
The developmental tendency of parabolic trough collector (PTC) is larger aperture area for energy harvest and novel optical design for higher solar concentration. Larger aperture faces a higher demand in tracking accuracy and lower tolerances with respect to wind loads, quality of mirrors, control, and mounting imprecisions. With the increase of reflection angle, the divergence size of concentrated focal spot gets enlarged on receiving surface, forming a Gaussian distribution. Receiving more energy and making best use of the concentrated solar power have become a key problem. In current study, a novel trough free-form solar concentrator (TFFC) has been developed by extending the aperture size with the aid of freeform optics and combined PV/thermal utilization. The structure model is composed by traditional parabolic trough with thermal tube, and extended freeform reflector with solar panel in slope configuration. The free-form surface is generated by geometric construction method for the sake of uniform heat flux distribution. The optical characteristics are validated by ray tracing method. The advantages will be revealed by compared with traditional system. The sensitivity analysis and error factors would be discussed as well. The initial results are promising and significant for the enhancement of trough type solar concentrator systems.
Citation: Xian-long Meng, Cun-liang Liu, Xiao-hui Bai, De-hai Kong, Kun Du. Improvement of the performance of parabolic trough solar concentrator using freeform optics and CPV/T design[J]. AIMS Energy, 2021, 9(2): 286-304. doi: 10.3934/energy.2021015
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The developmental tendency of parabolic trough collector (PTC) is larger aperture area for energy harvest and novel optical design for higher solar concentration. Larger aperture faces a higher demand in tracking accuracy and lower tolerances with respect to wind loads, quality of mirrors, control, and mounting imprecisions. With the increase of reflection angle, the divergence size of concentrated focal spot gets enlarged on receiving surface, forming a Gaussian distribution. Receiving more energy and making best use of the concentrated solar power have become a key problem. In current study, a novel trough free-form solar concentrator (TFFC) has been developed by extending the aperture size with the aid of freeform optics and combined PV/thermal utilization. The structure model is composed by traditional parabolic trough with thermal tube, and extended freeform reflector with solar panel in slope configuration. The free-form surface is generated by geometric construction method for the sake of uniform heat flux distribution. The optical characteristics are validated by ray tracing method. The advantages will be revealed by compared with traditional system. The sensitivity analysis and error factors would be discussed as well. The initial results are promising and significant for the enhancement of trough type solar concentrator systems.
Nonlinear equations arise always in electroanalytical chemistry with complex and esoteric nonlinear terms[1,2], though there are some advanced analytical methods to deal with nonlinear problems, for examples, the Gamma function method[3], Fourier spectral method[4], the reproducing kernel method[5], the perturbation method[6], the homotopy perturbation method[7,8], He's frequency formulation[9,10,11] and the dimensional method[12], chemists are always eager to have a simple one step method for nonlinear equations. This paper introduces an ancient Chinese algorithm called as the Ying Buzu algorithm[13] to solve nonlinear differential equations.
We first introduce the Taylor series method[14]. Considering the nonlinear differential equation:
d2udx2+F(u)=0. | (0.1) |
The boundary conditions are
dudx(a)=α, | (0.2) |
u(b)=β. | (0.3) |
If u(a) is known, we can use an infinite Taylor series to express the exact solution[14]. We assume that
u(a)=c. | (0.4) |
From (0.1), we have
u″(a)=−F(u(a))=−F(c), |
u‴(a)=−∂F(c)∂uu′(a)=−α∂F(c)∂u. |
Other higher order derivatives can be obtained with ease, and its Taylor series solution is
u(x)=u(a)+(x−a)u′(a)+12!(x−a)2u″(a)+13!(x−a)3u‴(a)+...+1N!(x−a)Nu(N)(a), |
the constant c can be determined by the boundary condition of (0.3).
The Ying Buzu algorithm[15,16] was used to solve differential equations in 2006[13], it was further developed to He's frequency formulation for nonlinear oscillators[13,17,18,19,20,21,22,23] and Chun-Hui He's algorithm for numerical simulation[24].
As c in (0.4) is unknown, according to the Ying Buzu algorithm[13,15,16], we can assume two initial guesses:
u1(a)=c1,u2(a)=c2. | (0.5) |
where c1 and c2 are given approximate values.
Using the initial conditions given in (0.2) and (0.5), we can obtain the terminal values:
u(b,c1)=β1,u(b,c2)=β2. |
According to the Ying Buzu algorithm[6,7,8,9,10,11,12], the initial guess can be updated as
u(a)est=c3=c1(β−β2)−c2(β−β1)(β−β2)−(β−β1), |
and its terminal value can be calculated as
u(b,c3)=β3. |
For a given small threshold, ε, |β−β3|≤ε, we obtain u(a)=c3 as an approximate solution.
Here, we take Michaelis Menten dynamics as an example to solve the equation. Michaelis Menten reaction diffusion equation is considered as follows[25,26]:
d2udx2−u1+u=0. | (0.6) |
The boundary conditions of it are as follows:
dudx(0)=0,u(1)=1. | (0.7) |
We assume
u(0)=c. |
From (0.6), we have
u″(0)=c1+c, |
u‴(0)=0, | (0.8) |
u(4)=c(1+c)3. |
The 2nd order Taylor series solution is
u(x)=u(0)+u′(0)1!x+u″(0)2!x2=c+c2(1+c)x2. |
In view of the boundary condition of (0.7), we have
u(1)=c+c2(1+c)=1, | (0.9) |
solving c from (0.9) results in
c=0.7808. |
So we obtain the following approximate solution
u(x)=0.7808+0.2192x2. |
Similarly the fourth order Taylor series solution is
u(x)=c+c2!(1+c)x2+c4!(1+c)3x4. |
Incorporating the boundary condition, u(1)=1, we have
c+c2!(1+c)+c4!(1+c)3=1. | (0.10) |
We use the Ying Buzu algorithm to solve c, and write (0.10) in the form
R(c)=c+c2(1+c)+c24(1+c)3−1. |
Assume the two initial solutions are
c1=0.8,c2=0.5. |
We obtain the following residuals
R1(0.8)=0.0279,R2(0.5)=−0.3271. |
By the Ying Buzu algorithm, c can be calculated as
c=R2c1−R1c2R2−R1=0.0279×0.5+0.3271×0.80.0279+0.3271=0.7764. |
The exact solution of (0.10) is
c=0.7758. |
The 4th order Taylor series solution is
u(x)=0.7758+0.2192x2+0.0057x4. |
Figure 1 shows the Taylor series solutions, which approximately meet the requirement of the boundary condition at x=1.
Now we use the Ying Buzu algorithm by choosing two initial guesses
u1(0)=0.5,u2(0)=1, |
which lead to u1=0.6726 and u2=1.2550, respectively, see Figure 2 (a) and (b).
It is obvious that the terminal value at x=1 deviates from u(1)=1 for each guess, according to the Ying Buzu algorithm, the initial guess can be updated as
u3(0)=0.5×(1−1.2550)−1×(1−0.6726)(1−1.2550)−(1−0.6726)=0.7810. | (0.11) |
The shooting process using (0.11) results in
u3(1)=1.0058, |
which deviates the exact value of u(1)=1 with a relative error of 0.5%, see Figure 3.
We can continue the iteration process to obtain a higher accuracy by using two following two guesses u1(0)=0.5, u3(0)=0.7810:
u4(0)=0.5×(1−1.0058)−0.7810×(1−0.6726)(1−1.0058)−(1−0.6726)=0.7761. |
Using this updated initial value, the shooting process leads to the result
u(1)=1.0001, |
so the approximate u(0)=0.7761 has only a relative error of 0.01%.
The above solution process couples the numerical method, and the ancient method can also be solved independently.
We assume that solution is
u(x)=c+(1−c)x2. | (0.12) |
Equation (0.12) meets all boundary conditions.
The residual equation is
R(x)=d2udx2−u1+u. |
It is easy to find that
R(0)=2(1−c)−c1+c. |
We choose two guesses:
c1=0.5,c2=1. |
We obtain the following residuals
R1(0)=2(1−0.5)−0.51+0.5=23, |
R2(0)=2(1−1)−11+1=−12. |
The Ying Buzu algorithm leads to the updated result:
c=c2R1(0)−c1R2(0)R1(0)−R2(0)=23×1+12×0.523+12=0.7857. |
The relative error is 1.2%, and the process can continue if a higher accuracy is still needed.
The ancient Chinese algorithm provides a simple and straightforward tool to two-point boundary value problems arising in chemistry, and it can be used for fast insight into the solution property of a complex problem.
The authors declare that they have no conflicts of interest to this work.
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