefficient or unfeasible (causing temperature cross), requiring multiple shells in series.

4. The Effectiveness-NTU ($\varepsilon$-NTU) Method

When the outlet temperatures of the fluid streams are unknown, the LMTD method requires tedious iterative trials. The $\varepsilon$-NTU method solves this analytically.

Heat capacity rates are defined for both streams: $C_h = \dot{m}_h c_{p,h}$ and $C_c = \dot{m}_c c_{p,c}$. The minimum capacity rate is $C_{min}$.

The Number of Transfer Units (NTU) represents the thermal size of the exchanger:

$$NTU = \frac{U A}{C_{min}}$$

Effectiveness ($\varepsilon$) is the ratio of actual heat transfer to the maximum possible heat transfer:

$$\varepsilon = \frac{Q}{Q_{max}} = \frac{Q}{C_{min}(T_{h,in} - T_{c,in})}$$

Using rigorous algebraic $\varepsilon$-NTU relations for specific TEMA geometries, the outlet temperatures can be resolved instantly without iteration.

5. Step-by-Step Worked Example: LMTD Rating of an Oil Cooler

Scenario: We are rating a 1-shell, 2-tube pass heat exchanger used to cool hot oil using cooling water.

• Hot Oil (Shell side): $T_{h,in} = 120^\circ\text{C}$, $T_{h,out} = 60^\circ\text{C}$, Heat duty $Q = 500\,\text{kW}$
• Cooling Water (Tube side): $T_{c,in} = 25^\circ\text{C}$, $T_{c,out} = 45^\circ\text{C}$
• Overall Heat Transfer Coefficient: $U = 450\,\text{W}/m^2\cdot\text{K}$

1. Calculate the Terminal Temperature Differences:

For a counter-current flow assumption:

$$\Delta T_1 = T_{h,in} - T_{c,out} = 120 - 45 = 75^\circ\text{C}$$ $$\Delta T_2 = T_{h,out} - T_{c,in} = 60 - 25 = 35^\circ\text{C}$$

2. Calculate the Log Mean Temperature Difference (LMTD):

$$\Delta T_{LM} = \frac{75 - 35}{\ln(75 / 35)} = \frac{40}{0.7621} = 52.48^\circ\text{C}$$

3. Apply the F-Factor Correction:

For a 1-2 shell and tube exchanger, using the standard TEMA charts based on $R$ and $P$ ratios, we find $F \approx 0.88$.

$$\Delta T_{effective} = 52.48 \times 0.88 = 46.18^\circ\text{C}$$

4. Calculate Required Heat Transfer Area ($A$):

$$A = \frac{Q}{U \cdot \Delta T_{effective}} = \frac{500,000}{450 \times 46.18} = 24.06\,\text{m}^2$$