Sandstone Formation

Roberta and I recently returned from a fantastic Grand Canyon Expeditions 8-Day Motorized Trip 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.

  

The "Big Dune" sand bar at mile 119 where we camped is backed by Tapeats Sandstone carved into flowing forms.

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?

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.

 

These photos from the USGS 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.

This photo and the detail below show the striations and their resemblance to the sandstone layers.

 

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.

 Names and mileages from:

"Belknap's Waterproof Grand Canyon River Guide"; New Edition 2014;
Buzz Belnap - Loie Belknap Evans;
Westwater Books, Evergreen Colorado US;
Library of Congress Control Number: 2006937059;
ISBN 978-0-916370-16-9;
ISBN 10:0-916370-16-X

Fan Calibration

Reflecting 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.

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.

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).


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.

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.

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.

The critical question is whether these non-linearities are properties of the fan-wind-tunnel system or artifacts of the anemometer.

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.

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.

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.

I set the fan speed so that under the plate
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!

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.

Ready for Forced convection Measurements

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.

 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.

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.


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.
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.

Some Mysteries Solved

http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements has been updated with photos of the new insulation, derivation of the model for the back surface, and measurements of natural convection.

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.

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!

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.

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.

With the upward natural convection measurement looking reasonably consistent with the model, it is time to mount the plate in the wind tunnel.

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.

Setbacks

Last month saw some setbacks.

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.

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.

The other setback is more serious.

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.

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.

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.

First Measurements

In "Insulating Both Sides", 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.

This story will evolve at http://people.csail.mit.edu/jaffer/SimRoof/Convection/Measurements#Integration".

The physical quantities to be measured are:

symbol		units		description
TF		K		Fluid (Air) Temperature
TS		K		Plate Surface Temperature
P		Pa		Fluid (Atmospheric) Pressure
V		m/s		Fluid Velocity
Φ		Pa/Pa		Relative Humidity
ΠH		W=J/s		Heater Power

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.

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.

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 h_R 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 U_B(V).

The equation of state for the plate in forced convection is:

ΠH = h(V) ⋅ AS ⋅ (TS − TF) + εS ⋅ εT ⋅ hR ⋅ As ⋅ (TS − TF) + UB(V) ⋅ (TS − TF) + C ⋅

d TS d t

symbol		value	units		description
C		4690	J/K		Plate Thermal Capacity
AS		0.090	m2		Convecting Area
εS		0.08			Plate Surface Emissivity
εT		0.95			Tunnel Surface Emissivity
hR		5.87	W/(m2K)		Radiative Surface Conductance
h			W/(m2K)		Convective Surface Conductance
UB(V)			W/K		Backside Surface Conductance (including radiative)

By collecting terms not dependent on T_S into U(V), the equation of state is simplified:

U(V) = h(V) ⋅ AS + εS ⋅ εT ⋅ hR ⋅ As + UB(V)

ΠH = U(V) ⋅ (TS − TF) + C ⋅

d TS d t

TS(t) = TF(t) +

ΠH(t) U(V)

−

C  U(V)

⋅

d TS(t) d t

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.

TS(t) = TF(t) +

ΠH(t) U(V)

−

C  ⋅ (TS(t) − TS(t')) U(V)  ⋅ (t−t')

TS(t) ⋅

C  + U(V)  ⋅ (t−t') U(V)  ⋅ (t−t')

= TF(t) +

ΠH(t) U(V)

+ TS(t') ⋅

C  U(V)  ⋅ (t−t')

TS(t) =

ΠH(t) ⋅ (t−t') + TS(t') ⋅ C  + TF(t) ⋅ U(V) ⋅ (t−t') C + U(V)  ⋅ (t−t')

When data is sampled every second, this simplifies to:

TS(t) =

ΠH(t) + TS(t') ⋅ C  + TF(t) ⋅ U(V) C + U(V)

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 C to compensate for the addition and removal of insulation; I won't clutter this article with that detail.

The image to the right shows the calculated T_S(t) (red) versus the measured T_S(t) (blue) and ambient temperature (black). Clearly there is a delay between the application of heat starting at 60 seconds and the plate temperature.

Introducing a 15 second delay for Π_H makes for a much better fit.

TS(t) =

ΠH(t−15) + TS(t') ⋅ C  + TF(t) ⋅ U(V) C + U(V)

The full equation of state could be solved for h(V), but division by T_S−T_F makes analysis of the noisy signals complicated. As in the delay case, simulation and visualization yield insights more readily than statistics.

My first measurements are of natural convection because V=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 h(0) on T_S is too weak to materially effect the finite difference equation.

Determining U_B(V) 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.

I constructed an insulated cover for the front of the plate. A rough estimate of its conductance is U_T(0)=134 mW/K.

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.

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 (V=0 in these first tests), the flow should be less.

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 U_B(V) could then be found by measuring the heat flow with the collar removed and subtracting 43% of the symmetrically insulated heat flow.

But the estimates are not close; in order to match the measured U(0) (blue trace) of the fully insulated plate, the red line on the graph is simulated at U(0)=420 mW/K, which is 36% larger than estimated.

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).

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.

The graph below shows the heating and cooling of the insulated plate facing up with the top cover but without the collar. The simulated U(0)=470 mW/K line (red) shows a bit too much curvature.

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.

So the estimates of conduction through the insulation were not good enough. However, if I simulate with U_B(0)=0.47 mW/K, the plate with the measured top-cover conductance, the plate upward convection matches beautifully!

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.

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 U(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.

The next step is to make a similar cut to the top-cover and additional cuts to the side insulation and rerun these measurements.

Temperature Sensor Calibration

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.

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.

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:

nREPS = 1	ambient	plate	free	offset
mean:	1696.75	1667.32	1734.46	-29
variance:	15.58	16.65	9.83
mean:	1.37.V	1.34.V	1.40.V	-24.mV
variance:	13.mV	13.mV	7.9.mV
mean:	20.84C	20.47C	21.30C	-0.36C
variance:	0.19C	0.20C	0.12C

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.

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.

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.

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:

nREPS = 1	ambient	plate	free	offset
mean:	1702.79	1669.57	1736.16	-33
variance:	3.04	3.03	1.47
mean:	1.37.V	1.35.V	1.40.V	-27.mV
variance:	2.4.mV	2.4.mV	1.2.mV
mean:	20.91C	20.50C	21.32C	-0.41C
variance:	0.04C	0.04C	0.02C

The variance is reduced by a factor of 5, which will greatly reduce the effect of noise on the time-derivative of plate temperature.

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.

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!

nREPS = 1	ambient	plate	free	offset
mean:	2173.75	2170.44	2230.43	-3
variance:	2.85	3.80	2.65
mean:	1.75.V	1.75.V	1.80.V	-2.7.mV
variance:	2.3.mV	3.1.mV	2.1.mV
mean:	26.69C	26.65C	27.39C	-0.04C
variance:	0.04C	0.05C	0.03C

A (304s) run near 20C also shows an offset smaller than the variance and guaranteed minimum offsets of the components.

nREPS = 1	ambient	plate	free	offset
mean:	1596.04	1598.53	1665.27	2
variance:	3.01	3.00	1.37
mean:	1.29.V	1.29.V	1.34.V	2.0.mV
variance:	2.4.mV	2.4.mV	1.1.mV
mean:	19.60C	19.63C	20.45C	0.03C
variance:	0.04C	0.04C	0.02C

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.