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
hR
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
UB(
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
TS
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
TS(
t) (red) versus the
measured
TS(
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
ΠH 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
TS−
TF
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
TS is
too weak to materially effect the finite difference equation.
Determining
UB(
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
UT(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
UB(
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
UB(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.
