Upward Natural Convection

There 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:

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.

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.

The streamlines in Fujii and Imura[76]'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.

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.

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.

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.

Insulating Both Sides

Because of edge effects, the heat leakage through the insulated back-of-the-plate is difficult to calculate.  It would be better to measure it; but that measurement must discount the heat transfer through the front side of the plate.  The .421 W which I estimate for leakage is less than the downward-facing convection (.525 W) for a plate at 5 K higher than ambient temperature.  Because the test surface of the plate is rough, I don't have confidence in convection calculations based on smooth plate measurements; and assuming what I am trying to measure means that I wouldn't be able to test the assertion that surface roughness has no effect on downward natural convection.

So I created insulation for the (rough) front surface of the plate. Constructed using the same materials and techniques as the backside insulation, its leakage should be roughly the same as the backside.

While cutting the extruded polystyrene foam I noticed that the board crowned in the center.  So I arranged the cut strips so that their centers pressed into the plate edges while the duct tape tensions their ends into corners.  It maintains a snug fit and doesn't need tape to seal it.  Because of the bevel, the backside foam is more complicated to cut; but I may at some point rebuild it using the same technique.

Because the expected downward-facing convection is close in magnitude to the leakage, the fully-insulated measurements should be made both level and at some inclinations (convection increases rapidly as the plate is tilted).

Insulating the Plate

The photo shows the thermal insulation for the aluminum plate's backside (covering the heaters).  The reflective square is 25 mm deep polyisocyanurate foam insulation (Dow TUFF-R) with foil on both sides.  The light green border is 19 mm thick extruded polystyrene foam which is beveled to match the back of the aluminum plate.  The corners and metal-to-foam seams are held together with duct tape.  The sensor cable and heater connections thread between the two types of insulation.  The pieces of insulation were machined using a radial-arm saw and a band-saw.  Knife cuts were not as clean.

While suspension by piano wire will be sufficient for performing forced convection measurements of a downward facing horizontal plate, we lack a way to support the plate with insulation at non-horizontal inclinations, which would allow interesting natural convection measurements.