US20150285534A1 - Solar collector with optimal profile for energy distribution on a tubular receiver - Google Patents

Abstract

The solar collector with optimal profile for energy distribution on a tubular receiver collects and distributes solar energy. Solar energy received from the sun can be modified by either being re-directed (reflection) or being redistributed. In the present invention energy is reflected and redistributed in a manner that yields a required energy variation over a surface. The receiver is a cylinder of known length and diameter. Longitudinal distribution of energy is specified by a user defined function. Circumferential distribution is assumed to be constant. Energy distribution is required to vary along the z axis of the receiver but remain constant in the circumferential direction. An axi-symmetric approach is used in which only one plane of the receiver in r and Z plane is considered. A geometric solution determines a reflecting surface that gives a required energy distribution along the z-axis. A complete reflector is designed by expanding the axi-symmetric behavior.

Description

BACKGROUND OF THE INVENTION
• 1. Field of the Invention
• The present invention relates to solar collectors, and particularly to a solar collector with optimal profile for energy distribution on a tubular receiver.
• 2. Description of the Related Art
• Solar energy has been used for producing steam for process heating, Rankin cycle, steam reforming, desalination processes and cooking. These processes take place at high temperatures which are usually obtained by the use of concentrating solar collectors. Amongst all, solar parabolic dish and solar parabolic trough are the prominent concentrating collectors and are used extensively for high temperature applications. The use of these concentrators for certain applications produces serious problems. Using a solar dish for a Stirling engine produces problems of hot spots. Production of steam within the receiver of a solar parabolic dish requires investigation of critical heat flux that is possible only if the energy distribution over the surface of the collector is known. A non-uniform temperature distribution over the reactor in steam methane reforming causes serious yield problems and is harmful for catalyst as well. Each of these processes requires a flux with specific distribution to take place in a controlled predictable environment.
• As for as the solar parabolic dish and parabolic trough are concerned, maximum energy concentration ratio is limited for these collectors. The extra restriction for a solar collector to be of imaging type was found to be irrelevant by Roland Winston. He proposed a non-imaging concentrating solar collector that can give higher concentration ratios than the imaging ones with the greater acceptance angle. After that, work was done by Ari Rabl and Winston to improve these collectors but still there was a question of flux distribution. The distribution of flux is critical in high temperature applications and its effects must be reported if not tackled carefully. Various attempts have been made to control the flux distribution over the surface of the absorber. The shape of the absorbing surface is optimized to get uniform flux distribution by using the ray tracing technique but the shape was far away from the usual conduits used for heat transfer. For a specific application of Stirling engine, the shape of absorber was also modified and a concave mirror like shape in the axi-symmetric plane of the absorber has been tried. Splitting the aperture's cross sectional area by using small reflectors dealing with a smaller wave front was also conducted. The collector had a complex geometry and reflecting surface was not continuous. Combinations of light path modification methods are also reported. In these methods, reflection and refraction are used together to control the flux absorbed by the absorber but the distribution of flux is not handled. Some detecting mechanism is also used to change the position of the receiver with respect to flux being focused. Some structures with a secondary reflector of parabolic shape is are reported where the light reflected from first reflector is placed at the focus of the second but small parabola. The diffused light is just scaled up or down but the way, it will be distributed, is not reported. Moving the focal point along the focal axis of parabola is also reported. The reflecting surface was imaging and can change its focus along the focal axis with change of sun's position. The use of cavity tubular receiver is also reported. The solar radiation at the entrance of tubular cavity is made to fall at certain angle so that the light keeps on reflecting in the cavity until it is absorbed. The path of the rays was not studied which does not allow one to understand the flux distribution along the length of absorber. A combination of Fresnel shaped reflectors was also used to pick a certain part of light wavefront and to reflect it separately. The achievement of uniform flux distribution was achieved by using the secondary collector. It can be seen from a literature survey that most of the present collectors do not handle the flux distribution. Some attempts are made to obtain some specific profiles of flux distribution by changing the shape of absorbers. There is a need to develop a collector that can give flux distribution in required pattern over a simple tubular absorbing surface.
• Thus, a solar collector with optimal profile for energy distribution on a tubular receiver solving the aforementioned problems is desired.
SUMMARY OF THE INVENTION
• A solar collector with optimal profile for energy distribution on a tubular receiver collects and distributes solar energy. Solar energy received from the sun can be modified by either being re-directed (reflection) or being redistributed. In the present invention energy is reflected and redistributed in a manner that yields a required energy variation over a surface. The receiver is a cylinder of known length and diameter. Longitudinal distribution of energy is specified by a user defined function. Circumferential distribution is assumed to be constant. Energy distribution is required to vary along the z axis of the receiver but remain constant in the circumferential direction. An axi-symmetric approach is used in which only one plane of the receiver in r and Z plane is considered. A geometric solution determines a reflecting surface that gives a required energy distribution along the z-axis. A complete reflector is designed by expanding the axi-symmetric behavior. The developed collector will concentrate as well as distribute the energy falling on its aperture in a user defined pattern. A specific most commonly used tubular heat transferring conduit is taken as absorber. The reflecting surface of the collector is designed in such a way that it produces a diffused focus of the range of the length of the tubular receiver. Inputs to an algorithm include the local solar radiation flux, initial radius of the reflector and a function of flux distribution in form of longitudinal dimension of the absorber. Assumptions made for the design method include (1) the collector is tracking the sun with 100% accuracy as nowadays photovoltaic based trackers are available that can track the sun with high accuracy; (2) solar flux incident on the aperture surface is taken to be constant (no variation in space); (3) steady state case is taken, meaning that the system takes no time to respond to the variation of solar flux (variation with respect to time); (4) solar radiations are parallel; (5) reflection is specular; and (6) absorbity of the reflecting surface is neglected but a constant value independent of the incident angle can be taken into account so that it can only act as a reducing factor without affecting the distribution. The complete reflector is formed from a parametric curve determined by the algorithm. The parametric curve is a single, continuous curve and the reflector formed therefrom has only one smooth reflecting surface.
• These and other features of the present invention will become readily apparent upon further review of the following specification and drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
• FIG. 1 is a perspective view of the solar collector according to the present invention.
• FIG. 2 is a flow chart of the manufacturing process of the solar collector according to the present invention.
• FIG. 3 is a diagram showing the relationship of local and global co-ordinates of the solar collector according to the present invention.
• FIG. 4 is a parametric curve showing the optimum design shape of the solar collector according to the present invention.
• FIG. 5 is a top plan view of the solar collector according to the present invention.
• FIG. 6 is a side view of the solar collector according to the present invention.
• FIG. 7 is a diagrammatic view of the cylindrical receiver of the solar collector according to the present invention.
• FIG. 8 is a plot comparing required and achieved flux according to the present invention.
• FIG. 9 is a plot showing the reflecting surfaces with various values or r_i according to the present invention.
• FIG. 10 is a plot showing surface area of reflecting surface for various values of r_i according to the present invention.
• FIG. 11 is a comparison plot showing achieved fluxes for b=500 and b=1000 according to the present invention.
• FIG. 12 is a plot showing energy distribution along the length of the receiver according to the present invention.
• FIG. 13 is a plot showing variation of incident angle of radiation along the length of the receiver according to the present invention.
• FIG. 14 is a plot showing variation of length of incident radiation along the length of the receiver according to the present invention.
• FIG. 15 is a plot showing final reflecting surface of the collector according to the present invention.
• FIG. 16 is a plot showing energy distribution for dish profile for constant temperatures according to the present invention.
• FIG. 17 is a plot showing incident angle variation along the length of the receiver for constant temperatures dish profile according to the present invention.
• FIG. 18 is a plot showing incident radiation length variation along the length of the receiver for constant temperatures dish profile according to the present invention.
• FIG. 19 is a plot showing final reflecting surface of collector for constant temperatures dish profile according to the present invention.
• Similar reference characters denote corresponding features consistently throughout the attached drawings.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
• The solar collector with optimal profile, for energy distribution on a tubular receiver collects and distributes solar energy. Solar energy received from the sun can be modified by either being re-directed (reflection) or being redistributed. In the present invention energy is reflected and redistributed in a manner that yields a required energy variation over a surface. The receiver is a cylinder of known length and diameter. Longitudinal distribution of energy is specified by a user defined function. Circumferential distribution is assumed to be constant. As shown in FIG. 1 the reflector is a smooth surfaced reflector 100 having an aperture through which the cylindrical tubular receiver 102 is disposed, extending outward therefrom. FIG. 5 is a top plan view showing the diameter 504 of tube 102 and the width 502 of the reflector 100. FIG. 6 is a side view that shows a length metric D of the reflector 100 and a length metric A of the tube 102. FIG. 7 shows the cylindrical shape of the receiver tube 102. The reflector 100 and tube 102 form a 3-D concentrating non-imaging collector. The aim of this collector is to concentrate as well as distributed the energy falling on its aperture in a user defined pattern. The reflecting surface of the collector is designed in such a way that it produces a diffused focus of the range of the length of the tubular receiver. The inputs from user are maximum energy required at one of the end of the receiver, local solar radiation flux, initial radius of the collector and a function in form of longitudinal dimension of the receiver. Moreover, the present invention contemplates that for a same function of energy distribution and for the same value of initial radius of the collector, the final shape of collector remains the same. It only stretches or contracts depending on the E_max/B (E_max is the maximum energy required at one end of the receiver, B is the local solar radiation flux). For developing this collector some general most common assumptions are made. The energy distribution is required to vary along the z axis of the receiver but it should be constant in circumferential direction. The simplest way of achieving this is to use an axi-symmetric approach in which we will consider only one plane of the receiver in r and Z plane and look for a reflecting surface that can give a required energy distribution along the z-axis then by expanding the axi-symmetric behavior we will get a complete reflector.
• Assumptions include the fact that the solar flux is constant and the maximum energy that is required by the user at the end of the receiver is also constant so the r and θ axis of the local co-ordinates are made separately the function of x which is the co-ordinate axis in which user will define the variation of energy. The variation of energy in form of x is split into r and θ axis and then this variation is transformed to the global axis to get the final curve.
• With respect to the mathematical formulation, Let's consider a tubular receiver of length L and outer radius r_ab. Let the solar flux falling on the aperture of collector is B. After reflection from desired solar surface, solar radiations shall fall on the reformer. Let the length of reflected ray, which falls on the reformer at the distance x, is r and it becomes r+dr at the length of x+dx of reformer as shown in plot 300 of FIG. 3. In this case, the energy falling on the small length of reformer ‘dx’ is
• 
  E _dx =πB((dr+r)² sin²(θ+dθ)−r ² sin²(θ))  (1)
• Let a specific profile of heat flux ‘B_x’ be required over the surface of reformer then the energy absorbed by the small length of reformer ‘dx’ will be,
• 
  E _dx=2πr _ab B _x dx  (2)
• Equating the Eq. (1) and Eq. (2) then simplifying, Eq. (3) is obtained as follows,
• $\begin{array}{cc}{r}_{\mathrm{ab}}B_x={\mathrm{<figure-callout\; id="2"\; label="Br"\; filenames="US20150285534A1-20151008-D00000.png,US20150285534A1-20151008-D00004.png"\; state="\{\{state\}\}">Br</figure-callout>}}^2\sin \left(\theta \right)\cos \left(\theta \right)\frac{\theta }{x}+\mathrm{Br}\frac{r}{x}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="US20150285534A1-20151008-D00000.png,US20150285534A1-20151008-D00004.png"\; state="\{\{state\}\}">sin</figure-callout>}}^2\left(\theta \right)& \left(3\right)\end{array}$
• To develop the desired solar collector, two coordinate systems are used which are global co-ordinates xx-yy axes and local co-ordinates that are (r-θ axes) moving along the length of reformer, where xx is the longitudinal direction of the collector and yy is the transverse direction of the collector. The equations of transformation between these co-ordinates systems are as follows,
• 
  xx=x−r cos(θ)  (4)
• 
  yy=r sin(θ)  (5)
• Now differentiating equation (4) and (5) with respect to r and then dividing to get dyy/dxx,
• $\begin{array}{cc}\frac{\mathrm{xx}}{r}=\frac{x}{r}-\cos \left(\theta \right)+r\sin \left(\theta \right)\frac{\theta }{r}\frac{\mathrm{yy}}{r}=\sin \left(\theta \right)+r\cos \left(\theta \right)\frac{\theta }{r}\frac{\mathrm{yy}}{\mathrm{xx}}=\frac{\sin \left(\theta \right)+r\cos \left(\theta \right)\frac{\theta }{r}}{\frac{x}{r}-\cos \left(\theta \right)+r\sin \left(\theta \right)\frac{\theta }{r}}& \left(6\right)\end{array}$
• Now from solar cone, we know that,
• $\begin{array}{cc}\frac{\mathrm{yy}}{\mathrm{xx}}=\tan \left(90-\theta /2\right)\frac{\mathrm{yy}}{\mathrm{xx}}=\cot \left(\theta /2\right)& \left(7\right)\end{array}$
• Equating equation (3) and (5), we get,
• $\begin{array}{cc}\frac{\sin \left(\theta \right)+r\cos \left(\theta \right)\frac{\theta }{r}}{\frac{x}{r}-\cos \left(\theta \right)+r\sin \left(\theta \right)\frac{\theta }{r}}=\cot \left(\theta /2\right)& \left(8\right)\end{array}$
• Thus Eq. (3) and Eq. (8) define the system and are solved to get final reflecting surface.
• In above analysis, r and θ axes of the local co-ordinates are made separately the function of x axis. User will define the required flux distribution in x axis. This distribution of flux is split into r and θ axes and then transformed to the global co-ordinates to get the final curve.
• Local solar flux density depends on the geographical location and weather. It should be selected accordingly to get accurate results. The polynomial of flux distribution and length of the receiver depend solely on the demand of user. The present algorithm can be used for heating the absorber of any length. Similarly, any distribution of required flux can be achieved. The value of initial radius for reflecting surface decides the space occupied by collector.
• The algorithm leaves a hole at the apex of the collector. It is due to the fact that when the point of intersection of solar radiations with the reflecting surface become in line with absorber's surface, no energy is absorbed. Thus, this hole can be left open as an air vent for cleaning purposes. It can also be closed to give strength to the structure.
• Numerical control method 200 utilizing the aforementioned system of equations to manufacture the tubular surface is shown in FIG. 2. The present method 200 accumulates polynomial energy distribution requirement inputs 202, local solar flux density input 204, initial radius of the reflecting surface input 206, and length of the reactor input 208. These inputs are then fed to a computer aided designing algorithm at step 210 wherein, inter alia, Eq. (8) is utilized to produce a set of reflecting surface profile data points at step 212 which are fed to a computer aided numerically controlled manufacturing system at step 214 which produces the desired reflecting surface at step 216.
• With respect to optimal heat profile for an exemplary SMR case, it is known in the art that the optimal heat flux profile for steam methane reforming (SMR) reaction can be approximated by a third order polynomial as,
• 
  B _x=0.1281x ³−0.871x ²+2.806x+49.7  (9)
• This flux distribution is used to develop a solar collector with user defined inputs as follows:
  Radius of tubular absorber, r_ab=0.00865 m,
  Length of tubular absorber, L=12 m,
  Initial radius, r_i=0.04 m,
  Solar flux, B=1 KW/m². The tubular absorber radius, r_ab, and B and r_i are the constants used in Eq. (3). L is used as upper boundary condition for x. By solving the Eq. (3) and Eq. (8) numerically, the following reflecting surface is obtained. The two dimensional axi-symmetric curve for developed solar collector is shown as plot 400 in FIG. 4 which represents the final curve of reflecting surface for optimal heat flux profile for SMR. Front and partially exploded, isometric views of the final collector are shown in FIGS. 5 and 1, respectively. FIG. 6 shows the side view of the developed solar collector 100.
• A comparison is made between required and achieved flux at the absorber surface. For the present example, required flux is the one of Eq. (9). Achieved flux is the flux transferred by the collector at the absorber surface and is calculated as follows,
• 
  πB(y ₂ ² −y ₁ ²)=2πr _ab B _x(x ₂ −x ₁)  (10)
• B_x is the achieved flux over the small length (x₂−x₁) of the absorber. Plot 800 of FIG. 8 shows the required and achieved flux over the surface of absorber.
• It can be seen that a great agreement is found between the required and achieved flux. To quantize the agreement achieved, a correlation factor is calculated between achieved and required flux. This correlation factor is a measure of fit of a straight line between required and achieved flux and is given as a correlation factor defined by the relation,
• $\begin{array}{cc}\mathrm{Correlation}\mathrm{factor}=\frac{\sum \left(x-\stackrel{\_}{x}\right)\left(y-\stackrel{\_}{y}\right)}{\sqrt{\sum {\left(x-\stackrel{\_}{x}\right)}^2\sum {\left(y-\stackrel{\_}{y}\right)}^2}}& \left(11\right)\end{array}$
• In above relation, x and y are the required and achieved flux respectively. A value of 1 indicates the perfect agreement between two series being compared. For the present comparison, a correlation factor of 0.999998 is achieved. Apart from achieving the required flux distribution, present invention is also capable of achieving more than one reflecting surfaces of different sizes and shapes. This is achieved by introducing a free variable r_i. By changing the value of r_i, reflecting surface of various shapes and sizes can be obtained. The exemplary case is solved for r_i values of 0.1, 0.5, 1 and 2 m. Plot 900 of FIG. 9 shows the various shapes of the reflecting surface that achieve the same distribution of flux. It can also be seen that as the value of r_i changes, the surface area of reflecting surfaces changes. Plot 1000 of FIG. 10 shows the variation of surface areas of reflecting surfaces with various values of r_i. Thus, by changing the value of r_i reflecting surface of various sizes and shapes can be achieved to satisfy the same flux distribution.
• Moreover, the present invention can secure the flux distribution even if the incoming flux changes. This property can be ensured by analyzing Eq. (3). Suppose a solar collector is constructed according to the inputs of the aforementioned exemplary case. Once the collector is constructed, all of its geometrical properties become constant and cannot be changed. Thus, the value of r_i,r_ab and variation of r and t with respect to x becomes constant. The only thing that can change is the incoming solar flux and the flux being achieved over the surface of absorber. Analyzing the Eq. (3) and Eq. (8), it can be seen that Eq. (8) is purely a geometric relationship that does not change once the collector is made. Thus, any change in the incoming solar flux is going to disturb Eq. (3) directly. Suppose incoming solar flux decreases to half of its value in the aforementioned exemplary case. This decrease is going to multiply the right hand side (R.H.S.) of Eq. (3) by 0.5 and this equation will not satisfy the collector obtained in the aforementioned exemplary case anymore. For this equation to satisfy the developed collector, left hand side (L.H.S.) must be multiplied by 0.5 as well. This multiplication on L.H.S. affects B_x directly as r_ab is constant. This multiplication of 0.5 with the B_x does not change the distribution at all but it will just decrease the achieved flux at each point by 0.5 securing the distribution. Thus, it is concluded that any change in the incoming solar radiation does not change the flux distribution. A case is solved by using inputs of the aforementioned exemplary case with incoming flux of 500 W/m². The required flux distribution is 0.5B_x. The reflecting surface obtained is exactly the one achieved for the exemplary parametric curve 400 shown in FIG. 4. Parametric curve 400 is a single, continuous parametric curve, the reflector being formed therefrom and having only one smooth reflecting surface. Comparison of non-dimensional achieved fluxes of this case and the aforementioned exemplary case is shown in plot 1100 of FIG. 11. It can be seen that flux distribution is completely secured.
• An exemplary dish profile for quadratic energy distribution where B/E_max=10, L=0.1 m and initial value of r_i is taken to be 3 m is shown in FIGS. 12, 13, 14 and 15. Plot 1200 of FIG. 12 shows the non-dimensional energy distribution along the receiver's length. Plot 1300 of FIG. 13 shows the variation of incident angle of radiation along the length of the receiver. Plot 1400 of FIG. 14 shows the variation of length of incident radiation along the length of the receiver. Plot 1500 of FIG. 15 shows the final reflecting surface of the collector where a horizontal lineal distance from an apex of the solar collector to an outer edge of the solar collector is approximately 6.7 meters to facilitate a quadratic energy distribution along the tubular receiver.
• With respect to an exemplary dish profile for constant temperature, as the solar flux falls on the surface of the receiver, the temperature achieved depends on many factors. Considering the simple one dimensional energy balance over a tubular receiver, the dependence of achieved temperature can be restricted to the mass flow rate of the heat transferring (HT) fluid, convective heat transfer co-efficient of the HT surface, radius of the receiver, heat capacity of the HT fluid, initial fluid temperature and constant value of required wall temperature. The values of these parameters taken to get the energy profile for constant temperature are as follows:
• Mass flow rate, m=0.001 Kg/s
• Convective heat transfer co-efficient, h=2 KW/m² C
• Radius of receiver=0.0475 m
• Heat capacity of HT fluid=4200 KJ/Kg C
• Initial temperature of HT fluid=30° C.
• Required wall temperature=400° C.
• B/E_max=10, L=0.1 m and initial value of r_i is taken to be 3 m.
• FIGS. 16 through 19 ( plots 1600, 1700, 1800, and 1900, respectively) illustrate the non-dimensional energy distribution along the length of the receiver, the variation of incident angle of radiation along the length of receiver, the variation of length of incident radiation along the length of receiver, and the final reflecting surface of the collector, respectively. In particular, plot 1900 of FIG. 19 shows the final reflecting surface of the collector where a horizontal lineal distance from an apex of the solar collector to an outer edge of the solar collector is approximately 0.043 meters to facilitate a constant energy distribution along the tubular receiver.
• As described earlier, we have just split the variation of energy so for constant value of initial radius ‘r_i’, the ratio of E_max/B will be the only changing parameter. Hence any change in this ratio, as it can be according to the needs of the user, will only stretch or contract the reflecting curve keeping the relationship between reflecting surface and receiver same. So for constant initial value of r_i, the shape of the reflecting surface will remain same and this will be true for every user defined function of energy variation.
• The present invention improves fulfillment of the need for a solar collector that can give a required flux distribution. An algorithm is developed that gives the data points for designing the reflecting surface. These data points can be used as input to numerically controlled manufacturing mechanism. This algorithm needs simple inputs such as local solar flux density, required profile of heat flux, initial radius of reflecting surface and length of absorber. Algorithm can develop various reflecting surfaces for achieving the same flux distribution allowing the user to select the best collector depending upon his needs. Developed collector also ensures the distribution even if the incoming solar flux changes. The absorbing surface is of standard tubular type. This algorithm works for all values of inputs. A representative example for optimal heat flux profile for SMR is solved. The data points for a reflecting surface are given.
• It is to be understood that the present invention is not limited to the embodiments described above, but encompasses any and all embodiments within the scope of the following claims.

Claims (10)

We claim:

1. A solar collector with optimal profile for energy distribution on a tubular receiver, comprising:

a reflector formed by a single, continuous parametric curve, the reflector having only one smooth reflecting surface;

a receiver through which heat carrying fluid circulates; and

wherein the receiver comprises at least one tube disposed along a focal axis defined by an aperture of the reflector to receive reflected solar energy directed by the reflector along the focal axis.

2. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein the single, continuous parametric curve of the reflector has a contour that conforms to a calculated solution using a system of equations characterized by the relations,

$r_{\mathrm{ab}}B_x={\mathrm{Br}}^2\sin \left(\theta \right)\cos \left(\theta \right)\frac{\theta }{x}+\mathrm{Br}\frac{r}{x}{\sin }^2\left(\theta \right),\mathrm{and}$ $\cot \left(\theta /2\right)=\frac{\sin \left(\theta \right)+r\cos \left(\theta \right)\frac{\theta }{r}}{\frac{x}{r}-\cos \left(\theta \right)+r\sin \left(\theta \right)\frac{\theta }{r}},$

where B is the solar flux falling on the aperture of the reflector, r_ab is the outer radius of a tubular receiver having length L, B_x is a specific profile of heat flux required over the surface of the tubular receiver, dx is a small length portion of the tubular receiver, r is the length of reflected solar ray, which falls on the tubular receiver at a distance x along the tubular receiver, becoming r+dr at a length of x+dx.

3. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein the tubular receiver is substantially cylindrical in shape.

4. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein the solar collector has an opening at its apex.

5. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein the heat carrying fluid is methane undergoing a steam methane reforming (SMR) reaction.

6. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein a horizontal lineal distance from an apex of the solar collector to an outer edge of the solar collector is approximately 6.7 meters to facilitate a quadratic energy distribution along the tubular receiver.

7. The solar collector with optimal profile for energy distribution on a tubular receiver according to claim 1, wherein a horizontal lineal distance from an apex of the solar collector to an outer edge of the solar collector is approximately 0.043 meters to facilitate a constant energy distribution along the tubular receiver.

8. A method for manufacturing a solar collector, the method comprising the steps of:

(a) accumulating polynomial energy distribution requirement inputs, local solar flux density input, initial radius of the reflecting surface input, and length of the reactor input;

(b) feeding the inputs of step (a) to a computer executing a procedure that calculates a solution of a system of equations characterized by the relations,

$r_{\mathrm{ab}}B_x={\mathrm{Br}}^2\sin \left(\theta \right)\cos \left(\theta \right)\frac{\theta }{x}+\mathrm{Br}\frac{r}{x}{\sin }^2\left(\theta \right),\mathrm{and}$ $\frac{\sin \left(\theta \right)+r\cos \left(\theta \right)\frac{\theta }{r}}{\frac{x}{r}-\cos \left(\theta \right)+r\sin \left(\theta \right)\frac{\theta }{r}}=\cot \left(\theta /2\right),$

where B is the solar flux falling on the aperture of the reflector, r_ab is the outer radius of a tubular receiver having length L, B_x is a specific profile of heat flux required over the surface of the tubular receiver, dx is a small length portion of the tubular receiver, r is the length of reflected solar ray, which falls on the tubular receiver at a distance x along the tubular receiver, becoming r+dr at a length of x+dx wherein said system of equations produces a set of reflecting surface profile data points; and

(c) feeding the set of reflecting surface profile data points to a computer aided numerically controlled manufacturing system, said computer aided numerically controlled manufacturing system producing a desired reflecting surface based on said inputs of step (a).

9. The solar collector manufacturing method according to claim 8, further comprising the step of determining B_x according to an optimal heat flux profile for a steam methane reforming (SMR) reaction wherein the SMR reaction is approximated by a third order polynomial characterized by the relation,

B _x=0.1281x ³−0.871x ²+2.806x+49.7.

10. The solar collector manufacturing method according to claim 9, wherein the input step (a) further comprises:

inputting radius of tubular absorber, r_ab=0.00865 m;

inputting length of tubular absorber, L=12 m;

inputting initial radius r_i=0.04 m; and

inputting solar flux, B=1 KW/m².

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