Fundamentals of the Steady-State Calculation

Overview

The steady-state calculation determines the settled thermo-hydraulic state of the heat network for individual operating points: pressures, mass fluxes, velocities and temperatures in all pipes, as well as pressure and temperature differences at energy plants and consumers, optionally including pipe heat losses. It is the central tool for hydraulic design: worst-point identification, verification of the pipe sizing and pump design. This page documents the calculation model; the operation is described in Performing the Calculation.

Access

Use the Steady-State Calculation button in the left-hand toolbar to switch to the steady-state calculation view, see Performing the Calculation.

Hydraulic Equation System

The network is represented as a graph of flow elements (pipes as well as pumps and heat exchangers of the plants) and nodes. The unknowns are the node pressures and the mass fluxes of all elements; the nonlinear equation system is solved iteratively with a Newton method. It consists of two types of equation:

Mass conservation at each node:

$$\sum_{j} \dot{m}_j = 0$$

Pressure equation for each flow element between inlet and outlet:

$$p_{in} - p_{out} = \Delta p(\dot{m})$$

Pressure Losses of the Pipes

Pipe friction is calculated according to Darcy-Weisbach; additional local resistances of the fittings are included via zeta values:

$$\Delta p = \left( \lambda(Re)\,\frac{L}{d_i} + \sum_k \zeta_k \right) \frac{\rho}{2}\, v\,\lvert v \rvert$$

with inner diameter $d_i$ in [m], pipe length $L$ in [m], density $\rho$ in [kg/m³] and flow velocity $v$ in [m/s]. The friction factor $\lambda$ is determined depending on the flow regime:

• laminar ($Re < 2300$): $\lambda = 64/Re$
• turbulent ($Re > 10000$): Colebrook-White equation with the pipe roughness from the pipe database
• transition range: linear interpolation between both values

Pressure Maintenance and Geodetic Height

The absolute pressure level is set by the pressure maintenance: the configured static pressure is applied at the selected reference node (inlet or outlet of an energy plant), and all remaining pressures follow relative to it. The geodetic pressure component from the height differences of the nodes

$$\Delta p_{geo} = \rho \, g \, \Delta z$$

is added in the result display via the option Consider geodetic height; without this option, only friction and component losses are evaluated.

Pumps

Pumps are part of the plants of energy plants or consumers and enter the equation system as a pressure increase. Depending on the component, different models are available: constant pressure difference, controlled pumps (e.g. differential-pressure control on the worst point), as well as characteristic-curve-based and power-limited pump models, see Components and Controllers. The supply type distinguishes between a central pump in the energy plant and decentralized pumps at the consumers — this also affects the worst-point identification.

Mass Flux of the Consumers

The mass flux of each consumer follows from its power at the operating point and the nominal temperature difference (spread) between supply and return:

$$\dot{m} = \frac{\dot{Q}}{c_p \cdot \Delta T}$$

Thermal Balance

If Consider pipe heat losses is enabled, each pipe exchanges heat with its surroundings:

$$\dot{Q}_{loss} = UA \cdot (T_m - T_u)$$

The UA value is formed from the length-specific U value $U'$ of the pipe wall including insulation (in W/(m·K), from the pipe database) and the internal convective heat transfer:

$$UA = \frac{L}{\dfrac{1}{\alpha_i \, \pi \, d_i} + \dfrac{1}{U'}}$$

The ambient temperature $T_u$ is reduced to a single value for the steady-state run from the heat-exchange boundary condition assigned to the pipe:

Boundary condition	Temperature in the steady-state run
Ground model	Annual mean of the outdoor air temperature from the climate data as a substitute for the undisturbed ground temperature (the dynamic ground model is not simulated)
Constant temperature	configured value
Time series / file	value at the governing hour of the operating point (hour of the maximum heating load)

If the option is disabled, all pipes are calculated adiabatically. The supply temperature of the energy plant follows from the heating curve at the outdoor temperature of the governing hour; for the operating point Connection load, the maximum of the heating curve (at the minimum annual outdoor temperature) is applied. Details on the boundary conditions are described in Heat Exchange Types.

Inputs

Input	Source / Unit
Power per consumer	Building demand profiles at the governing hour of the operating point or connection load; time shifts from simultaneity and part-load factors are applied
Nominal temperature difference	Spread at the consumers in [K]; determines the mass fluxes
Plants	Plant configurations of the energy plants and consumers (pumps, heat generators, heat exchangers, controllers)
Pressure maintenance	Pressure in [bar] and position (energy plant, inlet/outlet)
Maximum pipe pressure loss	in [Pa/m]; reference value for maximum capacity, utilization and reserve of the pipes
Fluid and pipe data	Density, heat capacity, viscosity of the network fluid; diameter, roughness and U value per pipe type from the databases

Calculation Sequence

For each operating point, VICUS Districts creates a standalone solver model and integrates it over the configured simulation time (default: 24 h) until the steady state has been reached. Multiple operating points are calculated in parallel. After the run, convergence is checked; non-converged nodes are reported as a warning.

Distinction from the Dynamic Simulation

	Steady-state calculation	Dynamic simulation
Consideration	individual operating points (e.g. peak load)	continuous period, e.g. one year
Thermal inertia	not considered (settled state)	heat capacities of pipes and plants are included
Ground	annual mean of the outdoor air temperature as a substitute value	dynamic ground model
Typical application	hydraulic design, worst point, pump design, verification of the pipe sizing	energy balances, temperature and load profiles, annual heat losses, control behavior

Related Topics

• Performing the Calculation
• Results of the Steady-State Calculation
• Calculation Method of the Pipe Sizing