== Overview ==
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[[NoteBox(note,This page is part of the ** Indoor climate and energy demand model ** (ICM) documentation. \\ It contains fundamental principles and the equations used in the model. \\ For an Overview of all ICM-related pages\, see the  **[wiki:doc/tec/indoor Indoor model main page]**.)]]

PALM offers an embedded indoor model. It takes account of the heat transfer through exterior walls, the shortwave solar gains and the heat transport by ventilation. It also considers internal heat gains, the energy demand for heating and cooling of the building. According to the building energy concept, the energy demand results in an (anthropogenic) waste heat, that is directly transferred to the urban environment.\\

The ICM has to work in tandem with the Urban surface model (USM) and the indoor model is only available if the USM activated. The used parameters for ICM can be find in the building database in the USM.\\

= Geometrical calculations =

For the initialization stage, it is important to calculate the vital main geometrics of the domain. Every grid point indicates if there is a building and what type. Furthermore, they store the presents of horizontal or vertical façade elements placed at the grid point. With the knowledge of these parameters from every grid point it is possible to assemble the complete façade of a building as sum of single horizontal and vertical façades elements.\\

A single façade element is the area of one grid point.\\

{{{
#!Latex
\begin{align*}
 & A_{fac,el}  =  dx * dy,
\end{align*}
}}}

The total area of the façade is the sum of all single areas where a façade is located.
{{{
#!Latex
\begin{align*}
  & A_{fac,tot} = A_{fac,el} \cdot \left( \sum_{i=0}^{n_h}\ n_{fac,h}i + \sum_{i=0}^{n_v} + n_{fac,v}i \right)
\end{align*}
}}}

To represent the opaque wall and transparent window areas in buildings, a window fraction for horizontal and vertical surfaces gives the ratio of window to wall at a single façade element. The sum  of each elemental ratio gives the ratio of the entire building. The ratio of the window area ''x'',,win,hv,, is a parameter of the USM.\\

{{{
#!Latex
\begin{align*}
  & A_{fac,win}= \sum_{i=0}^{n} \ A_{fac,el}i \cdot x_{win,hv}
\end{align*}
}}}

The total volume of a building is the sum of all elemental grid volumes where a building is located.

{{{
#!Latex
\begin{align*}
  & V_{tot}= \sum_{i=1}^{n} \ xi \cdot yi \cdot zi
\end{align*}
}}}

To take respect of an non rectangular building a virtual façade area specific indoor volume shown in figure 1, created.\\

 [[Image(virtual_volume.png,300px, border=0)]]

{{{
#!Latex
\begin{align*}
  & V_{fac,el}^{indoor} = \frac{V_{tot}}{\sum_{i=0}^{n_h}\ n_{fac,h}i \cdot xi \cdot yi  + \sum_{i=0}^{n_v} \ n_{fac,v}i \cdot xi \cdot zi }\
\end{align*}
}}}

To calculate all façade elements of the whole façade, it is necessary to create an indoor surface area per façade element.

{{{
#!Latex
\begin{align*}
  & A_{fac,floor} = \sum_{i=0}^n \ \frac{V_{fac,el}^{indoor}}{zi}\
\end{align*}
}}}

The complete ground surface of a building involves every storey gets a ground surface. To represent this, a net floor area is calculated. The height of the storey ''h'',,storey,, is a parameter of USM

{{{
#!Latex
\begin{align*}
  & A_{nfa} = \frac{V_{tot}}{h_{storey}}\
\end{align*}
}}}

The effective mass of a specific area takes respect of surfaces like ceilings, walls and furnishing.

{{{
#!Latex
\begin{align*}
  & A_{m} = a\cdot A_{nfa}\cdot \frac{A_{fac,el}}{A_{fac,tot}}\ \Lambda_{AT}
\end{align*}
}}}
The ratio of effective area Λ𝐴𝑇 and the dynamic parameter of specific effective surface 𝑎 are parameters of the USM\\

= Model scheme =

The ICM is based on an analytical solution of Fourier’s law considering a resistance model with five resistances ''R'' [K/W] and one heat capacity ''C'' [J/K] as seen in figure 2. 

[[Image(5R1C_scheme.png,400px, border=0)]]

The solution is based on a Crank-Nicolson scheme for a one-hour time step. Since the calculations are based on heat transfer coefficients, ''H'' [W/K] all figures and equations are based on heat transfer coefficients. This is the reciprocal value of ''R'' and takes short wave, long wave, convective and conductive heat transfer and heat transport (by air) into account.

'''Resistance and capacity calculations'''\\
From a numerical perspective, this network consists of five reciprocal resistances ''H'' and one heat storage capacity ''C'':\\
''H'',,v,, is the heat transport by ventilation between surface-near exterior air ''ϑ'',,n,, and indoor air ''ϑ'',,i,,. It is calculated with,

{{{
#!Latex
\begin{align*}
  & H_{v} = \left (c_{ACH,high} \cdot Z_{sched} + c_{ACH,low} \right) \cdot V_{fac,el}^{indoor} \cdot \rho_{air} \cdot c_p \cdot \left (1-\eta_v \right) 
\end{align*}
}}}

The volumetric heat capacity of air  ''ρ'',,air,,⋅''c'',,p,, is assumed as 0.33⋅W h K^−1^  m^-3^. The schedule on-time ''Z'',,sched,, , the airflow time of occupancy ''c'',,ACH,high,, , the airflow time of no occupancy ''c'',,ACH,low,, and the efficiency of heat recovery in the ventilation ''η'',,v,, are parameters of the USM.\\

''H'',,t,is,, is the connective heat transfer between indoor air ''ϑ'',,i,, and interior surface ''ϑ'',,s,, considering all room-enclosing surfaces.

{{{
#!Latex
\begin{align*}
  & H_{t,is} = A_{fac,tot}\cdot h_{is}
\end{align*}
}}}

''H'',,t,es,, is the heat transfer through windows between exterior air ''ϑ'',,e,, and interior surfaces ''ϑ'',,s,, .

{{{
#!Latex
\begin{align*}
  & H_{t,es} = A_{fac,win}\cdot h_{es}
\end{align*}
}}}
''h'',,es,, is the specific heat transfer coefficient through windows between exterior air and interior surface. Because of the model structure the specific heat transfer coefficient between indoor air and interior surface is removed.//
''H'',,t,ms,, is the conductive heat transfer between interior surface ''ϑ'',,s,, and interior mass node ''ϑ'',,m,, .

{{{
#!Latex
\begin{align*}
  & H_{t,ms} = A_{m}\cdot h_{ms}
\end{align*}
}}}
''H'',,t,wm,, is the conductive heat transfer between wall ''ϑ'',,w,, and interior mass node ''ϑ'',,m,, .

{{{
#!Latex
\begin{align*}
  & H_{t,wm} = \frac{1}{ \frac{1}{H_{t,wall}} \ - \frac{1}{H_{t,ms}} \ } \
\end{align*}
}}}

With ''H'',,t,wall,, as heat transfer of opaque components.

{{{
#!Latex
\begin{align*}
  & H_{t,wall} = \frac{1}{ \frac{1}{ \left(A_{fac,el}-A_{fac,win}\right)\cdot \frac {\lambda_{layer4}}{d_{layer4}} \ \cdot 0.5} + \frac{1}{H_{t,ms}} \ } \
\end{align*}
}}}
The thickness ''𝑑'',,𝑙𝑎𝑦𝑒𝑟4,, and the thermal heat conductivity ''𝜆'',,𝑙𝑎𝑦𝑒𝑟4,, of the fourth layer are a parameter of USM.\\
''H'',,t,1,, , ''H'',,t,2,, and ''H'',,t,3,, are auxiliary variables for calculation of the heat transport.

{{{
#!Latex
\begin{align*}
  & H_{t,1} = \frac{1}{ \frac{1}{H_{v}} \ + \frac{1}{H_{t,is}} \ } \ \\
  & H_{t,2} = H_{t,1} + H_{t,es} \\
  & H_{t,3} = \frac{1}{ \frac{1}{H_{t,2}} \ + \frac{1}{H_{t,ms}} \ } \
\end{align*}
}}}

''C'',,m,, is the internal heat capacity.
{{{
#!Latex
\begin{align*}
  & C_{m} = c \cdot A_{nfa} \cdot \frac{A_{fac,el}}{A_{fac,tot}} \
\end{align*}
}}}

'''Thermal load and temperature calculations'''\\

The internal air load is calculated with the internal heat gains with respect of occupancy of the building. The schedule is a parameter of the USM.
{{{
#!Latex
\begin{align*}
  & \Phi_{ia} = 0.5 \cdot \left ( \left(q_{int,high} \cdot Z_{sched} + q_{int,low}\right) \cdot A_{fac,floor}\right)
\end{align*}
}}}
''Φ'',,sol,, is the heat load from shortwave radiation through all windows in respect of automatic window shutters. At a value of 300 W m^-2^ shortwave radiation, the automatic window shutters are set as on. With activated the shutters the shading factor ''f'',,c,, of the sun protection take effect. The shading factor ''f'',,c,, and the g-value ''g'',,win,, are parameters of the USM.

{{{
#!Latex
\begin{align*}
  & \Phi_{sol} = \left ( \left(A_{fac,win} \cdot R_{net,sw} \cdot c_{sunprot,off} + A_{fac,win} \cdot R_{net,sw} \cdot f_c \cdot c_{sunprotec,on}\right) \cdot A_{fac,floor}\right)
\end{align*}
}}}

''Φ'',,st,, is the mass specific heat load (without thermal mass).
{{{
#!Latex
\begin{align*}
  & \Phi_{st} = \left ( 1- \frac{A_m}{A_{nfa}} \ - \frac {H_{t,es}}{9.1 \cdot A_{nfa}} \ \right) \cdot \left ( \Phi_{ia} \cdot \Phi_{sol} \right)
\end{align*}
}}}
''Φ'',,m,, is the mass specific heat load for internal and external heat sources of the inner node.
{{{
#!Latex
\begin{align*}
  & \Phi_{m} = \frac{A_m}{A_{nfa}}\ \cdot \left(\Phi_{ia}+\Phi_{sol}\right)
\end{align*}
}}}
''ϑ'',,ind,wall,win,, is the weighted temperature of innermost wall and window layer.
{{{
#!Latex
\begin{align*}
  & \vartheta_{ind,wall,win} = x_{wall,veg}\cdot\vartheta_{wall,veg}+x_{win,hv}\cdot\vartheta_{win}
\end{align*}
}}}
The fractions for wall/vegetation ''𝑥'',,𝑤𝑎𝑙𝑙,𝑣𝑒𝑔,, and window ''𝑥'',,𝑤𝑖𝑛,ℎ𝑣,, are parameters of the SURFACEMOD.
The temperatures for of wall/vegetation ''𝜗'',,𝑤𝑎𝑙𝑙,𝑣𝑒𝑔,, and for windows ''𝜗'',,𝑤𝑖𝑛,, are parameters of USM.\\
''Φ'',,𝑚,𝑡𝑜𝑡,, is the of total mass specific thermal load, internal and external.
{{{
#!Latex
\begin{align*}
  & \Phi_{m,tot}=\Phi_m+H_{t,wm}\cdot\vartheta_{ind,wall,win}+H_{t,3}\cdot \frac{\Phi_{st}+H_{t,es}\cdot\vartheta_{amb}+H_{t,1} \left( \frac{\Phi_{ia}+\Phi_{HC,nd}}{H_v} +\vartheta_{near,fac} \right)}{H_{t,2}} \
\end{align*}
}}}
The ambient temperature ''𝜗'',,𝑎𝑚𝑏,, is the undisturbed outside temperature and an input of PALM Model. The near façade temperature ''𝜗'',,𝑛𝑒𝑎𝑟,𝑓𝑎𝑐,, is the outside air temperature 10 cm away from the façade and an input of the Surface mod.\\

''ϑ'',,m,t,, is the (fictive) component temperature at actual time step.
{{{
#!Latex
\begin{align*}
  & \vartheta_{m,t}=\frac{\vartheta_{m,t,prev} \cdot \left( \frac{C_m}{3600} \ -0.5\cdot \left(H_{t,3}+H_{t,wm} \right) \right)+\Phi_{m,tot}}{ \frac{C_m}{3600} \ +0.5 \cdot \left(H_{t,3}+H_{t,wm}\right)} \
\end{align*}
}}}
''ϑ'',,m,t,prev,, is the (fictive) component temperature at previous time step.\\
''ϑ'',,s,, is the surface temperature at actual time step.
{{{
#!Latex
\begin{align*}
  & \vartheta_{s}=\frac{H_{t,ms}\cdot\vartheta_m+\Phi_{st}+H_{t,es}\cdot\vartheta_{amb}+H_{t,1}\cdot\left(\vartheta_{near,fac}+\frac{\Phi_{ia}+\Phi_{HC,nd}}{H_v}\ \right)}{H_{t,ms}+H_{t,es}+H_{t,1}}\
\end{align*}
}}}
''ϑ'',,air,, is the indoor air temperature.
{{{
#!Latex
\begin{align*}
  & \vartheta_{air}=\frac{H_{t,is}\cdot\vartheta_s+H_v\cdot\vartheta_{near,fac}+\Phi_{ia}+\Phi_{HC,nd}}{H_{t,is}+H_v} \
\end{align*}
}}}
''ϑ'',,op,, is the operative temperature. The operative temperature is a weighted average of the indoor air temperature and mean radiation temperature. 
{{{
#!Latex
\begin{align*}
  & \vartheta_{op}=0.3\cdot\vartheta_{air}+0.7\cdot\vartheta_s
\end{align*}
}}}

'''Heating and Cooling Demand'''\\

The heating and cooling demand ''Φ'',,HC,nd,, is disposed in 5 different stages as shown in figure 3. \\

[[Image(Phi_HCnd_scheme.png,400px, border=0)]]

Stage 1: No heating or cooling necessary, because room temperature ''ϑ'',,air,, is between the set comfort temperatures when heating ''ϑ'',,heat,set,, or cooling ''ϑ'',,cool,set,, is needed.
In this case the demand is:
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,nd}=0
\end{align*}
}}}
The calculated indoor air temperature is described as ''ϑ'',,air,0,, .\\
Stage 2: If the room temperature is outside the comfort threshold, heating or cooling are needed. Then the heating/cooling power is calculated with 10 W m^-2^ as ''Φ'',,HC,10,, .
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,10}=10\cdot A_{fac,floor}
\end{align*}
}}}
The indoor air temperature ''ϑ'',,air,10,, is calculated again with ''Φ'',,HC,10,, .\\
''ϑ'',,air,set,, is the intended air temperature depended of heating ''ϑ'',,h,set,, and cooling ''ϑ'',,c,set,, .

{{{
#!Latex
\begin{align*}
  & \vartheta_{air,set} \begin{cases} \vartheta_{h,set}   \\  \vartheta_{c,set}  \end{cases} 
\end{align*}
}}}
The intended air temperatures for heating ϑ_(h,set) and for cooling ''ϑ'',,c,set,, are parameters of USM. \\
To estimate the needed amount of heating/ cooling, the unlimited heating/cooling demand ''Φ'',,HC,nd,un,, is calculated without the consideration of the maximum thermal capacity.
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,nd,un}=\Phi_{HC,10}\cdot \frac {\vartheta_{air,set}-\vartheta_{air,0}}{\vartheta_{air,10}-\vartheta_{air,0}} \
\end{align*}
}}}
Stage 3: Checking if the unlimited heating/cooling demand ''Φ'',,HC,nd,un,, lower as the maximal heating ''Φ'',,heat,max,, or cooling ''Φ'',,cool,max,, power, than is the heat/cooling demand ''Φ'',,HC,nd,, equal the unlimited heating/cooling demand ''Φ'',,HC,nd,un,, .
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,nd}=\Phi_{HC,nd,un}
\end{align*}
}}}
Stage 4: If the unlimited heating or cooling demand is higher than the maximal heating ''Φ'',,heat,max,, or cooling ''Φ'',,cool,max,, power the heating demand is assumed as the maximum heating flux.
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,nd}=\Phi_{heat,max}
\end{align*}
}}}
And the cooling demand is maximum cooling heat flux.
{{{
#!Latex
\begin{align*}
  & \Phi_{HC,nd}=\Phi_{cool,max}
\end{align*}
}}}
The maximal heating ''Φ'',,heat,max,, and cooling ''Φ'',,cool,max,, power is calculated with the heat flux ''q'',,h,max,, and ''q'',,c,max,, which are parameters of USM.
{{{
#!Latex
\begin{align*}
  & \Phi_{heat,max}=q_{h,max} \cdot A_{fac,el} \\
  & \Phi_{cool,max}=q_{c,max} \cdot A_{fac,el} \\
\end{align*}
}}}
In this case, the set indoor temperature is not reachable. It will get higher than the requested indoor temperature in summer (cooling) cases and colder in winter (heating) cases. \\
'''Heat fluxes and waste heat'''\\
''q'',,wall,win,, is the heat flux through the walls and windows.
{{{
#!Latex
\begin{align*}
  & q_{wall,win}=H_{t,ms} \cdot \frac {\vartheta_s-\vartheta_m}{A_{fac,el}-A_{fac,win}} 
\end{align*}
}}}
''q'',,waste,, is the waste heat through walls/windows with respect on heating/cooling demand and the efficiency of heating/cooling technology. ''c'',,heat,on,, and ''c'',,cool,on,, are flags to separate heating and cooling technology. Whilst the possible values for ''c'',,heat,on,, are 0 and 1 the possible values for ''c'',,cool,on,, are 0 and -1 because the Cooling demand ''Φ'',,HC,nd,, is negative, but anthropogenic waste heat ''q'',,waste,, always be positive. 
{{{
#!Latex
\begin{align*}
  & q_{waste}=\frac {\Phi_{HC,nd} \cdot \left ( c_{waste,heat}\cdot c_{heat,on}+c_{waste,cool}\cdot c_{cool,on} \right) }{A_{fac,el}}
\end{align*}
}}}
The anthropogenic heat parameter for heating c_(waste,heat) and cooling c_(waste,cool) are parameters of USM.\\

TABLE 1