Mathematical Formulation

Energy Balance

For a steady heat exchanger,

\dot Q = \dot m_h c_{p,h}(T_{h,i}-T_{h,o})
= \dot m_c c_{p,c}(T_{c,o}-T_{c,i}).

LMTD Method

\dot Q = UA\Delta T_{lm}

\Delta T_{lm} =
\frac{\Delta T_1-\Delta T_2}
{\ln(\Delta T_1/\Delta T_2)}.

For shell-and-tube arrangements, a correction factor F is applied:

\dot Q = UAF\Delta T_{lm}.

Internal Convection

The reusable model evaluates

Re=\frac{\rho VD}{\mu}, \qquad
Pr=\frac{c_p\mu}{k_f}

and uses laminar or turbulent tube-flow correlations to determine Nu, then

h=\frac{Nu\,k_f}{D}.

Pressure Loss

The cold-side design constraint is evaluated with

\Delta p = f\frac{L}{D}\frac{\rho V^2}{2}.

The double-pipe design solves diameter and length so that heat duty and maximum pressure drop are satisfied simultaneously.

Boiler Laboratory Model

The notebook under labs/lab_3/ extends this methodology to a fire-tube boiler using: