Proceedings of Clima 2007 WellBeing Indoors
Modeling Of Non-Linear HVAC system using SIMBAD
K. N. Nagabhushan, K.N and D. C. Hittle, D.C.
Colorado State University
Corresponding email: hittle@colostate.edu
ABSTRACT
The term heating, ventilating and air conditioning (HVAC) covers a wide range of equipments.
An air conditioning system maintains desired environmental conditions within the zone. In
most cases term heating, ventilating and air conditioning (HVAC) systems, a central air supply
provides air at the controlled temperature and flow rate for heating or cooling the space. The
main objective of this project was to develop an accurate, dynamic and non-linear digital model
of an HVAC system using SIMBAD that could be implemented in the MATLAB/Simulink
environment [13]. HVAC system models have been implanted in many proprietary and public
domain programs. However, our goal was to implement a model in MATLAB/Simulink, a
software system and tool box widely used to study dynamic response and control systems. This
paper documents the model for the benefit of those interested in studying HVAC control. The
model was also verified by comparing its output to the dynamic response of an experimental
HVAC system (see [16]for a discussion of the experimental facility). Our point of departure
was the SIMBAD toolbox from CSTB Paris, used in the MATLAB/Simulink environment [12].
INTRODUCTION
The SIMBAD toolbox provided many of the components of HVAC systems used in the
modeling. The components of the system are modeled using standard Simulink block diagrams
and models compiled in C code. The main characteristics of the SIMBAD tool box is the
definition of connection vectors. The main two vectors used are :
The air vector : Air dry bulb temperature [C], Air humidity ratio [kg/kg], Static pressure [Pa],
Mass flow rate [kg/s]
The water vector : Water temperature [C], Static pressure [Pa], Mass flow rate [kg/s]
SIMBAD is obviously in a state of evolution (and like most software will probably always be in
that state). Therefore we developed or modified models to reflect our needs, to consider North
American versus European design practice, and to introduce options for multi input multi
output control.
MODELING
The literature is quite rich about dynamic modeling of HVAC systems using linear
approximation. This research is mainly concerned with the non-linear modeling and simulation
of various HVAC components and the overall HVAC system.
Proceedings of Clima 2007 WellBeing Indoors
Components that are modeled here are : Dynamic heating and cooling coil, Electrically heated
boiler, Variable speed pump, Two-way convergent air duct, Outside and return dampers, Threeway mixing valve, Variable speed fan, PI controllers.
The first four component models came from the SIMBAD toolbox. The next three were
developed in- house and the PI controller model was taken from the Simulink controls tool kit.
Modeling of heat exchanger
Almost every thermal environmental system involves heating or cooling of the atmospheric air.
It forms the part of both water and air cycle. The parameters for our heat exchanger modeled
are: 4 pass, 23 run, plate fin and counter flow type heat exchanger. The tube material is copper
and the tube core is 0.01 m in diameter. The fins have a pitch of 472/m,are 0.25 mm thick and
are made of aluminum. These parameters are by way of example. Any rational set of
parameters can be used in the model.
Simulink elements and five programs in “C” language calculate specific variables such as:
Geometry,
Exchange coefficients, Pressure calculations, Heat transfer, Temperature (C programs),
Pressure (C programs), Humidity ratio (C programs), and Mass flow rates (C programs).
Temperature calculations
This section deals with the calculations of output air temperature. The equations used are :
Tao = Tai – Qs / (Cpi * ma )
(1)
Tao = Outlet air temperature (C), Tai = Inlet air temperature (C), Qs = Sensible heat transfer rate
(kW), ma = Mass flow rate (kg/s).
The sensible heat transfer rate is the heat transfer rate which is solely manifested in raising the
temperature of the incoming air.
Cpi = Cpa + Cpg * Gi
(2)
Cpi = Inlet air specific heat capacity (kJ/kg K), Cpa = Specific heat capacity of air (kJ/kg K), Cpg
= Specific heat capacity of steam (kJ/kg K), Gi = Humidity ratio of inlet air (kg/kg).
Proceedings of Clima 2007 WellBeing Indoors
Figure 1 Dynamic model of the heat exchanger
Qs = Qa * SHR
(3)
Qs = Sensible heat transfer rate (kW), Qa = Heat transfer rate on air side (kW), SHR =
Sensible heat ratio (kg/kg).
Qa = ( Tai – T[0] ) / Rst1
(4)
Tai = Inlet air temperature (C), Rst1 = Air-side thermal resistance (K/kW), T[0] = Initial coil
temperature (C), Rst1 depends on the type of regime, either dry or wet . Here it is assumed to be
a dry regime.
Rst1 = Bf * Rst
(5)
Rst = Coil total thermal resistance (K/kW), Bf = By-pass factor for dry coil .
Rst = 1.0 / [ Effect * mcp]
(6)
Effect = coil effectiveness, mcp = minimum capacity rate (kW/K)
Cair = Cpi * ma
(7)
Cair = Air capacity rate (kW/K), Cpi = Inlet air specific heat capacity (kJ/kg.K), ma = Mass
flow rate of air (kg/s).
Cwater = Cpw * mw
(8)
Cwater = Water capacity rate (kW/K), Cpw = Water specific heat capacity (kJ/kg.K), mw = Flow
rate of water (kg/s).
Pressure Calculations
Here, we are introducing one independent variable u, and another dependent variable, x . Both
u and x are not used in the model. It is used here to explain the model clearly.
AD = (273.16 / (273.16 + Tai ) ) / 1.293
(9)
Proceedings of Clima 2007 WellBeing Indoors
AD = Air density (kg/m3), Tai = Inlet air temperature (C).
u = (1 / AD ) * ma
(10)
x = u2 * Resistance coefficient
(11)
u = Volumetric flow rate (m3/s), x = Pressure drop (Pa), ma = mass flow rate of air (kg/s) and
the resistance coefficient is an input parameter into the heat exchanger .
Po = Pi – x
(12)
Po = Output pressure (Pa), Pi = Inlet pressure (Pa).
Humidity ratio calculations :
The equation used to calculate humidity ratio is :
Go = Gi – ( Qa – Qsensi ) / ( HFG * ma )
(13)
Go = Outlet air humidity ratio (kg/kg), Gi = Inlet air humidity ratio (kg/kg), Qa= Heat transfer
rate on air side (kW), Qsensi = Sensible heat transfer rate (kW), HFG = Latent heat of
vaporization of water (kJ/kg), ma = Mass flow rate of air (kg/s).
Mass flow rate calculations
There is no change in the mass flow rate of air through the heat exchanger. On the water side
there are only two variables-temperature and flow rate. As the flow rate remains constant, the
only calculation is that of temperature.
Two = Twi + Qr / Cwater.
(14)
Two = Water output temperature (C), Twi = Water input temperature (C), Qr = Heat transfer
rate on water side (kW), Cwater = Water capacity rate (kW/K).
Qr = ( Tci –Twi ) / Rst2 .
(15)
Tci = Initial coil temperature (C), Rst2 = Water + metal thermal resistance (K/kW), Twi =
Water inlet temperature (C).
Rst2 = 0.001 * Rst
(16)
Rst = Total coil thermal resistance (K/kW). The value for Rst comes from program header files.
Cwater = Cpw * mw
(17)
Cpw = Specific heat capacity of water (kJ/kg.K), mw = Mass flow rate of water (kg/s).
Electrical Boiler
The boiler modeled here is an electric heat boiler. The design of electric boilers is largely
determined by the shape and heat release rate of electric heating elements used. The example
boiler has a capacity of 14 kilowatts. The heater is controlled through a PI control loop that
tracks the set point temperature under transient conditions as illustrated in figure 2. The boiler
output water temperature can also be calculated using steady state calculations.
Proceedings of Clima 2007 WellBeing Indoors
The boiler model has two main sections, the thermal cycle and hydraulic cycle. The thermal
cycle model calculates the output temperature of the water and the hydraulic cycle model
calculates output pressure depending on the flow rate.
Figure 2 Electrically heated boiler
Thermal Cycle
The inputs to the thermal model are:
Qint = Command from the controller. This entity depends on the required hot water temperature.
Win = Input water vector
Tamb = Ambient temperature (C)
The inputs are used to define a column matrix. The matrix will have 3 entities, A, B and U.
These three entities will be used to calculate outlet water temperature both in steady state and in
transient conditions.
Defining the matrix :
A = (( mwi * Cw /MC ) + ( Ahtank / MC ) )* Gain
(18)
mwi = Inlet flow rate of water (kg/s), Cw = Specific heat capacity of water (kJ/kg.K) Ahtank =
Surface heat exchange product (kW/K), MC = Total thermal mass (kJ/K).
MC = mCp_water + mCp_metal
(19)
mCp_water = Total thermal mass of water (kJ/K), mCp_metal = Total thermal mass of metal
(kJ/K)
mCp_water = V_water * ρwater * Cw
(20)
ρwater = Density of water (1000 kg/m3)
V_water = Volume of water in the boiler (m3), Cw = Specific heat capacity of water (kJ/kg.K),
mCp_metal = V_metal * ρcopper * Cpm
(21)
V_metal = Volume of metal on the boiler (m3), Cpm= Specific heat capacity of metal (kJ/kg.K),
ρcopper = Density of copper (8000 kg/m3) .
Proceedings of Clima 2007 WellBeing Indoors
As per Simbad, the value of the gain is constant at –1. As shown in the equation, the values of
Cw and Ahtank are normalized with MC.
The second element in the matrix is B. It has three entities, B1, B2, B3.
B1 = mwi * Cw / MC
(22)
B2 = Ahtank / MC
(23)
B3 = 1 / MC
(24)
The vector B can be defined as
B = [ B1 B2 B3 ]
(25)
The last and final element in the matrix U is also a vector consisting of
U = [ Twi Ta Qint * Pnom]
(26)
Twi = Inlet water temperature (C), Ta = Ambient temperature (C), Qint = Controller input,
Pnom = Total capacity of the coil (14 kW)
Subsequently the matrix will have three outputs: [ A B U ]. Here we have considered transient
calculations. The matrix defined above forms the input for the transient calculations. For this
purpose, we are defining two variables x and y.
Where
x = ∑ (B * U)
(27)
y= A+x
(28)
and
Now the output water temperature is calculated by the formula
Two = A * ∫ y
(29)
The output water temperature, in the transient calculation is determined by integrating the inlet
water temperature, with matrix A, over time, until the set point temperature is attained.
Hydraulic Cycle :
The hydraulic phenomena calculates output pressure depending on the pressure drop during the
nominal flow rate of the fluid. In this model we have assumed the pressure drop to be zero.
Proceedings of Clima 2007 WellBeing Indoors
Pump
Figure 3 Variable speed pump
One of the main methods for transporting thermal energy into, out of, and within a building is
with water. The pump inputs the required flow rate into the water circuit. The flow rate from
the pump depends on the command input given at the ‘com‘ input port as illustrated in figure 3.
The pump speed can be varied to vary the flow rate of the fluid as required up to the maximum
flow capacity of the pump.
The equation is given as
Fw = Com * Maximum flow rate. (kg/s)
(30)
Maximum flow rate is an input function of the pump.
It is predominantly a linear function. The output water vector will have the same temperature
and pressure as the input; only the flow rate will be changed depending on the command.
Two-way convergent air duct
The two-way convergent air duct gives the resultant air vector due to mixing of two air vectors
- the fresh air vector from the atmosphere that comes through the outside air damper and the
return air vector from the zone coming through the return damper. The flow rates of the two
vectors are added and no head loss is considered.
Mixing temperature
Tmix = ( Ta1 * ma1 + Ta2 * ma2 ) / (ma1 + ma2 )
(31)
Ta1 = Temperature of outside air (C), Ta2 = Temperature of return air (C), ma1 = Flow rate of
outside air (kg/s), ma2 = Flow rate of return (kg/s.
Mixing humidity
Umix = ( U1 * ma1 + U3 * ma2 ) / (ma1 + ma2 )
(32)
U1 = Humidity ratio of outside air (kg/kg), U3 = Humidity ratio of return air (kg/kg) ma1 = Flow
rate of outside air (kg/s), ma2 = Flow rate of return air (kg/s)..
Mixing pressure
Pmix = ( P1 * ma1 + P2 * ma2 ) / (ma1 + ma2 )
(33)
Proceedings of Clima 2007 WellBeing Indoors
P1 = Pressure of outside air (Pa),P2 = Pressure of return air (Pa), ma1 = Flow rate of outside
air (kg/s), ma2 = Flow rate of outside air (kg/s)
The values of both P1 and P2 are equal at the entrance of the mixing box. The resultant flow
rate will be the sum total of both the individual flow rates.
The following components were designed and built in-house. The same components in
SIMBAD were not working in accordance with the components in the experimental facility.
Three-way mixing valve
Figure 4 Three way mixing valve
The valve has three inputs and three outputs. The three inputs are: Command signal from the
controller depending on the set point temperature, Input port for water coming from boiler,
Input port for water retuning from the heat exchanger, The three outputs are: Main water vector
going into the heat exchanger, and Water vector going into the bypass circuit.
The resultant water port mixes water vectors from the heat exchanger and bypass circuit. The
bypass water vector and the water vector are mixed internally. Thus the bypass port is used for
monitoring purposes and it is not connected anywhere in the system.
The valve has three parts:
Flow fraction determination: Depending on the command from the controller, which in turn is
determined by the set point temperature, a look up table between the command and flow
fraction determines the percentage of total flow going into the heat exchanger. The command
from the controller is in effect the percentage opening of the valve.
Flow calculations: The input to this function is the flow fraction in the form of percentage of
the total flow going into the heat exchanger. The flow fraction in the by pass circuit is
determined by formula::
mw (bypass) = mw - mw (main)
(34)
mw = Total flow rate of water from the boiler (kg/s), mw (main) = Flow rate of water into the
main circuit (kg/s), mw (bypass) = Flow rate of water into the bypass circuit (kg/s).
Proceedings of Clima 2007 WellBeing Indoors
Temperature calculations: The main purpose of this block is to calculate the resultant water
temperature after mixing between return water and bypass water vectors. Inputs to this block
are flow fraction, temperature of water returning from the heat exchanger and temperature of
water in the bypass vector. The flow rate of the resultant water vector is the summation of the
two water vectors of bypass and return vector from the heat exchanger; no head loss is
considered.
The resultant temperature of the water is determined by the relation:
Tres = (The * mw_he + Tbypass * mw_bypass) / (mw_he + mw_bypass)
(35)
The = Temperature of water returning from heat exchanger (C), Tbypass = Temperature of water
in bypass circuit (C), mw_he = Flow rate of water returning from heat exchanger (kg/s), mw_bypass
= Flow rate of water in bypass circuit (kg/s)
Actuator
The actuating action required here is linear motion of a piston. Depending on the set-point
temperature, the valve regulates the flow of water into the heat exchanger. The actuator
modeled here has a transfer function :
F(s) = 0.005 / (3s+0.005)
(36)
Now the inverse laplace transform of the above transfer function is :
f(t) = 1/600 * e-1/ 600*t
(37)
From the equation the time constant of the actuator is τ = 600 s. The time constant indicates that
the response decays exponentially every 600 seconds.
Dampers :
Dampers are used for the control of airflow to maintain temperature and/or pressure. An
important entity in the design of dampers is the dimensionless entity loss coefficient, Cd. Over
most of the operating range, the loss coefficient is an exponential function of blade angle for
both parallel and opposed blade dampers. That is,
Cd=k1.ek2.θ
(38)
Where k1 and k2 are constants, and θ is the blade angle in degrees.
Now the fraction of full flow is calculated by using the formula
Fraction of full flow
=
Cdo
f . Cd + (1-f). Cdo
(39)
Cdo = loss coefficient for the wide-open damper, Cd = loss coefficient, a function of blade
angle, f = open damper resistance as a fraction of the total system resistance
The values of Cdo and f are obtained from ASHRAE fundamentals (1997). Since return and
outside dampers are used in tandem, one control loop is used to control both dampers
depending on the set point temperature.
Proceedings of Clima 2007 WellBeing Indoors
Actuator :
The main task of the actuator in a damper is to precisely open or close the damper blades
depending on the command from the controller. Since the action required is circular motion of
the damper blades, the actuator modeled here is a motor having the transfer function,
F(s) = 5 / (3s+5)
(40)
Now the inverse laplace transform of the above transfer function is :
f(t) = 5/3 * e-5/3t
(41)
From the equation the time constant of the actuator is τ = 0.6 s. This time constant indicates that
the response decays exponentially every 0.6 seconds.
Variable speed fan
The fan modeled here has one input and two outputs. The input to the fan is the air vector
coming from the heat exchanger . The outputs are: Air vector going into the zone and the total
power imparted by the fan on the air.
Figure 5 Damper
Fig 6 Variable speed fan
The fan model has two lookup tables : The relationship between volume flow and pressure loss
and the relationship between volume flow and RPM. Depending on the mass flow rate of the
input air vector the volume flow rate is calculated by:
Q = ma /ρ
(42)
Proceedings of Clima 2007 WellBeing Indoors
Q = Volume flow rate (m3/s ), ma = mass flow rate (kg/s), ρ = Density of the air (kg/m3)
The volume flow rate thus obtained will be used to get the RPM from the second lookup table
and the pressure loss from the first lookup table.
The inertial effects of the fan impeller are calculated to make the model dynamically more
accurate. The value of the RPM from the lookup is used to calculate the angular velocity
(radians/s).
θ’ = RPM * 2 * π / 60
(43)
The angular acceleration is:
θ” = θ’ / t.
(radian/s2), with t=time constant taken as 10 s
(44)
Now the total energy imparted by the fan to the air is given by:
W = Q*(P1-P2) + m * θ” + c * θ’
(45)
W = Total energy imparted by the fan on air (kW), Q = Volume flow rate (m3/s ), P1 = Set
point pressure in the zone (Pa), P2 = Pressure coming into the zone after all losses (Pa), m =
Mass of the fan impeller (kg), c = Coefficient of bearing friction, θ” = Angular acceleration
(m/s2), θ’ = Angular velocity (m/s)
Speed change time constant
The action required here is to change the circular motion of the fan impeller. Depending on the
set-point temperature the fan regulates the volume of air flow into the zone. The motor has a
transfer function:
F(s) = 1 / (10s+1)
(46)
Now the inverse laplace transform of the above transfer function is :
f(t) = 1/10 * e-1/10t
(47)
From the equation the time constant of the actuator is τ = 10 s.
The time constant indicates that the response decays exponentially every 10 seconds
RESULTS
Steady State
The following table gives simulation test results for steady state testing.
Twi
Two
Ts
Tai
Tao
m_w
m_a
Q_a
(C)
(C)
(C)
(C)
(C)
(kg/s) (kg/s)
(kW)
55
13.86
18
10.96
17.99
0.0298 0.7191 5.086
55
27.39
25
13.46
25.12
0.0829 0.8197 9.624
55
35.76
30
20.96
30.04
0.1569 0.1.38 12.66
55
38.41
35
25.37
35.00
0.2185 1.5662 15.18
55
42.27
40
30.09
39.99
0.2851 1.5249 15.2
55
46.491 45
35.97
45.00
0.3543 1.3898 12.63
Steady state energy balances were calculated using EES Software.
Q_w
(kW)
5.058
9.571
12.62
15.15
15.17
12.61
EB
0.027
0.053
0.03
0.030
0.031
0.027
Proceedings of Clima 2007 WellBeing Indoors
Dynamic testing :
Dynamic tests were performed with step changes given to a set point temperature. The
temperature set point was changed from 40 0C to 50 0C .
The following graphs show the comparison between experimental data and the simulation data.
In all the cases the temperature set points were 400C (initial value) and 500C (final value).
CONCLUSIONS
The primary objective of this project was to develop a Simulink model of an HVAC system to
allow simulation in a software environment that will allow the study of intelligent control
systems –MATLAB/Simulink are the preferred tools for studying dynamic response and
control. This paper documents the models.
A second objective, achieved and reported here was the experimental verification of the
models. Agreement is good. This builds confidence that users can apply the model to study
control problems. The “code” is available upon request – first visit our web site at
http://www.engr.colostate.edu/nnhvac/.
Fig 7 Water vector in main circuit of the 3-way mixing valve
Proceedings of Clima 2007 WellBeing Indoors
Fig 8 Water vector in the bypass circuit of 3-way mixing valve.
Fig 9 Temperature of air entering the heat exchanger.
Proceedings of Clima 2007 WellBeing Indoors
Fig 10 Temperature of air leaving the heat exchanger.
Fig 11 Graphs showing comparison between experimental data and simulation data for the same
set point temperatures.
Proceedings of Clima 2007 WellBeing Indoors
Fig 12 Figure showing simulation model of complete HVAC system
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