Ducts are represented as series of duct elements. Each element represents a section with distinct geometric properties. The goal of DuctDesigner is to estimate the overall pressure loss $\Delta p$ and heat transfer $\dot{Q}$. $\Delta p$ is expressed as the superposition of the flow rate $\dot{V}$ dependet pressure losses of the elements:
$$\Delta p=\sum_{i=1}^N \Delta p_i(\dot{V})$$The total heat flux is calculated of the element's individual heat fluxes:
$$\dot{Q}=\sum_{i=1}^N \dot{Q}_i=\sum_{i=1}^N \alpha_i(\dot{V})\cdot S_i\cdot \left(T_{fluid,i}-T_{wall,i}\right)$$$\alpha_i$ is the flow rate dependent heat transfer coefficient, $S_i$ the elements surface area, while $T_{fluid,i}$ and $T_{wall,i}$ are the elemnt's fluid and wall temperature.
The pressure losses are calculated as $$\Delta p_{i}(\dot{V})= \left(\zeta_{i}\cdot\frac{l_{i}}{D_{i}}+\zeta_{i\rightarrow i+1}\right)\cdot\frac{\rho_{fluid}\cdot \dot{V}^2}{2\cdot A_{i}^2}$$ where $l_i$ is the length of the element and $D_i$ its hydraulic diameter, $A_i$ is cross-sectional area and $\rho_{fluid}$ the fluid's density. $\zeta_{i}$ is the pressure loss coefficient of the element, while $\zeta_{i\rightarrow i+1}$ is the one of the change in the cross-section from the current to the next element. The later are calculated by the element model Fitting.
$\alpha_i$ is calculated based on the Nusselt-number $\text{Nu}$ and the heat transfer coefficient $\lambda_{fl}$ of the fluid: $$\alpha_{i}(\dot{V})=\text{Nu}_{i}(\dot{V})\cdot\frac{\lambda_{fluid}}{D_{i}}$$
Assuming $l_i\ll\sum_{i=1}^N l_i\,\forall\,i=1\ldots N$, the temperature of an element is aproximated by the temperature of the upstream element plus the temperature raise caused by the heat input and the frictional losses: $$T_{i}=T_{i-1}+\frac{\dot{Q}+\Delta p_i\cdot\dot{V}}{c_p\cdot\cdot\rho_{fluid}\cdot l_i \cdot A_i}$$ $c_p$ is the specific heat capacity of the fluid.
The details regarding the computation of $\zeta$ and $\text{Nu}$ are shown in the element model descriptions (see further readings). Some models define coefficient for laminar and for turbular flow. In the frame work, laminar flow considered if $\text{Re}\leq2'300$ and turbulent flow is considered if $\text{Re}\geq10'000$. If $2'300\lt\text{Re}\lt10'000$, linear interpolation is used as suggested by VDI: $$x=x_{lam}+(\text{Re}-2'300)\cdot\frac{x_{turb}-x_{lm}}{7'700}$$