Elements


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: $$x=x_{lam}+(\text{Re}-2'300)\cdot\frac{x_{turb}-x_{lm}}{7'700}$$

  1. Züst S (2017) Model Based Optimization of Internal Heat Sources in Machine Tools. ETH Diss-Nr. 24482.
  2. VDI Wärmeatlas

Further readings

  1. Arc
  2. Bypass
  3. DefinedValues
  4. DefinedValues2
  5. Drilling
  6. ElbowFitting
  7. Extension
  8. Fitting
  9. FlowAround
  10. Helix
  11. PartialFlowAround
  12. Pipe
  13. Reduction
  14. StaggeredHelix
  15. WireTurbulator