Changes between Version 8 and Version 9 of doc/tec/usm

Timestamp:

Oct 16, 2020 4:01:58 PM (4 years ago)

Author:

pavelkrc

Comment:

--

doc/tec/usm

@@ -33 +33 @@
 The physical properties of the urban surfaces and wall, window and green soil elements can be set using values from a building database where different types of buildings are defined. The insulation value of windows are there only characterized by the U-value and and the heat capacity and heat conductivity is evenly distributed (no real glas or gas layers are taken into account.
 
-The absorption of shortwave radiation inside the window layers is calculated using an logarithmic absorption function and the absorption coefficient is calculated using the overal window transmissivity value. The heat that is absorpted within each window layer is taken into account by the wall heat model.
-
 The green heat model calculates the transport of soil moisture but neglects the extraction of water from the respective soil layers.
 
+=== Window transmissivity: representation ===
+
+The radiant flux received by the window (incident radiant flux, Φ,,I,,)
+is partially reflected back (Φ,,R,,), partially absorbed
+by the mass of the glass (Φ,,A,, which is simulated
+by four discretized layers of window depth) and partially transmitted
+through the window, where the transmitted flux Φ,,T,,
+may be processed by the indoor model (if enabled), therefore
+{{{
+#!Latex
+\[
+\Phi_{\mathrm{I}}=\Phi_{\mathrm{R}}+\Phi_{\mathrm{A}}+\Phi_{\mathrm{T}}
+\]
+}}}
+Most of the reflection happens as specular reflection on the frontal
+and rear boundary between the glass and air. The radiant flux reflected
+at the rear boundary is partially reflected again at the frontal boundary,
+then partially at the rear boundary again and so on, however, these
+fluxes are typically negligible, as are the non-specular reflections,
+the absorption of the reflected fluxes and the scattering inside the
+glass; a bias can be avoided by adjusting the parameters of the non-neglected
+processes. The reflected radiant flux can thus be simplified as Φ,,R,,=Φ,,RF,,+Φ,,RR,,,
+where Φ,,RF,, is the radiant flux reflected at the frontal
+boundary and Φ,,RR,, is the radiant flux reflected at
+the rear boundary.
+
+The ''total transmissivity'' ''T''=Φ,,T,,/Φ,,I,,
+is the fraction of transmitted and received radiant flux, i.e. it
+includes loss by reflection and absorption together. The ''internal
+transmissivity'' ''T'',,I,,=Φ,,TT,,/Φ,,TI,,
+describes the loss by absorption by a single pass of the light through
+the glass, where Φ,,TI,,=Φ,,I,,-Φ,,RF,,
+is the radiant flux entering the glass after frontal boundary reflection
+and Φ,,TT,,=Φ,,TI,,-Φ,,A,, is
+the radiant flux leaving the glass before rear boundary reflection.
+The ''frontal reflectivity'' ''R'',,F,,=Φ,,RF,,/Φ,,I,,
+and ''rear reflectivity'' ''R'',,R,,=Φ,,RR,,/Φ,,TT,,
+express the fraction of radiant flux reflected at each boundary. Together,
+the radiant flux passing through the glass can be described sequentially
+as it is diminished by frontal reflection, absorption and rear reflection.
+(2) describes this process additively while (1)
+describes the fractions multiplicatively:
+{{{
+#!Latex
+\begin{align}
+T & =(1-R_{\mathrm{F}})T_{\mathrm{I}}(1-R_{\mathrm{R}})\\
+\Phi_{\mathrm{T}} & =\Phi_{\mathrm{I}}-\Phi_{\mathrm{RF}}-\Phi_{\mathrm{A}}-\Phi_{\mathrm{RR}}
+\end{align}
+}}}
+
+The internal transmissivity is described by the Beer–Lambert law.
+For a homogeneous material with width ''z'', it is equal to
+{{{
+#!Latex
+\[
+T_{\mathrm{I}}=e^{-az}
+\]
+}}}
+where ''a'' is the absorption (attenuation) coefficient.
+
+=== Window transmissivity: modelling ===
+
+The window fraction of surfaces in PALM is described by two parameters:
+`albedo` (total reflectivity in the respective band, ''R''=Φ,,R,,/Φ,,I,,)
+and `transmissivity` (total, ''T'').
+
+The frontal and rear reflectivities of glass are similar. From simple
+Fresnel equations they are equal, in reality the frontal reflectivity
+is slightly stronger. In PALM they are modelled as equal and they
+are calculated from the total reflectivity.
+{{{
+#!Latex
+\begin{align*}
+\Phi_{\mathrm{R}} & =\Phi_{\mathrm{RF}}+\Phi_{\mathrm{RR}}\\
+\Phi_{\mathrm{R}} & =\Phi_{\mathrm{I}}R_{\mathrm{F}}+(\Phi_{\mathrm{T}}+\Phi_{\mathrm{R}}-\Phi_{\mathrm{RF}})R_{\mathrm{R}}\\
+R & =R_{\mathrm{F}}+(T+R-R_{\mathrm{F}})R_{\mathrm{R}}
+\end{align*}
+}}}
+Using ''R'',,F,,=''R'',,R,, we get:
+{{{
+#!Latex
+\[
+R_{\mathrm{F}}=\frac{R+T+1-\sqrt{(R+T+1)^{2}-4R}}{2}
+\]
+}}}
+
+In order to simulate the absorption by the discretized window layers,
+the absorption coefficient has to be calculated from the parameters:
+{{{
+#!Latex
+\begin{align*}
+T_{\mathrm{I}} & =\frac{\Phi_{\mathrm{TT}}}{\Phi_{\mathrm{TI}}}\\
+T_{\mathrm{I}} & =\frac{\Phi_{\mathrm{T}}+\Phi_{\mathrm{R}}-\Phi_{\mathrm{RF}}}{\Phi_{\mathrm{I}}(1-R_{\mathrm{F}})}\\
+e^{-az} & =\frac{T+R-R_{\mathrm{F}}}{1-R_{\mathrm{F}}}\\
+a & =\frac{-\log\frac{T+R-R_{\mathrm{F}}}{1-R_{\mathrm{F}}}}{z}
+\end{align*}
+}}}
+
+In the prognostic equations, the absorbed flux is added to the temperature
+tendency in the Runge–Kutta method for each layer ''l'', depending
+on layer width and. The absorbed flux is equal to
+{{{
+#!Latex
+\[
+\Phi_{\mathrm{A},l}=\Phi_{\mathrm{I}}(1-R_{\mathrm{F}})(e^{-az_{l-1}}-e^{-az_{l}})
+\]
+}}}
+where ''z'',,''l''-1,, is the depth of the previous layer (cumulative width
+of all previous layers) and ''z'',,''l'',, is the depth of layer ''l''.
 
 === Boundary conditions ===