Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 121 (2015) 1764 – 1771
9th International Symposium on Heating, Ventilation and Air Conditioning (ISHVAC) and the 3rd International Conference on Building Energy and Environment (COBEE)
Fast Simulation Methods for Dynamic Heat Transfer through Building Envelope Based on Model-Order-Reduction Xiao Hea, Qiongxiang Konga,*, and Zhihua Xiaob a
School of Human Settlement and Civil Engineering, Xi’an Jiaotong University, Xi’an 710049, China b School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Abstract In this paper, fast numerical simulation methods for dynamic heat transfer of building envelope have been developed by using Krylov subspace and balanced truncation model order reduction (MOR) methods. A numerical example about the transient heat transfer of the building envelope is given. The decrement factor of the roof and the time lag of the peaks or troughs calculated by the MOR methods are quite consistent with the results from the harmonic response method, the maximum absolute errors of the temperature and the heat flux on the interior surface of the roof are less than 0.2oC and 1.5W/m2, respectively. All the simulation results prove that the MOR methods are reliable in solving heat transfer problem of the building envelope, and MOR method can effectively speed up the calculation. Krylov subspace MOR method has the advantage of quick calculating speed. Balanced truncation MOR method maintains the stability of the original system. © Published by Elsevier Ltd. This © 2015 2015The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility the organizing committee of ISHVACCOBEE 2015. under responsibility of theoforganizing committee of ISHVAC-COBEE 2015 Peer-review
Keywords: Dynamic heat transfer; Model order reduction (MOR); Krylov subspace; Balanced truncation
1. Introduction As improved requirements of building performance and increased focus on building energy efficiency continue, building energy analysis applying numerical simulation methods becomes more useful. Now many software tools, such as BLAST, DOE-2, EnergyPlus, ESP-r, HASP, DeST, etc, can implement building energy simulation. However,
* Corresponding author. Tel.: +86 13186002310. E-mail address:
[email protected]
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ISHVAC-COBEE 2015
doi:10.1016/j.proeng.2015.09.149
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their results may vary if different software tools are used to implement the calculation of the same problem [1,2]. Moreover, the inconsistencies between building energy predictions and actual, operational energy use data have been usually reported and cited in many articles. These software tools use variants of the energy simulation methods. The heat balance method, utilized in the EnergyPlus program which combines the best features of BLAST and DOE-2 programs and has various innovative capabilities, requires the simultaneous simulation of a number of equations, which must be solved with a digital computer and needs more computational time. Therefore, it is necessary to develop fast numerical methods. Recently, some new methods, such as parallel computing method [3,4,5,6], model order reduction (MOR) method [7,8] were investigated in building performance simulation. For building energy analysis, envelop thermal performance calculation is a critical component that calls for further development [3], thus extensive research was carried out dealing with the dynamic heat transfer in external walls [9,10,11]. The equations about transient conduction heat transfer through the building envelope are partial differential equations (PDE) related to time and space, which are usually discretized spatially into ordinary differential equations (ODE) first. For multi-dimensional envelop, the number of variables can extend from hundreds to several millions. Finding the transient behavior of such large systems would require excessive computational effort. MOR method can be used to address this issue [12]. MOR is an efficient technique to generate a small model that approximates the behaviour of the original large system, where this small one facilitates both the theoretic analysis and the computationally efficient analysis of the system. Model order reduction has received considerable attention in the past decades, and a number of techniques are proposed [13]. The developed methods can be classified into two main approaches: one is balanced truncation based on singular value decomposition (SVD) [14,15,16], and the other is Krylov subspace related [17,18,19]. Balanced truncation was first proposed by Moore B. [14]. The attractive aspects of balanced truncation methods are that computable error bounds are available for the reduced model, and the reduced model is preserving stability of the original system. In this methods, solving the Lyapunov equations is a key step. Direct solution of the Lyapunov equation is only possible for medium-scale systems because the computational complexity is O(n3), where n is the dimension of the system. Therefore, several methods have been proposed to extend the range of applicability of balanced truncation to large-scale systems[15,16]. The Krylov subspace-based method is the most commonly used method in model order reduction. This method is superior in numerical efficiency with the low cost of computation, but in general the stability of the original system may be lost, and there is no general error bound similar to balanced truncation except under some special conditions [20]. In this paper, the Krylov subspace and balanced truncation model order reduction methods are used to solve the transient conduction heat transfer equations of the building envelope, and the simulation results of different methods are compared and discussed. 2. Methods 2.1. Mathematical models The transient conduction heat transfer through the building envelope is a PDE related to time and space, which generally can be described as
Uc
wT wt
w § wT · w § wT · w § wT · ¸ ¨O ¨O ¸ ¨O ¸S wx © wx ¹ wy © wy ¹ wz © wz ¹
(1)
T ( oC ) is the temperature, t ( s) is the time, U (kg / m3 ) is the density, c( J / (kg oC )) is the specific heat o 2 capacity, O (W / (m C )) is the thermal conductivity, and S (W / m ) is the internal heat source of building envelope, S SC SPTP ( SP 0) . where
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The boundary conditions should be considered carefully. Generally, on both sides of building envelope Robin boundary conditions are presented. Therefore, at the interior and exterior surfaces, the boundary conditions are as follows
Oi
wTi wni
hi (Ti ,e Ti )
(2)
Oo
wTo wno
ho (To,e To )
(3)
where h(W / (m K )) is the heat transfer coefficient at the inside or the outside surface, n is the normal vector at the surfaces, and the subscripts i and o refer to the interior and the exterior surfaces of the building envelope, respectively. In addition, Ti ,e and To ,e are inside air temperature and outside solar-air temperature, respectively. 2
Fig. 1. the control volume of P.
Finite volume schemes are used and the diffusion term is handled by first-order central difference (Fig. 1.) for discretization in the Eq. (1). The Eq. (1) about transient conduction heat transfer through the roof needs to discretize spatially into ODEs, as follows
§ dT · ¨ ¸ © dt ¹ P where
aT
aW TW aETE aS TS aN TN aT TT aBTB aPTP
Sc ( U c) P
(4)
Ow Oe Os On 1 1 1 1 , aE , aS , aN , ( U c) P 'x G xw ( U c) P 'y G yn ( U c) P 'x G xe ( U c) P 'y G ys Ot Ob 1 1 SP , aB , aP aW aE aS aN aT aB . ( U c) P 'z G zb ( U c) P 'z G zt ( U c) P aW
Considering the boundary conditions as Eq. (2) and Eq. (3), the Robin boundary condition is converted into Dirichlet boundary condition by the additional source method. So the influence factors of the first or the last node in building envelop by the inside or outside temperature are respectively.
aE
1
G xe / Oi 1/ hi
or
aW
1
G xw / Oo 1/ ho
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While, the heat flux on the interior surface of the roof is expressed as
y(t ) hi (Ti ,e Ti (t ))
(5)
Generally, the outdoor temperature periodically varies in one day or one year, while the indoor air temperature is hoped to keep unchanged. Therefore, the initial condition of periodic heat transfer of building envelope is set to periodic or a given initial value. 2.2. Model-order-reduction methods Furthermore, a state-space form of the above differential Eq. (4) and Eq. (5) is presented for the convenience of control application, which approximates the original PDE model
dT (t ) AT (t ) Bu (t ) ° ® dt ° ¯ y (t ) CT (t ) where
A
nun
,
B
nu p
, C
mun
are the coefficient matrices (
(6)
is real number set), T (t )
n
is the state
variable, u (t ) is the input variable, y(t ) is the output variable. Depending on the desired spatial accuracy, the number of variables can extend from hundreds to several millions. Finding the transient behaviour of such large systems would require excessive computational effort. MOR can be used to address this issue, and it has been proved as a useful tool to obtain efficiency in simulations while ensuring desired accuracy. p
m
Krylov subspace model order reduction The Krylov subspace-based MOR method aims to construct projection matrices to reduce large systems. The main is to generate a projection matrix based on the Krylov subspace process
Kq ( A1 , A1B) colspan{A1B, A2 B, , A q B} , V
nur
(r
pq, r
n)
where
q
is a nonegative integer. If a projection matrix
is obtained by the block Arnoldi algorithm [21] based on this Krylov subspace, then the
following reduced system can be obtained by the transformation T (t ) | VT (t )
dT (t ) AT (t ) Bu(t ) ° ® dt ° ¯ y (t ) CT (t ) , y(t ) , and A V T AV rur , B V T B ru p , C CV where T (t ) The reduced system Eq. (7) matches the first q moments of the original system Eq. (6). r
m
(7)
mur
.
Balanced truncation model order reduction
H is first calculated, and then by the ˆ transformation T (t ) HT (t ) , the system Eq. (6) can be converted to the following form For the balanced truncation MOR method, a nonsingular matrix
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dTˆ (t ) ˆ ˆ ˆ (t ) AT (t ) Bu ° ® dt ° yˆ (t ) CT ˆ ˆ (t ) ¯ ˆ where A
H 1 AH
nun
, Bˆ
H 1B
nu p
, Cˆ
CH
mun
(8)
.
ˆ , Qˆ Corresponding to Eq. (8), the so-call controllability and observability Gramians P
nun
satisfy two
ˆ ˆ PA ˆ ˆ T BB ˆ ˆ T 0 and Aˆ T Qˆ QA ˆ ˆ Cˆ T Cˆ 0 . The transformation matrix H is called Lyapunov equations AP ˆ Qˆ diag{V , V , , V } , where V t V t t V t 0 are called as the Hankel a balanced matrix when P 1 2 n 1 2 n singular values, and the system Eq. (8) is called as a balanced system [14]. By truncating the smaller Hankel singular
ª Tˆ1 (t ) º « » , where Tˆ1 (t ) is the state variable of the reduced ˆ ¬«T2 (t ) ¼» ˆ º ªˆ ªˆ º ˆ , Bˆ and Cˆ as follows: Aˆ « A11 A12 » , Bˆ « B1 » , system. Correspondingly, we partition the matrices A ˆ ˆ ˆ ¬« B2 ¼» ¬« A21 A22 ¼» Cˆ ª¬Cˆ1 Cˆ 2 º¼ .
value of the system Eq. (8), we can partition Tˆ (t ) into
Then, we can obtain the reduced system Eq. (7), and the state variable and coefficient matrices, respectively, are
T (t ) Tˆ1 (t ), A Aˆ11 , B Bˆ1 , C Cˆ1 . The above reduced system can preserve the stability of the original system Eq. (6), and the error can be estimated by means of Hankel norm. Finally, the fully implicit scheme is used in temporal discretization, solving the low-order model can greatly reduce the amount of calculation. 2.3. Calculation example Take the heat transfer in a reinforced concrete roof with 15cm thickness for one-day period for example. The outside solar-air temperature of the roof each day is To,e=42+23.3cos(15t-191.6), oC, where tԖ[0,24], hour. The inside air temperature Ti,e keeps 28 oC consistently. Assume that there is no internal heat generation in the roof, and heat wave propagation is only in the roof thickness direction. The combined convection coefficient on the outside or inside of the roof is ho=23.3 W/(m2·K) or hi=7.0 W/(m2·K), and the thermal diffusivity of the roof is 2.79×10 -3 m2/h. The discretized spatially Eq. (4) is presented as a state-space form Eq. (6), n is the sum of nodes in the 1D domain (1500 in this study), οx is the spatial mesh size,οx=0.0001m, a=λ/ρc, A is the system matrix with dimension n, T is a n dimensional state vector containing each node temperature, and u is the air temperature. So matrix A tends to be
§ (1 H ) 1 · §H 0 · § T1 · ¨ ¸ ¨ ¸ ¨ ¸ 1 2 1 0 0 ¸ T2 ¸ ¨ ¨ ¨ ¸ a a ¸ ˈB ¨ ¸ , ¨ ¸ , and T t ¨ A 2 'x 2 ¨ 'x ¨ ¨ ¸ ¸ ¸ 1 2 1 ¨ Tn 1 ¸ ¨0 0 ¸ ¨ ¸ ¨ 0 H '¸ ¨ ¨T ¸ 1 (1 H ') ¸¹nun © ¹ nu 2 © © n ¹ nu1 1 1 , H' , n is the order of the system, n=1500, u is the outside and where H 1/ 2 O / ho 'x 1/ 2 O / hi 'x
Xiao He et al. / Procedia Engineering 121 (2015) 1764 – 1771
inside air temperature,
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§ To,e · u ¨ ¸ . © Ti ,e ¹2u1
For this system, the system matrix of original model can be reduced by using Krylov subspace or balanced truncation MOR methods. Then, the fully implicit scheme is used in the temporal discretization (the temporal mesh size is 1/100 hour). Finally, for comparison, the high-order model and its corresponding low-order models are solved, respectively. 3. Results & Discussion In the problem of dynamic heat transfer calculation for single-layer flat roof, using both the Krylov subspace and the balanced truncation MOR methods, the original model with 1500-order is reduced to a 15-order model. The calculation results for the roof are shown in Fig. 2. The Fig. 2 shows the temperature distribution in the roof in the day calculated by direct numerical simulation and two MOR methods, and it is clearly shown that the results of the three methods are quite consistent with each other. As the outside temperature varies from 65.5 oC to 18.7oC during the day and indoor temperature remains 28oC, the temperature gradient in the roof thickness direction is different, big or small, positive or negative. And the computing performance of MOR methods is shown in Table 1. The solution procedure of this problem can be divided into modelling and numerical solving. For the same problem, the follow-up solving can directly call the model, so only calculating the solving time speedup. The Krylov subspace and the balanced truncation MOR methods are solving faster than direct numerical simulation, while the method of Krylov subspace technique has the advantage of quick calculating speed. The modelling time of the balanced truncation MOR method is very large, because of this method can produce nearly optimal model, but it is more computationally expensive than the Krylov subspace MOR method.
(a) original model
(b) Krylov subspace
(c) balanced truncation
Fig. 2. temperature distribution in the roof for a day. (a) original model. (b) Krylov subspace. (c) balanced truncation. Table 1. The computing performance of MOR methods. Calculation time (s)
Model order reduction
Modeling
Solving
Speedup
Original model
1500-order
4.559
7.926
-
Krylov subspace
15-order
0.572
0.036
220
Balanced truncation
15-order
83.667
0.053
149
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Using harmonic response method [22] to calculate this problem, the decrement factor of the roof is 2.86, the time lag of the peaks or troughs is 4.14 hours. The comparison among the Krylov subspace MOR, the balanced truncation MOR and the harmonic response methods are shown in Fig. 3. The Fig. 3 shows that the peaks or troughs of the temperature and heat flux on the interior surface of the roof are 43.35oC and 107.43W/m2 or 27.05oC and -6.63W/m2, respectively, and the decrement factor of the roof is 2.88, the time lag of the peaks or troughs is 4.17 hours compared to that of the outside temperature. Compared with the harmonic response method, the maximum temperature differences of the MOR methods are 0.2oC, the relative errors are less than 1%, and the corresponding heat transfer density differences are 1.5W/m2. The results show that the Krylov subspace and the balanced truncation MOR methods are reliable for solving the heat transfer problem in the building envelope. Compared with the calculations of harmonic response method and the original model, the order-reduction models have the same accuracy but faster calculation speed. For the same problem, the higher system order of the governing equation, the more accurate results, but the calculation time will be extended. In this calculation example, it is clear that the computational efficiency of the Krylov subspace MOR method is obviously better than that of the balanced truncation MOR method. This is due to that the balanced truncation MOR method needs to solve two large-scale Lyapunov equations, so constructing a reduced order model maybe takes more time. However, the balanced truncation MOR method can produce nearly optimal reduced-order model.
Fig. 3. temperature and heat flux on the interior surface of the roof.
4. Conclusions Using the MOR methods of Krylov subspace and balanced truncation, the fast and accurately numerical simulation for dynamic heat transfer through the building envelope has been complemented. For the calculation example of the heat transfer through the roof, the speedup of the Krylov subspace and the balanced truncation MOR method is 220 or 149, respectively. The Krylov subspace MOR method has more high calculating speed than that of the balanced truncation MOR method. References [1] D. Sun, W. Xu, Y. Zou, X. Chen, Air conditioning cooling load calculation method and comparison and improvement of calculation software, HV&AC. 42 (2012) 54-60. (in Chinese)
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