José María Morelos y Pavón

Transcription

1 Secretarı a de Educacio n Pu blica Tecnolo gico Nacional de Me xico Instituto Tecnolo gico de Morelia José María Morelos y Pavón Laboratory Session 3 Modeling a Heating System supervised by MSc. Gerardo Cha vez-campos November 24, 2017

2 Introduction This guide shows how to model a dynamic system using Simulink software[1]. The model is a house heating system that includes a heater (plant), thermostat (controller), and room (environment). Contents Defining a House Heating System Determining the model goals Identify system components How the system will works Define System Equations Heat rate gain equation References Defining a House Heating System In order to define a complete system it is necessary to identify the requirements of the control and mathematical equations. To model the heating system it is vital to collect data for adjusting the model parameters and validate the simulation results. Determining the model goals Before designing a model, consider the goals and requirements. The goals for modeling the heating system are: Observe how the changing outdoor temperature affects the indoor temperature. Using experimentally measured outdoor and indoor temperatures, compare simulation results with measured values. Identify system components The house heating system defines a heating system and its relationship to a room. It includes: Thermal characteristics of a house Thermal characteristics of a heater A thermostat to control the heater Outdoor environment Indoor environment -1-

3 How the system will works After selecting the thermostat setting, the thermostat will turn the heater on and off depending on the difference between the outside temperature and the room temperature. The model for this system includes three components: heater, thermostat, and room. Define System Equations Three variables define the house heating model: Thermal energy transferred from the heater (Q gain ) to a room Thermal energy transferred from the room (Q loss ) to the outdoor environment Temperature of the room (T room ) A differential equation defines each of the variables, but since heat transfer is defined in terms of changing temperature, only room temperature is a state variable(ẋ). Heat rate gain equation To define the heat rate equation gained in the room, considers that the temperature of the air in the heater is T heater and the room temperature is T room. -2-

4 Also considering that the main phenomenon gain thermal energy at the room is by convection of heated air from the heater to air of the rorom. Thus, heat gain for a mass of air in the heater (m heaterair ) is: Q gain = m heaterair c air (T heater T room ) (1) However considers that a fan takes the room air, and passes it through the heater and then the air is take it back to the room. Therefore the rate of thermal energy gain from the heater is: dq gain = dm heaterair c air (T heater T room ) (2) Since the mass of air per unit time from the heater is constant, it is posible to replace dm heaterair / with a constant M heaterair and simplify the equation to: Q gain = M heaterair c air (T heater T room ) (3) Rate of heat loss equation The case of thermal energy losses from the room is mainly due to the conduction phenomenon through the walls and windows: Q loss = ka(t room T outside )t D then the rate of thermal energy loss is: (4) dq loss = ka(t room T outside ) D replacing ka/d with 1/R, where R is the thermal resistance, simplifies the equation to: (5) dq loss = T room T outside R (6) Room temperature equation Define the rate of temperature change in the room by subtracting the rate of heat loss from the rate of heat gain: ( dt room 1 dqgain = dq ) loss (7) m roomair c air -3-

5 Variable Description Units A Area of wall or windows (m 2 ) D Depth of the wall A wall=914 (m) Q Thermal energy transferred (J) dq/ Rate of thermal energy transferred J h 1 k Thermal conductivity; J/(m h C) k glass=2808 r Thermal resistivity; r = 1/k m h C/J R Thermal resistance; R = h C/J D/kA = (T 1 T 2 )Q; R wall = 1.599E 6, R window = 5.935E 7 m Mass of air in the room; m room = kg 1470 dm/ Rate of air mass passing through kg h 1 the heater M Constant rate of air mass passing through the heater kg h 1 c Specific heat capacity; C air = J kg 1 C T heater Constant air temperature from C heater; T heater = 50 T room Air temperature of room; initial conditiont toom = 20 C Collecting the data Most of the parameter values needed for the house heating model are published in standard property tables. The flow rate for the heater is from a manufacturer data sheet. List the variables and coefficients from your equations and check for dimensional consistency between the units. Since the unit of time for the model is hours, convert published values for the thermal property of materials from units of seconds to hours. Equation Variables and Constants. You can use the constant names and values in this table when building the model. -4-