Thermodynamic simulation of a detached house with district heating subcentral. Jonas Gustafsson, Jan van Deventer, Jerker Delsing January 11, 2008 *E-mail:
[email protected] Keywords: District heating, building simulation, simulink.
Abstract With the continuously increasing energy demand, with global warming as its biggest threat, big challenges arise. District heating (DH) is environmental friendly heat-energy source for residential, industry and commercial building. The heat-energy is usually a bi-product of industrial processes or power production [1, 2, 3]. This makes district heating a reliable and environmental friendly energy source. The heat is transported to its customers in water- or steam-filled extensive underground pipe network. At the customer the energy is usually transfered to a house internal radiator and tap water system (varies over the globe) in a so called district heating subcentral (DHC). This article focuses on thermodynamic simulation of a detached house connected to a district heating system with a district heating subcentral with plate heat exchangers.
1
Introduction
Combined Heat and Power plant.
Picture of combined heat and power plant
As the district heating system often is used as a cooling circuit for industrial processes, it is better the bigger the temperature drop in the cooling circuit is. In each house with a district heating central, the total energy usage is measured and billed for. So to maximize the profit for the energy 1
companies the temperature drop in the DH-system should be as big as possible over the subcentral. Todays control system does not consider the return temperature (primary temperature difference) as it is normally not monitored by the control system, but it is monitored by the energy meter. If the data in the energy meter would be shared with the control system, the primary return temperature can be considered and the temperature difference would be possible to increase. This would also improve the prerequisites for error detection. The financial winnings of increasing the temperature drop over the subcentral is with the approximate Swedish energy pricing XXX Mkr. For the industries it is hard to predict any number as it depends on many factors, but one can be sure that it will affect the finances in a positive way. To be able to test these theories, a computer model of a district heating subcentral installed in a detached house with water-borne heating have been R created in Mathworks Simulink°in ”collaboration” with Lund University 1 . To the writing of this article, big improvements have been made on the usability of the model, huge configuration files have been replaced with masked subsystems that are easy to handle. The overview of the model have also been improved by re-arranging the blocks in a more pedagogical way. The exterior walls, roof and floor have also been improved to act more realistic to the house it created to resemble. The control methods used in the model are regular electrical-powered PI-control systems that does NOT take the primary temperatures into account.
1
The model have been developed over several years by different people involved in several projects carried out at Lule˚ a University of Technology, ”www.ltu.se” and Lund University ”www.lth.se”
2
2
Theory
In this chapter the model is described from the fundamental thermal relations all the way to the complete model.
q k A T m m ˙ L cp h U n ρ σ Θm LM T D C R V I
Table 1: Nomenclature [W ] Heat current [W/m · K] Thermal conductivity [m2 ] Area [K, C] Temperature [kg] Mass [kg/s] Mass flow [m] Length/Thickness [J/kg · K] Constant pressure specific heat [W/m2 · K] Heat transfer coefficient 2 [W/m · K] Overall heat transfer coefficient [−] Radiator constant [kg/m3 ] Density [W/m2 · K 4 ] Stefan-Boltzmann constant [K] True temperature difference [K] Logarithmic Mean Temperature Difference [F ] Electric capacitance [K/W ,Ω] Thermal resistance, Electric resistance 3 [V , m ] Electric voltage, volume [A] Electric current Index in Inlet Indoor out Outlet Outdoor cd Conduction cv Convection r Radiation lc Lumped capacity f r Flow resistance i Numeric index c Cold side h Hot side w Water Wall int Interior sf Space heating forward sr Space heating return
3
2.1
Thermodynamic fundamentals
Here follows a short resume of the fundamental thermodynamic laws that has been used to form the model. 2.1.1
One-dimensional conduction
One-dimensional heat conduction can be derived from Fourier’s law in one dimension, equation 1, where dT dx is the temperature gradient, and as it is negative it brings the minus sign, k the thermal conductivity and A the area of the for the heat flow section. Equation 1 can be rewritten as equation 2, if the area and thermal conductivity is presumed constant. qcd = −kA qcd =
dT dx
kA∆T ∆T ∆T ¶ = =µ L L Rcd kA
(1) (2)
Equation 2 is actually Ohm’s law, where the heat current q corresponds to the electrical current I, section conductive resistance Rcd to the electrical resistance R and the temperature difference ∆T to the voltage V. I= 2.1.2
∆T V ≡ qcd = R Rcd
Lumped-capacity heating and cooling
The comparison to electrical calculations can also be used to show the similarity between thermal heat capacity and electrical capacity. The electrical current through a capacitor is described by equation 3. A heat current through a wall with theoretically no thermal resistance, but heat capacity capabilities is described in equation 4, by comparing it with 3, one can clearly see the similarities. i=
qlc =
dV C dt
dT dT cp m = cp ρLA dt dt
4
(3)
(4)
2.1.3
Convection
The thermal heat transport from a surface to a liquid or gas in motion or vice versa is described in 5. qcv = hA (T1 − T2 ) 2.1.4
(5)
Flow resistance
The power emitted by a flow of gas or liquid that does not undergo a phase change, but a change in temperature can be calculated by using equation 6. qf r = mc ˙ p (Tout − Tin ) 2.1.5
(6)
Radiation
The radiation between two surfaces or between a surface and its surroundings is not linearly dependent on the temperature difference like the conduction and convection. The mathematical expression for radiation can be seen in equation 7. For smaller temperature differences, equation 7 can be linearized to equation 8 regarding to [4]. There are also other methods to use such as the Logarithmic Mean Temperature Method that is describes in section 2.3. For more information regarding the thermodynamic fundamentals, see [4, 5, 6]. ³
qr = σAF T14 − T24
´
qr = hr A(T1 − T2 ) ¡
(7) (8)
¢
where hr = σF T12 + T22 (T1 + T2 )
2.2
Thermodynamic building
The thermodynamic behavior of a building depends several things, such as size, building material, and area of usage. 2.2.1
External area (walls, roof, floor)
The inner wall temperature can be calculated by using Kirchhoff’s Current Law (KCL) [7] as the heat currents can be transformed in to corresponding electrical currents, see figure 1 for an electrical equivalent for the complete wall. Equations 9 to 12 shows the temperature for the inner wall surface temperature is calculated. Inner wall temperature. 5
T1
Tindoor
T2
Ri
T3
R1
R2
C1/2
0
T4
C1/2
R3
C2/2
0
0
T5
C2/2
R4
C3/2
0
C3/2
0
Ro
C4/2
0
0
Toutdoor
C4/2
0
Figure 1: RC equivalent of wall.
IRin = IRi + I Ci
(9)
qcv = qcd + qlc
(10)
2
ki dTi hin (Tin − Ti ) = (Ti − Ti+1 ) + Li dt dTi = dt
µ
cp,i ρi Li 2
¶
(11)
ki (Ti − Ti+1 ) Li cp,i ρi Li 2
hin (Tin − Ti ) −
(12)
Where i = 1 Temperatures between the layers inside the wall. IRi = IRi+1 + I Ci + I Ci+1 2
ki ki+1 dTi (Ti−1 − Ti ) = (Ti − Ti+1 ) + di Li+1 dt
(13)
2
µ
cp,i ρi Li cp,i+1 ρi+1 Li+1 + 2 2
ki ki+1 (Ti−1 − Ti ) − (Ti − Ti+1 ) dTi Li Li+1 = cp,i ρi Li cp,i+1 ρi+1 Li+1 dt + 2 2
¶
(14)
(15)
Where i = 2 . . . (N − 1) Temperature of the outside wall. IRout = IRi + I Ci
(16)
2
ki dTi hout (Ti − Tout ) = (Ti−1 − Ti ) + Li dt 6
µ
cp,i ρi Li 2
¶
(17)
dTi = dt
ki (Ti−1 − Ti ) Li cp,i ρi Li 2
hout (Ti − Tout ) −
(18)
Where i = N Each of the above explained thermal relationships can be created in simulink, see figure 2 for an example of how a single layer is realized in simulink. The layers are connected together to form a complete model of the wall, this can be seen i figure 3, observe the second output of the first block that tells us what the current heat flow is. 1 T_1
-K1 s
-K-
1 T_2 [C]
-K-
2 T_3
Figure 2: One layer of the wall. Q_dot w [W/m^2]
2 To 1 Ti
1 T_a
T_o [C ]
T_c
Q _dot [W /m ^2]
T_1
T_1
T_1
T_2 [C ]
Indoor - Plaster
T_3
Plaster - Insulation 2
T_2 [C ] T_3
T_a T_o
T_2 [C ] T_3
Insulation - Air 2
Air - Brickwall 2
T_c
Brickwall - Outdoor
Figure 3: Four-layer wall model.
2.2.2
Internal mass thermodynamics.
The thermodynamic interior, like inner walls furniture can be compared with another RC-net, see figure 4. For readers having knowledge in electronics it is easy similarities between the charging of a capacitor and the heating of an object. The object is heated/cooled by the surrounding air by convection, it could also be discussed if the direct radiative heat from the radiators should be included, but in section 2.3 we see that all heat from the radiators transfers to the indoor air. The mathematical explanation can be seen in equations 19 to 21 and a figure of a simulink realization can be viewed in figure 5. qcv = qlc
7
(19)
Tindoor
R
C
0
Figure 4: RC-equivalent for thermodynamic interior.
hint A (Tin − Tint ) =
dTint (cp,int mint ) dt
(20)
dTint hint A (Tin − Tint ) = dt cp,int mint
1 Ti
(21)
1 s
Mb _int Cp_b
dTb ,i => Tbi 1 Ainner
hi -1
1 q [W]
Figure 5: Internal thermodynamics. 2.2.3
Complete building
When all the exterior and interior simulink block are connected, we get something like seen i figure 6. This represents the complete building of interest, in our case a two-floor detached house with sides 10 times 20 meters, which gives us an approximate total floor area of 200m2 . The house is a brick-wall house with bla bla bla . . . As the house is located in the far north of Sweden, and the interesting time of investigation is winter, we have not included sun-heating as a part of the heating. Internal heat sources like humans, computers, televisions etc. can be included by simply add the heat power the emit.
2.3
Radiator
Supplied power to the radiator is described by equation 6, this can be adapted to support our application in a better way by applying the Logarithmic Mean Temperature Difference (LMTD) method described in [4, 5]. The LMTD describes the temperature difference between the surrounding air and the water in the radiator along the radiator, see equation 22. By replacing the temperature difference (T1 − T2 ) in 8 with the LMTD we retain equation 23 where U is the overall heat transfer coefficient (see equation 24) and F is a correction factor depending on P, R and flow arrangement [4]. 8
Ti Q_dot w [W/m^2]
2 To
Wall
To
Wall model volume Ti
5
Q_dot g [W/m^2] T ground
T ground
Floor
Floor
Ti Q_dot r [W/m^2]
Roof
To
volume [m^3]
Roof
q [W]
Temp/t [C/s] Power [W]
P [W] Ti Q_dot [W/m^2]
Windows
Power to Temp ./time
To
1 s dTi => Ti
1 T indoor [C]
Windows House dimensions 150 Internal power [W] (humans , tv, etc.)
Ti
Q_dot[W]
Interior thermodynamics Ti Temp/t[C/s] To
1 Q radiator [W]
Air exchange rate
Figure 6: Complete building. The correction factor could also be expressed as an approximate radiator constant n which usually is around 1.3, see equation 25 [8]. In a static case the supplied power would be equal to the radiated power. But when dynamics is introduced the supplied power will not be equal to radiated power as there is a thermal inertia in the system. Hence the lumped capacity equation, eq. 4 can be used to describe this phenomenon. When setting the difference between the supplied power and radiated power equal to the lumped capacity, equation 27 is retained. Using equation 27, one can calculate the water return temperature from the radiator, see equation 27, 28 and figure 7. Θm = LM T D =
Tsf − Tsr ¶ Tsf − Tindoor ln Tsr − Tindoor µ
q = U AF Θm U=
1 1 1 + + Rdc + Rdh hc hh
Where in this case Rdc = Rdh = 0
9
(22)
(23) (24)
qr = U AΘnm
(25)
qlc = qf r − qr
(26)
dTsr ms cp,w = cp,w m ˙ s (Tsf − Tsr ) − U AΘnm dt
(27)
cp,w m ˙ s (Tsf − Tsr ) − U AΘnm dTsr = dt ms cp,w
(28)
cp_w
Cs
3
1
1 s
ms
Tout
Cs*dTs->Cs*Ts q_fr=m*cp*dT
u/Cs 2
1
q_fr
Tin [Tut Tin Ti]
2 Ti
q
3 q_lmtd
q_r=UA*LMTD ^radexp
Figure 7: Radiator model.
2.4
Heat exchangers
Heat exchangers are very similar to radiators, basically they are the same. But here the LMTD can not be calculated as we do not know the outlet temperature of the heat exchanger or the surrounding temperature. Hence the heat exchanger is split in to linear sections that can be connected to form a ”realistic” model of the heat exchanger. The heat supplied to one section of the heat exchanger is described in 6, the heat is ”absorbed” by convection in the separating wall, and then transfered to the second fluid. The temperature ”delay” can be described by equation 4. In equations 29 to 31 the resulting temperature from a fluid flow through a heat exchanger is shown. qlc = qf r − qcv
(29) µ
dTout Tin + Tout mcp,water = mc ˙ p,water (Tin − Tout ) − hA − Tw dt 2 µ
dTout dt
Tin + Tout mc ˙ p,water (Tin − Tout ) − hA − Tw 2 = mcp,water 10
¶
(30)
¶
(31)
The temperature of the wall separating the two heat carriers can be calculated in a similar way, see equation. qlc = qcv,h − qcv,c µ
dTw Tin,c + Tout,c mw cp,w = hc Ac Tw − dt 2 µ
dTw = dt
Tin,c + Tout,c Tw − 2
hc Ac
(32)
¶
µ
− hh Ah
¶
µ
− hh Ah
¶
Tin,h + Tout,h − Tw 2 (33)
Tin,h + Tout,h − Tw 2
¶
(34)
mw cp,w
A simulink block is created for each section of the heat exchanger, a block can be viewed in figure 8. Several of these blocks can be connected to form a more realistic heat exchanger, see figure 9, but the more sections, the heavier the calculations of the model becomes, so a trade-off has to be done. In this case we have used 3 sections. Th,out Transport delay hot side
Th ,in
f(u)
q_fr
mh f(u )
1 s
dQh /(Mh *cp _ w)
dThout > Thout 1
qh =m _ h *cp _ w*(Th ,in -Th ,out )
2 f(u )
Th,in
To
f(u)
h_h
Time Delay , h
q_cv
Th,out
q =h_ h *A *((Th ,in +Th ,out )/2 - Tw)
mc Tw
Tw f(u )
{h and m} q=h _ c*A *(Tw-(Tc,in +Tc,out )/2)
mh Tc ,in
h_c
f(u)
1 f(u )
dTw
dQwall /(Mwall *cp _ AISI )
1 s Integrator
q_cv Tc,out
f(u )
1 s
dQc /(Mc *cp _ w)3
dTcout > Tcout 2
Tc,in
To
Time Delay , c Transport delay cold side
2 Th ,out
qc =mc *cp _ w*(Tc,in -Tc ,out )
1 Tc ,out
mc f(u)
q_fr
Tc,out
Figure 8: Heat exchanger section.
2.5
Thermodynamic valve
The thermodynamic valve controls the flow through the radiator by sensing the room temperature, the colder the room temperature, the higher flow, see equation 35. p
V˙ = f (Terr. ) ∆Pvalve Kvs
(35)
where f (Terr. ) is the valve characteristics. m ˙ =
V˙ ρwater (T ) 3600 11
(36)
1 Tsf [C] Tc,in
Tpf [C]
Tc,out
Tc,in
Tc,out
Tc,in
Tc,out
1
Th,in Th,out
Th,in Th,out
Th,in Th,out
Tsr [C]
Hex_section _1
Hex_section _2
Hex_section _3
2 Tpr [C]
3 {allinone }
ms [kg/s] 2 uv
Ns_ch nr sec channels mp [kg/s]
Goto Tag c0
yhx {allinone }
exponent
4 Np_ch
uv
c0
nr prim channels
Figure 9: Complete heat exchanger.
2.6
PI controller and control valve
The space heating control system is set to supply the radiators with a pre set temperature that depends on the outdoor temperature. This should be adapted so the radiator system flow has to change as little as possible even at big temperature drops. By doing this, the thermostatic valve mounted at the radiators will not have to adjust a lot to keep the indoor temperature at a constant level. In this model a regular PI-control (eq. 37) system is used to control the radiator supply temperature. u(Terr. ) = Kr Terr. + Ki The valve is. . .
12
Z t 0
Terr. dt
(37)
3
Simulation results.
13
4
Conclusions
The model is fully functional by theory, and our research can now move on to verification and final calibration of the model.
14
References [1] I. D. E. Association, “http://www.districtenergy.org,” November 2007. [2] S. Fredriksen and S. Werner, Fjarrvarme, Teori, teknik och funktion. Lund, Sweden: Studentlitteratur, 1993. [3] T. S. D. H. Association, “http://www.svenskfjarrvarme.se,” November 2007. [4] A. Bejan and A. D. Kraus, Heat transfer handbook. Hoboken, New Jersey: John Wiley and Sons, Inc., 2003. [5] Y. A. Cengel, Thermodynamics and heat transfer. 2, McGraw-Hill, 1997. Mycket bra bok. [6] G. Sparr and A. Sparr, Kontinuerliga system. Studentlitteratur, 2 ed., 2000. [7] H. D. Young and R. A. Freedman, University Physics. Addison Wesley Longman, 10 ed., 2000. [8] P. Ljunggren, Optimal och robust drift av fjrrvrmecentraler. PhD thesis, Lund University, Division of energy economics and planning, Department of Heat and Power Technology, Lund Institute of Technology, Lund University, Sweden, August 2006.
15
