From SGreen@swri.edu Thu Feb 22 09:14:46 1996
To:
Subject: Solar Collector Test Cell
I have had a chance to look at your write-up and the data for your solar
collector test. You have used the term 'structured convection' in this
write-up to describe the heat transfer process in the collector as some new
phenomenon. You have said that it as least different from 'normal turbulent
convection'. In fact, the heat transfer process in place during most of the
time that the data are gathered is simple natural convection that is described
in most engineering heat transfer textbooks.
We could develop a mathematical model using computational fluid dynamics for
your test cell, but I'm not sure that it will tell us anything really new
about the flow field. The data you provided, including the smoke pattern
description and the 'anomaly', can all be described by a rough engineering
analysis. What follows is such a first-order assessment of the data based on
my interpretation of the test setup.
For natural convection from a vertical surface, only the fluid close to the
surface moves under the effect of bouyancy forces caused by the decreased
fluid density with respect to the fluid farther away from the surface. The
fluid in contact with the wall is stationary and the fluid far away from the
wall is approximately stationary. In the region just off the wall, however,
the fluid reaches a maximum velocity but decreases to zero at the wall and far
away from the wall. While the exact fluid velocity can be obtained by a
numerical solution to the appropriate differential equations, an equation
which closely approximates the velocity profile for turbulent flow is given as
u/umax = 1/0.537 * (y/d)^(1/7) * (1-y/d)^4
where u = local velocity parallel to the wall, umax = maximum velocity in the
profile, y = distance from the wall, d = local total boundary layer
thickness. By differentiating this equation, we can see that the maximum
velocity occurs at about 3.5% of the boundary layer thickness away from the
wall. When expressed this way (i.e., dimensionless), the velocity profile is
independent of the position along the wall.
The boundary layer thickness and the maximum velocity are both functions of
the distance along the wall. These have been reported in the literature by a
number of workers, but the solution of Eckert and Jackson (NACA TN 2207, 1950)
is still a good approximation:
umax = 0.636 * mu/x * (Gr)^(1/2) * (1+0.494*Pr^(2/3))^(-1/2)
d = 0.565 * x * Gr^(-1/10) * Pr^(-8/15) * (1+0.494*Pr^(2/3))^(1/10)
where x = distance up the wall from the bottom, mu = fluid dynamic viscosity,
Gr = Grashof Number (a heat transfer parameter), Pr = fluid Prandtl Number
We can uses these relationships to get an idea of the velocity and the
boundary layer thickness in your collector. From the graphs near noon, I
estimate that the wall temperature is 180 F and the air temperature is 100 F
at a position about halfway up the wall, which I assume is about 6 ft total
height, from these temperatures, the following air properties are taken from
a handbook:
[PFH ed. Note: measurements were taken at ~3 ft height]
100 F 180 F mean value
density = 0.071 0.062 0.067 lb/ft^3
viscosity = 1.285*10^-5 1.409*10^-5 1.347*10^-5 lb/ft*sec
B = 1.79*10^-3 1.57*10^-3 1.68*10^-3 1/F (thermal
expansion coefficient)
Pr = 0.72 0.72 0.72
The Grashof Number, a natural convection heat transfer parameter is
Gr = (density)^2*g*B/mu^2 * (Twall-Tair) * x^3
Using the arithmetic mean of the properties, the Grashof Number (a
dimensionless value) is
Gr = 2.89*10^9
The Gr*Pr product is known as the Rayleigh Number, Ra = Gr*Pr = 2.08*10^9 for
this case. If the Rayleigh Number is greater than 1*10^9, the flow is
turbulent; so the use of the above equations is warranted. The boundary layer
thickness and the maximum velocity are now computed for this position along
the wall as d=2.9" and umax=1.9 ft/sec, respectively. The maximum velocity is
found at 0.1" from the wall.
[PFH ed. Note: observed d<1/2", observed umax=~15 ft/sec]
This explains the nature of your smoke patterns. The fluid is relatively
quiescent at distances greater than 3 " from the wall, but the velocity
increases to almost 2 ft/sec near the surface. The layer of fluid moving near
the wall 'pulls' fluid from the bulk so that the boundary layer thickens as it
moves up the heated wall. This value of Ra is only 2 times the value for
laminar-turbulent transition; so the flow is not strongly turbulent. This is
evidenced by the wavy pattern of the smoke as opposed to a more chaotic smoke
pattern.
The heated fluid spills over the top of the absorber and is replaced by cooler
fluid entering at the botteom, thereby setting up a circulating flow in the
container - mostly up in the air space along the absorber, mostly down through
the rock storage. Because of the limited capated capacity of the rock
storage, the top of the rock pile is heated to such a temperature late in the
day that the driving force for flow up the storage is nearly as strong as for
flow up the wall. In that case the, air will circulate only within the space
in front of the absorber which will rapidly heat the volume of air between the
glass and the absorber. This is indicated in the data. When the air is
heated to a high enough temperature, or the rock pile cools sufficiently, the
normal circulation pattern is re-established.
I hope this assessment is useful to you. If necessary, I can expand it and
make it more formal, but I will have to charge you for that type of effort.
Steve Green
Southwest Research Institute
210-522-3519
sgreen@swri.edu