3. Results and discussion

To find the minimum mass of the ribs of the heat-exchange apparatus with the maximum of its heat productivity developed mathematical model of multi-criteria optimization problem

Ji

r ¼ r0; ϑ ¼ ϑ0; r ¼ r

Modified Bessel equation for the radial fin of rectangular profile is:

r <sup>2</sup> d<sup>2</sup> ϑ dr<sup>2</sup> <sup>þ</sup> <sup>r</sup>

selected criteria under the given constraints.

the outer surface of the ribs.

fin height, m.

number for air.

Given the heat transfer coefficient:

min <sup>≤</sup> Ji ð Þ xi <sup>≤</sup> Ji

0 0; dϑ

J1ðx1; … x21; U1; U2Þ ! max,

Thus, the problem can be formulated as follows: it is required to find such controlled parameters of the element heat exchanger (fins), which are optimal from the perspective of the

The basis of mathematical model equations Bessel, describing the temperature distribution on

dϑ dr � <sup>m</sup><sup>2</sup> r 2

<sup>m</sup> <sup>¼</sup> <sup>2</sup><sup>α</sup> λ<sup>с</sup><sup>2</sup> δ

where, m—the dimensionless complex; α—heat transfer coefficient from the outer surface to

<sup>α</sup> <sup>¼</sup> <sup>Ν</sup>u<sup>в</sup> � <sup>ν</sup><sup>в</sup>

<sup>в</sup> � <sup>d</sup><sup>н</sup> hp <sup>0</sup>, <sup>54</sup>

where R—step rib, m; d—outside diameter of pipe, m; h—height of fin, m; ReB—Reynolds

Fpc � θ<sup>1</sup>

þ Fп

<sup>α</sup><sup>2</sup>пр <sup>¼</sup> <sup>α</sup><sup>2</sup> � <sup>F</sup><sup>p</sup> � <sup>θ</sup><sup>0</sup>

� h hp �0, <sup>14</sup>

where Nuв—the Nusselt number for air; νв—coefficient of kinematic viscosity of air, m2

the air, W/(m<sup>2</sup> \* �); λ<sup>с</sup>2—thermal conductivity of fin, W/(m \* �); δ—thickness of fin, m.

The heat transfer coefficient from the outer surface to the air, W/(m<sup>2</sup> \*grad.):

The Nusselt number for turbulent regime of the air movement:

Nu<sup>в</sup> <sup>¼</sup> <sup>0</sup>, <sup>096</sup> � Re0,<sup>72</sup>

The task of searching for the optimal height of the ribs is to find xe D in cases, when

max, (3)

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93

ϑ ¼ 0, (6)

, (7)

<sup>h</sup> , (8)

Fpc , (10)

/s; h—

, (9)

dr , <sup>δ</sup>0, <sup>λ</sup><sup>с</sup><sup>2</sup> <sup>¼</sup> const: (4)

<sup>J</sup>2ðx1; … <sup>x</sup>21; <sup>U</sup>1; <sup>U</sup>2Þ ! min: (5)

The Solution of Private Problems of Optimization for Engineering Systems

Figure 1. Radial edge.

parameters heat-giving elements of the heat-exchange apparatus systems of air heating of buildings, which is solved using the method of nonlinear optimization [12, 13].

Formulation of the mathematical model has been made for the outer surface of the radial ribs (Figure 1).

The temperature on the surface of the ribs defines heat output, and the height of the rib - metal heat-exchange apparatus, so as the optimality criteria for staging a mathematical model selected [14]:


As unmanaged parameters taken: the radius of the carrier pipe, of a thickness of edges, the thermal conductivity of the ribs, the location of the beam in a heat exchanger, step ribs, the number of ribs, the number of the Nusselt number for the air, the Reynolds number for the air, the coefficient of heat transfer from the wall to the air, etc. (x1 … xn).

In work the estimation of influence of these parameters on the process of heat exchanger.

As of controlled parameters are: ambient air temperature, the temperature of the heat-carrier (U1 … Un) as the most influence on the heat transfer process.

In the process of heat exchange occurs between the warm water and the environment, which depend on parameters of water and air. Parameters that characterize these changes are within the permissible limits, established for the process.

The dependence of optimization criteria [15] and process parameters:

$$\begin{aligned} J\_1 &= J\_1(\mathbf{x}\_1, \dots \ x\_{21}; \mathcal{U}\_1, \mathcal{U}\_2) \to \max, \\ J\_2 &= J\_2(\mathbf{x}\_1, \dots \ x\_{21}; \mathcal{U}\_1, \mathcal{U}\_2) \to \min. \end{aligned} \tag{1}$$

Restrictions on the parameters of the process, is within the following limits:

$$\mathbf{x}\_{i}^{\text{min}} \le \mathbf{x} \le \mathbf{x}\_{i}^{\text{max}},\tag{2}$$

The Solution of Private Problems of Optimization for Engineering Systems http://dx.doi.org/10.5772/intechopen.80520 93

$$J\_i^{\min} \le J\_i \text{ ( $\chi\_i$ )} \le J\_i^{\max} \text{ .} \tag{3}$$

$$r = r\_0; \quad \mathfrak{B} = \mathfrak{B}\_0; \quad r = r'\_0; \quad \frac{d\mathfrak{B}}{dr'} \quad \delta\_{0\nu} \quad \lambda\_{c\_2} = \text{const.} \tag{4}$$

The task of searching for the optimal height of the ribs is to find xe D in cases, when

$$\begin{aligned} J\_1(\mathbf{x}\_1, \dots, \mathbf{x}\_{21}, \mathcal{U}\_1, \mathcal{U}\_2) &\to \max, \\ J\_2(\mathbf{x}\_1, \dots, \mathbf{x}\_{21}, \mathcal{U}\_1, \mathcal{U}\_2) &\to \min. \end{aligned} \tag{5}$$

Thus, the problem can be formulated as follows: it is required to find such controlled parameters of the element heat exchanger (fins), which are optimal from the perspective of the selected criteria under the given constraints.

The basis of mathematical model equations Bessel, describing the temperature distribution on the outer surface of the ribs.

Modified Bessel equation for the radial fin of rectangular profile is:

$$r^2\frac{d^2\mathfrak{G}}{dr^2} + r\frac{d\mathfrak{G}}{dr} - m^2r^2\mathfrak{G} = 0,\tag{6}$$

$$m = \frac{2a}{\lambda\_{c\_2}\delta'} \tag{7}$$

where, m—the dimensionless complex; α—heat transfer coefficient from the outer surface to the air, W/(m<sup>2</sup> \* �); λ<sup>с</sup>2—thermal conductivity of fin, W/(m \* �); δ—thickness of fin, m.

The heat transfer coefficient from the outer surface to the air, W/(m<sup>2</sup> \*grad.):

$$\alpha = \frac{\text{Nu}\_{\mathfrak{s}} \cdot \text{v}\_{\mathfrak{s}}}{h},\tag{8}$$

where Nuв—the Nusselt number for air; νв—coefficient of kinematic viscosity of air, m2 /s; h fin height, m.

The Nusselt number for turbulent regime of the air movement:

$$\mathrm{Nu}\_{\mathfrak{s}} = 0,096 \cdot \mathrm{Re}\_{\mathfrak{s}}^{0.72} \cdot \left(\frac{d\_{\mathrm{H}}}{h\_{\mathrm{P}}}\right)^{0.54} \cdot \left(\frac{h}{h\_{\mathrm{P}}}\right)^{-0.14} \text{.}\tag{9}$$

where R—step rib, m; d—outside diameter of pipe, m; h—height of fin, m; ReB—Reynolds number for air.

Given the heat transfer coefficient:

parameters heat-giving elements of the heat-exchange apparatus systems of air heating of

Formulation of the mathematical model has been made for the outer surface of the radial ribs

The temperature on the surface of the ribs defines heat output, and the height of the rib - metal heat-exchange apparatus, so as the optimality criteria for staging a mathematical model

As unmanaged parameters taken: the radius of the carrier pipe, of a thickness of edges, the thermal conductivity of the ribs, the location of the beam in a heat exchanger, step ribs, the number of ribs, the number of the Nusselt number for the air, the Reynolds number for the air,

In work the estimation of influence of these parameters on the process of heat exchanger.

As of controlled parameters are: ambient air temperature, the temperature of the heat-carrier

In the process of heat exchange occurs between the warm water and the environment, which depend on parameters of water and air. Parameters that characterize these changes are within

J<sup>1</sup> ¼ J1ðx1;… x21; U1; U2Þ ! max,

min ≤ x ≤ х<sup>i</sup>

<sup>J</sup><sup>2</sup> <sup>¼</sup> <sup>J</sup>2ðx1;… <sup>x</sup>21; <sup>U</sup>1; <sup>U</sup>2Þ ! min: (1)

max, (2)

buildings, which is solved using the method of nonlinear optimization [12, 13].

1. the temperature on the surface of the ribs (J1);

the coefficient of heat transfer from the wall to the air, etc. (x1 … xn).

The dependence of optimization criteria [15] and process parameters:

Restrictions on the parameters of the process, is within the following limits:

хi

(U1 … Un) as the most influence on the heat transfer process.

the permissible limits, established for the process.

(Figure 1).

92 HVAC System

Figure 1. Radial edge.

selected [14]:

2. the height of the ribs (J2).

$$
\alpha\_{2np} = \alpha\_2 \cdot \left(\frac{F\_\text{p} \cdot \theta\_0}{F\_{pc} \cdot \theta\_1} + \frac{F\_\text{in}}{F\_{pc}}\right) \tag{10}
$$

where α2—heat transfer coefficient, W/(m<sup>2</sup> \*�); θ0—the difference between the temperatures of the surfaces of the ribs and of air, degrees; θ1—the difference in temperature between the core tube surface and air, �; Fп—surface area between the ribs, m2 ; Fpc—area finned surface, m2 ; Fр—area of fins, m2 .

The general solution is determined by the ratio:

$$
\Theta = \mathbb{C}\_1 I\_0(mr) + \mathbb{C}\_2 K\_0(mr). \tag{11}
$$

selecting the data of heat exchangers. Solution of the task is carried out with the help of the method of conjugate gradients—iterative method for unconstrained optimization of the multidimensional space, of the solution of a quadratic optimization problem for a finite

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As a software complex for solving multiparameter nonlinear multiobjective optimization of

A feature of this complex is compatible with Microsoft Excel and other programs. Empirically obtained data are compiled in Excel. In the program IOSO are controllable and uncontrollable parameters, optimality criteria, restrictions on the parameters of the process. Next, the pro-

Preliminary procedure IOSO is the formation of the initial experiment plan that could be implemented as a passive way (using information about various parameters, the optimization criteria and constraints obtained earlier) and active way, when too much is generated in the initial search area in accordance with a given distribution law. For each vector of variable

gram IOSO generates optimal process parameters selection method from Excel [8].

Figure 2. The temperature of the substrate surface () from the thickness of the plate (m) (all solutions).

Figure 3. The temperature of the substrate surface () from a distance from the base of the ribs (m) (all solutions).

number of steps.

engineering systems of buildings used IOSO NМ [9].

The constant K1, K0, J0, J<sup>1</sup> is calculated in accordance with boundary conditions described above.

$$
\Theta\_0 = \mathbb{C}\_1 \mathbb{I}\_0(mr\_0) + \mathbb{C}\_2 \mathbb{K}\_0(mr\_0'), \tag{12}
$$

$$0 = \mathbb{C}\_1 I\_1(mr\_0) + \mathbb{C}\_2 K\_1(mr\_0). \tag{13}$$

Calculating С1, С<sup>2</sup> we find the temperature distribution along fin height for the radial fin of rectangular shape (grad.):

$$\Theta(r) = \frac{\Theta\_0 \left( K\_1 \left( mr\_0' \right) \cdot I\_0(mr), + I\_1 \left( mr\_0' \right) K\_0(mr) \right)}{I\_0(mr\_0) K\_1 \left( mr\_0' \right) + I\_1(mr\_0') K\_0(mr\_0)},\tag{14}$$

where θ<sup>0</sup> is the temperature on the surface of the carrier pipe, degrees.

To estimate the heat flux from the surface of the ribs used in the work the dependence proposed by Bessel:

$$q\_0 = 2\pi\_0 r \delta\_0 \lambda m \theta\_0 \left[ \frac{I\_1(mr\_0')K\_1(mr\_0) - K\_1(mr\_0')I\_1(mr\_0)}{I\_0(mr\_0)K\_1(mr\_0') + I\_1(mr\_0')K\_0(mr\_0)} \right]. \tag{15}$$

Thus, the problem can be formulated in the following way: it is required to find such managed parameters element of heat exchanger (the height of the edges), which are optimal from the point of view you are the chosen criteria under certain constraints.

The basis of mathematical model based on Bessel equation describing the distribution of temperature on the outer surface of the ribs [4].

For the staging of the mathematical model are determined boundary conditions of the process of heat transfer in the work of the air heating systems, described in the dependencies (2–4).

To create the most resource-efficient heat exchangers in the article the low temperature and middle temperature of the heating system of the buildings with the following parameters the heat-carrier from +45 to +95�. With these parameters the heat exchanger of the most metalconsuming.

To identify the most critical conditions of heat exchange on the surface of the ribs considered by the turbulent mode of movement of the heat-carrier temperature of ambient air from �35 up to 10�. The analysis of use of brands fans in systems of air heating of buildings. Set the maximum speed of the air entering the heat exchanger—7 kg/(m<sup>2</sup> •�), which would consider in selecting the data of heat exchangers. Solution of the task is carried out with the help of the method of conjugate gradients—iterative method for unconstrained optimization of the multidimensional space, of the solution of a quadratic optimization problem for a finite number of steps.

where α2—heat transfer coefficient, W/(m<sup>2</sup>

.

The general solution is determined by the ratio:

θð Þ¼ r

q<sup>0</sup> ¼ 2π0rδ0λmθ<sup>0</sup>

temperature on the outer surface of the ribs [4].

point of view you are the chosen criteria under certain constraints.

maximum speed of the air entering the heat exchanger—7 kg/(m<sup>2</sup>

θ<sup>0</sup> K<sup>1</sup> mr<sup>0</sup>

where θ<sup>0</sup> is the temperature on the surface of the carrier pipe, degrees.

0

J<sup>1</sup> mr<sup>0</sup> 0

J0ð Þ mr<sup>0</sup> K<sup>1</sup> mr<sup>0</sup>

Fр—area of fins, m2

rectangular shape (grad.):

above.

94 HVAC System

by Bessel:

consuming.

tube surface and air, �; Fп—surface area between the ribs, m2

\*�); θ0—the difference between the temperatures of

θ ¼ C1J0ð Þþ mr C2K0ð Þ mr : (11)

0 ¼ C1J1ð Þþ mr<sup>0</sup> C2K1ð Þ mr<sup>0</sup> : (13)

0 � �J1ð Þ mr<sup>0</sup>

0 � �K0ð Þ mr<sup>0</sup>

0

0

0 � �K0ð Þ mr<sup>0</sup>

; Fpc—area finned surface, m2

� �, (12)

, (14)

: (15)

•�), which would consider in

;

the surfaces of the ribs and of air, degrees; θ1—the difference in temperature between the core

The constant K1, K0, J0, J<sup>1</sup> is calculated in accordance with boundary conditions described

θ<sup>0</sup> ¼ C1J0ð Þþ mr<sup>0</sup> C2K<sup>0</sup> mr<sup>0</sup>

Calculating С1, С<sup>2</sup> we find the temperature distribution along fin height for the radial fin of

� � � <sup>J</sup>0ð Þ mr ; <sup>þ</sup>J<sup>1</sup> mr<sup>0</sup>

0 � � <sup>þ</sup> <sup>J</sup><sup>1</sup> mr<sup>0</sup>

To estimate the heat flux from the surface of the ribs used in the work the dependence proposed

J0ð Þ mr<sup>0</sup> K<sup>1</sup> mr<sup>0</sup>

Thus, the problem can be formulated in the following way: it is required to find such managed parameters element of heat exchanger (the height of the edges), which are optimal from the

The basis of mathematical model based on Bessel equation describing the distribution of

For the staging of the mathematical model are determined boundary conditions of the process of heat transfer in the work of the air heating systems, described in the dependencies (2–4).

To create the most resource-efficient heat exchangers in the article the low temperature and middle temperature of the heating system of the buildings with the following parameters the heat-carrier from +45 to +95�. With these parameters the heat exchanger of the most metal-

To identify the most critical conditions of heat exchange on the surface of the ribs considered by the turbulent mode of movement of the heat-carrier temperature of ambient air from �35 up to 10�. The analysis of use of brands fans in systems of air heating of buildings. Set the

� �K0ð Þ mr � �

� �K1ð Þ� mr<sup>0</sup> <sup>K</sup><sup>1</sup> mr<sup>0</sup>

" #

0 � � <sup>þ</sup> <sup>J</sup><sup>1</sup> mr<sup>0</sup> As a software complex for solving multiparameter nonlinear multiobjective optimization of engineering systems of buildings used IOSO NМ [9].

A feature of this complex is compatible with Microsoft Excel and other programs. Empirically obtained data are compiled in Excel. In the program IOSO are controllable and uncontrollable parameters, optimality criteria, restrictions on the parameters of the process. Next, the program IOSO generates optimal process parameters selection method from Excel [8].

Preliminary procedure IOSO is the formation of the initial experiment plan that could be implemented as a passive way (using information about various parameters, the optimization criteria and constraints obtained earlier) and active way, when too much is generated in the initial search area in accordance with a given distribution law. For each vector of variable

Figure 2. The temperature of the substrate surface () from the thickness of the plate (m) (all solutions).

Figure 3. The temperature of the substrate surface () from a distance from the base of the ribs (m) (all solutions).

parameters, the values of the optimization criteria and constraints are determined by direct appeal to the mathematical model of the investigated object. The number of points constituting the initial plan of the experiment depends on the dimension of the problem and the selected

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Figures 2–8 shows the Pareto-optimal set of values when solving multi-criteria parameter optimization of a fin heat exchanger by using design software IOSO NM obtained the optimal values. For ease of understanding the obtained values of the graphs for each controlled parameter.

Figure 7. The temperature of the substrate surface () from the thickness of the plate (m) (optimal solution).

Improving the energy efficiency of engineering systems of buildings and structures, reduction of energy consumption through the optimization of their operation, introduction of energy saving technologies and optimization of structural elements of engineering systems is important.

Figure 8. The temperature of the substrate surface () from a distance from the base of the ribs (m) (the optimal solution).

approximation functions.

4. Conclusion

Figure 4. The dependence of the distance from the base (m) of the fin from the thermal conductivity of the plate (W/m• ) (optimal solution).

Figure 5. The dependence of the distance (m) from the base of the fin the thickness of the plate (m) (optimal solution).

Figure 6. The temperature of the substrate surface () from the thermal conductivity of the plate (W/m• ) (optimal solution).

parameters, the values of the optimization criteria and constraints are determined by direct appeal to the mathematical model of the investigated object. The number of points constituting the initial plan of the experiment depends on the dimension of the problem and the selected approximation functions.

Figures 2–8 shows the Pareto-optimal set of values when solving multi-criteria parameter optimization of a fin heat exchanger by using design software IOSO NM obtained the optimal values. For ease of understanding the obtained values of the graphs for each controlled parameter.

Figure 7. The temperature of the substrate surface () from the thickness of the plate (m) (optimal solution).

Figure 8. The temperature of the substrate surface () from a distance from the base of the ribs (m) (the optimal solution).

#### 4. Conclusion

Figure 4. The dependence of the distance from the base (m) of the fin from the thermal conductivity of the plate (W/m• )

Figure 5. The dependence of the distance (m) from the base of the fin the thickness of the plate (m) (optimal solution).

Figure 6. The temperature of the substrate surface () from the thermal conductivity of the plate (W/m• ) (optimal solution).

(optimal solution).

96 HVAC System

Improving the energy efficiency of engineering systems of buildings and structures, reduction of energy consumption through the optimization of their operation, introduction of energy saving technologies and optimization of structural elements of engineering systems is important.

As applied problems of improvement of engineering systems of buildings considered an example of optimization of heat exchanger element air heating system of the building.

[4] Khrustalev BM, Nesenchuk AP, et al. Heat and Mass Transfer. Part 1. Minsk: Belarusian

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[5] EU. The Vilkas, Maiminas AS. Solution: Theory, Information, Modeling. Moscow: Radio

[6] Himmelblau D. Applied Nonlinear Programming. Translation from English Publishing

[7] Melekhin AA, Melekhin AG. Optimization of parameters of heat exchangers of air heating systems. Applied Mechanics and Materials. 2014;672–674:1471-1480; Switzerland; 1662-7482

[8] Salimpour MR, Bahrami Z. Thermodynamic analysis and optimization of air-cooled heat exchangers. International Journal of Heat and Mass Transfer. New York: Springer. 2011;47

[9] Egorov IN, Kretinin GV, Leschenko IA, Kuptcov SV. Multi-objective optimization using IOSO technology. In: 7th ASMO UK/ISSMO Conference on Engineering Design Optimi-

[10] The tool Microsoft Excel Solver uses an algorithm of nonlinear optimization Generalized Reduced Gradient (GRG2), developed by the Leon Lasdon (University of Texas at Austin)

[11] Burke EK, Landa Silva JD, Soubeiga E. Hyperheuristic approaches for multiobjective

[12] Kashevarova GG, Permyakova TB. Numerical Methods for Solving Problems of Construc-

[13] Kashevarova GG, Martirosyan AS. Software implementation of the algorithm the statistical straggling of the mechanical properties of materials in the design of structures. Advanced Materials Research. Switzerland: Trans Tech Publication. 2013;684:106-110 [14] Lapidus AS. Selection of criteria for the engineering and economical optimization of heat exchangers. International Journal of Chemical and Petroleum Engineering. New York:

[15] Sobol IM, Statnikov RB. The Choice of Optimal Parameters in Problems with Many

optimization. In: Proceeding of the MIC 2003; Kyoto, Japan; 2003

tion of the Computer. Perm: Publishing house of PSTU; 2007

National Technical University; 2007

House "World". Moscow: World; 1975

zation; Bath, UK; 7–8 July, 2008

Springer. 1977;13(2):160-165.

Criteria. Moscow: Science; 1981

and Allan Waren (Cleveland State University)

and Communication; 1981

(1):35-44

Finding the best managed of the parameters of the heat exchanger element air heating system of a building is possible with the developed by the author of a comprehensive method of research based on multi-criteria parameter optimization with the introduction of empirically obtained data.

When developing a mathematical model of the heat exchanger element air heating system of a building used for the basic equations of heat and mass transfer.

The decision of tasks of optimization carried out using the method of nonlinear optimization in design-software complexes IOZO.

The aim of the study is to increase the efficiency of engineering systems of buildings by optimizing the parameters of the elements of heat exchangers used in air heating systems of buildings.

To achieve this goal the author posed and solved the following tasks:

