Temperature Constraints With Series Connections

Component Template

The component template Temperature_Constraints_Series_Connections.e-ctpl is located in the Component template library folder Operational Side Conditions.

Integration Into a Scheme

The following figure shows as an example how to integrate the component into a scheme.

How the Component Works

The system is optimized according to the Temperature Constraints Series Connections. Tutorial 65 demonstrates how the component works.

High temperature heat \(HT\) and low temperature heat \(LT\) are distinguished.

When a temperature limit is applied to a heat generator in a series connection, the output power is compared to the required power. Because the mass flow rate \( \dot{m}\) is constant and the specific heat capacity \( c_{p}\) is assumed to be constant, the thermal power \( \dot{Q}\) is proportional only to the temperature rise \( \Delta T\).

\( \begin{aligned} \dot{Q} &= c_{p} \cdot \dot{m} \cdot \Delta T \\[.3cm]
\dot{m}_{total} &= \dot{m}_{1} = \dot{m}_{2} = …= \dot{m}_{n}\\[.3cm]
\Rightarrow \dot{Q} &\sim \Delta T \\[.3cm]
\\[.3cm]
\end{aligned}\)

The temperature rise is known from the heat demand. With \(T_{LT,0} = T_{in}\) the ratio \( \alpha \) of the temperatures of the generated heat and the heat demand can be calculated as a secondary power constraint:

\( \begin{equation} \begin{aligned} \alpha_{n} &= \dfrac{\Delta T_{total}}{\sum\nolimits_{k=1}^n \Delta T_{LT,k}} \\[.3cm]
&= \dfrac{T_{HT} -T_{in}}{\sum\nolimits_{k=1}^n\ (T_{LT,k}- T_{LT,k-1})} \\[.3cm]
&= \dfrac{T_{HT} -T_{in}}{T_{LT,n} – T_{in}} \\[.3cm] \end{aligned}\end{equation}\\\)

Because heat components compensate temperature rises of heat generators connected upstream, the ratio is fomulated by the sum of the low temperature heat generators. With the maximum number of low temperature heat generators in the system under consideration 𝑁 ∈ ℕ applies:

\( \begin{equation} \begin{aligned} \forall n \in \{1, …,N\}: \ \
\dot{Q}_{total}\geq \alpha_{n} \sum_{k=1}^n \dot{Q}_{LT,k},\ \ with\ \ \dot{Q}_{total}=\dot{Q}_{HT}+\sum_{n\in N}\dot{Q}_{LT,n}\\\end{aligned}\end{equation}\\\)

The constraint shown is defined for each heat component to be controlled based on user input in the component Temperature Constraints Series Connections. The heat component can then not deliver more power than the temperature control allows.

For more information, see the articles Operating Constraints, Series and Parallel Connection of Heat Components and Compliance With Temperature Specifications for Series Connections.

Input Data

The unit at the end of the series must have the highest Outlet temperature.

If you have activated the Material property Heat/Cooling Temperature dependent, check the box next to the Temperature-dependent material property. In this case, the supply temperatures (outlet temperatures) are specified directly in the heating and cooling components. In models with multiple demand components, the same outlet temperature must be set in all components, and one of the demand components can be selected under Reference to unit (see following figure).

Output Data