Optimised RC Networks Incorporated within Macro ... - CiteSeerX

Optimised RC Networks Incorporated within Macro-Elements for Modelling Thermally Activated Building Constructions SUBMITTED: REVISED: PUBLISHED:

April 2004. September 2004. December 2004.

Dietrich Schmidt, PhD Division of Building Technology, Department of Civil- and Architectural Engineering KTH-The Royal Institute of Technology SE-10044 Stockholm, Sweden email: [email protected] and Fraunhofer Institute for Building Physics Project Group Kassel DE-34127 Kassel, Germany email: [email protected] Gudni Jóhannesson, Prof. Division of Building Technology, Department of Civil- and Architectural Engineering KTH-The Royal Institute of Technology SE-10044 Stockholm, Sweden email: [email protected] KEYWORDS: Thermally activated building constructions, low temperature heating and cooling, optimised models, dynamic computer simulation. SUMMARY: This paper presents an effective modelling method for the thermal simulations of thermally activated building constructions or hybrid systems, and a scheme in which the resulting mathematical models can be implemented in a commonly used dynamic simulation computer program. The advantage of this method is the fact that only a limited number of nodes are required to obtain reliable results for the simulation. A general overview of the modelling method, based on an earlier conducted analysis (Schmidt and Jóhannesson 2002a), is provided and the modelling method is described in detail in presented case studies. A comparison between the new model and a derived analytic solution for a case example shows the quality of the model. The so-called macro element modelling method, MEM, is based on earlier research on the modelling dynamic heat flows in solid constructions with discrete resistances and capacitances (Jóhannesson 1981, Mao 1997, Akander 2000). Here, it has been expanded by the simultaneous modelling of heat carrier flows and used on a different class of constructions, thermally activated building constructions, such as hydronic floor and wall heating or hollow core slabs. As shown in an example in this paper, the MEM method is suitable for the dynamic simulations of thermally activated components in buildings. With the MEM model, the temperature variation of the heat carrier fluid is modelled correctly along the flow path with the assumption that the temperatures of the mass nodes of the construction vary stepwise linearly along the direction of the flow. The fluid’s temperature profile is calculated in quasi-steady state conditions. In the present stage this limits the model validity to cases in which the fluid’s flow-through time is shorter than the period of studied time variations. The results can be corrected when the actual flow through time is taken into account. The cases shown demonstrate the advantages of using MEM models to calculate even extreme time-dependent processes, such as switching the flow direction or steps in the inlet temperature or flow conditions. These are preferable to other known simplified models of thermally activated constructions in dynamic simulations. It has been demonstrated that the MEM method is generally suitable for modelling the dynamic behaviour of combined systems with a heat carrier flow and solid construction parts with substantial heat storage capacity.

1

Introduction

As part of the measures taken to reduce the emissions from energy utilisation processes, efforts have been made to reduce energy consumption in buildings as well, since buildings account for a major fraction of the world’s annual energy demand. This has been achieved by constructing heavily thermally insulated buildings, by improving the quality of window glazing, and by using the thermal storage of the construction itself. In order to

Nordic Journal of Building Physics Vol. 3, 2004, Schmidt, Jóhannesson

1

make the energy use in buildings even more efficient, low temperature heating and cooling systems have been implemented. A promising solution in this sector is the increasing use of thermally activated building components. Such solutions, which utilise low temperature differences, allow for an easier implementation of renewable energy sources in the built environment, as well as making way for a low exergy demand for space heating and cooling. “Hybrid systems” or “thermally activated constructions” are building components or a combination of different building components that utilise the heat transfer properties and the heat capacity of the whole construction to achieve room conditioning. Typical examples of this are embedded hydronic pipes in concrete slabs or air flow in hollow core slabs, in which case the entire slab construction is tempered to heat or cool the room. Another commonly known example is hydronic floor heating. The room surfaces are kept at a warm, fairly uniform temperature, providing a good level of thermal comfort. For the dynamic analyses of such a system, several heat transfer mechanisms are involved and must be treated simultaneously. For the design of a heating or cooling system which includes thermally activated building components, efficient and reliable tools are needed to analyse the performance of the system. The commonly used simplified models, like the one-dimensional or steady state models, are not sufficient for estimating the total performance of such system solutions. In addition, the already existing stand-alone-models, depicting some thermally activated heating and cooling systems, are not satisfactory for design purposes. Dynamic simulation models, including all three dimensions of the heat transfer and all possible heat transfer effects, have to be implemented in an effective way into known simulation programs such as IDA ICE (Bring et al 1999) or TRNSYS (TRNSYS 2000). Extended simulation of the dynamic behaviour of the overall system, taking into account realistic climate conditions, regulation strategies, and occupants, requires extensive computational resources, which should however not be used for a detailed simulation of only one system part, e.g. the thermally activated construction. There has been some attention paid to the modelling of thermally activated heating constructions such as heating and cooling systems. Some generalised design methods for the regarded systems have been developed and proposed (Rittelmann et al 1983, Evans et al 1985, Fort 1989). During the “Solar Heating and Cooling Programme” and the “Energy Conservation in Buildings and Community Service Programme” performed by the International Energy Agency, some focus was on the modelling (Jørgensen 1984, Scartezzini et al 1987). The modelling for whole building simulation has also been discussed intensively (Hauser 1977, Nakhi 1995, Sahlin 1996). More detailed models of thermally activated constructions, suitable for the implementation in dynamic simulation environments, have been published (Koschenz and Dorer 1999, Fort 1989, Caccavelli and Mounajed 1995, Caccavelli and Bedouani 1998, Strand and Pedersen 2002, Laouadi 2004). These models use constant temperatures during a time step in a calculation element and for some models, the cross section of the system has to be calculated via a finite difference model before estimating the temperature variation of the fluid along the flow path. Generally, this approach of calculating a cross section before being able to perform the simulation has already been widely used (Fort 1989, Strand 1995, Caccavelli and Mounajed 1995). This paper presents an alternative method of modelling these special building constructions in commonly used dynamic simulation environments, the so-called macro element method (MEM). It is the second paper in a series of articles describing the modelling method both in detail, and with different case studies. Research has been conducted on the optimisation of the resistances and capacitances (RC) network, which is the basic of the MEM method (Jóhannesson 1981, Akander 1995, Mao 1997, Akander 2000, Schmidt and Jóhannesson 2002a). In this paper, the methodology has been applied to a different class of constructions, i.e. thermally activated constructions. It is explicitly shown here how the RC networks can be combined to larger macro elements. The main ideas within this method are: 1. The transformation of the dynamic properties of the construction’s cross-section into a limited number of mass nodes. These are then represented in an optimised RC network. 2. The temperatures of the capacitive nodes vary linearly along the fluid’s flow path, e.g. the length of the duct or pipe. 3. The fluid’s temperature profiles are modelled accurately in a quasi-steady state based on the assumption that the surrounding mass node temperatures vary linearly along the flow path. 4. The models have been verified with analytical solutions of cases in the frequency domain.


2

2

Thermally activated building components

The term “thermally activated components” is commonly used for constructions in which a flow of medium heats or cools the entire construction and thereby, the spaces to be tempered. They are also called hybrid systems, since a building construction part merges together with a heat exchange component (Schmidt and Jóhannesson 2002a). In this way, these systems act as combinations of active and passive components and become hybrid systems (Gertis 1982, Johannesson 1984). These components utilise the entire construction with its heat storage capacity and the whole surface for heat transfer to the environment. The combination of heat storage inside the construction, and a combination of mass and heat transfer at low temperature differences between the room to be conditioned and the heating or cooling medium are special advantages of these systems. Hollow core slab (airborne) Source: Strängbetong

Thermally activated constructions

Co-axial flow heat storage (waterborne)

Double air gap wall (airborne)

Source: Weber

Embedded pipes in slab (waterborne)

Figure 1: Examples of water and air borne thermally activated components with counter flow characteristics. There is a variety of thermally activated building components available. Hydronic floor heating, which uses parts of the floor construction as a heat storage, is a typical example. Generally, these systems utilise air or water as the heat carrier medium. The heated or cooled water/air has to be transported from the source (boiler or chiller) to the emission system and through the emission system itself. The flow of the heat carrier medium inside these constructions is often designed in a counter flow pattern, comparable to a heat exchanger. In a heat exchanger, the exchange between the medium flows is of main concern, not the exchange to the surroundings. In thermally activated constructions it is the other way around, i.e. the heat emission to the room is of utmost relevance. However, heat exchange also takes place in the thermally activated components between neighbouring flows of the medium. This affects the overall efficiency of these heating or cooling systems. This paper focuses on the modelling of such systems. Some examples are shown in Figure 1, in which the small arrows indicate the counter flow pattern: The hollow core slabs use the warm or cool ventilation air also for tempering the surface of the construction, thereby heating or cooling the room (Ren and Wright 1997).


3

Another building part, although not in the narrow sense “building part”, is a heat storage with a co-axial piping in a bore hole. This system, which was already analysed in the first article of this series (Schmidt and Jóhannesson 2002a), is modelled in about the same way that other thermally activated constructions can be modelled. It is used here as an example to explain the macro element method because of its simple circular geometry and the availability of an analytical solution of the dynamic properties of the system in the frequency domain. Embedded pipes in concrete slabs are commonly used for thermal emission, mainly for the heating and cooling of office buildings. This system utilises relatively low temperature differences between the heat carrier and the room temperature, thus advantageously having a relatively low exergy demand (Weber et al 2000, Schmidt 2004). An example building where such a system has been implemented and monitored is the office building of the Centre for Sustainable Building in Kassel, Germany (Schmidt 2002b). The tempering of not only floor and ceiling constructions, but also walls, is common in some countries (NOVEM 2001). A possible solution for lightweight wooden buildings is the double air gap wall construction proposed by the LOWTE Company / Sweden. It is an airborne system, with no water flow inside the wooden wall construction. The wall surfaces are heated or cooled by a circulating air flow inside the wall which has a very low temperature difference to the room temperature. An analysis of this system set up is provided in (Schmidt and Jóhannesson 2001).

3

The macro element modelling method (MEM)

A method for mathematical modelling was developed by Jóhannesson for the analysis of the thermal conditions of a building component and for the description of multi-dimensional heat conduction (Jóhannesson 1981, Mao 1997 and Akander 2000). In this paper, this method, or analysis procedure for wall and ceiling constructions, has been extended to incorporate mass flow inside the construction. Therefore, it is even possible, using a limited number of macro elements, to model hybrid building components with an optimised resistance-capacitance (RC) network.

3.1

Approach

The aim of the proposed method is to obtain a practical and reliable model of a thermally activated construction. But still, the major dynamic characteristics of the construction should be modelled correctly. • First, a cross section of the regarded system is transformed into a limited number of mass nodes, i.e. an optimised network of discrete resistances and capacitances. This means that less nodes are needed compared to, for example, finite element method (FEM) or finite difference (FDM) models. For this transformation, the cross section has to be analysed in frequency domain, in which linear systems are assumed. In the transformation step, the solid parts are modelled correctly in a chosen frequency range with the main dynamics via a network of discrete resistances and capacitances. Deviation between model and analysis can be seen directly in the Bode diagram, a diagram showing the thermal properties of the construction in the frequency domain. The resistances and capacitances are optimised for a certain chosen frequency range in accordance with the later use of the model and the processes to be estimated. These so called RC networks, representing the solid construction, can be used in time domain applications for linear as well as non linear processes, such as varying the flow and switching the flow direction. • For the MEM model, a linear variation of the mass node temperatures along the fluid’s flow path is assumed. This makes it possible to significantly increase the element size compared to the solution with constant mass node temperatures along each flow element. • The temperature profile in the fluid flow is modelled accurately in quasi-steady state conditions and calculated with an assumed linear variation along the flow path of the neighbouring mass node temperatures. It is assumed that the temperature distribution of the fluid is, at any time, in equilibrium with the surrounding mass nodes. This assumption is valid when the velocity of the fluid is sufficiently high, i.e. when the fluid passes the construction in a fraction of the shortest period of the processes to be studied. • The model is built up along the flow direction. Heat transport to and from the fluid, orthogonal to the flow direction, is modelled via a network of discrete resistances and capacitances. • The time domain MEM model is verified using cases analysed in the frequency domain. The results in the time domain for time dependent harmonic inputs are compared with the analytical solution.


4

3.2

Included processes and assumptions

For the modelling of thermally activated components, a number of parallel heat transfer processes must be simultaneously regarded. These are taken into consideration for both the flows of the heat carrier medium and for the surrounding solid construction. Mathematical models of real processes use simplifications to allow for a more manageable resulting model. The method described here is aimed at achieving easy to handle, small mathematical models. • Convective surface heat transfer coefficients and possible radiative heat transfer coefficients between surfaces have to be set constant for each element. There is of course the possibility to vary these for each time step based on changes in the mass flow or temperatures. • The resulting temperature distribution along the flow channels is solved analytically for the stepwise linearly varying surrounding mass nodes temperatures. These calculations are done for quasi steady state conditions. The heat loss from the fluid along the element is calculated from the resulting temperature change. • All heat carrier properties and material properties of the solid constructions are set to be constant. This is valid in the expected temperature and pressure range close to standard conditions (20°C and 1013.25 hPa) (VDI 1991). It is possible to expand the modelling in such a way that the medium properties are estimated for each time step. • Heat transport by convection along the direction of the flow is a major part of the model, therefore the heat transfer capacity of the flowing fluid is taken into full consideration. However, the thermal storage capacity of the stagnant medium is neglected, i.e. the storage term is set to zero and the temperature distribution for the flow is, at each time point, in a steady state equilibrium with the surrounding mass nodes. • Dynamic heat conduction in the solid construction is modelled with an optimised resistance-capacitance network. Solar radiation hitting inner parts of the construction and being absorbed there is handled as heat generation in the modelling. • As a core of the MEM method, the temperatures of the mass nodes in the solid are seen as changing stepwise linearly with the co-ordinate of length. Similar to the temperature profile in the heat carrier flow, the real temperature variation at the surface of the construction would be an exponential function. In this model, the resulting surface temperature variation is to be a linear function of length for every macro element. • Heat exchange directly between two adjacent heat carrier flows can also be taken into consideration.

3.3

Background of the method

The MEM-approach can be divided into the following three basic steps: analysis, transformation and implementation. 3.3.1 Analysis of the system The analysis of the solid construction parts is done in the frequency domain. If possible, as for simple geometries, the system responses of a cross section are directly calculated using analytical methods. If this is not possible, the construction is implemented in a finite difference of finite element program in order to calculate the system responses of a cross section (Weber, Koschenz and Johannesson 2004). For the analysis, all possible simplifications (e.g. symmetry) are made (Schmidt and Jóhannesson 2002a). The analysis result of the heat transfer balance between two arbitrary isothermals 0 and 1 is in the form of a frequency dependent matrix (Carslaw and Jaeger 1959):  T~1   A B  ~  =   Q1  C D  ω

 T~  ⋅  ~0  Q0 

(1)

The admittance can be seen as a measure of the heat exchange between the system and its adjacent surroundings. It is defined as a relation between the heat flow through the wall and the temperature on the same side and can be expressed in the terms of the matrix in equation ( 1 ):


5

~ Q ~ Y1 = ~1 = D / B ; T0 = 0 T1

(2)

3.3.2 Transformation The heart of the MEM method is the transformation of the building construction’s properties into a network of discrete resistances and capacities. A RC-network is optimised in such a way that, for a given nodal configuration, the system responses from the RC-network are, for a certain frequency band, as close as possible to the response from the former calculations of the building construction’s cross section. For the chosen RCnetwork configuration of a given construction, a heat transfer matrix equation for each frequency can be expressed in terms of the resistive and capacitative parts (Beuken 1936, Rouvel and Zimmermann 1997, Akander 2000). The deviation of the system responses between the detailed model and the RC-network can then be calculated and the deviation for the simplified model can be studied in the frequency domain. To integrate the RC networks into macro elements, the heat balance of the mass flow inside the construction is described by a set of differential equations. By solving the equations for a linear change in (pipe/duct) surface temperatures in the direction of the flow in each macro element, it is possible to use considerably enlarged segments for a given discretisation error. Investigations carried out during the described project have demonstrated that the error made by applying a linear change for the surface temperature is reduced to about 1/3, in comparison to calculations with stepwise constant temperatures. 3.3.3 Implementation For the presented investigations, the resulting simplified model is formulated in the MathCad (Mathsoft 1997) environment and will be translated into Neutral Model Format (NMF) (Sahlin 1996). This format includes a model definition based on equations. A continuous component, which can be described by a system of differential-algebraic equations, can be modelled directly with its equations. The NMF-code is translated and the models thereby implemented in dynamic simulation programs such as TRNSYS (TRNSYS 2000) or IDA (Bring et al 1999). This provides a method of examining thermally activated building components and their interactions with the rest of the building. The resulting models can, in this way, be made available to a broader audience of researchers and designers.

3.4

The MEM modelling of counter flow elements

As described above, the MEM method shows how to combine a number of optimised RC networks to larger macro elements by setting up a system of differential equations for the heat carrier flows. The analysis and optimisation procedure of solid construction parts and the estimation of the RC network are handled in the preceding paper in this series (Schmidt and Jóhannesson 2002a). Firstly, a heat balance for an infinitesimally small element of every flow must be set up. In general, the heat balance of a system includes three major parts: the amount of heat stored in the system, the heat flow across the system’s boundary and the heat produced inside the system. d s dV ′ = − ∫ q dA′ + ∫ φ dV ′ dt V∫ A V

(3)

Storage = Transport + Source The time dependent storage term, which is neglected in this modelling, is thus represented on the left side, and the heat flow over the systems boundary plus the heat source term on the right. The heat flow q could be divided into one part for a convective transport q C for a heat transfer in combination with a mass flow, and one part for a diffusive / conductive transport q D for a heat transmission through a material.

0 = −∇q C − ∇q D + φ

(4)

An isotropic homogeneous system described in Cartesian co-ordinates (x, y and z) and implementing Fouriers law results in:


6

r q D = − λ ∇T

∂qC , y ∂qC , z  ∂q + 0 = − C , x + ∂y ∂z  ∂x

(5)  ∂ 2T ∂ 2T ∂ 2T    + λ ⋅  2 + 2 + 2  + φ ∂y ∂z    ∂x

(6)

The time dependent heat balance for the modelling in one dimension x, for example a circular pipe, consists for an infinitesimal channel element of two terms. The heat production is assumed to be zero. The changed content of the thermal energy of an infinitesimal channel element inside the heat carrier medium due to exchanged heat is the heat flux from the medium to the surroundings: u ⋅ ρ ⋅c ⋅ A⋅

∂T0 (x ) dx + ∫ q0 ( x) ⋅ dx = 0 ∂x P

(7)

convective transfer + lateral heat transfer = 0 The density of heat flow rate q0(x) over the boundary represents the heat transfer to the surrounding construction, as well as possible transport to other heat carrier medium flows.

3.5

Choice of time steps and stability problems

The time step used in dynamic calculations is chosen to give a good representation of the timely variation of the boundary conditions and to ensure the stability of the calculation results over time. For the boundary conditions, an interesting aspect for the choice of time step length is the application of the Nyqvist theorem. It states that the smallest period that contains information of a long term process is twice as long as the sampling period. If the simulation time step is set to be equal to the maximum time step, it can be stated that (Akander 2000):

τ Ny ≈ 2 ⋅ ∆t max

(8)

In other words, if fast processes or fast changing boundary conditions are modelled, the used time step has to be smaller. On the other hand, the choice of the length of the time step is dependent on the stability of the method applied to solve the involved differential equations. There is a variety of known and used methods. Hauser gives an overview of some methods commonly used in dynamic simulations on buildings (Hauser 1977). The three most commonly utilised methods to calculate time step tasks are: 1. Explicit method or forward Euler

Tt = f (Tt −1 )

2. Implicit method or backwards Euler

Tt = f (Tt )

3. Implicit Crank-Nicolson method

Tt =

1

2

( f (Tt −1 ) +

f (Tt ) )

All of these procedures have certain advantages and disadvantages. The first one, the forward Euler, is an explicit method to calculate time dependent problems. The new state of the system is calculated assuming that all system properties remain constant under the entire time step at the value of the end of the preceding step. Since it is an explicit method it is easy to implement, but the time step size is limited by the stability criterion. Compared to the explicit method, the implicit formulation, the backward difference method or backward Euler, has the important advantage of being unconditionally stable and has been chosen for the calculation of the network temperatures in the modelling presented here. The solution remains stable for all space and time intervals (Incropera and DeWitt 2000). Also, this method is based on the assumption that by temperatures in the network, generated heat flows remain constant during the entire time step and at the value at the end of the time step. The Crank-Nicolson method solves both the accuracy and stability problems, but there is the problem of “noise” or rapid oscillations in the solution when the used time step exceeds a critical value. The Crank-Nicolson scheme and its variants are probably the most frequently used procedures for time step problems (Reddy and Gartling 2001).


7

4

Verification of the MEM method for a case

The objective of this case study is to demonstrate how the MEM method can be applied to a simple case, and to verify the results obtained by calculations done using a MEM model. A concrete cylinder with an embedded pipe, a waterborne system with surrounding indoor air has been chosen. Here, it is chosen as an example with an analytical solution of the dynamic heat transfer matrix in frequency domain, because of the simple circular geometry. The analytical solution in frequency domain can be compared with time domain calculations of the MEM model. It is assumed that the concrete cylinder is surrounded by air at room temperature and the heat carrier at the inlet has a lower temperature than the room, i.e. a cooling case is regarded. Figure 2 shows the configuration of the system: Length L Ambient temperature Ti

Inlet temperature Tin

Emitted heat / cooling

and capacity flow m& c Figure 2: An embedded pipe with a mass flow of water inside a concrete cylinder.

As described in chapter 3.2, a number of processes and simplifications have to be applied to the physical system to model it according to the MEM approach. Figure 3 shows the important issues of the modelling. The system is divided into a limited number of macro elements. The part of the solid construction represented by a macro element is modelled via a RC network. The boundary temperatures between each macro element are assumed to vary linearly with the length of the construction.

Linear change of boundary temperature between macro elements

Real exponential profile of medium temperature

Optimised RC network

Macro element x

Figure 3: The MEM model of a pipe embedded in a concrete cylinder.


8

For the heat carrier medium flow, a differential equation must be set up. As a result, an exponential temperature profile of the flow can be calculated. For further modelling, one macro element from the mentioned case is described in more detail: T01(x)

T1

R1

Rhc1 Rw

T0 R4

Flow

T2

R2

C1 T3

R3

T4 RSi

C2 R5

Ti

C3

Figure 4: Detailed model of a macro element with one flow and a Π-T-link RC network.

For this case study, the heat carrier is assumed to be water. As shown in Figure 4, the resulting network consists of resistance-stars. A star of resistances connecting the points T1, T3 and T01(x) can be transformed to an equivalent triangle network, as shown in Figure 5. T1

T1 Rt1

Rw+Rhc1 T01(x)

R1 T0

Rt3 T01(x)

R4

Rt2

T3

T3

Figure 5: Star-triangle transformation for resistor-networks

Calculation of the resistances in the triangle formation according to Rt1 = (Rw + Rhc1 ) + R1 +

(Rw + Rhc1 ) ⋅ R1 ,

(9)

(Rw + Rhc1 ) ⋅ R4

( 10 )

Rt 2 = (Rw + Rhc1 ) + R4 +

and

4.1

Rt 3 = R1 + R4 +

R4

R1

R1 ⋅ R4

(Rw + Rhc1 )

( 11 )

Modelling the heat carrier flow

The modelling along the pipe is done via an energy balance for each element. The heat transfer balances can be set up in a differential equation: Convection in flow 1 u1 ρ1c1 A1

Lateral heat flow from flow 1 to network

1 1 ∂T 01( x) dx = ⋅ (T 1( x) − T 01( x) ) dx + ⋅ (T 3( x) − T 01( x) ) dx ∂x Rt1 Rt 2


( 12 )

9

The temperatures T1 and T3 are assumed to vary linearly within the macro elements: T 1( x) = C1 ⋅ x + D1

( 13 )

T 3( x) = C2 ⋅ x + D2

( 14 )

The differential equation can be formulated to: ∂T 01( x) 1 + ∂x u1 ρ 1 c1 A1

 1 1  1    R + R  ⋅ T 01( x) = + u ρ c A t2  1 1 1 1  t1

 C1 C 2  1    R + R ⋅ x + u ρ c A t2  1 1 1 1  t1

 D1 D 2    R +R  t2   t1

( 15 )

To improve readability, the constant factors X1, X2 and X3 have been introduced in the differential equation described above. So this equation is reformulated as: ∂T 01( x) + X 1 ⋅ T 01( x) = + X 3 ⋅ x + X 2 ∂x

( 16 )

The solution to the differential equation for the flow temperature is an exponential function. X3 X 2 X 3  − X 1⋅ x 1    + + + X 3⋅ x T 01( x) =  Tin − ⋅e X2− X1 X 1 X 12  X1   

4.2

( 17 )

Modelling the solid construction

For this circular problem, an analytical solution of the heat transfer matrix can be found. The thermal properties, and the dynamic admittance and transmittance were calculated for radial heat conduction (Carslaw and Jaeger 1959). In analysing the diagrams for the admittance and transmittance, three break frequencies can be found within the relevant frequency range. To establish an equivalent RC network at least three capacitances should be used. A Π-network parallel to a T-network, (see Figure 4), has been found to give an acceptable representation of the dynamic properties. The results for an optimised network compared to the analytical solution are given in Figure 6. 1 day

1 hour

6 Analytical transmittance

[W/m²K]

RC model transmittance Analytical admittance RC model admittance 1/Rtot

1 1E-6

1E-5

1E-4

1E-3 [rad/s] 1E-2

Angular frequency Figure 6: Amplitudes for the admittances and transmittances of the analytical solution and the optimised Π-T link RC network based on the geometric data in Table 1.


10

The quality of the optimisation of the chosen RC network configuration is shown in a Bode diagram, see Figure 6. It is demonstrated that the RC model is in accordance with the analytical solution for processes with time periods longer than 1 hour. The optimisation has been made for the transmittance and the admittance from the cylinder to the heat carrier flow, as shown in Figure 6. The admittance to the outside environment is not focussed on here. The heat carrier flow interacts with the RC network via the heat flows to the linear varying network node temperatures T1 and T3. This connection is calculated via the representative heat flows ∆Q&1 and ∆Q& 2 . ∆Q&1

∆Q& 2

R2

T1

R3

T2

C2

C1

T4 RSi

Ti

T3 R5 C3

Figure 7: Remaining part of the Π-T network to be calculated (compare Figure 4).

The heat flows from the heat carrier flow ∆Q&1 and ∆Q& 2 can be estimated explicitly for every macro element assuming a linear variation of the node temperatures T1 and T3 along the flow. The flow temperature is T01(x), connected to the RC network via a resistance of the pipe wall RW and the resistance of the convective heat transfer Rhc1. These are represented together with the resistances R1 and R4 by Rt1 and Rt1. ∆Q&1 =

∆Q& 2 =

l

1 Rt 1

∫ (T 01(x ) − C1 ⋅ x + D1 )dx

1 Rt 2

∫ (T 01(x ) − C 2 ⋅ x + D2 )dx

( 18 )

0

l

( 19 )

0

Since T01(x) is a simple function (see equation ( 17 )) it is possible to solve the integrals analytically with:

m& ⋅ c ∆Q& flow1 = ⋅ (T 01(l ) − T 01(0) ) l

( 20 )

and

∆Q& flow1 + ∆Q&1 + ∆Q& 2 = 0

( 21 )

An effective way of calculating the networks, using a matrix solution as an implicit solving method, has been applied here. Again, as for the flows, energy or heat balances for all nodes have to be set up according to the configuration shown in Figure 7. Here, ∆t is the chosen time step for the calculation in the time domain. The heat balances for the nodes become: node 1:

C T2 − T1 + (T1,t −1 − T1 ) ⋅ 1 + ∆Q&1 = 0 ∆t R2

( 22 )

 1 C1  T2 C  + − T1  + = −T1,t −1 ⋅ 1 − ∆Q&1 R ∆ t R ∆t 2  2 

( 23 )


11

node 2:

node 3:

node 4:

T1 − T2 T4 − T2 C + + (T2,t −1 − T2 ) ⋅ 2 = 0 ∆t R2 R3

( 24 )

 1 1 C 2  T4 C T1  + = −T2,t −1 ⋅ 2 − T2  + + ∆t R2 R R ∆ t R 3 3   2

( 25 )

C T4 − T3 + (T3,t −1 − T3 ) ⋅ 3 + ∆Q& 2 = 0 ∆t R5

( 26 )

 1 C 3  T4 C + = −T3,t −1 ⋅ 3 − ∆Q& 2 − T3  +  ∆t  R5 ∆t  R5

( 27 )

T2 − T4 T3 − T4 Ti − T4 + + =0 R3 R5 RSi

( 28 )

 1 T2 T3 1 1 + − T4  + + R3 R5  R3 R5 R Si

( 29 )

 T  = − i RSi 

Combining all balance equations into a matrix solution results in:  1 C1 − R − ∆t 2  1   R2   0   0  

1 R2 1 1 C2 − − − R2 R3 ∆t 0 1 R3

0 0 −

1 C3 − R5 ∆t 1 R5

 C1  &    − T1,t −1 ⋅ ∆t − ∆Q1   T   1 C2   1  −T  ⋅   2 , 1 t −  T2 R3 t ∆     = ⋅ C   1  T3  − T3,t −1 ⋅ 3 − ∆Q& 2   t ∆ R5  T4    Ti 1 1 1    − − − −    R Si R3 R5 R Si  0

( 30 )

The new temperatures T1 to T4 can be calculated by matrix inversion.

4.3

Results from the model verification

For verification of the suggested MEM model, a comparison has been made between the results in the frequency domain of an analytical solution of the problem and of the MEM model. For this embedded pipe in a concrete cylinder case, geometrical values that are somewhat corresponding to a real system have been chosen. Even though this case has been chosen on the basis of the possible analytical solution of the heat transfer matrix, the characteristic quantities originate from a real building project, which includes a thermally activated floor slab (Baumgartner 2002).


12

Table 1: Values for the embedded pipe calculation. Description

Variable

Value

Unit

Inner radius of pipe

r2

6.5

mm

Outer radius of pipe

r1

8.5

mm

Outer radius of concrete cylinder

rn

150

mm

Heat conductivity of concrete

λcon

1.8

Density of concrete

ρcon

2400

Specific heat capacity of concrete

ccon

1100

J

Heat conductivity of pipe

λp

0.22

W

Density of pipe

ρp

1350

Specific heat capacity of pipe

cp

1000

Pipe length

L

73.6

Mass flow in the piping

m&

100

kg

Resulting velocity of water flow in pipe

u1

0.21

m

Resulting convective heat transfer resistance

Rhc1

0.138

K ⋅m

Surface heat transfer resistance

Rsi

0.1

K ⋅m

Surrounding room temperature

Ti

20

°C

12

-

6.13

m

Number of macro elements

l

Resulting macro element length

W

m⋅K

kg

kg ⋅ K m⋅K

kg

J

m3

m3

kg ⋅ K

m h s W W

Table 2: Values of the optimised RC network according to Figure 6 for the regarded case (see Figure 4). Resistance

K ⋅m

W

Capacitance

J

m⋅K

R1

0.9368

C1

2.6261 . 103

R2

0.8257

C2

1.9656 . 104

R3

0.5

C3

4.8785 . 104

R4

0.4479

R5

0.1106

4.3.1 Step response to estimate steady state conditions For an easy to check process, the results of the steady state process must be controlled. As an example, the results of a temperature step from 20°C to 10°C, at the beginning of the process, are shown in the time domain. The outlet temperature and emitted cooling power response are shown for the configuration given in Table 1.


13

20

12

[°C]

[W/m²]

Cooling power

18

10

16

8

14

6 Outlet temperature

12

4

Inlet temperature 10

2

8 0

5

10

15

20

0 25 [h] 30

Time Figure 8: In- / outlet temperature and emitted cooling power for the embedded pipe system. Step of –10 K at inlet temperature at time 0.

With an outlet temperature of 16 °C after 100 hours, the heat gain for the fluid can be calculated as follows: ∆Q& flow1 = m& ⋅ c ⋅ (Toutlet − Tinlet ) = 695 W

( 31 )

This should be equal to the heat flow at the surface of the system: ∆Q& surface =

no of MEMs

∑ n =1

1 (Ti − T 4 n ) ⋅ 2πrn ⋅ L = 695 W Rse

( 32 )

The analytical steady state solution also gives an outlet temperature of 16°C (see Figure 9): ∆Q& flow1,analytical = m& ⋅ c ⋅ (Toutlet − Tinlet ) = 695 W

( 33 )

4.3.2 Frequency domain

Dynamic properties and performance in the frequency domain can be shown in a Bode diagram. For the case given in Table 1, the analytical solution and the analytical RC model solution in frequency domain and the time domain solution for MEM model with harmonic boundaries are compared.


14

1 day

1 hour 1 π/4

0,5 [-/-] J

J

J

J J

F F

F

F

F F F

J

F F F J

0

J J

0,2 1E-6

1E-5

J

-1π/4 1E-3 [rad/s] 1E-2

1E-4

Angular frequency Amplitudes

J

Phase shifts Analytical solution

Analytical solution

RC model analytical

RC model analytical

MEM Model

F

MEM Model MEM Model corrected

Figure 9: Results from the case. The analytical solution compared with the analytical solution with RC networks and with the MEM mode solution in the frequency domain.

The MEM model complies very well with the analytical solution for low frequencies, but the phase shift deviates for high frequencies, i.e. frequencies higher than approximately 2.10-4 rad/s, corresponding to time periods of about 9 hours. For moderate frequencies, the agreement between the analytical model and the MEM model is satisfying. The phase lag complies quite well with the one from the analytical solution and very well with the analytical solution of the optimised RC network. For frequencies higher than 2.10-5 rad/s, the phase lag of the MEM model and the RC analytical model are a little less than for the analytical solution and the amplitude of the MEM model is a bit higher than for the analytical model. At a frequency of approximately 2.10-4 rad/s, the phase of the MEM model starts to deviate from the analytical and RC model solution. If the phase shift of the MEM model solution is corrected taking the flow through time of the heat carrier through the construction into account, even the phase shift complies well with the analytical RC model solution and also with the analytical model. The amplitude of the MEM model complies well with the analytical RC model. By comparing Figure 9 and Figure 6 we see that the inaccuracy in the modelling for the cross section starts increasing at a higher frequency for the total solution than for the RC model for the construction. The flowthrough time of the heat carrier medium is, for the given velocity, 350 seconds or 0.1 hours. The time dependent model is built on the restriction of a fully developed temperature profile for the fluid in every time step and based on the assumption of a quasi steady state. For shorter periods of time or those in the same order of magnitude as the flow through time, the MEM modelling is no longer valid, and the results deviate from the analytical solution and from the real process. The amplitude is modelled correctly, the phase shift has to be corrected by taking the flow through time of the heat carrier into account. If faster processes are going to be regarded with varying flow directions and mass flow conditions, a modification of the model must be implemented and “packages” of heat carrier have to be traced in the calculation. This represents a further development of the modelling.


15

5

Application of the MEM method at a co-axial counter flow element

This part of the study demonstrates how the MEM method can be applied to thermally activated components. The chosen system is a coaxial flow pipe, a so-called C-pipe in a borehole. The configuration chosen has an analytical solution for the dynamic heat transfer in frequency domain. More complicated geometries can be analysed with FEM or FDM models.

x

adiabatic boundary

q2(x)

q1(x)

dx

L

Figure 10: Heat storage equipped with a C-pipe.

Figure 11: Heat storage as a counter flow element.

The model is made to investigate the dynamic properties of a heat store in the ground by means of a bore hole (Schmidt and Jóhannesson 2002a). Water, as the chosen heat carrier, flows downward in the inner pipe and back upward in the outer one, or the other way around. Some insulation may be applied between the flows (see Figure 10 and Figure 11). The surrounding area is represented by a cylindrical piece of soil, which is modelled via an optimised RC network construction. For further modelling, one macro element from the mentioned counter flow element is described in more detail: R11

T01(x) Rhc1i Rhc1o Rhc2i

T0

Riso

T02(x)

R12

T1

Flow

T5

C1

Rw

T2 R21

R22 C2

C5

Flow Figure 12: Detailed model of a macro element with two flows and a half 5-node RC network to model a halfinfinite body.

For this case study, the heat carrier is assumed to be water. Direct heat exchange between the pipe wall due to radiation does not take place. If a gas is used, which allows radiative heat exchange directly between two pipe walls, it can be modelled with the indicated resistance (dashed light grey line). A triangle-star transformation is then necessary to solve the heat transport equations for the Rhc1o, Rhc1i and the indicated resistance (see Schmidt and Jóhannesson 2001). For this heat storage model, an adiabatic boundary is considered. If other boundary conditions, a varying temperature or heat flow, are to be considered, the model is expanded, e.g. to a full 5-node RC network (indicated in light grey). As shown, all involved processes and effects have been taken into consideration in the modelling.

5.1

Modelling the heat carrier flows

The modelling along the pipe is done via an energy balance for each element. The heat transfer balances can be set up under the assumption that there is a heat exchange between the two flows as a set of differential equations. Heat balances for the flows:


16

Flow 1: Convection in flow 1 Lateral heat transfer between flows u1 ρ1c1 A1

∂T 01( x) 1 dx + ⋅ (T 02( x) − T 01( x) )dx + Riso ∂x

1 1 ⋅ (T 1( x) − T 01( x) )dx + ⋅ (T 2( x) − T 01( x) )dx = 0 Rt1 Rt 2

( 34 )

Convection/diffusion from flow 1 to network Flow 2: Convection in flow 2 − u 2 ρ 2 c2 A2

Convection/diffusion between flows

∂T 02( x) 1 dx + ⋅ (T 01( x) − T 02( x) )dx = 0 Riso ∂x

( 35 )

Where Riso is :

Riso =

r + d iso 1 1 1 + ln 2 + hc1i ⋅ 2(r2 + d iso ) 2λiso r2 hc 2i ⋅ 2r2

( 36 )

π

The boundary temperatures are assumed to vary linearly within the macro elements: T 1( x) = C1 ⋅ x + D1

( 37 )

T 2( x) = C2 ⋅ x + D2

( 38 )

The two heat balances can be reformulated in a matrix:

( )

( )

( )

1 1 1 R R  ∂T 01( x)   R + +  ∂x   u1 ρ1c1 A1 u1 ρ1c1 A1 u1 ρ1c1 A1  ∂T 02( x)  +  1 R    −  ∂x   u 2 ρ 2 c2 A2 t1

t2

iso

( ) iso

( )

( )

( )

( )

  u1 ρ1c1 A1  T 01( x)  ⋅ = 1  T 02( x) R  u 2 ρ 2 c2 A2 

−

1 Riso

( ) iso

( )

1 1 C2  D2   R1 D1  R1 C1 R R + +  ⋅x+    u1 ρ1c1 A1 u1 ρ1c1 A1   u1 ρ1c1 A1 u1 ρ1c1 A1  0 0     t1

t2

t1

t2

( 39 )

To improve the readability of the formulas in the following, two-dimensional matrices are denoted by bold capital letters, e.g. A, and one-dimensional matrices (the vectors) by small bold letters, e.g. t (according to Bronstein et al 1991). ∂t + A⋅t = b⋅ x +c ∂x

( 40 ) with

T 01( x)  t=  T 02( x)

( 41 )

This integration-problem (equation ( 39 )) can be solved via the following formulation (see also Bergqvist et al 1980):


17

A = S ⋅ D ⋅ S −1

( 42 )

where, D = 2 x 2 matrix, where the diagonal elements λ1 and λ2 are the eigenvalues of matrix A. The other elements are zero. and,

S = also a 2 x 2 matrix, where the columns are the eigenvectors of matrix A. ∂t + S ⋅ D ⋅ S −1 ⋅ t = b ⋅ x + c ∂x

(

( 43 )

)

∂ −1 S ⋅ t + D ⋅ S −1 ⋅ t = S −1 ⋅ (b ⋅ x + c ) ∂x

( 44 )

The resulting solution for this integration problem is as follows: e λ1⋅L   0

L −1 T 01(L ) −1 T 01(0 ) ⋅ (S ) ⋅  − (S ) ⋅  = ∫ e D⋅x S −1 ⋅ b ⋅ x + e D⋅x S −1 ⋅ c dx   T 02(L ) T 02(0 ) 0 

0 

e

(

λ2 ⋅L 

 λ ⋅L λ1 ⋅ L − 1 1 + 2 e ⋅ λ1 2 λ1 =  0  

(

)

 λ1 e λ ⋅L − 1  0  1

1

1

λ1

(e

0 λ1 ⋅ L

( )

e λ ⋅L 2

( 45 )

( )

 1 C2   R1 C1  R + − 1   ⋅S ⋅ u ρ c A u ρ c A  + 1 1 1 1  1 1 1 1 λ ⋅ L −1 1 ⋅ 2 2 + 2  0    λ2 λ2  0

1

)

t1

( )

t2

( )

1 1  −1  R D1 + R D2   ⋅ S ⋅  u1 ρ1c1 A1 u1 ρ1c1 A1  − 1  0  

)

t1

t2

( 46 )

With the introduction of the boundary conditions that T 02(0) is given and T 01(L ) = T 02(L ) , the solution, the temperature at the outlet T 01(0) , can be calculated via a matrix formulation. The result is then obtained via a matrix inversion, which can be easily and quickly calculated in computer programmes.

5.2

Modelling the solid construction

For the considered problem, an analytical solution can be found, as presented in (Schmidt and Jóhannesson 2002a). The thermal properties of the construction, the dynamic admittances of the whole perimeter, were calculated as a solution for radial heat conduction in cylinders (Carslaw and Jaeger 1959). When the frequency response has been established, an optimised RC network can be developed. To derive a model for a half-infinite body with good agreement for even high and low frequencies, a “half 5 node” network is chosen (Akander 2000). Starting from two main optimisation frequencies (period times of one year and one hour), each of the two parallel T-chains (two resistances and one capacitance) are estimated, in order to provide a good agreement, for the high frequency and the other one for the low frequency. C5 is given as the remaining part of the total heat capacity (see Figure 12). The solution is then improved by a number of iteration steps. The deviation between the optimised RC network model and the analytical solution is estimated for a number of relevant frequencies. This optimisation procedure for so-called optimised RC networks has been taken from Akander (Akander 2000). The best suitable configuration or set of resistances and capacitances with the least calculated error according to the following equation is chosen. n

∑ Err =

i

Yanalytic (ω i ) − YRC _ configuration (ω i ) Yanalytic (ω i ) nω

( 47 )

i

Where n is the number of frequencies. For the chosen configuration, a control is done in the frequency domain, as shown in Figure 13.


18

1 year

1 hour

1E+4 [W/m2K] 1E+3 1E+2 1E+1 1E+0 1E-1 1E-2 1E-3 single mass solution

1E-4 1E-5

infinite body solution

1E-6

RC network solution

1E-7

analytical solution

1E-8 [rad/s]

Angular frequency

Figure 13: Amplitude of the admittance for a borehole wall in soil giving the analytical and the RC network solution and the asymptotes for single mass and an adiabatic external boundary.

The heat carrier flow interacts with the RC network via the heat flows to the linear varying network node temperatures T1 and T2. This connection is calculated via the representative heat flows ∆Q1 and ∆Q2 as shown in Figure 14. ∆Q&1

∆Q& 2

R12

T1

T5

C1 T2 R22 C2

C5

Figure 14: Remaining part of the half 5 node network to be calculated (compare Figure 12).

The heat flows from the heat carrier flow 1 to the network ∆Q1 and ∆Q2 can be estimated explicitly for every macro element: ∆Q&1 =

∆Q& 2 =

1 Rt 1

1 Rt 2

l

∫ (T 01(x ) − C1 ⋅ x + D1 )dx

( 48 )

0

l

∫ (T 01(x ) − C 2 ⋅ x + D2 )dx

( 49 )

0

These integrals are difficult to solve analytically, but can be estimated by summing up the temperature difference between the heat carrier flow and the linearly changing network temperature for a number of steps. The heat balance for every macro element has to be calculated:


19

with:

∆Q& flow1 + ∆Q& flow 2 + ∆Q&1 + ∆Q& 2 = 0

( 50 )

∆Q& flow1 = m& ⋅ c ⋅ (T 01(l ) − T 01(0) )

( 51 )

∆Q& flow 2 = m& ⋅ c ⋅ (T 02(l ) − T 02(0) )

( 52 )

Similar to the case presented in chapter 4.2, the network temperatures T1, T2 and T5 can be found by a matrix inversion.  1 C − 1 − R  12 ∆t  0   1  R12 

5.3

0 −

C 1 − 2 R22 ∆t 1 R22

 1 C1  &    − T1,t −1 ⋅ ∆t − ∆Q1  R12  T1    C2 1  ⋅ T  = − T & 2 Q ⋅ − ∆ − 2 2 , t 1     R22 ∆t     C5  C 5  T5   − T 1 1 ⋅ − − − 5,t −1   t ∆   R12 R22 ∆t 

( 53 )

Demonstration of the MEM model applications for time varying processes

As mentioned above, an analysis of the regarded counter flow element has been published (Schmidt and Jóhannesson 2002a). An analytical solution for the problem can be obtained in the frequency domain. The MEM model is used in the time domain to demonstrate possible applications for this model. Table 3: Values for the counter flow duct calculation. Description

Variable

Value

Unit

Inner radius of inner pipe

r2

16.3

mm

Inner radius of outer pipe

r1

61.25

mm

Thickness of pipe wall

d

2

mm

Heat conductivity of pipe walls

λp

0.6

W

Heat conductivity of insulation

λiso

0.04 (0.6)

W

Heat conductivity of ground

λgr

2

W

Specific heat capacity of ground

cgr

1000

J

Density of ground

ρgr

2000

Total thickness of insulation between pipes

di

13.2

mm

Pipe length, bore hole depth

L

170

m

Water flow in the piping

V&

0.5

l

Resulting velocity of water flow in outer pipe

u1

0.06

m

Resulting velocity of water flow in inner pipe

u2

0.59

m

m⋅K m⋅ K m⋅ K kg ⋅ K

kg

m3

s s s

In the following, two cases of an application are shown. The chosen cases have non-linear input boundaries, whereas most other models are not able to handle such non-linear model processes. The use of RC networks makes it possible in this case. The major dynamics are modelled correctly. Both cases have been calculated with twelve macro elements modelling the entire system.


20

5.3.1 Application with a varying inlet temperature

Changing the inlet temperature of the system is a typical application. In this way, it can be shown how heat is stored in the storage mass. To demonstrate the effect of the heat exchange between the two adjacent flows, the conductivity of the insulation material is set to 0.6 W/mK. This value corresponds to the value for piping material, showing a situation in which no insulation is applied between the flows, as done in the case of normal so-called U-pipes.

Table 4: Data for the case with varying inlet temperature, cross section and other data according to Table 3. Description

Value

Unit

Time step

0:15

h

Number of time steps for the calculation

50

-


12

-

Total length of construction

170

m

- For time 0:15h to 6:00h

5

°C

- For time 6:15h to 12:30h

10

°C

Inlet temperature:

Inlet at

Outer pipe / flow 1

Initial temperature of all nodes

°C

10

The temperature profiles are calculated in quasi steady state. An exponential varying temperature is calculated for each time step, depending on the temperature of the adjacent mass nodes, here, the network temperature. These temperatures vary linearly between the macro element centres, as indicated in Figure 3. 11 Flow 1 at 0h

[°C]

Flow 2 at 0h

10

Flow 1 at 0:15h

9

Flow 2 at 0:15h Flow 1 at 6h

8

Flow 2 at 6h Flow 1 at 6:15h

7

Flow 2 at 6:15h

6

5 0

20

40

60

80

100

120

Length of construction

140 160 [m]

Figure 15: Temperature profiles at different times for the showcase with varying inlet temperature. Inlet at the outer pipe / flow 1, indicated in solid lines. The profile of the inner pipe / flow 2 in dashed lines. In the beginning (0:15 h), the profile shows the typical exponential shape, where the boundary temperatures are constant. After a while, the environment is cooled down at the beginning of the construction and the temperature profile evens out. The return flow in the inner pipe, flow 2, is affected by the one in the outer pipe. What’s remarkable is the profile right after the change of inlet temperature (at 6:15 h). The inlet environment is cooled due to the preceding flow and now, the environment cools the new flow. But, after about 80 m, the environment


21

is warmer than the flow and now heats the heat carrier. The charging and discharging of the heat storage is demonstrated. This can be seen more clearly when observing the development of the node temperatures in time. The macro element 1 extends itself from 0 to 14.2 m, macro element 6 from 70.8 to 85 m, and the last one, macro element 12, from 155.8 to 170 m. T1 at macro element 1

10 [°C]

T1 at macro element 6 T1 at macro element 12

9


8


7

T5 at macro element 6

6

5 0


2

4

6

8

10 [h] 12

Time Figure 16: Development of node temperatures of the RC networks in the macro elements. Node 1 is delineated by dashed lines, node 2 by solid ones and node 5 by dotted ones. The node temperatures of macro element 1 are in dark black, of macro element 6 in grey and macro element 12 in light grey. Again, the charging and discharging of the heat storage is clearly shown. Only the fast responding node 2’s take part in this process. This process is too slow to influence the other nodes, i.e. node 1’s for medium term processes and node 5’s for long term processes. Most changes happen close to the inlet, in macro element 1. The profiles in Figure 15 show the same behaviour. 5.3.2 Application with reverse flow

An important application for this type of model is for the study of reverse flow processes, where the direction of the flow can be changed. This is relevant for fluid ducts in a heat storage to store heat and cold at opposite ends of the storage, as exemplified in Table 5, but also for floor heating systems, that are operated with reversed flow to get a more favourable temperature distribution on the floor surface.


22

Table 5: Data for the case with reverse flow pattern, cross section and other data, according to Table 3. Description

Value

Unit

Time step

0:15

h

Number of time steps for the calculation

200

-


12

-

Total length of construction

170

m

5

°C

Inlet temperature: Inlet at: - For time 0 to 12:15 h

Outer pipe / flow 1

- For time steps 12:30 to 37:15 h

Inner pipe / flow 2

- For time steps 37:30 to 50 h

Outer pipe / flow 1

Initial temperature of all nodes

10

°C

The temperatures are similar to the previous case calculated in quasi steady state and shown in Figure 17 as profiles at different times. 11 Flow 1 at 0:15h

[°C]

Flow 2 at 0:15h

10

Flow 1 at 12:15h

9

Flow 2 at 12:15h Flow 1 at 12:30h

8

Flow 2 at 12:30h Flow 1 at 37:15h

7

Flow 2 at 37:15h

6

5 0

20

40

60

80

100

120

Length of construction

140 160 [m]

Figure 17: Temperature profiles at different times for the case with reverse flow pattern. The profiles in outer pipe / flow 1 are indicated by solid lines, the profiles of the inner pipe / flow 2 are indicated by dashed lines. The temperature profile from time 0 to 12:15 h is equal to the one in Figure 15. However, with the changing of the flow direction from inlet at the outer pipe 1 to the inner pipe 2, the profile changes dramatically. Similar to the varying inlet temperature, the environment at the beginning of the construction is cooled down. The inlet is now at the inner pipe 2 (dashed profiles). A minimal heat transfer between the inner and outer flow takes place and the inner fluid is heated. When the fluid leaves the inner pipe and turns to the outer one it is heated by the environment. Later, before exiting the construction, it is cooled again due to the coolness stored in the solid parts during the preceding time. After a time, the reverse flow pattern becomes similar to that of the start profile. The charging and discharging process is shown in more detail in Figure 18.


23


10 [°C]


9


8


7

T5 at macro element 6 6

5 0


5

10

15

20

25

30

35

40

Time

45

50 [h]

Figure 18: Development of node temperatures of the RC networks in the macro elements. Node 1 is in dashed lines, node 2 in solid ones and node 5 in dotted ones. The temperatures of nodes of macro element 1 are in dark black, of macro element 6 in grey and the nodes of macro element 12 are in light grey. Like in Figure 16, the charging and discharging of the heat storage is displayed and again, mainly the fast responding node 2’s take part in this process. The process is too fast for the others to contribute. Because of the change in the inlet, both macro element 1 and macro element 12 node temperatures show the largest amplitudes.

6 Conclusions The presented paper offers a description of how the macro element modelling method can be applied to thermally activated components. The aim of the proposed method is to obtain implementable and reliable models for a thermally activated construction. Nonetheless, the major dynamic characteristics of the construction should be modelled as correctly as possible. The MEM method requires only a few macro elements for the setting up a model, even for large constructions. This means fewer nodes must be calculated compared to, for example, direct coupling of finite element (FEM) or finite difference (FDM) models. The approach of the linear variation of the mass node temperatures makes it possible to increase the element size. For validation, the MEM model has been checked against an analytically derived solution in the frequency domain. The MEM model deviates from the analytical solution for high frequencies, due to the assumption of quasi steady state calculation of the heat carrier temperature profile. The modelling error is larger than what can be expected from the modelling error for the cross section. The solution deviates from the analytical solution when the flow-through time of the fluid above or in the same order of magnitude as the analysed time period. If the results from the MEM model are corrected by taking the actual flow through time into account the MEM model complies well with an analytical solution. Most of the thermally activated components have shorter flowthrough times than the regarded cases. For example, for an airborne double air gap construction the flow through time is about 9 seconds. Most dynamic simulation environments utilise time steps of about 1 hour, so the MEM model limitation of quasi steady state calculations of the flow temperature profiles is no hindrance for its use in commonly used simulation programmes. Yet to expand its validity, the MEM model should, in future developments, be improved by a module which traces “packages” of heat carrier fluid as they travel through the construction. Then, a modelling of the stagnant heat carrier medium, with its heat capacity, is also possible. The cases shown demonstrate the capabilities of MEM models to calculate even extreme time variant processes, such as switching the flow direction or steps in the inlet temperature or flow conditions. These are advantages over other known simplified models of thermally activated constructions in dynamic simulations. It has been demonstrated that the MEM method is generally suitable for modelling the dynamic behaviour of combined systems with a heat carrier flow and solid construction parts with substantial heat storage capacity.


24

7

Further work

The remaining step of implementation into a common simulation environment will be conducted in future work, in which the MEM model will be programmed in the equation based NMF-code (Neutral Model Format) (Sahlin 1996). In this way, the calculation in the time domain and the influence of the examined construction, the thermal heat storage, on the entire building system can be evaluated in a commonly used simulation environment. Moreover, the MEM method will be applied to other thermally activated systems. In Europe, embedded pipes in concrete slab systems are becoming more and more popular as heating and cooling systems. Based on an analysis undertaken in the frequency domain, utilising a triangular FEM formulation (Weber, Koschenz and Johannesson 2004 and Weber and Johannesson 2004), discrete RC networks for a cross section of a floor slab can be derived and the MEM method applied for modelling the temperature distribution in the heat carrier flows. The results of the MEM model are to be verified with measurements from ongoing monitoring projects (Baumgartner 2002, Weber and Jóhannesson 2004, Schmidt 2002). A more strategic distribution of element sizes along the flow paths could further improve the accuracy of a given number of elements. If fast processes or low velocities of the heat carrier are to be modelled with the MEM model, the carrier must be expanded by a module which traces “packages” of heat carrier fluid as they make their way through the construction. This further development is to be outlined in a future paper.

8

List of symbols

Notation

Dimension

Meaning

ω

rad/s

Cycle frequency

ρ

kg/m³

Density

φ

3

W/m

∆

-

λ

W/mK

A

-

Factor

B

-

Factor

C

-

Factor

D

-

Density of heat production Difference Heat conductivity

Factor ²

A

m

c

J/kgK

Specific heat capacity

C

J/Km

Capacitance

d

m

Diameter

Err

-

Error, model deviation

i

-

Imaginary part

i

-

Counting variable

l

m

Length of macro element

L

m

Length, bore hole depth

m

kg

Mass

P

m

Perimeter

q

W/m²

Q&

W

Specific heat flow

r

m

Radius

s

J/m3

Area

Specific heat flow per area

Specific storage capacity


25

R

Km/W

t

s

Time

T

K

Temperature

u

m/s

V

3

m

Volume

x

m

Co-ordinate of place

y

m

Co-ordinate of place

Y

W/Km²

z

m

Resistance

Velocity

Admittance Co-ordinate of place

Indices Notation

c con

Meaning

Convective part Concrete

d

Diffusive part

gr

Ground

hc

Convective heat transfer by medium flow

i

Indoor

in

Inlet

iso

Insulation

max

Maximum

Ny

Nyqvist

p

Pipe

Si

Surface inside

W

Wall

Superscript, “~”, indicates a quantity in harmonic oscillation.

9

References

Akander J. (1995): Efficient Modelling of Energy Flow in Building Components. Tekn. Lic. Thesis, KTH Department of Building Science, Stockholm, Sweden. Akander J. (2000): The ORC Method – Effective Modelling of thermal Performance of Multilayer Building Components, bulletin No 180, KTH – Division of Building Technology, Stockholm, Sweden. Baumgartner T. (2002): Thermoaktive Betondecke. Neubau Büro- und Wohngebäude „Weissadlergasse“ Helvetia Versicherungen. Final report. Hevetia Versicherungen Frankfurt, Germany. Beuken D.L. (1936): Wärmeverluste bei periodisch betriebenen Öfen, PhD Thesis, Freiburg, Germany. Bring A., Sahlin P., Voulle M. (1999): Models for Building Indoor Climate and Energy Simulation. IEA SHC Task 22 Building Energy Analysis Tool Report. Bronstein I. N., Semendjajew K. A. (1991): Taschenbuch der Mathematik. 25. Edition, B.G. Teubner Verlagsgesellschaft, Leipzig, Germany. Caccavelli D., Bedouani B. (1995): Thermal analysis of a slab on ground heating floor in buildings – numerical approach. MARC European user meeting. October 18-19, 1995. Düsseldorf, Germany. Caccavelli D., Bedouani B. (1998): Modélisation et validation expérimentale du comportement thermique d’un plancher chauffant sur terre-plein. IBSPA France 98. December 1998. Sophia-Antipolis, France.


26

Caccavelli D., Mounajed G. (1995): Coupled thermo-mechanical finite element analysis for heating inforced slab in buildings – simulation and comparison with experimental results. MARC European user meeting. October 18-19, 1995. Düsseldorf, Germany. Carslaw H.S., Jaeger J.C. (1959): Conduction of Heat in Solids, Oxford University Press, London, England. Claesson J. (2003): Dynamic thermal networks: a methodology to account for time-dependent heat conduction. Proceedings of the 2nd International Conference on Building Physics, September 14-18, 2003, Leuven, Belgium. Evans B.L., Klein S.A., Duffie J.A. (1985): Design Method for Active –Passive Hybrid Space Heating Systems. Solar Energy, Vol. 35, No. 2, pp. 189-197. Fort K. (1989): Dynamisches Verhalten von Fussbodenheizungen, PhD Thesis, Diss. Nr. 8893, Eidgenössische Technische Hochschule Zürich, Switzerland. Gertis K. (1982): Solarenergienutzung-mit passiven und aktiven Maßnahmen. HLH, Heizung Lüftung/Klima Haustechnik. Vol. 33, No. 5, pp. 169-173. Hauser G. (1977): Rechnerische Vorherbestimmung des Wärmeverhaltens großer Bauten. PhD Thesis. University of Stuttgart, Germany. Incropera F., DeWitt D. (2000): Fundamentals of Heat and Mass Transfer. Updated 4th edition. Wiley publisher, New York, USA. Jóhannesson G. (1981), Active Heat Capacity, Models and Parameters for the Thermal Performance of Buildings, report TVBH – 1003, Lund, Sweden. Jóhannesson G. (1984): Lågenergihus-nuläge och forskningsbehov. Report BFR R31:1984, Stockholm, Sweden. Jørgensen O.C (1984): Evaluation and Development of Thermal Analysis Models within the IEA Solar Heating and Cooling Programme, Task VIII. Passive and Hybrid Solar Low Energy Buildings. EUR Report. Commission of the European Communities. Koschenz M., Dorer V. (1999): Interaction of an air system with concrete core conditioning. Energy and Buildings, Vol. 30, No. 2, pp. 139-145. Laouadi, A. (2004): Development of a heating and cooling model for building energy simulation software. Building and Environment, Vol. 39, No. 4, pp. 421-431. Mao G. (1997): Thermal Bridges, Efficient Models for Energy Analysis in Buildings. PhD Thesis, KTH – Division of Building Technology, Stockholm, Sweden. Mathsoft (1997): Mathcad 7 professional. Mathsoft Inc. Cambridge Massachusetts. International Thomson Publishing, Bonn, Germany. Nakhi A.E. (1995): Adaptive construction modelling within whole building dynamic simulation. PhD Thesis, Department of Mechanical Engineering, Energy Systems Research Unit, University of Strathclyde, Glasgow, UK. NOVEM (publisher) (2001): LTV voor nieuwbouw en renovatie. Méér comfort met minder energie. NOVEM The Netherlands Agency for Energy and the Environment, Sittard, The Netherlands. Reddy J.N., Gartling D.K. (2001): The Finite Element Method in Heat Transfer and Fluid Dynamics. CRC Press, New York, USA. Ren M.J., Wright J.A. (1998): A Ventilated Slab Thermal Storage System Model. Building and Environment. Vol. 33, No. 1, pp 43-52 Rittelmann R.P., Davy J.E., Ahmed S.F. (1983): Survey of Design Tools for Passive and Hybrid Low Energy Buildings. Conference Papers. Progress in Passive Solar Energy Systems Vol. 8, Santa Fe, USA. Rouvel L., Zimmermann F. (1997): Ein regelungstechnisches Modell zur Beschreibung des thermisch dynamischen Raumverhaltens, Teil 1, HLH Vol. 48, No 10, Springer VDI Verlag, Düsseldorf, Germany. Sahlin P. (1996): Modelling and Simulation Methods for Modular Continuous Systems in Buildings, PhD Thesis, ISRN KTH/IT/M –39-SE, Stockholm, Sweden.


27

Scartezzini J.L., Faist A., Liebling T. (1987): Using Markovian Stochastic Modelling to Predict Energy Performances and Thermal Comfort of Passive Solar Systems. Energy and Buildings, Vol. 10, No. 2, pp. 135-150. Schmidt D (2002): The Centre of Sustainable Building (ZUB)- A Case Study. Proceedings of the 3rd International Sustainable Building Conference, September 23-25, 2002, Oslo, Norway. Schmidt D. (2001): Models for Coupled Heat and Mass Transfer Processes in Buildings – Applications to Achieve Low Exergy Room Conditioning, Tekn. Lic. Thesis, KTH Department of Building Science, Stockholm, Sweden. Schmidt D. (2004): Design of Low Exergy Buildings – Method and Pre-Design Tool. The International Journal of Low Energy and Sustainable Buildings, Vol. 3 (2004), pp. 1-47. Schmidt D., Jóhannesson G. (2001): Model for the Thermal Performance of a Double Air Gap Wall Construction. Nordic Journal of Building Physics, Vol. 2 (2001), pp. 59-78. Schmidt D., Jóhannesson G. (2002): Approach for Modelling of Hybrid Heating and Cooling Components with Optimised RC Networks. Nordic Journal of Building Physics, Vol. 3 (2002), pp. 13-47. Strand R.K. (1995): Heat Source Transfer Functions and their Application to Low Temperature Radiant Heating Systems. PhD thesis. University of Illinois at Urbana-Champaign, USA. Strand R.K., Pedersen C.O. (2002): Modelling Radiant Systems in an Integrated Heat Balance Based Energy Simulation Program. Ashrae Transactions, Vol. 108. TRNSYS (2000): TRNSYS 15, Transient System Simulation Program, Solar Energy Laboratory (SEL), University of Wisconsin Madison, USA. VDI (publisher) (1991): VDI-Wärmeatlas – Berechnungsblätter für den Wärmeübergang. VDI-Verlag. Düsseldorf, Germany. Weber T., Jóhannesson G., Koschenz M. (2004): Frequency Domain Solution for dynamic Heat Conduction Using Triangular FEM Formulation. To be published in: Building and Environment. Weber T., Jóhannesson G. (2004): An Optimised RC-Network for Thermally Activated Building Components. To be published in: Building and Environment. Weber T., Schmidt D., Jóhannesson G. (2000): Concrete Core Cooling and Heating – A Case Study about Exergy Analysis on Building Components. Proceedings to the International Building Physics Conference 2000, September 18-21, 2000, Eindhoven, The Netherlands. pp. 91-98. Woodson R.D. (1999): Radiant floor heating. McGraw-Hill. New York, USA.  Nordic Journal Building Physics, 2004 Available at http://www.byv.kth.se/avd/byte/bphys/


28

Optimised RC Networks Incorporated within Macro ... - CiteSeerX

Optimised RC Networks Incorporated within Macro ... - CiteSeerX

Suggest Documents

optimised flooding algorithms for ad hoc networks - CiteSeerX

Dynamic diffusion in evolutionary optimised networks Takanori

Silver Nanoparticles Incorporated Within ... - Wiley Online Library

An Optimised Structure Optimised Structure Optimised ... - IJRIT

mmc .ASSOCTES RC. - HIE Networks

mmc .ASSOCTES RC. - HIE Networks

Security within Ad hoc Networks - CiteSeerX

Managed networks within cancer services: an ... - CiteSeerX

Computing signal delay in general RC networks by tree ... - CiteSeerX

within the Urban Macro Region of Chile

Macro-Star Networks: Efficient Low-Degree Alternatives to ... - CiteSeerX

Networks within Networks: Service Link Overlap ...

MACRO-REGIONS - CiteSeerX

Optimised Embedded Distributed Controller for Automated ... - CiteSeerX

GAPS: Genetic Algorithm Optimised Parallelisation - CiteSeerX

OPTIMISED EIGENSTRUCTURE ASSIGNMENT BY ANT ... - CiteSeerX

Distributed Macro Calibration in Sensor Networks

Macro-and micro-structure of trust networks

Macro-and micro-structure of trust networks

Optimised Virtual Keyboards with and without ... - CiteSeerX

Complementary Credit Networks and Macro-Economic ... - ewp.rpi.edu

Low Temperature Ionic Conductor: Ionic Liquid Incorporated within a ...

Optimised local caching in cellular mobile networks - Amazon Web ...

RC-RC-RC-emblems-sp.pdf - Google Drive