J. Govardhan et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 1, 013 - 024
Design and Experimental Implementation of an Electro-Mechanical Cam operated Valve for Oscillating Combustion J. Govardhan*
Department of Mech. Engineering, AVN I E T, Ibrahimpatnam (M), A.P., India
G.V.S. Rao
S. Rajesham
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Department of Mech. Engineering, PIRM E C, Chevella, A.P., India Department of Mechanical Engineering, PRRM EC, Shabad, A.P., India,
J. Narasaiah
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Department of Mech. Engineering, PRRM EC, Shabad, A.P., India
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Abstract Oscillating combustion is a simple, innovative, low-cost technology can be applied as a retrofit in the heat transfer industries such as steel mills, glass plants, forging shops and foundry process furnaces to enhance the performance characteristics. Different kinds of oscillating mechanism to create oscillations in the combustion used earlier in the research were electrostrictive actuators, cyclic valves, solenoid based (EGR) exhaust gas recirculation valves, rotatory plug valves. This paper explains the design and development of an oscillating valve developed by the author to incorporate and to study the influence of amplitude and frequency of oscillations in an oscillating combustion. Unlike other oscillating valves used earlier a cam operated electromechanical valve was used to introduce oscillations in the liquid fuel flow at ambient conditions. The experiments were carried out on a crucible furnace both at steady state and oscillating combustion mode and compared the two modes of combustion. The
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investigations confirm the benefits of introducing the oscillations during the combustion and effects of oscillating combustion on performance characteristics such as heat transfer rate, melting time, specific energy consumption (SEC) and furnace efficiency. Keywords: Oscillating combustion, oscillating valve, crucible furnace, heat transfer rate, specific energy consumption 1. Introduction Oscillating combustion is a gaining importance these days and attracted significant attention as an efficient technology to meet future fuel economy, energy utilization factor, reduction in emissions and improved thermal efficiency. Experiments on the oscillating combustion with different oscillating valves are ongoing. Furnaces are used with retrofit valves for more promising technological improvements. A clean energy combustion system, Inc. has developed
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modifications were carried out. These include stream lining the flue gas passage in the furnace, incorporating manometers, 3-way cock along with burette, piezometer tubes, thermo-couples with digital temperature indicators and sensing probe. The oscillating valve has been tested at a frequency of 5 and 10 Hz, amplitude of 100 and 200, different airfuel ratios varying from 13:1 to 17:1 above and below the stoichiometric ratio using 10, 15 and 20 kg of aluminum loads. The results indicate the optimization at certain parameters used in the tests. The main focus in this paper is on the proposed oscillating valve on the kinematics profile of the cam, variable speed actuator, system modeling, design and control considerations, fluid mechanical and thermal considerations and variation in fuel flow during the fuel-rich and fuel-lean zones.
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environmental friendly technologies to burn wide range of fuels using oscillating combustion. The principle on which the valve works to introduce oscillations in the combustion is based on cyclical perturbation of the fuel line. GTI tests in late 1980â€&#x;s used a solenoid valve or solenoid-based exhaust gas recirculation valve. Air Liquide Chicago research centre used rotatory plug valve. These valves did not have enough life and were expensive, thereby unsuitable for industrial applications. Ceramphysics, Inc. Ohio, developed a low cost long life valve known as Solid State Proportionate valve with a flow capacity of only 40 scf/h operating at high frequency of 20 Hz was used and found virtually noise free. GTI laboratories are producing SSP valves with variable flow capacity. Experiments on the oscillating combustion technology were conducted on natural gas with variable conditions. In this article, the author developed indigenously an electro-mechanical cam operated valve which is simple, low-cost, reliable in operation and incorporated in a crucible furnace to experiment the oscillating combustion especially on liquid fuel. The oscillations of the fuel create alternately successive fuel-rich and fuel-lean zones within the furnace. The fuel-rich zones are more luminous, longer in length and causes more heat transfer from flame to the load. Both the zones mix in the furnace only when the heat has been transferred from the fuelrich zone to the load thereby resulting in low peak temperature in the furnace. The effects of oscillating combustion are low melting time, increased productivity rate, reduction in emissions, low specific energy consumption and increase in furnace efficien Before presenting the experimental data results, a brief mechanism description is presented. The oscillating valve was introduced as a retrofit in a fuel fired crucible furnace and tested on both steady state and oscillating combustion. Typical furnace
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Nomenclature
A = ampere a.c. = alternating current A/F = air-fuel ratio D = diameter d.c = direct current EGR = exhaust gas recirculation F = force f = friction factor, frequency g = acceleration due to gravity, gram GTI = gas technology institute kg/h = kilogram per hour k = spring constant kg = kilogram m = mass mA = milli ampere N= speed P = pressure, power rpm.= revolutions per minute scf/h = standard cubic feet per hour SEC = specific energy consumption SHM = simple harmonic motion SSP = solid state proportionate T = Torque V = volt
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J. Govardhan et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 1, 013 - 024
v = velocity Z = datum
Θ = uniform angular displacement θ0 = cam angle for out stroke μ = viscosity e = density τ θ, = torque ωn = circular frequency of SHM ω = angular velocity ∆ = small distance Π = radian
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Greek symbol
Fig. 1. Displacement, velocity and acceleration diagrams.
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Subscripts
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Fn = natural frequency Fo = maximum acceleration of follower out stroke Fr = maximum acceleration of follower return stroke HL = loss of head Hz = hertz , cycles To = time of outstroke Tr = time of return stroke Vo = maximum velocity of follower out stroke. Vr = maximum velocity of the follower return stroke
2. Kinematics Profile of Cam 2.1. Uniform velocity
Fig.2. Modified displacement, velocity and acceleration diagrams
For the uniform motion or uniform velocity of the follower, the slope of the displacement curve will be constant because the displacement is directly proportional to time and time is directly proportional to θ, for the cam running at uniform angular velocity .
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It is seen from the Figure 1 that, during the period θd1 the displacement remains unchanged and so is the case during θd2 Thus, during θd1 and θd2 the velocity of the follower is zero. During θ0 interval, velocity has a definite value and again during θd2 it is zero.θ0+ θd1+θr+θd2 = 3600 or one complete revolution of the cam. It may be pointed out here that at point A, the velocity of the follower is changed from to a finite value in an infinitely small interval of time therefore the acceleration to be produced will be infinitely large. For any small mass of the follower the inertia forces produced will be infinitely large causing the high stress levels and wear. Therefore, uniform velocity of the cam is not a practical proposition. It is therefore, necessary to modify the conditions which govern the follower motion, reducing the values of acceleration to finite value this is accomplished by rounding the sharp corners at A,B,C and D, so that the follower reaches the desired velocity gradually at the beginning of the stroke. Shorter rounding the radius the nearer to the undesirable conditions of the constant velocity profile.
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Fig.3. S.H.M of the follower diagrams
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Figure 6. Profile of the cam
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Fig.4. Cam with angles
Figure 5. Profile of SHM of the follower
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A-B = θ0 = angle of the cam for out stroke (1200) B-C = θd1 = dwell angle (400) C-D = θr = angle of the cam for return stroke (1200) D-E = θd2 = dwell angle (800)
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In the above Figure 2, the follower motion is assumed with uniform acceleration for these small intervals of time. A radius equal to the follower displacement is chosen often in practice. Unfortunately, the modified straight line profile does not exhibit very attractive characteristics. The derivative of the acceleration called jerk or pulse will have infinite spikes in modified straight line case. This is a measure of rate of change of inertia force and thus give impact levels impact causes noise, shortens life due to wear and fatigue.
peripheral speed = π S/2 *1/ to or = π S /2*(ω / θ0)
(1)
or the maximum velocity on out stroke along diameter is equal to peripheral velocity of point along the cumference of the circle on which the point is assumed to move for its projection to execute S.H.M. Vo = π ω/ θ0 *S/2
(2)
and it occurs at point when cam has turned by an angle θ = θ0/2 also the centripetal acceleration of the point by which SHM defined is given by (during out stroke)
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2.3. Simple Harmonic Motion
For outstroke we have peripheral speed of the point moving along the circle with stroke as diameter
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2.2. Uniform acceleration:
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Figure 3 shows that, velocity curve is a sine curve and acceleration curve is a cosine curve. Obviously, the velocity and acceleration on the return stroke are higher than those on the outward stroke. The same displacement is completed during return stroke in the angular rotation of the cam rotation which is half of that on the out stroke in the example into consideration.
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Let θ = uniform angular displacement; θ0 = angle of the cam for out stroke θr = angle of the cam for return stroke To = time to perform outstroke θ0/ω; Tr = time to perform return stroke= θr/ω Vo = velocity of follower out stroke; Vr = velocity of the follower return stroke; Fo = acceleration of follower out stroke; Fr = acceleration of follower return stroke ω = uniform angular velocity; S = Stroke of the follower.
SHM is defined as by the projection on the diameter of point moving at a uniform speed round the circumference or the periphery of a circle with the stroke as the diameter.
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Fo = V02 / (S/2) = (πω/θ S/2)2 */S/2
(3)
= π2 ω2 / θ0 2 *(S/2) (4) And this is maximum positive at θ =0 and maximum negative at θ = θ0 and at θ ≤ θ0 /2 it is zero. Discussing on the same lines for the return stroke we have Vr = πω / θ*(S/2)
(5)
Fr = ω2 π 2 / θr2 * S/2
(6)
2.2. Non-linear mechanical relations of spring The natural frequency and time period of the spring may be derived from “Equilibrium Method”. This is based on the principle that whenever a vibrating system is in equilibrium, algebraic sum of all the forces and moments acting on it is zero. This is in accordance with
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J. Govardhan et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 1, 013 - 024
D`Alembert‟s principle, that the sum of the inertia forces and external forces on a body in equilibrium must be zero.
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The kinematic profile of the cam position versus time (amplitude), swivel disc speed versus time (frequency) are of fixed shape and are timed relative to the oscillating valve‟s swivel disc position. An electromechanically operated motor system can control the amplitude and frequency of the valve. The swivel disc positioned at an angle or partially in closed position and allows the amount of fuel required as per the air-fuel ratio. From this position it can be further partially closed of fully closed to suit to the furnace operation requirement. In effect, the oscillating combustion valve should have some features of flexibility. - The system must operate with minimum power consumption. - It should not have excessive wear of the cam and follower. - The system should have soft contact between the cam and follower and should be noise free.
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2.2.1. Variable speed actuator
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The follower is connected to the swivel disc and is held by a spring in its equilibrium position. The equilibrium position for this mass-spring system is in the middle of the valve stroke. Such a system possesses its natural frequency (fn), mass (m), spring constant (k) and frequency ratio (S). In this, an initial displacement of the valve in the direction of the spring would result in sustained oscillations in the valve at the systems natural frequency considering any requirement of damping. Relatively small current is consumed during the operation.
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Fig.7. Schematic view of the cam profile with spring (above) and the oscillating valve In the Figure 7 as shown, the spring supports the follower with its free end at A-O. When the follower is shifted by the cam profile the spring is stretched by a distance ∆ (for sustained damping) and B-O becomes the equilibrium position. This ∆ is the static deflection of the spring by the cam. Let
S = Stiffness of the spring – Force required for the unit deflection. In the static equilibrium position, (considering the outward movement as upward stroke and inward movement as downward stroke) Upward force = power of the motor P=Tω (7) (T = Torque of the motor, ω = angular velocity) = T x (2ПN/60) = (F x (D/2)) x (2ПN/60)
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J. Govardhan et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 1, 013 - 024
(8)
2.3. Non-linear Mechanical Transformation Relations
(9)
„θ‟ is a function of „S‟ and vice-versa. It is easy to show that the use of non-linear mechanical transformations implies that the following relations hold between θ and S.
Upward force = Downward force; or (2П DNF) / (120)
=
S∆
Now when the spring is pulled during the operation by the follower through a distance (Z) m, the forces acting on the spring will be, Inertia force = downward force ..
= (2П DNF) / (120) X
(10)
As the sum of inertia and the external force on the body in any direction is zero (D‟ Alembert‟s principle). Inertia force + spring force = 0 ..
[(2П DNF) / (120) ] X
+
S.*X = 0
(12)
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When the motor starts, the spring would start oscillating above and below the equilibrium position. These oscillations would tend to continue till the motor is stopped. The above equation can be written as,
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.. X + { S /[(2П DNF)/ (120)] } *X = 0.
(13)
The above equation is of simple harmonic .. motion and is analogous to X + (ωn)2.x = 0, Where ωn = natural frequency or circular frequency of SHM. and ωn = √ { S / [(2П DNF) / (120)] } and linear frequency of the vibration system fn =. ωn/ 2П (14)
and the Time period T =1/ fn
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Therefore the velocity of the follower is given by, V=dS/dt V=dS/ dθ * dθ/dt V=dS/d θ *ω V= ω dS/dθ (17)
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Spring force or restoring force = S.*X downward (11)
Let S= Stroke length or linear displacement of the follower. θ= cam angle of the rotation in radians. S = f (θ) → θ= f -1(S) (16)
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= (2П DNF) / (120 ) Downward force = S*∆*g or S∆ (g=negligible)
(15)
(Slope of the displacement curve at angle θ or time t) Since the cam rotates with uniform angular velocity of ω radians/sec. dθ/dt =ω Acceleration of the follower is given by F=dV/dt = dV/d θ * d θ/dt = ω dV/dθ = ω d2S/d θ2 m/sec2 = ω dV/d θ (18) = (Slope of the velocity curve at angle θ or time t) Also, pulse or jerk is given by, P=ω3 d3y/d θ3 (19) = (Slope of the acceleration curve at angle θ or time t) The non-linear mechanical transformer provides a desirable relation between S and θ domains. By equating the power in the S and θ domains and using the non-linear mechanical transformer characteristics, the following relation results. τ θ = dS/ dθ. Fs (20)
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1. The fuel flow regulations are to be keenly monitored. The amount of the discharge of fuel before installation of the valve should be maintained after installation too. This was ensured by calculating the flow before and after the valve‟s installation. 2. There was a slight pressure drop in the fuel system when the valve was positioned. This was taken care by adjusting the fuel flow lever into higher position and also the fuel level in the fuel drum. 3. Importance is given to precise control of the cam operation with the follower for soft raising and lowering in turn the movement of swivel disc in the valve. 4. The friction between the cam and follower is at its minimum as there are no much of variable loads during the operation. 5. The motor can rotate steadily due to the reason for this characteristic is that a flat slope is used for the cam profile. 6. Oscillating valve is carefully designed and extra care was taken to avoid any kind of fuel leakages and unwanted noise during the operation. The motor can rotate past for 60o of the cam in θ domain where as the follower can sway for about 12 mm in the S domain. [θ domain is in the angle of cam and „S‟ domain is the shift of follower from its equilibrium position]. Since the mass and inertia of the moving components in the oscillating valve apparatus are as small as possible, the required spring constants and forces grow proportionally with the mass and inertia of any components of any system. Since the mass and inertia of the swivel disc which restricts the fuel flow in the valve happened to be very small, soft and light spring was employed with low spring constant. This was enabled to employ a small
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The kinematics of the cam are described by Inertial force + Spring force = 0 The inertia represents the mass of the cam and actuator‟s inertia. When the inertia and spring forces are linear, the force balance becomes (21) m d2s/dt2 + ks = 0
There are number of important issues in the design of the oscillating combustion valve.
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Where τ θ, is torque in the θ domain and Fs is the force in the S domain. At the end of the stroke, the slope of the cam characteristic dS/ dθ is very small. This characteristic makes it easier to control the motor velocity near the end of the stroke. A mechanical design in which the smooth operation follows natural dynamic trajectories require no longer actuator forces to apply control is the required means of reducing peak actuator power.
where „m‟ is the mass inertia and „k‟ is spring stiffness, respectively. A cam operated electro-mechanical valve with small spring forces at both ends of the stroke results in smooth cam kinematics and less jerks without large driving forces.
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3. Design and control considerations
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The views of experimental setup with the temperature measuring apparatus, oscillating combustion valve and swivel disc positions inside the valve are shown in Figures 8-12. The input supply is fed to the motor through a potential differentiometer or through variable resistance as per the requirement of the speed. The motor sets into motion and makes the cam to rotate (cam is fixed on the spindle of the motor). The cam which is in contact with the follower which operates against the massspring mechanism acts as non-linear mechanical transformer. The swivel valve positioned in the fuel flow chamber is actuated by the non-linear mechanical transformer thereby restricting the fuel flow inducing the oscillations.
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J. Govardhan et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 2, Issue No. 1, 013 - 024
Fig. 10. Oscillating valve with cam
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9V d.c. motor, 4 mA current with 600 rpm. The motor was controlled by a potentiometer to get the required operating frequency (5 to 10 cycle/sec). 2 modes of frequency and the amplitude of the swivel disc were chosen for the experimental analysis. The oscillating combustion valve along with its accessories was placed on a stand and coupled to the fuel line system. A converter used to convert 220V a.c. input into 9V d.c and a digital tachometer was used to ensure the required rpm of the motor to be set in.
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Fig.8. Experimental setup
Fig. 11. Schematic of oscillating valve 3 D view
a) closed
b) 300
c) 450
Fig. 9. Oscillating valve on fuel line d) 600
e) 750
f)900
Fig. 12. Schematic of swivel disc positions inside the oscillating valve
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The pressure drop or loss of head in the system due to the installation of the oscillating valve in the system may not be so important but is considered. According to Hagen-Poiseuille law the loss of heat or pressure drop for a laminar flow (P1 – P2 )/ ω
= 32 μvl / ωd2
(21)
and Darcy- WeischBack gives HL = (4flv2) / 2gd
(22)
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Here, the pressure drop is directly proportional to the length of the pipe which is used is a major concern. Since the pipe used for the oscillating valve was very small. Thereby pressure drop can be neglected. Since the fuel is incompressible applying the law of Bernoulli‟s Equation and choosing a reference line between the oscillating valve and the fuel drum, the pressure loss can be found. (23) i.e. (P1 – P2 )/ ω= (V22 / 2g) + Z2 ;
position, due to the cam and follower action it causes reduction in the volume. Thereby, restricts the fuel flow through the valve to the burner. When the cam resumes its normal position, the spring attached to the follower brings back the swivel disc to its original position or to its starting point. The swivel valve could open and close in 1/10th of a second, and can vary according to the control by potentiometer. The oscillations of the swivel disc are adjusted electromechanically and the amplitude of the swivel disc is adjusted according to the size of the cam or cam profile. When the oscillating valve was tested at different oscillations, the fuel flow was scaled up from 3.0 kg/h average flow to 4.50 kg/h depending upon the air-fuel ratios and furnace loads. The valve in the position but without oscillations it was tested from 3.0 kg/h to 5.0 kg/h at ambient conditions. When oscillations occur, the pressure amplitude is sufficient enough to produce significant variations in axial velocity within the nozzle annulus. These axial velocities can vary during the oscillating combustion. The swirl vanes on the surface of the fuel gun of the burner would provide combustion air with a tangential velocity of high swirl. Due to this the flow around the nozzle‟s annulus is having high and low regions of tangential velocity convected along the main axial flow of the fuel. The magnitude of heat release depends upon the variations in the axial velocity of the fuel due to the variations in amplitude and frequency of oscillations introduced by the oscillating valve and the variations in the tangential velocity of combustion air. The load heats up faster since heat transfer rate from flame to load increases due to more luminous fuel-rich zones. The increased turbulence and high luminous flames created by the flow oscillations break up the thermal boundary layer In oscillating modes of operation the oscillating valve is able to open and close steadily at higher amplitude and lower
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4. Fluid Mechanical Considerations
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( where ω = eg ) e = density of fluid ; g = acceleration due to gravity V2 = velocity at the oscillating valve; Z2 = datum line above height.
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The calculations gave small amount of pressure drop which was taken care by adjusting the fuel lever at the entry and by increasing the fuel level in the fuel drum. 5. Results and comments
The proposed oscillating valve has a swivel disc incorporated on the fuel flow pipe. The axis of the swivel disc is perpendicular to the axis of fuel flow through the pipe. When the swivel valve is actuated, it rotates either side of its axis and controls the volume of the pipe. When the swivel disc is oscillated from its
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(a) (b) (c) (d) (e) (f)
Radiant heat in furnace Oscillating fuel-rich flame Steady state flame Melting operation Molten metal Furnace with sensing probe
Conclusions
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The objective of the development of the oscillating valve is to integrate into electromechanical controlled experimental test stand supported by thermo-couples, digital temperature indicators and sensing probe along with mechanical apparatus, and to carry out experiments to improve performance characteristics of a furnace from steady state mode to oscillating mode of combustion. The oscillating valve developed was found to be ideal for the oscillating combustion as the experiments carried out on liquid fuel at varying air-fuel ratios, amplitude, frequency and load have shown promising results. The amplitude of the flow rate, the oscillations produced by this valve were adjusted mechanically and electrically, thereby the valve is considered to be flexible and appear to be easier to scale up for any furnace. To help the heat transfer industry to switch on to newer combustion concepts, especially the oscillating combustion which is an advanced technology, introduction of oscillating valve finds answers to many questions in terms of low melting time, low fuel consumption and low specific energy consumption, increased productivity rate with reduced emissions and increase in thermal efficiency of the furnace. The theoretical aspects of systems modeling, design and control considerations, fluid mechanical considerations were analyzed. Based on these considerations the embodiment of the valve was realized.
(b)
(c)
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(a)
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frequency facilitating break up of thermal boundary layer which shortens heat up time. In the oscillating combustion mode, within a reasonable short time the furnace wall temperature becomes more or less uniform because time scale of the flame propagation is less and its velocity is faster due to more luminous flame from the fuel-rich zone of the flame. Fuel consumption tends to become low due to less time taken for the load to melt. This is due to the oscillations created during the operation by the oscillating valve oscillates the air-fuel ratio of 13:1 into above and below the stoichiometric ratio, producing alternatively fuel-rich and fuel-lean zones in the flame resulting in improved efficiency. Some of the visual observations made after the retrofit of oscillating valve and experimentation are shown here. Distinct difference can be noticed between the steady state combustion to oscillating combustion flames. The oscillating combustion flame was found to be highly turbulent, radiative and more luminous to that of the steady state flame.
(d)
(e)
(f)
Fig.13. Different types of flames and status of molten metal
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The author is grateful to the Management of A.V.N. Institute of Engineering & Technology, Ibrahimpatnam for their support and P.R.R.M.
Engineering College, Shabad, R. R. Dist., Andhra Pradesh, India for providing the facilities for the execution of this experimental analysis in the P. T. Laboratory of the Department of Mechanical Engineering. References
3. 4.
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Clean Energy Combustion Systems, Technologies Delabroy O, Louedin O. et al. 2001 “Oxy combustion for reheat furnace”. Joint International Combustion Symposium 2001: 5: 217-240. Energy Matters, Spring 2002 – Office of Industrial Technologies. J. Govardhan and GVS. Rao, An Experimental investigation to study Temperature Distribution and Thermal Transfer in a Crucible Furnace with Oscillating Combustion, International Journal of Applied Environmental Sciences (IJAES), ISSN 0973- 6077 Volume 4, Number 4, pp. 485-498, 2009. J. Govardhan and GVS. Rao, Break up of Thermal Boundary Layer- An Energy Improvement Strategy using Oscillating Combustion Technology, IEEE- Xplore (CPS), 978-0-7695-3884-6/09 IEEE , pp 1025-1030.in the proceedings of the 2009 Second International Conference on Emerging Trends in Engineering & Technology, ICETET-09), Nagpur, India. Stretcher, eric et al. 2001, “oscillating combustion technology boosts furnace efficiency”, Industrial heating. John ,C and Wagner, 2002, “ Demonstration of oscillating combustion on a reheat furnace”. Final Report, Reports, Publications and Software ; 56: GRI-02/0144. John ,C and Wagner, 2004 “NOx emission reduction by oscillating combustion”, GTI. T.A. Parlikar, W.S.Chang et al, 2005“, Design and Experimental Implementation of an Electromagnetic Engine Valve Drive”, IEEE Transactions on Mechatronics.VOL. 10.No. 5.
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10. Ronald J. Pierik and James F. Burkhard, 2000, “ Design and Implementation of a Mechanical Variable Actuation System”, SAE 2000 World Congress, Detroit, Michigan, SAE Technical paper series, 2000-01-1221. 11. Theory of Machines by Thomas Bevan M.Sc, M.Tech (Manchester) AMI Mech.E. Senior 12. Theory of Machines and Mechanisms P.L. Ballaney, 13. Theory of Machines by S.S.Rattan
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Acknowledgements
6.
7.
8. 9.
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