Models¶

Private tour through all models contained in the WaterHub package.

EndUses¶

The EndUses package clusters all appliances within an household that allow interaction with a water consumer. They are of special interest because they are associated with a consumer-generated flow of water that acts as a trigger for the rest of the system’s building blocks.

BaseEndUses¶

Model Name	BasicValve	Partial Model for the definition of end-uses
Type	Partial Model
Inlets	`inletCold`	cold water inlet
	`inletHot`	hot water inlet
	`flowInput`	data input from e.g. block HydrographFromFile
Outlets	`outlet`	water outlet

Partial model acting as a basic valve: two water inlets mixing into one water outlet. The flow is triggered by the flowInput port.

Showers¶

Base Shower¶

Model Name	BaseShower	Partial Model for showers
Type	Partial Model
Parameters	`T_wanted`	Targeted Temperature

Base model for showers. It simply makes sure \(T_{cold} \leq T_{wanted} \leq T_{hot}\) using a trivial algorithm:

algorithm
  if T_wanted < inletCold.T then
    T_achieved := inletCold.T;
  elseif T_wanted > inletHot.T then
    T_achieved := inletHot.T;
  else
    T_achieved := T_wanted;
  end if;

Classic Shower¶

Model Name	ClassicShower	Lossless shower model
Type	Model	extends partial model `BaseShower` and partial model `BasicValve`
Inlets	`inletCold`	cold water inlet
	`inletHot`	hot water inlet
	`flowInput`	data input from e.g. block HydrographFromFile
Outlets	`outlet`	water outlet
Parameters	`T_wanted`	Targeted Temperature

Simple model for showers. The flow is triggered through the flowInput port, connected to e.g. an HydrographFromFile block. Energy and mass balance equations describe the thermal behavior:

\[T_{out} m_{out} = T_{in}^{cold} m_{in}^{cold} + T_{in}^{hot} m_{in}^{hot}\]

\[m_{in}^{cold} + m_{in}^{hot} = m_{out}\]

Taps¶

Model Name	ClassicShower	Lossless tap model
Type	Model	extends partial model `BasicValve`
Inlets	`inletCold`	cold water inlet
	`inletHot`	hot water inlet
	`flowInput`	data input from e.g. block HydrographFromFile
Outlets	`outlet`	water outlet
Parameters	`T_wanted`	Targeted Temperature

Analogous to Classic Shower.

RecoverySystems¶

Models specialized in energy recovery. Heat exchanger and heat pump models are the most obvious examples.

Heat Exchangers¶

Simple Heat Exchanger¶

Model Name	SimpleHeatExchanger	Black box heat exchanger that retrieves energy with user defined efficiency
Type	Model
Inlets	`inlet`	water inlet
Outlets	`outlet`	water outlet
Outlets	`heat_out`	heat flow outlet
Parameters	`efficiency`	Efficiency factor for the heat recovery

Not So Simple Heat Exchanger¶

Model Name	NotSoSimpleHeatExchanger	Water-water Heat exchanger model
Type	Model
Inlets	`inletCold`	cold water inlet
Inlets	`inletHot`	hot water inlet
Outlets	`outletCold`	cold water outlet
Outlets	`outletHot`	hot water outlet
Parameters	`alphaFactor`	Efficiency factor for the heat recovery. Depends on length of tube, heat exchange coefficient, flows and contact areas
Parameters	`flowHE`	1 if parallel flow, -1 if counterflow

The NotSoSimpleHeatExchanger Model has been inspired by a derivation from the Wikipedia page on heat exchangers, in the section “A model of a simple heat exchanger”. This derivation is based on the Book “Fluid Mechanics and Transfer Processes”, Cambridge University Press, Kay J.M. and Nedderman R.M.

The simplest heat exchanger consist of two straight pipes with fluid flows. Let the pipe be of length \(L\), with fluid capacities \(C_i\), flow rates \(j_i\), and temperature profiles along the pipes \(T_i(x)\). Assume the heat transfer occurs only transversely between the two fluids and not along the pipe. From Newton’s law of cooling:

\[\begin{split}\label{eq:NewtonCooling} \frac{\partial u_1}{\partial t} & = \gamma (T_2-T_1) \nonumber \\ \frac{\partial u_2}{\partial t} & = \gamma (T_1-T_2)\end{split}\]

where u_i(x) is the thermal energy profile. It must be noted here that this is for parallel flows. Counterflows heat exchangers require a negative sign in the second equation. \(\gamma\) is the thermal connection constant, a function of the heat exchange coefficient and the contact area. The time change in thermal energy for a fluid unit volume being carried along the pipe can be also written as

\[\begin{split}\label{eq:ThermalEnergyTimeRate} \frac{\partial u_1}{\partial t} & = C_1j_1 \frac{\partial T_1}{\partial x} \nonumber \\ \frac{\partial u_2}{\partial t} & = C_2j_2 \frac{\partial T_2}{\partial x}\end{split}\]

Here, \(C_i j_i\) are the thermal flow rates. So, equating above equations results in a steady-state, x-only differential equation, that can be solved with

\[\begin{split}\label{eq:diffEq} T_1(x) & = A - \frac{Bk_1}{k} e^{-kx} \nonumber \\ T_2(x) & = A + \frac{Bk_2}{k} e^{-kx}\end{split}\]

where \(k_i = \gamma/(C_ij_i)\), \(k = k_1 + k_2\) and \(A,B\) being integration constants. Knowing the input temperatures at (x = 0) \(T_{10}\) and \(T_{20}\), we can derive (for parallel flows)

\[\begin{split}\label{eq:temps} B & = (T_{20} - T_{10}) & = \Delta T \nonumber \\ A & = T_{10} + \frac{\Delta T}{(1+j_1/j_2)} & = T_{20} - \frac{\Delta T}{(1 + j_2/j_1)} \nonumber\end{split}\]

\[\begin{split}T_{1L} & = &T_{10} + \frac{\Delta T}{(1 + j_1/j_2)} (1 - e^{\frac{\gamma}{j_1+j_2}L}) \nonumber \\ T_{1L} & = &T_{20} - \frac{\Delta T}{(1 + j_2/j_1)} (1 - e^{\frac{\gamma}{j_1+j_2}L})\end{split}\]

Letting \(\alpha = (1-e^{\frac{\gamma}{j_1+j_2}L})\), this term thus describes the efficiency of the heat-exchanger, depending on many parameters such as the heat exchange coefficients, exchange surface area and length of pipes. With \(\alpha = 0\), no heat is transferred between the pipes, while all the available heat is transferred when \(\alpha = 1\).

Pipes & Carriers¶

Models of water pipes, electric wires and other carrier systems.

Water Pipes¶

PipeLossesAtRest¶

Model Name	PipeLossesAtRest	Loses energy to environment when water flow is zero
Type	Model
Inlets	`inlet`	water inlet
Outlets	`outlet`	water outlet
Outlets	`heatOutlet`	heat flow outlet
Parameters	`triggerValue`	Triggers modeling of heat losses when flow get below
	`pipeLength`	Length of water pipe in meters
	`pipeDiameterInside`	Inside diameter of pipe in meters
	`pipeThickness`	Thickness of pipe walls in meters
	`material`	Roll menu to choose material thermal properties
	`tEnvironment`	Temperature of external air in Kelvin

Model of a water pipe that loses energy to its environment when the flow is zero (i.e when water is stagnating in the pipe). The model computes the UA-value, i.e. the total thermal conductance of the pipe, using:

\[\frac{1}{UA} = \frac{1}{h_{ci}A_i} + \sum\frac{s_n}{k_nA_n} + \frac{1}{h_{co}A_o}\]

where \(h_{ci}\) and \(h_{co}\) are the convection heat transfer coefficients of the inside, respectively outside fluid. \(A_i\) and \(A_o\) are the inside, respectively outside contact areas. \(s_n\) is the thickness, \(k_n\) the thermal conductivity and \(A_n\) the mean area of pipe layer n.

UA is then used in the ODE:

\[V C_{v} \frac{dT}{dt} = -UA(T-T_{env})\]

to compute the time-dependent fluid temperature.