Passivhaus Compendium for Exam and Daily Use

Sources: Passivhaus Institut (PHI), Passivhaus Dienstleistung GmbH (PHD)

v2.6m | page 1 | © André Harrmann | Not liable for any errors and omissions. All references are made to PHPP Manual Version 8 (2013): [PHPP:page]

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Annual Space Heat Demand

Transmission Heat Losses

Ventilation Heat Losses

Utilization factor for free heat gains

Solar Gains

Internal Heat Gains

QH = QT + QV – ƞ × ( QS + QI ) 2,211 kWh/a 5,835 kWh/a 759 kWh/a 0.94 2,937 kWh/a 1,722 kWh/a

qH = qT + qV – ƞ × ( qS + qI ) 14 kWh/m2a 37.4 kWh/m2a 4.9 kWh/m2a 0.94 18.8 kWh/m2a 11.0 kWh/m2a

QH = [PHPP:124] Amount of heat (fuel) required per year to keep building at 20˚C; Specific Annual Heat Demand qH = QH / ATFA ≤ 15kWh/m2a in a PH QH,window = QT,window - QS = Window energy balance; QL = QT + QV = Total heat losses; QF = QI + QS = Free heat (heat gains); QG = QF x ƞG = useful heat gains ƞ = Defined as the fraction of free heat that can be utilized for space heating. (Surplus heat, e.g. excess solar gains are not or only partially usable.) ƞG = (1-(QF/QL)

5) / (1-(QF/QL)6) = (1-(4,659/6,595)5)/(1-(4,659/6,595)6) = (1-0.175)/(1-0.124) = 0.94 (QF/QL=1 ƞ=0.8, QF/QL=2 ƞ=0.5) [PHPP:122,124]


Area of envelope / building element U-value

Temperature correction factor

Heating degree hours

QT = A × U × ft × Gt 2,075 kWh/a 184.3 m2 0.138 W/m2K 1.0 81.9 kKh/a

QT = Calculated for each individual building element (exterior dimensions) [PHPP:115]. QT,window = Aw × U × Gt [PHPP:87] QT,thermal bridge = l × Ψ × ft × Gt = 116.9m × -0.030W/mK × 1.0 × 81.9kKh/a = -285kWh/a -285kWh/a / 156m2 = -1.83kWh/m2a [PHPP:118] Ψ (psi) = Linear thermal bridge heat loss coefficient, relative to the exterior dimensions, can be negative. l = length of thermal bridge χ (chi) = Point thermal bridge loss coefficient. [PHPP:66,74,118] ft = 1.0 if exposed to ambient air (worst case) [PHPP:55] ft < 1.0 if element is below ground or against unheated basement (reduction for reduced temperature difference against ground is calculated [PHPP:76], typical 0.5-0.7) or adjoining other buffer zones [PHPP:55] GT = Time integral of temperature differences between interior and outside air GT = ΔT × hours GT,Germany,PHPP-Default = 82 kKh/a GT,monthly,5˚C = (20-5)K × (31d×24h)/1,000 = 11.16 kKh/month GT,Vancouver = 70, GT,Yellowknife = 213, GT,New York = 72, GT,San Francisco = 28 kKh/a

Ventilation &

Infiltration Losses Ventilated

volume Energetically effective

air exchange rate Volumetric heat capacity of air

Heating degree hours

QV = VV × nV,Q × cp,air × Gt 759 kWh/a 390 m3 0.072 h-1 0.33 Wh/m3K 81.9 kKh/a

VV = TFA × average room height = Reference volume of the ventilation system = 156m2 × 2.5m = 390m3 (A standard residential room height of 2.5m is used for calculation purposes – larger values would result in excessive exchanged air volume.) [PHPP:119] nV,Q = Energetically effective air change rate (for Heat Demand calculation) = nequiv. equivalent air exchange = ventilation + leakage nV,Q = nV,System × (1 - ϕHR) + nV,Rest,Q = 0.300h-1 × (1 - 0.82) + 0.019h-1 = 0.072h-1 ϕHR = Overall heat recovery efficiency [PHPP:119,124] nV,system = Average air exchange rate of the ventilation system = 0.4h-1 default value for residences [PHPP:119] or calculated [PHPP:105] nV,Rest,Q = Infiltration air change through envelope leakage = 0.6×0.07 = 0.042h-1 default value at 0.6ACH [PHPP:103]

Solar Gains

Reduction factor

g-value (= SHGC)

Gross window area

Global solar irradiation energy

QS = r × g × AW × G 2,489 kWh/a 0.44 0.5 30.4 m2 370 kWh/m2a

g = SHGC = Total solar energy transmission coefficient for the glazing at a normal to the irradiated surface. From Windows worksheet. AW = Rough opening of window; G = Total solar radiation energy (diffuse and direct) during heating period, averaged over all allocated windows with the same orientation [PHPP:81,121]. Calculated on Windows sheet, based on deviation from cardinal points [PHPP:81]. r = Reduction / attenuation factor r = rShading × rDirt × rincidence-angle × rFrame [PHPP:90] rshading = rH × rR × rO × rother [PHPP:91-98] rDirt = 0.95 Constant, rincidence-angle = 0.85 Constant for reflection (non-perpendicular incident radiation), rFrame = AGlass / AWindow = glazing fraction (0.6 …0.7 are typical values; higher value = less frame)

rShading = 0.75 Default value or calculated on shading worksheet with these values: rH = Continuous horizontal obstruction, rR = Vertical (reveal, vertical shading, lateral wall), rO = Horizontal (overhang, balcony), rother = Additional shading. (The larger the shading factor the less shaded the window is!)

Internal

Heat Gains Length of

heating period Spec. internal

heat gains Treated Floor Area

(TFA) QI = tHEAT × qi × ATFA

1,722 kWh/a 219 d/a x 0.024 kh/d 2.1 W/m2 156.0 m2 tHEAT = HT × 0.024 kh/a HT = Heating days per year [PHPP:120] HT,Germany,PHPP-Default = 219d, HT,Vancouver = 208d, HT,Yellowknife = 243d, HT,New York = 181d, HT,San Francisco = 107d Default average internal heat gains qi = 2.1 W/m2 for residential project (PHPPv9: qi = 2.1 … 4.1W/m2 depending on size of dwelling units) qi = 4.1 W/m2 for assisted living, qi = 3.5 W/m2 for offices, qi = 2.8 W/m2 for schools [PHPP:120], or calculated on IHG worksheet [PHPP:186]

Losses QT Transmission Windows QT Transmission Opaque Elements QV Ventilation

& Infiltration ‘ƞ’

Energy Balance

South ...other Roof Walls Ground

Gains QS Solar Gains Windows South QS ...other QI Internal Heat Gains QH Heating Demand

Free Heat (not usable heat gains are considered a loss ‘ƞ’)

U-VALUE

VENTILATION

THERMAL BRIDGES

TFA

TFA



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Space Heating Load



Solar Gains

Internal Heat Gains

PH = PT + PV – ( PS + PI ) 1,558 W 2,188 W 264 W 645 W 250 W

pH = pT + pV - ( pS + pI ) 9.99 W/m2 14.03 W/m2 1.69 W/m2 4.13 W/m2 1.60 W/m2

PH = Size of heating system (maximum daily mean power) required to keep the building at 20˚C in 2 worst-case weather scenarios: ① cold, clear day (with higher heat losses and solar gains) or ② moderate, cloudy day (with lower heat losses and limited solar gains) [PHPP:132] Specific Heating Load pH = PH / ATFA < 10W/m2 in a PH = 1,558W / 156m2 = 9.99W/m2 PH,window = PT,window - PS = Window Energy Balance Pcandle ≈ 35W, Phuman ≈ 80W (55% of that is considered internal heat source as per PHPP)


Area of envelope or building element U-value


Temperature difference

PT = A × U × ft × Δt1 or t2 774 W 184.3 m2 0.138 W/m2K 1.0 30.6 K

PT = Calculated for each individual building element (exterior dimensions) [PHPP:129]. PT,window = Aw × U × Δt1 or t2

PT,thermal bridge = l × Ψ × ft × Δt = 116.9m × -0.030W/mK × 1.0 × 30.6K = -106W Ψ (psi) = Linear thermal bridge heat loss coefficient, relative to the exterior dimensions, can be negative. l = length of thermal bridge χ (chi) = Point thermal bridge loss coefficient. [PHPP:66,74,118] ft = 1.0 if exposed to ambient air (worst case) [PHPP:55] ft < 1.0 if element is below ground or against unheated basement (reduction for reduced temperature difference against ground is calculated [PHPP:76], typical 0.5-0.7) or adjoining other buffer zones [PHPP:55] Δt1 or t2 = Difference between 20˚C and outside temperature for worst case (of the two daily averages per PHPP climate data)

Ventilation &





PV = VV × nV,P × cp,air × Δt1 or t2 264 W 390 m3 0.068 h-1 0.33 Wh/(m3K) 30.6 K

VV = TFA × average room height = Reference volume of the ventilation system = 156m2 × 2.5m = 390m3 (A standard residential room height of 2.5m is used for calculation purposes – larger values would result in excessive exchanged air volume.) [PHPP:119] PV = Heat losses via leakage through the envelope and through the HRV system. nV,P = Energetically effective air change rate (for Heat Load design condition) [PHPP:129,132] nV,P = nV,System × (1 - ϕHR1 or HR2) + nV,Rest,P = 0.300h-1 × (1 - 0.93) + 0.047h-1 = 0.068h-1 nV,Rest,P = Infiltration air change through envelope leakage = 2.5 times the value of the average of the heating period (worst case scenario) = 2.5 × nV,Rest,Q = 2.5 × 0.019h-1 = 0.047h-1

Solar Gains

Reduction factor

g-value (= SHGC)

Gross window area

Global solar irradiation power

PS = r × g × AW × G1 or 2 605 W 0.44 0.5 30.4 m2 90 W/m2

g = SHGC = Total solar energy transmission coefficient for the glazing at a normal to the irradiated surface. From Windows sheet. AW = Rough opening of window. G = Daily mean global irradiation. Solar radiation power dependent on orientation for weather condition 1 & 2. [PHPP:130] r = Reduction / attenuation factor r = rShading × rDirt × rincidence-angle × rFrame [PHPP:90] rshading = rH × rR × rO × rother [PHPP:91-98] rDirt = 0.95 Constant, rincidence-angle = 0.85 Constant for reflection (non-perpendicular incident radiation), rFrame = AGlass / AWindow = glazing fraction (0.6 …0.7 are typical values; higher value = less frame)


Internal

Heat Gains Internal


(TFA) PI = qi,P × ATFA

250 W 1.6 W/m2 156.0 m2 qi,p = 1.6 W/m2 default for residential projects (Reduced to simulate unoccupied building, cannot be carried over from QH because annual heat demand calculates average for the entire heating period.) [PHPP:129] (new with PHPPv9: qi,P = qi,Q - 0.5 = 2.1-0.5 = 1.6 W/m2)

Losses PT Transmission Windows PT Transmission Opaque Elements PV Ventilation

& Infiltration

Energy Balance for two scenarios


Gains PS Solar Gains Windows South PS ...other PI Internal Heat Gains PH Heating Load

Free Heat (PI is calculated with 0.5W/m2 lower gains than QI)

HEATING via SUPPLY AIR

U-VALUE

THERMAL BRIDGES

TFA

VENTILATION

TFA

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Space Heating Load



Solar Gains

Internal Heat Gains

PH = PT + PV – ( PS + PI ) 1,558 W 2,188 W 264 W 645 W 250 W

pH = pT + pV - ( pS + pI ) 9.99 W/m2 14.03 W/m2 1.69 W/m2 4.13 W/m2 1.60 W/m2

PH = Size of heating system (maximum daily mean power) required to keep the building at 20˚C in 2 worst-case weather scenarios: ① cold, clear day (with higher heat losses and solar gains) or ② moderate, cloudy day (with lower heat losses and limited solar gains) [PHPP:132] Specific Heating Load pH = PH / ATFA < 10W/m2 in a PH = 1,558W / 156m2 = 9.99W/m2 PH,window = PT,window - PS = Window Energy Balance Pcandle ≈ 35W, Phuman ≈ 80W (55% of that is considered internal heat source as per PHPP)


Area of envelope or building element U-value



PT = A × U × ft × Δt1 or t2 774 W 184.3 m2 0.138 W/m2K 1.0 30.6 K

PT = Calculated for each individual building element (exterior dimensions) [PHPP:129]. PT,window = Aw × U × Δt1 or t2

PT,thermal bridge = l × Ψ × ft × Δt = 116.9m × -0.030W/mK × 1.0 × 30.6K = -106W Ψ (psi) = Linear thermal bridge heat loss coefficient, relative to the exterior dimensions, can be negative. l = length of thermal bridge χ (chi) = Point thermal bridge loss coefficient. [PHPP:66,74,118] ft = 1.0 if exposed to ambient air (worst case) [PHPP:55] ft < 1.0 if element is below ground or against unheated basement (reduction for reduced temperature difference against ground is calculated [PHPP:76], typical 0.5-0.7) or adjoining other buffer zones [PHPP:55] Δt1 or t2 = Difference between 20˚C and outside temperature for worst case (of the two daily averages per PHPP climate data)

Ventilation &





PV = VV × nV,P × cp,air × Δt1 or t2 264 W 390 m3 0.068 h-1 0.33 Wh/(m3K) 30.6 K

VV = TFA × average room height = Reference volume of the ventilation system = 156m2 × 2.5m = 390m3 (A standard residential room height of 2.5m is used for calculation purposes – larger values would result in excessive exchanged air volume.) [PHPP:119] PV = Heat losses via leakage through the envelope and through the HRV system. nV,P = Energetically effective air change rate (for Heat Load design condition) [PHPP:129,132] nV,P = nV,System × (1 - ϕHR1 or HR2) + nV,Rest,P = 0.300h-1 × (1 - 0.93) + 0.047h-1 = 0.068h-1 nV,Rest,P = Infiltration air change through envelope leakage = 2.5 times the value of the average of the heating period (worst case scenario) = 2.5 × nV,Rest,Q = 2.5 × 0.019h-1 = 0.047h-1

Solar Gains

Reduction factor

g-value (= SHGC)

Gross window area

Global solar irradiation power

PS = r × g × AW × G1 or 2 605 W 0.44 0.5 30.4 m2 90 W/m2

g = SHGC = Total solar energy transmission coefficient for the glazing at a normal to the irradiated surface. From Windows sheet. AW = Rough opening of window. G = Daily mean global irradiation. Solar radiation power dependent on orientation for weather condition 1 & 2. [PHPP:130] r = Reduction / attenuation factor r = rShading × rDirt × rincidence-angle × rFrame [PHPP:90] rshading = rH × rR × rO × rother [PHPP:91-98] rDirt = 0.95 Constant, rincidence-angle = 0.85 Constant for reflection (non-perpendicular incident radiation), rFrame = AGlass / AWindow = glazing fraction (0.6 …0.7 are typical values; higher value = less frame)


Internal

Heat Gains Internal


(TFA) PI = qi,P × ATFA

250 W 1.6 W/m2 156.0 m2 qi,p = 1.6 W/m2 default for residential projects (Reduced to simulate unoccupied building, cannot be carried over from QH because annual heat demand calculates average for the entire heating period.) [PHPP:129] (new with PHPPv9: qi,P = qi,Q - 0.5 = 2.1-0.5 = 1.6 W/m2)

Losses PT Transmission Windows PT Transmission Opaque Elements PV Ventilation

& Infiltration

Energy Balance for two scenarios


Gains PS Solar Gains Windows South PS ...other PI Internal Heat Gains PH Heating Load

Free Heat (PI is calculated with 0.5W/m2 lower gains than QI)

HEATING via SUPPLY AIR

U-VALUE

THERMAL BRIDGES

TFA

VENTILATION

TFA

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U-VALUE CALCULATION – OPAQUE ELEMENT [PHPP:45]

U = 1

RT=

1

Rsi + d1λ1

+ d2λ2

+ d3λ3

+ Rse

Surface Film Thermal Resistances Rsi and Rse [m2K/W]: * Considered horizontal if heat flow is up to ±30˚

from the horizontal ** Interior values might be used for ventilated

rainscreens (e.g. 0.13) and crawlspaces (e.g. 0.17) and ventilated roofs (e.g. 0.10) [PHPP:48]

U= 1

0.13 m2KW + 0.30m

0.035 WmK

+ 0.24m

0.79 WmK

+ 0.015m

0.70 WmK

+ 0.04 m2KW

= 1

9.07 m2KW

= 0.1103 W

m2K

Example above is: EIFS | Brick wall | interior plaster U [W/m2K] = Heat transfer coefficient (heat flow in W through 1m2 of a structure at ΔT = 1˚C) U-value for composite building elements (e.g. framed wall) is calculated [PHPP:47-48] RT [m2K/W] = Total thermal resistance Ri = di / λi = Thermal resistance of each layer [m2K/W] λ [W/mK] = Thermal conductivity [PHPP:46] d [m] = thickness of each layer Rsi [m

2K/W] = Thermal resistance of the interior surface [PHPP:48] Rse [m2K/W] = Thermal resistance of the exterior surface and below ground Rsi and Rse are already included in Uglass and Uframe for windows. Rsi is typically larger than Rse due to lower ΔT and less air movement on the interior surface

WINDOWS: U-Values and Surface Temperatures

Uwindow,installed = (Aglass × Uglass) + (Aframe × Uframe) + (Lspacer × Ψspacer) + (Linstall × Ψinstall)

Awindow

Uwindow = (1.224m2 × 0.6W/m2K) + (0.596m2 × 1.6W/m2K) + (4.45m × 0.08W/mK)

1.820m2 = 1.123W/m2K

Uw,installed = Uwindow + Linstall × Ψinstall

Awindow = 1.123W/m2K +

5.42m × 0.15W/mK1.820m2 = 1.123W/m2K + 0.447W/m2K = 1.569W/m2K

Awindow = wwindow × hwindow = Total window area (rough opening) = 1.23m × 1.48m = 1.820m2 [PHPP:78,87] Aglass = wglass × hglass = Glazing area = (1.23-0.117-0.117)m × (1.48-0.117-0.134)m = 0.996m × 1.229m = 1.224m2 Aframe = Awindow - Aglass = Total window frame area = 1.820m2 - 1.224m2 = 0.596m2

Lspacer = Lglass = 2 × wglass + 2 × hglass = Glazing perimeter (= spacer length) = (0.996+1.229)m × 2 = 4.45m

Linstall = Lframe = 2 × wwindow + 2 × hwindow = Window frame perimeter (install.) = (1.23+1.48)m × 2 = 5.42m Ψspacer = Ψglazing edge = Average thermal bridge heat loss coefficient of the glazing edge seal, can be ~0.02 [PHPP:84] Ψinstall = Average thermal bridge heat loss coefficient of the installation (~0.00 W/mK can be achieved with window installed in insulation layer and 60mm ‘over-insulation’), PHPP default is 0.04W/mK, more precise values may be obtained from window certification document, or calculated (e.g. THERM software) [PHPP:78,83-85] Ψ for windows is not a material specific parameter, but depends on the type of installation and type of spacer Criteria for glazing:

Comfort: Ug ≤ 0.80 W/m2K Energy: Ug - (S × g) < 0 S = radiation gain coefficient = 1.6W/m2K for Central Europe

Inside Surface Temperature of a Window (or Wall) Surface temperatures determine comfort level + risk of mould. Tsi = Ti - (U × Rsi × ΔT) = 20˚C - (2.8W/m2K × 0.13m2K/W × 30˚C) = 9.08˚C Tsi = Surface temperature inside Ti = Inside air temperature Te = Exterior air temperature Rsi = Surface thermal resistance inside U = U-value of the component ΔT = Temperature difference inside and outside ΔT = Ti - Te = 20˚C - (-10˚C) = 30˚C

H-VALUE [PHPP:59] THERMAL BRIDGES [PHPP:47,66,74,118]

H = Temperature specific transmission heat losses

H = A × U = 184.28m2 × 0.138W/m2K = 25.3W/K HΨ = l × Ψ = 116.85m × -0.03W/mK = -3.5W/K Hχ = χ = 0.77W/K

∑H = 22.57W/K

QT = ∑H × ft × Gt = 22.57W/K × 1 × 81.9kKh/a = 1,848kWh/a PT = ∑H × ft × Δt1 or t2 = 22.57W/K × 1 × 30.6K = 690W

The linear transmittance Ψ and point transmittance χ coefficients represent the increased heat flow at thermal bridges compared to adjoining building components (using 2D modelling of the heat flow, based on exterior dimensions). Compliance Definition ① (requires calculation of all thermal bridges): Thermal bridge free if there is no increase in the building envelope’s average U-value due to ΔUTB ≤ 0W/m2K (actual transmission losses of all thermal bridges ≤ losses of building elements alone, calculated using the external surfaces and regular U-values.) HTB = ∑(l × Ψ) + ∑(χ) = -3.5W/K + 0.77W/K = -2.73W/K ΔUTB = HTB / ATotal thermal envelope = -2.73W/K / 392.07m2 ≤ 0W/m2K thermal bridge free Compliance Definition ② (pragmatic approach): Thermal bridge free if for each linear thermal bridge Ψ < 0.01W/mK (to avoid heat losses); and change in U-value for each point thermal bridge ΔUTB = χ/AElement < 0.01W/m2K (to be considered for condensation), larger χ may be considered for transmission loss calculation [see example PHPP:88]. Thermal bridges for window openings are accounted for in the U-value calculation for windows Uw,installed. Thermal Bridge Rules: • Avoidance (do not penetrate insulation) • Geometry (avoid sharp angles, keep simple building form) • Pierce-through (if disturbance of insulation layer is unavoidable, use materials with high thermal resistance) • Connection (transfer insulation layers without gaps at connection details, connect the entire cross area) Repeating thermal bridges in composite/inhomogeneous opaque building elements (e.g. timber stud walls) can be approximated on the PHPP U-Values worksheet (recommended approach only if the calculation error resulting from the variation of the λ values in the different wall sections is less than 10%). [PHPP:47]

h win

dow

= 1.

48m

wwindow = 1.23m

h gla

ss =

1.2

29m

wglass = 0.996m

Aglass

A

frame

0.117m

0.134m

0.11

7m

inte

rior exterior

below ground 0.00 0.17 downward 0.00**

0.13 horizontal 0.04**

SEE ABOVE

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U-VALUE CALCULATION – OPAQUE ELEMENT [PHPP:45]

U = 1

RT=

1

Rsi + d1λ1

+ d2λ2

+ d3λ3

+ Rse

Surface Film Thermal Resistances Rsi and Rse [m2K/W]: * Considered horizontal if heat flow is up to ±30˚

from the horizontal ** Interior values might be used for ventilated

rainscreens (e.g. 0.13) and crawlspaces (e.g. 0.17) and ventilated roofs (e.g. 0.10) [PHPP:48]

U= 1

0.13 m2KW + 0.30m

0.035 WmK

+ 0.24m

0.79 WmK

+ 0.015m

0.70 WmK

+ 0.04 m2KW

= 1

9.07 m2KW

= 0.1103 W

m2K

Example above is: EIFS | Brick wall | interior plaster U [W/m2K] = Heat transfer coefficient (heat flow in W through 1m2 of a structure at ΔT = 1˚C) U-value for composite building elements (e.g. framed wall) is calculated [PHPP:47-48] RT [m2K/W] = Total thermal resistance Ri = di / λi = Thermal resistance of each layer [m2K/W] λ [W/mK] = Thermal conductivity [PHPP:46] d [m] = thickness of each layer Rsi [m

2K/W] = Thermal resistance of the interior surface [PHPP:48] Rse [m2K/W] = Thermal resistance of the exterior surface and below ground Rsi and Rse are already included in Uglass and Uframe for windows. Rsi is typically larger than Rse due to lower ΔT and less air movement on the interior surface

WINDOWS: U-Values and Surface Temperatures

Uwindow,installed = (Aglass × Uglass) + (Aframe × Uframe) + (Lspacer × Ψspacer) + (Linstall × Ψinstall)

Awindow

Uwindow = (1.224m2 × 0.6W/m2K) + (0.596m2 × 1.6W/m2K) + (4.45m × 0.08W/mK)

1.820m2 = 1.123W/m2K

Uw,installed = Uwindow + Linstall × Ψinstall

Awindow = 1.123W/m2K +

5.42m × 0.15W/mK1.820m2 = 1.123W/m2K + 0.447W/m2K = 1.569W/m2K

Awindow = wwindow × hwindow = Total window area (rough opening) = 1.23m × 1.48m = 1.820m2 [PHPP:78,87] Aglass = wglass × hglass = Glazing area = (1.23-0.117-0.117)m × (1.48-0.117-0.134)m = 0.996m × 1.229m = 1.224m2 Aframe = Awindow - Aglass = Total window frame area = 1.820m2 - 1.224m2 = 0.596m2

Lspacer = Lglass = 2 × wglass + 2 × hglass = Glazing perimeter (= spacer length) = (0.996+1.229)m × 2 = 4.45m

Linstall = Lframe = 2 × wwindow + 2 × hwindow = Window frame perimeter (install.) = (1.23+1.48)m × 2 = 5.42m Ψspacer = Ψglazing edge = Average thermal bridge heat loss coefficient of the glazing edge seal, can be ~0.02 [PHPP:84] Ψinstall = Average thermal bridge heat loss coefficient of the installation (~0.00 W/mK can be achieved with window installed in insulation layer and 60mm ‘over-insulation’), PHPP default is 0.04W/mK, more precise values may be obtained from window certification document, or calculated (e.g. THERM software) [PHPP:78,83-85] Ψ for windows is not a material specific parameter, but depends on the type of installation and type of spacer Criteria for glazing:

Comfort: Ug ≤ 0.80 W/m2K Energy: Ug - (S × g) < 0 S = radiation gain coefficient = 1.6W/m2K for Central Europe

Inside Surface Temperature of a Window (or Wall) Surface temperatures determine comfort level + risk of mould. Tsi = Ti - (U × Rsi × ΔT) = 20˚C - (2.8W/m2K × 0.13m2K/W × 30˚C) = 9.08˚C Tsi = Surface temperature inside Ti = Inside air temperature Te = Exterior air temperature Rsi = Surface thermal resistance inside U = U-value of the component ΔT = Temperature difference inside and outside ΔT = Ti - Te = 20˚C - (-10˚C) = 30˚C

H-VALUE [PHPP:59] THERMAL BRIDGES [PHPP:47,66,74,118]

H = Temperature specific transmission heat losses

H = A × U = 184.28m2 × 0.138W/m2K = 25.3W/K HΨ = l × Ψ = 116.85m × -0.03W/mK = -3.5W/K Hχ = χ = 0.77W/K

∑H = 22.57W/K

QT = ∑H × ft × Gt = 22.57W/K × 1 × 81.9kKh/a = 1,848kWh/a PT = ∑H × ft × Δt1 or t2 = 22.57W/K × 1 × 30.6K = 690W

The linear transmittance Ψ and point transmittance χ coefficients represent the increased heat flow at thermal bridges compared to adjoining building components (using 2D modelling of the heat flow, based on exterior dimensions). Compliance Definition ① (requires calculation of all thermal bridges): Thermal bridge free if there is no increase in the building envelope’s average U-value due to ΔUTB ≤ 0W/m2K (actual transmission losses of all thermal bridges ≤ losses of building elements alone, calculated using the external surfaces and regular U-values.) HTB = ∑(l × Ψ) + ∑(χ) = -3.5W/K + 0.77W/K = -2.73W/K ΔUTB = HTB / ATotal thermal envelope = -2.73W/K / 392.07m2 ≤ 0W/m2K thermal bridge free Compliance Definition ② (pragmatic approach): Thermal bridge free if for each linear thermal bridge Ψ < 0.01W/mK (to avoid heat losses); and change in U-value for each point thermal bridge ΔUTB = χ/AElement < 0.01W/m2K (to be considered for condensation), larger χ may be considered for transmission loss calculation [see example PHPP:88]. Thermal bridges for window openings are accounted for in the U-value calculation for windows Uw,installed. Thermal Bridge Rules: • Avoidance (do not penetrate insulation) • Geometry (avoid sharp angles, keep simple building form) • Pierce-through (if disturbance of insulation layer is unavoidable, use materials with high thermal resistance) • Connection (transfer insulation layers without gaps at connection details, connect the entire cross area) Repeating thermal bridges in composite/inhomogeneous opaque building elements (e.g. timber stud walls) can be approximated on the PHPP U-Values worksheet (recommended approach only if the calculation error resulting from the variation of the λ values in the different wall sections is less than 10%). [PHPP:47]

h win

dow

= 1.

48m

wwindow = 1.23m

h gla

ss =

1.2

29m

wglass = 0.996m

Aglass

A

frame

0.117m

0.134m

0.11

7m

inte

rior exterior

below ground 0.00 0.17 downward 0.00**

0.13 horizontal 0.04**

SEE ABOVE

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Fresh air [e.g. -10˚C] ODA

Exhaust air EHA

Extract air [20˚C]

Supply air

heat generator

cold

water (or... DHW

hot water storage tank

HR Air-to-air plate heat exchanger

com

bust

ion

air

chim

ney [≤ 52˚C]

post-heater

optional: solar hot water system

Frost protection

(cold water)

(or)

oil tank

[≥ 16.5˚C]

HEATING Central Heating Boiler: System Design

Supply Air ≤ 52˚C to avoid dust smoldering

Factors for thermal comfort: • Air temperature • Surface temperate • Local temperature difference • Draughts • Relative air humidity • Clothing and degree of activity (Thermal comfort is achieved if losses from human body are equal to heat production of body.)

Heat generation characteristics in PH (in addition to PH criteria) Heat Demand for DHW = 12-35kWh/m2a (dominance over heating demand in PH) Typical distribution losses = 15kWh/m2a (not useful) 5kWh/m2a (useful) It does not matter how and where heat is delivered (could be through supply air).

VENTILATION [PHPP:29] Dimensioning of Air Quantities nV,system and nV,Res [PHPP:105-108] (Too much ventilation leads to dry air - optimal 35-55% rel. humidity) ① Supply Air = 20 to 30m3/h per person = 4.5 person × 30m3/h/p = 134m3/h average airflow, distributed within the whole apartment

(The CO2 emissions of a person at average activity require 30m3/h for good air quality.) Typical values used for energy modeling: dwellings & offices 30m3/h/p, schools 15-20m3/h/p, sport halls 60m3/h/p

② Extract Air = kitchen 60m3/h, bathroom 40m3/h, WC and storage 20m3/h 60m3/h + 40m3/h + 20m3/h + 20m3/h = 140m3/h (Not permanently required if larger than calculated supply air – follow ① more closely.)

③ Minimum air change = 0.30h-1 x Vv 0.30h-1 × 390m3 = 117m3/h (average air flow) nV,system = Vaverage air flow / VV = 117m3h-1/390m3h-1 = 0.30h-1

m3/h ≈ cfm 20 12 30 18 40 24 50 29 60 35 100 59 150 88 200 118

Design Air Flow = max. of ① or ② or ③ [Vv × 0.3h-1 × 1.3] 390m3 × 0.30h-1 × 1.3 = 152m3/h (at 100%) Normal flow rate = 152 × 77% = 117m3/h

Infiltration

nV,Res= n50 × e × Vn50

VV ≈ 10% of n50 = 0.22h-1 × 0.07 ×

480m3

390m3 = 0.019h-1 n50 = V50

VAir =

measured air flow

net interior air volume =

106m3/ℎ480m3 = 0.22h-1

A50,Leakage ≈ 0.5cm2h/m3 × V50

≈ 0.5 × 300m3/h = 150cm2 nV,Rest = Infiltration air change through envelope n50 = Air change rate at pressure test e = exposure coefficient for screening class [PHPP:103] Vn50 = VAir = Pressure test reference volume, net air volume “visible air” to underside of suspended ceiling. VV = Ventilated volume (see under QV) V50 = Measured air flow rate at 50Pa measurements VP taken at a different pressure P can be approximately corrected to 50Pa by V50 ≈ (VP / P) × 50

Efficiency of Heat Recovery (HRV) [PHPP:78,100,110] ƞHR,eff [PHPP:106] Maximum heating load transportable via the supply air [PHPP:130]

ηHR= TETA- TEHA+ Pel

m × cp

TETA- TODA =

20˚C - 8.5˚C + 37W120 m3 h⁄ × 0.33 Wh m3K⁄20˚C - 4.0˚C

= 78%

Psupply,max = (Tsupply,max - Tsupply,min) × cp,air × VV,system Psupply,max = (52˚C - 18˚C) × 0.33Wh/m3K × 117m3/h = 1,314W PH ≤ Psupply,max 1,558W ≥ 1,314W not suitable for supply air heating Psupply,max = Maximum heating power which can be delivered in supply air cp,air = Volumetric heat capacity of air = 0.33Wh/m3K constant Tsupply,max = Max. supply air temp., ≤ 52˚C (downstream of post-heater) Tsupply,min = Supply air temperature, ≥ 16.5˚C (upstream of post-heater) Tsupply,min = TODA + ƞHR × (TETA - TODA) = -10˚C + 0.93 × (20˚C + 10˚C) = 18˚C VV,system = Average air flow rate through the ventilation system VV,system = VV × nV,system = 390m3 × 0.30h-1 = 117m3

ƞHR = Efficiency of HRV (≥ 75% so that SUP ≥ 16.5˚C) ṁ × cp = Vflow × cp,air

ṁ = Mass flow [kg/s] cp = Specific heat capacity of air [Ws/kg] cp,air = 0.33 Wh/m3K = Volumetric heat capacity of air at density 1.19 kg/m3

Electricity Demand = Pel / Vflow = 37W / 120m3/h = 0.31Wh/m3 (max. 0.45) Pel = electrical power (fans+controls) Vflow = Balanced air volume flow

ODA = Outdoor Air EHA = Exhaust Air ETA = Extract Air (20˚C) SUP = Supply Air (≥ 16.5˚C)

10W/m2 derivation:

pheating= VA

× ∆T × cp,air = 30m3/(h × person)

30m2/person × 30K × 0.33

Whm3K

≈ 10W/m2

Duct Diameter Openings for the transferred air

Duct diameter = 2×�V

velocity × π × 3,600 = 2×�

150m3/h2m/s × 3.14159 × 3,600

= 0.163m To allow air travel from delivery to exhaust zone keep

pressure loss < 1Pa (~ 1m/s). Guidelines: • Extract air rooms with 60m3/h 150cm2 total opening

gross section • Living rooms with 40m3/h 1.5-2cm gap under or

through door, or via lintel detail

V = Volumetric flow rate at standard rate (77%); Velocity = Speed of air flow in the duct, ideally max. 2m/s (to avoid turbulences), but could be 1.5-2.5m/s Typical: 100mm with ≤55m³/h, 150mm ≤120m³/h (at 2.0m/s) ≤160m³/h (at 2.5m/s)

Heating via Supply Air

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Fresh air [e.g. -10˚C] ODA

Exhaust air EHA

Extract air [20˚C]

Supply air

heat generator

cold

water (or... DHW

hot water storage tank

HR Air-to-air plate heat exchanger

com

bust

ion

air

chim

ney [≤ 52˚C]

post-heater

optional: solar hot water system

Frost protection

(cold water)

(or)

oil tank

[≥ 16.5˚C]

HEATING Central Heating Boiler: System Design

Supply Air ≤ 52˚C to avoid dust smoldering

Factors for thermal comfort: • Air temperature • Surface temperate • Local temperature difference • Draughts • Relative air humidity • Clothing and degree of activity (Thermal comfort is achieved if losses from human body are equal to heat production of body.)

Heat generation characteristics in PH (in addition to PH criteria) Heat Demand for DHW = 12-35kWh/m2a (dominance over heating demand in PH) Typical distribution losses = 15kWh/m2a (not useful) 5kWh/m2a (useful) It does not matter how and where heat is delivered (could be through supply air).

VENTILATION [PHPP:29] Dimensioning of Air Quantities nV,system and nV,Res [PHPP:105-108] (Too much ventilation leads to dry air - optimal 35-55% rel. humidity) ① Supply Air = 20 to 30m3/h per person = 4.5 person × 30m3/h/p = 134m3/h average airflow, distributed within the whole apartment

(The CO2 emissions of a person at average activity require 30m3/h for good air quality.) Typical values used for energy modeling: dwellings & offices 30m3/h/p, schools 15-20m3/h/p, sport halls 60m3/h/p

② Extract Air = kitchen 60m3/h, bathroom 40m3/h, WC and storage 20m3/h 60m3/h + 40m3/h + 20m3/h + 20m3/h = 140m3/h (Not permanently required if larger than calculated supply air – follow ① more closely.)

③ Minimum air change = 0.30h-1 x Vv 0.30h-1 × 390m3 = 117m3/h (average air flow) nV,system = Vaverage air flow / VV = 117m3h-1/390m3h-1 = 0.30h-1

m3/h ≈ cfm 20 12 30 18 40 24 50 29 60 35 100 59 150 88 200 118

Design Air Flow = max. of ① or ② or ③ [Vv × 0.3h-1 × 1.3] 390m3 × 0.30h-1 × 1.3 = 152m3/h (at 100%) Normal flow rate = 152 × 77% = 117m3/h

Infiltration

nV,Res= n50 × e × Vn50

VV ≈ 10% of n50 = 0.22h-1 × 0.07 ×

480m3

390m3 = 0.019h-1 n50 = V50

VAir =

measured air flow

net interior air volume =

106m3/ℎ480m3 = 0.22h-1

A50,Leakage ≈ 0.5cm2h/m3 × V50

≈ 0.5 × 300m3/h = 150cm2 nV,Rest = Infiltration air change through envelope n50 = Air change rate at pressure test e = exposure coefficient for screening class [PHPP:103] Vn50 = VAir = Pressure test reference volume, net air volume “visible air” to underside of suspended ceiling. VV = Ventilated volume (see under QV) V50 = Measured air flow rate at 50Pa measurements VP taken at a different pressure P can be approximately corrected to 50Pa by V50 ≈ (VP / P) × 50

Efficiency of Heat Recovery (HRV) [PHPP:78,100,110] ƞHR,eff [PHPP:106] Maximum heating load transportable via the supply air [PHPP:130]

ηHR= TETA- TEHA+ Pel

m × cp

TETA- TODA =

20˚C - 8.5˚C + 37W120 m3 h⁄ × 0.33 Wh m3K⁄20˚C - 4.0˚C

= 78%

Psupply,max = (Tsupply,max - Tsupply,min) × cp,air × VV,system Psupply,max = (52˚C - 18˚C) × 0.33Wh/m3K × 117m3/h = 1,314W PH ≤ Psupply,max 1,558W ≥ 1,314W not suitable for supply air heating Psupply,max = Maximum heating power which can be delivered in supply air cp,air = Volumetric heat capacity of air = 0.33Wh/m3K constant Tsupply,max = Max. supply air temp., ≤ 52˚C (downstream of post-heater) Tsupply,min = Supply air temperature, ≥ 16.5˚C (upstream of post-heater) Tsupply,min = TODA + ƞHR × (TETA - TODA) = -10˚C + 0.93 × (20˚C + 10˚C) = 18˚C VV,system = Average air flow rate through the ventilation system VV,system = VV × nV,system = 390m3 × 0.30h-1 = 117m3

ƞHR = Efficiency of HRV (≥ 75% so that SUP ≥ 16.5˚C) ṁ × cp = Vflow × cp,air ṁ = Mass flow [kg/s] cp = Specific heat capacity of air [Ws/kg] cp,air = 0.33 Wh/m3K = Volumetric heat capacity of air at density 1.19 kg/m3

Electricity Demand = Pel / Vflow = 37W / 120m3/h = 0.31Wh/m3 (max. 0.45) Pel = electrical power (fans+controls) Vflow = Balanced air volume flow

ODA = Outdoor Air EHA = Exhaust Air ETA = Extract Air (20˚C) SUP = Supply Air (≥ 16.5˚C)

10W/m2 derivation:

pheating= VA

× ∆T × cp,air = 30m3/(h × person)

30m2/person × 30K × 0.33

Whm3K

≈ 10W/m2

Duct Diameter Openings for the transferred air

Duct diameter = 2×�V

velocity × π × 3,600 = 2×�

150m3/h2m/s × 3.14159 × 3,600

= 0.163m To allow air travel from delivery to exhaust zone keep

pressure loss < 1Pa (~ 1m/s). Guidelines: • Extract air rooms with 60m3/h 150cm2 total opening

gross section • Living rooms with 40m3/h 1.5-2cm gap under or

through door, or via lintel detail

V = Volumetric flow rate at standard rate (77%); Velocity = Speed of air flow in the duct, ideally max. 2m/s (to avoid turbulences), but could be 1.5-2.5m/s Typical: 100mm with ≤55m³/h, 150mm ≤120m³/h (at 2.0m/s) ≤160m³/h (at 2.5m/s)

Heating via Supply Air

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Evaluation Criteria for residential buildings [PHPP:19] Space Heating Demand QH ≤ 15 kWh/m2a or alternatively Peak Heating Load PH ≤ 10 W/m2 (small building, large surface) Useful Cooling Demand ≤ 15 kWh/m2a or alternatively Peak Cooling Load ≤ 10 W/m2 + see [PHPP:19] Primary Energy Demand PE ≤ 120 kWh/m2a + see [PHPP:19] (heating, ventilation, cooling, DHW, aux electricity, household electricity) Building Airtightness n50 ≤ 0.6h-1 (0.649 is still OK) EN 13829 Method A: Envelope in the same condition as it is when heating or ventilation are being used. Difference between test results at positive and negative pressure <10%. Design temperature 20˚C (Can be different in justified cases.) [PHPP:32] Excess temperature ≤ 10% of total yearly hours with indoor air >25˚C Occupancy rate for Verification of residential buildings = 35 m2/person (PHPPv9: Based on typical occupancy rates for specific dwelling unit sizes); for Planning = 20-50 m2/p; entered manually for non-residential [PHPP:33]

New Evaluation Criteria 2015 with PHPPv9 (not yet mandatory in 2015) All of the above, except alternative methodology for Primary Energy PE and introduction of new classes to address renewable energy generation. Primary Energy Renewable PER is the demand profile of the individual energy application and locally available renewable primary power production from PV and wind (and hydro power).

PER = Edir + EMS

ηMS + ESS

ηSS + EDL

Edir + EMS + ESS

PER Factor for each source and application [kWhPER/kWh] Edir Electricity generated by RES used directly EMS Electricity from short/medium term storage ESS Electricity generated from energy in seasonal storage EDL Distribution and other losses ηMS and ηSS Efficiencies of storage processes (whole chain) New Passive House Classes: Classic Plus Premium Renewable PE Demand: ≤60 ≤45 ≤30 kWhPER/m2

TFAa Renewable Energy Generation: n/a ≥60 ≥120 kWhPER/m2

grounda

Cross Ventilation: Supply air zone Transferred air zone Extract air zone Coanda-Effect: Vacuum created by moving air from supply jet nozzle 5-20cm under ceiling pulls secondary air towards the ceiling. The ceiling acts as a half-sided duct and can move supply air up to 6m into room. Ductwork – insulation thickness ETA SUP SUP ODA (~20˚C) (16-20˚C) (heated) / EHA Outside thermal envelope: 100mm 100mm 150mm 0 Transfer/exhaust room inside: 0 0 20-30mm 50-100mm Supplied room inside: 0 0 0 50-100mm Insulation on cold ducts inside must be vapour impermeable. Use silencers. Keep all ducts as short as possible – in particular cold ducts inside. Treated Floor Area (TFA) The rules are based on German WoflV and DIN 277 and incentivise designing efficient plans with high-quality spaces within the thermal envelope. Rules for residential buildings (WoflV) [PHPP:63]: Included: Floor areas of rooms measured from clear width between building elements, in particular: living and circulation areas, washrooms, auxiliary rooms (storage, service, and utility rooms), stair heads and landings, floor to ceiling window reveals which are ≥0.13m deep. Included with 60%: Auxiliary rooms and circulation areas outside dwelling or on floors of detached houses in which less than 50% of the floor area is considered living space (e.g. in the basement). Excluded: Stairs with more than 3 risers, walls and other elements >1.5m high, shafts and chimneys and pillars >0.1m2, doorways, window reveals (unless see above), areas outside thermal envelope. Rule for all areas with reduced ceiling height: The TFA is reduced by 50% if clear room height is 1-2m. Areas <1m high are excluded from TFA.

The necessity of an airtight building envelope: (Infiltration and exfiltration are caused by wind and buoyancy, due to leakages in envelope. Design one airtight layer all around the building.) • Prevention of condensation in the construction (exfiltration most critical:

360g water/day can condense through 1mm x 1m leak, when outside 0˚C, 80%RH and inside 0˚C, 50%RH)

• Prevention of drafts • Prevention of cold floors in the ground floor • Preventing air pollution of the room air • Securing the sound insulation of building components • Securing the operation and effectiveness of the ventilation system • Securing the insulation effect of the external building components • Reduction of ventilation heat losses (infiltration) • For HRV to work efficiently, airtightness is important

Passive House Components (Quality Criteria) Heat protection: U ≤ 0.15 W/m2K thermal envelope, opaque elements (typically 0.10-0.15 W/m2K); thermal bridge free (to reduce heat loss and avoid cold interior surfaces) Heat Recovery Ventilation (HRV, MVHR): ηHR ≥ 75% efficiency (to maintain min. 16.5˚C supply air temp. at -10˚C, prefer 85-92%); Low velocity; Electricity demand max. 0.45 Wh/m3; Max. 25dB(A) in habitable rooms, 30dB(A) in functional rooms, 35dB(A) in room with ventilation unit; Balanced (≤ 10% during operation between ODA and EHA) and Controlled operation (basic / normal / purge: 54 / 77 / 100%, summer bypass); Filters for outdoor air ≥ F7 exhaust air ≥ G4; Frost protection to protect plate HE on exhaust side and post-heater if extraction fan is broken (e.g. air subsoil, brine loop, electric); Condensate drain in exhaust air, airtight and insulated. Windows*: Uwindow ≤ 0.80 W/m2K** and Uw,installed ≤ 0.85 W/m2K to keep radiant temperature asymmetry <4.2K (comfort criterion***), typically <3K with PH windows; Triple glazing Uglass ≤ 0.80 W/m2K, 0.60 W/m2K is typical; g-value = SHGC = 50-55% typical for PH windows * Values shown are for cool-temperate climate. (Transparent compo-

nent certification critera for other climate zones and efficiency classes phA+, phA, phB and phC can be found on www.passiv.de)

** For PH certificate: Uw = 0.80 W/m2K verified with Uglass = 0.70 W/m2K *** Other comfort criteria met by PH: Air speed < 0.08m/s; Room air

temperature stratification between head and ankles of seated person < 2K; Felt temperature difference in a room from place to place less than 0.8˚C

What happens if there is a problem (any problem) with the Passive House? • Passive House Standard not met • Level of comfort will decrease • Heating demand increases • Heating load increases • Supplementary heating might be required • No longer able to heat with supply air alone • Risk of mould increases • Exfiltration of internal air into structure leads to interstitial condensation (n50)

Miscellaneous see [PHPP] for symbols & definitions ϕ (phi), η (eta) = efficiency λ (lambda) = thermal conductivity ϑ (theta), T = temperature ΔT = temperature difference Ψ (psi) = linear thermal transmittance χ (chi) = point thermal transmittance

Acircle = (π × d2) / 4 = π × r2 ≈ 0.7854 × d2

Equilateral triangle = all sides and angles (60˚) equal 1 year = 365 days = 8,760 hours = 8.76 kh/a 1 hour = 3,600 seconds

Deviation from North [PHPP:81]

North: 0° Northeast: 45° East: 90° Southeast: 135° South: 180° Southwest: 225° West: 270° Northwest: 315°

THERMAL BRIDGES

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Evaluation Criteria for residential buildings [PHPP:19] Space Heating Demand QH ≤ 15 kWh/m2a or alternatively Peak Heating Load PH ≤ 10 W/m2 (small building, large surface) Useful Cooling Demand ≤ 15 kWh/m2a or alternatively Peak Cooling Load ≤ 10 W/m2 + see [PHPP:19] Primary Energy Demand PE ≤ 120 kWh/m2a + see [PHPP:19] (heating, ventilation, cooling, DHW, aux electricity, household electricity) Building Airtightness n50 ≤ 0.6h-1 (0.649 is still OK) EN 13829 Method A: Envelope in the same condition as it is when heating or ventilation are being used. Difference between test results at positive and negative pressure <10%. Design temperature 20˚C (Can be different in justified cases.) [PHPP:32] Excess temperature ≤ 10% of total yearly hours with indoor air >25˚C Occupancy rate for Verification of residential buildings = 35 m2/person (PHPPv9: Based on typical occupancy rates for specific dwelling unit sizes); for Planning = 20-50 m2/p; entered manually for non-residential [PHPP:33]

New Evaluation Criteria 2015 with PHPPv9 (not yet mandatory in 2015) All of the above, except alternative methodology for Primary Energy PE and introduction of new classes to address renewable energy generation. Primary Energy Renewable PER is the demand profile of the individual energy application and locally available renewable primary power production from PV and wind (and hydro power).

PER = Edir + EMS

ηMS + ESS

ηSS + EDL

Edir + EMS + ESS

PER Factor for each source and application [kWhPER/kWh] Edir Electricity generated by RES used directly EMS Electricity from short/medium term storage ESS Electricity generated from energy in seasonal storage EDL Distribution and other losses ηMS and ηSS Efficiencies of storage processes (whole chain) New Passive House Classes: Classic Plus Premium Renewable PE Demand: ≤60 ≤45 ≤30 kWhPER/m2

TFAa Renewable Energy Generation: n/a ≥60 ≥120 kWhPER/m2

grounda

Cross Ventilation: Supply air zone Transferred air zone Extract air zone Coanda-Effect: Vacuum created by moving air from supply jet nozzle 5-20cm under ceiling pulls secondary air towards the ceiling. The ceiling acts as a half-sided duct and can move supply air up to 6m into room. Ductwork – insulation thickness ETA SUP SUP ODA (~20˚C) (16-20˚C) (heated) / EHA Outside thermal envelope: 100mm 100mm 150mm 0 Transfer/exhaust room inside: 0 0 20-30mm 50-100mm Supplied room inside: 0 0 0 50-100mm Insulation on cold ducts inside must be vapour impermeable. Use silencers. Keep all ducts as short as possible – in particular cold ducts inside. Treated Floor Area (TFA) The rules are based on German WoflV and DIN 277 and incentivise designing efficient plans with high-quality spaces within the thermal envelope. Rules for residential buildings (WoflV) [PHPP:63]: Included: Floor areas of rooms measured from clear width between building elements, in particular: living and circulation areas, washrooms, auxiliary rooms (storage, service, and utility rooms), stair heads and landings, floor to ceiling window reveals which are ≥0.13m deep. Included with 60%: Auxiliary rooms and circulation areas outside dwelling or on floors of detached houses in which less than 50% of the floor area is considered living space (e.g. in the basement). Excluded: Stairs with more than 3 risers, walls and other elements >1.5m high, shafts and chimneys and pillars >0.1m2, doorways, window reveals (unless see above), areas outside thermal envelope. Rule for all areas with reduced ceiling height: The TFA is reduced by 50% if clear room height is 1-2m. Areas <1m high are excluded from TFA.

The necessity of an airtight building envelope: (Infiltration and exfiltration are caused by wind and buoyancy, due to leakages in envelope. Design one airtight layer all around the building.) • Prevention of condensation in the construction (exfiltration most critical:

360g water/day can condense through 1mm x 1m leak, when outside 0˚C, 80%RH and inside 0˚C, 50%RH)

• Prevention of drafts • Prevention of cold floors in the ground floor • Preventing air pollution of the room air • Securing the sound insulation of building components • Securing the operation and effectiveness of the ventilation system • Securing the insulation effect of the external building components • Reduction of ventilation heat losses (infiltration) • For HRV to work efficiently, airtightness is important

Passive House Components (Quality Criteria) Heat protection: U ≤ 0.15 W/m2K thermal envelope, opaque elements (typically 0.10-0.15 W/m2K); thermal bridge free (to reduce heat loss and avoid cold interior surfaces) Heat Recovery Ventilation (HRV, MVHR): ηHR ≥ 75% efficiency (to maintain min. 16.5˚C supply air temp. at -10˚C, prefer 85-92%); Low velocity; Electricity demand max. 0.45 Wh/m3; Max. 25dB(A) in habitable rooms, 30dB(A) in functional rooms, 35dB(A) in room with ventilation unit; Balanced (≤ 10% during operation between ODA and EHA) and Controlled operation (basic / normal / purge: 54 / 77 / 100%, summer bypass); Filters for outdoor air ≥ F7 exhaust air ≥ G4; Frost protection to protect plate HE on exhaust side and post-heater if extraction fan is broken (e.g. air subsoil, brine loop, electric); Condensate drain in exhaust air, airtight and insulated. Windows*: Uwindow ≤ 0.80 W/m2K** and Uw,installed ≤ 0.85 W/m2K to keep radiant temperature asymmetry <4.2K (comfort criterion***), typically <3K with PH windows; Triple glazing Uglass ≤ 0.80 W/m2K, 0.60 W/m2K is typical; g-value = SHGC = 50-55% typical for PH windows * Values shown are for cool-temperate climate. (Transparent compo-

nent certification critera for other climate zones and efficiency classes phA+, phA, phB and phC can be found on www.passiv.de)

** For PH certificate: Uw = 0.80 W/m2K verified with Uglass = 0.70 W/m2K *** Other comfort criteria met by PH: Air speed < 0.08m/s; Room air

temperature stratification between head and ankles of seated person < 2K; Felt temperature difference in a room from place to place less than 0.8˚C

What happens if there is a problem (any problem) with the Passive House? • Passive House Standard not met • Level of comfort will decrease • Heating demand increases • Heating load increases • Supplementary heating might be required • No longer able to heat with supply air alone • Risk of mould increases • Exfiltration of internal air into structure leads to interstitial condensation (n50)

Miscellaneous see [PHPP] for symbols & definitions ϕ (phi), η (eta) = efficiency λ (lambda) = thermal conductivity ϑ (theta), T = temperature ΔT = temperature difference Ψ (psi) = linear thermal transmittance χ (chi) = point thermal transmittance

Acircle = (π × d2) / 4 = π × r2 ≈ 0.7854 × d2

Equilateral triangle = all sides and angles (60˚) equal 1 year = 365 days = 8,760 hours = 8.76 kh/a 1 hour = 3,600 seconds

Deviation from North [PHPP:81]

North: 0° Northeast: 45° East: 90° Southeast: 135° South: 180° Southwest: 225° West: 270° Northwest: 315°

THERMAL BRIDGES

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Material Thermal

conductivity λ W / mK

Umetric �

Wm2K

� = 5.678

Rimp. �ft2F hBtu �

U-value RSI U-value R-value

W / m2K

m2K / W

Btu / ft2 F h

ft2 F h / Btu

typi

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0.055 18.18 0.010 103.24 0.060 16.67 0.011 94.64 0.065 15.38 0.011 87.36 0.070 14.29 0.012 81.12 0.075 13.33 0.013 75.71 0.080 12.50 0.014 70.98 0.085 11.76 0.015 66.80 0.090 11.11 0.016 63.09 0.095 10.53 0.017 59.77 0.100 10.00 0.018 56.78 0.105 9.52 0.018 54.08 0.110 9.09 0.019 51.62 0.115 8.70 0.020 49.38 0.120 8.33 0.021 47.32 0.125 8.00 0.022 45.43 0.130 7.69 0.023 43.68 0.135 7.41 0.024 42.06 0.140 7.14 0.025 40.56 0.145 6.90 0.026 39.16 0.150 6.67 0.026 37.86

0.20 5.00 0.035 28.39 0.25 4.00 0.044 22.71 0.30 3.33 0.053 18.93 0.35 2.86 0.062 16.22 0.40 2.50 0.070 14.20 0.45 2.22 0.079 12.62 0.50 2.00 0.088 11.36 0.55 1.82 0.097 10.32

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0.60 1.67 0.106 9.46 0.61 1.64 0.107 9.31 0.62 1.61 0.109 9.16 0.63 1.59 0.111 9.01 0.64 1.56 0.113 8.87 0.65 1.54 0.114 8.74 0.66 1.52 0.116 8.60 0.67 1.49 0.118 8.48 0.68 1.47 0.120 8.35 0.69 1.45 0.122 8.23 0.70 1.43 0.123 8.11 0.71 1.41 0.125 8.00 0.72 1.39 0.127 7.89 0.73 1.37 0.129 7.78 0.74 1.35 0.130 7.67 0.75 1.33 0.132 7.57 0.76 1.32 0.134 7.47 0.77 1.30 0.136 7.37 0.78 1.28 0.137 7.28 0.79 1.27 0.139 7.19 0.80 1.25 0.141 7.10 0.81 1.23 0.143 7.01 0.82 1.22 0.144 6.92 0.83 1.20 0.146 6.84 0.84 1.19 0.148 6.76 0.85 1.18 0.150 6.68 0.86 1.16 0.151 6.60 0.87 1.15 0.153 6.53 0.88 1.14 0.155 6.45 0.89 1.12 0.157 6.38 0.90 1.11 0.158 6.31

0.95 1.05 0.167 5.98 1.00 1.00 0.176 5.68 1.05 0.95 0.185 5.41 1.10 0.91 0.194 5.16 1.15 0.87 0.203 4.94 1.20 0.83 0.211 4.73 1.25 0.80 0.220 4.54 1.30 0.77 0.229 4.37 1.35 0.74 0.238 4.21 1.40 0.71 0.247 4.06 1.45 0.69 0.255 3.92 1.50 0.67 0.264 3.79 Find converter tool app for mobile

devices on www.15kwh10w.com

Copper 3802 Aluminium 1601,2 - 2002 Mild Steel 402 - 501 - 802

λmetric �W

mK� =

0.1442

Rper inch �ft2F h

Btu inch�

Stainless Steel 171 Concrete (Reinforced) 1.42 - 2.11 - 2.62 Cement Screed 1.41 Lightweight Concrete 0.151 - 0.31 Quinn Lite aerated concrete 0.1213 - 0.1913 Annual Energy Demand

Natural Stone 1.51 - 3.51 kWh / m2a

kBtu / ft2a

kWh / ft2a

Sand-Lime Masonry 11 Solid Clay Brick Masonry 0.81 - 1.21 1 0.317 0.093

Vertically Perforated Lightweight Masonry 0.31 - 0.451 15* 4.755 1.394 Adobe 0.41 - 0.82 25* 7.925 2.323 Float Glass 11 30** 9.510 2.787 Solid Plastic (Typical) 0.171 - 0.31 45** 14.26 4.181 Rubber 0.171 60** 19.02 5.574 Linoleum 0.171 120** 38.04 11.15 Carpet 0.061 * Heating /cooling criteria for PH and

EnerPHit ** Primary Energy criteria

Gypsum Plaster 0.181 - 0.561 Gypsum Plasterboard 0.251 For wood and wood products the thermal conductivity is to be multiplied by a factor of 2.2 when the heat flow is parallel to the direction of the fibres.1

Heating Load W /

m2 Btu / h.ft2

Hardwood 0.181

Softwood 0.131 1 0.317 Chipboard 0.101 - 0.181 10* 3.171 Oriented Strand Board (OSB) 0.092 - 0.131 * Heating load criterion for PH Plywood 0.082 - 0.112 Medium Density Fibreboard (MDF) 0.071 - 0.181 North American Softwood

Dimensional Lumber sizes

Wood Wool Lightweight Building Board 0.0651 - 0.0901 Fibre Insulating Material 0.0351 - 0.0501 nominal actual actual Wooden Softboard 0.0401 - 0.0701 1" ¾" 19 mm Agepan DWD Protect 0.0908 2" 1-½" 38 mm Agepan THD Insulating Wood Fibre Board 0.0468 - 0.0508 3" 2-½" 64 mm Agepan THD Static 0.0558 4" 3-½" 89 mm

Corkboard 0.0422 5" 4-½" 114 mm Coconut Fibre 0.0404 - 0.0504 6" 5-½" 140 mm Flax / hemp board 0.0406 7" 6-¼" 159 mm Mineral wool (rock wool, fibreglass batts) 0.0351 - 0.0451 8" 7-¼" 184 mm Roxul ComfortBoard CIS 0.0369 10" 9-¼" 235 mm

Fibreglass (blown fibres) 0.0383 - 0.0393 12" 11-¼" 286 mm Expanded perlite (EPB) 0.0454 - 0.0704 Source: http://en.wikipedia.org/wiki/lumber Sheep wool 0.0356 - 0.0452 1" = 25.4 mm 1' = 12" = 0.3048 m Cellulose (blown fibres) 0.0392,3 - 0.0506 Strawbale 0.0602 - 0.0752 1 Passive House Planning Package PHPP; Version 8;

Darmstadt, 2013 [PHPP:46] 2 Building Science for Building Enclosures; Straube,

Burnett; Building Science Press; 2005 3 ASHRAE Handbook; Parsons; 2005 4 Dämmstoffe: Grundlagen, Materialien,

Anwendungen; DETAIL; Munich; 2007 5 Building Enclosure Design Guide; HPO; 2011 6 www.wecobis.de; July 2015 7 www.u-wert.de/daemmstoffe; July 2015 8 Agepan System Brochure; July 2015 9 www.roxul.com; July 2015 10 www.geocell-schaumglas.eu; July 2015 11 www.jackon-insulation.com; July 2015 12 www.building-int.foamglas.com; July 2015 13 www.quinn-lite.com; July 2015 14 www.kingspaninsulation.de; July 2015 15 Schaumglasschotter als Wärmedämmung;

Fraunhofer-IBP Additional data is available in EN 12524 and national standards. For modelling and project certification use rated design values declared on technical data sheets.

Cellular Glass 0.0451 - 0.0601 Foamglas Perinsul loadbearing 0.05012

Foam glass gravel (dry) 0.08015 - 0.09515 Foam glass gravel (design value) 0.11015 - 0.14015 Geocell Foam Glass Gravel (design value) 0.11010

Expanded Rigid Polystyrene Foam (EPS) 0.0351 - 0.0401

Extruded Rigid Polystyrene Foam (XPS) 0.0301 - 0.0401 Jackodur Atlas (load bearing XPS) 0.03511 - 0.03811

Rigid Polyurethane Foam boards (PUR) 0.0236 - 0.0401 Low density open-cell spray foam (PUR) 0.0405 - 0.0385 High density closed-cell spray foam (PUR) 0.0285 - 0.0245 Rigid Polyisocyanurate 0.0202 - 0.0242 Rigid Phenolic foam (closed cell) 0.0172,3 - 0.0202 Kingspan Kooltherm phenolic foam board 0.02114 - 0.02214

Aerogel 0.0174 - 0.0214 Vacuum Insulated Panel (VIP) 0.0024 - 0.0084 Air space depending on thickness and heat flow [PHPP:49]

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Material Thermal

conductivity λ W / mK

Umetric �

Wm2K

� = 5.678

Rimp. �ft2F hBtu �

U-value RSI U-value R-value

W / m2K

m2K / W

Btu / ft2 F h

ft2 F h / Btu

typi

cal v

alue

s for

opa

que

elem

ents

0.055 18.18 0.010 103.24 0.060 16.67 0.011 94.64 0.065 15.38 0.011 87.36 0.070 14.29 0.012 81.12 0.075 13.33 0.013 75.71 0.080 12.50 0.014 70.98 0.085 11.76 0.015 66.80 0.090 11.11 0.016 63.09 0.095 10.53 0.017 59.77 0.100 10.00 0.018 56.78 0.105 9.52 0.018 54.08 0.110 9.09 0.019 51.62 0.115 8.70 0.020 49.38 0.120 8.33 0.021 47.32 0.125 8.00 0.022 45.43 0.130 7.69 0.023 43.68 0.135 7.41 0.024 42.06 0.140 7.14 0.025 40.56 0.145 6.90 0.026 39.16 0.150 6.67 0.026 37.86

0.20 5.00 0.035 28.39 0.25 4.00 0.044 22.71 0.30 3.33 0.053 18.93 0.35 2.86 0.062 16.22 0.40 2.50 0.070 14.20 0.45 2.22 0.079 12.62 0.50 2.00 0.088 11.36 0.55 1.82 0.097 10.32

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0.60 1.67 0.106 9.46 0.61 1.64 0.107 9.31 0.62 1.61 0.109 9.16 0.63 1.59 0.111 9.01 0.64 1.56 0.113 8.87 0.65 1.54 0.114 8.74 0.66 1.52 0.116 8.60 0.67 1.49 0.118 8.48 0.68 1.47 0.120 8.35 0.69 1.45 0.122 8.23 0.70 1.43 0.123 8.11 0.71 1.41 0.125 8.00 0.72 1.39 0.127 7.89 0.73 1.37 0.129 7.78 0.74 1.35 0.130 7.67 0.75 1.33 0.132 7.57 0.76 1.32 0.134 7.47 0.77 1.30 0.136 7.37 0.78 1.28 0.137 7.28 0.79 1.27 0.139 7.19 0.80 1.25 0.141 7.10 0.81 1.23 0.143 7.01 0.82 1.22 0.144 6.92 0.83 1.20 0.146 6.84 0.84 1.19 0.148 6.76 0.85 1.18 0.150 6.68 0.86 1.16 0.151 6.60 0.87 1.15 0.153 6.53 0.88 1.14 0.155 6.45 0.89 1.12 0.157 6.38 0.90 1.11 0.158 6.31

0.95 1.05 0.167 5.98 1.00 1.00 0.176 5.68 1.05 0.95 0.185 5.41 1.10 0.91 0.194 5.16 1.15 0.87 0.203 4.94 1.20 0.83 0.211 4.73 1.25 0.80 0.220 4.54 1.30 0.77 0.229 4.37 1.35 0.74 0.238 4.21 1.40 0.71 0.247 4.06 1.45 0.69 0.255 3.92 1.50 0.67 0.264 3.79 Find converter tool app for mobile

devices on www.15kwh10w.com

Copper 3802 Aluminium 1601,2 - 2002 Mild Steel 402 - 501 - 802

λmetric �W

mK� =

0.1442

Rper inch �ft2F h

Btu inch�

Stainless Steel 171 Concrete (Reinforced) 1.42 - 2.11 - 2.62 Cement Screed 1.41 Lightweight Concrete 0.151 - 0.31 Quinn Lite aerated concrete 0.1213 - 0.1913 Annual Energy Demand

Natural Stone 1.51 - 3.51 kWh / m2a

kBtu / ft2a

kWh / ft2a

Sand-Lime Masonry 11 Solid Clay Brick Masonry 0.81 - 1.21 1 0.317 0.093

Vertically Perforated Lightweight Masonry 0.31 - 0.451 15* 4.755 1.394 Adobe 0.41 - 0.82 25* 7.925 2.323 Float Glass 11 30** 9.510 2.787 Solid Plastic (Typical) 0.171 - 0.31 45** 14.26 4.181 Rubber 0.171 60** 19.02 5.574 Linoleum 0.171 120** 38.04 11.15 Carpet 0.061 * Heating /cooling criteria for PH and

EnerPHit ** Primary Energy criteria

Gypsum Plaster 0.181 - 0.561 Gypsum Plasterboard 0.251 For wood and wood products the thermal conductivity is to be multiplied by a factor of 2.2 when the heat flow is parallel to the direction of the fibres.1

Heating Load W /

m2 Btu / h.ft2

Hardwood 0.181

Softwood 0.131 1 0.317 Chipboard 0.101 - 0.181 10* 3.171 Oriented Strand Board (OSB) 0.092 - 0.131 * Heating load criterion for PH Plywood 0.082 - 0.112 Medium Density Fibreboard (MDF) 0.071 - 0.181 North American Softwood

Dimensional Lumber sizes

Wood Wool Lightweight Building Board 0.0651 - 0.0901 Fibre Insulating Material 0.0351 - 0.0501 nominal actual actual Wooden Softboard 0.0401 - 0.0701 1" ¾" 19 mm Agepan DWD Protect 0.0908 2" 1-½" 38 mm Agepan THD Insulating Wood Fibre Board 0.0468 - 0.0508 3" 2-½" 64 mm Agepan THD Static 0.0558 4" 3-½" 89 mm

Corkboard 0.0422 5" 4-½" 114 mm Coconut Fibre 0.0404 - 0.0504 6" 5-½" 140 mm Flax / hemp board 0.0406 7" 6-¼" 159 mm Mineral wool (rock wool, fibreglass batts) 0.0351 - 0.0451 8" 7-¼" 184 mm Roxul ComfortBoard CIS 0.0369 10" 9-¼" 235 mm

Fibreglass (blown fibres) 0.0383 - 0.0393 12" 11-¼" 286 mm Expanded perlite (EPB) 0.0454 - 0.0704 Source: http://en.wikipedia.org/wiki/lumber Sheep wool 0.0356 - 0.0452 1" = 25.4 mm 1' = 12" = 0.3048 m Cellulose (blown fibres) 0.0392,3 - 0.0506 Strawbale 0.0602 - 0.0752 1 Passive House Planning Package PHPP; Version 8;

Darmstadt, 2013 [PHPP:46] 2 Building Science for Building Enclosures; Straube,

Burnett; Building Science Press; 2005 3 ASHRAE Handbook; Parsons; 2005 4 Dämmstoffe: Grundlagen, Materialien,

Anwendungen; DETAIL; Munich; 2007 5 Building Enclosure Design Guide; HPO; 2011 6 www.wecobis.de; July 2015 7 www.u-wert.de/daemmstoffe; July 2015 8 Agepan System Brochure; July 2015 9 www.roxul.com; July 2015 10 www.geocell-schaumglas.eu; July 2015 11 www.jackon-insulation.com; July 2015 12 www.building-int.foamglas.com; July 2015 13 www.quinn-lite.com; July 2015 14 www.kingspaninsulation.de; July 2015 15 Schaumglasschotter als Wärmedämmung;

Fraunhofer-IBP Additional data is available in EN 12524 and national standards. For modelling and project certification use rated design values declared on technical data sheets.

Cellular Glass 0.0451 - 0.0601 Foamglas Perinsul loadbearing 0.05012

Foam glass gravel (dry) 0.08015 - 0.09515 Foam glass gravel (design value) 0.11015 - 0.14015 Geocell Foam Glass Gravel (design value) 0.11010

Expanded Rigid Polystyrene Foam (EPS) 0.0351 - 0.0401

Extruded Rigid Polystyrene Foam (XPS) 0.0301 - 0.0401 Jackodur Atlas (load bearing XPS) 0.03511 - 0.03811

Rigid Polyurethane Foam boards (PUR) 0.0236 - 0.0401 Low density open-cell spray foam (PUR) 0.0405 - 0.0385 High density closed-cell spray foam (PUR) 0.0285 - 0.0245 Rigid Polyisocyanurate 0.0202 - 0.0242 Rigid Phenolic foam (closed cell) 0.0172,3 - 0.0202 Kingspan Kooltherm phenolic foam board 0.02114 - 0.02214

Aerogel 0.0174 - 0.0214 Vacuum Insulated Panel (VIP) 0.0024 - 0.0084 Air space depending on thickness and heat flow [PHPP:49]

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Which final value Kn does a current capital K0 have at a future date t?

Which present value K0 does one future capital Kn have?

Which present value K0 does a constant payment A have?

How high is the annuity A, that is to be paid from a present value K0?

Kn = K0 × (1 + p)t K0 = Kn × (1 + p)-t K0 = A × 1 - (1 + p)-n

p A = K0 ×

p1 - (1 + p)-n

Accumulation factor Discount factor: equals the reciprocal value of the accumulation factor = 1 / (1 + p)t

Present value factor B: equals the accumulated discount

factors of the considered time period

Annuity factor a = 1/B: reciprocal value of the

present value factor

t = time index = interval from point of reference (t0 = starting date) n = useful life = number of periods = repayment period = length of mortgage p = interest rate [decimal] (use real interest rate for investment considerations) initial repayment rate = a - p a = annuity factor A = annuity = stream of payments (or income) with a fixed amount Ki B = present value factor Kn = capital at a given time tn = future value / final value annuity A = present value K0 / present value factor B = K0 × annuity factor a K0 = capital at a given time t0 = net present value NPV of annuity = current value of a stream of payments discounted by the interest rate

Net Present Value of an Annuity K0 = A × 1 - (1 + p)-n

p

Capital to invest today for 3 years, at an interest rate of 3.5%, to be able to withdraw 500$ at the end of each year?

K0 = 500$ × 1 - (1+0.035)-3

0.035 = 500$ × 2.802 = 1,401 $

What additional mortgage could be supported by annual savings of 1,500$ on heating cost, at an interest rate of 3% borrowed for 25 years?

K0 = 1,500$ × 1 - (1+0.03)-25

0.03 = 1,500$ × 17.413 = 26,119 $

Annuity Calculation A = K0 × p

1 - (1 + p)-n

Which amount can be taken at the end of each year for the next 4 years, from an initial capital of 3,000$ at an interest rate of 3.5%?

A = 3,000$ × 0.035

1 - (1+0.035)-4 = 3,000$ × 0.272 = 816.75 $

A client borrows 250,000$ for construction, at an interest rate of p = 4.5% for a repayment period n = 30a.

How high is monthly annuity (interest and repayment)? What is the initial repayment rate?

A = 250,000$ × 0.045

1 - (1+0.045)-30 = 250,000$ × 0.0614 = 15,347.89 $/a

monthly charge = 15,347$ / 12months = 1,278.99 $/month initial repayment rate = annuity factor a – interest rate p = 0.0614 – 0.045 = 1.64%

Nominal and real interest rates: pnominal = nominal interest rate (e.g. 7.5%) i = inflation rate (e.g. 4%) preal = real interest rate (inflation adjusted)

preal = 1 + pnominal

1 + i - 1 At low inflation and interest rates the result is approximately:

preal = pnominal - i

preal = (1+0.075) (1+0.04) ⁄ - 1 = 0.034 = 3.4% preal = 0.075 - 0.04 = 0.035 = 3.5%

Profitability of energy saving measures calculated with Pactual = 0.055 $/kWh: Psaved = 0.0142 $/kWh:

Annual cost without energy saving measures:

Aexist = P × Eexist Aexist = 0.055$/kWh × [250m2 × 1.03W/m2K × 1 × 84kKh/a / 0.90] Aexist = 0.055$/kWh × 24,033kWh/a = 1,321 $/a

Aexist = 0.0142 × 24,033 = 341 $/a

Annual cost with saving measure:

Anew = P × Enew + aloan × (Iadd - R) + Z Anew = 0.055$/kWh × [250m2 x 0.150W/m2K × 1 × 84kKh/a / 0.90] + 0.0672 × ($7,500$ - 3,170.79$) + 0$

Anew = 0.055$/kWh × 3,500 kWh/a + 291$ + 0$ = 483 $/a

Anew = 0.0142 × 3,500 + 291 + 0

= 341 $/a annual energy cost annuity of new with saving measure investment

Profitability if: Anew < Aexist 483 $ < 1,321 $ measure pays off measure just pays off

Equivalent price of saved energy: aloan(20years,3%) = 0.0672 Iadd = 250m2 × 1.50$/cm/m2 × 20cm = 7,500 $ R = (1- a50years,3% x B20years,3%) × Iadd = (1- 0.0388 × 14.877) × 7,500$ = 3,170.79 $ Esaved = 250m2 × (1.03-0.15)W/m2K × 1.0 × 84kKa/a / 0.90 = 250m2 × 82.13kWh/a = 20,533 kWh/a

Psaved = 0.0672 × (7,500$ - 3,170.79$)+ 0$

20,533kWh/a= 0.0142 $/kWh

Psaved = aloan × (Iadd - Rcomponent) + Z

Esaved

Investment worthwhile if:

aloan × (Iadd - R) + Z ≤ (P × E)saved 0.0672 × (7,500$ - 3,170.79$) + 0$ ≤ 0.055$/kWh × 20,533kWh/a 291 $ ≤ 1,129 $ worthwhile a = annuity factor B = present value factor Iadd = additional cost of investment for saving measures R = residual value of component R = (1 - alife expectancy × Binvestment) × Iadd

(A building component with an expected lifetime of 50 years has a residual value of 39% after 20 years at preal = 3.5%.)

Z = possible additional cost for operational and maintenance cost resulting from the saving measure (e.g. for mechanical systems, not applicable for insulation)

P = price per energy unit Psaved = equivalent price for saved energy

Example above: new 20cm EIFS (1.50$/m2/cm) on existing 250m2 wall, resulting in improved U-value from 1.03W/m2K to 0.150W/m2K lifetime of EIFS L = 50a ƞheating = 90% time period under consideration n = 20a real interest rate i = 3%

Enew = annual energy consumption after taking energy saving measure Eexist = annual energy consumption without taking measure Esaved = Eexist - Enew = annual energy savings after taking measure Esaved = Acomponent × q = Acomponent × Usaved × ft × Gt / ƞ Usaved = (Uexist – Unew) ƞ = marginal annual efficiency of heating system

In a Passive House the investment becomes more important – and base prices for energy supply systems are significant. But energy costs become almost insignificant because of the low consumption.

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SEE NEXT PAGES

Which final value Kn does a current capital K0 have at a future date t?

Which present value K0 does one future capital Kn have?

Which present value K0 does a constant payment A have?

How high is the annuity A, that is to be paid from a present value K0?

Kn = K0 × (1 + p)t K0 = Kn × (1 + p)-t K0 = A × 1 - (1 + p)-n

p A = K0 ×

p1 - (1 + p)-n

Accumulation factor Discount factor: equals the reciprocal value of the accumulation factor = 1 / (1 + p)t

Present value factor B: equals the accumulated discount

factors of the considered time period

Annuity factor a = 1/B: reciprocal value of the

present value factor

t = time index = interval from point of reference (t0 = starting date) n = useful life = number of periods = repayment period = length of mortgage p = interest rate [decimal] (use real interest rate for investment considerations) initial repayment rate = a - p a = annuity factor A = annuity = stream of payments (or income) with a fixed amount Ki B = present value factor Kn = capital at a given time tn = future value / final value annuity A = present value K0 / present value factor B = K0 × annuity factor a K0 = capital at a given time t0 = net present value NPV of annuity = current value of a stream of payments discounted by the interest rate

Net Present Value of an Annuity K0 = A × 1 - (1 + p)-n

p

Capital to invest today for 3 years, at an interest rate of 3.5%, to be able to withdraw 500$ at the end of each year?

K0 = 500$ × 1 - (1+0.035)-3

0.035 = 500$ × 2.802 = 1,401 $

What additional mortgage could be supported by annual savings of 1,500$ on heating cost, at an interest rate of 3% borrowed for 25 years?

K0 = 1,500$ × 1 - (1+0.03)-25

0.03 = 1,500$ × 17.413 = 26,119 $

Annuity Calculation A = K0 × p

1 - (1 + p)-n

Which amount can be taken at the end of each year for the next 4 years, from an initial capital of 3,000$ at an interest rate of 3.5%?

A = 3,000$ × 0.035

1 - (1+0.035)-4 = 3,000$ × 0.272 = 816.75 $

A client borrows 250,000$ for construction, at an interest rate of p = 4.5% for a repayment period n = 30a.

How high is monthly annuity (interest and repayment)? What is the initial repayment rate?

A = 250,000$ × 0.045

1 - (1+0.045)-30 = 250,000$ × 0.0614 = 15,347.89 $/a

monthly charge = 15,347$ / 12months = 1,278.99 $/month initial repayment rate = annuity factor a – interest rate p = 0.0614 – 0.045 = 1.64%

Nominal and real interest rates: pnominal = nominal interest rate (e.g. 7.5%) i = inflation rate (e.g. 4%) preal = real interest rate (inflation adjusted)

preal = 1 + pnominal

1 + i - 1 At low inflation and interest rates the result is approximately:

preal = pnominal - i

preal = (1+0.075) (1+0.04) ⁄ - 1 = 0.034 = 3.4% preal = 0.075 - 0.04 = 0.035 = 3.5%

Profitability of energy saving measures calculated with Pactual = 0.055 $/kWh: Psaved = 0.0142 $/kWh:

Annual cost without energy saving measures:

Aexist = P × Eexist Aexist = 0.055$/kWh × [250m2 × 1.03W/m2K × 1 × 84kKh/a / 0.90] Aexist = 0.055$/kWh × 24,033kWh/a = 1,321 $/a

Aexist = 0.0142 × 24,033 = 341 $/a

Annual cost with saving measure:

Anew = P × Enew + aloan × (Iadd - R) + Z Anew = 0.055$/kWh × [250m2 x 0.150W/m2K × 1 × 84kKh/a / 0.90] + 0.0672 × ($7,500$ - 3,170.79$) + 0$

Anew = 0.055$/kWh × 3,500 kWh/a + 291$ + 0$ = 483 $/a

Anew = 0.0142 × 3,500 + 291 + 0

= 341 $/a annual energy cost annuity of new with saving measure investment

Profitability if: Anew < Aexist 483 $ < 1,321 $ measure pays off measure just pays off

Equivalent price of saved energy: aloan(20years,3%) = 0.0672 Iadd = 250m2 × 1.50$/cm/m2 × 20cm = 7,500 $ R = (1- a50years,3% x B20years,3%) × Iadd = (1- 0.0388 × 14.877) × 7,500$ = 3,170.79 $ Esaved = 250m2 × (1.03-0.15)W/m2K × 1.0 × 84kKa/a / 0.90 = 250m2 × 82.13kWh/a = 20,533 kWh/a

Psaved = 0.0672 × (7,500$ - 3,170.79$)+ 0$

20,533kWh/a= 0.0142 $/kWh

Psaved = aloan × (Iadd - Rcomponent) + Z

Esaved

Investment worthwhile if:

aloan × (Iadd - R) + Z ≤ (P × E)saved 0.0672 × (7,500$ - 3,170.79$) + 0$ ≤ 0.055$/kWh × 20,533kWh/a 291 $ ≤ 1,129 $ worthwhile a = annuity factor B = present value factor Iadd = additional cost of investment for saving measures R = residual value of component R = (1 - alife expectancy × Binvestment) × Iadd

(A building component with an expected lifetime of 50 years has a residual value of 39% after 20 years at preal = 3.5%.)

Z = possible additional cost for operational and maintenance cost resulting from the saving measure (e.g. for mechanical systems, not applicable for insulation)

P = price per energy unit Psaved = equivalent price for saved energy

Example above: new 20cm EIFS (1.50$/m2/cm) on existing 250m2 wall, resulting in improved U-value from 1.03W/m2K to 0.150W/m2K lifetime of EIFS L = 50a ƞheating = 90% time period under consideration n = 20a real interest rate i = 3%

Enew = annual energy consumption after taking energy saving measure Eexist = annual energy consumption without taking measure Esaved = Eexist - Enew = annual energy savings after taking measure Esaved = Acomponent × q = Acomponent × Usaved × ft × Gt / ƞ Usaved = (Uexist – Unew) ƞ = marginal annual efficiency of heating system

In a Passive House the investment becomes more important – and base prices for energy supply systems are significant. But energy costs become almost insignificant because of the low consumption.

SEE NEXT PAGES

andrehar

Text Box


andrehar

Text Box

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Present Value Factor B = 1 - (1 + p)-n

p

n↓

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

←p

10.0

%

0.90

91

1.73

55

2.48

69

3.16

99

3.79

08

4.35

53

4.86

84

5.33

49

5.75

90

6.14

46

6.49

51

6.81

37

7.10

34

7.36

67

7.60

61

7.82

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8.02

16

8.20

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8.36

49

8.51

36

8.64

87

8.77

15

8.88

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8.98

47

9.07

70

9.16

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9.23

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9.30

66

9.36

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9.42

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9.47

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9.52

64

9.56

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9.60

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9.64

42

9.67

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9.70

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9.73

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9.75

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9.77

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9.79

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9.81

74

9.83

40

9.84

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9.86

28

9.87

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9.88

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9.89

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9.90

63

9.91

48

10.0

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9132

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9346

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5.

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9 13

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5 13

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9390

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2.

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4.

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6.

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8 14

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6.

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5%

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1.

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0.

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1.

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6.

0021

6.

7327

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9 22

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6 23

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5%

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0.

9709

1.

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4.

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5.

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7.

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8 21

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8 23

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9 24

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7 24

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7 25

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7 25

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7 25

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0%

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0.

9756

1.

9274

2.

8560

3.

7620

4.

6458

5.

5081

6.

3494

7.

1701

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10

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9 15

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9 22

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3 25

.102

8 25

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1 25

.820

6 26

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4 26

.503

8 26

.833

0 27

.154

2 27

.467

5 27

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2 28

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4 28

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5%

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0.

9804

1.

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2.

8839

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8077

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4720

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8.

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10

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0 18

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9 19

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.121

0 20

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9 21

.281

3 21

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4 22

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5 22

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7 23

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6 24

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6 24

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6 25

.488

8 25

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5 26

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6 26

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6 27

.355

5 27

.799

5 28

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8 28

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6 29

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0 29

.490

2 29

.892

3 30

.286

6 30

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1 31

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1 31

.423

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0.

9852

1.

9559

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9122

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8544

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7826

5.

6972

6.

5982

7.

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10

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6 16

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.719

6 21

.398

6 22

.067

6 22

.726

7 23

.376

1 24

.015

8 24

.646

1 25

.267

1 25

.879

0 26

.481

7 27

.075

6 27

.660

7 28

.237

1 28

.805

1 29

.364

6 29

.915

8 30

.459

0 30

.994

1 31

.521

2 32

.040

6 32

.552

3 33

.056

5 33

.553

2 34

.042

6 34

.524

7 34

.999

7 1.

5%

1.0%

0.

9901

1.

9704

2.

9410

3.

9020

4.

8534

5.

7955

6.

7282

7.

6517

8.

5660

9.

4713

10

.367

6 11

.255

1 12

.133

7 13

.003

7 13

.865

1 14

.717

9 15

.562

3 16

.398

3 17

.226

0 18

.045

6 18

.857

0 19

.660

4 20

.455

8 21

.243

4 22

.023

2 22

.795

2 23

.559

6 24

.316

4 25

.065

8 25

.807

7 26

.542

3 27

.269

6 27

.989

7 28

.702

7 29

.408

6 30

.107

5 30

.799

5 31

.484

7 32

.163

0 32

.834

7 33

.499

7 34

.158

1 34

.810

0 35

.455

5 36

.094

5 36

.727

2 37

.353

7 37

.974

0 38

.588

1 39

.196

1 1.

0%

p→

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

↑n

andrehar

Text Box

available at:

andrehar

Text Box

15kwh10w.com


www.15kwh10w.com

Present Value Factor B = 1 - (1 + p)-n

p

n↓

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

←p

10.0

%

0.90

91

1.73

55

2.48

69

3.16

99

3.79

08

4.35

53

4.86

84

5.33

49

5.75

90

6.14

46

6.49

51

6.81

37

7.10

34

7.36

67

7.60

61

7.82

37

8.02

16

8.20

14

8.36

49

8.51

36

8.64

87

8.77

15

8.88

32

8.98

47

9.07

70

9.16

09

9.23

72

9.30

66

9.36

96

9.42

69

9.47

90

9.52

64

9.56

94

9.60

86

9.64

42

9.67

65

9.70

59

9.73

27

9.75

70

9.77

91

9.79

91

9.81

74

9.83

40

9.84

91

9.86

28

9.87

53

9.88

66

9.89

69

9.90

63

9.91

48

10.0

%

9.5%

0.

9132

1.

7473

2.

5089

3.

2045

3.

8397

4.

4198

4.

9496

5.

4334

5.

8753

6.

2788

6.

6473

6.

9838

7.

2912

7.

5719

7.

8282

8.

0623

8.

2760

8.

4713

8.

6496

8.

8124

8.

9611

9.

0969

9.

2209

9.

3341

9.

4376

9.

5320

9.

6183

9.

6971

9.

7690

9.

8347

9.

8947

9.

9495

9.

9996

10

.045

3 10

.087

0 10

.125

1 10

.159

9 10

.191

7 10

.220

7 10

.247

2 10

.271

5 10

.293

6 10

.313

8 10

.332

2 10

.349

0 10

.364

4 10

.378

5 10

.391

3 10

.403

0 10

.413

7 9.

5%

9.0%

0.

9174

1.

7591

2.

5313

3.

2397

3.

8897

4.

4859

5.

0330

5.

5348

5.

9952

6.

4177

6.

8052

7.

1607

7.

4869

7.

7862

8.

0607

8.

3126

8.

5436

8.

7556

8.

9501

9.

1285

9.

2922

9.

4424

9.

5802

9.

7066

9.

8226

9.

9290

10

.026

6 10

.116

1 10

.198

3 10

.273

7 10

.342

8 10

.406

2 10

.464

4 10

.517

8 10

.566

8 10

.611

8 10

.653

0 10

.690

8 10

.725

5 10

.757

4 10

.786

6 10

.813

4 10

.838

0 10

.860

5 10

.881

2 10

.900

2 10

.917

6 10

.933

6 10

.948

2 10

.961

7 9.

0%

8.5%

0.

9217

1.

7711

2.

5540

3.

2756

3.

9406

4.

5536

5.

1185

5.

6392

6.

1191

6.

5613

6.

9690

7.

3447

7.

6910

8.

0101

8.

3042

8.

5753

8.

8252

9.

0555

9.

2677

9.

4633

9.

6436

9.

8098

9.

9629

10

.104

1 10

.234

2 10

.354

1 10

.464

6 10

.566

5 10

.660

3 10

.746

8 10

.826

6 10

.900

1 10

.967

8 11

.030

2 11

.087

8 11

.140

8 11

.189

7 11

.234

7 11

.276

3 11

.314

5 11

.349

8 11

.382

3 11

.412

3 11

.439

9 11

.465

3 11

.488

8 11

.510

4 11

.530

3 11

.548

7 11

.565

6 8.

5%

8.0%

0.

9259

1.

7833

2.

5771

3.

3121

3.

9927

4.

6229

5.

2064

5.

7466

6.

2469

6.

7101

7.

1390

7.

5361

7.

9038

8.

2442

8.

5595

8.

8514

9.

1216

9.

3719

9.

6036

9.

8181

10

.016

8 10

.200

7 10

.371

1 10

.528

8 10

.674

8 10

.810

0 10

.935

2 11

.051

1 11

.158

4 11

.257

8 11

.349

8 11

.435

0 11

.513

9 11

.586

9 11

.654

6 11

.717

2 11

.775

2 11

.828

9 11

.878

6 11

.924

6 11

.967

2 12

.006

7 12

.043

2 12

.077

1 12

.108

4 12

.137

4 12

.164

3 12

.189

1 12

.212

2 12

.233

5 8.

0%

7.5%

0.

9302

1.

7956

2.

6005

3.

3493

4.

0459

4.

6938

5.

2966

5.

8573

6.

3789

6.

8641

7.

3154

7.

7353

8.

1258

8.

4892

8.

8271

9.

1415

9.

4340

9.

7060

9.

9591

10

.194

5 10

.413

5 10

.617

2 10

.806

7 10

.983

0 11

.146

9 11

.299

5 11

.441

4 11

.573

4 11

.696

2 11

.810

4 11

.916

6 12

.015

5 12

.107

4 12

.192

9 12

.272

5 12

.346

5 12

.415

4 12

.479

4 12

.539

0 12

.594

4 12

.646

0 12

.693

9 12

.738

5 12

.780

0 12

.818

6 12

.854

5 12

.887

9 12

.919

0 12

.947

9 12

.974

8 7.

5%

7.0%

0.

9346

1.

8080

2.

6243

3.

3872

4.

1002

4.

7665

5.

3893

5.

9713

6.

5152

7.

0236

7.

4987

7.

9427

8.

3577

8.

7455

9.

1079

9.

4466

9.

7632

10

.059

1 10

.335

6 10

.594

0 10

.835

5 11

.061

2 11

.272

2 11

.469

3 11

.653

6 11

.825

8 11

.986

7 12

.137

1 12

.277

7 12

.409

0 12

.531

8 12

.646

6 12

.753

8 12

.854

0 12

.947

7 13

.035

2 13

.117

0 13

.193

5 13

.264

9 13

.331

7 13

.394

1 13

.452

4 13

.507

0 13

.557

9 13

.605

5 13

.650

0 13

.691

6 13

.730

5 13

.766

8 13

.800

7 7.

0%

6.5%

0.

9390

1.

8206

2.

6485

3.

4258

4.

1557

4.

8410

5.

4845

6.

0888

6.

6561

7.

1888

7.

6890

8.

1587

8.

5997

9.

0138

9.

4027

9.

7678

10

.110

6 10

.432

5 10

.734

7 11

.018

5 11

.285

0 11

.535

2 11

.770

1 11

.990

7 12

.197

9 12

.392

4 12

.575

0 12

.746

5 12

.907

5 13

.058

7 13

.200

6 13

.333

9 13

.459

1 13

.576

6 13

.687

0 13

.790

6 13

.887

9 13

.979

2 14

.065

0 14

.145

5 14

.221

2 14

.292

2 14

.358

8 14

.421

4 14

.480

2 14

.535

4 14

.587

3 14

.635

9 14

.681

6 14

.724

5 6.

5%

6.0%

0.

9434

1.

8334

2.

6730

3.

4651

4.

2124

4.

9173

5.

5824

6.

2098

6.

8017

7.

3601

7.

8869

8.

3838

8.

8527

9.

2950

9.

7122

10

.105

9 10

.477

3 10

.827

6 11

.158

1 11

.469

9 11

.764

1 12

.041

6 12

.303

4 12

.550

4 12

.783

4 13

.003

2 13

.210

5 13

.406

2 13

.590

7 13

.764

8 13

.929

1 14

.084

0 14

.230

2 14

.368

1 14

.498

2 14

.621

0 14

.736

8 14

.846

0 14

.949

1 15

.046

3 15

.138

0 15

.224

5 15

.306

2 15

.383

2 15

.455

8 15

.524

4 15

.589

0 15

.650

0 15

.707

6 15

.761

9 6.

0%

5.5%

0.

9479

1.

8463

2.

6979

3.

5052

4.

2703

4.

9955

5.

6830

6.

3346

6.

9522

7.

5376

8.

0925

8.

6185

9.

1171

9.

5896

10

.037

6 10

.462

2 10

.864

6 11

.246

1 11

.607

7 11

.950

4 12

.275

2 12

.583

2 12

.875

0 13

.151

7 13

.413

9 13

.662

5 13

.898

1 14

.121

4 14

.333

1 14

.533

7 14

.723

9 14

.904

2 15

.075

1 15

.237

0 15

.390

6 15

.536

1 15

.674

0 15

.804

7 15

.928

7 16

.046

1 16

.157

5 16

.263

0 16

.363

0 16

.457

9 16

.547

7 16

.632

9 16

.713

7 16

.790

2 16

.862

8 16

.931

5 5.

5%

5.0%

0.

9524

1.

8594

2.

7232

3.

5460

4.

3295

5.

0757

5.

7864

6.

4632

7.

1078

7.

7217

8.

3064

8.

8633

9.

3936

9.

8986

10

.379

7 10

.837

8 11

.274

1 11

.689

6 12

.085

3 12

.462

2 12

.821

2 13

.163

0 13

.488

6 13

.798

6 14

.093

9 14

.375

2 14

.643

0 14

.898

1 15

.141

1 15

.372

5 15

.592

8 15

.802

7 16

.002

5 16

.192

9 16

.374

2 16

.546

9 16

.711

3 16

.867

9 17

.017

0 17

.159

1 17

.294

4 17

.423

2 17

.545

9 17

.662

8 17

.774

1 17

.880

1 17

.981

0 18

.077

2 18

.168

7 18

.255

9 5.

0%

4.5%

0.

9569

1.

8727

2.

7490

3.

5875

4.

3900

5.

1579

5.

8927

6.

5959

7.

2688

7.

9127

8.

5289

9.

1186

9.

6829

10

.222

8 10

.739

5 11

.234

0 11

.707

2 12

.160

0 12

.593

3 13

.007

9 13

.404

7 13

.784

4 14

.147

8 14

.495

5 14

.828

2 15

.146

6 15

.451

3 15

.742

9 16

.021

9 16

.288

9 16

.544

4 16

.788

9 17

.022

9 17

.246

8 17

.461

0 17

.666

0 17

.862

2 18

.050

0 18

.229

7 18

.401

6 18

.566

1 18

.723

5 18

.874

2 19

.018

4 19

.156

3 19

.288

4 19

.414

7 19

.535

6 19

.651

3 19

.762

0 4.

5%

4.0%

0.

9615

1.

8861

2.

7751

3.

6299

4.

4518

5.

2421

6.

0021

6.

7327

7.

4353

8.

1109

8.

7605

9.

3851

9.

9856

10

.563

1 11

.118

4 11

.652

3 12

.165

7 12

.659

3 13

.133

9 13

.590

3 14

.029

2 14

.451

1 14

.856

8 15

.247

0 15

.622

1 15

.982

8 16

.329

6 16

.663

1 16

.983

7 17

.292

0 17

.588

5 17

.873

6 18

.147

6 18

.411

2 18

.664

6 18

.908

3 19

.142

6 19

.367

9 19

.584

5 19

.792

8 19

.993

1 20

.185

6 20

.370

8 20

.548

8 20

.720

0 20

.884

7 21

.042

9 21

.195

1 21

.341

5 21

.482

2 4.

0%

3.5%

0.

9662

1.

8997

2.

8016

3.

6731

4.

5151

5.

3286

6.

1145

6.

8740

7.

6077

8.

3166

9.

0016

9.

6633

10

.302

7 10

.920

5 11

.517

4 12

.094

1 12

.651

3 13

.189

7 13

.709

8 14

.212

4 14

.698

0 15

.167

1 15

.620

4 16

.058

4 16

.481

5 16

.890

4 17

.285

4 17

.667

0 18

.035

8 18

.392

0 18

.736

3 19

.068

9 19

.390

2 19

.700

7 20

.000

7 20

.290

5 20

.570

5 20

.841

1 21

.102

5 21

.355

1 21

.599

1 21

.834

9 22

.062

7 22

.282

8 22

.495

5 22

.700

9 22

.899

4 23

.091

2 23

.276

6 23

.455

6 3.

5%

3.0%

0.

9709

1.

9135

2.

8286

3.

7171

4.

5797

5.

4172

6.

2303

7.

0197

7.

7861

8.

5302

9.

2526

9.

9540

10

.635

0 11

.296

1 11

.937

9 12

.561

1 13

.166

1 13

.753

5 14

.323

8 14

.877

5 15

.415

0 15

.936

9 16

.443

6 16

.935

5 17

.413

1 17

.876

8 18

.327

0 18

.764

1 19

.188

5 19

.600

4 20

.000

4 20

.388

8 20

.765

8 21

.131

8 21

.487

2 21

.832

3 22

.167

2 22

.492

5 22

.808

2 23

.114

8 23

.412

4 23

.701

4 23

.981

9 24

.254

3 24

.518

7 24

.775

4 25

.024

7 25

.266

7 25

.501

7 25

.729

8 3.

0%

2.5%

0.

9756

1.

9274

2.

8560

3.

7620

4.

6458

5.

5081

6.

3494

7.

1701

7.

9709

8.

7521

9.

5142

10

.257

8 10

.983

2 11

.690

9 12

.381

4 13

.055

0 13

.712

2 14

.353

4 14

.978

9 15

.589

2 16

.184

5 16

.765

4 17

.332

1 17

.885

0 18

.424

4 18

.950

6 19

.464

0 19

.964

9 20

.453

5 20

.930

3 21

.395

4 21

.849

2 22

.291

9 22

.723

8 23

.145

2 23

.556

3 23

.957

3 24

.348

6 24

.730

3 25

.102

8 25

.466

1 25

.820

6 26

.166

4 26

.503

8 26

.833

0 27

.154

2 27

.467

5 27

.773

2 28

.071

4 28

.362

3 2.

5%

2.0%

0.

9804

1.

9416

2.

8839

3.

8077

4.

7135

5.

6014

6.

4720

7.

3255

8.

1622

8.

9826

9.

7868

10

.575

3 11

.348

4 12

.106

2 12

.849

3 13

.577

7 14

.291

9 14

.992

0 15

.678

5 16

.351

4 17

.011

2 17

.658

0 18

.292

2 18

.913

9 19

.523

5 20

.121

0 20

.706

9 21

.281

3 21

.844

4 22

.396

5 22

.937

7 23

.468

3 23

.988

6 24

.498

6 24

.998

6 25

.488

8 25

.969

5 26

.440

6 26

.902

6 27

.355

5 27

.799

5 28

.234

8 28

.661

6 29

.080

0 29

.490

2 29

.892

3 30

.286

6 30

.673

1 31

.052

1 31

.423

6 2.

0%

1.5%

0.

9852

1.

9559

2.

9122

3.

8544

4.

7826

5.

6972

6.

5982

7.

4859

8.

3605

9.

2222

10

.071

1 10

.907

5 11

.731

5 12

.543

4 13

.343

2 14

.131

3 14

.907

6 15

.672

6 16

.426

2 17

.168

6 17

.900

1 18

.620

8 19

.330

9 20

.030

4 20

.719

6 21

.398

6 22

.067

6 22

.726

7 23

.376

1 24

.015

8 24

.646

1 25

.267

1 25

.879

0 26

.481

7 27

.075

6 27

.660

7 28

.237

1 28

.805

1 29

.364

6 29

.915

8 30

.459

0 30

.994

1 31

.521

2 32

.040

6 32

.552

3 33

.056

5 33

.553

2 34

.042

6 34

.524

7 34

.999

7 1.

5%

1.0%

0.

9901

1.

9704

2.

9410

3.

9020

4.

8534

5.

7955

6.

7282

7.

6517

8.

5660

9.

4713

10

.367

6 11

.255

1 12

.133

7 13

.003

7 13

.865

1 14

.717

9 15

.562

3 16

.398

3 17

.226

0 18

.045

6 18

.857

0 19

.660

4 20

.455

8 21

.243

4 22

.023

2 22

.795

2 23

.559

6 24

.316

4 25

.065

8 25

.807

7 26

.542

3 27

.269

6 27

.989

7 28

.702

7 29

.408

6 30

.107

5 30

.799

5 31

.484

7 32

.163

0 32

.834

7 33

.499

7 34

.158

1 34

.810

0 35

.455

5 36

.094

5 36

.727

2 37

.353

7 37

.974

0 38

.588

1 39

.196

1 1.

0%

p→

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

↑n

andrehar

Text Box


andrehar

Text Box

15kwh10w.com


www.15kwh10w.com

Annuity Factor a = p

1 - (1 + p)-n

n↓

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

←p

10.0

%

1.10

00

0.57

62

0.40

21

0.31

55

0.26

38

0.22

96

0.20

54

0.18

74

0.17

36

0.16

27

0.15

40

0.14

68

0.14

08

0.13

57

0.13

15

0.12

78

0.12

47

0.12

19

0.11

95

0.11

75

0.11

56

0.11

40

0.11

26

0.11

13

0.11

02

0.10

92

0.10

83

0.10

75

0.10

67

0.10

61

0.10

55

0.10

50

0.10

45

0.10

41

0.10

37

0.10

33

0.10

30

0.10

27

0.10

25

0.10

23

0.10

20

0.10

19

0.10

17

0.10

15

0.10

14

0.10

13

0.10

11

0.10

10

0.10

09

0.10

09

10.0

%

9.5%

1.

0950

0.

5723

0.

3986

0.

3121

0.

2604

0.

2263

0.

2020

0.

1840

0.

1702

0.

1593

0.

1504

0.

1432

0.

1372

0.

1321

0.

1277

0.

1240

0.

1208

0.

1180

0.

1156

0.

1135

0.

1116

0.

1099

0.

1084

0.

1071

0.

1060

0.

1049

0.

1040

0.

1031

0.

1024

0.

1017

0.

1011

0.

1005

0.

1000

0.

0995

0.

0991

0.

0988

0.

0984

0.

0981

0.

0978

0.

0976

0.

0974

0.

0971

0.

0970

0.

0968

0.

0966

0.

0965

0.

0964

0.

0962

0.

0961

0.

0960

9.

5%

9.0%

1.

0900

0.

5685

0.

3951

0.

3087

0.

2571

0.

2229

0.

1987

0.

1807

0.

1668

0.

1558

0.

1469

0.

1397

0.

1336

0.

1284

0.

1241

0.

1203

0.

1170

0.

1142

0.

1117

0.

1095

0.

1076

0.

1059

0.

1044

0.

1030

0.

1018

0.

1007

0.

0997

0.

0989

0.

0981

0.

0973

0.

0967

0.

0961

0.

0956

0.

0951

0.

0946

0.

0942

0.

0939

0.

0935

0.

0932

0.

0930

0.

0927

0.

0925

0.

0923

0.

0921

0.

0919

0.

0917

0.

0916

0.

0915

0.

0913

0.

0912

9.

0%

8.5%

1.

0850

0.

5646

0.

3915

0.

3053

0.

2538

0.

2196

0.

1954

0.

1773

0.

1634

0.

1524

0.

1435

0.

1362

0.

1300

0.

1248

0.

1204

0.

1166

0.

1133

0.

1104

0.

1079

0.

1057

0.

1037

0.

1019

0.

1004

0.

0990

0.

0977

0.

0966

0.

0956

0.

0946

0.

0938

0.

0931

0.

0924

0.

0917

0.

0912

0.

0907

0.

0902

0.

0898

0.

0894

0.

0890

0.

0887

0.

0884

0.

0881

0.

0879

0.

0876

0.

0874

0.

0872

0.

0870

0.

0869

0.

0867

0.

0866

0.

0865

8.

5%

8.0%

1.

0800

0.

5608

0.

3880

0.

3019

0.

2505

0.

2163

0.

1921

0.

1740

0.

1601

0.

1490

0.

1401

0.

1327

0.

1265

0.

1213

0.

1168

0.

1130

0.

1096

0.

1067

0.

1041

0.

1019

0.

0998

0.

0980

0.

0964

0.

0950

0.

0937

0.

0925

0.

0914

0.

0905

0.

0896

0.

0888

0.

0881

0.

0875

0.

0869

0.

0863

0.

0858

0.

0853

0.

0849

0.

0845

0.

0842

0.

0839

0.

0836

0.

0833

0.

0830

0.

0828

0.

0826

0.

0824

0.

0822

0.

0820

0.

0819

0.

0817

8.

0%

7.5%

1.

0750

0.

5569

0.

3845

0.

2986

0.

2472

0.

2130

0.

1888

0.

1707

0.

1568

0.

1457

0.

1367

0.

1293

0.

1231

0.

1178

0.

1133

0.

1094

0.

1060

0.

1030

0.

1004

0.

0981

0.

0960

0.

0942

0.

0925

0.

0911

0.

0897

0.

0885

0.

0874

0.

0864

0.

0855

0.

0847

0.

0839

0.

0832

0.

0826

0.

0820

0.

0815

0.

0810

0.

0805

0.

0801

0.

0798

0.

0794

0.

0791

0.

0788

0.

0785

0.

0782

0.

0780

0.

0778

0.

0776

0.

0774

0.

0772

0.

0771

7.

5%

7.0%

1.

0700

0.

5531

0.

3811

0.

2952

0.

2439

0.

2098

0.

1856

0.

1675

0.

1535

0.

1424

0.

1334

0.

1259

0.

1197

0.

1143

0.

1098

0.

1059

0.

1024

0.

0994

0.

0968

0.

0944

0.

0923

0.

0904

0.

0887

0.

0872

0.

0858

0.

0846

0.

0834

0.

0824

0.

0814

0.

0806

0.

0798

0.

0791

0.

0784

0.

0778

0.

0772

0.

0767

0.

0762

0.

0758

0.

0754

0.

0750

0.

0747

0.

0743

0.

0740

0.

0738

0.

0735

0.

0733

0.

0730

0.

0728

0.

0726

0.

0725

7.

0%

6.5%

1.

0650

0.

5493

0.

3776

0.

2919

0.

2406

0.

2066

0.

1823

0.

1642

0.

1502

0.

1391

0.

1301

0.

1226

0.

1163

0.

1109

0.

1064

0.

1024

0.

0989

0.

0959

0.

0932

0.

0908

0.

0886

0.

0867

0.

0850

0.

0834

0.

0820

0.

0807

0.

0795

0.

0785

0.

0775

0.

0766

0.

0758

0.

0750

0.

0743

0.

0737

0.

0731

0.

0725

0.

0720

0.

0715

0.

0711

0.

0707

0.

0703

0.

0700

0.

0696

0.

0693

0.

0691

0.

0688

0.

0686

0.

0683

0.

0681

0.

0679

6.

5%

6.0%

1.

0600

0.

5454

0.

3741

0.

2886

0.

2374

0.

2034

0.

1791

0.

1610

0.

1470

0.

1359

0.

1268

0.

1193

0.

1130

0.

1076

0.

1030

0.

0990

0.

0954

0.

0924

0.

0896

0.

0872

0.

0850

0.

0830

0.

0813

0.

0797

0.

0782

0.

0769

0.

0757

0.

0746

0.

0736

0.

0726

0.

0718

0.

0710

0.

0703

0.

0696

0.

0690

0.

0684

0.

0679

0.

0674

0.

0669

0.

0665

0.

0661

0.

0657

0.

0653

0.

0650

0.

0647

0.

0644

0.

0641

0.

0639

0.

0637

0.

0634

6.

0%

5.5%

1.

0550

0.

5416

0.

3707

0.

2853

0.

2342

0.

2002

0.

1760

0.

1579

0.

1438

0.

1327

0.

1236

0.

1160

0.

1097

0.

1043

0.

0996

0.

0956

0.

0920

0.

0889

0.

0862

0.

0837

0.

0815

0.

0795

0.

0777

0.

0760

0.

0745

0.

0732

0.

0720

0.

0708

0.

0698

0.

0688

0.

0679

0.

0671

0.

0663

0.

0656

0.

0650

0.

0644

0.

0638

0.

0633

0.

0628

0.

0623

0.

0619

0.

0615

0.

0611

0.

0608

0.

0604

0.

0601

0.

0598

0.

0596

0.

0593

0.

0591

5.

5%

5.0%

1.

0500

0.

5378

0.

3672

0.

2820

0.

2310

0.

1970

0.

1728

0.

1547

0.

1407

0.

1295

0.

1204

0.

1128

0.

1065

0.

1010

0.

0963

0.

0923

0.

0887

0.

0855

0.

0827

0.

0802

0.

0780

0.

0760

0.

0741

0.

0725

0.

0710

0.

0696

0.

0683

0.

0671

0.

0660

0.

0651

0.

0641

0.

0633

0.

0625

0.

0618

0.

0611

0.

0604

0.

0598

0.

0593

0.

0588

0.

0583

0.

0578

0.

0574

0.

0570

0.

0566

0.

0563

0.

0559

0.

0556

0.

0553

0.

0550

0.

0548

5.

0%

4.5%

1.

0450

0.

5340

0.

3638

0.

2787

0.

2278

0.

1939

0.

1697

0.

1516

0.

1376

0.

1264

0.

1172

0.

1097

0.

1033

0.

0978

0.

0931

0.

0890

0.

0854

0.

0822

0.

0794

0.

0769

0.

0746

0.

0725

0.

0707

0.

0690

0.

0674

0.

0660

0.

0647

0.

0635

0.

0624

0.

0614

0.

0604

0.

0596

0.

0587

0.

0580

0.

0573

0.

0566

0.

0560

0.

0554

0.

0549

0.

0543

0.

0539

0.

0534

0.

0530

0.

0526

0.

0522

0.

0518

0.

0515

0.

0512

0.

0509

0.

0506

4.

5%

4.0%

1.

0400

0.

5302

0.

3603

0.

2755

0.

2246

0.

1908

0.

1666

0.

1485

0.

1345

0.

1233

0.

1141

0.

1066

0.

1001

0.

0947

0.

0899

0.

0858

0.

0822

0.

0790

0.

0761

0.

0736

0.

0713

0.

0692

0.

0673

0.

0656

0.

0640

0.

0626

0.

0612

0.

0600

0.

0589

0.

0578

0.

0569

0.

0559

0.

0551

0.

0543

0.

0536

0.

0529

0.

0522

0.

0516

0.

0511

0.

0505

0.

0500

0.

0495

0.

0491

0.

0487

0.

0483

0.

0479

0.

0475

0.

0472

0.

0469

0.

0466

4.

0%

3.5%

1.

0350

0.

5264

0.

3569

0.

2723

0.

2215

0.

1877

0.

1635

0.

1455

0.

1314

0.

1202

0.

1111

0.

1035

0.

0971

0.

0916

0.

0868

0.

0827

0.

0790

0.

0758

0.

0729

0.

0704

0.

0680

0.

0659

0.

0640

0.

0623

0.

0607

0.

0592

0.

0579

0.

0566

0.

0554

0.

0544

0.

0534

0.

0524

0.

0516

0.

0508

0.

0500

0.

0493

0.

0486

0.

0480

0.

0474

0.

0468

0.

0463

0.

0458

0.

0453

0.

0449

0.

0445

0.

0441

0.

0437

0.

0433

0.

0430

0.

0426

3.

5%

3.0%

1.

0300

0.

5226

0.

3535

0.

2690

0.

2184

0.

1846

0.

1605

0.

1425

0.

1284

0.

1172

0.

1081

0.

1005

0.

0940

0.

0885

0.

0838

0.

0796

0.

0760

0.

0727

0.

0698

0.

0672

0.

0649

0.

0627

0.

0608

0.

0590

0.

0574

0.

0559

0.

0546

0.

0533

0.

0521

0.

0510

0.

0500

0.

0490

0.

0482

0.

0473

0.

0465

0.

0458

0.

0451

0.

0445

0.

0438

0.

0433

0.

0427

0.

0422

0.

0417

0.

0412

0.

0408

0.

0404

0.

0400

0.

0396

0.

0392

0.

0389

3.

0%

2.5%

1.

0250

0.

5188

0.

3501

0.

2658

0.

2152

0.

1815

0.

1575

0.

1395

0.

1255

0.

1143

0.

1051

0.

0975

0.

0910

0.

0855

0.

0808

0.

0766

0.

0729

0.

0697

0.

0668

0.

0641

0.

0618

0.

0596

0.

0577

0.

0559

0.

0543

0.

0528

0.

0514

0.

0501

0.

0489

0.

0478

0.

0467

0.

0458

0.

0449

0.

0440

0.

0432

0.

0425

0.

0417

0.

0411

0.

0404

0.

0398

0.

0393

0.

0387

0.

0382

0.

0377

0.

0373

0.

0368

0.

0364

0.

0360

0.

0356

0.

0353

2.

5%

2.0%

1.

0200

0.

5150

0.

3468

0.

2626

0.

2122

0.

1785

0.

1545

0.

1365

0.

1225

0.

1113

0.

1022

0.

0946

0.

0881

0.

0826

0.

0778

0.

0737

0.

0700

0.

0667

0.

0638

0.

0612

0.

0588

0.

0566

0.

0547

0.

0529

0.

0512

0.

0497

0.

0483

0.

0470

0.

0458

0.

0446

0.

0436

0.

0426

0.

0417

0.

0408

0.

0400

0.

0392

0.

0385

0.

0378

0.

0372

0.

0366

0.

0360

0.

0354

0.

0349

0.

0344

0.

0339

0.

0335

0.

0330

0.

0326

0.

0322

0.

0318

2.

0%

1.5%

1.

0150

0.

5113

0.

3434

0.

2594

0.

2091

0.

1755

0.

1516

0.

1336

0.

1196

0.

1084

0.

0993

0.

0917

0.

0852

0.

0797

0.

0749

0.

0708

0.

0671

0.

0638

0.

0609

0.

0582

0.

0559

0.

0537

0.

0517

0.

0499

0.

0483

0.

0467

0.

0453

0.

0440

0.

0428

0.

0416

0.

0406

0.

0396

0.

0386

0.

0378

0.

0369

0.

0362

0.

0354

0.

0347

0.

0341

0.

0334

0.

0328

0.

0323

0.

0317

0.

0312

0.

0307

0.

0303

0.

0298

0.

0294

0.

0290

0.

0286

1.

5%

1.0%

1.

0100

0.

5075

0.

3400

0.

2563

0.

2060

0.

1725

0.

1486

0.

1307

0.

1167

0.

1056

0.

0965

0.

0888

0.

0824

0.

0769

0.

0721

0.

0679

0.

0643

0.

0610

0.

0581

0.

0554

0.

0530

0.

0509

0.

0489

0.

0471

0.

0454

0.

0439

0.

0424

0.

0411

0.

0399

0.

0387

0.

0377

0.

0367

0.

0357

0.

0348

0.

0340

0.

0332

0.

0325

0.

0318

0.

0311

0.

0305

0.

0299

0.

0293

0.

0287

0.

0282

0.

0277

0.

0272

0.

0268

0.

0263

0.

0259

0.

0255

1.

0%

p→

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

↑n

andrehar

Text Box

available at:

andrehar

Text Box

15kwh10w.com


www.15kwh10w.com

Annuity Factor a = p

1 - (1 + p)-n

n↓

1 2 3 4 5 6 7 8 9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

←p

10.0

%

1.10

00

0.57

62

0.40

21

0.31

55

0.26

38

0.22

96

0.20

54

0.18

74

0.17

36

0.16

27

0.15

40

0.14

68

0.14

08

0.13

57

0.13

15

0.12

78

0.12

47

0.12

19

0.11

95

0.11

75

0.11

56

0.11

40

0.11

26

0.11

13

0.11

02

0.10

92

0.10

83

0.10

75

0.10

67

0.10

61

0.10

55

0.10

50

0.10

45

0.10

41

0.10

37

0.10

33

0.10

30

0.10

27

0.10

25

0.10

23

0.10

20

0.10

19

0.10

17

0.10

15

0.10

14

0.10

13

0.10

11

0.10

10

0.10

09

0.10

09

10.0

%

9.5%

1.

0950

0.

5723

0.

3986

0.

3121

0.

2604

0.

2263

0.

2020

0.

1840

0.

1702

0.

1593

0.

1504

0.

1432

0.

1372

0.

1321

0.

1277

0.

1240

0.

1208

0.

1180

0.

1156

0.

1135

0.

1116

0.

1099

0.

1084

0.

1071

0.

1060

0.

1049

0.

1040

0.

1031

0.

1024

0.

1017

0.

1011

0.

1005

0.

1000

0.

0995

0.

0991

0.

0988

0.

0984

0.

0981

0.

0978

0.

0976

0.

0974

0.

0971

0.

0970

0.

0968

0.

0966

0.

0965

0.

0964

0.

0962

0.

0961

0.

0960

9.

5%

9.0%

1.

0900

0.

5685

0.

3951

0.

3087

0.

2571

0.

2229

0.

1987

0.

1807

0.

1668

0.

1558

0.

1469

0.

1397

0.

1336

0.

1284

0.

1241

0.

1203

0.

1170

0.

1142

0.

1117

0.

1095

0.

1076

0.

1059

0.

1044

0.

1030

0.

1018

0.

1007

0.

0997

0.

0989

0.

0981

0.

0973

0.

0967

0.

0961

0.

0956

0.

0951

0.

0946

0.

0942

0.

0939

0.

0935

0.

0932

0.

0930

0.

0927

0.

0925

0.

0923

0.

0921

0.

0919

0.

0917

0.

0916

0.

0915

0.

0913

0.

0912

9.

0%

8.5%

1.

0850

0.

5646

0.

3915

0.

3053

0.

2538

0.

2196

0.

1954

0.

1773

0.

1634

0.

1524

0.

1435

0.

1362

0.

1300

0.

1248

0.

1204

0.

1166

0.

1133

0.

1104

0.

1079

0.

1057

0.

1037

0.

1019

0.

1004

0.

0990

0.

0977

0.

0966

0.

0956

0.

0946

0.

0938

0.

0931

0.

0924

0.

0917

0.

0912

0.

0907

0.

0902

0.

0898

0.

0894

0.

0890

0.

0887

0.

0884

0.

0881

0.

0879

0.

0876

0.

0874

0.

0872

0.

0870

0.

0869

0.

0867

0.

0866

0.

0865

8.

5%

8.0%

1.

0800

0.

5608

0.

3880

0.

3019

0.

2505

0.

2163

0.

1921

0.

1740

0.

1601

0.

1490

0.

1401

0.

1327

0.

1265

0.

1213

0.

1168

0.

1130

0.

1096

0.

1067

0.

1041

0.

1019

0.

0998

0.

0980

0.

0964

0.

0950

0.

0937

0.

0925

0.

0914

0.

0905

0.

0896

0.

0888

0.

0881

0.

0875

0.

0869

0.

0863

0.

0858

0.

0853

0.

0849

0.

0845

0.

0842

0.

0839

0.

0836

0.

0833

0.

0830

0.

0828

0.

0826

0.

0824

0.

0822

0.

0820

0.

0819

0.

0817

8.

0%

7.5%

1.

0750

0.

5569

0.

3845

0.

2986

0.

2472

0.

2130

0.

1888

0.

1707

0.

1568

0.

1457

0.

1367

0.

1293

0.

1231

0.

1178

0.

1133

0.

1094

0.

1060

0.

1030

0.

1004

0.

0981

0.

0960

0.

0942

0.

0925

0.

0911

0.

0897

0.

0885

0.

0874

0.

0864

0.

0855

0.

0847

0.

0839

0.

0832

0.

0826

0.

0820

0.

0815

0.

0810

0.

0805

0.

0801

0.

0798

0.

0794

0.

0791

0.

0788

0.

0785

0.

0782

0.

0780

0.

0778

0.

0776

0.

0774

0.

0772

0.

0771

7.

5%

7.0%

1.

0700

0.

5531

0.

3811

0.

2952

0.

2439

0.

2098

0.

1856

0.

1675

0.

1535

0.

1424

0.

1334

0.

1259

0.

1197

0.

1143

0.

1098

0.

1059

0.

1024

0.

0994

0.

0968

0.

0944

0.

0923

0.

0904

0.

0887

0.

0872

0.

0858

0.

0846

0.

0834

0.

0824

0.

0814

0.

0806

0.

0798

0.

0791

0.

0784

0.

0778

0.

0772

0.

0767

0.

0762

0.

0758

0.

0754

0.

0750

0.

0747

0.

0743

0.

0740

0.

0738

0.

0735

0.

0733

0.

0730

0.

0728

0.

0726

0.

0725

7.

0%

6.5%

1.

0650

0.

5493

0.

3776

0.

2919

0.

2406

0.

2066

0.

1823

0.

1642

0.

1502

0.

1391

0.

1301

0.

1226

0.

1163

0.

1109

0.

1064

0.

1024

0.

0989

0.

0959

0.

0932

0.

0908

0.

0886

0.

0867

0.

0850

0.

0834

0.

0820

0.

0807

0.

0795

0.

0785

0.

0775

0.

0766

0.

0758

0.

0750

0.

0743

0.

0737

0.

0731

0.

0725

0.

0720

0.

0715

0.

0711

0.

0707

0.

0703

0.

0700

0.

0696

0.

0693

0.

0691

0.

0688

0.

0686

0.

0683

0.

0681

0.

0679

6.

5%

6.0%

1.

0600

0.

5454

0.

3741

0.

2886

0.

2374

0.

2034

0.

1791

0.

1610

0.

1470

0.

1359

0.

1268

0.

1193

0.

1130

0.

1076

0.

1030

0.

0990

0.

0954

0.

0924

0.

0896

0.

0872

0.

0850

0.

0830

0.

0813

0.

0797

0.

0782

0.

0769

0.

0757

0.

0746

0.

0736

0.

0726

0.

0718

0.

0710

0.

0703

0.

0696

0.

0690

0.

0684

0.

0679

0.

0674

0.

0669

0.

0665

0.

0661

0.

0657

0.

0653

0.

0650

0.

0647

0.

0644

0.

0641

0.

0639

0.

0637

0.

0634

6.

0%

5.5%

1.

0550

0.

5416

0.

3707

0.

2853

0.

2342

0.

2002

0.

1760

0.

1579

0.

1438

0.

1327

0.

1236

0.

1160

0.

1097

0.

1043

0.

0996

0.

0956

0.

0920

0.

0889

0.

0862

0.

0837

0.

0815

0.

0795

0.

0777

0.

0760

0.

0745

0.

0732

0.

0720

0.

0708

0.

0698

0.

0688

0.

0679

0.

0671

0.

0663

0.

0656

0.

0650

0.

0644

0.

0638

0.

0633

0.

0628

0.

0623

0.

0619

0.

0615

0.

0611

0.

0608

0.

0604

0.

0601

0.

0598

0.

0596

0.

0593

0.

0591

5.

5%

5.0%

1.

0500

0.

5378

0.

3672

0.

2820

0.

2310

0.

1970

0.

1728

0.

1547

0.

1407

0.

1295

0.

1204

0.

1128

0.

1065

0.

1010

0.

0963

0.

0923

0.

0887

0.

0855

0.

0827

0.

0802

0.

0780

0.

0760

0.

0741

0.

0725

0.

0710

0.

0696

0.

0683

0.

0671

0.

0660

0.

0651

0.

0641

0.

0633

0.

0625

0.

0618

0.

0611

0.

0604

0.

0598

0.

0593

0.

0588

0.

0583

0.

0578

0.

0574

0.

0570

0.

0566

0.

0563

0.

0559

0.

0556

0.

0553

0.

0550

0.

0548

5.

0%

4.5%

1.

0450

0.

5340

0.

3638

0.

2787

0.

2278

0.

1939

0.

1697

0.

1516

0.

1376

0.

1264

0.

1172

0.

1097

0.

1033

0.

0978

0.

0931

0.

0890

0.

0854

0.

0822

0.

0794

0.

0769

0.

0746

0.

0725

0.

0707

0.

0690

0.

0674

0.

0660

0.

0647

0.

0635

0.

0624

0.

0614

0.

0604

0.

0596

0.

0587

0.

0580

0.

0573

0.

0566

0.

0560

0.

0554

0.

0549

0.

0543

0.

0539

0.

0534

0.

0530

0.

0526

0.

0522

0.

0518

0.

0515

0.

0512

0.

0509

0.

0506

4.

5%

4.0%

1.

0400

0.

5302

0.

3603

0.

2755

0.

2246

0.

1908

0.

1666

0.

1485

0.

1345

0.

1233

0.

1141

0.

1066

0.

1001

0.

0947

0.

0899

0.

0858

0.

0822

0.

0790

0.

0761

0.

0736

0.

0713

0.

0692

0.

0673

0.

0656

0.

0640

0.

0626

0.

0612

0.

0600

0.

0589

0.

0578

0.

0569

0.

0559

0.

0551

0.

0543

0.

0536

0.

0529

0.

0522

0.

0516

0.

0511

0.

0505

0.

0500

0.

0495

0.

0491

0.

0487

0.

0483

0.

0479

0.

0475

0.

0472

0.

0469

0.

0466

4.

0%

3.5%

1.

0350

0.

5264

0.

3569

0.

2723

0.

2215

0.

1877

0.

1635

0.

1455

0.

1314

0.

1202

0.

1111

0.

1035

0.

0971

0.

0916

0.

0868

0.

0827

0.

0790

0.

0758

0.

0729

0.

0704

0.

0680

0.

0659

0.

0640

0.

0623

0.

0607

0.

0592

0.

0579

0.

0566

0.

0554

0.

0544

0.

0534

0.

0524

0.

0516

0.

0508

0.

0500

0.

0493

0.

0486

0.

0480

0.

0474

0.

0468

0.

0463

0.

0458

0.

0453

0.

0449

0.

0445

0.

0441

0.

0437

0.

0433

0.

0430

0.

0426

3.

5%

3.0%

1.

0300

0.

5226

0.

3535

0.

2690

0.

2184

0.

1846

0.

1605

0.

1425

0.

1284

0.

1172

0.

1081

0.

1005

0.

0940

0.

0885

0.

0838

0.

0796

0.

0760

0.

0727

0.

0698

0.

0672

0.

0649

0.

0627

0.

0608

0.

0590

0.

0574

0.

0559

0.

0546

0.

0533

0.

0521

0.

0510

0.

0500

0.

0490

0.

0482

0.

0473

0.

0465

0.

0458

0.

0451

0.

0445

0.

0438

0.

0433

0.

0427

0.

0422

0.

0417

0.

0412

0.

0408

0.

0404

0.

0400

0.

0396

0.

0392

0.

0389

3.

0%

2.5%

1.

0250

0.

5188

0.

3501

0.

2658

0.

2152

0.

1815

0.

1575

0.

1395

0.

1255

0.

1143

0.

1051

0.

0975

0.

0910

0.

0855

0.

0808

0.

0766

0.

0729

0.

0697

0.

0668

0.

0641

0.

0618

0.

0596

0.

0577

0.

0559

0.

0543

0.

0528

0.

0514

0.

0501

0.

0489

0.

0478

0.

0467

0.

0458

0.

0449

0.

0440

0.

0432

0.

0425

0.

0417

0.

0411

0.

0404

0.

0398

0.

0393

0.

0387

0.

0382

0.

0377

0.

0373

0.

0368

0.

0364

0.

0360

0.

0356

0.

0353

2.

5%

2.0%

1.

0200

0.

5150

0.

3468

0.

2626

0.

2122

0.

1785

0.

1545

0.

1365

0.

1225

0.

1113

0.

1022

0.

0946

0.

0881

0.

0826

0.

0778

0.

0737

0.

0700

0.

0667

0.

0638

0.

0612

0.

0588

0.

0566

0.

0547

0.

0529

0.

0512

0.

0497

0.

0483

0.

0470

0.

0458

0.

0446

0.

0436

0.

0426

0.

0417

0.

0408

0.

0400

0.

0392

0.

0385

0.

0378

0.

0372

0.

0366

0.

0360

0.

0354

0.

0349

0.

0344

0.

0339

0.

0335

0.

0330

0.

0326

0.

0322

0.

0318

2.

0%

1.5%

1.

0150

0.

5113

0.

3434

0.

2594

0.

2091

0.

1755

0.

1516

0.

1336

0.

1196

0.

1084

0.

0993

0.

0917

0.

0852

0.

0797

0.

0749

0.

0708

0.

0671

0.

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0.

0609

0.

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0.

0559

0.

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0.

0517

0.

0499

0.

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0.

0467

0.

0453

0.

0440

0.

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0.

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0.

0406

0.

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0.

0386

0.

0378

0.

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0.

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0.

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0.

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0.

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0.

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0.

0328

0.

0323

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0312

0.

0307

0.

0303

0.

0298

0.

0294

0.

0290

0.

0286

1.

5%

1.0%

1.

0100

0.

5075

0.

3400

0.

2563

0.

2060

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1725

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1486

0.

1307

0.

1167

0.

1056

0.

0965

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0888

0.

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0.

0769

0.

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0610

0.

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0.

0530

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0454

0.

0439

0.

0424

0.

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0.

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0.

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0.

0259

0.

0255

1.

0%

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