ANALYSES & EXPERIMENTATION REPORT Authors Kevin Dailey (kevindai@ buffalo.edu), Carter Twombly ([email protected]) Advisors Professor Joseph C. Mollendorf, PhD, Paul E. DesJardin, PhD Institution University At Buffalo, The State University of New York Grant “Combustion Characterization and Optimization of Outdoor Wood Burning Hydronic Heaters”, DesJardin, P (PI), JC Mollendorf (COPI) QUADRAFIRE PELLET STOVE PROJECT MAE 499 – Independent Research Study Summer 2016 ABSTRACT The continued analysis of Quadra-Fire Classic Bay 1200’s heat exchanger (HX) is annotated by this report. Throughout the second term of our grant, we have furthered theoretical reasoning and expanded on experimental fronts. The target of our study has remained the same; to increase the efficiency and reduce emissions of a residential pellet stove. Methods include characterizing efficiency based on the 1 st law of thermodynamics; energy balances carried through on the basis of differential control volumes. While boundary conditions were declared to be average temperatures [of air entering the room and exiting the flue stack] throughout the first term, we have significantly increased the reliability of our data to this respect, one item in a general overhaul of experimental utility. Here, we relax certain assumptions made initially, increasing viability of theory and testing methods alike. Chief theoretical modifications outline the relaxation of a former assumption which neglected multi-modal transfer possibilities; this report introduces quantification of radiative influence on heat output. In terms of experimentation, several modifications now facilitate new metrics and enhance acquisition of original species. Account expenditures have increased accessibility of instrumentation, supporting intuitive changes in solving problems which are hardware- sensitive. Summarily, this project combined thermal sensory upgrades, conservative theoretical modelling, and numerical computation improvements in order to provide a comprehensive foundation for overarching efforts. Results include a model to meter energy transfer as a continuous function of HX transmission. The model inherently accounts for radiation with respect to advanced convention; radiation is discovered to significantly influence stove’s achievement of basic heating purposes. Coded in MATLAB, the primary model is accompanied by several additional scripts to facilitate reproducibility of data logging. DAQ is initially compiled with Labview techniques, and the deliverables now automate various tiers of subsequent processing, rendering relationships which are accurate and easily interpreted. Ultimately, short term results pertain to an overall redesign by uncovering gaps between imperfect and ideal conditions. An essential finding was the notably air-rich reaction in the firepot, which directly impedes performance due to incomplete combustion of volatiles in baseline mixture. Research has included parametric analyses of digital logic behind stock operation; the wiring harness is thoroughly depicted in this report. Along with other parametric aspects, this leg of development outlines practical concerns for redesign; attributes drawn from both thermal science and circuit concept.
Transcript
ANALYSES & EXPERIMENTATION REPORT
Authors Kevin Dailey (kevindai@ buffalo.edu), Carter Twombly ([email protected])
Advisors Professor Joseph C. Mollendorf, PhD, Paul E. DesJardin, PhD
Institution University At Buffalo, The State University of New York
Grant “Combustion Characterization and Optimization of Outdoor Wood Burning
Hydronic Heaters”, DesJardin, P (PI), JC Mollendorf (COPI)
QUADRAFIRE PELLET STOVE PROJECT MAE 499 – Independent Research Study
Summer 2016
ABSTRACT
The continued analysis of Quadra-Fire Classic Bay 1200’s heat exchanger (HX) is annotated by this report. Throughout the second term of
our grant, we have furthered theoretical reasoning and expanded on experimental fronts. The target of our study has remained the same; to
increase the efficiency and reduce emissions of a residential pellet stove. Methods include characterizing efficiency based on the 1st law of
thermodynamics; energy balances carried through on the basis of differential control volumes. While boundary conditions were declared to
be average temperatures [of air entering the room and exiting the flue stack] throughout the first term, we have significantly increased the
reliability of our data to this respect, one item in a general overhaul of experimental utility. Here, we relax certain assumptions made
initially, increasing viability of theory and testing methods alike. Chief theoretical modifications outline the relaxation of a former
assumption which neglected multi-modal transfer possibilities; this report introduces quantification of radiative influence on heat output. In
terms of experimentation, several modifications now facilitate new metrics and enhance acquisition of original species. Account
expenditures have increased accessibility of instrumentation, supporting intuitive changes in solving problems which are hardware-
sensitive. Summarily, this project combined thermal sensory upgrades, conservative theoretical modelling, and numerical computation
improvements in order to provide a comprehensive foundation for overarching efforts. Results include a model to meter energy transfer as a
continuous function of HX transmission. The model inherently accounts for radiation with respect to advanced convention; radiation is
discovered to significantly influence stove’s achievement of basic heating purposes. Coded in MATLAB, the primary model is
accompanied by several additional scripts to facilitate reproducibility of data logging. DAQ is initially compiled with Labview techniques,
and the deliverables now automate various tiers of subsequent processing, rendering relationships which are accurate and easily interpreted.
Ultimately, short term results pertain to an overall redesign by uncovering gaps between imperfect and ideal conditions. An essential
finding was the notably air-rich reaction in the firepot, which directly impedes performance due to incomplete combustion of volatiles in
baseline mixture. Research has included parametric analyses of digital logic behind stock operation; the wiring harness is thoroughly
depicted in this report. Along with other parametric aspects, this leg of development outlines practical concerns for redesign; attributes
drawn from both thermal science and circuit concept.
1
TABLE OF CONTENTS
Introduction (2)
o Fundamental Objective
o Means
o Conclusive Projection
Modeling-Recall (3)
Modeling-New Considerations (5)
Theory- Recall (7)
Assumptions (8)
o Heat Output Values
o Energy Conservation
o Fluid Property Evaluation
o Cold Side Heat Transfer Coefficient
Theory – Continuous Convection Solution (10)
Theory – Additional Heat Loads (16)
Theory – Radiation (20)
o Assumptions
- Opaque, Gray, and Diffuse Behavior
- Re-Radiating Surface 3
- Non-Participating Medium
- Infinite Plane and Row of Cylinders
- Negligible End Effects
o Analysis
Experimentation – Introduction (29)
o Intentions
o Overview of Documentation
Experimentation – Baseline Design (30)
o Instrumentation and Materials
- Hardware
- Software
o Procedure
o Results
Experimentation – Final Design (33)
o Additional Instrumentation and Materials
o Methodology
o Procedure
o Results
Conclusions – Introduction (45)
Conclusions – Results vs Manufactures (46)
Conclusions – Digital Logic (49)
Appendix I (52)
Appendix II (61)
References (75)
2
INTRODUCTION
Fundamental Objective
Our research is conducted in an effort to redesign the Quadrafire Classic Bay 1200 Pellet
Stove. Redesign is characterized by two intuitive criteria:
Increasing energy efficiency
Decreasing harmful emissions
Specifically, this research focused on the objective of accurately characterizing the stove’s
power, or heating load, such that design stages can proceed with comprehensive understanding
of mechanisms currently employed to achieve the baseline load.
Means
Our approach begins at the definition of heating load Q = UA ∆𝑇𝐿𝑀 as function of the
heat exchanger’s longitudinal direction, x. This is ultimately achieved on the basis of mass and
energy balances for an arbitrary element with differential length dx, carried out according to
fundamental laws thermodynamics. Previously characterized on the basis of log mean difference
analysis with convection as the sole mode of transfer, our new approach accounts for radiation
transfer. Found to significantly influence the effective transfer load, the addition of a radiation
term renders a multi-modal transmission approach, relaxing previous neglect of this mechanism
which idealized model to an unrealistic extent.
Conclusive Projection
The equation of power as a continuous function of system geometry, with radiation
accounted for, yields a solution that is both precise and accurate. Properly oriented in
conjunction with acquisition of empirical data and numerical software techniques, the model is
integral to understanding the effects of parameter adjustments made to improve overall viability,
and allows us to continue with an with direct interactive feedback on such modifications.
3
MODELLING – RECALL
Figures 1 & 2 recall previous visuals depicting basic flow convention of either fluid.
Figure 3 recalls CAD rendering of one half of the system’s total volume, with basic geometry
highlighted for reference.
Figure 1 Figure 2
Figure 3
4
Tables 1 & 2 relay values obtained for constant geometric parameters, conducive to geometry
required for transfer analysis. Parameters in Table 2 were derived from raw dimensions, and are
compiled for referential convenience.
The entire cold fluid cross sectional area is characterized by the blue fill (Fig 4) and is:
𝐴𝑐,𝑡𝑜𝑡𝑎𝑙 = 4 ∗ (𝜋𝐷𝑖
2
4)
The cross sectional area of the cold fluid flowing through a singular tube is then:
𝐴𝑐,1 𝑡𝑢𝑏𝑒 =𝜋𝐷𝑖
2
4
The hot fluid cross sectional area is characterized by the red fill (Fig 4) and is:
𝐴ℎ = (𝑊𝑖 ∗ 𝐻𝑖) − (4((𝜋𝐷𝑖2)/4))
Hydraulic Diameter; Wetter Perimeter:
𝐷ℎ =4∗𝐴ℎ
𝑃𝑤=
4∗((𝑊𝑖∗𝐻𝑖)−(4(𝜋𝐷𝑖
2
4)))
(2∗𝐻𝑖)+(2∗𝑊𝑖)+(4∗𝜋𝐷𝑜𝐿)
Heat Transfer Surface Areas:
𝐴𝑠𝑢𝑟𝑓,𝑐 = 4(𝜋𝐷𝑖𝐿)
𝐴𝑠𝑢𝑟𝑓,ℎ = 4(𝜋𝐷𝑜𝐿)
Geometry [m]
Outer HeightHo 0.065405
Inner HeightHi 0.060325
Outer WidthWo 0.1905
Inner WidthWi 0.18542
LengthL 0.508
Outer DiameterDo 0.0381
Inner Diameter Di 0.03429
Derived Geometry
Ah 0.006625094
Ac, total 0.003693898
Ac, 1 tube 0.000923474
Wetted Perimeterhot fluid [m] Pw 0.97026872
Hydraulic Diameterhot fluid [m] Dh 0.02731241
Cold Side Surface Area [m2] Asurf, c 0.218897631
Hot side Surface Area [m2] Asurf, h 0.24321959
Cross Sectional Areas [m2]
Table 1 Table 2
Figure 4
5
MODELLING – NEW CONSIDERATIONS
Figure 5 depicts control volume of system with
further visual annotation.
Thin-lined arrows represent convective
paths of hot (red) and cold (blue)
airflows, while larger arrows roughly
indicate location of initial boundary
temperatures which were metered with
thermocouples.
The dimension labeled ‘L’ is the
longitudinal direction to which our new
approach quantifies load q(x). Thus for
any given distance x along this axial
direction, power is calculated using log
mean difference analysis for a counterflow
configuration with characteristic length x.
Figure 6 relays a visual extracted from an external
source, which is useful in description of ideal
locations of temperature metering for accurate
calculation of transfer efficiency. In the case of the
stove shown, calculations would relay such efficiency
taken across the entire system.
Location 1 analogous with 𝑇ℎ𝑖
Location 2 analogous with 𝑇ℎ𝑜
Location 3 analogous with 𝑇𝑐𝑖
𝑇𝑐𝑜 (not shown) ideally located where cold
fluid exits exchanger to room through diffuser
Figure 5
Figure 6
6
Figure 7
The frames above in Figure 7 depict concept visualization of thermal profiles due to radiative
heat energy. It is important to note the chief source of radiation in our case, which, for either half
of the control volume, is a 0.2” thick metal plate. Spanning the entire bottom face of the system
up until the point where tubes exceed length of combustion chamber, the left frame shows this
plate’s orientation; right plate has been removed in the picture while left plate is seen behind
imposed thermal profile.
Thermal concept in left frame depicts hypothetical temperature distribution for system’s
bottom face boundary with respect to location of firepot. In other words, a rough visual of
radiation from heat source to fire-side plate face.
Right frame then depicts thermal profile of a singular tube surface, resulting from
subsequent radiation transfer from the tube-side plate face to outer tube surface area.
7
THEORY – RECALL
*The heat transfer theory cited from this point is majorly predicated on material from the
text ‘Introduction to Heat Transfer, Sixth Edition’, by Theodore L. Bergman, Adrienne S.
Lavine, Frank P. Incropera, and David P. DeWitt. Any other sources will be cited periodically
throughout the remainder of this report as appropriate, and all sources utilized are available in
REFERENCES.*
Principle equations which continue to govern fundamental theory behind analysis:
𝑞 = 𝑈𝐴∆𝑇𝐿𝑀
𝑞 = 𝒎ℎ𝑐𝑝ℎ(𝑇ℎ𝑖 − 𝑇ℎ𝑜)
𝑞 = 𝒎𝑐𝑐𝑝𝑐(𝑇𝑐𝑜 − 𝑇𝑐𝑖)
Where terms are recounted in Table 3 below:
Variable Description Units
q Heat output of stove as
given by stove specification manual
[kW]
U Overall heat transfer
coefficient [
𝑊
(𝑚2)𝐾]
A Heat transfer surface
area [𝑚2]
∆𝑇𝐿𝑀 Log mean temperature
difference [K] (degrees Kelvin)
𝒎ℎ Mass flow rate of hot
fluid [kg/s]
𝒎𝑐 Mass flow rate of cold
fluid [kg/s]
𝑇ℎ𝑖 Hot fluid inlet temperature
[K]
𝑇ℎ𝑜 Hot fluid outlet
temperature [K]
𝑇𝑐𝑖 Cold fluid inlet temperature
[K]
𝑇𝑐𝑜 Cold fluid outlet
temperature [K]
Table 3
8
ASSUMPTIONS – RECALL vs NEW CONSIDERATIONS
Heat Output Values
The report which documents previous research checkpoint focused on modelling the system and
respective solutions around calculation of an effective inlet temperature for the hot fluid. Stated
in previous assumptions, available information at the time dictated a workable knowledge of
lower and upper power limits, which were specified in the stove’s operation manual. With such
limits, loads for 4 stove setting permutations in between the limits were iterated using governing
equations and interpolation techniques. Any validity of known values for heating loads, however,
have since been voided; through experimental methods, research unveiled performance metrics
of combustion and convection blowers which significantly differed from prescribed volumetric
flow rates. Frames in Figure 8 annotate methods used to both void assumption of previously
prescribed values, and accurately meter new values in real time.
‘Initial Test Rig’ shows convection blower dislodged from stove, rigged to a Bosch flow
metering device with leaks eliminated. Methods eventually employed an parallel method of
measuring airflow to desirable accuracy; depicted in the right two frames is a venturi meter.
Carefully machined, the venturi meter ultimately uncovered a slight deviation in Bosch readings,
leading to our re-calibration of the Bosch and reaffirmation reading accuracy.
Figure 8
9
Energy Conservation
Another major difference in underlying assumptions incurs relaxation of 𝑞𝑅𝑎𝑑 = 0. Our newly
established approach models radiative transfer through the system. Based on geometry, metered
surface temperatures, and material properties, this mechanism is documented through intuitive
superposition of enclosure conditions. Change in mechanical energy components, however, is
still ignored due to infinitesimal nature of changes in fluid parameters, including specific weight
and free-stream velocity, across heat exchanger. (∆𝑃𝐸𝑠𝑦𝑠 = ∆𝐾𝐸𝑠𝑦𝑠 = 0)
Fluid Property Evaluation
In the preceding report, experimental readings for inlet and outlet temperatures were quite
limited such cold fluid properties were evaluated at a constant room temperature of 298 K, and
hot fluid properties were evaluated at a film temperature between the hot fluid inlet temperature,
which was incrementally guessed, and the average flue stack temperature respective to each
setting, of which there were 3 (one for each heat setting) available from a single baseline run.
Our updated research now dictates that fluid properties are evaluated at a film temperature
between inlet and outlet values for either fluid for any given run. This evaluation is much more
accurate; we have now ascertained viable methods for continuously monitoring inlet and outlet
temperature readings (either fluid) using carefully calibrated thermocouples throughout runs
conducted over approximately 3 hours at each trial.
Cold-Side Transfer Coefficient
New data associated with upgrades surrounding airflow sensory hardware (Bosch & venture
meters) have voided the validity of assuming fully developed conditions for the cold fluid. In
light of a parametric lens into internal flow situations to which this fluid is subject, we have
uncovered more realistic interpretation of hydrodynamic entry lengths inside the exchanger’s
tubes, and a cold-side heat transfer coefficient ℎ(𝑥)𝑐 which is a function of distance x as a direct
result. As such, our numerical solutions include a function which parameterizes this coefficient
during post-DAQ processing. This analysis is available in detail in APPENDIX 2.
10
THEORY – CONTINUOUS CONVECTION SOLUTION
Figure 9 relays basis of our approach in ascertaining load q as a continuous function of distance
Figure 9
This is the first building block for a revitalized interpretation of stove power based on log mean
temperature difference. Our energy balance follows the succeeding series of equations for an
ultimate solution of fundamental form q(x).
𝑑𝑞 = −𝒎ℎ𝑐𝑝ℎ(𝑇(𝑥)ℎ) (1)
𝑑𝑞 = 𝒎𝑐𝑐𝑝𝑐(𝑇(𝑥)𝑐) (2)
The heat capacity rates simply combine the first two terms in either of these equations:
𝐶ℎ = −𝒎ℎ𝑐𝑝ℎ (3)
𝐶𝑐 = 𝒎𝑐𝑐𝑝𝑐 (4)
Substitute heat capacities & either 𝑇 term as apparent from Figure 9
𝑑𝑞 = −𝐶ℎ (𝑇(𝑥)ℎ − (𝑇(𝑥)ℎ + 𝑑𝑇(𝑥)ℎ)) (5)
𝑑𝑞 = 𝐶𝑐 ((𝑇(𝑥)𝑐 + 𝑑𝑇(𝑥)𝑐) − 𝑇(𝑥)𝑐) (6)
From here, we solve for heat capacities
𝐶ℎ = −𝑑𝑞
𝑑𝑇(𝑥)ℎ
(7)
11
𝐶𝑐 =𝑑𝑞
𝑑𝑇(𝑥)𝑐 (8)
And obtain
(𝒅𝑻(𝒙)𝒉 − 𝒅𝑻(𝒙)𝒄) = −𝑑𝑞 ((1
𝐶ℎ) + (
1
𝐶𝑐)) (9)
Which is self-consistent with differential temperature difference term obtained through:
Where 𝑇𝑡𝑢𝑏𝑒 𝑜𝑟𝑖𝑔𝑖𝑛 – Outer tube surface temperature @ cold fluid outlet
𝑇𝑡𝑢𝑏𝑒 𝑃 – Outer tube surface temperature at x = P
47
Solving for total heat transferred to the room due to convection:
𝑄𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 = 8∫ (𝑈(𝑥)[𝑇ℎ(𝑥) − 𝑇𝑐(𝑥)])𝑑𝑥𝐿
𝑜
Where the integral for one tube is multiplied by 8 to account for all eight tubes in the heat
exchanger. Inputting U(x), 𝑇ℎ(𝑥), 𝑇𝑐(𝑥) according to inlet and outlet values from Table 11, and
integrating with respect to x-distance yields the overall heat transfer from the hot to cold fluid due
to convection:
𝑄𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 = 8∫
(
1
[(1
ℎ(𝑥)𝑐(0.1077)
) + 0.00112 + (1
ℎ(𝑥)ℎ(0.1197)
)]
[𝑒(−.5045(𝑥)+ 6.3245) − 𝑒(−0.1673𝑥+5.7805)]
)
𝑑𝑥.508
𝑜
𝑸𝒄𝒐𝒏𝒗𝒆𝒄𝒕𝒊𝒐𝒏 = 𝟔𝟎𝟗. 𝟏𝟏𝟏𝟖 (𝑾𝒂𝒕𝒕𝒔)
Solving for heat transferred to the room due to radiation from plate divider to tubes:
𝑄1↔2(𝑥) = 8∫((5.67𝑒 − 8) (𝑇1
4 − 𝑇24))
(1 − 𝜀1
0.1197𝜀1) + (8.4655) + (
1 − 𝜀2
𝟎. 0413𝜀2)(𝑑𝑥)
𝑃
0
Inputting 𝑇𝑝𝑙𝑎𝑡𝑒(𝑥) & 𝑇𝑡𝑢𝑏𝑒(𝑥) as 𝑇1 and 𝑇2 (where distribution were found in MATLAB based
on average steady-state temperatures in Table 11), respectively, and integrating with respect to x-
distance yields the theoretical overall heat transfer from the hot to cold fluid. Remember, the
radiation occurs from x = 0 to x = P and will be integrated over that range. Also the emissivity’s
are taken to be 1.
𝑄𝑝𝑙𝑎𝑡𝑒↔𝑡𝑢𝑏𝑒 = 8∫((5.67𝑒 − 8) (𝑇𝑝𝑙𝑎𝑡𝑒(𝑥)
4 − 𝑇𝑡𝑢𝑏𝑒(𝑥)4))
(1 − 10.1197) + (8.4655) + (
1 − 10.0413)
(𝑑𝑥).305
0
𝑸𝒑𝒍𝒂𝒕𝒆↔𝒕𝒖𝒃𝒆 = 𝟓𝟕𝟓. 𝟔𝟎𝟖𝟔 (𝑾𝒂𝒕𝒕𝒔)
So , the total heat transferred is:
𝑄𝑐 = 𝑄𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 + 𝑄𝑝𝑙𝑎𝑡𝑒→𝑡𝑢𝑏𝑒
𝑸𝒄 = 1184.7 (Watts)
48
The manufacturer advertises a total heat output ranging from 13,500 to 35,600 BTU/hr which
corresponds 3,956.5 to 10,433 in units of Watts. Our model and data suggest a high heat output
of 1184.7 Watts. For further assurance of the deviation between theoretical data and advertised,
we examine the simple relation:
𝑄 = 𝒎𝑐𝑐𝑝𝑐(𝑇𝑐)
Where the flow energy per unit time of the cold fluid is integrated over the entire exchanger
length for all eight tubes, and, because the temperature change of the cold air is small, the
specific heat of the air barely changes. Hence, with the specific heat taken at the film temperature
between (average) the cold fluid inlet and outlet, and the mass flow rate as twice that given for
the ‘high; high’ setting in Table 10:
𝑸 = (0.0545 [𝑘𝑔
𝑠]) (1047.4 [
𝐽
𝑘𝑔𝐾]) (26.3921[𝐾]) = 1506.8 (Watts)
Thus, the range of heat output advertised by the manufacturer does not even eclipse our highest
predicted values according to theory. Beyond comparing such values, the conclusions we may
draw from the deviations are limited in that very little is known about how the manufacturer
arrives at advertised values. They are perhaps utilizing efficiency values given by TESTO’s
oxygen probing, and backing out corresponding output values for heating load based on a
combustion efficiency. This could partially explain such high advertised values of heat output, as
certain recent studies have
49
DELIVERABLES – DIGITAL LOGIC
~Overview & Intentions~
In order to gain a better understanding of stove operation, the circuit board was an offshoot focus
of research. The purpose of this section is to document a comprehensive review of operating
procedure, with respect to digital logic, such that ongoing research may benefit from our
findings. Ultimately, stove redesign requires knowledge of the current circuitry; only then can
adjustments to aspects such as combustion intake, convection fan speed, and pellet feed control
be wired appropriately. The circuit board is hardwired to various components from the control
box; the control box located behind the switch module shown in Figure 25.
Along with the above figure, the installation manual [2] provides an electrical schematic, shown
in Figure 26.
Figure 26
Figure 25
50
~Operation~
We have simplified an understanding of electrical operation with the visual in Figure 27.
Figure 27
Figure 27 relays a schematic similar to that provided in the manual, however many modifications
have been imposed to aid analysis. A typical operation cycle is chronologically segmented by the
following steps:
1. When the stove is turned on, the thermostat block calls for heat for an amount of time
depending on the desired room temperature (set by user). During this time, the red call
light, located on the external switch module, is illuminated.
2. Regardless of the switch settings, the appliance begins on high heat output for at least 4
minutes and 15 seconds, at which point it then automatically defaults to the specified
setting. If set to medium or low, the stove runs quieter, but will take longer to heat the
room to a desired temperature.
3. The manual originally specified one air flow rate for the combustion blower, while a
range was given for the convection blower. It has been confirmed that the convection
blower actually yields six different possible volumetric flow rates depending on the
combined heat output and fan speed settings. The combustion blower operates based on a
relationship with the feed motor which is not completely understood in terms of electrical
logic as of this point. Regardless, once the temperature of the room satisfies the
thermostat block, the feed motor and combustion blower shut off.
4. If the thermostat does not call for heat for a certain duration of time, the appliance will
completely shut off, but the convection fan will continue to run until the stove is cool
enough ( until snap disc 1 detects a temperature under 145ºF).
51
Certain component are now annotated for clarity of function.
Thermocouple: located directly above the firepot, the thermocouple is radiation shielded as
it is subject to immediate contact with the flames. When the temperature reading above the
firepot drops below 200ºF during operation, the thermocouple sends a 15.7 mV signal to the
control box causing an internal light to turn green; when the reading is above 600ºF, the light
turns red.
Heat Output Switch: located on the external control module, this switch regulates burn rate
by communicating with both the convection speed and the pellet motor feed.
Convection Blower Switch: Called ‘Fan Speed’ on the external control module, this switch
toggles the convection fan between high and low magnitudes for the given heat output
setting. These magnitudes are based on differing rpm levels reached by the blower rotor.
Vacuum Switch: Found on the left inside of the firebox behind the inner shield, this switch
allows the feed system to turn on only when there is a vacuum present inside the chamber.
(turns system off if door is opened)
Hopper Switch: Found near the upper right hand corner of the hopper, this switch shuts off
the feed motor when the hopper lid is open.
Feed Motor: The feed motor system primarily consists of a mount, collar, the motor itself,
fasteners, and a coil spring assembly, or auger, which rotates in place to move pellets from
the hopper down to the firepot through a chute between the two halves of the heat exchanger.
It is worthy to note that a feed system disassembly would most likely merit in the form of
useful information in the future, as research has not yet uncovered how the feed motor
communicates with other electrical components.
Capacitor: The capacitor is connected directly to the feed motor; its function is to reverse
the polarity of the induction coil (auger) if spring action is impeded by pellet buildup in the
feed opening. If there is a pellet bind inhibiting the spring assembly from rotating, the
buildup of friction causes a local rise in temperature; if this temperature is sufficiently high,
the capacitor’s charge toggles auger rotation. The spring assembly then turns in the opposite
direction to free up pellet feed until the impediment is lifted, at which point the auger is
reversed back into normal rotation.
Snap Discs: Snap discs are devices equipped with thermal sensors which carry out various
functions depending on temperature reading. Snap disc 1 is located on the right side of the
heat exchanger (two purple wires), and ensures the convection blower is running so long as it
reads a temperature above 145ºF. This expedites cooling of the appliance during shut down
mode. Snap disc 2 is further back towards the convection blower on the right side of the heat
exchanger (two yellow wires), and shuts off the feed system if it reads a temperature above
200ºF. This disc responds to cases such as over-fire or convection fan failure. Snap disc 3 is
mounted on the back the auger tube, and shuts the entire system off if a temperature over
250ºF is detected. This prevents further failure in the case of burn-back from the chamber
into the rest of the appliance.
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APPENDICES – APPENDIX 1
Overall Quadrafire Code
%% Solve for Convective Heat Transfer Coefficient % (1) Load Data from Excel and declare frequency Test = 'C:\Users\Carter\Desktop\Quadrafire\Test Data\August
% (2) Synchronize Data Time-Step for Testo and LabVIEW % - Find time step, implement into data sets [ n, m, p, v ] = Experiment_Interval_Sync( Test, Testo, freq ); %
% - Preform moving mean on data sets, Labview & Testo Lab = movingmean(Lab,10,[],[]); % Do
moving averages of entire data set Tes = movingmean(Tes,10,[],[]); % (3) Define Time Intervals For Labview and Testo Data time_lab = 0:(1/freq):((m-n)/freq); %
Define time intervals time_tes = (0:1:v-1); % (4) Define Labview data set to measurement names %
Declare data from excel sheet % - Tfire = Temperature of Exhaust Intake Tfire = labview(:,1)+273; % - Thotin = Temperature of Cold air intake Thotin = labview(:,3)+273; % - Thotout = Temperature of Cold air out Tcoldin = labview(:,5)+273; % - Tccoldin = Temperature of Cold air pipe surface out Thotout = labview(:,7)+273; % - Tcoldout = Temperature of radiation plate Tcoldout = labview(:,9)+273; % - Tpoco = Temperature of radiation plate Tpoco = labview(:,11)+273; % - Tpor = Temperature of radiation plate Tpor = labview(:,13)+273; % - Tplate5 = Temperature of radiation plate #5 (top) Tplate5 = labview(:,15)+273; % - Tplate4 = Temperature of radiation plate #4 Tplate4 = labview(:,17)+273; % - Tplate3 = Temperature of radiation plate #3 Tplate3 = labview(:,19)+273; % - Tplate2 = Temperature of radiation plate #2 Tplate2 = labview(:,21)+273; % - Tplate1 = Temperature of radiation plate #1 (bottom) Tplate1 = labview(:,23)+273; % - Loadv = Load cell data in volts Loadv = labview(:,25);
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Loadvs = movingmean(Loadv,50,[],[]); %
Preform additional moving average for more smoothing % - Boschv = Bosch Flow Meter Data in volts Boschvs = labview(:,27); Boschv = movingmean(Boschvs,50,[],[]); % (5) Define Labview data set to measurement names % - Tstack = Temperature of Exhaust Tstack = movingmean(Tes(:,2),20,[],[]); % - Tamb = Temperature of Ambient Air Tamb = movingmean(Tes(:,12),20,[],[]); % - ppmNo = part per million of No ppmNo = movingmean(Tes(:,3),50,[],[]); % - ppmNox = part per million of Nox ppmNox = movingmean(Tes(:,4),20,[],[]); % - ppmCo = part per million of Co ppmCo = movingmean(Tes(:,7),50,[],[]); % - perO2 = part per million of No perO2 = movingmean(Tes(:,5),20,[],[]); % (6) Data Calibration for Data Sets Labview and Testo % - Load = weight of stove in volts Load = Loadvs.*25; %
Change load volts to load in lbs % - MAF_h = mean mass flow rate of hot fluid x = Boschv; MAF_ht = (29.93.*x.^3)-(122.9.*x.^2)+(231.1.*x)-138.1; %
change bosch volts to mass flow rate in kg/hr MAF_h = mean(MAF_ht); % (7) Declare fixed mass flow rate of cold fluid % - MAF_c = mass flow rate of cold fluid at all six speeds MAF_c = ([4.0343 26.3691 37.6045 46.1173 63.9859 91.1183
98.1463]).*2; % known mass flow rates
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% DEFINE PROPERTIES OF AIR
%% (1) Define the temperatures for hot and cold fluid inlets/outlets % - Declare range of steady state data to average ran1 = 5000; ran2 = 8000; % - Tci = Temperature of cold fluid in Tci = Tcoldin(ran1:ran2) ; % - Tco = Temperature of cold fluid out Tco = Tcoldout(ran1:ran2); % - Thi = Temperature of hott fluid in Thi = Thotin(ran1:ran2) ; % - Tho = Temperature of hott fluid out Tho = Thotout(ran1:ran2) ;
% (2) Develop functions for hot/cold fluid temperatures as func of x % - Define parameters of cold fluid tube L = .508; x = linspace(0,L,1000); Dp = .03430; Ac = (pi*Dp^2)/4;
%% Air properties vs temperature % (1) Define air properties vs temperature % - Stanadrd air propertie are inputed vs temperature % - Use these inputs to determine air properties at certain temps T = 250:.01:1100; Temp_air = 250:50:1100 ; row_air = [1.3947;1.1614;.9950;.8711;.7740;.6964;.6329;.5804;... .5356;.4975;.4643;.4354;.4097;.3868;.3667;.3482;.3324;.3166]; cp_air = [1.0061;1.007;1.009;1.014;1.021;1.030;1.040;1.051;1.063... ;1.075;1.087;1.099;1.110;1.121;1.131;1.141;1.150;1.159]; mu_air = [159.6;184.6;208.2;230.1;250.7;270.1;288.4;305.8;322.5... ;338.8;354.6;369.8;384.3;398.1;411.3;424.4;436.7;449]; k_air = [22.3;26.3;30;33.8;37.3;40.7;43.9;46.9;49.7;52.4;54.9... ;57.3;59.6;62;64.3;66.7;69.1;71.5]; Pr_air =
% (3) Declare all properties of air at mean temperatures % (For Tci, Tco, Thi, Tho mean air properties) % Hot Fluid Inlet Air Properties row_Thi = row_air_sp(T == Thi_ss_avg); cp_Thi = cp_air_sp(T == Thi_ss_avg); mu_Thi = mu_air_sp(T == Thi_ss_avg); k_Thi = k_air_sp(T == Thi_ss_avg); Pr_Thi = Pr_air_sp(T == Thi_ss_avg); % Hot Fluid Outlet Air Properties row_Tho = row_air_sp(T == Tho_ss_avg); cp_Tho = cp_air_sp(T == Tho_ss_avg); mu_Tho = mu_air_sp(T == Tho_ss_avg); k_Tho = k_air_sp(T == Tho_ss_avg); Pr_Tho = Pr_air_sp(T == Tho_ss_avg); % Cold Fluid inlet Air Properties row_Tci = row_air_sp(T == Tci_ss_avg); cp_Tci = cp_air_sp(T == Tci_ss_avg); mu_Tci = mu_air_sp(T == Tci_ss_avg); k_Tci = k_air_sp(T == Tci_ss_avg); Pr_Tci = Pr_air_sp(T == Tci_ss_avg); % Cold Fluid Outlet Air Properties row_Tco = row_air_sp(T == Tco_ss_avg);
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cp_Tco = cp_air_sp(T == Tco_ss_avg); mu_Tco = mu_air_sp(T == Tco_ss_avg); k_Tco = k_air_sp(T == Tco_ss_avg); Pr_Tco = Pr_air_sp(T == Tco_ss_avg); % (4) Declare mean fluid properties for cold fluid and then hot fluid % Find mean air properties for hot and cold fluids row_Tc = (row_Tco + row_Tci)/2; cp_Tc = (cp_Tco + cp_Tci)/2; mu_Tc = (mu_Tco + mu_Tci)/2; k_Tc = (k_Tco + k_Tci)/2; Pr_Tc = (Pr_Tco + Pr_Tci)/2; % Find mean air properties for hot and cold fluids row_Th = (row_Tho + row_Thi)/2; cp_Th = (cp_Tho + cp_Thi)/2; mu_Th = (mu_Tho + mu_Thi)/2; k_Th = (k_Tho + k_Thi)/2; Pr_Th = (Pr_Tho + Pr_Thi)/2;
%% Determine convective heat transfer coefficents % (1) Define parameters of physical system (dimensions) % - Dpo = cold fluid outer pipe diameter Dpo = .0383; % - Dpi = cold fluid inner pipe diameter Dpi = .0343; % - Dh = hydraulic diameter of hot fluid s = .0403; h = .060325; %Dh = (h*s*4)./((2*s)+(2*h)); Dh = .02731241; % (2) Convert air properties to standard units and rename for easy
declaration % - muc = mean dynamic viscosity of cold fluid muc = (mu_Tc)*(10^-7); % - muh = mean dynamic viscosity of hot fluid muh = (mu_Th)*(10^-7); % - Red_c = Reynolds number of cold fluid Red_c = (4*MAF_c)./(8*3600*pi*Dpi*muc); % - Red_h = Reynolds number of hot fluid Red_h = (MAF_h*Dh)./(2*3600*.006625094*muh); % - prc = prandlt number of cold fluid prc = Pr_Tc; % - prh = prandlt number of hot fluid prh = Pr_Th; % - kc = thermal conductivity of cold fluid kc = (k_Tc)*(10^-3); % - kc = thermal conductivity of hot fluid kh = (k_Th)*(10^-3);
% (3) Solve for heat transfer coefficent in cold pipe % - Nudcx = Nusselt number of cold fluid in pipe % - idx = interval where Nud reaches maximum limit [Nudcx,idx] = Heat_Transfer_Coefficent_Pipes(Red_c,7,x,prc,Dpi); %
Function generates Nudcx curve % - hcx = heat transfer coefficent of cold fluid as function of x hcx = (Nudcx.*kc')./Dpi;
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% (4) Solve for heat transfer coefficent in hot fluid flow % - Nudhx = heat transfer coefficent of hot fluid Nudhx = 3.96; %
Constant due to geometry % - hhx = heat transfer coefficent of hot fluid hhx = (Nudhx*kh)/Dh; n = 1000; for i = 1 : n h(i) = hcx((n+1)-i); end %% Define parameters of temperature distributions through out heat exchanger % (1) Estimated Temperature Distributions % (a) Th(x) = Temperature of Hot Fluid % (b) Tc(x) = Temperature of Cold Fluid % (c) Tpo(x) = Temperature of Outer Pipe Surface % (d) Tpi(x) = Temperature of Inner Pipe Surface % (2) Define parameters of the system % (a) L = length of heat exchanger (meters) L = .508; % (b) x = correlates to distancces downstream from hot inlet (meters) n = 1000; %
Step Size (non-dimensional) step = L/n; %
incremental distance (dx) x = linspace(0,L,n); %
Distance as an array % (c) Api = cross sectional area of inner pipe (meters^2) Api = 2*pi*((.0343)/2); %
Cross sectional area % (d) R = distance from x = 0 to end of radiating plate (meters) R = .305 ; %
R is .75 the length of L
%% Determine Cold fluid temperature distribution as func of x % (1) Assumptions % - Assume the the properties of air are a mean value of mean inlet and % outlet temperatures of cold fluid % - Assume that the radiated plate transfers heat to the pipes outer % surface and thus to the cold fluid % - Assume that system is adibatic with control volume surrounding entire % heat exchanger and isothermal plate % - Because of these assumptions, assume that the hot fluid temperature % distribution is logrythmic from cold fluid inlet to just before the % isothermal plate. Also, heat transfer into the pipe (thus cold fluid) % is from convection of the hot fluid. % - Because of these assumptions heat transfer to pipe (thus cold fluid) % over the isothermal plate is due to convection of hot fluid and % radiating isothermal plate
% (2) Find temperature of cold fluid out w/o radiation into system % Tco_conv = Temp of cold fluid out w/o radiation a = 7; Tco_prime = (((MAF_h)*(cp_Th))/((MAF_c(a))*(cp_Tc)))*... % Temp
of cold fluid out w/o raadiation/ (Kelvin) (mean(Thi) - mean(Tho)) + mean(Tci); % (3) Define new temperature distribution w/o radiation
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% m = interval in x distance that correlates to distance R m = round((R*n)/L); % Tco_conv = Now the temperature dist of Tco w/o radiation Tcox_pr= exp(((log(mean(Tci))-log(Tco_prime))./L).*x' +
log(Tco_prime)); % Temp dist of cold fluid w/o radiation/ (Kelvin) % Tcr = Temp of cold fluid at x = R w/o radiation Tcr = Tcox_pr(m); % Tco_convx = Cold fluid temp dist w/o radiation in system xr = linspace(0,L-R,n-m); Tcxr = exp(((log(mean(Tci))-log(Tcr))./(L-R)).*xr' + log(Tcr)); dTcxr = ((log(mean(Tci))-log(Tcr))./(L-R)).*exp(((log(mean(Tci))-
log(Tcr))./(L-R)).*xr' + log(Tcr)); % (4) Define actual temp dist of cold fluid over plate % xpl = range in x-distance from x=0 to x=R xpl = x(1,1:m); % Tco_pl = Temp dist of cold lfuid from x=0 to x=R Tcoxpl = exp(((log(Tcr)-log(mean(Tco)))/R).*xpl + log(mean(Tco))); dTcoxpl = ((log(Tcr)-log(mean(Tco)))/R).*exp(((log(Tcr)-
log(mean(Tco)))/R).*xpl + log(mean(Tco))); % Tcox = actual temperature distribution of cold fluid Tcx = vertcat(Tcoxpl',Tcxr)'; dTcx = vertcat(dTcoxpl',dTcxr)'; %% Determine hot fluid temperature distribution as func of x % (1) Assumptions % - Assume the the properties of air are a mean value of mean inlet and % outlet temperatures of hot fluid % - Assume that the radiated plate has negligable heat transfer to the hot % fluid % - Because of these assumptions, assume that the hot fluid temperature % distribution is logrythmic
% (2) Define hot fluid temp distribution % Thx = Temp dist of hot fluid (Kelvin) Thx = exp((((log(mean(Tho))-log(mean(Thi)))/L).*(x)) +
log(mean(Thi)))/L).*(x)) + log(mean(Thi))); %% Determine Outer pipe temperature distribution as func of x % (1) Assumptions % - Assume that function of outer pipe surface temp vs distance is nearly % logrythmic, and resembles the temp dist of cold and hot fluid % - Reasonable because temperature gradient is small across pipe % surface % - Also, hot and cold fluid temperature as function of distance is % logryhtmic so w/o any radiation transfer into the pipe, then pipe % surface temperature as function of distance would be logrythmic % - Assume the pipe has constant physical parameters % - Assume constant parameters are the same for every pipe % - For now, assume temp distribution of single pipe is a mean % distribution of every pipe in system
% (2) Define mean temperatures of outer pipe surface % Tpoco = mean temp of outer pipe surface at x = 0 TPoco = mean(Tpoco); % Tpor = mean temp of outer pipe surface at x = R TPor = mean(Tpor);
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% (3) Declare Temperature Distribution of Outer Pipe Surface from 0<x<R Tpox = exp((((log(TPor)-log(TPoco))/R).*(xpl)) + log(TPoco)); dTpox = ((log(TPor)-log(TPoco))/R).*exp((((log(TPor)-
log(TPoco))/R).*(xpl)) + log(TPoco)); %% Determine Radiation Plate Temperature Distribution as func of x % (1) Assumptions % - Assume the pipe has constant physical parameters % - Assume that temperature distribution of plate is constant acrross % entire surface % - Meaning, the temp dist found is function of z (height) and x % (distance downstream). So the temp dist does not change with y % - Currently do not have means to find temp distribution in all % directions % - Assume that temp dist can be found be finding a single point at % maximum of known temperatures of plate surface
% (2) Declare mean temperatures of Tplate#'s (1-5) & corresponding distance % Tpl1 = average temperature of Tplate1 Tpl1 = mean(Tplate1); % Tpl2 = average temperature of Tplate1 Tpl2 = mean(Tplate2); % Tpl3 = average temperature of Tplate1 Tpl3 = mean(Tplate3); % Tpl4 = average temperature of Tplate1 Tpl4 = mean(Tplate4); % Tpl5 = average temperature of Tplate1 Tpl5 = mean(Tplate5); % displ = array of distances downstream from x = 0 to x = R pl_num = 5; %
number of thermocouple on plate displ = [.025 .0912 .1435 .1952 .25864]; %
distance increment of thermocouples % Tplate = row vector of plate temperatures Tplate = [Tpl5 Tpl4 Tpl3 Tpl2 Tpl1]; %
array of plate temperatures from Tpl5-->Tpl1
% (3) Find cubic function to fit data points of mean plate temperatures p = polyfit(displ,Tplate,3); %
fit data to a cubic function Tplate_fun = (p(1).*(xpl.^3)) + (p(2).*(xpl.^2)) + (p(3).*xpl)+p(4);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DETERMIINE HEAT TRANSFERS IN AND OUT OF
SYSTEM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Determine heat transfer between hot/cold fluid w/o radiation % (1) Define all the resistances in the system Rconvc = (h.*pi.*Dpi).^-1; Rcond = (log(Dpo/Dpi))/(2.*pi.*18); %
18 - thermal conductivity of pipe Rconvh = (pi.*hhx.*Dpo).^-1;
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Rtot = Rconvc + Rcond + Rconvh;
% (2) Define overall heat transfer coefficent U = Rtot.^-1; % (3) Declare dqconv dq = U.*(Thx-Tcx); % (4) Integrate to find total heat along L dq_spl = spline(x,dq); Q = integral(@(x)ppval(dq_spl,x),0,L);
%% Determine heat transfer from radiation of plate % (1) View Factor % s = distance between center of pipes s = .0403; F = 1-(1-((Dpo/s)^2))^.5+((Dpo/s)*atan(((s^2-Dpo^2)/Dpo^2)^.5)); % sigma = stephan boltzs constant sigma = 5.67e-8; % Apl = Surface area of plate Apl = s; % Apo = surface area of outer diameter plate Apo = pi*Dpo; % epl = emissivity of plate epl = 1; % epo = emissivity of epo = 1; % (2) Find differential of radiation % find resistance of radiation in enclosure Rpl = (1-epl)/(Apl*epl); RF = (1/(Apl.*F))+((1/(Apl*(F-1)))+(1/(Apo*(F-1))))^-1; Rpo = (1-epo)/(Apo*epo); Rrad = Rpl+RF+Rpo; % (3) dqr = radiation as function of x dqr_dx= ((Rrad.^-1).*sigma).*(Tplate_fun.^4 - Tpox.^4); % (4) Integrate over (0,L) to find total heat dqr_spl = spline(xpl,dqr_dx); Qr = integral(@(x)ppval(dqr_spl,x),0,R);
%% Determine heat transfer from hot fluid % (1) solve for heat transfer as function of x dTh = ((log(mean(Tho))-log(mean(Thi)))/.508).*... exp((((log(mean(Tho))-log(mean(Thi)))/.508).*(x))... +log(mean(Thi))); dqh = (MAF_h/(8*3600).*cp_Th.*1000.*dTh); dqh_spl = spline(x,dqh); for i = 1: n-1 Qhx(i) = integral(@(x)ppval(dqh_spl,x),x(i),x(i+1)); %
heat transfer as func of x end % (2) Integrate over (0,L) to find total heat transfer Qhint = integral(@(x)ppval(dqh_spl,x),0,L); %
total heat transfer by integration % (3) Solve for heat transfer total of hot fluid by mass transfer Qh = (MAF_h/(8*3600)).*cp_Th.*1000.*(mean(Thi-Tho));
% total heat transfer by integration
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%% Determine total heat transfer from cold fluid % (1) solve for heat transfer as function of x dqc = (MAF_c(7)/(8*3600)).*cp_Tc.*1000.*dTcx; dqc_spl = spline(x,dqc); for i = 1: n-1 Qcx(i) = integral(@(x)ppval(dqc_spl,x),x(i),x(i+1)); %
heat transfer as func of x end % (2) Integrate over (0,L) to find total heat transfer Qcint = integral(@(x)ppval(dqc_spl,x),0,L);
% total heat transfer by integratio Qc = (MAF_c(a)/(8*3600))*cp_Tc*1000*(mean(Tco-Tci));
The heat transfer coefficient function below is used for a system where…
Circular pipes
Sharp Corner Entry Length
Fully underdeveloped turbulent flow
Constant parameters throughout entire pipe length
Smooth pipe interior
function [ Nudx,idx ] = Heat_Transfer_Coefficent_Pipes(Red_c,p,x,prcm,Dp) % Red = Reynolds number % C & m & f = coefficents depending on Reynolds % p = Which flow is currently being obserevd % [1 = ll , 2 = ml , 3 = hl, 4 = lh, 5= mh, 6 = hh] % Frictiona Coefficent (Assume smooth) fd_c = (.79.*log(Red_c(p)) - 1.64).^(-2); % Nusselt number as an average Nud_fud = ((fd_c./8).*(Red_c - 1000).*(prcm))./(1 -
(12.7.*(fd_c./8).^(1/2).*(prcm.^(2/3)-1))); % Nuseelt number as a limit Nud_fud_lim = ((fd_c./2).*(Red_c - 1000).*(prcm))./(1 -
(12.7.*(fd_c./2).^(1/2).*(prcm.^(2/3)-1))); % Nusselt number actual C = 34.*(Red_c(p).^-.23); m = .815 - (2.08e-6.*Red_c(p)); Nudx = Nud_fud(p).*(1 + (C./((x/Dp).^m))); tmp = abs(Nudx-Nud_fud_lim(p)); [idx idx] = min(tmp); Nudx(1:1:idx)= Nud_fud_lim(p);
end
This code solves for the nusselt number as a function of distance and implements this value to
find the heat transfer coefficient in the “Overall Code of the Quadrafire”.
The equations on the following page were used finding of the cold-side transfer coefficient ℎ(𝑥)𝑐
when it was uncovered that this fluid does not experience fully developed flow regime
conditions.
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Moving Mean Function
Use this function in the overall Quadrafire code
function result = movingmean(data,window,dim,option) %Calculates the centered moving average of an n-dimensional matrix in any
% Inputs: % 1) data = The matrix to be averaged. % 2) window = The window size. This works best as an odd number. If an
even % number is entered then it is rounded down to the next odd number. % 3) dim = The dimension in which you would like do the moving average. % This is an optional input. To leave blank use [] place holder in % function call. Defaults to 1. % 4) option = which solution algorithm to use. The default option works % best in most situations, but option 2 works better for wide % matrices (i.e. 1000000 x 1 or 10 x 1000 x 1000) when solved in the % shorter dimension. Data size where option 2 is more efficient will % vary from computer to computre.This is an optional input. To leave % blank use [] place holder in function call. Defaults to 1. % % Example:
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% Calculate column moving average of 10000 x 10 matrix with a window size % of 5 in the 1st dimension using algorithm option 1. % d=rand(10000,10); % dd=movingmean(d,5,1,1); % or % dd=movingmean(d,5,[],1); % or % dd=movingmean(d,5,1,[]); % or % dd=movingmean(d,5,1); % or % dd=movingmean(d,5,[],[]); % or % dd=movingmean(d,5); % % Moving mean for each element uses data centered on that element and % incorporates (window-1)/2 elements before and after the element. % % Function is broken into two parts. The 1d-2d solution, and the % n-dimensional solution. The 1d-2d solution is the fastest that I have % been able to come up with, whereas the n-dimensional solution trades % speed for versatility. % % Function includes some code at the end so that the user can do their % own speed testing using the TIMEIT function. % % Has been heavily tested in 1d-2d case, and lightly tested in 3d case. % Should work in n-dimensional space, but has not been tested in more % than 3 dimensions other than to make sure it does not return an error. % Has not been tested for complex inputs.
%errors for leaving out data or window inputs if ~nargin error('no input data') end
if nargin<2 error('no window size specified') end
%defaults the dimension and option to 1 if it is not specified if nargin<3 || isempty(dim) dim=1; end
if nargin<4 || isempty(option) option=1; end
%rounds even window sizes down to next lowest odd number if mod(window,2)==0 window=window-1; end
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%Calculates the number of elements in before and after the central element %to incorporate in the moving mean. Round command is just present to deal %with the potential problem of division leaving a very small decimal, ie. %2.000000000001. halfspace=round((window-1)/2);
%calculates the size of the input data set n=size(data);
%returns error messages for incorrect inputs if mod(dim,1) error('dimension number must be integer') end
if ((ndims(data)<=2 && (option<1 || option>3)) || (ndims(data)>=3 &&
(option<1 ||option>2))) ... || mod(option,1) error('invalid algorithm option') end
if mod(window,1) error('window size must be integer') end
if dim>ndims(data) error('dimension number exceeds number of dimensions in input matrix') end
if dim<1 error('dimension number must be >=1') end
if n(dim)<window error('window is too large') end
if ndims(data)<=2 %Solution for 1d-2d situation. Uses vector operations to optimize %solution speed.
%To simplify algorithm the problem is always solved with dim=1. %If user input is dim=2 then the data is transposed to calculate the %solution in dim=1 if dim==2 data=data'; end
%The three best solutions I came up to for the 1d-2d problem. I kept %them in here to preserve the code incase I want to use some of it %again. I have found that option 1 is consistenetly the fastest. %option=1;
65
switch option case 1 %option 1, works best for most data sets
%Computes the beginning and ending column for each moving %average compuation. Divide is the number of elements that are %incorporated in each moving average. start=[ones(1,halfspace+1) 2:(n(dim)-halfspace)]; stop=[(1+halfspace):n(dim) ones(1,halfspace)*n(dim)]; divide=stop-start+1;
%Calculates the moving average by calculating the sum of elements %from the start row to the stop row for each central element, %and then dividing by the number of elements used in that sum %to get the average for that central element. %Implemented by calculating the moving sum of the full data %set. Cumulative sum for each central element is calculated by %subtracting the cumulative sum for the row before the start row %from the cumulative sum for the stop row. Row references are %adusted to take into account the fact that you can now %reference a row<1. Divides the series of cumulative sums for %by the number of elements in each sum to get the moving %average. CumulativeSum=cumsum(data); temp_sum=CumulativeSum(stop,:)-CumulativeSum(max(start-1,1),:);
case 2 %option 2, Can be faster than option 1 in very wide matrices %(i.e. 100 x 1000 or l000 x 10000) when solving in the shorter %dimension, but in general it is slower than option 1. %Uses a for loop to calculate the data from the start row to %the stop row, and then divides by the number of rows %incorporated.
result=zeros(size(data)); for i=1:n(dim) start=max(1,i-halfspace); stop=min(i+halfspace,n(dim)); result(i,:)=sum(data(start:stop,:),1)/(stop-start+1); end
case 3 %option 3, Creates a sparse matrix to indicate which rows to
include %data from for the moving average for each central element. %Uses matrix multiplication to calculte the cumulative sum for %each central element. Then divides by the number of elements %used in each cumulative sum to get the average for each %central element. %Elegant but slow. I think this is because bsxfun has to do %a lot of matrix replication. Also, it is easy for the use
matrix to %exceed the maximum matrix dimensions for MATLAB.
%Transposes the matrix if the problem was solved in dimension=2 so that %the ouput is correctly oriented. Undoes the initial transposition. if dim==2 result=result'; end else %Solution of the n-dimensional problem. This solution is %slower than the 1d-2d solution in a 1d-2d problem, but depending on the %dimensions of the input array can be faster for the same number of %elements in a 3d+ problem (ie. 1000000 x 1 vs 100 x 100 x 100). %Either way it is much more versatile. The algorithm can solve for a %moving average in any dimension of an n-dimension input matrix.
%Two solutions of the problem. In general option 1 is faster than %option 2, but I think that for some matrix sizes option 2 will be %faster. %option=1; switch option
case 1 %Option 1 is based on option 1 for the 1d-2d solution. In %general this is the faster solution. For a more extensive %explanation of the code see option 1 in the 1d-2d section.
%Calculates the start and stop positions, and creates a vector %of how many elements are in each moving sum. start=[ones(1,halfspace+1) 2:(n(dim)-halfspace)];
%Reorients the start, stop, and divide vectors in the direction %of the specified dimension. %Creates a vector to use to reshape the start, stop, and divide %vectors. Puts one in for the each dimension excpet for the %specified dimension, which is set to the size of the input %matrix in that direction. ex. [1 1 100 1] for a %100 x 100 x 100 x 100 input matrix, solving in the 3rd %dimension. temp_dims=ones(size(n)); temp_dims(dim)=n(dim);
%inv_temp_dims=n; %inv_temp_dims(dim)=1;
%Uses the vector created above to reorient the start, stop, and %divide vectors in the correct direction. start=reshape(start,temp_dims); stop=reshape(stop,temp_dims); divide=reshape(divide,temp_dims);
%Calculates the cumulative sum of the data in the specified %direction. CumulativeSum=cumsum(data,dim);
%I have found the most versatile way of writing an algorithm %that will solve in any dimension of an unknown dimensional %input matrix is to built some functions with text that is %based on the input parameters.
%This text builds the function that subtracts the cumulative %sum at the stop position from the cumulative sum at one %element before the start position for each central element. %Compensates for not have an index less than 1. function_text=['@(CumulativeSum,start,stop) CumulativeSum('
%Converts the string above into an anonymous function with the %handle "difference". difference=str2func(function_text);
%Runs the function built above to determine the cumulative sum %between start and stop. temp_sum=difference(CumulativeSum,start,stop);
%Text to make a function that corrects for the fact that the %cumulative sum for moving sums that start at position 1 have %the first data position subtracted out. This is becasue you
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%can not have a matrix index less than 1. Uses bsxfun to %multiply start==1 by the first plane of data (in the %appropriate dimension) and then adding that to temp_sum. Need %to do it this way vs using something based on reshaping %data(repmat(start,inv_temp_dims)==1) because the correction %needs to be the same size as temp_sum. This is needed b/c I %do not know how to use text to set the variable elements that %will accept the addition (i.e. %temp_sum(:,:,1:sum(start==1))=). correction_text=['@(temp_sum,start,data)
%Converts the above text into a function. correction=str2func(correction_text);
%Applies the correction describe previously to the temp_sum %matrix temp_sum=correction(temp_sum,start,data);
%Divides the temp_sum matrix by the divide vector to %calculate the moving average for each central elemant. result=bsxfun(@rdivide,temp_sum,divide);
case 2 %An older solution to the n-dimensional moving average problem.
%Based on the approach in option 2 in the 1d-2d solution. %Depending on matrix dimensions this might be faster than %option 1 in certain situations (ex input matrix size is 10 x %1000 x 1000 and the average is done in dimension 1).
%Builds a string for the function used in the for loop. %Basiscally it builds something like
sum(data(:,:,start:stop,:),3) but %is able to work in any dimensions and have start:stop be in %any of those dimensions. function_text='@(start,stop,dim) sum(data('; for i=1:ndims(data) if i==dim if i~=1 function_text=[function_text ',start:stop']; else function_text=[function_text 'start:stop']; end else if i~=1 function_text=[function_text ',:']; else function_text=[function_text ':']; end end end
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function_text=[function_text '),dim)'];
%Converts the text above into a function moving_sum=str2func(function_text);
%For loop determines the start position, stop position, and %number of elements for the moving average of each central %element. Calculates the moving sum for each central element %in that position, and divides by the number of elements to get %the moving average. for i=1:n(dim) start=max(1,i-halfspace); stop=min(i+halfspace,n(dim));
temp_m=moving_sum(start,stop,dim)/(stop-start+1);
%This was the most versatile way of reassembling all of the %moving average matrices back into the final output matrix, %although it is slow b/c the result matrix keeps resizing. if i==1 result=temp_m; else result=cat(dim,result,temp_m); end end
end end
%code for speed testing %generates a series of random vectors with 10 to 10000000 cells and then %times the moving average at each size. Time for each input size is %stored.
Use this function to make testo and labview operate on same real time interval.
Function [ n, m, p, v, Test_Time, Testo_Time ] = Experiment_Interval_Sync(
Test, Testo, freq ) clc % Declare and Write Time Sheet [test_num,test_time]=xlsread(Test,’A:A’); [testo_num,testo_time]=xlsread(Testo,’A:A’); % Filter and convert Date and Time to dd-MMM-YYYY HH:MM:SS test_strn = test_time(2:1:end); testo_strn = testo_time(2:1:end); formatOut = ‘mm/dd/yyyy HH:MM:SS:FFF AM’; test_str = datestr(test_strn,formatOut); testo_str
=datestr(testo_strn,formatOut); % Find First Common Point In Time % n = index to offset test % p =index to offset test % m = index to offset testo % v = index to offset testo freq = 10; w = 1/freq; [C,idx_test,idx_testo] = intersect(test_str,testo_str,’rows’);
n = idx_test(1); m = idx_test(end); p = idx_testo(1); v = idx_testo(end);
This will give mass flow rate of convection fan, these numbers are already in overall Quadrafire
code but is useful if the experiment needs to be repeated (It won’t need to be repeated).
% CONVECTION FAN DATA AND RESULTS ON MASS FLOW RATE (BOSCH) % Load all raw data from labview for each convection fan trial % - Six total trials for each respective heat/fan speed setting % (1) Load all data from each Excel sheet, only bosch voltage % (2) Define all parameters and convert from voltage to flow % (3) Plot all raw data on same figure(1) % (4) Preform moving average each data set
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% (5) Plot smoothed data all on same figure(2) clc %% (1) Load Excel Data from First and Second trial % - excell data loaded file = 'C:\Users\Carter\Desktop\Quadrafire\Test Data\Convection
Fan\Combined.xlsx'; data = xlsread(file); % - Filter NAN out of data ind = find(isnan(data)); data(ind)=0; n = ind(1)-1; m = size(data) ; m = m(2); conv = data(1:n,1:m); for j= 1:m for i = 1:n mfr(i,j) = ((31.42.*conv(i,j).^3)-
(122.9.*conv(i,j).^2)+(231.1.*conv(i,j))-138.1);%kg/hr end end % - Load Second Trial Data filename = 'C:\Users\Carter\Documents\MATLAB\VenturiDATA__5.txt'; startRow = 24; endRow = 1824; %% [Time,BoschMAF_V,Pressure1_V,Pressure2_V,Volts3,Volts4,Tamb_F,Volts5,DegF1,De
%% Define Mass Flow Rate of Second Trial x = BoschMAF_V; Bosch_MAF_2 = (31.42.*x.^3)-(124.37.*x.^2)+(235.25.*x)-142.31; Bosch_MAF_Cal_2 = (29.93.*x.^3)-(122.9.*x.^2)+(231.1.*x)-138.1; n = 240; m = 40; Bosch_MAF_Avg_2 = [mean(Bosch_MAF_2(m:1:n-m)) mean(Bosch_MAF_2(n+m:1:2*n-m))
6,VOLTS7,VOLTS8,VOLTS9] % = IMPORTFILE(FILENAME, STARTROW, ENDROW) Reads data from rows STARTROW % through ENDROW of text file FILENAME. % % Example: %
6,Volts7,Volts8,Volts9] = importfile('VenturiDATA__5.txt',24, 1824); % % See also TEXTSCAN.
% Auto-generated by MATLAB on 2016/07/18 20:58:46
%% Initialize variables. delimiter = '\t'; if nargin<=2 startRow = 24; endRow = inf; end
%% Format string for each line of text: % column1: double (%f) % column2: double (%f) % column3: double (%f) % column4: double (%f) % column5: double (%f) % column6: double (%f) % column7: double (%f) % column8: double (%f) % column9: double (%f) % column10: double (%f) % column11: double (%f) % column12: double (%f) % column13: double (%f) % column14: double (%f) % For more information, see the TEXTSCAN documentation. formatSpec = '%f%f%f%f%f%f%f%f%f%f%f%f%f%f%*s%[^\n\r]';
%% Open the text file. fileID = fopen(filename,'r');
%% Read columns of data according to format string. % This call is based on the structure of the file used to generate this % code. If an error occurs for a different file, try regenerating the code % from the Import Tool. dataArray = textscan(fileID, formatSpec, endRow(1)-startRow(1)+1,
'ReturnOnError', false); for col=1:length(dataArray)
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dataArray{col} = [dataArray{col};dataArrayBlock{col}]; end end
%% Close the text file. fclose(fileID);
%% Post processing for unimportable data. % No unimportable data rules were applied during the import, so no post % processing code is included. To generate code which works for % unimportable data, select unimportable cells in a file and regenerate the % script.
