Surface
Heat Balance With Moveable Insulation[LINK]
Basic Heat Balance
Cases[LINK]
A heat balance must exist at the outside surface-air
interface. The incoming conductive, convective, and radiative
fluxes must sum up to zero:
\[Conductive + Convective +
Radiative = 0\]
In contrast to the internal surface heat balance that
treats all surfaces simultaneously, the external thermal
balance for each surface is performed independent of all other
surfaces. This implies that there is no direct interaction
between the individual surfaces.
TARP includes four possible representations for the basic
outside surface heat balance. The first two depend on which of
the optimal surface conductance algorithms the user selects.
The simple outside surface conductance that includes both the
convective and thermal interchange between the surface and the
environment in a single coefficient, is represented by the
thermal network in Figure .
Equation can also be expressed as:
\[\left[
{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {{\rm{Y}}_{\rm{0}}}\cdot
{\rm{T}}{{\rm{I}}_t} - {{\rm{X}}_{\rm{0}}}\cdot
{\rm{T}}{{\rm{O}}_{\rm{t}}}} \right]{\rm{ + }}\left[
{{\rm{HO}}\cdot \left( {{{\rm{T}}_{\rm{a}}} -
{\rm{T}}{{\rm{O}}_t}} \right)} \right]{\rm{ +
QSO = 0}}\]
This can be solved for the outside surface temperature.
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + HO}}\cdot
{{\rm{T}}_{\rm{a}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HO}}}}}
\right]{\rm{ }}\]
The detailed outside surface conductance model considers
convection and radiant interchange with the sky and with the
ground as separate factors. Its use in the outside thermal
balance is shown in Figure .In
this case, equation can be expanded to give
\[\left[
{{\rm{KO}}{{\rm{P}}_{\rm{t}}}{\rm{ +
}}{{\rm{Y}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ -
}}{{\rm{X}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}}
\right]{\rm{ + }}\left[ {{\rm{HA}}\cdot \left(
{{{\rm{T}}_{\rm{a}}}{\rm{ - T}}{{\rm{O}}_{\rm{t}}}}
\right){\rm{ + HS}}\cdot \left( {{{\rm{T}}_{\rm{s}}}{\rm{ -
T}}{{\rm{O}}_{\rm{t}}}} \right){\rm{ + HG}}\cdot \left(
{{{\rm{T}}_{\rm{g}}}{\rm{ - T}}{{\rm{O}}_{\rm{t}}}} \right)}
\right]{\rm{ + QSO = 0 }}\]
This can be solved for the outside surface temperature:
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + HA}}\cdot
{{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{
+ HG}}\cdot {{\rm{T}}_{\rm{g}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ +
HA + HS + HG}}}}} \right]{\rm{ }}\]
The third and fourth representations occur when the outside
surface has been covered with movable insulation. The
insulation has a conductance of UM. The thermal network in Figure
represents this case.The insulation must be mass-less because
it is not generally possible to perform a correct thermal
balance at the juncture of two surfaces each modeled by
CTF.
The equation for the thermal balance between the surface
and the insulation is
\[\left[
{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {{\rm{Y}}_{\rm{0}}}\cdot
{\rm{T}}{{\rm{I}}_t} - {{\rm{X}}_{\rm{0}}}\cdot
{\rm{T}}{{\rm{O}}_{\rm{t}}} + {\rm{UM}}\cdot \left( {{\rm{TM -
T}}{{\rm{O}}_{\rm{t}}}} \right)} \right]{\rm{ +
QSO = 0}}\]
Which can be rewritten to solve for TO :
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + UM}}\cdot
{\rm{TM}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM}}}}}
\right]{\rm{ }}\]
Depending on whether or not the detailed or simple
algorithm for surface conductance is being used, there are two
expressions for TM, the outside temperature of the insulation.
For the simple conductance:
\[{\rm{TM}} = \left[
{\frac{{{\rm{QSM + UM}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{
+ HO}}\cdot {{\rm{T}}_{\rm{a}}}}}{{{\rm{UM + HO}}}}}
\right]\]
For the detailed conductance:
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{QSM}} + {\rm{UM}}\cdot
{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ + HA}}\cdot
{{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{
+ HG}}\cdot {{\rm{T}}_{\rm{g}}}}}{{{\rm{UM + HA + HS + HG}}}}}
\right]{\rm{ }}\]
In this case the values of HA, HS and HG must be found by
using an estimated value of TM in place of TO.
Thermal Network for Simple
Outside Surface Coefficient
Thermal Network for Detailed
Outside Surface Coefficient
Thermal Network for Outside
Moveable Insulation
Heat Balance Cases[LINK]
TO\(_{t}\) and TI\(_{t}\) are related through the
Y\(_{0}\) CTF. However TI\(_{t}\) is also unknown. While it
is possible to combine the outside and the inside surface heat
balances to compute TO\(_{t}\) and TI\(_{t}\) simultaneously, TARP uses
a simpler procedure where TO\(_{t}\) is based on a previous
value of TI. When Y\(_{0}\)
is small, as occurs in well insulated or very massive
surfaces, TI\(_{t}\) can be
replaced by TI\(_{t-1}\)
(which is known for the previous hour’s heat balance) without
significantly effecting the value of TO\(_{t}\) When Y\(_{0}\) is large, TO and TI can so
strongly be coupled that separate outside and inside heat
balances do not work because the environment and zone
temperatures have negligible influence on the heat balances.
The TARP uses the inside surface heat balance to couple
TO\(_{t}\) with TZ and TR.
These two temperatures are less strongly influenced by TO and
allow a reasonable heat balance. On the first heat balance
iteration, TZ and TR are the values at time t-1. The user may
optionally require that TO\(_{t}\) be recomputed with every
iteration of TI\(_{t}\) . In
this case TZ and TR have values from the previous iteration
and a true simultaneous solution is achieved. In most
conventional constructions, recomputing TO\(_{t}\) does not significantly
change the computed zone loads and temperatures. The inside
surface heat balance is given by
\[{\rm{T}}{{\rm{I}}_{\rm{t}}} =
\left[ {\frac{{KI{P_t} + QSI + HC\cdot TZ + HR\cdot TR +
{Y_0}\cdot TO}}{{{Z_0} + HC + HR}}} \right]\]
The surface heat balances can be combined in eight ways
according to conditions for calculations of the outside
surface temperature
\[{F_1} = \left[
{\frac{{{Y_0}}}{{{Z_0} + HI + HR}}} \right]\]
\[{F_2} = \left[
{\frac{{UM}}{{UM + HO}}} \right]\]
\[{F_3} = \left[
{\frac{{UM}}{{UM + HA + HS + HG}}} \right]\]
Case1:
Y\(_{0}\) small, simple
conductance, no movable insulation:[LINK]
From Equation
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ +
HO}}\cdot {{\rm{T}}_{\rm{a}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ +
HO}}}}} \right]{\rm{ }}\]
Case2:
Y\(_{0}\) not small, simple
conductance, no movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{\rm{HO}}\cdot {{\rm{T}}_{\rm{a}}} + {{\rm{F}}_1}\cdot \left(
{{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ +
HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HO
- }}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_0}}}}
\right]{\rm{ }}\]
Case3:
Y\(_{0}\) small, detailed
conductance, no movable insulation:[LINK]
From Equation
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ +
HA}}\cdot {{\rm{T}}_{\rm{a}}} + {\rm{HS}}\cdot
{{\rm{T}}_{\rm{s}}} + {\rm{HG}}\cdot
{{\rm{T}}_{\rm{g}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HA + HS +
HG}}}}} \right]{\rm{ }}\]
Case4:
Y\(_{0}\) not small, detailed
conductance, no movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{\rm{HA}}\cdot {{\rm{T}}_{\rm{a}}} + {\rm{HS}}\cdot
{{\rm{T}}_{\rm{s}}} + {\rm{HG}}\cdot {{\rm{T}}_{\rm{g}}} +
{{\rm{F}}_{\rm{1}}}\cdot \left(
{{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ +
HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HA
+ HS + HG - }}{{\rm{F}}_1}\cdot {{\rm{Y}}_0}}}}
\right]{\rm{ }}\]
Case5:
Y\(_{0}\) small, simple
conductance, with movable insulation:[LINK]
From Equations and \(TO_t =
\left[ \frac{ KOP_t + QSO + HA \cdot T_a + HS \cdot T_s + HG
\cdot T_g + F_1 \cdot \left( KIP_t QS1 + HI \cdot TZ + HR
\cdot TR \right) } { X_0 + HA + HS + HG - F_1 \cdot Y_0 }
\right]\)
Case6:
Y\(_{0}\) not small, simple
conductance, with movable insulation:[LINK]
From Equations , and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{F}}_2}\cdot \left( {{\rm{QSM + HO}}\cdot
{{\rm{T}}_{\rm{a}}}} \right) + {{\rm{F}}_1}\cdot \left(
{{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ +
HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}} + {\rm{UM
- }}{{\rm{F}}_{\rm{2}}}\cdot {\rm{UM -
}}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_{\rm{0}}}}}}
\right]{\rm{ }}\]
Case7:
Y\(_{0}\) small, detailed
conductance, with movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ +
}}{{\rm{F}}_{\rm{3}}}\left( {{\rm{QSM + HA}}\cdot
{{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{
+ HG}}\cdot {{\rm{T}}_{\rm{g}}}}
\right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM -
}}{{\rm{F}}_{\rm{3}}}\cdot {\rm{UM}}}}}
\right]{\rm{ }}\]
Case8:
Y\(_{0}\) not small, detailed
conductance, with movable insulation:[LINK]
From Equations , and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ =
}}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} +
{{\rm{F}}_1}\cdot \left( {{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ +
QSI + HI}}\cdot {\rm{TZ + HR}}\cdot {\rm{TR}}} \right){\rm{ +
}}{{\rm{F}}_{\rm{3}}}\left( {{\rm{QSM + HA}}\cdot
{{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{
+ HG}}\cdot {{\rm{T}}_{\rm{g}}}}
\right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM -
}}{{\rm{F}}_{\rm{3}}}\cdot {\rm{UM -
}}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_{\rm{0}}}}}}
\right]{\rm{ }}\]
Fortran Algorithm
Examples[LINK]
Case5:
Y\(_{0}\) small, simple
conductance, with movable insulation:[LINK]
From Equation
! Outside heat balance case: Movable insulation, slow conduction, simple convection
F2 = DBLE(HmovInsul) / ( DBLE(HmovInsul) + DBLE(HExtSurf(SurfNum)) )
TH(SurfNum,1,1) = (-CTFConstOutPart(SurfNum) &
+DBLE(QRadSWOutAbs(SurfNum) ) &
+Construct(ConstrNum)\%CTFCross(0)*TempSurfIn(SurfNum) &
+F2* ( DBLE(QRadSWOutMvIns(SurfNum)) &
+ DBLE(HExtSurf(SurfNum))* DBLE(TempExt) ) ) &
/( Construct(ConstrNum)\%CTFOutside(0) + DBLE(HmovInsul) &
- F2* DBLE(HMovInsul))Case6:
Y\(_{0}\) not small, simple
conductance, with movable insulation:[LINK]
From Equation
! Outside heat balance case: Movable insulation, quick conduction, simple convection
F2 = DBLE(HmovInsul) / ( DBLE(HmovInsul) + DBLE(HExtSurf(SurfNum)) )
TH(SurfNum,1,1) = (-CTFConstOutPart(SurfNum) &
DBLE(QRadSWOutAbs(SurfNum)) &
+F2*( DBLE(QRadSWOutMvIns(SurfNum)) &
+DBLE(HExtSurf(SurfNum))* DBLE(TempExt) ) &
+F1*( CTFConstInPart(SurfNum) &
+ DBLE(QRadSWInAbs(SurfNum)) &
+ DBLE(QRadThermInAbs(SurfNum)) &
+ DBLE(HConvIn(SurfNum))*MAT(ZoneNum) &
+ DBLE(NetLWRadToSurf(SurfNum)) ) ) &
/( Construct(ConstrNum)\%CTFOutside(0) + DBLE(HmovInsul) &
-F2* DBLE(HMovInsul )- F1*Construct(ConstrNum)\%CTFCross(0) )Case7:
Y\(_{0}\) small, detailed
conductance, with movable insulation:[LINK]
From Equation
! Outside heat balance case: Movable insulation, slow conduction, detailed convection
F2 = DBLE(HMovInsul)/ ( DBLE(HMovInsul) + DBLE(HExtSurf(SurfNum)) &
+DBLE(HSky) + DBLE(HGround) )
TH(SurfNum,1,1) = (-CTFConstOutPart(SurfNum) &
+DBLE(QRadSWOutAbs(SurfNum)) &
+Construct(ConstrNum)\%CTFCross(0)*TempSurfIn(SurfNum) &
+F2*( DBLE(QRadSWOutMvIns(SurfNum)) &
+DBLE(HExtSurf(SurfNum))*DBLE(TempExt) &
+DBLE(HSky)*DBLE(SkyTemp) &
+DBLE(HGround)*DBLE(OutDryBulbTemp) ) ) &
/( Construct(ConstrNum)\%CTFOutside(0) &
+DBLE(HMovInsul) - F2*DBLE(HMovInsul) )Case8:
Y\(_{0}\) not small, detailed
conductance, with movable insulation:[LINK]
From Equation
! Outside heat balance case: Movable insulation, quick conduction, detailed convection
F2 = DBLE(HMovInsul)/ ( DBLE(HMovInsul) + DBLE(HExtSurf(SurfNum)) &
+DBLE(HSky) + DBLE(HGround) )
TH(SurfNum,1,1) = (-CTFConstOutPart(SurfNum) &
+DBLE(QRadSWOutAbs(SurfNum)) &
+F1*( CTFConstInPart(SurfNum) &
+DBLE(QRadSWInAbs(SurfNum)) &
+DBLE(QRadThermInAbs(SurfNum)) &
+DBLE(HConvIn(SurfNum))*MAT(ZoneNum) &
+DBLE(NetLWRadToSurf(SurfNum)) ) &
+F2*( DBLE(QRadSWOutMvIns(SurfNum)) &
+DBLE(HExtSurf(SurfNum))*DBLE(TempExt) &
+DBLE(HSky)*DBLE(SkyTemp) &
+DBLE(HGround)*DBLE(OutDryBulbTemp) ) &
/( Construct(ConstrNum)\%CTFOutside(0) &
+DBLE(HMovInsul) - F2*DBLE(HMovInsul) &
-F1*Construct(ConstrNum)\%CTFCross(0) )Fortran Variable
Descriptions[LINK]
Fortran Variables and Descriptions
FORTRAN Variable
Description
Tarp Variable
Units
Description
TH(SurfNum,1,1)
Temperature History(SurfNum,Hist
Term,In/Out), where: Hist Term (1 = Current Time,
2-MaxCTFTerms = previous times), In/Out (1 = Outside, 2 =
Inside)
TO
C
Temperature of outside of
surface I at time t
Construct(ConstrNum) %
CTFCross(0)
Cross or Y term of the CTF
equation
Y0
W/m K
Cross CTF term
Construct(ConstrNum) %
CTFInside(0)
Inside or Z terms of the CTF
equation
Z0
W/m K
Inside CTF term
Construct(ConstrNum) %
CTFOutside(0)
Outside or X terms of the CTF
equation
X0
W/m K
Outside CTF term
CTFConstInPart(SurfNum)
Constant inside portion of the
CTF calculation
KIP
W/m
Portion of inward conductive
flux based on previous temperature and flux history terms
CTFConstOutPart(SurfNum)
Constant Outside portion of the
CTF calculation
KOP
W/m
Portion of outward conductive
flux based on previous temperature and flux history terms
F1, F2, F3
Intermediate calculation
variables
F1, F2, F3
Radiation interchange factor
between surfaces
GroundTemp
Ground surface temperature
T
C
Temperature of ground at the
surface exposed to the outside environment
HConvIn(SurfNum)
Inside convection
coefficient
HI
W/m K
Inside convection
coefficient
HExtSurf(SurfNum)
Outside Convection
Coefficient
HO, HA
W/m K
Overall outside surface
conductance
HGround
Radiant exchange (linearized)
coefficient
HG
W/m K
Radiative conductance (outside
surface to ground temperature
HmovInsul
Conductance or “h” value of
movable insulation
UM
W/m K
Conductance of Movable
insulation
HSky
Radiant exchange (linearized)
coefficient
HS
W/m K
Radiative conductance (outside
surface to sky radiant temperature
MAT(ZoneNum)
Zone temperature
TZ
C
Temperature of zone air
NetLWRadToSurf(SurfNum)
Net interior longwave radiation
to a surface from other surfaces
HR*TR
W/m
Net surface to surface radiant
exchange
QRadSWInAbs(SurfNum)
Short-wave radiation absorbed on
inside of opaque surface
QSI
W/m
Short wave radiant flux absorbed
at inside of surface
QRadSWOutAbs(SurfNum)
Short wave radiation absorbed on
outside opaque surface
QSO
W/m
Short wave radiant flux absorbed
at outside of surface
QRadSWOutMvIns(SurfNum)
Short wave radiation absorbed on
outside of movable insulation
QSM
W/m
Short wave radiant flux absorbed
at surface of movable insulation
QRadThermInAbs(SurfNum)
Thermal Radiation absorbed on
inside surfaces
W/m
Longwave radiant flux from
internal gains
SkyTemp
Sky temperature
T
C
Sky temp
TempExt
Exterior surface temperature or
exterior air temperature
TM, T
C
Temperature of external surface
of movable insulation or outside ambient air temperature
TempSurfIn(SurfNum)
Temperature of inside surface
for each heat transfer surface
TI
C
Temperature of inside of surface
I at time t-1
Walton, G.N. 1983. “The Thermal Analysis Research Program
Reference Manual Program (TARP)”, National Bureau of Standards
(now National Institute of Standards and Technology).
Surface Heat Balance With Moveable Insulation[LINK]
Basic Heat Balance Cases[LINK]
A heat balance must exist at the outside surface-air interface. The incoming conductive, convective, and radiative fluxes must sum up to zero:
\[Conductive + Convective + Radiative = 0\]
In contrast to the internal surface heat balance that treats all surfaces simultaneously, the external thermal balance for each surface is performed independent of all other surfaces. This implies that there is no direct interaction between the individual surfaces.
TARP includes four possible representations for the basic outside surface heat balance. The first two depend on which of the optimal surface conductance algorithms the user selects. The simple outside surface conductance that includes both the convective and thermal interchange between the surface and the environment in a single coefficient, is represented by the thermal network in Figure. Equation can also be expressed as:
\[\left[ {{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {{\rm{Y}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{I}}_t} - {{\rm{X}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}} \right]{\rm{ + }}\left[ {{\rm{HO}}\cdot \left( {{{\rm{T}}_{\rm{a}}} - {\rm{T}}{{\rm{O}}_t}} \right)} \right]{\rm{ + QSO = 0}}\]
This can be solved for the outside surface temperature.
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + HO}}\cdot {{\rm{T}}_{\rm{a}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HO}}}}} \right]{\rm{ }}\]
The detailed outside surface conductance model considers convection and radiant interchange with the sky and with the ground as separate factors. Its use in the outside thermal balance is shown in Figure.In this case, equation can be expanded to give
\[\left[ {{\rm{KO}}{{\rm{P}}_{\rm{t}}}{\rm{ + }}{{\rm{Y}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ - }}{{\rm{X}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}} \right]{\rm{ + }}\left[ {{\rm{HA}}\cdot \left( {{{\rm{T}}_{\rm{a}}}{\rm{ - T}}{{\rm{O}}_{\rm{t}}}} \right){\rm{ + HS}}\cdot \left( {{{\rm{T}}_{\rm{s}}}{\rm{ - T}}{{\rm{O}}_{\rm{t}}}} \right){\rm{ + HG}}\cdot \left( {{{\rm{T}}_{\rm{g}}}{\rm{ - T}}{{\rm{O}}_{\rm{t}}}} \right)} \right]{\rm{ + QSO = 0 }}\]
This can be solved for the outside surface temperature:
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + HA}}\cdot {{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{ + HG}}\cdot {{\rm{T}}_{\rm{g}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HA + HS + HG}}}}} \right]{\rm{ }}\]
The third and fourth representations occur when the outside surface has been covered with movable insulation. The insulation has a conductance of UM. The thermal network in Figure represents this case.The insulation must be mass-less because it is not generally possible to perform a correct thermal balance at the juncture of two surfaces each modeled by CTF.
The equation for the thermal balance between the surface and the insulation is
\[\left[ {{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {{\rm{Y}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{I}}_t} - {{\rm{X}}_{\rm{0}}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}} + {\rm{UM}}\cdot \left( {{\rm{TM - T}}{{\rm{O}}_{\rm{t}}}} \right)} \right]{\rm{ + QSO = 0}}\]
Which can be rewritten to solve for TO :
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{\rm{t}}}{\rm{ + UM}}\cdot {\rm{TM}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM}}}}} \right]{\rm{ }}\]
Depending on whether or not the detailed or simple algorithm for surface conductance is being used, there are two expressions for TM, the outside temperature of the insulation. For the simple conductance:
\[{\rm{TM}} = \left[ {\frac{{{\rm{QSM + UM}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ + HO}}\cdot {{\rm{T}}_{\rm{a}}}}}{{{\rm{UM + HO}}}}} \right]\]
For the detailed conductance:
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{QSM}} + {\rm{UM}}\cdot {\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ + HA}}\cdot {{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{ + HG}}\cdot {{\rm{T}}_{\rm{g}}}}}{{{\rm{UM + HA + HS + HG}}}}} \right]{\rm{ }}\]
In this case the values of HA, HS and HG must be found by using an estimated value of TM in place of TO.
Heat Balance Cases[LINK]
TO\(_{t}\) and TI\(_{t}\) are related through the Y\(_{0}\)CTF. However TI\(_{t}\) is also unknown. While it is possible to combine the outside and the inside surface heat balances to compute TO\(_{t}\) and TI\(_{t}\) simultaneously, TARP uses a simpler procedure where TO\(_{t}\) is based on a previous value of TI. When Y\(_{0}\) is small, as occurs in well insulated or very massive surfaces, TI\(_{t}\) can be replaced by TI\(_{t-1}\) (which is known for the previous hour’s heat balance) without significantly effecting the value of TO\(_{t}\) When Y\(_{0}\) is large, TO and TI can so strongly be coupled that separate outside and inside heat balances do not work because the environment and zone temperatures have negligible influence on the heat balances. The TARP uses the inside surface heat balance to couple TO\(_{t}\) with TZ and TR. These two temperatures are less strongly influenced by TO and allow a reasonable heat balance. On the first heat balance iteration, TZ and TR are the values at time t-1. The user may optionally require that TO\(_{t}\) be recomputed with every iteration of TI\(_{t}\). In this case TZ and TR have values from the previous iteration and a true simultaneous solution is achieved. In most conventional constructions, recomputing TO\(_{t}\) does not significantly change the computed zone loads and temperatures. The inside surface heat balance is given by
\[{\rm{T}}{{\rm{I}}_{\rm{t}}} = \left[ {\frac{{KI{P_t} + QSI + HC\cdot TZ + HR\cdot TR + {Y_0}\cdot TO}}{{{Z_0} + HC + HR}}} \right]\]
The surface heat balances can be combined in eight ways according to conditions for calculations of the outside surface temperature
\[{F_1} = \left[ {\frac{{{Y_0}}}{{{Z_0} + HI + HR}}} \right]\]
\[{F_2} = \left[ {\frac{{UM}}{{UM + HO}}} \right]\]
\[{F_3} = \left[ {\frac{{UM}}{{UM + HA + HS + HG}}} \right]\]
Case1: Y\(_{0}\) small, simple conductance, no movable insulation:[LINK]
From Equation
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ + HO}}\cdot {{\rm{T}}_{\rm{a}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HO}}}}} \right]{\rm{ }}\]
Case2: Y\(_{0}\) not small, simple conductance, no movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {\rm{HO}}\cdot {{\rm{T}}_{\rm{a}}} + {{\rm{F}}_1}\cdot \left( {{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ + HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HO - }}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_0}}}} \right]{\rm{ }}\]
Case3: Y\(_{0}\) small, detailed conductance, no movable insulation:[LINK]
From Equation
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ + HA}}\cdot {{\rm{T}}_{\rm{a}}} + {\rm{HS}}\cdot {{\rm{T}}_{\rm{s}}} + {\rm{HG}}\cdot {{\rm{T}}_{\rm{g}}}}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HA + HS + HG}}}}} \right]{\rm{ }}\]
Case4: Y\(_{0}\) not small, detailed conductance, no movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {\rm{HA}}\cdot {{\rm{T}}_{\rm{a}}} + {\rm{HS}}\cdot {{\rm{T}}_{\rm{s}}} + {\rm{HG}}\cdot {{\rm{T}}_{\rm{g}}} + {{\rm{F}}_{\rm{1}}}\cdot \left( {{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ + HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + HA + HS + HG - }}{{\rm{F}}_1}\cdot {{\rm{Y}}_0}}}} \right]{\rm{ }}\]
Case5: Y\(_{0}\) small, simple conductance, with movable insulation:[LINK]
From Equations and \(TO_t = \left[ \frac{ KOP_t + QSO + HA \cdot T_a + HS \cdot T_s + HG \cdot T_g + F_1 \cdot \left( KIP_t QS1 + HI \cdot TZ + HR \cdot TR \right) } { X_0 + HA + HS + HG - F_1 \cdot Y_0 } \right]\)
Case6: Y\(_{0}\) not small, simple conductance, with movable insulation:[LINK]
From Equations , and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{F}}_2}\cdot \left( {{\rm{QSM + HO}}\cdot {{\rm{T}}_{\rm{a}}}} \right) + {{\rm{F}}_1}\cdot \left( {{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ + HR}}\cdot {\rm{TR}}} \right)}}{{{{\rm{X}}_{\rm{0}}} + {\rm{UM - }}{{\rm{F}}_{\rm{2}}}\cdot {\rm{UM - }}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_{\rm{0}}}}}} \right]{\rm{ }}\]
Case7: Y\(_{0}\) small, detailed conductance, with movable insulation:[LINK]
From Equations and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{Y}}_0}\cdot {\rm{T}}{{\rm{I}}_{{\rm{t - 1}}}}{\rm{ + }}{{\rm{F}}_{\rm{3}}}\left( {{\rm{QSM + HA}}\cdot {{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{ + HG}}\cdot {{\rm{T}}_{\rm{g}}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM - }}{{\rm{F}}_{\rm{3}}}\cdot {\rm{UM}}}}} \right]{\rm{ }}\]
Case8: Y\(_{0}\) not small, detailed conductance, with movable insulation:[LINK]
From Equations , and
\[{\rm{T}}{{\rm{O}}_{\rm{t}}}{\rm{ = }}\left[ {\frac{{{\rm{KO}}{{\rm{P}}_{\rm{t}}} + {\rm{QSO}} + {{\rm{F}}_1}\cdot \left( {{\rm{KI}}{{\rm{P}}_{\rm{t}}}{\rm{ + QSI + HI}}\cdot {\rm{TZ + HR}}\cdot {\rm{TR}}} \right){\rm{ + }}{{\rm{F}}_{\rm{3}}}\left( {{\rm{QSM + HA}}\cdot {{\rm{T}}_{\rm{a}}}{\rm{ + HS}}\cdot {{\rm{T}}_{\rm{s}}}{\rm{ + HG}}\cdot {{\rm{T}}_{\rm{g}}}} \right)}}{{{{\rm{X}}_{\rm{0}}}{\rm{ + UM - }}{{\rm{F}}_{\rm{3}}}\cdot {\rm{UM - }}{{\rm{F}}_{\rm{1}}}\cdot {{\rm{Y}}_{\rm{0}}}}}} \right]{\rm{ }}\]
Fortran Algorithm Examples[LINK]
Case5: Y\(_{0}\) small, simple conductance, with movable insulation:[LINK]
From Equation
Case6: Y\(_{0}\) not small, simple conductance, with movable insulation:[LINK]
From Equation
Case7: Y\(_{0}\) small, detailed conductance, with movable insulation:[LINK]
From Equation
Case8: Y\(_{0}\) not small, detailed conductance, with movable insulation:[LINK]
From Equation
Fortran Variable Descriptions[LINK]
References[LINK]
Walton, G.N. 1983. “The Thermal Analysis Research Program Reference Manual Program (TARP)”, National Bureau of Standards (now National Institute of Standards and Technology).
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