Constructal Design of T-Shaped Water Distribution Networks

Abstract

More than one billion people do not have access to clean drinking water both in urban and rural areas in the world. This is why, according to a number of scientists involved in drinking water engineering, hundreds of kilometers of networks for water distribution will be constructed or rehabilitated in the coming decades because of both population growth and the crucial need of water services. Therefore the need to optimally design these systems will be increasing everywhere (North and South) and will keep on increasing, in particular in developing countries, very poorly equipped and in need to make strong efforts for the construction of such systems. According to literature, the three main design constraints that are generally met in water distribution projects are water quality, pumping energy, and investment cost [1–4]. In most of the cases, the design problem formulates as follows: “a certain population of density ${\sigma_{\rm{p}}}$ grouped in households, distributed over a given area, needs to be supplied in drinking water through a distribution network. A lot of technically acceptable solutions can be found to this problem. Some methods differ from the others in terms of total head losses, overall residence time, initial or total investment due to the design” [5]. The challenge for engineers relies on how to optimally design the network in terms of pumping energy and the evolution of water quality in the network.

Appendices

Appendix: First-Order Construct

We consider the network shown in Fig. 7.1. The Lagrangian function written to minimize the head losses subject to the residence time and the total surface writes as follows:

$$ {L_{\rm{g}}} = a\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{H}{{{2^1}}} + a\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{h}{{{2^2}}} + {\lambda_1}\left( {\frac{{\pi D_{{{H_1}}}^2}}{{4{Q_{{{H_1}}}}}}\frac{H}{{{2^1}}} + \frac{{\pi D_{{{h_1}}}^2}}{{4{Q_{{{h_1}}}}}}\frac{h}{{{2^2}}} - t} \right) + {\lambda_2}\left( {hH - {S_T}} \right). $$

(7.1)

By differentiating L _g with respect to ${D_{{{h_1}}}}$, ${D_{{{H_1}}}}$, h, H, ${\lambda_1}$, and ${\lambda_2}$, and cancelling all the derivatives, we obtain

$$ - am\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^{{m + 1}}}}\frac{H}{{{2^1}}} + 2\frac{\pi }{4}{\lambda_1}\frac{{{D_{{{H_1}}}}}}{{{Q_{{{H_1}}}}}}\frac{H}{{{2^1}}} = 0, $$

(7.2a)

$$ - am\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^{{m + 1}}}}\frac{h}{{{2^2}}} + 2\frac{\pi }{4}{\lambda_1}\frac{{{D_{{{h_1}}}}}}{{{Q_{{{h_1}}}}}}\frac{h}{{{2^2}}} = 0, $$

(7.2b)

$$ a\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} + {\lambda_1}\frac{{\pi D_{{{H_1}}}^2}}{{4{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} + {\lambda_2}h = 0, $$

(7.2c)

$$ a\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} + {\lambda_1}\frac{{\pi D_{{{h_1}}}^2}}{{4{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} + {\lambda_2}H = 0, $$

(7.2d)

$$ \frac{{D_{{{H_1}}}^2}}{{{Q_{{{H_1}}}}}}\frac{H}{{{2^1}}} + \frac{{D_{{{h_1}}}^2}}{{{Q_{{{h_1}}}}}}\frac{h}{{{2^2}}} = \frac{{4t}}{\pi }, $$

(7.2e)

$$ \frac{{\partial {L_{\rm{g}}}}}{{\partial {\lambda_2}}} = 0 \Rightarrow hH - {S_{\rm{T}}} = 0. $$

(7.2f)

From (7.2a) and (7.2b), we can write

$$ \frac{{am\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^{{m + 1}}}}\frac{H}{{{2^1}}}}}{{am\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^{{m + 1}}}}\frac{h}{{{2^2}}}}} = \frac{{2\frac{\pi }{4}{\lambda_1}\frac{{{D_{{{H_1}}}}}}{{{Q_{{{H_1}}}}}}\frac{H}{{{2^1}}}}}{{2\frac{\pi }{4}{\lambda_1}\frac{{{D_{{{h_1}}}}}}{{{Q_{{{h_1}}}}}}\frac{h}{{{2^2}}}}}. $$

(7.3)

This yields in

$$ \frac{{D_{{{h_1}}}^{{m + 2}}}}{{D_{{{H_1}}}^{{m + 2}}}} = \frac{{Q_{{{h_1}}}^{{n + 1}}}}{{Q_{{{H_1}}}^{{n + 1}}}}. $$

(7.4)

And, noting that because of the symmetry in the structure $2{Q_{{{h_1}}}} = {Q_{{{H_1}}}}$, then we have

$$ \frac{{{D_H}_{{_1}}}}{{{D_h}_{{_1}}}} = {2^{\gamma }}\;{\text{with}}\quad \gamma = (n + 1)/(m + 2). $$

(7.5)

From (7.2a) and (7.2b), ${\lambda_1}$ can be expressed as follows:

$$ {\lambda_1} = \frac{{2am}}{\pi }\frac{{Q_{{{H_1}}}^{{n + 1}}}}{{D_{{{H_1}}}^{{m + 2}}}} = \frac{{2am}}{\pi }\frac{{Q_{{{h_1}}}^{{n + 1}}}}{{D_{{{h_1}}}^{{m + 2}}}}. $$

(7.6)

By introducing ${\lambda_1}$ in (7.2c) and in (7.2d) we obtain

$$ a\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} + \left( {\frac{{2am}}{\pi }\frac{{Q_{{{H_1}}}^{{n + 1}}}}{{D_{{{H_1}}}^{{m + 2}}}}} \right)\frac{{\pi D_{{{H_1}}}^2}}{{4{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} = - {\lambda_2}h, $$

(7.7)

$$ a\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} + \left( {\frac{{2am}}{\pi }\frac{{Q_{{{h_1}}}^{{n + 1}}}}{{D_{{{h_1}}}^{{m + 2}}}}} \right)\frac{{\pi D_{{{h_1}}}^2}}{{4{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} = - {\lambda_2}H. $$

(7.8)

This yields in

$$ \left( {a + \frac{{am}}{2}} \right)\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} = - {\lambda_2}h, $$

(7.9)

$$ \left( {a + \frac{{am}}{2}} \right)\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} = - {\lambda_2}H. $$

(7.10)

By dividing member by member (7.9) by (7.10), we obtain

$$ \frac{h}{H} = 2\frac{{Q_{{{H_1}}}^n}}{{Q_{{{h_1}}}^n}}\frac{{D_{{{h_1}}}^m}}{{D_{{{H_1}}}^m}}. $$

(7.11)

Considering (7.5) and the fact that 2Q _h1 = Q _H1 , (7.11) gives

$$ \frac{h}{H} = {2^{{n + 1 - m\gamma }}} = {f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}. $$

(7.12)

Considering (7.12) and (7.2f), we have

$$ \left\{ \begin{array}{lll} hH - {S_{\rm{T}}} = 0, \hfill \cr \frac{h}{H} = {2^{{n + 1 - m\gamma }}} ={f_{{{\rm{op}}{{\rm{t}}_1}}}}. \end{array} \right. $$

(7.13)

This allow writing, by eliminating h,

$$ {H^2} = \frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}}} \Rightarrow H = \sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_1}}}}} \right)}^{{ - 1}}}}}. $$

(7.14)

And then, with (7.12), we can determine h as follows:

$$ h = {f_{{{\rm{op}}{{\rm{t}}_1}}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_1}}}}} \right)}^{{ - 1}}}}}. $$

(7.15)

From (7.2e), we can write (by multiplying (7.2e) by the quantity Q _h1/D² _h1)

$$ \frac{{D_{{{H_1}}}^2}}{{D_{{{h_1}}}^2}}\frac{{{Q_{{{h_1}}}}}}{{{Q_{{{H_1}}}}}}\frac{H}{{{2^1}}} + \frac{h}{{{2^2}}} = \frac{{4t{Q_{{{h_1}}}}}}{{\pi D_{{{h_1}}}^2}}. $$

(7.16)

This yields, by considering (7.15) and (7.13), in

$$ {2^{{2\gamma - 2}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_1}}}}} \right)}^{{ - 1}}}}} + {2^{{ - 2}}}{f_{{{\rm{op}}{{\rm{t}}_1}}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_1}}}}} \right)}^{{ - 1}}}}} = \frac{{4t{Q_{{{h_1}}}}}}{{\pi D_{{{h_1}}}^2}}. $$

(7.17)

And then

$$ {D_{{{h_1}}}} = \sqrt {{\frac{{4t{Q_{{{h_1}}}}/\pi }}{{\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}} \right)}^{{ - 1}}}}} \left( {{2^{{2\gamma - 2}}} + {2^{{ - 2}}}{f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}} \right)}}}}; {D_{{{H_1}}}} = {2^{\gamma }}\sqrt {{\frac{{4t{Q_{{{h_1}}}}/\pi }}{{\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}} \right)}^{{ - 1}}}}} \left( {{2^{{2\gamma - 2}}} + {2^{{ - 2}}}{f_{{{\rm{op}}{{\rm{t}}_{{1}}}}}}} \right)}}}}. $$

(7.18)

Second-Order Construct

In the same manner as the previous case, the Lagrangian function writes as follows:

$$ \begin{array}{lll} {L_{\rm{g}}} = \left( {a\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} + a\frac{{Q_{{{H_2}}}^n}}{{D_{{{H_2}}}^m}}\frac{1}{{{2^2}}}} \right)H + \left( {a\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} + a\frac{{Q_{{{h_2}}}^n}}{{D_{{{h_2}}}^m}}\frac{1}{{{2^3}}}} \right)h + \hfill \cr {\lambda_1}\left( {\left( {\frac{{\pi D_{{{H_1}}}^2}}{{4{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} + \frac{{\pi D_{{{H_2}}}^2}}{{4{Q_{{{H_2}}}}}}\frac{1}{{{2^2}}}} \right)H + \left( {\frac{{\pi D_{{{h_1}}}^2}}{{4{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} + \frac{{\pi D_{{{h_2}}}^2}}{{4{Q_{{{h_2}}}}}}\frac{1}{{{2^3}}}} \right)h - t} \right) + {\lambda_2}(hH - {S_{\rm{T}}}). \end{array}$$

(7.19)

By differentiating L _g with respect to ${D_{{{H_1}}}}$, ${D_{{{H_2}}}}$, ${D_{{{h_1}}}}$, ${D_{{{h_2}}}}$, h, H, ${\lambda_1}$, and ${\lambda_2}$, and cancelling the derivatives, we obtain

$$ \left( {a\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} + a\frac{{Q_{{{H_2}}}^n}}{{D_{{{H_2}}}^m}}\frac{1}{{{2^2}}}} \right) + {\lambda_1}\left( {\frac{{\pi D_{{{H_1}}}^2}}{{4{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} + \frac{{\pi D_{{{H_2}}}^2}}{{4{Q_{{{H_2}}}}}}\frac{1}{{{2^2}}}} \right) = - {\lambda_2}h, $$

(7.20a)

$$ \left( {a\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} + a\frac{{Q_{{{h_2}}}^n}}{{D_{{{h_2}}}^m}}\frac{1}{{{2^3}}}} \right) + {\lambda_1}\left( {\frac{{\pi D_{{{h_1}}}^2}}{{4{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} + \frac{{\pi D_{{{h_2}}}^2}}{{4{Q_{{{h_2}}}}}}\frac{1}{{{2^3}}}} \right) = - {\lambda_2}H, $$

(7.20b)

$$ ma\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^{{m + 1}}}}\frac{1}{{{2^1}}} = 2{\lambda_1}\frac{{\pi {D_{{{H_1}}}}}}{{4{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}}, $$

(7.20c)

$$ ma\frac{{Q_{{{H_2}}}^n}}{{D_{{{H_2}}}^{{m + 1}}}}\frac{1}{{{2^2}}} = 2{\lambda_1}\frac{{\pi {D_{{{H_2}}}}}}{{4{Q_{{{H_2}}}}}}\frac{1}{{{2^2}}}, $$

(7.20d)

$$ ma\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^{{m + 1}}}}\frac{1}{{{2^2}}} = 2{\lambda_1}\frac{{\pi {D_{{{h_1}}}}}}{{4{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}}, $$

(7.20e)

$$ ma\frac{{Q_{{{h_2}}}^n}}{{D_{{{h_2}}}^{{m + 1}}}}\frac{1}{{{2^3}}} = 2{\lambda_1}\frac{{\pi {D_{{{h_2}}}}}}{{4{Q_{{{h_2}}}}}}\frac{1}{{{2^3}}}, $$

(7.20f)

$$ \left( {\frac{{D_{{{H_1}}}^2}}{{{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} + \frac{{D_{{{H_2}}}^2}}{{{Q_{{{H_2}}}}}}\frac{1}{{{2^2}}}} \right)H + \left( {\frac{{D_{{{h_1}}}^2}}{{{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} + \frac{{D_{{{h_2}}}^2}}{{{Q_{{{h_2}}}}}}\frac{1}{{{2^3}}}} \right)h = \frac{{4t}}{\pi }, $$

(7.20g)

$$ hH = {S_{\rm{T}}}. $$

(7.20h)

From (7.20c), (7.20d), (7.20e), and (7.20f), one can express ${\lambda_1}$ as follows:

$$ {\lambda_1} = \frac{{2ma}}{\pi }\frac{{Q_{{{H_i}}}^{{n + 1}}}}{{D_{{{H_i}}}^{{m + 2}}}} = \frac{{2ma}}{\pi }\frac{{Q_{{{h_i}}}^{{n + 1}}}}{{D_{{{h_i}}}^{{m + 2}}}}\quad i = 1,2. $$

(7.21)

By combining terms of (7.21), one can write (given that Q _H1/Q _h1 = 2, Q _H2/Q _h2 = 2, and Q _h1/Q _H2 = 2 because of the symmetry)

$$ \frac{{{D_{{{H_1}}}}}}{{{D_{{{h_1}}}}}} = \frac{{{D_{{{h_1}}}}}}{{{D_{{{H_2}}}}}} = \frac{{{D_{{{H_2}}}}}}{{{D_{{{h_2}}}}}} = {2^{\gamma }}. $$

(7.22)

By introducing these expressions of ${\lambda_1}$ defined in (7.21), (7.20a), and (7.20b), we obtain

$$ \left\{ \eqalign{ \left( {a + \frac{{ma}}{2}} \right)\left( {\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^2}}} + \frac{{Q_{{{h_2}}}^n}}{{D_{{{h_2}}}^m}}\frac{1}{{{2^3}}}} \right) = - {\lambda_2}H, \hfill \\\left( {a + \frac{{ma}}{2}} \right)\left( {\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}} + \frac{{Q_{{{H_2}}}^n}}{{D_{{{H_2}}}^m}}\frac{1}{{{2^2}}}} \right) = - {\lambda_2}h. \hfill \\}<!endgathered> \right. $$

(7.23)

From (7.23), when we eliminate ${\lambda_2}$, we can obtain by multiplying the numerator and denominator by the quantity ${{{D_{{{h_2}}}^m}} \left/ {{Q_{{{h_2}}}^n}} \right.}$

$$ \left( {\frac{h}{H}} \right) = \frac{{{{\left( {\frac{{{D_{{{h_2}}}}}}{{{D_{{{H_1}}}}}}} \right)}^m}{{\left( {\frac{{{Q_{{{H_1}}}}}}{{{Q_{{{h_2}}}}}}} \right)}^n}\frac{1}{{{2^1}}} + {{\left( {\frac{{{D_{{{h_2}}}}}}{{{D_{{{H_2}}}}}}} \right)}^m}{{\left( {\frac{{{Q_{{{H_2}}}}}}{{{Q_{{{h_2}}}}}}} \right)}^n}\frac{1}{{{2^2}}}}}{{\frac{1}{{{2^3}}} + \frac{1}{{{2^2}}}{{\left( {\frac{{{D_{{h2}}}}}{{{D_{{{h_1}}}}}}} \right)}^m}{{\left( {\frac{{{Q_{{{h_1}}}}}}{{{Q_{{{h_2}}}}}}} \right)}^n}}}. $$

(7.24)

By considering (7.22), we finally obtain

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = \frac{{{2^{{3n - 1 - 3m\gamma }}} + {2^{{n - m\gamma - 2}}}}}{{{2^{{ - 3}}} + {2^{{2n - 2 - 2m\gamma }}}}} = {f_{{{\rm{op}}{{\rm{t}}_2}}}}. $$

(7.25)

By considering (20 h), and (7.25), the expression of H can be obtained:

$$ H = \sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}}. $$

(7.26)

And then

$$ h = {f_{{{\rm{opt}}2}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}}. $$

(7.27)

By multiplying (20 g) by the quantity Q _h2/D² _h2 we obtain

$$\begin{array}{llll} \left( {\frac{{D_{{{H_1}}}^2}}{{D_{{{h_2}}}^2}}\frac{{{Q_{{{h_2}}}}}}{{{Q_{{{H_1}}}}}}\frac{1}{{{2^1}}} + \frac{{D_{{{H_2}}}^2}}{{D_{{{h_2}}}^2}}\frac{{{Q_{{{h_2}}}}}}{{{Q_{{{H_2}}}}}}\frac{1}{{{2^2}}}} \right)\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}} + \left( {\frac{{D_{{{h_1}}}^2}}{{D_{{{h_2}}}^2}}\frac{{{Q_{{{h_2}}}}}}{{{Q_{{{h_1}}}}}}\frac{1}{{{2^2}}} + \frac{1}{{{2^3}}}} \right){f_{{{\rm{op}}{{\rm{t}}_2}}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}} = \frac{{4t{Q_{{{h_2}}}}}}{{\pi D_{{{h_2}}}^2}}.\end{array}$$

(7.28)

From this, we can write

$$ \left( {{2^{{2 \times 3\gamma - 4}}} + {2^{{2\gamma - 3}}}} \right)\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_{{2}}}}}}} \right)}^{{ - 1}}}}} + \left( {{2^{{2 \times 2\gamma - 4}}} + {2^{{ - 3}}}} \right){f_{{{\rm{opt}}2}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}} = \frac{{4t{Q_{{{h_2}}}}}}{{\pi D_{{{h_2}}}^2}}. $$

(7.29)

And then

$$ {D_{{{h_2}}}} = {\left( {4t{Q_{{{h_2}}}}/\pi } \right)^{{1/2}}}{\Psi_2}^{{ - 1/2}}. $$

(7.30)

We finally obtain D _h2, D _H1, and D _H2 from (7.22) and from (7.30):

$$ {D_{{{H_2}}}} = {2^{\gamma }}{\left( {4t{Q_{{{h_2}}}}/\pi } \right)^{{1/2}}}{\Psi_2}^{{ - 1/2}}, $$

(7.31)

$$ {D_{{{h_1}}}} = {2^{{2\gamma }}}{\left( {4t{Q_{{{h_2}}}}/\pi } \right)^{{1/2}}}{\Psi_2}^{{ - 1/2}}, $$

(7.32)

$$ {D_{{{H_1}}}} = {2^{{3\gamma }}}{\left( {4t{Q_{{{h_2}}}}/\pi } \right)^{{1/2}}}{\Psi_2}^{{ - 1/2}}, $$

(7.33)

$$ {\Psi_2} = \sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_2}}}}} \right)}^{{ - 1}}}}} \left( {\left( {{2^{{2 \times 3\gamma - 4}}} + {2^{{2\gamma - 3}}}} \right) + {f_{{{\rm{op}}{{\rm{t}}_2}}}}\left( {{2^{{2 \times 2\gamma - 4}}} + {2^{{ - 3}}}} \right)} \right). $$

(7.34)

The kth-Order Construct

Consider the situation presented in Fig. 7.2, but taken in a more general form where the network is more and more ramified and has D ₀, D ₁ until D _k . From the dimension h and H, of Fig. 7.3, one can group the pipes into two groups of index h (those in the direction of h) and of index H (those in the direction of H). To understand, see the first and second constructs that are displayed in Figs. 7.1 and 7.2.

In the same way as in previous cases, the Lagrangian function writes

$$ {L_{\rm{g}}} = Ha\sum\limits_{{i = 1}}^k {\frac{{Q_{{{H_i}}}^n}}{{D_{{{H_i}}}^m}}} \frac{1}{{{2^i}}} + ha\sum\limits_{{i = 1}}^k {\frac{{Q_{{{h_i}}}^n}}{{D_{{{h_i}}}^m}}} \frac{1}{{{2^{{i + 1}}}}} + {\lambda_1}\left( {H\sum\limits_{{i = 1}}^k {\frac{{\pi D_{{{H_i}}}^2}}{{4{Q_{{{H_i}}}}}}\frac{1}{{{2^i}}}} + h\sum\limits_{{i = 1}}^k {\frac{{\pi D_{{{h_i}}}^2}}{{4{Q_{{{h_i}}}}}}} \frac{1}{{{2^{{i + 1}}}}} - t} \right) + {\lambda_2}\left( {hH - {S_{\rm{T}}}} \right). $$

(7.35)

Note that k is the number of pipes in direction of H, or in the direction of h. It is important to mention that, in order to facilitate calculus, we did not consider constructions in which there are i pipes in the direction of H and i + 1 pipes in the direction of h (and vice versa). In this view, on the first construction of Fig. 7.1, k = 1 and on the second construction of Fig. 7.2, k = 2.

By differentiating L _g with respect to ${D_{{{h_i}}}}$, ${D_{{{H_i}}}}$, h, H, ${\lambda_1}$, and ${\lambda_2}$, and cancelling all the derivatives, we obtain

$$ ma\frac{{Q_{{{H_i}}}^n}}{{D_{{{H_i}}}^{{m + 1}}}} = 2{\lambda_1}\frac{{\pi {D_{{{H_i}}}}}}{{4{Q_{{{H_i}}}}}}, $$

(7.36a)

$$ ma\frac{{Q_{{{h_i}}}^n}}{{D_{{{h_i}}}^{{m + 1}}}} = 2{\lambda_1}\frac{{\pi {D_{{{h_i}}}}}}{{4{Q_{{{h_i}}}}}}, $$

(7.36b)

$$ a\sum\limits_1^k {\frac{{Q_{{{H_i}}}^n}}{{D_{{{H_i}}}^m}}\frac{1}{{{2^i}}}} + {\lambda_1}\sum\limits_1^k {\frac{{\pi D_{{{H_i}}}^2}}{{4{Q_{{{H_i}}}}}}\frac{1}{{{2^i}}}} = - {\lambda_2}h, $$

(7.36c)

$$ a\sum\limits_1^k {\frac{{Q_{{{h_i}}}^n}}{{D_{{{h_i}}}^m}}\frac{1}{{{2^{{i + 1}}}}}} + {\lambda_1}\sum\limits_1^k {\frac{{\pi D_{{{h_i}}}^2}}{{4{Q_{{{h_i}}}}}}\frac{1}{{{2^{{i + 1}}}}}} = - {\lambda_2}H, $$

(7.36d)

$$ H\sum\limits_1^k {\frac{{D_{{{H_i}}}^2}}{{{Q_{{{H_i}}}}}}\frac{1}{{{2^i}}}} + h\sum\limits_1^k {\frac{{D_{{{h_i}}}^2}}{{{Q_{{{h_i}}}}}}\frac{1}{{{2^{{i + 1}}}}}} = \frac{{4t}}{\pi }, $$

(7.36e)

$$ hH = {S_{\rm{T}}}. $$

(7.36f)

From (7.36a), and (7.36b), we can express ${\lambda_1}$ as follows:

$$ {\lambda_1} = \frac{{2ma}}{\pi }\frac{{Q_{{{H_1}}}^{{n + 1}}}}{{D_{{{H_1}}}^{{m + 2}}}} = \frac{{2ma}}{\pi }\frac{{Q_{{{H_2}}}^{{n + 1}}}}{{D_{{{H_2}}}^{{m + 2}}}} = \cdots = \frac{{2ma}}{\pi }\frac{{Q_{{{h_1}}}^{{n + 1}}}}{{D_{{{h_1}}}^{{m + 2}}}} = \frac{{2ma}}{\pi }\frac{{Q_{{{h_2}}}^{{n + 1}}}}{{D_{{{h_2}}}^{{m + 2}}}} = \cdots $$

(7.37)

By combining terms of (7.37), we obtain (given that Q _H1/Q _h1 = 2, Q _H2/Q _h2 = 2, …, because of the symmetry of the structure)

$$ \frac{{{D_{{{H_1}}}}}}{{{D_{{{h_1}}}}}} = \frac{{{D_{{{h_1}}}}}}{{{D_{{{H_2}}}}}} = \frac{{{D_{{{H_2}}}}}}{{{D_{{{h_2}}}}}} = \frac{{{D_{{{h_2}}}}}}{{{D_{{{H_3}}}}}} = \cdots = \frac{{{D_{{{H_k}}}}}}{{{D_{{{h_k}}}}}} = {2^{\gamma }}. $$

(7.38)

By introducing the expressions of ${\lambda_1}$ (defined in (7.37), (7.36c), and (7.36d)), we obtain

$$ \left( {a + \frac{{ma}}{2}} \right)\sum\limits_1^k {\frac{{Q_{{{H_i}}}^n}}{{D_{{{H_i}}}^m}}\frac{1}{{{2^i}}}} = - {\lambda_2}h, $$

(7.39)

$$ \left( {a + \frac{{ma}}{2}} \right)\sum\limits_1^k {\frac{{Q_{{{h_i}}}^n}}{{D_{{{h_i}}}^m}}\frac{1}{{{2^{{i + 1}}}}}} = - {\lambda_2}H. $$

(7.40)

By dividing member by member, the terms of (7.39) and (7.40), we obtain

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = \frac{{\sum\limits_1^k {\frac{{Q_{{{H_i}}}^n}}{{D_{{{H_i}}}^m}}\frac{1}{{{2^i}}}} }}{{\sum\limits_1^k {\frac{{Q_{{{h_i}}}^n}}{{D_{{{h_i}}}^m}}\frac{1}{{{2^{{i + 1}}}}}} }}. $$

(7.41)

By multiplying both the numerator and denominator of (7.41) by the quantity ${D_{{{h_k}}}}^m/{Q_{{{h_k}}}}^n$, as follows

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = \frac{{\frac{{Q_{{{H_1}}}^n}}{{D_{{{H_1}}}^m}}\frac{1}{{{2^1}}}\frac{{D_{{{h_k}}}^m}}{{Q_{{{h_k}}}^n}} + \cdots + \frac{{Q_{{{H_k}}}^n}}{{D_{{{H_k}}}^m}}\frac{1}{{{2^k}}}\frac{{D_{{{h_k}}}^m}}{{Q_{{{h_k}}}^n}}}}{{\frac{{Q_{{{h_1}}}^n}}{{D_{{{h_1}}}^m}}\frac{1}{{{2^{{1 + 1}}}}}\frac{{D_{{{h_k}}}^m}}{{Q_{{{h_k}}}^n}} + \cdots + \frac{{Q_{{{h_k}}}^n}}{{D_{{{h_k}}}^m}}\frac{1}{{{2^{{k + 1}}}}}\frac{{D_{{{h_k}}}^m}}{{Q_{{{h_k}}}^n}}}}, $$

(7.42)

and by taking into account (7.38), we obtain

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = \frac{{\sum\limits_{{i = 1}}^k {{2^{{\left( {2k - \left( {2i - 1} \right)} \right)n - m\left( {2k - \left( {2i - 1} \right)} \right)\gamma - i}}}} }}{{\sum\limits_{{i = 1}}^k {{2^{{\left( {2k - \left( {2i} \right)} \right)n - m\left( {2k - \left( {2i} \right)} \right)\gamma - \left( {i + 1} \right)}}}} }} = f{}_{{{\rm{op}}{{\rm{t}}_k}}}. $$

(7.43)

Verifications

Construction 1: k = 1 leads to

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = {2^{{n + 1 - m{ }\gamma }}}. $$

(7.44)

Construction 2: k = 2 leads to

$$ {\left( {\frac{h}{H}} \right)_{\rm{opt}}} = \frac{{{2^{{3n - 1 - 3m\gamma }}} + {2^{{n - 2 - m\gamma }}}}}{{{2^{{2n - 2 - 2m\gamma }}} + {2^{{ - 3}}}}}. $$

(7.45)

By posing that ${\left( {{{h} \left/ {H} \right.}} \right)_{\rm{opt}}} = {f_{{{\rm{op}}{{\rm{t}}_k}}}}$, we obtain from (7.43) and from (7.36f) the expressions of H and h as follows:

$$ H = \sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_k}}}}} \right)}^{{ - 1}}}}}, $$

(7.46)

$$ h = {f_{{{\rm{op}}{{\rm{t}}_k}}}}\sqrt {{{S_{\rm{T}}}{{\left( {{f_{{{\rm{op}}{{\rm{t}}_k}}}}} \right)}^{{ - 1}}}}}. $$

(7.47)

In order to make ${Q_{{{h_k}}}}/{Q_{{{H_i}}}},\;{Q_{{{h_k}}}}/{Q_{{{h_i}}}},\;{D_{{{H_i}}}}/{D_{{{h_k}}}}$ and ${D_{{{h_i}}}}/{D_{{{h_k}}}}$ appear, we multiply (7.36e) by ${Q_{{{h_k}}}}/D_{{{h_k}}}^2$, and we obtain

$$ \sqrt {{\frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_k}}}}}}}} \sum\limits_{{i = 1}}^k {\frac{{D_{{{H_i}}}^2}}{{D_{{{h_k}}}^2}}\frac{{{Q_{{{h_k}}}}}}{{{Q_{{{H_i}}}}}}\frac{1}{{{2^i}}}} + {f_{{{\rm{op}}{{\rm{t}}_k}}}}\sqrt {{\frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_k}}}}}}}} \sum\limits_{{i = 1}}^k {\frac{{D_{{{h_i}}}^2}}{{D_{{{h_k}}}^2}}\frac{{{Q_{{{h_k}}}}}}{{{Q_{{{h_i}}}}}}} \frac{1}{{{2^{{i + 1}}}}} = \frac{{4t{Q_{{{h_k}}}}}}{{\pi D_{{{h_k}}}^2}}. $$

(7.48)

Noting that

$$ \frac{{D_{{{H_i}}}^2}}{{D_{{{h_k}}}^2}} = \left( {\frac{{D_{{{H_i}}}^2}}{{D_{{{h_i}}}^2}} \times \frac{{D_{{{h_i}}}^2}}{{D_{{{H_{{i + 1}}}}}^2}} \times \cdots \times \frac{{D_{{{h_{{k - 1}}}}}^2}}{{D_{{{H_k}}}^2}} \times \frac{{D_{{{H_k}}}^2}}{{D_{{{h_k}}}^2}}} \right) = {2^{{2\left( {2\left( {k - i} \right) + 1} \right)\gamma }}}, $$

(7.49)

$$ \frac{{D_{{{h_i}}}^2}}{{D_{{{h_k}}}^2}} = \frac{{D_{{{h_i}}}^2}}{{D_{{{H_{{i + 1}}}}}^2}} \times \frac{{D_{{{H_{{i + 1}}}}}^2}}{{D_{{{h_{{i + 1}}}}}^2}} \times \cdots \times \frac{{D_{{{h_{{k - 1}}}}}^2}}{{D_{{{H_k}}}^2}} \times \frac{{D_{{{H_k}}}^2}}{{D_{{{h_k}}}^2}} = {2^{{2\left( {2\left( {k - i} \right)} \right)\gamma }}}, $$

(7.50)

$$ \frac{{{Q_{{{H_i}}}}}}{{{Q_{{{h_k}}}}}} = \frac{{{Q_{{{H_i}}}}}}{{{Q_{{{h_i}}}}}} \times \frac{{{Q_{{{h_i}}}}}}{{{Q_{{{H_{{i + 1}}}}}}}} \times \cdots \times \frac{{{Q_{{{h_{{k - 1}}}}}}}}{{{Q_{{Hk}}}}} \times \frac{{{Q_{{Hk}}}}}{{{Q_{{{h_k}}}}}} = {2^{{2\left( {k - i} \right) + 1}}}, $$

(7.51)

and

$$ \frac{{{Q_{{{h_i}}}}}}{{{Q_{{{h_k}}}}}} = \frac{{{Q_{{{h_i}}}}}}{{{Q_{{{H_{{i + 1}}}}}}}} \times \frac{{{Q_{{{H_{{i + 1}}}}}}}}{{{Q_{{{h_{{i + 1}}}}}}}} \times \cdots \times \frac{{{Q_{{{h_{{k - 1}}}}}}}}{{{Q_{{{H_k}}}}}} \times \frac{{{Q_{{{H_k}}}}}}{{{Q_{{{h_k}}}}}} = {2^{{2\left( {k - i} \right)}}}, $$

(7.52)

equation (7.48) can be rewritten as follows:

$$\begin{array}{llll} \sqrt {{\frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_k}}}}}}}} \sum\limits_{{i = 1}}^k {{2^{{2\left( {2\left( {k - i} \right) + 1} \right)\gamma - \left( {2\left( {k - i} \right) + 1} \right) - i}}}} + {f_{{{\rm{op}}{{\rm{t}}_k}}}}\sqrt {{\frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_k}}}}}}}} \sum\limits_{{i = 1}}^k {{2^{{2\left( {2\left( {k - i} \right)} \right)\gamma - \left( {2\left( {k - i} \right)} \right) - \left( {i + 1} \right)}}}} = \frac{{4t{Q_{{{h_k}}}}}}{{\pi D_{{{h_k}}}^2}}.\end{array}$$

(7.53)

From this, we finally obtain

$$ {D_{{{h_k}}}} = {\left( {4t{Q_{{{h_k}}}}/\pi } \right)^{{1/2}}}{\Psi_k}^{{ - 1/2}}, $$

(7.54)

where

$$ {\Psi_k} = \sqrt {{\frac{{{S_{\rm{T}}}}}{{{f_{{{\rm{op}}{{\rm{t}}_k}}}}}}}} \left( {\sum\limits_{{i = 1}}^k {{2^{{2\left( {2\left( {k - i} \right) + 1} \right)\gamma - \left( {2\left( {k - i} \right) + 1} \right) - i}}}} + {f_{{{\rm{op}}{{\rm{t}}_k}}}}\sum\limits_{{i = 1}}^k {{2^{{2\left( {2\left( {k - i} \right)} \right)\gamma - \left( {2\left( {k - i} \right)} \right) - \left( {i + 1} \right)}}}} } \right). $$

(7.55)

Then, considering (7.38), and for any j ranging from 1 to k, we obtain

$$ {D_h}_{{_j}} = {2^{{2\left( {k - j} \right)\gamma }}}{\left( {4t{Q_{{{h_k}}}}/\pi } \right)^{{1/2}}}{\Psi_k}^{{ - 1/2}} $$

(7.56)

and

$$ {D_H}_{{_j}} = {2^{{\left( {2\left( {k - j} \right) + 1} \right)\gamma }}}{\left( {4t{Q_{{{h_k}}}}/\pi } \right)^{{1/2}}}{\Psi_k}^{{ - 1/2}}. $$

(7.57)