Heat exchanger duty (LMTD)

The log-mean-temperature-difference rating equation, and the sharply nonlinear sensitivity of duty to the terminal temperature approach.

Governing equation

$Q$

Heat duty (W) — an unknown.

$\Delta T_1$

Terminal temperature difference at end 1 (K) — an unknown.

$\Delta T_2$

Terminal temperature difference at end 2 (K).

$U$

Overall heat transfer coefficient (W m⁻² K⁻¹).

$A$

Heat transfer area (m²).

Derivation

1. 1
   $$dQ = U\,(T_h - T_c)\,dA$$

   Local rate equation across a differential area, with the driving force the local hot-cold temperature difference ΔT.

2. 2
   $$\frac{d(\Delta T)}{dA} = -U\,\Delta T\left(\frac{1}{\dot m_h c_h} + \frac{1}{\dot m_c c_c}\right)$$

   For constant capacity rates the driving force decays exponentially along the exchanger.

3. 3
   $$Q = U A\,\frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$

   Integrating over the area replaces the varying ΔT with its log-mean — exact for constant U and capacity rates.

Jacobian

The matrix $J_{ij} = \partial F_i / \partial x_j$ of the residual vector with respect to the unknowns, at the representative operating point. Every value below is computed by the exact forward-mode autodiff engine the solver uses — click a non-zero cell to see the symbolic derivative, each one verified against that engine in our test suite. This is the block a Newton step factorizes.

References

• Incropera & DeWitt, Fundamentals of Heat and Mass Transfer, 6th ed., §11.3.
• Perry's Chemical Engineers' Handbook, 8th ed., §11 (heat-transfer equipment).