Solar Thermal Cogeneration
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Solar Collector


The Solar Thermal Cogeneration (STC) system uses a tracking solar collector composed of a receiver, which is a pipe containing water/steam, and a tracking concentrator, which is a reflector system that tracks the sun and concentrates sunlight on the receiver.  Design goals include high absorptivity of solar radiation, low heat loss, accurate tracking, mechanical strength, long life, an economical design, simple to fabricate and maintain, with a low profile for hiding behind a flat roof parapet.

Low Profile Requirement

Low profile roof-mounting provides several advantages for a residential solar collector.  Roof-mounting hides the collector from view and provides shortest connections to points-of-use.  Roof-mounting minimizes space utilization and reduces cooling costs in hot climates by reducing sunlight on the roof. 

To address the low-profile requirement, various collector geometries must be understood.  A line-focus collector focuses sunlight on a line and a point-focus collector focuses sunlight on a point.  A simple line-focus concentrator forms a parabola in two dimensions (trough). A simple point-focus concentrator forms a parabola in three dimensions (dish).  To keep the trough low profile, its length may be extended to increase capacity.  The only way to keep the dish low profile while increasing capacity is to replicate a small dish.

The main advantage of the trough is that it can be extended in length with simple mechanics for easier fabrication.  The main advantage of the dish is its compactness making it more appropriate for small spaces.  Replicating dishes also offers modularity and opportunity for incremental development.  But its compactness implies high temperature differentials and expensive materials.  Replicating the complex mechanisms of two-dimensional tracking can add further expense, and risk of defects.  If roof area is available, and low-temperature receiver materials are available at reasonable cost, the trough probably delivers better performance/cost.

Fresnel is another collector geometry which slices a parabola into discrete sections of flat-surface reflectors and mounts them in a plane.  Flat mirrors have several advantages over the continuously-curved metal parabolic concentrator.  Flat mirrors are inexpensive, have superior reflectivity (silver protected by glass, 95 to 97%), and don't oxidize, requiring less maintenance (polishing or other treatment). 

The line-focus fresnel geometry is known as linear fresnel while the point-focus fresnel geometry is known as power-tower.   As with parabolas, line-focus fresnel requires single-dimensional tracking and point-focus requires two.  The discrete sectioning also increased the number of shafts, bearings and gears.  Although the fresnel concentrators are low-profile, the fresnel receivers must be elevated, violating the STC's low-profile requirement.  Linear fresnel receivers may be more cost-effective than parabolic trough receivers in limiting receiver heat loss.

The various geometries are feasible for various applications depending on geometric considerations, cost/reliability of fabrication, and cost/availability of materials.  The parabolic trough line-focus geometry was chosen for the STC's initial development to maximize the solar collection area for a flat residential roof with parapets while minimizing mechanical complexity, materials requirements, and to accommodate the energy conversion approach (low-temperature steam).  The linear fresnel collector is briefly covered next for some context.  Lots of info on various collector types may be found in Power From The Sun.

Linear Fresnel Geometry

Think of one giant parabolic trough directed vertically and then slice the trough long ways, bring the slices down to a flat plane, and fit each with its own axis of rotation.  There are three potential advantages of the fresnel design over the parabolic trough.  First, the reflectors can be flat glass mirrors offering superior reflection with less maintenance.  However there will be more heat loss due to the wider receiver needed to compensate the lack of curvature in the reflectors.  Second, the reflectors can be easily hidden behind the flat roof parapet.  However, this still leaves the receiver above the parapet and the reflectors being below misses an opportunity to "scoop" the light that falls on the rear parapet during the winter.  Third, wider sets of reflectors can be used with fewer rows of receivers, boosting the concentration relative to the cost of receiver materials, however this again puts the receiver high above the parapet. 

More reflectors add more complexity to the fresnel's tracking system, and more axles, more bearings, and more motor mechanisms.  This can add up to a significant cost and maintenance disadvantage. Another issue is that the receiver requires an secondary reflector pointing down.  It must be a highly reflective precision compound parabola, which is costly to fabricate.  Offsetting this disadvantage is the ability use inexpensive flat glass instead of glass tube. 

Slightly curved sheet metal, reflectors instead of flat glass mirrors would enable a narrower receiver but not eliminate the secondary reflector because the curvature of the primary would need to change with the sun's incident angle to maintain a point focus like the parabolic trough.  The surest way to find the best of the parabolic trough and fresnel approaches is to proceed through the design and parts costing of both. 

The tracking task for linear fresnel is single dimensional, like the parabolic trough.  Although the fresnel design calls for many more individual reflectors axles, and gears than the parabolic trough, they handle less force so they can be made from lighter materials.  There is less opportunity to use large diameter gears.  This may be compensated by putting all the worm gears on the same shaft and driving them with one motor connected with a large diameter gear.  If individual reflectors need to turn at different rates the gears can be made at different diameters and the the worm gear shaft be positioned at an angle to the array plane.  If the size differences are non-linear, a flexible worm gear axles can be used.  This complexity is one of the drawbacks of the fresnel approach.

Parabolic Trough Geometry


The parabolic trough concentrator, or trough, makes the shape of a parabola in its cross-section.  The trough focuses solar radiation on a line, i.e. the receiver pipe, at the parabola's origin.  The trough is made long in proportion to the parabola dimensions for increased capacity.

Reflecting Material

The reflecting material should have high reflectivity, high corrosion resistance, reasonable strength, and minimum costs.  Silver-backed mirror glass provides the best reflectivity (97%) across the solar spectrum, but the reflecting material must be curved for a parabolic trough so sheet metal becomes most feasible.  Two basic options are aluminum and stainless steel.  Aluminum has a maximum reflectivity of around 94% in the solar spectrum.   Stainless steel has a maximum reflectivity of around 90%.  Oxidation may lower reflectivity to 65% with 80% more typical for both materials.  A wax or a more permanent coating may reduce or eliminate the oxidation.  Aluminum telescope mirrors are typically treated with a half wavelength thick overcoat of silicon monoxide but it's vulnerable to abrasion.

Aluminum is very lightweight for its strength and has relatively low cost.  Stainless steel is about twice as heavy/strong and has  about twice the cost, so the two metals are about equal on strength versus cost.  Virgin aluminum has massive embodied energy, but recycled aluminum has 5% of that *, and less than stainless steel due to its lower melting point.  With an edge in reflectivity and probably embodied energy, aluminum seems superior to stainless steel for this application. 

Sheet metal is available in a wide range of thicknesses.  The cost of metal is usually closely related to weight but the price/weight goes up for thinner sheet.  Depending on the price/weight of thinner and thicker sheet available at the time of construction, the number of support members may be chosen for the thickness sheet with the best market value.  A cold rolling mill may be used to decrease the thickness and increase the length of a thicker roll.

Virgin aluminum production is responsible for 1% of man-made greenhouse gases, according to South Asian Action Network.  Only 40% of aluminum is recycled in the U.S., so it seems beneficial to recycle scrap aluminum.  Aluminum is found in beverage cans, automotive engine blocks and other components, window frames and heavy gauge electrical conductors.  See Al Recycling in the US in 2000 (pdf), Metalcasting/Machining.

The SEGS plant concentrators, mounted directly on the Mojave desert surface in California are washed every two weeks.  An STC's concentrators mounted on a residential roof might get by with a washing every two months being more isolated from dust.  Polishing/waxing would occur once at the start of each winter and maybe twice a year *.

Electromagnetic Radiation
Solar Radiation in the Atmosphere

Trough Dimensions

The figure below shows the plane perpendicular to the receiver pipe, with the pipe as the origin, and the trough facing in the positive y or vertical direction.  The radius, r, is the vertical distance from the pipe to the bottom of the trough.   The width, w, is the distance from trough end point to end point in the x or horizontal direction.  The depth, d, is the distance from trough end points to trough bottom in the y direction.  The trough circumference, c, is the distance traversed along the trough from end point to end point.

parabolaThe standard form of the equation for a parabola symmetric about the y axis and offset below the origin with radius r is:  y = ax² - r, where a is a scale factor determining the parabola's width.  But the width needs to be set in relation to the radius, r, so that that all light rays moving parallel to the y axis will reflect from the inner surface of the trough to the pipe, i.e. the origin.  Require at y = 0, the parabola's slope s = 1 (45° angle) so that a vertical ray reaching the parabola there reflects horizontally to the origin.  The slope of the parabola is its derivative, dy/dx = 2ax.  Setting this derivative to 1 (for 45°) gives x = 1/(2a) and y = 1/(4a) - r.  At y = 0, a = 1/(4r) and the parabola's equation becomes:  y = x²/(4r) - r

The trough end point coordinates are (-w/2, d-r) and (w/2, d-r) while the bottom point coordinates are (0, -r). The relationships between width, depth and radius are found by adding r to the equation with x set to w/2d = w²/(16r), and w = 4*sqrt(dr). Any two of r, d, w are specified and the third is derived.  Circumference, c, is derived here

Solar energy concentrated is proportional to trough width.  For a fixed radius, linearly increasing the width exponentially increases the trough depth.  Too much depth means excess material (circumference), excess reflection error (slope) and reduced maintenance access.  Too little depth means an excessively narrow angular range of light concentration around the pipe circumference.  The trough width determines its maximum vertical profile when the tracker points the trough at the horizon. The trough's vertical profile should be minimized for hiding it from ground view.

Sizing the Trough

The trough width and row spacing determine how much of the field area is available for collection.  If the trough width equals the row spacing maximum collection is achieved for any solar altitude angle (1).  But this results in excessive row overlap and waste of materials for a large percentage of the year, especially at winter in higher (-/+) latitudes.  If the trough width is much less than the row spacing the roof area is under-utilized at high solar altitude angle, i.e. during the hot season.  For this reason it is important to formulate an energy budget to use in making the choice of trough width for a given number of collector field rows and roof width.  If there is a use for extra energy during the hot season, a wider trough for a given roof area is appropriate. 

(1) angle of the sun above the horizon from the position of the collector field

Solar Irradiance

A common clear-sky estimate for the sun-normal ground-level irradiance, Isn, is 1000 W/m², which is the average exo-atmospheric solar irradiance (1380W/m²) corrected with the average clear-sky atmospheric transmittance (0.7).  Due to the Earth's orbit, the value increases in winter and decrease in summer each about 3%.  Variance in ground temperature has little effect on the heat transfer because of the much higher temperature of the sun.  There are analytic models that attempt to further account for atmospheric effects.  One such model developed by Campbell and Norman is presented in Solar Radiation at the Earth Surface (pdf).  This model parameterizes the location's latitude and altitude.  The STC's worksheet implements its sun-normal direct-only component but not its diffuse component because the parabolic trough captures only a negligibly thin slice of the sky as a source of diffuse radiation.  The most accurate information for specific locations is usually by measurement, e.g. see hourly data, which includes a sun-normal direct-only component relevant for tracking solar concentrators.

Solar Aperture

A point-focus concentrator (e.g. parabolic dish) maintains a constant solar aperture in the sun-normal plane through two-dimensional tracking so Isn may be used directly to calculate its energy rate for a given day and time, and the total radiation for a given day.  But a line-focus concentrator requires a sun-normal to earth-normal translation.  The parabolic trough, in the dimension perpendicular to the field rows, maintains a constant solar aperture by the tracking mechanism until the solar altitude angle drops below the point where the rows overlap in the solar aperture.  Below that point the solar aperture in that dimension varies with the sine of the altitude angle.  The solar aperture in the dimension parallel to the field rows varies with the sine of the altitude angle all the time.  The parabolic trough also has end loss to account for when the field length is relatively short (see figure ).

Day Radiation

To calculate the total solar radiation (watt-hours) available on a given day to a parabolic trough collector field, first let Lr = row length, Nr = number of rows, Sr = row spacing, Wt = trough width (see figure ), and let trough row orientation be east-west.  Let Tsa be the solar altitude angle (angle of the sun above the horizon from the position of the collector field), let Tnz be Tsa's projection in the north-south zenith plane, and let Tez be Tsa's projection in the east-west zenith plane.  The north-south zenith plane angle where the troughs in successive rows start to overlap, as seen by the sun (in the solar aperture), is the row-overlap angle, Tro = asin(Wt/Sr).

When Tnz is above Tro, the row-perpendicular field width seen by the sun is fixed at Nr * Wt by the tracker.  In the row-parallel direction the row length seen by the sun varies according to sin(Tez).  The end-loss correction for the row length, Cel = abs(Rt / tan(Tez)), where Rt is the trough radius.  So when Tnz is above Tro, the solar aperture area Asa = (Lr - Cel) * sin(Tez) * Nr * Wt. 

When Tnz is below Tro, in the row-perpendicular direction the rows are seen by the sun to overlap so a correction of sin(Tnz) is applied to all rows except the first one.  So Asa = (Lr - Cel) * sin(Tez) * [Wt+ (Nr-1) * Sr * sin(Tnz)].

Given the sun-normal ground-level irradiance Isn for a location and given that Asa has been calculated for a given day and time, the collector energy rate (watts) can be calculated:  E(W) = Isn * Asa.  The total radiation for a given day requires an integration of E from sunrise to sunset:  Day radiation, Rd (Wh) = sum [E(day,time) * dt] where dt = time increment in hours (see code listing).

These estimates for energy rate at any time of day and the total radiation at any day of year can be used to make various design decisions although a good margin of error is needed in any weather-related estimation.  The following is a day radiation example using the implemented Campbell and Norman model (altitude= 20m, latitude= 32°, Nr= 4, Lr= 32 ft, Sr= 6 ft):

 Rt (ft)
 Wt (ft)
 Rd (Wh)
winter solstice 0.75 3 121500
summer solstice 0.75 3 258600
winter solstice 1 4 138700
summer solstice 1 4 343000
The SEGS plant's Sr is 2.5 x Wt.  

Tracking Mechanism

The tracking mechanism rotates the trough around its axle, concentric with the receiver pipe, to track the sun.  It must be low-power, low-cost, high-reliability, simple, accurate and quiet.  Trough range of motion depends on latitude but should be 140° at most.  Wind and rain can put considerable forces on the concentrators and precision adjustment is needed to stabilize and maximize the radiation on the receiver pipe.  The trough should be rotationally balanced to minimize the load on the tracking mechanism.  A possible design consist of a cable attached to the two trough edges, a cable pulley, a worm wheel, a worm gear, and an electric motor, as in the figure below.   The pulley and worm gear secure the concentrator against wind forces. 

tracking mechanismThe receiver pipe sections must be short enough to prevent pipe sag from stressing the glass tubes (see Structural Support) so each receiver section might have an independent tracking mechanism.  The advantages of independent over ganging tracking include focus precision, simplified design, ability to adjust individual sections, e.g. if the system is running too hot for the load demand, and to smooth out the tracking load on the power source.  The disadvantage is a larger number of small components that could harbor defects.

Here's how to make a worm gear on a lathe.  The material should be hardened.  The  worm gear mount should allow axial adjustment to a fresh positions as the threads wear.  To get even wear on both the wheel and worm gears, first account for the relative hardness of each, then make the worm gear length times circumference equal the wheel gear width times circumference.  The wheel attaches to a cable pulley.

The cable pulley may be small in diameter to maximize the gear ratio, to reduce the load on the motor.  Given long power distribution lines, higher speed (voltage) and lower torque (current) motors are preferred for efficiency.  Also, with a small diameter cable pulley, wind force transmits less to the worm gear and more to the pulley axle.  But a larger diameter pulley provides the pulley more leverage and provides the cable more traction and less strain (1).  For increasing traction, contact area is preferred over tension, so the number of loops is increased.  A certain amount of slip allows high winds to push the trough to the end-stops, offloading the tracking mechanism.

Manila and hemp are the strongest natural ropes, while synthetics are 1.6 to 3 times stronger.  Flax and jute are also used.  Cotton tolerates more bending.  Nylon is unique in having a lot of stretch; the others may need spring-loading to absorb wind shock and shrinkage when wet.  Wire rope is stiffer and thus requires more tension to wind onto a small diameter pulley.  The greatest stress on the rope is from the pulley winding tension.  Sand/grit is the other major stressor in arid regions, followed by temperature changes.  Occasional washing seems to be beneficial for natural ropes.  Ropes are treated with water repellent/lubricating oil during manufacture and pine tar has been used for protecting sail rigging from moisture/decay.  Maintenance treatments are probably beneficial.

It's very important to protect the mechanisms from the elements.  The cable and pulley must be exposed but the worm wheel/gear and motor should be enclosed in sheet metal.  The shaft bearing and seal can be mount in a sheet metal plate that becomes one side of the enclosure, with the worm wheel on the inside and the pulley on the outside.

(1) Pulley diameter less than about eight cable diameters is a significant strain.

Manila Rope  Hemp Rope

Rotational Balance

Rotational balance prevents gravity from loading the tracking mechanism and is necessary to minimize motor load and mechanical stresses.  Imagine the x-y plane described above suspended in gravity by a string at the origin.  The trough and its support structure will tilt the plane in the negative y direction.  Placing appropriate weights at the trough end points in the positive y half of the plane will correct the tilt and place the rotational center of gravity at the origin, or receiver pipe, providing rotational balance. 

The trough end points should be near y=0 to facilitate attaching the weights which must extend into the positive y space.  Although there are diminishing returns in terms of solar aperture captured (parabola width, w) versus sheet metal requirements (parabola circumference, c), as parabola depth, d, is increased, this adds not more than 15% sheet metal compared to shallower depths.

A force-bearing mechanism such as a screw-gear telescopic actuator might be a cost-effective alternative to balancing with weights.  Another alternative is fixing the pipe to the trough and allowing the entire assembly to rotate about its natural rotational center of gravity.  But this calls for very expensive flexible steam pipe at the trough ends and results in a higher overall profile and more receiver exposure to the wind.  Wind greatly increases the heat loss and can carry abrasive particles that scratch the glass tubes. 

Tracking Motors

The tracking motors should be most efficient and reliable to minimize power draw and full costs.  In a collector field with R rows of S sections, the total number of tracking motors needed is R*S =N (for residential, typically 4*5 =20). The number of motors might be halved by driving pairs of troughs with one motor using a shaft that extends to the center of each trough.  Given long wire lengths, higher voltage and lower current are preferred to minimize resistive loss in the wires.  Given low cost electronic components, variable frequency drive (VFD) is probably feasible.  Speed bursts are required to quickly defocus the troughs to help maintain steam cycle stability.  Operating the motors at slightly different speeds may be useful, plus smooth starts/stops can greatly reduce stresses and power consumption. 

The N motors may be wired in a matrix with 1 VFD and R + S solid state switches, or a hybrid of R VFDs and S switches, or the N motors may be driven by N dedicated VFDs.  If a single VFD is used, the switch losses become an issue.  If switch losses are low, light motors may be used, and driven simultaneously at a slow rate by the single VFD.  The tracking is continuous with occasional synchronizations.  But if switch losses are significant, the VFD must adjust each trough individually, and the relatively fast and frequent starts and stops require heavier motors, with more stress and energy loss to static and kinetic friction

By running N dedicated VFDs with light motors, the switches may be eliminated.  The VFDs would be much smaller than the single VFD, probably equalizing the component costs.  The power wire mass needed in each case is nearly equal, as a function of average power draw.  If the VFDs are mounted in the motors they may all share a power bus, and have individual control wires.  But indoors is a better environment for the VFDs and this eliminates the control wires, replaced by individual power wires, with probably more reliable connections than the shared-bus.  It's the same wire mass for the same power.  But there is an increased risk of defects in such a large number of small wires and components, so if switch losses are low, a single VFD is probably the best approach.

The motors may be three-phase permanent magnet AC types, driven at a high voltage to minimize current and wire losses.  The motor might have a large diameter with a large number of small magnets on its rotor and a small number of coils (1) driven by a low frequency power waveform.  A three-phase motor might have 6 coils, with one at the 8, 9, 10, and 2, 3, 4 o'clock positions on the rotor diameter.  The rotor would have twenty magnets for ten power waveform cycles per revolution.  If the rotor diameter is 5" and the worm gear pitch is 0.05", the worm wheel diameter is 5", the cable pulley diameter is 1" and the cable run is 4" per hour, then the rotor would turn (4/60)*5/0.05 = 6 rpm, driven by a 1 Hz power waveform.  At such low speed the mechanical stresses and electrical losses would be minimum. 

(1) Fewer coils of higher inductance reduces number of connections, increasing reliability. 

Tracking Controller

The parabolic trough has only a single dimension tracking task, updating quite often, and possibly include a tracking offset to limit the receiver temperature when necessary.  The STC's control/monitor subsystem simultaneously monitors and coordinates electrical power conversion, steam system regulation and also concentrator tracking for optimum performance. 

angle detector The solar focus sensor may use two omnidirectional photodetectors attached at the end of the receiver pipe, facing the sky and separated by a thin black semi-circular disk parallel to the pipe.  When the concentrator points directly at the sun, the sun reaches both photodetectors, else the sun reaches just one photodetector while the disk shades the other. 

There is probably a need to detect the sun's position through clouds. proposes a polarizing optical detector.  The controller must be able to respond to system commands, for example, in switching to night sky cooling mode, or when the system detects overheating, or in shutting down for part of the day. 


A parabolic trough receiver is a long pipe containing transfer fluid with receiving energy
focused on the outside of the pipe by the parabolic trough concentrator.   The fluid transfers the energy to the plant, which draw energy from the fluid and returns the fluid to the receiver in a closed circuit.  Efficient energy transfer, low energy loss, and low thermal stress on the receiver pipe are high priorities for the receiver design.

Receiver as Steam Boiler

The receiver serves as the steam boiler, one of the basic Rankine cycle components.  Steam is generated from water in the boiler, releases its energy in the turbine, returns to water in the condenser and is pumped back to the boiler by the feedpump.  A low profile collector on a flat rooftop calls for a long length, small diameter receiver pipe.  This most resembles the monotube, flash, or once-through boiler type, which operates with a relatively small volume of fluid and a relatively fast response to source energy and load changes. 

The receiver pipe is conceptually divided into three segments:  The water segment where the water is heated to boiling, the evaporator segment where the water boils and the superheater segment where the steam is further heated.  Precise control of the feedpump and loads are necessary to maintain the evaporator segment in a steady position and maintain stable/bounded fluid temperature/pressure in response to rapid source energy and load changes.  It's preferable to attach temperature sensors to the pipe's outside surface instead of on the inside to avoid compromising boiler integrity.  The measure of fluid temperature is thereby delayed.  It's also preferable to avoid safety valves and pressure gauges as these also compromise boiler integrity.  The system must therefore rely on the the rpm and flowrate sensors together with the temperature sensors to detect transients and maintain stable/bounded fluid temperature/pressure. 

For added boiler temperature/pressure control, a collector tracking offset may be employed to reduce source energy rate to limit cloud-induced transients.  The sun's movement is slow enough such that normal source energy changes may be adequately detected by the temperature sensors.  But intermittent clouds present a challenge by causing rapid source energy transients.  An inexpensive image sensor and recognition process could be implemented in the control/monitor subsystem to monitor the sky and during periods of intermittent cloud cover the system could introduce a collector tracking offset to limit the transients.

Boiler Materials

The boiler pipe should be strong to withstand the high temperature, pressure and turbulence inherent with the monotube boiler type.  The pipe should conduct heat well, not sag or deteriorate or corrode over time, and should be available at a minimum cost and minimum embedded energy. 

Typical boiler pipe material is carbon steel.  The most common problem with carbon steel is oxygen corrosion from water having greater than 1 ppm oxygen content.  Corrosion at the condenser is also a problem when there is too high a carbon dioxide content. Corrosion problems increase with temperature and with rapid temperature changes.  Normally, a small amount of oxygen in the water will react with the inner pipe surface to form an oxide layer which protects the pipe from further oxidation. But rapid temperature changes create mechanical stress that fractures the oxide layer, exposing the pipe to further oxidation, leading eventually to breaches in the pipe.  Excessive turbulence will also affect the integrity of the oxide layer, but some turbulence is necessary for efficient heat transfer.

To maximize the life of carbon steel pipe the steam system should be free from debris/contaminants, free from pits/scratches/defects, have maximum radius turns, minimum mechanical stresses, minimum discontinuities in the inner wall surface, the air drawn out of the system, the temperature and pressure changes be as smooth as possible and maintained within rated limits.

Only pure water, i.e. distilled/deaerated, should be used with carbon steel.  At high temperature dissolved mineral bicarbonates decompose into much less soluble carbonates and deposit onto the inner pipe walls reducing heat transfer and causing the pipe to overheat leading to premature failure.  Without deaeration, the slow formation of oxygen bubbles creates deep pits on the pipe walls and a similar corrosion by carbon dioxide takes place in the condenser.  An oxygen scavenger such as sodium sulfite and an alkalizer such as sodium hydroxide may be added to reduce oxygen corrosion * but it's better to avoid additives when possible.
The receiver's glass tubes, seals, and the threaded pipe ends are all vulnerable to stress from pipe sag.  Thicker pipe walls provide more mechanical strength, reduce noise and vibration from boiling turbulence, provide more protection against corrosion failure, and provide more thermal mass to help smooth out thermal stress created by intermittent clouds.  The trade-off includes higher cost, lower heat transfer rate (due to thickness), higher thermal stress (due to higher temp. diff.), and greater radiative loss (due to surface area).  Common grades for carbon steel include Schedule 40, and Schedule 80 which is thicker and more expensive.

The receiver joints connect the receiver pipe sections to form the boiler circuit and provide the points of mechanical support for the pipe circuit.  To allow removing pipe sections for maintenance, the threads at each end of a section are cut in opposite directions and the joint threads are tapped to the middle of the joint.  There the section ends butt against each other so the fluid sees only one transition per joint.  Appropriate tools support the various components as the section is screwed in and out of the joints.

If the receiver pipe is welded-seam the seam should be placed away from the concentrator.  This means that the threads should be tapped so that the seam is oriented correctly when two pipe sections meet inside a joint.  The pipe threads reduce the pipe thickness and are likely to corrode through before the rest of the pipe.  Sometimes commercially cut threads are deeper than half the pipe thickness.  Instead, an increased number of finer, shallower threads may seal as effectively if they are cut with precision and are clean, defect-free and aligned well.

In the superheater section of the boiler, spiral channels (rifling) might be cut in the inside surface of the boiler pipes.  This creates centrifugal force which slings water vapor in the steam to the pipe surface increasing heat transfer and lowering the pipe temperature * (pdf).  The advantage has to be weighed against both the cost of cutting the channels and the increase in turbulence stress on the pipe.

Corrosion Testing Laboratories

Boiler Stresses

Thermal/mechanical stresses, along with corrosion, determine the lifespan of the boiler pipe.  The temperature magnitude, rate of change, and number of temperature cycles all contribute thermal stresses.  A lower operating temperature reduces thermal stress but this sacrifices energy transfer efficiency.  A higher operating temperature is tolerated by stronger metal alloys but these can be very expensive.  A buildable design calls for inexpensive materials and thus a lower operating temperature, even if collector area must increase. 

The thermal conductivities of water and steam are very different and the pipe segment holding water stays relatively cool while the pipe segment holding steam stays relatively hot.  This hot segment, or superheater segment, receives greater thermal stresses than the water segment and may require a higher grade material.  The evaporator segment of the pipe faces phase-change turbulence and intensely fluctuating temperature gradients, which may lead to thermal shock and failure in material with inadequate toughness.  The evaporator segment is likely to require a tougher material.  The total length of the evaporator segment depends on the system's ability to control the location and length of the fluid flow undergoing the phase-change.  It is probably best to specify stainless steel for the evaporator and superheater segments and (4x lower-cost) carbon steel for the water segment.  The DISS (Direct Solar Steam) [*|*] (pdf) program provides more detail.

Given the temperature and pressure range in a day cycle, largely determined by the fluid phase and flow rate, a pipe manufacturer may provide data to estimate the lifetime of the pipe.  Eddy-current testing may be used to check for cracks during periodic maintenance.

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Mechanical failure modes

Pipe Standards

Pipes are manufactured according to temperature and pressure standards.  Some are listed below.
SME B31.9 Building Services Piping  maximum 366°F (186°C), 150 psig (1000 kPa).
ASTM A53 B Carbon Steel Pipes - Working Pressure for 400°F:  Sch 40:  214  psig,  Sch 80:  753 psig
Bursting internal pressure of STM A312 Stainless Steel Pipes, 1/2", Sch 5:  11,607 psi
ASTM A269 Welded & Bright Annealed Stainless Steel Tubing  (304L/316L) 1/2", 0.02", 1500 psi

A106 Grade B Carbon Steel Pipes - Pressure and Temperature Ratings
ASTM A53 B Carbon Steel Pipes - Working Pressure
Bursting and Collapsing Pressures of ASTM A312 Stainless Steel Pipes
Steel Tubes - Working Pressures
Stainless Steel Pipes - Pressure Ratings

Temp. Drop / Friction Loss

The pipe temperature limit and the temp. drop due to the thermal resistance of the receiver pipe/fluid together place an upper limit on the fluid temp., and thereby on the overall system capacity.  The heat transfer rate from the outer pipe wall to the fluid inside is Q (W) = [ 2π L (To - Tf) ] / [ ln(Ro/Ri) / C  +  1 / RiH ] where L = pipe length (m), To = pipe outer surface temp. (°K), Tf = fluid temp. (°K), Ro = outer pipe radius (m), Ri = inner pipe radius (m), C = wall conductivity (W/m°K) and H = fluid heat transfer coef. (W/m2°K).

For a solar collector field of 20 6 ft sections, (120ft * 0.3048 = 36.6m), pipe thickness 3 mm, outer diameter 3/4" * 0.0254 = 0.02 m, outer temp. 575°K and inner temp. 500°K, stainless steel pipe wall thermal conductivity 25 W/(m°K), steam heat transfer coefficient 500 W/m2°K, the heat transfer rate is: Q (W) = [ 2π * 36.6m * ( 575°K - 500°K ) ] / [ ln(0.02/0.017) / 25  +  1 / (0.017*500) ] = 139 kW.

Fluid friction in the receiver pipe reduces steam pressure/temperature at the turbine nozzle.  According to, friction is a fucntion of fluid viscosity and its velocity gradient set up by the no-slip condition at the wall *.  Friction loss in steam pipes is mostly a function of pipe length and diameter although bends and fittings also create friction, turbulence and noise.  From / steam pipe sizing:  steam pressure loss due to pipe friction is: 

pressure loss (psia) = 14.7 + 1.306e-4*L*q2*(1+3.6/D)/(3600*d*D5),
   where L = length (ft), q = mass flow rate (lb/h), D = diameter of pipe (in), d = density of steam (lb/ft3).  
velocity (ft/s) = volume flow (ft3/h) /  ((sec/hour) * π * radius (ft)2)

pressure loss (bar) = 0.06895*(14.7 + 4.284e-4*L*(2.2*q)2*(1+3.6/(0.03937*D))/(224.64*d*(0.03937*D)5)),
   where L = length (m), q = mass flow rate (kg/h), D = diameter of pipe (mm), d = density of steam (kg/m3).
velocity (m/s) = volume flow (m3/h) /  ((sec/hour) * π * radius (m)2)

For example,  L = 5 m, q = 25 kg/h, D = 12 mm, d = 4.16 kg/m3 (P = 8 bar a, T = 76°C),
pressure loss (bar) = 0.06895*(14.7 + 4.284e-4*5*(2.2*25)2*(1+3.6/(0.03937*12))/(224.64*4.16*(0.03937*12)5)) =0.43 bar.

given, from above, specific volume = 1/d = 0.24 m3/kg,
velocity = volume flow  / area = (25 kg/h * 0.24 m3/kg) / (3600 s/h * π * 0.0062 m2) =14.7 m/s

Also see Spirax Sarco Example 10.2.2.  Spirax Sarco's steam table calculation generally agreed with Engineering Toolbox.  Spirax Sarco says the general rule for steam velocity is 25 to 40 m/s.  Above 40 can create noise and erosion.  Engineering Toolbox suggests that superheated steam is ok up to 60 m/s.

Friction loss formulas for the pumping of fluids, which are incompressible, include the Darcy-Weisbach Equation, Moody diagram, and Hazen-Williams Equation and also steel pipe friction loss and velocity diagrams: Sch 40, Sch 80.

Sizing The Receiver

First, the trough size, the number of rows in the collector field, and the length of the series circuit are determined from the roof dimensions and amount of solar energy to be collected.  Next, the receiver pipe diameter is considered.  A smaller diameter pipe increases feedpump load through friction loss, but also reduces heat loss and increases heat transfer by turbulence through increased fluid velocity.  Turbulence should be limited to avoid excessive thermal stresses, corrosion (1), and noise.

Next, a steam temperature setpoint is determined to meet the turbine target efficiency, up to the pipe thermal stress limit.  Then a volume flowrate that transfers the required energy is determined from the temperature setpoint, pipe diameter, and steam specific heat.  A number of flowrates may be calculated for a number of pipe diameters and the materials costs considered and a final decision made on the pipe diameter.  The calculations must be made for water, saturated steam and superheated steam.  The percentage of the receiver pipe occupied by these phases are determined in System Design.  A larger diameter pipe for the evaporator and superheater sections might yield better overall performance (2).

(1) Excess turbulence can break up the protective oxide layer that forms on the inside of metal pipes, causing more oxidation and faster erosion.
(2) Phase change creates high turbulence in the evaporator section and high steam velocity maintains considerable turbulence in the superheater section.  So a larger diameter pipe for these sections may provide adequate heat transfer from the wall to the steam while reducing friction drag.  Friction drag is reduced by decreasing the ratio of pipe surface area to fluid volume and by reducing fluid velocity.

Receiver Emissivity

During operation, the receiver pipe surface temp. remains significantly higher than the steam temp. due to the thermal resistance of the pipe material.  This means potentially high heat loss from the pipe through convection to the air *, and radiation to the sky *, a low temperature radiation sink.  A selective coating on the pipe surface is required to limit radiation loss from the pipe to the sky.  The selective coating should have solar absorptance of >0.9 and infrared emittance of <0.15 over the full range of incident angles, and endure 600°F over the system lifespan.

Most of the energy radiated by the pipe is in the infrared region of the radiation spectrum while most of the energy from the sun is in the visible region.  A wavelength-selective coating on the pipe can limit infrared emission while absorbing visible radiation.  One approach is to first electroplate [1] the pipe with nickel, an excellent reflector, to limit infrared emission.  Then a coating is applied to cause absorption of the solar radiation without affecting the infrared emission.  The coating accomplishes the absorption geometrically [2], through wavefront discrimination by the particles. 

Selective paints composed of metal oxide pigments in a silicon binder have solar absorptance of 0.92 and infrared emittance of 0.13 [2].  These numbers are good but it appears that the temperature limit for such paints are below 600°F.  Chrome is deposited [1] in a microscopic pattern onto a nickel coating have solar absorptance of 0.95 and infrared emittance of less than 0.25 [3]. 

Magnetron sputtering is an alternative to electroplating that doesn't involve chemicals.  The SEGS plants in Southern California used Mo-Al2O3 cermet solar coatings to achieve solar absorptance of 0.96 and infrared emittance of 0.16 at 660°F [4].  The electroplating method is probably more appropriate for the STC given its low cost, assuming the coating can handle the high temperatures.

[1] electroplating [*|*], thin film deposition
[2] Solar Energy - State of the art (pdf)
[3] Solar Collectors - Power From The Sun
[4] Sputtered Solar Absorbing Coatings
Powder Coater's Manual

Glass Tube, Vacuum Seals

The receiver pipe is enclosed in a glass tube to maintain a vacuum that reduces convection/conduction heat loss from the pipe to the surrounding air.  The glass tube should have very high transmission in the solar spectrum, high reliability, and a reasonable cost.

Glass with high iron content (a green tint in the edge) has a transmittance of around 0.8 and absorbs some 20% of solar radiation so low-iron glass should be specified.  Borosilicate glass with transmittance as high as 0.93 across the solar spectrum for thicknesses of several mm, is probably the most appropriate for the application, given its strength at high temperatures, and high availability.  Most glass has around 4% reflectance across the solar spectrum at each interface.  A magnesium fluoride coating can reduce this to 2% for much of the spectrum and more elaborate layering can reduce it close to zero for most of the solar spectrum.  This coating may have issues with abrasion and temperature. 

The glass tube will absorb most of the infrared emission from the receiver pipe * and then emit that to the colder environment.  Convection/conduction heat loss from the pipe is determined by the temperature, pressure, and the types of residual gases in the vacuum (off-gassing from inside surfaces, e.g. water, oil vapors).  The heat loss rate for an ideal gas may be calculated but the residuals may not behave as ideal gases.  The pressure is chosen to balance insulating quality against costs, including the cost of vacuum equipment.  Convection across a 1 cm gap becomes insignificant below 10 torr.  Conduction across a 1 cm gap is roughly constant with pressure above 0.1 torr, proportional to pressure between 0.1 torr and 0.001 torr, and insignificant below 0.001 torr *, which is about the limit of mechanical vacuum pumps (see creating a vacuum). 

The vacuum seals at the ends of the tube should withstand thermal stresses and should not off-gas into the vacuum.  The seals should hold the vacuum while accommodating the glass tube's and metal pipe's different rates of thermal expansion.  Linear expansion equals length times the material's linear expansion coefficient times the change in temperature.  For example, a receiver is 10 ft long, the linear expansion coefficient for borosilicate glass is 0.32e-5 per °C and for carbon steel, 1.24e-5 per °C.  At a temperature increase of 300°C the the glass length expands by 10 ft x 0.32e-5 x 300 = 0.12" and the steel by 10 ft x 1.24e-5 x 300 = 0.45".  The difference in expansion length is 0.33". 

At high temperatures, metal is probably the best vacuum seal material.  One approach to sealing concentric pipes to accommodate linear expansion is a bellow (accordion) seal.  Bellow seals are made by welding a stack of sheet metal donut shaped disks together at their inner edges and at their outer edges and welding a pipe and tube seal at the ends of the assembly.  The bellow disks must be strong enough to support the weight of the tube and withstand the vacuum at high temperature, and thin enough to flex with expansion/contraction. 

The pipe seal can be copper and the tube seal can be a nickel-iron alloy that matches the glass's expansion coefficient.  To install the seals, the contact surfaces are polished and cleaned, then the seals are pressed on with the glass expanded by high heat, the seals expanded by medium heat and the pipe kept relatively cool, but above dew point.

SEGS Parabolic Trough Receivers:  Schott, the company that made the SEGS receivers provides considerable information on their PTR 70 receiver.  The glass is borosilicate with an anti-reflection coating that has been strengthened to withstand abrasion.  The tube is steel with a selective coating for high absorption and low emission at the operating temperature of 700°C.  The seals are a nickel alloy with a thermal expansion that maintains a seal with both the glass and the steel.

Basics of Design Engineering - Engineering Materials - Glass
Metal bellows sealing

Heat Losses

The receiver heat loss prevention schemes are critical to the STC's overall efficiency.  The receiver components, particularly the coatings on the glass and pipe and the vacuum seals, must be tested to ensure they will maintain the target efficiency over the system service life. Several variation of the design should be tested together.  The receiver's lifespan in the target environment is estimated through accelerated time tests and extrapolation of test data.  For example, the number of diurnal temperature cycles that the receiver can tolerate is estimated by extrapolating the results of a series of different short period cycle tests.

The receiver pipe transfers energy to the flowing steam at a rate equal to the incident solar energy minus the heat loss from the receiver pipe to the environment.  For a collector field of 20 4ft * 6ft sections, and a clear-sky estimate for the sun-normal ground-level irradiance of 1000 W/m², the total incident power is 20 * 4ft * 6ft * 0.0929 ft²/m² * 1000 W/m² = 44.6 kW.  For a given pipe temperature, heat loss calculations through the glass tube involve selecting the glass tube inner and outer surface temperatures (Tglis, Tglos) that equate the pipe-to-glass heat transfer, the glass heat transfer, and the glass-to-environment heat transfer.  When the pipe temperature is below roughly 1000°F, practically 100% of the radiation will be absorbed by the glass instead of transmitted *.

pipe-to-glass radiation: If the pipe temp. is 600°F, pipe diameter is 1/2", pipe emissivity is 0.15, the pipe-to-glass radiation loss is:  pipe emissivity 0.15 * Stefan's constant 5.67e-8 W/(m^2·K^4) * pipe surface area (20 * 6 ft * pi * 1/24 ft * 0.0929 ft²/m²) * ( Tpipe 589°K^4 - Tglis(?)°K^4 ).

pipe-to-glass conduction: Thermal conduction in air is approximately linear with air pressure below .01 torr and is * approximately (2.66 * torr) W/(m·K).  If the air pressure in the vacuum between the pipe and glass is 0.005 torr, the pipe-to-glass spacing is 0.5 cm, then pipe-to-glass conduction loss is: [ conduction coefficient (2.66 * 0.005 W/(m·°K)) / spacing 0.005 m ] * pipe surface area (20 * 6 ft * pi * 1/24 ft * 0.0929 ft²/m²) * ( Tpipe 589°K - Tglis(?)°K ).
glass conduction: The thermal conductivity of glass is around 1 W/(m*K) *.  Given a glass thickness of 3 mm, and diameter 1", the glass conduction rate is: [ thermal conductivity 1 W/(m*K) * glass surface area (20 * 6 ft * pi * 1/12 ft * 0.0929 ft²/m²)) / glass thickness 0.003 m ] * ( Tglis(?)°K - Tglos(?)°K ).

glass-to-sky radiation: At wavelengths above 5 um (less than 1000°F glass temp.) the emissivity of borosilicate glass appears to be near 1 *.  If the winter day min. effective sky temp. is 10°F, and the glass diameter is 1", the glass-to-sky radiation loss is: Stefan's constant 5.67e-8 W/(m^2·K^4) * glass surface area (20 * 6 ft * pi * 1/12 ft * 0.0929 ft²/m²) * ( Tglos(?)°K^4 - Tsky 261°K^4 ).

glass-to-air convection: Using a high-wind convection heat transfer coefficient and winter day min. air temp. of 40°F, the glass-to-air convection loss is: convection coefficient 100 W/m²°K * glass surface area (20 * 6 ft * pi * 1/12 ft * 0.0929 ft²/m²) * (Tglos(?)°K - Tair 278°K). 

Next, the glass surface temps. are adjusted in the calculations to equate the total pipe-to-glass transfer, glass transfer, and the total glass-to-environment transfer:

pipe-to-glass radiation: 0.15 * 5.67e-8 * 35/24 * ( 589^4 - 288^4 ) = 1400 W
pipe-to-glass conduction: (2.66 * 0.005 / 0.005) * 35/24 * ( 589 - 288 ) = 1100 W
glass conduction: (1/0.003) * 35/12 * ( 288 - 285.4 ) = 2500 W
glass-to-sky radiation: 5.67e-8 * 35/12 * ( 285.4^4 - 261^4 ) = 300 W
glass-to-air convection: 100 * 35/12 * ( 285.4 - 278 ) = 2200 W

Verifying the transfers match: 1400 W + 1100 W = 2500 W = 300 W + 2200 W = 2.5 kW, the resulting glass temps are Tglis =  288°K = 58°F, and Tglos = 285.4°K = 54°F.  So the resulting heat loss through the glass for a receiver temp. of 600°F and worst-case environmental parameters is 2.5 kW, or 5.6% of the total incident power, 44.6 kW.  This equals a heat transfer efficiency of 94%, better than the receiver target efficiency of 85%.  Add the heat loss through seals, structural supports and connector pipes for the total receiver heat loss.

Collector Field


The ideal collector field configuration is two trough rows forming a single loop but four rows more likely for typical roofs. Fewer/longer troughs minimize connector losses and end losses at low sun angle.  Troughs may be oriented east-west or north-south (1).  East-west orientation better fits  the longer east-west roof dimension of building structures optimized for passive solar heating, and requires two to three times less tracking work.  North-south orientation minimizes trough end loss and better fit roofs that are not longer in the east-west dimension, however gravity drainage imposes limitations on the use of this orientation.

For east-west orientation, in latitudes far from the equator, equator side parapet height and trough position should minimize the parapet's blocking the winter sun.  The opposite side trough should be close to the opposite side parapet.  East/west side parapets blocking of morning/evening sun on the trough ends should be minimized.  In equatorial latitudes, north/south side troughs may be close to the parapets. 

For north-south orientation, in latitudes far from the equator, equator side parapet blocking of the winter sun on the trough ends should be minimized but trough ends on the opposite side should be close to the parapet.   East/west side parapets blocking of morning/evening sun should be minimized.  In equatorial latitudes, trough ends may be close to the north/south parapets. 

The following sections assume trough east-west orientation.  The troughs should be spaced so that the sun sees no overlap/underlap between trough rows at the sun's winter solstice high point. 

With four rows, a parallel circuit may be considered.  If certain pipe diameters are more available than others, parallel could yield better performance/cost by reducing pipe friction. But the first trough row will receive a different shadow than the three troughs behind it, creating a pressure imbalance between the two parallel halves of the circuit.  Compensation would require two feedpumps and more complex control, so a series circuit is probably better.

The connecting pipes between trough rows should remain at the receiver height and the turns should be wide radius to minimize pressure and turbulence.  The connectors should be well-insulated.  The plant should be at the corner of the field at the end of the circuit to minimize the steam connector length.
(1) The true north-south line is along the solar noon shadow of a straight rod planted in the ground aligned with a plumb-bob string.  Solar noon is halfway between sunrise and sunset.
(2) The ratio of summer to winter insolation intensifies toward the poles.  In fact, the summer insolation peak is greatest at the poles.

Gravity Drain

In freezing climates, the receiver pipe should drain by gravity to avoid water freezing in the pipe at night and to avoid water hammer on startup each morning.  It is recommended for horizontal pipework to have a 1:100 slope for drainage.  This means every 8 feet of run should have 1 inch of rise.  So, for example, a trough field with four 32 foot rows and 6 foot connectors requires a total rise of 18 inches. 

If the trough rows are east-west-oriented, the equator-near row should be the lowest so that the drainage incline may increase the exposure of the other rows when the sun is lower in the sky.  Ideally the roof would be sloped toward the equator-facing wall for rainwater drainage and in new construction this would provide the needed slope for steam system drainage.  The drainage slope may be included in calculations as a tilt angle for the earth-normal plane.

If the trough rows are north-south oriented the drainage incline may also face the equator to benefit winter exposure except when the number of rows is four or more because the equator side loops cannot drain, so the incline will have to face the east or west instead.

The condenser reservoir holds all of the steam system water when the system is not operating, and is mounted such that the reservoir fillpoint is lower than all of the pipework outside the plant housing.  For the above trough field, with 1/2" inside pipe diameter, and water filling 3/4 of the circuit during daytime operation, the reservoir has to hold an extra (3/4)* pi * (0.25/12)ft^2 * (5 * 6ft + 4 * 32ft) * 7.48 gal./ft^3 = 1.2 gal. beyond its daytime operation fillpoint.  To protect the water from freezing in the reservoir, the plant housing is insulated and the reservoir may be placed in thermal contact with the building interior. 

Thermal Expansion

receiver expansion imageLong lengths of pipe undergoing large temperature changes require allowances for thermal expansion and contraction.  The linear thermal expansion coefficient for carbon steel is 8e-6 per °F, so a 30 ft pipe heated 300°F expands about 0.85 inch in length.  The roof and support frame will stay relatively cool and dimensionally stable under white paint and concentrator shadow.   Since the receiver pipes are fixed at the plant, the supports at the opposite end of the collector field should allow the pipe to slide in the axial direction but hold in the lateral direction.  This prevent lateral stresses on the receiver joints.  This lateral tension on the pipe might call for a roller support but the pipe might need tighter support to dampen turbulence vibration (see Structural Support).  With lateral bracing on the pipe supports, the wide-radius pipe connectors bend to absorb their own expansion.  The wide-radius connectors also minimize turbulence, wear and noise by eliminating elbow joints.

Structural Support

The solar collector field must be strong enough to withstand wind, hail, snow accumulation and ice formation.  A load of ice 3 inches deep, 8 inches wide, and 6 feet long weighs (0.036 lbs/sqin) x 3 in. x 8 in. x 72 in. = 62 lbs.  A load of snow 1 foot deep, 3 feet wide, and 6 feet long weighs (0.012 lbs/sqin) x 12 in. x 36 in. x 72 in. = 373 lbs.  Structures in high snow areas are typically built to withstand a 1.5 to 2 feet deep snow load (30 to 40 lbs/sq ft).  See Meteorological Data.

The maximum length of a collector section is determined mostly by the strength of the receiver materials. Heat causes the steel receiver pipe to lose some of its strength.  According to Spirax Sarco, for steam application, a small diameter steel pipe needs support every six feet.  The receiver's glass tube/seal stress ratings may further reduce the practical length of a collector section.  However bellow seals may relieve much tube/seal stress caused by pipe sag.

On each end of the concentrator, a horizontal strut extends the concentrator width from corner to corner above the receiver pipe, connecting to the hinge joint (around the receiver pipe) with a U-clamp/sleeve.  This helps move the concentrator's center of gravity to the hinge joint.  A vertical strut extends from the concentrator bottom to the U-clamp, forming a "T" with the horizontal strutThese end struts must be high strength for minimum thickness and shadow on the concentrator.  Two long struts each made of four stacked 1/4" U-channel rods extend along the long concentrator edges and attach to the horizontal strut ends at the concentrator corners.  These channels may be filled with weights to help move the concentrator center of gravity to the hinge joint. 

The main support struts should be very rigid to avoid transmitting wind forces on the concentrators to the receivers.  The receiver joint sleeves must allow for receiver thermal expansion, mainly in the axial direction, but should also dampen or resist fluid turbulence to prevent transmission to the concentrators and roof frame (see Noise and Vibration Control (pdf).  The sleeves should also have low thermal conductivity to minimize abrupt temperature gradients along the receiver pipe.

Steel rods with circular cross section should be used for the collectors' main support struts, being the strongest geometry against bending forces at multiple angles.  The main struts should be the minimum diameter to minimize blockage of radiation on the concentrators.  The main struts must provide full stability to the concentrators along the axis of the receiver pipe without any aid from the receiver pipe.  They are unlikely to align with roof beams so a framework resting on the roof is needed, probably wood.

Many building codes call for buildings to withstand 50 year wind events.  Such events may produce ten times average wind pressures on a building, or a wind pressure on a roof of 30 lbs/ft2.  Given a concentrator section 6 ft long and 4 ft wide, the total wind force on the concentrator and on each main support strut (one per concentrator), F = 720 lbs.  For a beam of length L, modulus of elasticity E, and area moment of inertia I, the elastic deflection D = FL3 / (3EI).  Further given the main support strut is made from a circular carbon steel rod, of diameter d = 1 in, length L = 2 ft, I = πd4/64 = 0.0031 in4, and E = 29e6 lbs/in2, then D = 720 lbs x (24 in)3 / (3 x 29e6 lbs/in2 x 0.0031 in4) = 2.3 in. This deflection puts far too much bending stress on the receiver pipe.   Shortening the effective strut length through reinforcement to 12 in. reduces the deflection to 0.29 in., which is probably ok.  The 1" diameter of such a solid steel rod would make it very costly.  The area moment of inertia for a hollow pipe is comparable to the solid rod.  However, the large diameter also casts a shadow on the concentrators, so maybe a different geometry is more effective. 

The wind load on the concentrator is unlikely to be as great in the axial direction (along the receiver pipe) as it would be in the lateral direction.  A pair of small I-beams forming a truss (triangle) with the roof member is likely to provide the most lateral strength for the amount of material and shadow.  The area moment of inertia of the I-beam cross-section may provide substantial strength in the receiver-axial direction, while the tensile-compressive strength of the truss provides the much greater strength needed in the lateral direction.  See Structural Mechanics.

For the concentrator's structural strength, it's probably better to rely mostly on a frame instead of the reflecting material.  The frame can be 1"x1" wood strips.  Wood is available from municipal trees and may be sawn with a portable sawmillGreen wood is more pliable than dried wood and can be dried after bending.  To minimize splitting, strips with grain runout should be avoided and the ratio of bend radius to wood thickness should be from 12:1 to 16:1 depending on species and grain.  To make the strips pliable for bending, a long enclosure is made to allow steam to circulate around the strips before exiting through a small outlet.  The inlet connects the enclosure to a heated water kettle, allowing condensate to drain back.  Roughly 60 minutes of steaming is required per inch thickness for average density species.  The resulting heat and humidity from the steamer should be kept in the room. Within seconds out of the steamer, a strip is bent over a parabolic shaped form using a metal strap.  This strap can have handles on the ends and provide compression to the ends of the strip to further reduce the risk of splitting. The form may be made of metal rods the length of a concentrator section and also serve as the section assembly form.  The wood may be dried to 8% moisture in several weeks in a solar kiln.  There is usually a small amount of springback after the form is removed, depending on species and grain.  This springback may be canceled with an overbend built into the form figured after a trial bending and 24 hour cool down. 

To protect wood components, a penetrating finish such as raw linseed or tung oil may be used.  The oil penetrates into the wood where it slows water absorption, thereby slowing decay and swelling/shrinking that can lead to warping and cracking.  A water repellent [*|*] (pdf) is a penetrating finish that includes 10 to 20% paraffin wax in the oil to further slow water absorption.  In contrast to penetrating finishes, film-forming finishes are prone to trapping moisture, accelerating wood decay.  The wood surface should be rough and the oil should be heated to 80°F (but not higher) for better penetration.  The components may be dipped or the oil painted on with a natural bristle brush.  End grains and areas subject to restricted airflow should be coated liberally.  Give the coat a week to dry in a moderate temperature in the shade, then clean if dirty for outer coat application.  An outer coat may contain 50% pigment to reflect radiation, reducing heat absorption, to help protect the wood.  This outer coat may be sanded and reapplied on the five or ten year maintenance schedule.  Clean up brushes and containers with soap and water.  Cleaning the finish with a soft bristle brush and mild soap and water, and a coat of wax every couple of years is probably good. *  *  *

Wind on Rooftops

When the wind encounters a flat-roofed structure, the wind streamlines are elevated by the edges of the roof.  Above the roof but below the streamlines, a vortex is created that presents an uplifting force on the roof.  If the streamlines remain elevated across the roof, the turbulent vortex below is relatively mild and uniform across the roof.  But if the streamlines drop down in the middle of the roof, then the smaller isolated vortex near the windward edge exerts a much more intense (2x to 4x) local uplift, especially at the corners.  This uplift may stress sections of the solar collector field.  The much greater turbulence of this vortex may increase heat loss from the receiver pipe.  Wind can also carry sand and grit that scratches the receiver glass and the much greater turbulence only makes it worst.  The parapet height may be calculated, based in the structure dimensions, to ensure that the wind streamlines remain elevated across the roof instead of dropping down after clearing the parapet.  And the collector field may also be kept away from the roof edges to minimize uplift stress if space permits.  Wind data may be found here.  

Wind Pressures and Suctions on Roofs

Design/Modeling Worksheet

Eventually include all of the STC's systems.  Worksheet in Gnumeric XML-based spreadsheet format.


exo-atmo solar irradiance Eo W/m² 1380
collector altitude Ac m
altitudinal pressure ratio Rap
air mass number m
Rap/sind (Tsa)
clear-sky atmo transmittance Tack
atmospheric transmittance Ta
direct solar irrad sun-norm Irsn W/m² Eo* Ta
direct solar irrad earth-norm Iren W/m² Eo * sind (Tsa) * Ta
diffuse solar irrad earth-norm Ifen W/m² Eo * sind (Tsa) * 0.3 * (1 - Ta)
total solar irrad earth-norm Iten W/m² Iren + Ifen

collector latitude Lc degrees s pole -90, eq 0, n pole +90

field area Af ft²
trough radius Rt ft
trough width Wt ft
row length Lr ft
number rows Nr

row spacing Sr ft
row overlap angle Tro degrees asind(Wt / Sr)

insolation-day aperture Aid
2 / pi()
sunset angle Ts degrees acosd(-(sind(Lc)*sind(Td))/(cosd(Lc)*cosd(Td)))
hours of collection Hc hours 2*Ts/15
sunset Ht hours 12+Ts/15
sunrise Hr hours Ht-Hc
day of year day

declination angle Td degrees -23.45 * cosd ( 360 * (day+10)/365 )
hour of day hour

hour angle Th degrees 15*(12-hour)
solar altitude angle Tsa degrees asind(sind(Lc)*sind(Td)+cosd(Lc)*cosd(Td)*cosd(Th))
solar azimuth angle Tsz degrees asind(-cosd(Td)*sind(Th)/cosd(Tsa));
if(cosd(Th)>=tand(Td)/tand(Lc)) Tsz=pi()-Tsz;
else if(Tsz<0) Tsz=pi()+Tsz;
solar nz altitude angle Tnz degrees asind(sind(Tsa)/sqrt(pow(cosd(Tsa)*cosd(Tsz),2)+pow(sind(Tsa),2)))
solar ez altitude angle Tez degrees asind(sind(Tsa)/sqrt(pow(cosd(Tsa)*sind(Tsz),2)+pow(sind(Tsa),2)))

sq. feet to sq. meters fstoms
end loss correction Cel
abs(Rt / tand(Tez))
field energy rate above Tro Efa W Irsn * fstoms * (Lr - Cel) * sin(Tez) * Nr * Wt
field energy rate below Tro Efb W Irsn * fstoms * (Lr - Cel) * sin(Tez) * (Nr+1) * Sr * sin(Tsa)
model day radiation Id Wh moddayrad( day, Ac, Lc, Nr, Rt, Wt, Lr, Sr)


Day Radiation Code Listing (see worksheet and figure)
// moddayrad calculates the day radiation (Wh) using the sun-normal direct radiation
// component of the Campbell and Norman model for atmospheric transmission for a given
// day of year, altitude Ac (m), latitude Lc (deg), number of parabolic trough rows Nr,
// trough radius Rt (ft), trough width Wt (ft), row length Lr (ft) and row spacing Sr (ft).
// the functions are added into /usr/src/rpm/SOURCES/gnumeric-0.67/src/functions/fn-math.c
// and compiled into gnumeric, the build instructions are in the file above the functions
// to use the functions in gnumeric, first click on the cell with the fucntion,
// then remove the apostrophe from the start of the string, and press return
// the returned value should be in the cell, also data is printed to stdio

static float
moddayrad (float day, float Ac, float Lcd, float Nr, float Rt, float Wt, float Lr, float Sr)
float i, j=0, sum=0.0;
float Lc= Lcd*M_PI/180;
float Tro= asin(Wt/Sr);
float Td= -23.45*cos(2*M_PI*(day+10)/365)*M_PI/180;
float Ts= acos(-(sin(Lc)*sin(Td))/(cos(Lc)*cos(Td)));
float sunset= 12+(Ts/15)*(180/M_PI);
float Hc= 2*(Ts/15)*(180/M_PI);
float sunrise= sunset-Hc;
float fstoms =0.09290304; //ft^2 to m^2
float hrfrac=2048;

for(i=1;i<Hc*hrfrac-1;i+=1) {

float hour=sunrise+i/hrfrac;
float Th=(15*(12-hour))*M_PI/180;
float Tsa=asin(sin(Lc)*sin(Td)+cos(Lc)*cos(Td)*cos(Th));
float Eo=1380; // W/m^2
float Rap=pow(10,(5-Ac/15500))/100000;
float Ta=pow(0.7,Rap/sin(Tsa));
float Cel,Tez,Tnz,Tsz=asin(-cos(Td)*sin(Th)/cos(Tsa));

if(cos(Th)>=tan(Td)/tan(Lc)) Tsz=M_PI-Tsz;
else if(Tsz<0) Tsz=M_PI+Tsz;

Cel=abs(Rt / tan(Tez));

if(Tnz>Tro) sum += Eo * Ta * fstoms * (Lr - Cel) * sin(Tez) * Nr * Wt;
else sum += Eo * Ta * fstoms * (Lr - Cel) * sin(Tez) * (Wt+ (Nr-1)*Sr*sin(Tnz));


return sum / hrfrac;

Circumference of Parabola

The circumference c (or length) of a function y = f(x) in the x-y plane equals the integral, between the two relevant bounding values of x, of the hypotenuse of the right triangle formed by increments of x and y = f(x)

The hypotenuse dh = sqrt( dx² + dy² )
but another form is sqrt( 1 + dy²/dx² ) = sqrt( 1 + f'(x)² )

Given the function for the parabola: y = x²/4r - r, and y' = x/2r,
then dh = sqrt( 1 + x²/4r² )
and c = 2 * integral[0:w/2] sqrt( 1 + x²/4r² ) dx.

Substituting u = x/2r,
c = 4r * integral[0:w/4r] sqrt( 1 + u² ) du

and from integral tables,
c = 4r * { u/2 * sqrt(1+u²) + 1/2 * ln(u + sqrt(1+u²)) }   [0:w/4r]

= 4r *
{ w/8r * sqrt(1+(w/4r)²) + 1/2 * ln(w/4r + sqrt(1+(w/4r)²)) }

w/2 * sqrt(1+(w/4r)²) + 2r * ln(w/4r + sqrt(1+(w/4r)²)).

Solar Aperture Area Illustration

Solar Aperture Area Illustration


Copyright (c) 2005-2009 Robert Drury
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2
or any later version published by the Free Software Foundation;
with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
See "GNU Free Documentation License". 

Disclaimer:  This information may contain inaccuracies and is provided
without warranty.  Safety first when working with high temperatures,
pressures, potentials, speeds, energies, various tools and materials.