analytical model for predicting whole building

7/24/2019 Analytical Model for Predicting Whole Building

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Abstract—A new analytical model is developed which providesclose-formed solutions for both transient indoor and envelopetemperature changes in buildings. Time-dependent boundarytemperature is presented as Fourier series which can approximate realweather conditions. The final close-formed solutions are simple,concise, and comprehensive. The model was compared withnumerical results and good accuracy was obtained. The model can

be used as design and control guidelines in engineering applicationsfor analysing mechanical heat transfer properties for buildings.

Keywords—Analytical model, heat transfer, whole building.

I. I NTRODUCTION

ODELLING whole building's heat transfer is important forunderstanding and predicting building thermal behaviours

in order to provide techniques for designing and analysing problems such as energy demands, passive design,environmental comfort and the response of control in a

building. Over the decades, there have been numerous building heat transfer models for such purposes [1]. Mostwhole building heat transfer programs are numerical models.This is due to the fact that even for building's multiple layeredenvelope alone, the heat transfer problem already remains too

big and complex to analytically solve. However, analyticalmodels are very useful in design, analysis and optimisation ofcomplex interactions of the physical processes in buildings.Furthermore, an important final step to promote the practicaluse of any model, including numerical models, is to condenseit for engineering applications. And this can only beaccomplished by identifying the fundamental physical

processes by means of analytical methods.One of the difficulties, as described previously, with the

analytical models for whole building's heat transfer is relatingheat conduction through building envelope which is oftenconstructed with multiple layers. In general, heat conductionin a multiple layered composite slab does not accept simpleand closed-formed solutions. Mathematically, four classes ofanalytical techniques are often used to solve for a heatconduction problem: finite integral transform which is oftenadopted for single layer material, Green's functions,orthogonal expansions and Laplace's transform [2,3].Concerning the techniques of Green's function and orthogonal

Xiaoshu Lu is with Department of Civil and Structural Engineering, Schoolof Engineering, Aalto University, PO Box 12100, FIN-02150, Espoo, Finlandand Finnish Institute of Occupational Health, Finland (corresponding author;e-mail: xiaoshu@ cc.hut.fi).

Martti Viljanen is with Department of Civil and Structural Engineering,School of Engineering, Aalto University, PO Box 12100, FIN-02150, Espoo,

Finland (e-mail: [email protected])..

expansions, an eigenvalue problem often accompanied. Theassociated eigenvalue problem may become much morecomplicated especially for a slab with many layers. Theapplication of Laplace transform often yields residuecomputation. The calculation procedure is tedious if the slabhas more than two layers [4].

Therefore, the eigenvalue or residue computation hasalways posed a challenge to analytical methods on solvingheat conduction in composite slab. Above all, the heat transferequation on building space adds complexity andcomputational expense to the system of simultaneous heat

transfer equations for the whole building system. Therefore,most of the existing methods do not directly tackle thegoverning heat equations. Rather, response factor methodsand conduction transfer function methods are most widelyutilised for solving building heat transfer problems [5,6].

Numerical iteration is often needed to apply such methods. Itis well known that a risk of numerical instability exists inusing iterative programs.

More common models simplify the calculations byassuming steady-state heat transfer and zero thermal capacityof the building envelope [7]. These models lose the ability tosimulate transient behaviours of the building system which are

most important to characterise building physical processes ofheat transfer. This paper presents an analytical model for

predicting whole building heat transfer process. The model issimple, flexible, and driven by real weather conditions. The

building setting dealt with by the developed model is similaras the one described in Boland's paper [8]. Comparison

between the analytical and the numerical results shows a goodaccuracy of the developed analytical model.

.

II. ASSUMPTIONS

In practice, heat transfer process for building has a strong

three-dimensional character. However, the entire three-dimensional heat transfer for buildings is too complex to be performed by any analytical method and the physical processmight be too complex for analysis and interpretation.Therefore, most analytical heat transfer models havesimplified the building system. Even with the simplification,analytical methods are still troubled by the multiple layeredenvelope which does not permit analytical solutions to theheat conduction problem. In this paper, the followingassumptions are made to the modelled building system:

• Building has only one zone.

Analytical Model for Predicting Whole BuildingHeat TransferXiaoshu Lu and Martti Viljanen

M

World Academy of Science, Engineering and Technology 52 2011

601



• Envelope consists of the walls, the roof, the windows andthe floor foundation. The door belongs to a part of wallcomponent.• One-dimensional geometry is assumed for all the envelopecomponents.

• Indoor air is well mixed with time-dependent temperature T +(t ).• Outdoor temperature is accounted by the combined effect ofthe outdoor dry bulb temperature and the solar radiation whichis denoted as T −(t ).• Each component is an n-layer composite slab havingconstant thermal conductivity, diffusivity and density for eachlayer.• Ventilation rate is constant throughout the calculationduration.• Only steady-state heat transfer is considered for windowsthroughout the calculation duration.• Boundaries for envelope components are the time-dependentindoor and outdoor temperatures. In particular, the boundariesfor the floor and foundation are the indoor temperature andthe time-dependent temperature which is placed under thefoundation in the ground and denoted as T

ground (t ).

A schematic picture of the modelled building configurationis displayed in Fig. 1.

Fig. 1 Schematic picture of the modelled building configuration

III. MODEL SOLUTION

A. General Model Equation for Building Envelope

Using the proceeding assumptions, each buildingcomponent is an n-layer composite slab having constantthermal conductivity, diffusivity and density for each layer(Fig. 1). The thermal conductivity, diffusivity and thicknessare presented as λ j, k j and l j, j = 1…n. Just for a

demonstrational convenience, we provide the heat equationfor a general n-layer slab first.

For an n-layer slab, we set the following notations l0 = 0

and L j = l0+…+l j, j = 1…n. Then the layer boundaries are [ L0

= 0 , L1], [ L1 , L2] and [ Ln-1 , Ln] (Fig. 1). The general heat

conduction in an n-layer slab can be described by thefollowing equations for the jth layer temperature T j(t,x):

t

T j

∂

∂=

2

2

x

T k

j

j∂

∂, x∈[ L j-1, L j], j = 1...n (1a)

with boundary and initial conditions

−λ 1 x

T

∂

∂ 1 (t,L0)=−α +(T 1(t,L0)− T +(t )) (1b)

T j(t,L j)=T j+1(t,L j), j = 1…n-1 (1c)

λ j x

T j

∂∂ (t,L j)=λ j+1

x

T j

∂∂ +1

(t,L j), j = 1…n-1 (1d )

−λ n x

T n

∂

∂(t,Ln)=−α − (T −(t )− T n(t,Ln)) (1e)

T j(0 ,x)=0, x∈[ Ln-1 , Ln], j = 1…n (1 f )

For simplicity, zero initial temperature and perfect contact between layers are assumed. Surface convective and radiativeheat transfer coefficients are α + and α − .

Heat transfer equations for all the building envelopecomponents can be obtained from equation (1). The subscriptsand superscripts are used to present different envelopecomponents, for example, T

wall(t ) for walls, T roof (t ) for roof,

and T floor (t ) for floor and foundation in equation (1). For

T floor (t ), the equations exhibit somewhat differently in equation

(1) where equation (1e) is changed as

T n floor (t , Ln) =T

ground (t ) (1g)

where T ground (t ) is the ground temperature beneath the building

floor.

B. Equation for Indoor Air

For indoor temperature T +(t ), the heat transfer equation ismodelled as

C air V dt

dT +=∑

wall

Awallα +(T 1wall(t , L0)− T +)

+ Aroof α +(T 1

roof (t , L0)− T +) + A floor α +(T 1

floor (t , L0)− T +)

+ ∑window

Awindow

U window(T −(t )− T +) + η (T −(t )− T +) (2a)

T +(0) = 0 (2b)

where V presents the volume of the room, A the inner area of

the component, C air the thermal mass of the indoor air

x

T +(t )

layer

[ L j-1 L j]

T −(t )

l j

roof

wall

floor

windo

T ground

(t )

T −(t )

T −(t )


602



(product of density and specific heat), U the glazing thermaltransmittance coefficient, and η the ventilation rate of the building.

C. Weather Conditions

It is known that any function can be approximated asFourier series in an extended interval. Therefore, for any realweather condition of outdoor temperatures, we canapproximate it using Fourier series. Just for the sake ofcalculation simplification, we assume that both the outdoorand the ground temperatures change periodically as

T −(t ) = cos(ω t+ϕ ) (3a)

T ground (t ) = cos(ϖ t+φ ) (3b)

Note that for general boundary temperatures T − (t ) and

T

ground

(t ), they can be presented as

T − (t )=a0+∑∞

=

+1

)cos(k

k k k t a ϕ ω (4a)

T ground (t )=b0+∑

∞

=

+1

)cos(k

k k k t b φ ϖ (4b)

Hence the corresponding solution can be expressed as the sumof solutions with boundary temperature of equation (3).

Furthermore, assume the boundary temperatures are as thecomplex form

T −(t ) = eiω t + iϕ (5a)T

ground (t ) = eiϖ t + iφ (5b)

Clearly, the solution is the real part of the sought-aftersolution.

D. Model Solution

Applying Laplace transform on equation (1) we get

s jT

=2

2

x

T

k

j

j∂

∂ x∈[ L

j-1, L

j], j = 1...n (6a)

with boundaries

−λ 1 x

T

∂

∂ 1 (s,L0)=−α +( 1T (s,L0)− +T (s)) (6b)

jT (s,L j)= 1+ jT (s,L j), j = 1…n-1 (6c)

λ j x

T j

∂

∂(s,L j)=λ j+1

x

T j

∂

∂ +1(s,L j) j = 1…n-1 (6d )

−λ n x

T n

∂

∂(s,Ln)=−α − ( −T (s)− nT (s,Ln)) (6e)

where jT presents the Laplace transform of the building

envelope components jT wall and jT roof , for instance.

A bar over function f (t ) designates its Laplacetransformation on t as [2]

)(s f = τ τ τ

∫∞

−

0

)( d f e s (7)

The convolution property of Laplace transformation is givenas [2]

)(s f = )(1 s f )(2 s f

then f (t ) = f 1(t )∗ f 1(t ) = τ τ τ ∫ −t

d t f f 0

21 )()( (8)

Similarly, the Laplace transformation of equation (2) reads

s +T (s) =∑wall

β wall( 1T wall(s, L0)− +T (s))

+ β roof ( 1T roof (s, L0)− +T (s)) + β floor ( 1T floor (s, L0)− +T (s))

+ ∑window

β window( −T (s)− +T (s)) + η ( −T (s)− +T (s)) (9)

where β =V C

A

air

+α with the corresponding superscripts and

subscripts, but β window =V C

U A

air

windowwindow

.

Such system of simultaneous equations has been studied in[9]. Without showing the details, we copy the results here: forany jth layer, denote

q j =

jk

s, ξ j =q jl j, h0=λ nqncoshξ n+α − sinhξ n,

h j= j

j

λ

λ 1+

1+ j

j

k

k

, j=1…n− 1, hn=λ nqnsinhξ n+α − coshξ n (10a)

Δ(s)=

n

nnn

nn

hh

h

h

q

0

111

11

111

11

11

0...............

0sinhcosh............

10coshsinh............

........................

........................

............0sinhcosh

............10coshsinh

............00

−−−

−−

+

−

−

−

−

−

ξ ξ

ξ ξ

ξ ξ

ξ ξ

α λ

(10b)


603



Δ1(s)=−α +

)(

121)(

s

delete jcolumnrowbut sassame

Δ

−−−Δ (10c)

Δ2(s)= −α −

)(

122)(

s

delete jcolumnnrowbut sassame

Δ

−−−Δ (10d )

Δ3(s)=α +

)(

21)(

s

delete jcolumnrowbut sassame

Δ

−−Δ (10e)

Δ4(s)=α −

)(

22)(

s

delete jcolumnnrowbut sassame

Δ

−−Δ (10 f )

M j(s,x)=Δ1sinh(q j( x− L j-1))+Δ3cosh(q j( x− L j-1)) (10g)

N j(s,x)=Δ2 sinh(q j( x− L j-1))+Δ4cosh(q j( x− L j-1)) (10h)

=)(sG

∑∑∑∑∑∑

−−−+++++

+++

),(),(),(

)),(),((

010101

0101

Ls M Ls M Ls M s

Ls N Ls N

floor floor roof roof wallwallwindow floor roof wall

windowroof roof wallwall

β β β β β β β η

β β β η

(11a)

=)(sF

),(),(),(

),(

010101

01

∑∑∑∑ −−−+++++ Ls M Ls M Ls M s

Ls N floor floor roof roof wallwallwindow floor roof wall

floor floor

β β β β β β β η

β

(3.11)

(11b)

T +(t ) =G (iω )T − + F (iω )T ground (12)

The final solution is the real part of T +(t ), Re(T +(t )).The above close-formed solution Re(T +(t )) presents the

transient temperature variation indoors. It shows globally atthe response of internal temperature in a building under freeand real outdoor boundary condition. It is not difficult to seethat the close-formed solutions for building envelope can beobtained in a similar way [9].

IV. VALIDATION

To assess the accuracy of the developed analytical method,the results from the analytical model will be compared to theresults obtained from numerical studies. Heat transfer in one-

room house is simulated. The house's dimension is 6×6×2 m3

.

Its envelope consists of three layers. Such envelope structurehas been adopted in our test building. The main material of thewall is mineral wool (200 mm). Boundary layers consist ofwall-paper (25 mm) and gypsum board (13 mm). Fig. 2 showsa schematic picture of the house and Table 1 the physical

properties of its envelope.

Fig. 2 Schematic diagram of the simulated house

Boundary temperature was taken from the measurement andthen fitted with periodic functions with periods 30, 5, 2 and 1days as

T − (t ) = a0+ ))(2cos(4

1i

ii

it a ϕ

ω

π −∑=

(13)

where fitting parameters are listed in Table 2. The convectiveand radiative heat transfer coefficients are α − = 25W/m2/K andα += 6W/m2/K.

The calculated transient indoor temperatures by analyticaland numerical models are displayed in Fig. 3. The maximaldiscrepancy is about 0.9°C (relative error about 6%).Agreement between numerical and analytical results is good.The validation of the numerical program can be found in [10].

V. CONCLUSION

An analytical model was developed in this paper which

gives close-formed solutions for both transient indoortemperature and construction temperatures in a building

T t T

( t)

13mm/200mm/25m

sensor

TABLE 1MATERIAL PROPERTIES AND DIMENSIONS FOR THE ENVELOPE OF THE

SIMULATED HOUSE

Thermalconductivity(W/m/K)

Thermaldiffusivity (m2/s)

Thickness(mm)

gypsum board 0.23 4.1×10-7 13mineral wool 0.147 1.5×10-6 200

paper 0.12 1.5×10-7 25


604



system. No restriction on the layer number and the thermal physical properties of the construction was needed. Themethod is free of numerical program. In general literatureworks for such problem, however, there usually existslimitation on layer number of the construction, or associated

numerical iteration is often necessitated. Additionally,mathematical expression for closed-formed solutions is simpleand concise with good accuracy. The model can be used toanalyse the thermal process in relation to physical parameters.

ACKNOWLEDGMENT This research was supported by the Academy of Finland.

.

R EFERENCES

[1] X. Lu, D. Clements-Croome and M. Viljanen, "Past, present and futuremathematical models for buildings: focus on intelligent buildings (I)",Intelligent Building International Journal , vol 1, pp23-38, 2009.

[2]

H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solid (2nd ed.),Oxford University Press: Oxford, UK,1959.

[3] M.D. Mikhailov and M.N. Özisik, Unified Analysis and Solutions of

Heat and Mass Diffusion, Dover Publications Inc.: Mineola, N.Y., 1994.

[4]

T.P. Fredman, "An analytical solution method for composite layerdiffusion problem with an application in metallurgy", Heat and MassTransfer, vol. 39, pp. 285-295, 2003.

[5]

G.P. Mitalas, "Calculation of transient heat flow through walls androofs", ASHRAE Transactions, vol. 74, pp. 182–188, 1968.

[6] D.G. Stephenson and G.P. Mitalas, "Calculation of heat conductiontransfer functions for multilayer slabs", ASHRAE Transaction, vol. 77,

pp. 117–126, 1971.[7] M.S. Bhandari and N. Bansal, "Solar heat gain factors and heat loss

coefficients for passive heating concepts", Solar Energy, vol. 53, pp.199–208, 1994.

[8]

J. Boland, "The analytical solution of the differential equationsdescribing heat flow in houses", Building and Environment, vol. 37, pp.1027-1035, 2002.

[9] X. Lu, T. Lu and M. Viljanen, "A new analytical method to simulate heattransfer process in buildings", Applied Thermal Engineering, vol. 26,

pp. 1901-1909, 2006.[10]

X. Lu, "Modelling heat and moisture transfer in buildings: (I) Model program", Energy and Buildings, vol. 34, pp. 1033-1043, 2002.

-4

-1

2

5

8

11

14

0 5 10 15 20 25 30

Time (day)

T e m p e r a t u r e ( ° C )

Outdoor

Analytical

Numerical

Fig. 3 Comparison of numerical and analytical results

TABLE IIPARAMETERS IN EQUATION (13)

ω 130.0

ω 25.0

ω 32.0

ω 41.0

ϕ 1 5.607506 ϕ 213.59596

ϕ 31.451539

ϕ 45.418717

a0 5.0

a1 2.72217

a2 -5.019664

a3 1.084058

a4 0.4648


605

analytical model for predicting whole building

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