tag:blogger.com,1999:blog-68061182512415433442017-12-08T18:45:55.107-05:00VoluntocracyGovernance by those who do the work.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.comBlogger79125tag:blogger.com,1999:blog-6806118251241543344.post-87997096879204664552017-12-08T18:45:00.000-05:002017-12-08T18:45:55.161-05:00The Physics of MarblingLast week I spoke on <a href="https://www.newton.ac.uk/seminar/20171128111011501" target="_blank">The Physics of Marbling</a> at the <a href="http://www.newton.ac.uk/event/gfsw04" target="_blank"><i>Form in art, toys and games</i></a> workshop at the Isaac Newton Institute for Mathematical Sciences in the University of Cambridge!<br /><br /><i></i> <table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://www.newton.ac.uk/files/events/group-photos/gfsw04-1125491.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="533" data-original-width="800" height="425" src="https://www.newton.ac.uk/files/events/group-photos/gfsw04-1125491.jpg" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">I am the one in the bright blue jacket.</td></tr></tbody></table>The four day workshop had many fascinating presentations on a wide variety of topics. <a href="https://www.newton.ac.uk/webseminars?field_when_value_1[min]&field_when_value_1[max]&title_2=&title=&combine=&page=1" target="_blank">Videos for most of the talks are available.</a><br /><i></i>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com020 Clarkson Rd, Cambridge CB3 0EH, UK52.2095075 0.1028499999999894552.0539495 -0.21987350000001055 52.3650655 0.42557349999998945tag:blogger.com,1999:blog-6806118251241543344.post-66529973492165367892017-08-06T09:40:00.000-04:002017-08-06T21:15:26.835-04:00Mixed Convection from an Isothermal Rough Plate<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ln11F1sqai8/WYcbjp2W80I/AAAAAAAACXc/_BsJAliN8eYsAx_H688LKpDRTdlTcvXoACLcBGAs/s1600/mixed-dn-correlation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="554" data-original-width="792" height="447" src="https://2.bp.blogspot.com/-ln11F1sqai8/WYcbjp2W80I/AAAAAAAACXc/_BsJAliN8eYsAx_H688LKpDRTdlTcvXoACLcBGAs/s640/mixed-dn-correlation.png" width="640" /></a></div><br />In March 2017 the roughness of the aluminum plate was reduced from 3mm to 1mm. I installed it in the wind-tunnel and started running experiments. The forced convection measurements were nearly 30% higher than expected!<br /><br />I examined nearly every aspect of the physical device and its mathematical model. The fan-speed calibration was found to be sensitive to the distance between the test surface and the wind-tunnel wall. Conditioning the rpm-to-speed conversion on the plate's orientation improved the earlier data taken with the plate with 3mm roughness.<br /><br />When the plate is not parallel to the wind-tunnel, the forced and mixed measurements are affected. With the 3mm roughness plate, the alignment had been controlled within a couple of millimeters over the plate's 305mm length. The 1mm roughness plate seemed to require stricter tolerances. Using a caliper, I am able to control the alignment to better than 1mm.<br /><br />The primary cause of the measured excess was that, when the height of the posts had been reduced, the size and spacing of the posts had not been reduced. At high wind-speeds the convection from the flat post tops was exceeding the "fully-rough" mode of convection. The model incorporating this phenomena is developed in the "Rough to Smooth Turbulence Transition" section of my "<a href="http://people.csail.mit.edu/jaffer/convect" target="_blank">Mixed Convection from an Isothermal Rough Plate</a>" paper.<br /><br />Writing a paper forces one to revisit all the questions and anomalies that occurred during research and experiment. Understanding, resolving, and testing all of these issues has taken months. I would appreciate any proof-reading or critiquing that others might provide before I submit it for publication.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-49074233041447885792017-02-08T19:17:00.001-05:002017-02-08T19:17:45.443-05:00Oseen Flow in Ink Marbling<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-TluAsF1Q-Q8/WJuWEuexpOI/AAAAAAAACSs/Ys2TSG_01Q0s45tuI14dkoAeOdi4Pq0kQCLcB/s1600/marble-arc-66.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-TluAsF1Q-Q8/WJuWEuexpOI/AAAAAAAACSs/Ys2TSG_01Q0s45tuI14dkoAeOdi4Pq0kQCLcB/s1600/marble-arc-66.jpg" /></a></div><br />Mathematical modeling of ink marbling has long been a fascination of mine. My <a href="http://people.csail.mit.edu/jaffer/Marbling">Ink Marbling</a> web pages have presented emulations of a number of marbling techniques. But the raking techniques modeled were either paths across the whole tank or circular paths.<br /><br />Pictorial ink marbling designs are created using short strokes, where a stylus is inserted into the tank; moved a short distance; then extracted. There seems to be no way to adapt the line or circle draws to short strokes with endpoints.<br /><br />Having bought a copy of <i>Boundary-layer theory </i>(Hermann Schlichting et. al.) for my convection project, I started reading from the beginning. It didn't take long until I found a description of Oseen flow on page 115 (chapter IV, very slow motion). Its streamline figure looked very promising. After further research I have written: <i><a href="https://arxiv.org/abs/1702.02106">Oseen Flow in Ink Marbling</a></i><a href="https://arxiv.org/abs/1702.02106"> arXiv:1702.02106 [physics.flu-dyn]</a>.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-SntAaKPl62M/WJuzdf78yDI/AAAAAAAACTU/oA58Y0uueVM7ROukFo6jNgwZeM0Nz_0XwCLcB/s1600/stroke-4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-SntAaKPl62M/WJuzdf78yDI/AAAAAAAACTU/oA58Y0uueVM7ROukFo6jNgwZeM0Nz_0XwCLcB/s1600/stroke-4.png" /></a></div><br /><br />Unlike a cylinder in a 2-dimensional flow, the velocity field induced by an infinitesimally thin stylus can be exactly solved in closed form. In this sense, marbling is the purest form of Oseen flow.<br /><br />The partial differential equations solved include conservation of mass (divergence=0), but not conservation of momentum (Navier-Stokes). It's not clear how much momentum is imparted by the stylus, or how that imparting momentum changes with time.<br /><br />Liquid marbling is sensitive to the speed of a stylus moving through the tank. At low speeds the induced flow is laminar; at high speeds the flow becomes turbulent. Both are used by marblers, but only the laminar flow is possible to solve in closed form.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FwsStge0S7g/WJueNXursUI/AAAAAAAACTE/2Vf6ornH3ycsY6FLrIUWqa1Fkg87gvJrQCLcB/s1600/flower.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://2.bp.blogspot.com/-FwsStge0S7g/WJueNXursUI/AAAAAAAACTE/2Vf6ornH3ycsY6FLrIUWqa1Fkg87gvJrQCLcB/s1600/flower.jpg" /></a></div><br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-31267179671321035682016-10-02T23:11:00.000-04:002016-10-09T22:28:12.860-04:00Mixed Convection from a Rough PlateIts been a long time since my last blog entry; there are new developments.<br /><br />After completing the vertical convection measurements, I returned the Convection Machine to its horizontal orientation and did some runs to make sure everything was as before. But things were not the same.<br /><br />With the rough side facing down, the transition where mixed convection dropped below the linear asymptote had disappeared. Varying parameters did not restore the dip. Changing inclination of the plate; tilting the wind-tunnel; resealing the cardboard around the fan; nothing I tried restored the dip. Something has permanently changed in the wind-tunnel or the plate.<br /><br />So I reran the measurements of horizontal upward and downward facing mixed convection. The new curves match simple L2 and L4 norms of the forced and natural convection components. My guess of what changed has to do with the suspension of the plate. The vertical suspension was a single long wire which, after hooking around the a top corner post, wrapped across the back and hooked around the other top corner post. Wrapping around the back compressed the back sheet and insulation against the back side of the rough plate. Perhaps the pressure closed gaps in the glue between aluminum and insulation.<br /><a href="https://3.bp.blogspot.com/-1BsiigYLhb0/V_HJ123i_FI/AAAAAAAACQ0/5_GwCwLdV0sWj2JPvELV4aFY4x82iOvhQCLcB/s1600/12-point-suspension.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="245" src="https://3.bp.blogspot.com/-1BsiigYLhb0/V_HJ123i_FI/AAAAAAAACQ0/5_GwCwLdV0sWj2JPvELV4aFY4x82iOvhQCLcB/s320/12-point-suspension.jpg" width="320" /></a><br />Having data for horizontal forced flow with 3 plate orientations, it was time to measure downward forced flow with a vertical plate. Because vibration of the plate had caused excess convection with the single wire suspension, I added two wrap-around wires pulling in opposite directions to the plate suspension. This new suspension is quite rigid and works with the wind-tunnel in any orientation.<br /><a href="https://2.bp.blogspot.com/-gfWat5R9C_Y/V_HKELJdUcI/AAAAAAAACQ8/AEToT9fZVoIFJPG9zdtq9RVyh896JExHQCEw/s1600/vertical-wind-tunnel.jpg" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-gfWat5R9C_Y/V_HKELJdUcI/AAAAAAAACQ8/AEToT9fZVoIFJPG9zdtq9RVyh896JExHQCEw/s320/vertical-wind-tunnel.jpg" width="134" /></a><br />I added four legs to support the wind-tunnel upright with the fan drawing downward. I was expecting either L2-norm mixing or for convection to drop below the natural level when the natural and forced components were equal. But it was neither! I devised a model which transitions between L2 and L4 norm that matched the measurements well; it is detailed in my paper.<br /><a href="https://3.bp.blogspot.com/-TwaZuUIVUmI/V_HJ8OkSocI/AAAAAAAACQ4/scVoay2J9Y493KrOu8oROD8eo1oH-kPxACEw/s1600/verticalwindtunnelup.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-TwaZuUIVUmI/V_HJ8OkSocI/AAAAAAAACQ4/scVoay2J9Y493KrOu8oROD8eo1oH-kPxACEw/s320/verticalwindtunnelup.jpg" width="127" /></a><br />Because the opposed mixing was unexpected, aided mixing had to be tried. It also turned out to involve a transition between L2 and L4 norms, but with a gentler transition.<br /><br />I have finished writing the article and put it and the supplementary data on <a href="http://people.csail.mit.edu/jaffer/convect">http://people.csail.mit.edu/jaffer/convect</a><br /><br />As described in the paper, the next step is to shave 4.mm off the rough side of the plate and repeat the measurements.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-18709460407742024142016-06-13T21:52:00.000-04:002016-07-05T19:34:25.011-04:00Mixed Convection from a Vertical Rough SurfaceI turned the wind-tunnel on its side and hung the plate vertically as shown in the photograph.<br /><br /><a href="https://3.bp.blogspot.com/-LEv7pPe1Zbk/V19fCLjbaQI/AAAAAAAACM8/Bcxg2cXcMhIkQjJ7XtKJznyPAXMWOR2eQCLcB/s1600/WT-vertical.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://3.bp.blogspot.com/-LEv7pPe1Zbk/V19fCLjbaQI/AAAAAAAACM8/Bcxg2cXcMhIkQjJ7XtKJznyPAXMWOR2eQCLcB/s640/WT-vertical.jpg" width="480" /></a><br /><br />These graphs show that my mixed convection model is successful from natural through forced convection for horizontal and vertical rough plates with forced flow perpendicular to the natural flow. The leftmost red dot in each graph is the natural convection (Re=0) for that orientation; it is placed at Re=1000 so that it can appear on the graph.<br /><a href="https://2.bp.blogspot.com/-PTe14KbQw_0/V19eO46PAoI/AAAAAAAACMo/qnAUX6FrH6YpMD42G5BaKda0Tcbu3H-agCKgB/s1600/downward-correlation-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="https://2.bp.blogspot.com/-PTe14KbQw_0/V19eO46PAoI/AAAAAAAACMo/qnAUX6FrH6YpMD42G5BaKda0Tcbu3H-agCKgB/s640/downward-correlation-2.png" width="640" /></a><br /><a href="https://2.bp.blogspot.com/-70FSBcj5j6Y/V19eO6eOCPI/AAAAAAAACMs/FHxBSW2TkV41_myRjAujMeEb5toxG_huQCKgB/s1600/vertical-correlation-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="https://2.bp.blogspot.com/-70FSBcj5j6Y/V19eO6eOCPI/AAAAAAAACMs/FHxBSW2TkV41_myRjAujMeEb5toxG_huQCKgB/s640/vertical-correlation-2.png" width="640" /></a><a href="https://4.bp.blogspot.com/-T5KNdjvdjfQ/V19eOxMOFEI/AAAAAAAACMk/-KH1ROQur-ULwwz1DeDKAA1mHQFU0xPwwCKgB/s1600/upward-correlation-2.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="https://4.bp.blogspot.com/-T5KNdjvdjfQ/V19eOxMOFEI/AAAAAAAACMk/-KH1ROQur-ULwwz1DeDKAA1mHQFU0xPwwCKgB/s640/upward-correlation-2.png" width="640" /></a><br />The L4-norm for downward convection indicates that the interaction between the downward mode and forced convection is more competitive than the fairly cooperative L2-norm of the vertical and upward modes. <br /><br />Horizontal downward and vertical natural convection from the rough plate match that expected from a smooth plate. Horizontal upward convection matches assuming that the upper (rough) surface convection is reduced by 93% of the non-forced convection from the four adjoining sides. In order to test if horizontal upward convection is the same for rough and smooth, I will cover the rough surface with a flat sheet of aluminum and repeat the test.<br /><br />I have started writing a paper titled "Mixed Convection from a Rough Plate". Which journal should I submit it to?<br /><span id="goog_1985933606"></span><span id="goog_1985933607"></span>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-85851419824257422502016-06-04T19:16:00.000-04:002016-06-10T09:45:51.008-04:00Fan WindspeedWith the wind-tunnel fan being phase-locked now, the speed variability which plagued earlier speed measurements should be reduced or eliminated.<br /><br />The process of conducting the measurements for these graphs finds this to be the case. Although some variability remains above 3 m/s (1000 r/min), at slower speeds the anemometer readings are steady after the phase-lock-loop settles. The measured traces are in blue; the black curve is that used in convection calculations. This first graph is for the wind tunnel with horizontal plate.<br /><br /><br /><a href="https://1.bp.blogspot.com/-pTJjwrIikdk/V1NfuE478-I/AAAAAAAACMI/B4AzTw_I06Yl506WdBcK1wavCrHf-vR9gCKgB/s1600/20160528T202907-pllcal.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="250" src="https://1.bp.blogspot.com/-pTJjwrIikdk/V1NfuE478-I/AAAAAAAACMI/B4AzTw_I06Yl506WdBcK1wavCrHf-vR9gCKgB/s320/20160528T202907-pllcal.png" width="320" /> </a><br /><br />This second graph is with the wind tunnel laying on its side with the plate vertical. <br /><br /><a href="https://2.bp.blogspot.com/-natk9Mzr-F8/V1NfuMYsMaI/AAAAAAAACME/tqv30_T7GSo7x0mqApzL0JKdYeEyuC2LQCKgB/s1600/20160603T201759-pllcal.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="250" src="https://2.bp.blogspot.com/-natk9Mzr-F8/V1NfuMYsMaI/AAAAAAAACME/tqv30_T7GSo7x0mqApzL0JKdYeEyuC2LQCKgB/s320/20160603T201759-pllcal.png" width="320" /></a><br /><br />An earlier post found that the kink just above 2 m/s is due to the anemometer; it remains. Friction in the anemometer makes the measurements below fan speeds of 700 r/min unreliable; below fan speeds of 200 r/min the anemometer reads 0.0.<br /><span id="goog_2062932961"></span><span id="goog_2062932962"></span>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-14400455297440197672016-05-07T12:14:00.003-04:002016-05-07T21:30:05.566-04:00How to Phase-Lock a FanUsing an auto-transformer to reduce the voltage to the wind-tunnel fan<br />in order to reduce its speed didn't work below 45 r/min (it ran for a<br />while and stopped). So I modified The Convection Machine to toggle<br />the fan power with a solid-state relay controlled by micro-processor.<br /><br />Consider the shaft of the wind-tunnel fan. Every full rotation of the<br />fan results in 3 micro-processor interrupts. A phase-accumulator<br />register is incremented by the desired rotation rate (in r/min) 1200<br />times a second and decreased by 24000 every time a fan blade crosses a<br />light beam. If the fan rotates at the desired rotation rate, then the<br />average phase-accumulator value is constant. If the fan is too slow,<br />then the phase-accumulator value increases with time; if it is too<br />fast, then the phase-accumulator value decreases with time.<br /><br />Phase-locking is the process of controlling the fan-speed so that the<br />average phase-accumulator value is 0. Such feedback systems are<br />tricky to stabilize. My fan controller operates in one mode when the<br />desired speed is less than 400 r/min and a different one otherwise.<br /><br />At high speeds, the fan speed is roughly proportional to the average<br />voltage applied, which is proportional to the duty cycle of applied<br />voltage. The phase accumulator operates as described above. Its<br />instantaneous value is compared with a variable which decrements from<br />the upper phase range bound to 0 ten times a second. If greater, the<br />fan is turned on, otherwise it is turned off. Some of the<br />instabilities of the fan speed may be due to a centripetal switch<br />disconnecting the starter capacitor and hooking in the running<br />capacitor, which increases the loop gain of the system. The change in<br />gain causes the system to overshoot and undershoot the desired r/min<br />with long settling times.<br /><br />At low speeds each pulse of power incrementally increases the fan<br />speed while friction continually slows it. The solid-state relay has<br />"zero-crossing" control, so only complete half-cycles of 60 Hz power<br />are applied to the fan motor. The combination of the motor windings<br />and phase capacitor stores energy, so the acceleration of the rotor is<br />delayed from the application of power. At low speeds the rotational<br />inertia of the rotor introduces 90 degrees of phase shift. The<br />microprocessor clock is not synchronized to the line voltage, so the<br />minimum pulse width varies with the relative phase, another source of<br />loop gain variation.<br /><br /><a href="https://3.bp.blogspot.com/-2dk9lCr7dOY/Vy4SFJ3iiGI/AAAAAAAACLA/oBaYJPfBgDwatGv7F7lmpwFcuZPP5q_mwCLcB/s1600/speed-control.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://3.bp.blogspot.com/-2dk9lCr7dOY/Vy4SFJ3iiGI/AAAAAAAACLA/oBaYJPfBgDwatGv7F7lmpwFcuZPP5q_mwCLcB/s320/speed-control.jpg" width="320" /></a><br /><br />This photo shows the new fan-speed control. The number on the<br />7-segment displays is the rotation rate in r/min measured every<br />second. The right 3 dial switches set the desired rotation rate.<br /><br /><iframe allowFullScreen='true' webkitallowfullscreen='true' mozallowfullscreen='true' width='320' height='266' src='https://www.blogger.com/video.g?token=AD6v5dwyn1qDNYopI2_v3yDujehIPpvF--PWvJhHqQsLBbKZiYXzwwMktYyuvSZqNHCkVrfdDWZgikvbNPCDvdA8BA' class='b-hbp-video b-uploaded' FRAMEBORDER='0' /> <br /><br />This video which shows the phased-locked fan in<br />operation at a variety of speeds. The low light level was necessary<br />so that the stroboscopic interaction of camera shutter with the<br />scanned 7-segment display didn't render the numbers unreadable. If<br />you turn up the audio volume you can hear the fan chugging as its<br />power is switched on and off.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-85286346488872023422016-03-19T13:59:00.000-04:002016-03-19T13:59:33.881-04:00Make Square Opening by Drilling Five Round HolesIn making a speed control for an off-the-shelf electric fan, I needed to install a square power receptacle in the phenolic box I am using for the speed dial switches and 7-segment displays. Phenolic is brittle and does not machine well with the woodworking tools that I have, chipping instead of cutting.<br /><br />A straightforward way to cut the hole would be to drill a hole, disassemble a coping saw and reassemble it with the blade through the hole, and sawing. But the small size of the box would limit the saw strokes to a few centimetres. <br /><br />A reciprocating "Sabre" saw might do the job but is hard to control.<br /><br />Twist drill bits seemed to work the best of my tools on the material, and I have a good selection of sizes. How close to a square opening can one create by drilling a small number of round holes? It turns out that I can come fairly close. There are two ways the problem can be posed, the largest opening bounded within the square or the smallest opening just larger than the square. I am interested in the latter; the other solution can be had by scaling.<br /><br />The idea is to drill holes on the diagonals near the 4 corners such that the corner touches the rim of the hole. One then drills a hole in the center which is enough larger than the square so that its points of contact with the square are points of contact with the 4 smaller holes. If the diameter of the 4 corner holes is reduced, then the diameter of the center hole must be increased in order for it to intersect the smaller holes and the square.<br /><br />Let L be the length of one side of the square, R be the radius of the center hole, and r be the radius of the corner holes. The center of each corner hole is L/sqrt(2)-r from the center of the square so that the rim lies on the corner point. The furthest that the corner holes exceed the desired square is<br /><br /> r+(L/sqrt(2)-r)/sqrt(2)-L/2 = r-r/sqrt(2) = r*(1-sqrt(1/2))<br /><br />The furthest that the large center hole exceeds the desired square is R-L/2. Desired is:<br /><br /> R-L/2 = r*(1-sqrt(1/2))<br /><br />The other constraint is that the rim of the center hole and the rim of the corner hole intersect on the side of the square. The distance from the middle of the side to the intersection of the side with the large hole is x<br /><br /> R^2 = (L/2)^2 + x^2<br /><br />The distance from the corner to the intersection is r*sqrt(2). So:<br /><br /> L/2 = r*sqrt(2) + x<br /><br />R, r, and x scale with L. Let L = 1.<br /><br /> R^2 = 1/4 + x^2<br /><br /> 1/2 = r*sqrt(2) + x<br /><br /> R-1/2 = r*(1-sqrt(1/2))<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-ACTJz1iPfjg/Vu2PtaGcBlI/AAAAAAAACKQ/krEPZnka3C4gsmc-RSysH--IKmeesQIRg/s1600/hole.jpg" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><a href="https://1.bp.blogspot.com/-BPcWmHjuOgA/Vu2Pta-zzaI/AAAAAAAACKM/9vyP8Hccu6oIhJGzdJGtrQrmRKP3SuqDQ/s1600/square5.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="https://1.bp.blogspot.com/-BPcWmHjuOgA/Vu2Pta-zzaI/AAAAAAAACKM/9vyP8Hccu6oIhJGzdJGtrQrmRKP3SuqDQ/s1600/square5.png" /></a><br />Solving this system:<br /><br /> r = 1/(4+sqrt(2))<br /><br /> r = L * 184.69903125906464e-3<br /><br /> R = L * 554.097093777194e-3<br /><br />The diameters when L = 1.125 are:<br /><br /> d = 415.57282033289544e-3<br /><br /> D = 1.2467184609986863<br /><br />I made the holes with a 3/8 inch twist drill and a 1.25 inch hole saw. A 7/16 bit would have been closer in size.<br /><a href="https://3.bp.blogspot.com/-U0LxmJIfEeA/Vu2PtQomwqI/AAAAAAAACKU/TLUl-qWfixc_-VEiEjAzCnBbb77xjGcUw/s1600/socket.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-U0LxmJIfEeA/Vu2PtQomwqI/AAAAAAAACKU/TLUl-qWfixc_-VEiEjAzCnBbb77xjGcUw/s320/socket.jpg" width="160" /></a><a href="https://1.bp.blogspot.com/-ACTJz1iPfjg/Vu2PtaGcBlI/AAAAAAAACKQ/krEPZnka3C4gsmc-RSysH--IKmeesQIRg/s1600/hole.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-ACTJz1iPfjg/Vu2PtaGcBlI/AAAAAAAACKQ/krEPZnka3C4gsmc-RSysH--IKmeesQIRg/s320/hole.jpg" width="160" /></a><br />The flange on the outlet covers the non-square parts of opening.<br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-49496981675242058082016-02-18T22:03:00.000-05:002016-02-19T19:46:36.545-05:00Vertical Natural Convection<img align="right" alt="" src="http://people.csail.mit.edu/jaffer/SimRoof/Convection/vert-susp8.jpg" height="576" width="423" /> Unexpected results for downward convection at small-angles raised the question of whether vertical natural convection is the same for rough and smooth plates. This photo shows the plate suspended vertically by steel wire from the two boards above. The ambient temperature sensor is taped to the table leg. I measured the natural convection over three temperature ranges as was done in the other natural convection runs. <br /> If there is less convection than expected, then it could be due to heat from one side reducing the convection of a side above it, as happens in the upward facing case. <br /> But slightly more convection than expected was measured. As the plate is no longer in the wind tunnel, modeling the emissivity of the room as 0.9 (versus 0.8 for the wind tunnel) brings the simulation into reasonable agreement with measurement. It thus appears that, at least for laminar flows from rectangular plates, natural convection from a rough surface has the same magnitude as convection from a smooth surface. <br /> The graph below is linked to a <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/natural.pdf">pdf of the measurements and simulations of natural convection</a> in level and vertical orientations. <br /> <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/natural.pdf"><img alt="natural vertical convection correlation" border="1" src="http://people.csail.mit.edu/jaffer/SimRoof/Convection/vertical-correlation.png" height="447" width="640" /></a> <br /><a href="https://www.blogger.com/null" name="Future Experiments"></a> <h2></h2>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-4440403163821786802016-02-06T22:32:00.000-05:002016-02-06T22:47:28.936-05:00Mixed ConvectionAs my blog post <a href="http://voluntocracy.blogspot.com/2015/12/upward-natural-convection.html">Upward Natural Convection</a> details, there is no guarantee that natural convection of my plate with the rough surface facing up behaves in a manner which can be modeled. The air warmed by the insulated back and sides rises adjacent to the heated rough surface which draws air towards its center. So how strong is this effect? Measured with peak temperature differences of 15 K, 10 K, and 5 K, the measured upward natural convection is about half that predicted. But looking at the natural convection components, the deficit is roughly equal to sum of the back and side convective heat flows! <br />With the rough surface facing down, the mix of convective and radiative heat loss from the four sides matters little because both are subtracted from the overall heat loss. But when the rough surface faces upward it does matter; side convection reduces the rough surface convection while thermal radiation does not. Creating a plot of downward natural convection at a range of temperature differences allows evaluation of simulated mixtures. As the fraction of simulated radiative heat loss increases, the slope connecting the measured points increases. The graph below shows the fit when the effective radiative height of the side is 41% of its actual height, and the effective convective surface area is adjusted to fit: <br /><img alt="Downward Natural Convection" src="http://people.csail.mit.edu/jaffer/SimRoof/Convection/downward-correlation.png" height="447" width="640" /><br />With this rough estimate of the relative strengths of convective and radiative heat loss, we are now ready to see whether the effect of the sides on the top surface can be reasonably modeled. <br />The plot below compares upward convection correlations with total non-radiative heat flow minus 77% of the (modeled) sides and back natural convection. There is less variation from point to point because upward natural convection is three times stronger than downward natural convection. <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/index#HHT">"Horizontal Hot Top"</a> is my generalization of the four conventional upward convection correlations. <br /><img alt="Upward Natural Convection" src="http://people.csail.mit.edu/jaffer/SimRoof/Convection/upward-correlation.png" height="447" width="640" /><br />The unexpectedly close match above lends support to the idea that air heated by the sides is drawn over the upper surface of the plate, reducing the effective temperature difference between the plate and air, and can be modeled as a reduction of upward convection by an amount proportional to the side convection. <br />Now that I have convection measurements at low fan speeds which are comparable in magnitude to natural convection, the next step is to evaluate mixed convection (with the rough surface facing upward). <br />Just as there was no guarantee of a workable model for the interaction of the sides with the top in still air, there is no guarantee of a model of that interaction in forced air. Needed is a generalization matches the model developed for <i>V</i>=0 and matches the forced correlation as <i>V</i> grows. <br />The graph below shows the correlation I have arrived at for upward convection (<a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/forced-up.pdf">see simulations</a>). It assumes that mixed convection for the four short sides is the L<sup>4</sup>-norm of the natural and forced convections and that 77% of only the natural component of the back and sides is absorbed by the convection of the (upward-facing) rough surface. <br /><img alt="Forced Mixed with Upward Natural Convection" src="http://people.csail.mit.edu/jaffer/SimRoof/Convection/mixed-up-correlation-2.png" height="447" width="640" /><br />If the L<sup>4</sup>-norm mixing for the short sides is instead L<sup>2</sup>, the increase in natural convection from the sides reduces the top surface convection, spoiling the upward-convection match with L<sup>2</sup>-norm (gray dashed line) and other L-norm exponents (best is about L<sup>2.3</sup>). <br />Here are the calculated values of convection from all six sides of the plate at windspeeds from 0 to 4 m/s and Δ<i>T</i>=11K: <br /><blockquote><table> <tbody><tr><th>insulated<br />back</th><td>+ 2 ⋅</td><th>parallel<br />side</th><td>+ 2 ⋅</td><th>windward<br />leeward</th><td>=</td><th>total</th><td>vs</td><th>rough</th><td>@</td><th>windspeed</th></tr><tr><td>53.1mW/K</td><td><br /></td><td>55.3mW/K</td><td><br /></td><td>55.3mW/K</td><td><br /></td><td>0.272W/K</td><td><br /></td><td>0.467W/K</td><td><br /></td><td>0.0m/s</td></tr><tr><td>55.2mW/K</td><td><br /></td><td>55.4mW/K</td><td><br /></td><td>55.4mW/K</td><td><br /></td><td>0.274W/K</td><td><br /></td><td>0.484W/K</td><td><br /></td><td>0.12m/s</td></tr><tr><td>60.4mW/K</td><td><br /></td><td>55.5mW/K</td><td><br /></td><td>55.4mW/K</td><td><br /></td><td>0.280W/K</td><td><br /></td><td>0.520W/K</td><td><br /></td><td>0.25m/s</td></tr><tr><td>65.0mW/K</td><td><br /></td><td>55.9mW/K</td><td><br /></td><td>55.9mW/K</td><td><br /></td><td>0.286W/K</td><td><br /></td><td>0.666W/K</td><td><br /></td><td>0.50m/s</td></tr><tr><td>68.5mW/K</td><td><br /></td><td>57.3mW/K</td><td><br /></td><td>57.6mW/K</td><td><br /></td><td>0.296W/K</td><td><br /></td><td>1.18W/K</td><td><br /></td><td>1.0m/s</td></tr><tr><td>71.0mW/K</td><td><br /></td><td>61.5mW/K</td><td><br /></td><td>63.4mW/K</td><td><br /></td><td>0.318W/K</td><td><br /></td><td>2.22W/K</td><td><br /></td><td>2.0m/s</td></tr><tr><td>72.6mW/K</td><td><br /></td><td>72.4mW/K</td><td><br /></td><td>78.4mW/K</td><td><br /></td><td>0.371W/K</td><td><br /></td><td>4.36W/K</td><td><br /></td><td>4.0m/s</td></tr></tbody></table></blockquote>I had expected L<sup>4</sup>-norm mixing for upward convection. So these results will require modifications to my theory of mixed convection. Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-84897551122760393062016-01-27T15:08:00.000-05:002016-01-27T15:08:45.940-05:00Forced Convection on Convection Machine 2.0 The new back eliminated the dual modes seen earlier with peak Δ<i>T</i>=15K. Reducing the r/min to m/s conversion by 2% (which is within the ± 3% error band of the anemometer) and adjusting the breakpoint results in a beautiful fit! <br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zcRWQCUtfFA/VqkhoY6zPII/AAAAAAAACI8/DV0bTMOmGB8/s1600/forced-correlation-2D15K.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-zcRWQCUtfFA/VqkhoY6zPII/AAAAAAAACI8/DV0bTMOmGB8/s640/forced-correlation-2D15K.png" width="640" /></a></div><br /> The points at Re=3545 and Re=5231 could be following either the Smooth Turbulent Asymptote or the Transition Model; it's too close to tell. The reason that the Transition Model has less convection than the Smooth Turbulent Asymptote below Re=6000 is because the leading edge of the boundary layer is modeled as thinner than the roughness; but division by the local characteristic length, 0, is undefined at the leading edge. Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-75484329849130706292016-01-08T22:22:00.000-05:002016-01-08T22:22:10.031-05:00Slightly Tilted Downward Convection<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div>With the current configuration, I am measuring the natural convection from a plate inclined a few degrees from facing downward. Turning the front tuning pegs shortens the support wires and tilts the plate. <br /> <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/--szrS31Ta0c/VpB5u2ahzDI/AAAAAAAACHA/vg1jnaYW2as/s1600/downward-angle-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/--szrS31Ta0c/VpB5u2ahzDI/AAAAAAAACHA/vg1jnaYW2as/s640/downward-angle-1.png" width="640" /></a></div><br /> This graph shows that downward convection does not blend with other convective modes. With <i>θ</i> between 87° and 90° downward convection is in control; between 0° and 87° the vertical mode is dominant. I have modified the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/index#UCT3">UCT3</a> formula in the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/index#Downward">Downward Natural Convection</a> section with this new knowledge. <br /> <blockquote> <table> <tbody><tr> <td nowrap="nowrap"><i>R</i> = min(<i>L<sub>H</sub></i>, <i>L<sub>W</sub></i>)/2</td> </tr><tr> <td nowrap="nowrap"><i>h</i> = <i>k</i>⋅max<b>(</b>Nu<sub>9.3</sub>(Ra(<i>L<sub>H</sub></i>) cos <i>θ</i>)/<i>L<sub>H</sub></i> <b>,</b> Nu<sub>45</sub>(Ra(<i>R</i>) sin <i>θ</i>)/<i>R</i>) </td> <td align="right" width="25%">0°≤<i>θ</i>≤+90°</td> <td align="right" width="20%">UCT3</td> </tr></tbody></table></blockquote>The random variation in the plot above makes it less convincing than it might be. I modified the firmware to increase the heating threshold from 5 K to 15 K, which increases the average Δ<i>T</i> from 3.7 K to 11 K and reduces the relative variation by about a factor of 3. <br /> <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-fb2kjth7Ih8/VpB5u5q3MsI/AAAAAAAACHE/_3mi3kUq4CA/s1600/downward-angle-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="447" src="http://3.bp.blogspot.com/-fb2kjth7Ih8/VpB5u5q3MsI/AAAAAAAACHE/_3mi3kUq4CA/s640/downward-angle-2.png" width="640" /></a></div><br /> Running the tilt tests at the higher temperature difference results in markedly reduced variation as can be seen by the proximity of the half-interval measurements (black dots) to the full-interval measurements (red dots). But the slope below 87.5° clearly does not match that derived from Nu<sub>9.3</sub>, the natrual convection from a smooth vertical plate! More about this below.<br /> I also ran a series of forced convection trials with the 15 K peak temperature difference: <br /> <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/forced-D15.pdf"></a><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-KT1xQa3KWag/VpB5ug90GEI/AAAAAAAACG8/Biaz0h8x1iU/s1600/forced-correlation-2D15.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-KT1xQa3KWag/VpB5ug90GEI/AAAAAAAACG8/Biaz0h8x1iU/s640/forced-correlation-2D15.png" width="640" /></a></div><br /> There are several things to notice about this graph. The tests where only part of the plate had roughness penetrating the boundary layer seemed to have two distinct breakpoints: Re=16666 and Re=22222. For nearly all the measurements, the half-interval measurements (black dots) are very close to the full-interval measurements (red dots). There are two which are not so close. The black dots around Re=7440 lie close to the line, indicating the wind speed change which occured partway through the run. This can be seen in the bottom panel on page 8 of the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/forced-D15.pdf">PDF</a>. The dots at Re=8237 had the same windspeed, perhaps indicating that the system spent time in both breakpoint modes. <br /> Below Re=7500 the measurements track a slope of turbulent convection (Re<sup>4/5</sup>). Unfortunately the wind-tunnel can't produce wind speeds below 0.25 m/s (Re=4900), so we can't see the transition to natural downward convection. <br /> <hr /> <a href="https://www.blogger.com/null" name="Natural Tilt"></a> Returning to natural convection, either the natural convection from a rough vertical surface differs from that of a smooth vertical surface or the formula combining the downward and vertical convection is wrong. The former can be tested by suspending the plate vertically. I plan to do this by wrapping the wire suspending the plate around its (insulated) back. In order to protect the insulation from being cut by the wire, a sheet of aluminum should be glued to the back. I was planning to do this already in order to test upward natural convection.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Wmv6bVT-1Nk/VpB5-ciBeAI/AAAAAAAACHY/Ui1zuU-QIxY/s1600/dscf1178.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://1.bp.blogspot.com/-Wmv6bVT-1Nk/VpB5-ciBeAI/AAAAAAAACHY/Ui1zuU-QIxY/s640/dscf1178.jpg" width="640" /></a></div><br /> Here is the back side of the plate ready to have the 0.65 mm sheet of aluminum glued to it. Notches have been cut (using the "nibbler" on the sheet) near the sheet corners to catch the wires which will suspend it. <br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-KSKMA_wt4os/VpB5-UbR45I/AAAAAAAACHc/yu7GA4HtfYY/s1600/dscf1179.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://4.bp.blogspot.com/-KSKMA_wt4os/VpB5-UbR45I/AAAAAAAACHc/yu7GA4HtfYY/s640/dscf1179.jpg" width="640" /></a></div><br /> Gluing completed. <br /> <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-veNBVmw2NR8/VpB5-WrZg1I/AAAAAAAACHU/iU0ZhlpSDWs/s1600/dscf1180.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://3.bp.blogspot.com/-veNBVmw2NR8/VpB5-WrZg1I/AAAAAAAACHU/iU0ZhlpSDWs/s640/dscf1180.jpg" width="640" /></a></div><br />Here the rough surface of the plate is facing up in the wind tunnel. But first I must rerun all the downward facing measurements.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-5TNsfmmVLOg/VpB5-lFqeWI/AAAAAAAACHg/Q_eWFACCB_Y/s1600/dscf1181.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="480" src="http://3.bp.blogspot.com/-5TNsfmmVLOg/VpB5-lFqeWI/AAAAAAAACHg/Q_eWFACCB_Y/s640/dscf1181.jpg" width="640" /></a></div><br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-20741780438598611312015-12-20T22:49:00.000-05:002015-12-20T22:49:01.240-05:00Upward Natural ConvectionThere was a mistake in my simulation of the (insulated) back surface. It doesn't affect the measurement or simulation of the test surface. Here is a proper simulation of the plate with the rough side facing down: <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-lh90PunK_1Q/Vndq6d3mYPI/AAAAAAAACGc/QNDFwRFT74w/s1600/20151220T205655-downward.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://4.bp.blogspot.com/-lh90PunK_1Q/Vndq6d3mYPI/AAAAAAAACGc/QNDFwRFT74w/s640/20151220T205655-downward.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-KNgY9FzKnFQ/VnVrlvBNG7I/AAAAAAAACGI/4CiMposR-9k/s1600/20151009T134206-downward.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div>The red lines are the simulations; the others are measured. The backside simulation expects more convection than was measured, resulting in the measured back face temperature being higher than its simulated temperature.<br /><br />Because they are not thermally conductive over their whole surfaces, natural convection from the four vertical sides can't be modelled with established theory. My simulation models these sides as having 3.2 times the natural convection from the 52 mm × 305 mm metal surfaces not covered by insulation. The 3.2 factor was arrived at from natural convective runs on the Convection Machine.<br /><br />The streamlines in Fujii and Imura<a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/#76">[76]</a>'s figure 14(f) show air from the edges of the plate moving toward the rising column at the plate's centre. But in the Convection Machine this air has already been heated by the rough (downward) face and four sides. Thus the convection from the (upward) back side would be reduced and its temperature higher than if the other faces were not convecting.<br /><br />With the rough test surface facing downward its convection is not affected by the other faces. With the plate 5 K hotter than ambient, convection from the bottom face is about 0.6 W and about .348 W for each vertical face. Their combined 2 W dwarfs the .297 W expected through the insulation for the back face (upward). So it is not surprising that the back face convection is reduced.<br /><br />If the rough test surface faces up, then its expected 20.3 W convection will experience reduction from the 1.63 W of convection from the other faces. Unfortunately, in this case it does affect the measurements. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-KNgY9FzKnFQ/VnVrlvBNG7I/AAAAAAAACGI/4CiMposR-9k/s1600/20151009T134206-downward.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-14074010114152975292015-12-18T20:19:00.000-05:002015-12-18T21:04:39.795-05:00Scaled Colburn Analogy Asymptote<h3><span style="font-weight: normal;">I have rewritten the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/index#Scaled%20Colburn%20Analogy%20Asymptote">Scaled Colburn Analogy Asymptote</a> section to address the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements">Convection Machine</a> measurements at low Reynolds numbers:</span></h3><h3><span style="font-weight: normal;"></span> <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-4Zlz2q7M9Fw/VnSvKijQiaI/AAAAAAAACFk/oalqbouytsM/s1600/forced-correlation-1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-4Zlz2q7M9Fw/VnSvKijQiaI/AAAAAAAACFk/oalqbouytsM/s640/forced-correlation-1.png" width="640" /></a></div><div class="" style="clear: both; text-align: left;"><span style="font-weight: normal;">The logarithmic scale amplifies the variances below Re=10000. The graph below shows Nu/Pr<sup>1/3</sup> on a linear scale where variations can be seen as comparable in magnitude to the variation between experiments where Re is larger than 10000.</span></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-V-7F1WkJozo/VnSpp61XUxI/AAAAAAAACFE/qmqcoLcA8OM/s1600/linear-correlation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-V-7F1WkJozo/VnSpp61XUxI/AAAAAAAACFE/qmqcoLcA8OM/s640/linear-correlation.png" width="640" /></a><span style="font-weight: normal;"> </span></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><span style="font-weight: normal;">The revised formula, which includes the Re range over which the surface should be treated as rough, is:</span></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://1.bp.blogspot.com/-XAVdWXZTvjk/VnS2CTEjfII/AAAAAAAACF0/Hjyzx1JGChQ/s1600/formula.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="160" src="http://1.bp.blogspot.com/-XAVdWXZTvjk/VnS2CTEjfII/AAAAAAAACF0/Hjyzx1JGChQ/s640/formula.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></h3>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-31363043517131981392015-12-05T22:41:00.000-05:002015-12-05T22:45:16.353-05:00A Convection SurpriseYears ago I heard advice to experimental physicists to continue refining their experiments, even after they yield a hoped-for result.<br /><br />Last week I realized that if I switched the wind-tunnel fan to the lower speed setting, it might be able to run at speeds lower than 100r/min. I tried it and it worked! This updated plot has points at Re values less than 10000 and they show significantly less convection than expected. As in an earlier post, there would be many possible explanations for finding too much convection, but not for finding too little.<br /><br />At these low wind speeds the fan speed wanders through a +/-10% range and the convection measurements are averaged over time. I have rewritten my program to compute the convection over smaller intervals than the whole trial and included both the whole trial and smaller runs in the graph. My program also averages measurements through a trapezoidal window, which reduces the variation between the smaller intervals within an experimental trial.<br /><br /><div class="separator" style="clear: both; text-align: center;"><img border="0" height="444" src="http://3.bp.blogspot.com/-hJPFMAfcRW0/VmOrMxLORhI/AAAAAAAACEY/oHzRP3ioNsU/s640/Nu-vs-Re-2.png" width="640" /></div><br /><a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/Nu-vs-Re-s.pdf">Here is a link to a pdf of plots splitting the measurements in 2 through 8 pieces</a>. Each trial's dots are distributed vertically, reflecting the noisy temperature readings.<br /><br />The four lowest trials cluster near the asymptote for turbulent convection from a smooth surface, even though the rough surface asymptote would convect more heat. In the "Measured over time split into 4 intervals" chart it is seen that the plate spends most of its time over the range between the two asymptotes.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-yiyjhDAZKr4/VmOtSrLhM9I/AAAAAAAACEk/vTju0G5wiMw/s1600/Nu-vs-Re-4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://4.bp.blogspot.com/-yiyjhDAZKr4/VmOtSrLhM9I/AAAAAAAACEk/vTju0G5wiMw/s640/Nu-vs-Re-4.png" width="640" /></a></div><br />If the switching between asymptotes were correlated with fan speed variations, we would expect to see the dots for each trial on a slanted line; but they are roughly vertical. The transition between laminar and turbulent forced convective modes remains an unsolved problem for theory. Modelling the transition between these two turbulent forced convective modes may be as intractable.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-9926994986777439662015-11-25T22:31:00.001-05:002015-11-26T13:48:26.211-05:00Simplified Convection ModelWith the asymptote established, it was time to work on the lower speeds. My models were stubbornly predicting too much convection for 0109rpm-20151031T023629. It turns out that the apparent deficit for rough upward natural convection was nearly equal to the apparent deficit for rough downward convection. Thus it was likely that I was underestimating the radiative cooling and natural convection from the four sides, which act in both orientations. Increasing the modeled side surface conductances yields values for the rough surface matching smooth natural convection within the experimental variations.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-noCWHyskDMQ/VlZ3x1eueMI/AAAAAAAACD0/RkcFWJ5PROo/s1600/Nu-vs-Re.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://2.bp.blogspot.com/-noCWHyskDMQ/VlZ3x1eueMI/AAAAAAAACD0/RkcFWJ5PROo/s640/Nu-vs-Re.png" width="640" /></a></div><br />I managed to coax the wind-tunnel fan to run at 93 r/min for a data run, extending test coverage down to Re=8000. Where my earlier plot showed points clustering around the L4-norm of the Scaled-Colburn-Analogy-Asymptotes, the new point aligns with the others along the asymptotes themselves, as predicted by my SCAA formula:<br /><br /><table> <tbody><tr><td nowrap=""> Nu<sub>SCAA</sub> = max(Nu<sub>8.9</sub><b>,</b> Nu<sub>8.11</sub><b>,</b> Nu<sub>RS</sub>)</td> </tr></tbody></table><br />The small convection from 0093rpm-20151125T170700 and 0109rpm-20151031T023629 also confirm that mixed convection uses a high degree norm (the highest being the "max" function). This essentially says that natural and forced convection do not mix; it is either one or the other. When the Convection Machine gets fitted with a smaller fan, I will be able to test at the transition point.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-27775167894919700962015-11-21T09:30:00.001-05:002015-11-26T13:45:13.305-05:00Forced Convection From a Rough Plate<div class="separator" style="clear: both; text-align: center;">In October I collected data from experiments over the full range of wind speeds possible in my wind tunnel: 0.45m/s to 4.5m/s. More time was consumed in making a simulation which matched the dozen datasets. In this speed range all convection is a blend of natural and forced convection. I built the simulation to support variable L-norms for combining the natural and forced convection components. The L-norm that allowed match to the data was L4, which was also the only working L-norm for combining the three speed ranges of forced convection. The L4-norm looks like: (N<sup>4</sup>+F<sup>4</sup>)<sup>1/4</sup>. The main goal was to test my correlation for forced convection from a rough plate:</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-YX9L5fm4Sb0/Vk_HpNrWt4I/AAAAAAAAB74/lr95-w0BO-U/s1600/formula.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="http://2.bp.blogspot.com/-YX9L5fm4Sb0/Vk_HpNrWt4I/AAAAAAAAB74/lr95-w0BO-U/s400/formula.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;">This graph shows the experimental support for my "Scaled Colburn Analogy Asymptote" formula for forced convection from a rough plate. Here is a link to a pdf of <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/forced.pdf">graphs comparing simulation with experiment</a>.</div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-s14q1_nHBlI/VlHYTy637MI/AAAAAAAACAM/HTH2YwB5v3I/s1600/Nu-vs-Re.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://2.bp.blogspot.com/-s14q1_nHBlI/VlHYTy637MI/AAAAAAAACAM/HTH2YwB5v3I/s640/Nu-vs-Re.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;">The largest uncertainty in the measurements is the wind-speed (proportional to Re), which the anemometer gives as 3% at 4m/s. As the speed reduces, the uncertainty grows quickly. The "fan law" says that wind-speed is proportional to fan rotation rate. This turned out to not hold at speeds above 3m/s, which had a lesser slope than at low speeds. At high speeds, the anemometer readings inside the wind-tunnel are erratic. This link to the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/fancal.pdf">fan wind-speed curve</a> shows the formula used in simulations versus 5 measured wind-speed data runs.</div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><img border="0" height="499" src="http://1.bp.blogspot.com/-odx5PO8lb88/VlB3mnQp4pI/AAAAAAAAB8o/aNcnYKiANls/s640/20150801T183831-fancal.png" width="640" /></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Some of my assumptions were wrong. Natural convection from a rough surface turns out to be greater than convection from a smooth surface both upward and downward facing. In order to obtain the matches (red versus blue, green) shown below, the downward convection is multiplied by 2.44; the upward convection by 1.56. The rough plate surface has an area 2.44 times that of a smooth surface and 1.56 is its square-root. There is latitude in these numbers and convection from the four sides; experiments with the rough surface covered by a sheet of aluminum would refine the model.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Jjd8JWJI8gw/VlHYotv_s-I/AAAAAAAACAU/F3K0naDWK7M/s1600/0000rpm-20151024T231700-forced.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-Jjd8JWJI8gw/VlHYotv_s-I/AAAAAAAACAU/F3K0naDWK7M/s640/0000rpm-20151024T231700-forced.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;"><br /></div><div style="text-align: left;"> The blue trace is the measured temperature from the smooth back. The simulated red trace is a good match when the back is on top. The poor match below when the smooth back is facing down is probably because the plate was not suspended, but sitting on small wooden blocks without much clearance. I will perform the measurement again with more clearance in the future.</div><div style="text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-HOfSj_rEgwA/VlHYynb92aI/AAAAAAAACAc/fFRh6B4vDo4/s1600/20150711T112609-upward.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-HOfSj_rEgwA/VlHYynb92aI/AAAAAAAACAc/fFRh6B4vDo4/s640/20150711T112609-upward.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><br /></div>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-64958350717996652832015-10-05T20:27:00.001-04:002015-10-05T20:36:19.611-04:00Forced Convection Success!When I replaced the plate insulation, I covered the back surface (but not the sides) with aluminum foil. An isothermal surface, I put a temperature sensor at the center of the back foil and added modeling for this back surface to the simulation, which is described in <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements">Measurements of Convection From a Rectangular Plate</a><br /><br /><a href="http://voluntocracy.blogspot.com/2015/08/fan-calibration.html"><span id="goog_988989384"></span>After calibrating the fan speed in the wind-tunnel<span id="goog_988989385"></span></a>, the next task was to model the forced convective component of the heat transfer from the parts of the heated plate other than the surface under test: the back and sides. The isothermal back can be modeled using the standard formula in series with the block of insulation. Unlike natural convection, forced convection can be modeled for non-isothermal surfaces by integrating the local convection in the direction of the fluid flow. There are three calculations for the four sides: the sides whose long dimensions are parallel to the flow, the side facing into the flow, and the side facing away from the flow.<br /><br />For the sides parallel to the flow, the local convective surface conductance in series with the local insulation conductance is integrated in the direction of the flow.<br /><br />For the side facing the flow, the fluid velocity along the long center-line line is zero, increasing to V at the long edges. So I integrate from the center-line to the long edges.<br /><br />For the side facing away from the flow, the average fluid velocity along the long center-line is also zero. I integrate from the long edges to the center-line, but only the turbulent component (the chart below shows the difference from windward is insignificant).<br /><br /><pre>ins_back long_side windward leeward total rough<br />65.6mW/K + 2*77.2mW/K + 77.2mW/K + 77.2mW/K = 0.374W/K vs 0.572W/K @ 0.0m/s<br />73.9mW/K + 2*98.3mW/K + 0.108W/K + 0.101W/K = 0.479W/K vs 1.20W/K @ 1.0m/s<br />74.9mW/K + 2*0.111W/K + 0.120W/K + 0.117W/K = 0.535W/K vs 2.17W/K @ 2.0m/s<br />75.5mW/K + 2*0.123W/K + 0.132W/K + 0.131W/K = 0.584W/K vs 3.25W/K @ 3.0m/s<br />76.0mW/K + 2*0.133W/K + 0.145W/K + 0.144W/K = 0.631W/K vs 4.33W/K @ 4.0m/s</pre><pre> </pre>4 m/s simulations with the new model were predicting way too much convection when compared with measurements. After checking all the calculations, it was clear that something basic was wrong. The formula which the apparatus was built to test is the <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/#Forced%20Convection%20from%20a%20Rough%20Plate">Scaled Colburn Analogy</a>. Originally (it has been updated) it was scaling only the characteristic length in the Colburn Analogy. Scaling both the characteristic length and mean-height-of-roughness yields this:<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-rpjv6mMxwiY/VhMTw101-NI/AAAAAAAAB6M/fqQwmdwaQAw/s1600/20150731T004012-mixed.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-rpjv6mMxwiY/VhMTw101-NI/AAAAAAAAB6M/fqQwmdwaQAw/s1600/20150731T004012-mixed.png" /></a></div><br />The green trace is the measured plate temperature; black is the ambient temperature; and blue is the temperature of the back foil. The thin red traces are the simulated plate and back temperatures. Because 4 m/s flow has high convective surface conductance, the back and ambient temperatures are nearly the same. The ambient bumps occured at times when a dehumidifier in the room turned on.<br /><br />To have such close match before tweaking is exciting! A run at 3 m/s also shows excellent match:<br /><br /><a href="http://1.bp.blogspot.com/-SrQKDhdgFyA/VhMTwyz5JmI/AAAAAAAAB6I/hJZ8RGq48Qc/s1600/20150804T032743-mixed.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-SrQKDhdgFyA/VhMTwyz5JmI/AAAAAAAAB6I/hJZ8RGq48Qc/s1600/20150804T032743-mixed.png" /></a><br /><br />Does the model work for natural convection?<br /><br /><a href="http://4.bp.blogspot.com/-ShMvY0c2kY4/VhMTw6XKE_I/AAAAAAAAB6U/U2WOlZMbQ3c/s1600/20150806T222131-mixed.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-ShMvY0c2kY4/VhMTw6XKE_I/AAAAAAAAB6U/U2WOlZMbQ3c/s1600/20150806T222131-mixed.png" /></a><br /><br />This match was unexpected because I didn't have a natural convection model for the four sides. But this match leads to an explanation: the air heated by the downward-facing plate rises past the four sides, so there is little temperature difference through the sides to drive additional convection.<br /><br />This natural convection match is perhaps too good. The measurements were taken in the wind tunnel whose top panel will somewhat impede the ascent of the heated air. The missing difference could be because the simulation has the thermal radiation from the sides cooling the plate, but most of that radiation would be due to heat rising from the (bottom) test surface.Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-30792835383213377992015-09-06T21:39:00.004-04:002015-09-07T10:19:55.565-04:00Sandstone Formation<div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: left;"><a href="http://1.bp.blogspot.com/-zubcmam0mWI/Veznybm9zFI/AAAAAAAAB4s/EqSj0ybFB0k/s1600/rafting.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-zubcmam0mWI/Veznybm9zFI/AAAAAAAAB4s/EqSj0ybFB0k/s320/rafting.jpg" width="320" /></a>Roberta and I recently returned from a fantastic <a href="http://www.gcex.com/motorized.html">Grand Canyon Expeditions 8-Day Motorized Trip</a> down the Colorado River through the Grand Canyon. The Geology is fascinating and quite unlike the glacier-scoured granite mountains in New Hampshire with which I am familiar.</div><br /><br /><div class="separator" style="clear: both; text-align: center;"> <a href="http://1.bp.blogspot.com/-ffxDtKXpq40/VezhXHXbG8I/AAAAAAAAB3U/rX55tweSybc/s1600/sandstone1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-ffxDtKXpq40/VezhXHXbG8I/AAAAAAAAB3U/rX55tweSybc/s320/sandstone1.jpg" width="320" /> </a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: left;">The "Big Dune" sand bar at mile 119 where we camped is backed by Tapeats Sandstone carved into flowing forms.</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-B-8vC3XPrXI/Vezhi1Lr6bI/AAAAAAAAB3c/_7LbPyj7r8c/s1600/sandstone2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-B-8vC3XPrXI/Vezhi1Lr6bI/AAAAAAAAB3c/_7LbPyj7r8c/s320/sandstone2.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-t_jIFkfuko4/Vezi-JPfSOI/AAAAAAAAB3o/IlFi2uZnYO4/s1600/BigDune20131001.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div>Sandstone is obviously deposited in layers. Is this process observable at human time scales; or would one need to spend years at the bottom of a shallow sea?<br /><br />The sand banks on which we camped were deposited by varying water flows in the Colorado river. The Glen Canyon Dam having reduced the Colorado's flow and variations, in 2013 the flow was increased fourfold for 6 days in an effort to replenish the sandbanks which have been diminishing.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://137.227.239.42/Collections/hfe2013/RC1194R_20131001_1359.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://2.bp.blogspot.com/-t_jIFkfuko4/Vezi-JPfSOI/AAAAAAAAB3o/IlFi2uZnYO4/s320/BigDune20131001.jpg" width="320" /> </a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://137.227.239.42/Collections/hfe2013/RC1194R_20131118_1316.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="http://1.bp.blogspot.com/-NiHEPUiMUA4/Vezi-F3jB8I/AAAAAAAAB3s/cwVb4EZviJA/s320/BigDune20131118.jpg" width="320" /></a></div><br />These photos from the <a href="http://www.gcmrc.gov/gis/sandbartour2013/index.html" target="_blank">USGS</a> show the Big Dune sandbar before and after the flood. Sand was deposited. While walking along a small vertical face in the sand bar, I noticed horizontal striations. So the sandbar is deposited in layers.<br /><br />This photo and the detail below show the striations and their resemblance to the sandstone layers.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-gWWN6oC6ge4/VezkQqpkE7I/AAAAAAAAB4E/rBIXbIJXsDs/s1600/sand1-stone.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-gWWN6oC6ge4/VezkQqpkE7I/AAAAAAAAB4E/rBIXbIJXsDs/s320/sand1-stone.jpg" width="240" /></a></div><div class="separator" style="clear: both; text-align: center;"> <a href="http://4.bp.blogspot.com/-pXC6y7mggto/VezkQKCDnyI/AAAAAAAAB4A/Yaj9ngWFNmQ/s1600/sand1-det.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-pXC6y7mggto/VezkQKCDnyI/AAAAAAAAB4A/Yaj9ngWFNmQ/s320/sand1-det.jpg" width="240" /></a></div><br />This would have merely been an amusing coincidence had not our next camp site, "OC's" at mile 137, had an even finer example of sand layering. Not only does the sandstone face show both highly parallel layers and more fluid flows, the sand displays them as well.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-JxVQKD1wxiU/VezlO155-tI/AAAAAAAAB4U/QVsaA5Oz2Yw/s1600/sand2-stone.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://3.bp.blogspot.com/-JxVQKD1wxiU/VezlO155-tI/AAAAAAAAB4U/QVsaA5Oz2Yw/s320/sand2-stone.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-fSteb7hT5l4/VezlOtin97I/AAAAAAAAB4Y/GWk3yNKvupU/s1600/sand2-det.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-fSteb7hT5l4/VezlOtin97I/AAAAAAAAB4Y/GWk3yNKvupU/s320/sand2-det.jpg" width="320" /></a></div> Names and mileages from:<br /><br />"Belknap's Waterproof Grand Canyon River Guide"; New Edition 2014;<br />Buzz Belnap - Loie Belknap Evans;<br />Westwater Books, Evergreen Colorado US;<br />Library of Congress Control Number: 2006937059;<br />ISBN 978-0-916370-16-9;<br />ISBN 10:0-916370-16-XAubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-87368242712116377262015-08-04T00:21:00.002-04:002015-08-04T22:29:23.432-04:00Fan CalibrationReflecting the infrared beam off of the fan blades to gauge its rotation rate was receiving spurious ticks. I moved the photo-transistor opposite the LED so that the blades interrupt the beam, which solved the problem.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-21tSUvworAo/Vb_7LAbBVGI/AAAAAAAAB00/5UGRN9PNFsU/s1600/fancalib.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-21tSUvworAo/Vb_7LAbBVGI/AAAAAAAAB00/5UGRN9PNFsU/s1600/fancalib.jpg" width="320" /></a></div><br />I wrote a program which logs the fan rotation rate every second while capturing the anemometer readings I type in. Because there is delay between rotational speed changes and air speed in the chamber, my test runs start at full speed, ramp down to 300 r/min, then ramp back up to full speed. The delay will affect the two halves in opposite directions. When the delay is properly modeled, the two lines will converge.<br /><br clear="all" /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-audM0rCobQQ/VcFzwUtfDUI/AAAAAAAAB2U/8gyObRZRiqY/s1600/20150802T193625-fancal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><br />From the graphs it was determined that the delay ranged from less than 1 second at full speed to 5 seconds around 300 r/min. The graph to the right shows airspeed versus rotation rate (compensating for this delay).<br /><br /><a href="http://1.bp.blogspot.com/-F-dtejD3Dfw/VcFzvxV7-VI/AAAAAAAAB2M/O8RmOyqeTIg/s1600/20150731T214836-fancal.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="250" src="http://1.bp.blogspot.com/-F-dtejD3Dfw/VcFzvxV7-VI/AAAAAAAAB2M/O8RmOyqeTIg/s320/20150731T214836-fancal.png" width="320" /></a><br />The graph below is a time-series of the same run (with the delay compensated). Around 2 m/s the airspeed vs rotation rate curve is not monotonic. The time-series shows this is true both with decreasing and increasing rotation-rate. And this feature appears in all of the measurement sequences I captured.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-dQZSUEn3kBI/VcFzv6jWNNI/AAAAAAAAB2E/KoY0uaPGvS0/s1600/20150731T214836-fancal-time.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="250" src="http://2.bp.blogspot.com/-dQZSUEn3kBI/VcFzv6jWNNI/AAAAAAAAB2E/KoY0uaPGvS0/s320/20150731T214836-fancal-time.png" width="320" /></a></div><br />At rates above 1200 r/min it was difficult to keep up with the larger airspeed variations. But in all the datasets I captured, the slope above 1200 r/min is less steep than below 1200 r/min.<br /><br />A quick search of the Internet finds engineering "fan laws" saying that air speed is proportional to rotational speed. The black line on the graph is the sort predicted by fan laws. It has a slope of 0.191 m/r.<br /><br />The critical question is whether these non-linearities are properties of the fan-wind-tunnel system or artifacts of the anemometer.<br /><br clear="all" /><a href="http://2.bp.blogspot.com/-SdU5_N9P9PI/VcFzv6P3bjI/AAAAAAAAB2I/vOECcoiMolg/s1600/20150727T014758-fancal.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="250" src="http://2.bp.blogspot.com/-SdU5_N9P9PI/VcFzv6P3bjI/AAAAAAAAB2I/vOECcoiMolg/s320/20150727T014758-fancal.png" width="320" /></a>We removed the plate from the chamber and ran the tests again. Because the plate is not reducing the cross-section of the chamber, the airspeed and slope of the black line is less, 0.172 m/r. The negative slope around 2.1 m/s persists. This does not appear to be hysteresis or flow state because the negative slope happens with both increasing and decreasing speed. While the rotational speed around which this occurs increases by 100 r/min, the airspeed range is unchanged. So this test doesn't provide a razor for identifying the source of the non-linearity.<br /><br clear="all" /><a href="http://1.bp.blogspot.com/-i7mkIq1DJCg/Vb_7O9xscWI/AAAAAAAAB08/zNumpg_wqIA/s1600/diffuser1.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-i7mkIq1DJCg/Vb_7O9xscWI/AAAAAAAAB08/zNumpg_wqIA/s320/diffuser1.jpg" width="320" /></a>While standing 3 meters in front of the fan running at full speed I felt erratic puffs of air. Could these puffs be the same phenomena as the 10% airspeed variations found inside the chamber at high speeds? I repeatedly folded a length of plastic deer fence to fashion a diffuser and strapped it to the front of the fan. Remember that the fan sucks air through the chamber; the diffuser is on the exhaust side of the fan.<br /><br clear="all" /><a href="http://2.bp.blogspot.com/-audM0rCobQQ/VcFzwUtfDUI/AAAAAAAAB2U/8gyObRZRiqY/s1600/20150802T193625-fancal.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="250" src="http://2.bp.blogspot.com/-audM0rCobQQ/VcFzwUtfDUI/AAAAAAAAB2U/8gyObRZRiqY/s320/20150802T193625-fancal.png" width="320" /></a>This resulted in a dramatic decrease in the erratic puffing and the airspeed variation inside the chamber. The high-speed slope variation is reduced, indicating that at least part of this slope is a property of the wind-tunnel. The 2 m/s non-linearity kink is unaffected.<br /><br />I set the fan speed so that under the plate<br />the anemometer read just above 2.1m/s. I slowly pulled the anemometer forward so that it was no longer under the plate. The airspeed should have then dropped, and it does so at other speeds, but not at 2.1 m/s. So the anemometer is responsible for the kink!<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-vX8JgKbGH6E/VcFzwFmxQ9I/AAAAAAAAB2Q/r6v0gabkZ8Y/s1600/20150802T193625-fancal-time.png" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="250" src="http://4.bp.blogspot.com/-vX8JgKbGH6E/VcFzwFmxQ9I/AAAAAAAAB2Q/r6v0gabkZ8Y/s320/20150802T193625-fancal-time.png" width="320" /></a></div><br />Anemometers perform poorly at low speeds and are calibrated at the upper end of their ranges. If the "fan laws" hold, I can rely on the more accurate high speed readings and interpolate the rest. With the diffuser the upper curves are close enough to the straight line to use the straight line in calculations.<br /><br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-28266559069384116242015-07-18T23:15:00.000-04:002015-07-18T23:15:29.002-04:00Ready for Forced convection Measurements<div class="separator" style="clear: both; text-align: center;"></div><a href="http://4.bp.blogspot.com/-oFPh4THtn8M/VaqqTDY81gI/AAAAAAAABz0/K6QTXnLd1_Q/s1600/suspend1.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-oFPh4THtn8M/VaqqTDY81gI/AAAAAAAABz0/K6QTXnLd1_Q/s320/suspend1.jpg" style="cursor: move;" width="320" /></a> Martin Jaffer helped suspend the plate in the wind tunnel. The plate is suspended from four 0.38 mm-diameter lengths of steel piano wire terminated at eight zither tuning pins in wooden blocks above the test chamber. The wire is sheathed by close-fitting Teflon tubing where it would contact the plate. The plate is not centered, but shifted halfway towards the front of test chamber. This gives more clearance from the turbulent transition regions spreading from the tunnel walls. <br /><br /> There is a strip of duct-tape covering the metal edge between the insulation and the machined surface. Although it is more difficult to model (what is the thermal conductivity of duct-tape?), measurement shows that it reduces the heat flow from the non-test surfaces. <br /><br clear="all" /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-y8EMxrTh8to/VaqqTKO-XPI/AAAAAAAABz4/_CAVaI87MBw/s1600/suspend2.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-y8EMxrTh8to/VaqqTKO-XPI/AAAAAAAABz4/_CAVaI87MBw/s320/suspend2.jpg" width="320" /></a></div>This photo shows the electronics board, cables, and fan speed control. The ambient sensor board (temperature, pressure, humidity) can be seen hanging off the left side of the tunnel. <br /><br clear="all" /><a href="http://1.bp.blogspot.com/-ga-Vzwo4J_o/VarlABhoENI/AAAAAAAAB0I/dg4QZh8yODk/s1600/fanspeed.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="224" src="http://1.bp.blogspot.com/-ga-Vzwo4J_o/VarlABhoENI/AAAAAAAAB0I/dg4QZh8yODk/s320/fanspeed.png" width="320" /></a> <br /> I finished the interrupt-driven code to measure the rotation rate of the fan. Each second it divides the number of times a blade has passed the sensor by the length of time between the zeroth and last blade passing. This reduces quantization error such that most readings are stable to within a couple of r/min. Occasionally there is an extra interrupt causing one reading to spike. It may be possible to correct the spike in software. <br /> Measurements with the impeller anemometer are used correlate the fan rotation rate with the wind speed inside the tunnel. But the two significant digits of the anemometer are limiting. We tried to interpolate the dithering of the low order digit in collecting the data shown in this graph. Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-86612328695838287752015-07-14T22:42:00.000-04:002015-07-14T22:42:38.413-04:00Some Mysteries Solved<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-o3tszXpnLmI/VaXHCHG8HLI/AAAAAAAABzQ/cdriz1ArOYk/s1600/opto.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div><a href="http://4.bp.blogspot.com/-o3tszXpnLmI/VaXHCHG8HLI/AAAAAAAABzQ/cdriz1ArOYk/s1600/opto.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements">http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements</a> has been updated with photos of the new insulation, derivation of the model for the back surface, and measurements of natural convection.<br /><br />In the first version of the electronics the plate and ambient temperature sensors had 3 times the variance of the back surface sensor. I also noticed that, in most of the runs, the ambient temperature would have an initial ramp up in the first 5 minutes.<br /><br />The LM35CZ temperature sensor datasheet warns of self-heating for sensors not attached to heatsinks. So I added circuitry and modified the program to power on the sensor 2ms before it is read and turn it off after. Surprisingly, it didn't eliminate the ramp, but it reduced the variance to a level comparable with the back surface sensor!<br /><a href="http://3.bp.blogspot.com/-Qg5j3RbdLew/VaXGsXW-WGI/AAAAAAAABzI/9BLsCQ0NXf8/s1600/sensors.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://3.bp.blogspot.com/-Qg5j3RbdLew/VaXGsXW-WGI/AAAAAAAABzI/9BLsCQ0NXf8/s320/sensors.jpg" width="320" /></a><br />I checked the other components and found that the MPXH6115A6U atmospheric pressure sensor dissipates 30mW (and has a long warm-up time in its specification). The LM35CZ was mounted next to the MPXH6115A6U on the same header; heat conduction through their leads was responsible for the warmup. I moved the LM35CZ off of the header and connected it with thin wires, which solved the problem.<br /><br />I am still left with the mystery of why the LM35CZ in the plate has the large variance. If the thermal adhesive which fastens it in a hole in the aluminum plate is broken, is self-heating creating the variance? Disassembling the insulation to examine it may damage the insulation, requiring it to be replaced; so I will leave it as it is for now.<br /><br />With the upward natural convection measurement looking reasonably consistent with the model, it is time to mount the plate in the wind tunnel.<br /><a href="http://4.bp.blogspot.com/-o3tszXpnLmI/VaXHCHG8HLI/AAAAAAAABzQ/cdriz1ArOYk/s1600/opto.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-o3tszXpnLmI/VaXHCHG8HLI/AAAAAAAABzQ/cdriz1ArOYk/s320/opto.jpg" width="320" /></a><br />When I rebuilt the electronics, I improved the sensitivity of the opto-interrupter. Its sensitivity is now high enough that it responds to reflection of the fan blade. This allows the led and phtotransistor to be on the same side of the fan, which simplifies mounting. The photo shows the blue board with the LED adjacent to the board with the phototransistor.<br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-12247398237830877992015-06-21T09:48:00.000-04:002015-06-21T09:48:56.565-04:00Setbacks<br />Last month saw some setbacks.<br /><br />While measuring a voltage, my probe slipped and shorted the 20 volt supply to a signal pin, burning out the STM32F3Discovery board and many of the electronic components. The damaged components are now replaced; the electronics are working as before.<br /><br />The analog-to-digital converters on the STM32F303 have multiple inputs. While checking voltages on the repaired board, I noticed that for each ADC, the first input to be converted was converted accurately, but the others were not. Increasing the converter sampling-time from ADC_SAMPLETIME_1CYCLES_5 to ADC_SAMPLETIME_19CYCLES_5 results in all readings being reasonably accurate. The noise on converted signals is also reduced.<br /><br />The other setback is more serious.<br /><br />Heat flow measurements are of heat flow from both sides of the plate. To infer the undesired heat flow from measurements of total heat flow requires that the heat flow from the front of the plate be precisely known. But the point of this experiment is to measure the heat flow from the front.<br /><br />An insulating cover for the front of the plate turned out to interact strongly with the convection from the short sides. The only option remaining is to analyze and estimate the heat flow through the back of the plate.<br /><br />I am making progress in calculating this, but the estimates are significantly smaller than measured by the experimental apparatus. I am rebuilding the insulation to match the estimate model.<br />Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-82415654470933073132015-05-07T20:40:00.000-04:002015-05-07T20:40:10.268-04:00First MeasurementsIn <a href="http://voluntocracy.blogspot.com/2015/02/insulating-both-sides.html"> "Insulating Both Sides"</a>, the large squares of (polyisocyanurate) foam insulation are faced with aluminum foil, but the borders were extruded polystyrene foam. Their unfaced surfaces would be subject to radiative transfer, which depends on the temperature of surfaces in the room, and is difficult to isolate from convection. I realized that by facing all the insulation with aluminum foil (with emissivity around .08), the radiative transfer would be cut to negligible levels. But there were unintended consequences which are discussed below.<br /><br />This story will evolve at <a href="http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements#Integration">http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements#Integration"</a>. <br /><br />The physical quantities to be measured are: <br /><table> <tbody><tr><th>symbol</th><td rowspan="99" width="5%"></td><th>units</th><td rowspan="99" width="5%"></td><th>description</th></tr><tr><td><i>T<sub>F</sub></i></td><td>K</td><td>Fluid (Air) Temperature</td></tr><tr><td><i>T<sub>S</sub></i></td><td>K</td><td>Plate Surface Temperature</td></tr><tr><td><i>P</i></td><td>Pa</td><td>Fluid (Atmospheric) Pressure</td></tr><tr><td><i>V</i></td><td>m/s</td><td>Fluid Velocity</td></tr><tr><td><i>Φ</i></td><td>Pa/Pa</td><td>Relative Humidity</td></tr><tr><td><i>Π<sub>H</sub></i></td><td>W=J/s</td><td>Heater Power</td></tr></tbody></table><br />Previous laboratory measurements of forced convection have been performed by starting the fluid flow and plate heater, waiting until the system reaches equilibrium (as indicated by a stable plate temperature), then recording the measurements of the physical quantities. <br /><br />Because the heat flow for forced convection over a rough surface is expected to be larger than for a smooth surface, our test plate is more massive than previous experiments in order to maintain uniform temperature across the plate. This results in settling times of minutes in the best case and times approaching and hour at low airflow rates. <br /><br />Radiative heat transfer is not directly measurable separately from convective heat transfer. The approach here is to minimize radiative transfer by making the test plate and its insulated backside have low emissivity (roughly 8%). The <i>h<sub>R</sub></i> of the plate is estimated assuming that the emissive surfaces (mostly the inside of the wind-tunnel) are at ambient temperature. The insulated backside will be at a lower temperature than the plate. With a 5K temperature difference between the plate and ambient, the backside is about 1K above ambient in still air; in moving air this difference will be less. The small temperature difference the backside and ambient, combined with its low emissivity mean that its radiative transfer can be lumped with the backside convection <i>U<sub>B</sub></i>(<i>V</i>). <br /><br />The equation of state for the plate in forced convection is: <br /><table> <tbody><tr> <td nowrap=""><i>Π<sub>H</sub></i> = <i>h</i>(<i>V</i>) ⋅ <i>A<sub>S</sub></i> ⋅ (<i>T<sub>S</sub></i> − <i>T<sub>F</sub></i>) + ε<sub><i>S</i></sub> ⋅ ε<sub>T</sub> ⋅ <i>h<sub>R</sub></i> ⋅ <i>A<sub>s</sub></i> ⋅ (<i>T<sub>S</sub></i> − <i>T<sub>F</sub></i>) + <i>U<sub>B</sub></i>(<i>V</i>) ⋅ (<i>T<sub>S</sub></i> − <i>T<sub>F</sub></i>) + <i>C</i> ⋅ </td> <td align="center" nowrap=""><i>d T<sub>S</sub></i> <br /><hr noshade="" /><i>d t</i> </td> </tr></tbody></table><table> <tbody><tr><th>symbol</th><td rowspan="99" width="5%"></td><th>value</th><th>units</th><td rowspan="99" width="5%"></td><th>description</th></tr><tr><td><i>C</i></td><td>4690</td><td>J/K</td><td>Plate Thermal Capacity</td></tr><tr><td><i>A<sub>S</sub></i></td><td>0.090</td><td>m<sup>2</sup></td><td>Convecting Area</td></tr><!-- <tr><td><i>T<sub>T</sub></i></td><td><i>T<sub>F</sub></i></td><td>K</td><td>Tunnel Surface Temperature</td></tr>--><tr><td>ε<sub>S</sub></td><td>0.08</td><td></td><td>Plate Surface Emissivity</td></tr><tr><td>ε<sub>T</sub></td><td>0.95</td><td></td><td>Tunnel Surface Emissivity</td></tr><tr><td><i>h<sub>R</sub></i></td><td>5.87</td><td>W/(m<sup>2</sup>K)</td><td>Radiative Surface Conductance</td></tr><tr><td><i>h</i></td><td></td><td>W/(m<sup>2</sup>K)</td><td>Convective Surface Conductance</td></tr><tr><td><i>U<sub>B</sub></i>(<i>V</i>)</td><td></td><td>W/K</td><td>Backside Surface Conductance (including radiative)</td></tr></tbody></table><br />By collecting terms not dependent on <i>T<sub>S</sub></i> into <i>U</i>(<i>V</i>), the equation of state is simplified: <br /><table> <tbody><tr> <td nowrap=""><br /><i>U</i>(<i>V</i>) = <i>h</i>(<i>V</i>) ⋅ <i>A<sub>S</sub></i> + ε<sub><i>S</i></sub> ⋅ ε<sub>T</sub> ⋅ <i>h<sub>R</sub></i> ⋅ <i>A<sub>s</sub></i> + <i>U<sub>B</sub></i>(<i>V</i>) </td> </tr></tbody></table><table> <tbody><tr> <td nowrap=""><i>Π<sub>H</sub></i> = <i>U</i>(<i>V</i>) ⋅ (<i>T<sub>S</sub></i> − <i>T<sub>F</sub></i>) + <i>C</i> ⋅ </td> <td align="center" nowrap=""><i>d T<sub>S</sub></i> <br /><hr noshade="" /><i>d t</i> </td> </tr></tbody></table><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) = <i>T<sub>F</sub></i>(<i>t</i>) + </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>) <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> </td> <td nowrap="">− </td> <td align="center" nowrap=""><i>C<sub> </sub></i> <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> </td> <td nowrap="">⋅ </td> <td align="center" nowrap=""><i>d T<sub>S</sub></i>(<i>t</i>) <br /><hr noshade="" /><i>d t</i> </td> </tr></tbody></table><br />This linear differential equation could be solved analytically, but because the independent variables are measured at discrete times, it make more sense to solve the analogous finite difference equation.<br /><br /><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) = <i>T<sub>F</sub></i>(<i>t</i>) + </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>) <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> </td> <td nowrap="">− </td> <td align="center" nowrap=""><i>C<sub> </sub></i> ⋅ (<i>T<sub>S</sub></i>(<i>t</i>) − <i>T<sub>S</sub></i>(<i>t</i>')) <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> ⋅ (<i>t</i>−<i>t</i>') </td> </tr></tbody></table><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) ⋅ </td> <td align="center" nowrap=""><i>C<sub> </sub></i> + <i>U</i>(<i>V</i>)<sup> </sup> ⋅ (<i>t</i>−<i>t</i>') <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> ⋅ (<i>t</i>−<i>t</i>') </td> <td nowrap="">= <i>T<sub>F</sub></i>(<i>t</i>) + </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>) <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> </td> <td nowrap="">+ <i>T<sub>S</sub></i>(<i>t</i>') ⋅ </td> <td align="center" nowrap=""><i>C<sub> </sub></i> <br /><hr noshade="" /><i>U</i>(<i>V</i>)<sup> </sup> ⋅ (<i>t</i>−<i>t</i>') </td> </tr></tbody></table><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) = </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>) ⋅ (<i>t</i>−<i>t</i>') + <i>T<sub>S</sub></i>(<i>t</i>') ⋅ <i>C<sub> </sub></i> + <i>T<sub>F</sub></i>(<i>t</i>) ⋅ <i>U</i>(<i>V</i>) ⋅ (<i>t</i>−<i>t</i>') <br /><hr noshade="" /><i>C</i> + <i>U</i>(<i>V</i>)<sup> </sup> ⋅ (<i>t</i>−<i>t</i>') </td> </tr></tbody></table><br />When data is sampled every second, this simplifies to:<br /><br /><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) = </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>) + <i>T<sub>S</sub></i>(<i>t</i>') ⋅ <i>C<sub> </sub></i> + <i>T<sub>F</sub></i>(<i>t</i>) ⋅ <i>U</i>(<i>V</i>) <br /><hr noshade="" /><i>C</i> + <i>U</i>(<i>V</i>)<sup> </sup> </td> </tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-EgqqmvQWgsg/VUv_fMsQMHI/AAAAAAAABxQ/_YPp26IVDQo/s1600/knee-0.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="223" src="http://4.bp.blogspot.com/-EgqqmvQWgsg/VUv_fMsQMHI/AAAAAAAABxQ/_YPp26IVDQo/s320/knee-0.png" width="320" /></a></div><h2></h2>I had originally planned that my controller program would servo the plate temperature in a narrow range. But when I saw the first datasets from the plate being heated, I realized that having the plate temperature slew through a temperature span enables me to separate the dynamics of heating from convection. On the basis of the heating slope I make slight adjustments to <i>C</i> to compensate for the addition and removal of insulation; I won't clutter this article with that detail. <br /><br />The image to the right shows the calculated <i>T<sub>S</sub></i>(<i>t</i>) (red) versus the measured <i>T<sub>S</sub></i>(<i>t</i>) (blue) and ambient temperature (black). Clearly there is a delay between the application of heat starting at 60 seconds and the plate temperature. <br /><br clear="all" /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-4IxS29tq5xE/VUv_tsBUYyI/AAAAAAAABxY/ci48gvpHLV0/s1600/knee-15.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="223" src="http://4.bp.blogspot.com/-4IxS29tq5xE/VUv_tsBUYyI/AAAAAAAABxY/ci48gvpHLV0/s320/knee-15.png" width="320" /></a></div><h2></h2>Introducing a 15 second delay for <i>Π<sub>H</sub></i> makes for a much better fit.<br /><br /><table> <tbody><tr> <td nowrap=""><i>T<sub>S</sub></i>(<i>t</i>) = </td> <td align="center" nowrap=""><i>Π<sub>H</sub></i>(<i>t</i>−15) + <i>T<sub>S</sub></i>(<i>t</i>') ⋅ <i>C<sub> </sub></i> + <i>T<sub>F</sub></i>(<i>t</i>) ⋅ <i>U</i>(<i>V</i>) <br /><hr noshade="" /><i>C</i> + <i>U</i>(<i>V</i>)<sup> </sup> </td> </tr></tbody></table><br />The full equation of state could be solved for <i>h</i>(<i>V</i>), but division by <i>T<sub>S</sub></i>−<i>T<sub>F</sub></i> makes analysis of the noisy signals complicated. As in the delay case, simulation and visualization yield insights more readily than statistics. <br /><br />My first measurements are of natural convection because <i>V</i>=0 eliminates one of the variables; also because the formulas for natural convection are well established (assuming that it is the same for rough surfaces as for smooth). In the temperature ranges tested here the dependence of <i>h</i>(0) on <i>T<sub>S</sub></i> is too weak to materially effect the finite difference equation. <br /><br clear="all" />Determining <i>U<sub>B</sub></i>(<i>V</i>) is a bit involved. In order to not have insulation projecting out of the four sides of the plate, the back edges of the plate were beveled and this space filled with wedges of insulation. The heat flow through these sides of the plate is larger than the heat flow through the insulation on the backside of the plate; and theory is insufficient to calculate the heat flow through the sides. So it must be measured. But measurement can only be of the heat transfer from all of the plate. Theory does predict the convection from the front of the plate, but the point of this experiment is to measure that. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-hTOwBii-JaI/VUv_9tUdBjI/AAAAAAAABxo/JaJj2hFRutc/s1600/fullinsulation.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="237" src="http://4.bp.blogspot.com/-hTOwBii-JaI/VUv_9tUdBjI/AAAAAAAABxo/JaJj2hFRutc/s320/fullinsulation.jpg" width="320" /></a></div><h2></h2><br />I constructed an insulated cover for the front of the plate. A rough estimate of its conductance is <i>U<sub>T</sub></i>(0)=134 mW/K. <br /><br />By adding a collar of insulation to the sides of the back (seen to the right), the plate is symmetrically encapsulated in insulation. The conductance through the insulation can be estimated by assuming that the conductance through each truncated pyramid (of insulation) is the same as the conductance through a brick of insulation having dimensions which are the means of the truncated pyramid dimensions. <br /><br />For a fully insulated plate in moving air, the outer surface of the insulation should be close to ambient temperature. I estimate it should conduct 190 mW/K through its front and back and 117 mW/K through its 4 sides, for a total of 308 mW/K between the plate and the insulation envelope. In still air (<i>V</i>=0 in these first tests), the flow should be less.<br /><br />If these estimates are good, then the heat flow through the front cover will be 43% of the measured heat flow through the full insulation. The value of <i>U<sub>B</sub></i>(<i>V</i>) could then be found by measuring the heat flow with the collar removed and subtracting 43% of the symmetrically insulated heat flow. <br /><br clear="all" />But the estimates are not close; in order to match the measured <i>U</i>(0) (blue trace) of the fully insulated plate, the red line on the graph is simulated at <i>U</i>(0)=420 mW/K, which is 36% larger than estimated. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-T4-yL3JGF_0/VUwAJGMsqkI/AAAAAAAABxw/cCRSROX0sUM/s1600/fullinsup.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://3.bp.blogspot.com/-T4-yL3JGF_0/VUwAJGMsqkI/AAAAAAAABxw/cCRSROX0sUM/s640/fullinsup.png" width="640" /></a></div><h2></h2>The graph above shows the heating and cooling of the plate symmetrically encased in insulation. The black trace is ambient temperature, green the envelope temperature underneath the plate, blue the measured plate temperature, and red the simulated plate temperature (the red trace overlays the blue).<br /><br />The graph below shows the heating and cooling of the insulated plate facing down (shown in the photograph above). It cools slightly faster than when facing up, perhaps because of the unsealed seams facing upward. With the temperature sensor (visible in the photograph) being on top, its green trace is much closer to ambient because upward convection is more efficient than downward convection. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-2MObrY5Ogck/VUwAhAP-xZI/AAAAAAAABx4/6iXoeR9Ct_k/s1600/fullinsdown.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://3.bp.blogspot.com/-2MObrY5Ogck/VUwAhAP-xZI/AAAAAAAABx4/6iXoeR9Ct_k/s640/fullinsdown.png" width="640" /></a></div><h2></h2>The graph below shows the heating and cooling of the insulated plate facing up with the top cover but without the collar. The simulated <i>U</i>(0)=470 mW/K line (red) shows a bit too much curvature. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-zajUJ7o20V0/VUwApXKPE9I/AAAAAAAAByA/-5K4ZvTSkt4/s1600/frontcover.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-zajUJ7o20V0/VUwApXKPE9I/AAAAAAAAByA/-5K4ZvTSkt4/s640/frontcover.png" width="640" /></a></div><h2></h2>The graph below shows the convection from the plate facing up (without the top cover). The simulation (subtracting 43% of the full-insulation heat flow from the top-cover heat flow) shows significantly less convection than measured. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-YmjruIhkyYs/VUwBFdUGEiI/AAAAAAAAByI/9KTSqpYa1RY/s1600/upward2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://1.bp.blogspot.com/-YmjruIhkyYs/VUwBFdUGEiI/AAAAAAAAByI/9KTSqpYa1RY/s640/upward2.png" width="640" /></a></div><h2></h2>So the estimates of conduction through the insulation were not good enough. However, if I simulate with <i>U<sub>B</sub></i>(0)=0.47 mW/K, the plate with the measured top-cover conductance, the plate upward convection matches beautifully! <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-KQD8BQarcKs/VUwBOcsYcuI/AAAAAAAAByQ/pcugN4pepG4/s1600/upward.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://3.bp.blogspot.com/-KQD8BQarcKs/VUwBOcsYcuI/AAAAAAAAByQ/pcugN4pepG4/s640/upward.png" width="640" /></a></div><h2></h2>One explanation would be that the top cover reduces the convection from the (four) plate sides by the same amount as conduction through the top cover insulation. The top-cover overhanging the plate sides would somewhat obstruct convection. But coincidences are toxic to science experiments.<br /><br />Covering the side insulation with aluminum tape (to reduce emissivity) provided a heat conduction path from the areas where the conduction is thinnest to the rest of the back. So I made a cut through the aluminum tape on the sides, and it reduces the full-insulation <i>U</i>(0) from 0.42 to 0.38 W/K (shown below), about 10%. Recall that adding the collar to the plate with top-cover reduced the conduction from 470 mW/K to 420 mW/K, about 11%. This lends support to the idea that the aluminum tape was spreading plate heat to the foil envelope. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-tiW5WvQ90pk/VUwBcreZdjI/AAAAAAAAByY/9Ub-I_l9NPQ/s1600/fullins-cut.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="446" src="http://2.bp.blogspot.com/-tiW5WvQ90pk/VUwBcreZdjI/AAAAAAAAByY/9Ub-I_l9NPQ/s640/fullins-cut.png" width="640" /></a></div><br clear="all" />The next step is to make a similar cut to the top-cover and additional cuts to the side insulation and rerun these measurements. Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0tag:blogger.com,1999:blog-6806118251241543344.post-37593401597656139642015-04-10T20:25:00.000-04:002015-04-10T20:25:07.506-04:00Temperature Sensor Calibration<P>Having gotten the digital-to-analog and analog-to-digital conversions working while connected to each other, I plugged the STM32F3DISCOVERY board into my apparatus and started to debug the program to do analog-to-digital conversions of the LM35CZ temperature sensors. The initial readings were quite noisy; I traced this back to power supply noise caused by current fluctuations from the STM32F3DISCOVERY LEDs being switched on and off. I changed the program to keep the LEDs off during conversions and improved the power supply conditioning. The STM32F3DISCOVERY board consumes about 90.mA while running. </P><P>I found that the plate heater was slightly heating even though my program had the DAC controlling it set to 0. It turns out that when its output buffer is enabled, the STM32F303VCTx isn't specified to drive that output below 0.2V. Disabling the buffer reduced the output to a few millivolts. </P><P>The next task was to calibrate the temperature sensors. The calculations for measurement of convection are most sensitive to the time-derivative of the plate temperature and the difference between temperatures of the plate and ambient air. I attached the two ambient sensors ("ambient" and "free") to the back of the plate with thermal adhesive so that they would be at the same temperature. The statistics of (triple) 621 samples taken once per second are: </P><TABLE CELLPADDING=5 BORDER=1><TR><TH>nREPS = 1</TH><TH>ambient</TH><TH>plate</TH><TH>free</TH><TH>offset</TH></TR><TR><TH>mean:</TH><TD>1696.75</TD><TD>1667.32</TD><TD>1734.46</TD><TD>-29</TD></TR><TR><TH>variance:</TH><TD>15.58</TD><TD>16.65</TD><TD>9.83</TD><TD></TD></TR><TR><TH>mean:</TH><TD>1.37.V</TD><TD>1.34.V</TD><TD>1.40.V</TD><TD>-24.mV</TD></TR><TR><TH>variance:</TH><TD>13.mV</TD><TD>13.mV</TD><TD>7.9.mV</TD><TD></TD></TR><TR><TH>mean:</TH><TD>20.84C</TD><TD>20.47C</TD><TD>21.30C</TD><TD>-0.36C</TD></TR><TR><TH>variance:</TH><TD>0.19C</TD><TD>0.20C</TD><TD>0.12C</TD><TD></TD></TR></TABLE><P>The plate-ambient offset is about -29 LSB = -24.mV = -0.36C. The 13.mV variance is rather large and matches measurements made by a RMS voltmeter at the ADC inputs; this bodes ill for the time-derivative of the plate temperature. </P><P>Note that voltage is measuread at the ADC input. There is a voltage gain of 6.56 from each LM35CZ (10.mV/C) output to the ADC input. </P><P>The STM32F303VCTx datasheet specifies a total unadjusted error of +/-4.5 LSB for the single-ended ADC at 25C and with a 3.3V supply. </P><P>The 16 LSB variance is much larger than the ADC error; thus it will act as a dither and smooth discontinuities in the ADC transfer function. So I rewrote the program to sum the result of converting each signal 16 times (per second). The statistics of 943 readings averaged over 16 conversions apiece are: </P><TABLE CELLPADDING=5 BORDER=1><TR><TH>nREPS = 1</TH><TH>ambient</TH><TH>plate</TH><TH>free</TH><TH>offset</TH></TR><TR><TH>mean:</TH><TD>1702.79</TD><TD>1669.57</TD><TD>1736.16</TD><TD>-33</TD></TR><TR><TH>variance:</TH><TD>3.04</TD><TD>3.03</TD><TD>1.47</TD><TD></TD></TR><TR><TH>mean:</TH><TD>1.37.V</TD><TD>1.35.V</TD><TD>1.40.V</TD><TD>-27.mV</TD></TR><TR><TH>variance:</TH><TD>2.4.mV</TD><TD>2.4.mV</TD><TD>1.2.mV</TD><TD></TD></TR><TR><TH>mean:</TH><TD>20.91C</TD><TD>20.50C</TD><TD>21.32C</TD><TD>-0.41C</TD></TR><TR><TH>variance:</TH><TD>0.04C</TD><TD>0.04C</TD><TD>0.02C</TD><TD></TD></TR></TABLE><P>The variance is reduced by a factor of 5, which will greatly reduce the effect of noise on the time-derivative of plate temperature. </P><P>This time the plate-ambient offset is about -33 LSB = -27.mV = -0.41C. All three temperatures are higher in the averaged test than in the single conversion test. This and subsequent sample runs in a thermostatically controlled building show that ambient and plate temperatures change enough to hamper calibration. The problem was worse when the plate was heated because the ambient sensor had only a small contact area with the plate, causing a temperature gradient which would be indistinguishable from a gain error. </P><P>So I fitted some insulation over the temperature sensors to protect them from air currents. This reduced the plate-ambient offset by an order of magnitude! </P><TABLE CELLPADDING=5 BORDER=1><TR><TH>nREPS = 1</TH><TH>ambient</TH><TH>plate</TH><TH>free</TH><TH>offset</TH></TR><TR><TH>mean:</TH><TD>2173.75</TD><TD>2170.44</TD><TD>2230.43</TD><TD>-3</TD></TR><TR><TH>variance:</TH><TD>2.85</TD><TD>3.80</TD><TD>2.65</TD><TD></TD><TD></TD></TR><TR><TH>mean:</TH><TD>1.75.V</TD><TD>1.75.V</TD><TD>1.80.V</TD><TD>-2.7.mV</TD></TR><TR><TH>variance:</TH><TD>2.3.mV</TD><TD>3.1.mV</TD><TD>2.1.mV</TD><TD></TD></TR><TR><TH>mean:</TH><TD>26.69C</TD><TD>26.65C</TD><TD>27.39C</TD><TD>-0.04C</TD></TR><TR><TH>variance:</TH><TD>0.04C</TD><TD>0.05C</TD><TD>0.03C</TD><TD></TD></TR></TABLE><P>A (304s) run near 20C also shows an offset smaller than the variance and guaranteed minimum offsets of the components. </P><TABLE CELLPADDING=5 BORDER=1><TR><TH>nREPS = 1</TH><TH>ambient</TH><TH>plate</TH><TH>free</TH><TH>offset</TH></TR><TR><TH>mean:</TH><TD>1596.04</TD><TD>1598.53</TD><TD>1665.27</TD><TD>2</TD></TR><TR><TH>variance:</TH><TD>3.01</TD><TD>3.00</TD><TD>1.37</TD><TD></TD></TR><TR><TH>mean:</TH><TD>1.29.V</TD><TD>1.29.V</TD><TD>1.34.V</TD><TD>2.0.mV</TD></TR><TR><TH>variance:</TH><TD>2.4.mV</TD><TD>2.4.mV</TD><TD>1.1.mV</TD><TD></TD></TR><TR><TH>mean:</TH><TD>19.60C</TD><TD>19.63C</TD><TD>20.45C</TD><TD>0.03C</TD></TR><TR><TH>variance:</TH><TD>0.04C</TD><TD>0.04C</TD><TD>0.02C</TD><TD></TD></TR></TABLE><P>So the measurement program will not need to apply offset and scale corrections to the plate-ambient temperature difference. The ambient and plate sensors came from the same lot, the free sensor from a different batch. To convert the free temperature to the equivalent ambient temperature, multiply by 1.023 and subtract 1.318C. </P>Aubrey Jafferhttp://www.blogger.com/profile/14575029475024146146noreply@blogger.com0