Analysis Parameterization, & Simulation of a Heat Gun
Submitted by
Thomas A. Bowers December 10, 2002
2.141: Modeling and Simulation of Dynamic Systems Fall 2002 Massachusetts Institute of Technology
1 Introduction
This paper discusses the dynamic analysis and simulation of a heat gun. The system consists of an electric heating coil and a universal AC electric motor that drives a centrifugal fan in order to produce airflow. A diagram of the system is shown in Figure 1. Although the system looks relatively simple there are complex interactions between electrical, mechanical, thermal, and fluid domains.
Heating Coil
+
AC Motor Centrifugal Fan
Figure 1: Schematic of Heat Gun
2 System Model
Because the system operates in four domains there are several couplers required to convert energy from one domain into energy in another domain. 2.1 Electro-mechanical Coupling The coupling between electrical and mechanical domains is the universal AC motor. A diagram of the universal motor is shown in Figure 2. Because the windings on the rotor are connected in series with windings on the two poles of the stator, this motor is able to
Schematic and graph removed due to copyright considerations. See reference [1].
Figure 2: Universal Motor Wiring Diagram and -N Curve
operate with an AC or DC power supply [1]. This allows the motor to be treated similarly to a simple DC motor, which is modeled as a linear gyrator. The motor constant, Km, can be determined experimentally by measuring the input voltage and current when driving the motor at a known speed. It is evident from part b of Figure 2 that AC operation is 1
even more linear than DC operation for this type of motor adding validity to the use of a linear gyrator. 2.2 Electro-thermal Coupling The interaction between the electrical domain and the fluid domain is a thermal coupling. To transfer energy to the fluid, the material of the electrical resistor must first heat up. The coupling is modeled as a non-conservative two-port resistor. This is due to the fact that electrical energy is converted to thermal energy, but thermal energy does not create electrical energy. The power dissipated in the resistor is equal to e2/R. This power is converted to thermal energy through the generation of entropy. Thermal energy is stored in the resistor, which acts as a thermal capacitor, and transferred to the air by convection. 2.3 Thermo-Fluid Coupling The thermal energy that is transferred through convection can be modeled using the HRS macro element that is presented in Brown [2]. This bond is used to model heat exchangers and allows the heater temperature to be much larger than the temperature of the fluid at the inlet or outlet port. 2.4 Mechanical-Fluid Coupling The centrifugal fan provides the coupling between mechanical and fluid domains. The geometry of the fan used in this heat gun is a forward-curved blade centrifugal blower, which is also known as a sirocco fan. As with the coupling between electrical and fluid domains, the mechanical to fluid transmission requires a non-conservative coupler modeled as a two-port resistor. Losses in the fan are due to vorticity, friction, and turbulence. An efficiency coefficient, , can be used to account for these losses in the fan. 2.5 System Bond Graph After determining the necessary transmission elements of the system, it is possible to create the bond graph of the system, which is shown below in Figure 3. p eline eback motor GY Se 0 1 1 I:J imotor icoil
fan
RS : Rcoil SR T 0
R : Rmotor Q
Rf
P
1 Pamb Ploss Se
S loss
Se Pamb 0 P,h2 HRS
SC
P,h1
C
R
Figure 3: Bond Graph Representation of Heat Gun
2
2.6 System Equations Because the heat gun has a parallel circuit, it is possible to analyze much of the system as two separate networks. The output of the motor-fan system, which is the airflow through the heat gun, can be determined independent of the thermal characteristics of the system. The thermal response of the system is dependent on the transient in the flow rate; however, the flow rate transient is much faster than the heat coil transient. The equations for the fan are the most difficult to derive, but they can be determined based on conservation of momentum [3]. The rate of change of fluid angular momentum is equal to the torque applied on it:
fan =
dH 0 d = dt dt
(V r )dm = m(V
2 2
r - V 1 r1 )
For the system blade geometry the velocity diagrams shown in Figure 4 are used to determine V2 and V1.
Vr2 V2
2
W2 V2 U2
W1
V1 = Vr1 U1
Figure 4: Velocity Diagrams for Inlet and Outlet Flow of Centrifugal Fan
From Figure 4 it is evident that the V2 is equal to:
V 2 = U 2 + V r 2 cot 2 = r2 + Q cot 2 2r2 b2
where 2r2 b2 is the area of the fan outlet and Q is the volumetric flowrate. The flow into the fan is assumed to be purely radial, which gives V1 = 0. Substituting these velocities back into the torque equation results in the following expression:
fan = mr2 r2 + Q cot 2 2r2 b2 Q cot 2 = Qr2 r2 + 2r2
b2
From this is clear that the fan torque is dependent both on volumetric flowrate and angular velocity. This is consistent with non-conservative two-port resistors, which supports the use of this type of coupling in the system bond graph. However, the torque equation does not account for losses in the fan. The forward-curved blade geometry is not 3
as efficient as a backward-curved blade or an airfoil [4]. Therefore, the fan torque needs to be divided by the fan efficiency in order to model its non-conservative nature.
fan =
Qr2
Q cot 2 r2 + 2r2 b2
The pressure rise in the fan is determined from the conservation of energy (with the efficiency loss taken into account).
fan = PQ P =
r2
Q cot 2 r2 + 2r2 b2
The pressure rise in the fan is equal to the pressure lost as the air travels through the heat gun. An adjustable orifice varies the inlet area allowing for control of the flowrate. Additional restrictions in the flow path include the heating coil and wall friction along the length of the flow path [5].
2 2 2 V22 - V12 Q Q n Li Q - +
P = + hl = fi 2 2 A2 A1 ( ) i = 2
2 Di Ai ( ) L 1 Q2 A2 - A2 n P = Q 2 2 2 2 1 + f i
i 2 = Di Ai 2 Ae 2
2 A2 A1 i=2
where, fi, Li, and Di are the friction factor, length, and diameter of the ith restriction
section, respectively, and Ae is the effective area of the entire system.
The ratio L/D, also known as the effective length, is provided for various pipe geometries,
valves, and other types of restrictions.
The preceding expression for P is in terms of Q only; however, P found from the
energy balance equation also included the angular velocity of the fan, . Therefore, Q
can be solved in terms of the angular velocity, :
r2
Q cot 2
Q 2 = r2 + 2 A 2
2r A 2 2r2 b2 2 e e 2 + 2 Ae
2
2 cot 2 Q - Q - r2 = 0 2r2 b2
Q=
cot 2 cot 2 + 2r b 2r2 b2 2
2
r2 Ae2
The angular velocity is related to the state variable, p, by the following equation:
4
=
p
J
The remaining equations needed to solve for the system dynamics are provided by the motor equations and the state equation for angular momentum, p:
eback = K m i motor
= eline - eback R motor
motor = K m i motor
p = motor - fan
The thermal component of the heat gun behavior is determined from the equations for the HRS macro element and the conversion of electrical energy into thermal energy. The thermal equations are as follows:
Tcoil = T0 e
P=
S loss
SC - S 0 mc
2 2 eline eline = Tcoil S R S R = Rcoil Tcoil Rcoil 1- Tamb / Tcoil 1- Tamb / Tcoil = = 1/ H + 1/ c p m 1/ H + 1/ c p Q
S C = S R - S loss
3 System Parameterization
In order to simulate the heat gun, the system parameters were determined through measurement and experimentation. The following subsections discuss the methods used to determine the system parameters.
3.1 Fan Parameters Most of the parameters associated with the fan involved its geometry, as indicated by the velocity diagrams in Figure 4. The dimensions of the fan and its blades were measured, and the blade angle at the outlet estimated by assuming that the blade formed the arc of a circle. According to Logan [6], the maximum efficiency of centrifugal fans varies from 70 to 90 percent. Logan also provides a table relating centrifugal fan efficiency to volumetric flow rate. The logarithmic regression from his table provides the following relationship between flow rate and efficiency:
= 0.0722 ln(Q) + 0.5638
5
For this application the flow rate is probably on the order of 0.5 L/s, which leads to an efficiency of about 50%. Therefore, the fan parameters are as follows:
%Fan Parameters r1=.03066; B1=59.27*pi/180; b1=.01; r2=.0375; B2=10*pi/180; b2=.01; eta=.50; %Inlet Blade Raidus, m %Inlet Blade Angle, rad %Inlet Height, m %Outlet Blade Radius, m %Outlet Blade Angle, rad %Outlet Height, m %Fan Efficiency
3.2 Flow Parameters The flow parameters were the most difficult to determine by measurement. While the geometries of the inlet and exit were easy to measure, the internal restrictions in the heat gun posed a problem. The inlet and outlet were treated as lossless elements in the flow path, only contributing to the calculation of velocity into and out of the system for Bernoulli's equation. To determine the pressure losses due to the flow within the heat gun, two major sources of loss were considered. The first was the loss associated with the flow directional change due to the fan volute. Fox and McDonald [5] provide a graphical estimation for losses associated with 90 pipe bends that is based on the ratio of the radius of curvature to the diameter of the pipe, r/D. For this system the ratio is ~3, which results in an effective length, Le/D, of about 13. This value is multiplied by 3 to account for 270 of volute bend. The volute pressure loss is also proportional to the loss coefficient, which is determined by the Reynolds number and the Blasius correlation for turbulent flow in smooth pipes:
f = 0.316 Re 0.25
The second, and more significant loss of pressure within the heat gun, is the restriction due to the heating coil. The coil significantly reduces the diameter of the pipe for a length of 10cm just before the outlet. In addition to this significant reduction in diameter, the coil also creates fully turbulent flow in this section of the heat gun. Both of these effects result in huge losses, which are difficult to determine analytically. Therefore, this parameter, which was identified as the effective length of the coil, was left as a variable to be optimized in the simulation. This parameter also accounts for other losses in the fan including the losses at the inlet and outlet that were neglected. The complete set of flow parameters used in the simulation are as follows:
%Flow Parameters Douter=.0670; %Intake Outer Diameter, m Dinner=.0320; %Intake Inner Diameter, m Ain=pi*(Douter^2-Dinner^2)/4%Maximum Intake Area, m^2 A1=cos(6*theta)*Ain/2; %Restricted Intake Area, m^2 D2=.029; %Outlet Diameter, m A2=pi*D2^2/4; %Outlet Area, m^2 rho=1.19; %Assume Constant Density, kg/m^3 mu=2e-5; %Dynamic Viscosity of Air, Ns/m^2 Dduct=.0145; %Volute Diameter, m Aduct=pi*Dduct^2/4; %Duct Area, m^2 Re=rho*.01/Aduct*Dduct/mu; %Reynolds Number in Duct (Assume 1 L/s flow)
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fduct=.316/Re^.25; %Duct Friction Factor (Blasius) fcoil=.015; %Coil Friction Factor Lduct=13*3; %Effective Length of Duct, L/D Lcoil=550; %Effective Length of Coil, L/D Acoil=A2/2; %Effective Coil Area, m^2 A=(A2^2-A1^2)/(A1^2*A2^2)+fduct*Lduct/Aduct^2+fcoil*Lcoil/Acoil^2;
3.3 Motor Parameters The motor parameters were fairly easy to determine, though measurements were made under several different operating conditions to fully characterize the system. The easiest parameter to measure was the resistance of the motor windings, though this varied from 70 to 75 depending on the angle of the motor shaft. Because the system incorporated a universal motor, its parameterization was simplified by utilizing a DC power supply. By applying a known voltage and recording steady-state current draw and shaft speed, it was possible to determine the motor constant, Km, and therewith the motor torque. The motor speed was measured with a timing gun. Km is plotted versus applied voltage in Figure 5 below for two operating conditions: no restriction (cover removed), and minimum restriction (A1=Ain).
0.6
No Restriction (Cover Removed) Minimum Restriction (A1=Ain)
0.5
Motor Constant, Km
0.4
0.3
0.2
0.1
0 0 10 20 30 40 50 60 70 80 90
DC Voltage (V)
Figure 5: Motor Constant, Km, versus DC Voltage
For increasing voltage the value of Km decays to about 0.9 Nm/A. However, it was shown in Figure 2b that AC operation yields slightly different values for Km. Therefore, the steady-state current and speed were also measured for 120VAC. This resulted in Km equal to 0.114 Nm/A, which is about 25% larger than the motor constant corresponding to DC operation. In reality there is a thermal transient in the motor as the rotor and stator heat up due to electrical losses. This results in a slight decay in the current draw that was observed to stabilize completely after about a minute. The decrease in current during this transient was not included in the simulation because it was very small (~10mA). Additionally, the
7
motor resistance was measured immediately after running in order to capture the true steady-state characteristics of the system. The DC measurements were also used in characterizing the low-speed characteristics of the motor. The steady-state operating torque was found for each of the points on Figure 5 and plotted versus the motor speed. This resulted in the torque-speed curve of the load, which is shown below in Figure 6.
0.12
No Restriction (Cover Removed) Minimum Restriction (A1=Ain)
0.1
0.08
Torque (Nm)
0.06
0.04
0.02
0 0 1000 2000 3000 4000 5000 6000
Motor Speed (rpm)
Figure 6: Load Torque versus Motor Speed
This figure is very informative because of the information it contains about motor friction. At low speed operation, one would expect the contribution of torque due to flow rate to be negligible. Therefore, at low speeds the load torque is dominated by motor friction. Judging from the figure, the load torque at 500 to 1500 rpm is entirely due to kinetic friction; beyond this range, the air flow begins to add to the motor load at a quadratic rate. This provides the parameter for motor friction, which is assumed to provide a constant resistive torque (independent of motor speed). This value is not shown on the bond graph, but it would be represented by a constant effort source applied at the 1-junction for motor>friction. The motor parameters are as follows:
%Motor Parameters eline=120; Rmotor=72.5; Km=.114; J=.0001; tau0=.03; %Line Voltage, Vrms %Motor Coil Resistance, Ohms %Motor Constant, Nm/Amp %Rotatioinal Inertia, kg/m^2 %Motor Friction, Nm
3.4 Heat Coil Parameters The heat coil parameters were very easy to measure. An ohmmeter was used to find the resistance of the coil, which was then weighed to determine its mass. The coil was assumed to be Nichrome (80%-Ni, 20%-Cr) and the specific heat was found in Incropera
8
and DeWitt [7]. The value for the initial entropy of the coil does not affect the result. The only value dependent on the coil entropy is the coil temperature, and since temperature is dependent on the difference between the instantaneous entropy and the initial entropy the initial value is arbitrary.
%Heat Coil Parameters Rcoil=6; m=.05; c=385; S0=0; %Heat Coil Resistance, Ohms %Heat Coil Mass, kg %NiChrome Specific Heat, J/kg_K %Initial Entropy Condition
3.5 Fluid Convection Parameters The inlet air was assumed to be at standard temperature and pressure. Therefore, the only variable that was needed for the simulation was the convection coefficient. In order to determine this constant, the thermal behavior of the heat gun was measured using a thermocouple. The following two plots, Figure 7 and Figure 8, show the temperature of the exhaust air for two operating conditions: maximum inlet area and minimum inlet 1
400
Trial area.
450
Trial 2
Trial 3
Average
350
300
Temperature ( C)
o
250
200
150
100
50
0
0
5
10
15
20
25
30
35
40
Time (s)
Figure 7: Exhaust Air Temperature with Maximum Inlet Area
9
600
Trial 1
500
Trial 2
Trial 3
Average
400
T em p eratu re ( o C )
300
200
100
0 0 5 10 15 20 25 30 35 40
Time (s)
Figure 8: Exhaust Air Temperature with Minimum Inlet Area
From the preceding figures it is clear that reducing the inlet area increases the output temperature. This is due to the fact that convection is dependent on the mass flow of air and the flow rate depends on the inlet area. These two figures can be used to adjust the value of H in the simulation.
%Fluid Convection Parameters Tamb=297; %Ambient Temperature, K Pamb=101325; %Ambient Pressure, Pa cp=1004; %Air Specific Heat @ Constant Pressure, J/kg_K H=4.5; %Bulk Convection Constant, J/K
4 System Simulation
Using the parameters defined in Section 3 and the constitutive equations from Section 2.6, the following MATLAB input was used to simulate the dynamic behavior of the heat gun:
%2.141 Term Project %Simualtion of a Heat Gun clear all global r2 eline Rmotor Rcoil Km Tamb S0 m c H cp rho J Vt2 eta A omegaout tauout tau0 %j=1; for j=1:2 theta=14*(j-1)*pi/180; %theta=input('What is the Restriction Plate Angle (0-14)?')*pi/180; Q1(j)=fzero(@solveQ,[1e-20 .02]);
10
omega1(j)=omegaout; tau1(j)=tauout; tau=[Km*eline/Rmotor 0]; omega=[0 eline/Km*30/pi]; %ODE Solver t=0:.1:40; [T,X]=ode45('heatgun_dot',t,[1e-10 S0]); %Shaft Angular Speed Omega(:,j)=X(:,1)/J; for i=1:length(X) %Volumetric Flow Rate Q(j,i)=(r2*Vt2+sqrt((r2*Vt2)^2+2*eta*A*r2^2))/(eta*A*J/X(i,1)); %Fan Load tau_fan(j,i)=rho*Q(j,i)*(r2*(r2*Omega(i,j)+Q(j,i)*Vt2))/eta; %Exhaust Air Temperature Tcoil(j,i)=Tamb*exp((X(i,2)-S0)/(m*c));
Sloss_dot(i)=(1-Tamb/Tcoil(j,i))/(1/H+1/(cp*rho*Q(j,i)));
Qdot(i)=Tcoil(j,i)*Sloss_dot(i);
Tair2(j,i)=Tamb+Qdot(i)/(cp*rho*Q(j,i))-273.15;
%Motor Current eback(i)=Km*Omega(i,j);
imotor(j,i)=(eline-eback(i))/Rmotor;
end close all if j==2 figure plot(omega1(1)*30/pi,tau1(1),'bo',omega1(2)*30/pi,tau1(2),'ro') hold on plot(omega,tau,'k--')
xlabel('Shaft Speed, (rpm)')
ylabel('Torque, (Nm)')
title('Torque-Speed Curve for Motor Indicating Steady-State Operation Points')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure
plot(Omega(:,1)*30/pi,tau_fan(1,:),'b',Omega(:,2)*30/pi,tau_fan(2,:),'r',omega,tau-
tau0,'k--')
xlabel('Fan Speed, (rpm)')
ylabel('Load Torque, Nm')
title('Load Torque versus Speed for Minimum and Maximum Inlet Area')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure
plot(T,Omega(:,1)*30/pi,'b',T,Omega(:,2)*30/pi,'r')
xlabel('Time, (s)')
ylabel('Shaft Speed, (rpm)')
title('Motor Speed versus Time')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure
plot(T,Q(1,:)*100,'b',T,Q(2,:)*100,'r')
xlabel('Time, (s)')
ylabel('Volumetric Flowrate, (L/s)')
title('Volumetric Flowrate, Q, versus Time')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure
plot(T,Tcoil(1,:)-273.15,'b',T,Tcoil(2,:)-273.15,'r')
xlabel('Time, (s)')
ylabel('Heat Coil Temperature, (decC)')
11
title('Heat Coil Temperature for Minimum and Maximum Inlet Area')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure
plot(T,Tair2(1,:),'b',T,Tair2(2,:),'r')
xlabel('Time, (s)')
ylabel('Exhaust Air Temperature, (degC)')
title('Exhaust Air Temperature for Minimum and Maximum Inlet Area')
legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)')
figure plot(T,imotor(1,:),'b',T,imotor(2,:),'r') xlabel('Time, (s)') ylabel('Motor Current, (A)') title('Motor Current for Minimum and Maximum Inlet Area') legend('\theta = 0 deg (max area)','\theta = 14 deg (min area)') end end
The first function called on by the preceding file was an early solution to the problem that ignored the inertia of the motor and fan. It was, therefore, a zero order system providing the steady-state behavior of the motor and fan. The function `solveQ' is shown below:
function solveQ=f(Q) global r2 eline Rmotor Km rho Vt2 eta A omegaout tauout tau0 tauout=(r2^2*eline/Km+Q*r2*Vt2)/(eta/(rho*Q)+r2^2*Rmotor/Km^2)+tau0; omegaout=eline/Km-Rmotor*tauout/Km^2; P=rho*omegaout*(r2*(r2*omegaout+Q*Vt2))/eta; solveQ=sqrt((1/A)*2*P/rho)-Q;
The second function that is solved by the main function is the set of differential equations for the system:
function heatgun_dot=f(t,x) global r2 eline Rmotor Rcoil Km Tamb S0 m c H cp rho J Vt2 eta A tau0 heatgun_dot=[0 0]'; p=x(1); Scoil=x(2); w=p/J; Q=(r2*Vt2+sqrt((r2*Vt2)^2+2*eta*A*r2^2))/(eta*A/w); tau_fan=rho*Q*(r2*(r2*w+Q*Vt2))/eta; tau_motor=Km*(eline-Km*w)/Rmotor; p_dot=tau_motor-tau_fan-tau0; mdot=Q*rho; Tcoil=Tamb*exp((Scoil-S0)/(m*c)); Sloss_dot=(1-Tamb/Tcoil)/(1/H+1/(cp*mdot)); Sr_dot=eline^2/(Rcoil*Tcoil); Scoil_dot=Sr_dot-Sloss_dot; heatgun_dot(1)=p_dot; heatgun_dot(2)=Scoil_dot;
4.1 Simulation Results The simulation was run for equal to 0 and 14, which correspond to the maximum and minimum inlet areas achievable by adjustment of the restriction plate, respectively. The
12
following results show the outstanding correlation between the mathematical simulation and the recorded data. The first simulation output, Figure 9 below, shows the motor torque versus speed. This line can be drawn without simulation using the parameter Km, which was found through experimentation.
Torque-Speed Curve for Motor Indicating Steady-State Operation Points 0.2
0.18
= 0 deg (max area) = 14 deg (min area)
0.16 0.14
0.12 Torque, (Nm)
0.1
0.08
0.06
0.04
0.02
0
0
2000
4000
6000 Shaft Speed, (rpm)
8000
10000
12000
Figure 9: Motor Torque-Speed Curve (Linear Km) Including Steady-State Speeds for Minimum and Maximum Inlet Area
The blue and red circles on the plot are also determined from the zero dynamics of the system. Because the system is first order, with only a very small inertance due to the motor rotor and fan, the majority of the system operation is at steady-state. The blue circle on the plot, which indicates the steady-state motor speed and torque for the maximum inlet area configuration, is at 6669 rpm. This is less than 0.3% larger than the measured steady-state speed of 6650 rpm. The red circle corresponds to a smaller inlet area and therefore a lower flow rate. Because less momentum is transferred from motor to fluid due to the reduced flow, the resulting torque is lower and the speed is higher. An experimental value of this motor speed could not be measured for this geometry because the restriction plate completely covered the fan preventing the use of the timing gun. However, the current drawn by the motor was measured for both of these conditions (it was used to determine the motor constant for the first configuration) and can be used to calculate the steady-state torque and speed. For the high speed operation with the restriction fully closed, the current draw was 530mA. This results in 0.06043 Nm of torque at a speed of 6832 rpm. The MATLAB output for this configuration was 6970 rpm, which is only 2% larger than the value calculated by the current measurement.
13
The following two plots, Figure 10 and Figure 11, show the transients in the motor speed and current. Because of the low motor and fan inertia the system is able to reach steadystate speed in just a few seconds.
Motor Speed versus Time 9000
= 0 deg (max area) = 14 deg (min area)
8000
7000
6000 Shaft Speed, (rpm)
5000
4000
3000
2000
1000
0
0
5
10
15
20 Time, (s)
25
30
35
40
Figure 10: Motor Speed for Dynamic Simulation of Heat Gun
Motor Current for Minimum and Maximum Inlet Area 2
= 0 deg (max area) = 14 deg (min area)
1.8
1.6
1.4 Motor Current, (A)
1.2
1
0.8
0.6
0.4
0
5
10
15
20 Time, (s)
25
30
35
40
Figure 11: Motor Current for Dynamic Simulation of Heat Gun
Figure 12 shows the volumetric flow rate of the heat gun in Liters per seconds for both operating regimes. Substituting the steady-state flow values back into the equation for 14
centrifugal fan efficiency from Section 3.1 gives efficiency values of 52% for the fully restricted fan and 53% for the fully opened fan. These values are only a fraction higher than the value of 50%, which was used in the simulation.
Volumetric Flowrate, Q, versus Time 0.8
= 0 deg (max area) = 14 deg (min area)
0.7
0.6
Volumetric Flowrate, (L/s )
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20 Time, (s)
25
30
35
40
Figure 12: Volumetric Flow Rate for Dynamic Simulation of Heat Gun
While an accurate measurement of flow rate was not made, the experimental load versus speed curve shown previously in Figure 6 illustrated how flow rate contributes quadratically to the load. A similar result is seen in Figure 13, which shows both the shaft load and the motor torque versus speed, which is directly proportional to flow rate.
Load Torque versus Speed for Minimum and Maximum Inlet Area 0.2
0.18 0.16 0.14
= 0 deg (max area) = 14 deg (min area)
Load Torque, Nm
0.12
0.1 0.08
0.06 0.04
0.02
0
0
2000
4000
6000 Fan Speed, (rpm)
8000
10000
12000
Figure 13: Load Curve and Motor Torque-Speed Curve
15
Finally, the thermal aspects of the heat gun were addressed. Figure 14 shows the simulated exhaust air temperature for each of the system configurations considered. The time constant of the thermal behavior was modified by adjusting the convection coefficient. The temperature could be scaled by changing the effective length of the flow restrictions in order to decrease volumetric flow rate. Of course this affects the steadystate current and speed, which were known values. The following plot represents the optimized model of the system, which was constrained to meet the known speed and current parameters while also attempting to simulate the thermal data shown in Figure 15.
Exhaust A ir Temperature for Minimum and Max imum Inlet Area 500
450 400
= 0 deg (max area) = 14 deg (min area)
Exhaust Air Temperature, (degC)
350 300
250 200 150 100 50 0
0
5
10
15
20 Time, (s)
25
30
35
40
Figure 14: Exhaust Air Temperature for Minimum and Maximum Inlet Area
500 450 400 350
Temperature (oC)
300 250 200 150 100 50 0 0 5 10 15 20 25 30 35 40
Time (s)
Figure 15: Experimental Thermal Data for Minimum and Maximum Inlet Area
16
It is apparent from Figure 14 that the magnitude of the simulated temperature was only about 25-30C less than the recorded temperature--an error of 5-7%. The simulation demonstrates a remarkable correlation with the data. There are several possible sources for the error in the simulation with the most likely source being temperature rise in the fluid due to flow friction. This explanation seems very feasible especially when examining the thermocouple data. In the data presented in Figure 7, which provided the thermal data for the fully opened configuration, it was seen that the initial temperature of the air was about 50C. Each of these tests was run with the motor already at steady state, which suggests that the temperature rise of the fluid as it flowed through the heat gun was about 25C before it reached the heating element. This would provide the 25C bias that is seen in the simulation versus the actual data. This hypothesis is also supported by the use of dissipative elements in the model of the fluid circuit without also including their entropy contribution to the system. With a 5-7% effect on the result of the simulation, it appears that these losses are important to the thermal dynamics of the system. The final variable considered in order to validate the model was the temperature of the coil. Although no data was taken for the coil temperature, the color of the wire can be used to qualitatively approximate the steady-state coil temperature. It was observed during thermal testing that the coil glowed in the medium to light orange color range. The following table provided by Process Associates of America indicates the possible range of temperatures corresponding to this color range [8].
Table 1: Metal Color versus Temperature
Table removed due to copyright considerations. See reference [8].
From this table it is evident that the temperature of the coil was probably in the range of 890-940C. The temperature of the coil as predicted by the mathematical model is shown in Figure 16. With a bulk heat transfer coefficient of 4.5 W/K the simulation predicts coil temperatures of 850 and 910C for the fully opened and fully restricted flows respectively. According to Table 1, these temperatures would result in a coil color in the range of salmon to medium orange, which is indeed the case.
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Heat Coil Temperature for Minimum and Maximum Inlet Area 1100 1000 900 800 Heat Coil Temperature, (decC) 700 600 500 400 300 200 100 0
= 0 deg (max area) = 14 deg (min area)
0
5
10
15
20 Time, (s)
25
30
35
40
Figure 16: Temperature of Heating Coil for Minimum and Maximum Inlet Area
5 Conclusion
Although the heat gun initially seemed very complicated due to the coupling between multiple domains, it was found that the lack of energy storage elements in the system resulted in a deficiency of system dynamics. From the constitutive equations for the heat gun it is apparent that the fluid flow exhibits first order behavior and that the thermal behavior depends on both the time constant of the flow rate and on the thermal time constant of the heat coil. However, the time constant of the flow rate is so small compared to the thermal time constant that its contribution is not even apparent in the thermal results of the system model. Because of this, the mechanical, electrical, and fluid flow elements of the system can essentially be treated as a zero order system with complicated static coupling between domains. While the dynamics of the system were relatively uninteresting, the coupling between domains led to a rigorous analysis of domain interactions and provided tremendous insight into non-conservative couplers. In addition, the inability of the model to fully predict the exhaust gas temperatures led to further insight about system dynamics that were not included in the model. A more complete model would attempt to alleviate this problem by including the entropy generated by flow restrictions in the overall temperature increase in the system. Overall, the dynamic model was very successful at demonstrating the behavior of this device in all four domains. Unfortunately, the system could not be characterized entirely through measurement and analysis and some values, such as the effective length of the
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flow restriction and the convection coefficient, were determined simply through trial and error. However, experimental results agreed completely with the final simulated results, with the only major discrepancy being the exhaust air temperature as discussed previously. With this major discrepancy between the results accounted for, the model appears to provide a complete picture of the system.
References
[1] "Universal Motors". University of Michigan, Mechanical Engineering, ME 350 Webpage. Accessed 12/04/2002. http://www.engin.umich.edu/labs/csdl/ME350/motors/ac/universal/index.html. [2] Brown, Forbes T. Engineering System Dynamics: A Unified Graph-Centered Approach. New York: Marcel Dekker Inc. 2001. [3] Wright, Terry. Fluid Machinery: Performance, Analysis, and Design. New York: CRC Press. 1999. [4] "Centrifugal Wheel Designs". Lau Industries Inc., Barry Blower Central Webpage. Accessed 12/08/2002. http://www.barryblower.com/centrifugal.htm. [5] Fox, R.W. and A.T. McDonald. Introduction to Fluid Mechanics: Fifth Edition. New York: John Wiley and Sons, Inc. 1997. [6] Logan, Earl Jr. Turbomachinery: Basic Theory and Applications, Second Edition. New York: Marcel Dekker, Inc. 1993. [7] Incropera, F.P and D.P. DeWitt. Fundamentals of Heat and Mass Transfer: Fourth Edition. New York: John Wiley & Sons. 1996. [8] "Metal Temperature by Color," Process Associates of America. 1995-2002. Website accessed 12/09/02. http://www.processassociates.com/process/heat/metcolor.htm
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