$\text{d}{Q}_{1con}+\text{d}{Q}_{1rad}=\text{d}{Q}_2=\text{d}{Q}_{3con}+\text{d}{Q}_{3rad}$ (5)

where $\text{d}{Q}_{1con},\text{d}{Q}_{1rad}$—Convective and radiant heat flows on the inner surface of the pipe, $\text{d}{Q}_2$—heat flux by heat conduction through the pipe wall, $\text{d}{Q}_{3con},\text{d}{Q}_{3rad}$—Convective and radiant heat fluxes on the outer surface of the pipe, where

$\text{d}{Q}_2=\pi D\text{d}l\lambda /\delta \left(T_{wi}-T_{wo}\right)=\pi D\text{d}l{c}_0{\epsilon }_w\left(T_{wo}^4-T_o^4\right){10}^{-8}+\pi D\text{d}l{\alpha }_2\left(T_{wo}-T_o\right)$ (6)

$\text{d}{Q}_2=\pi D\text{d}l{\alpha }_1\left(T-T_{wi}\right)+\pi D\text{d}l{c}_0\epsilon \left(T^4-T_{wi}^4\right){10}^{-8}=\pi D\text{d}l\lambda /\delta \left(T_{wi}-T_{wo}\right)$ (7)

To obtain a quasi-dimensional mathematical model of the thermal and hydraulic modes of the TGN, it is proposed to consider the temperature dependence of the heater surface as a function of linear and angular coordinates:

$T_{wi}=T_{wi}\left(l,\varphi \right)$

$T_{wо}=T_{wо}\left(l,\varphi \right)$ (8)

$T_{о}=const$,

where $\varphi$—angular coordinate of the heater surface; $T_o$—absolute ambient temperature.

$\text{d}{Q}_{3con}=\frac{D}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\alpha }_2}\left[T_{wo}\left(l,\varphi \right)-T_o\right]\text{d}\varphi \text{d}l$ (9)

$\text{d}{Q}_{3rad}=\frac{D}{2}{с}_0{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\epsilon }_2}\left[T_{wo}^4\left(l,\varphi \right)-T_o^4\right]\text{d}\varphi \text{d}l{10}^{-8}$ (10)

A set of parameters affecting the temperature distribution around the heater perimeter due to convective motion can be identified and represented as follows ${\theta }_{wi}=\psi \left({\theta }_{wo},Pr,Re,Gr\right)$, where ${\theta }_{wo}$—average surface temperature of the heater, $Pr,Re,Gr$—Prandtl, Reynolds and Grasshoff numbers. On the basis of the experimental results, the regression dependence was obtained in the form of

${\theta }_{wi}=T_{wi}\left(\varphi \right)/T_{wo}=k_1+k_2{\left(Pr\cdot Gr\right)}^{k_3}\cdot R{e}^{k_4}\cos \left(\varphi \right)$ (11)

where $T_{wi}\left(\varphi \right)$ is the temperature on the surface of the heater cross-section with angular coordinate $\varphi$, $T_{wo}$ is the average temperature across the heater cross-section, and $k_1,k_2,k_3,k_4$ are the regression constants. The regression dependence (11) is obtained as a result of minimizing the deviation of the calculated values of the surface temperature of the tubular heater from the experimental values presented in [9] . The error of this dependence on the experimental values is 0.07, i.e., the dependence corresponds to the process of complex heat transfer for a tubular heater and can be used for its mathematical modelling.

Following [10] , optimisation is not just a tool in the arsenal of machine learning and artificial intelligence, it is the foundation upon which the efficiency and effectiveness of these fields results. Optimisation algorithms, discussed in the study [11] , are key tools in machine learning and artificial intelligence. Each algorithm uses a different strategy to navigate the complex and treacherous terrain of the Rosenbrock function, a function known for its complex optimisation landscape characterised by narrow valleys and steep ridges. Gradient descent (GD) is one of the most fundamental and widely used optimisation algorithms, but the algorithm may overshoot the minimum, leading to divergence. Conversely, the algorithm can converge very slowly, requiring an impractical number of iterations to reach the minimum [12] . Stochastic Gradient Descent (SGD) is based on the principles of standard gradient descent, but introduces an element of randomness by updating the model parameters not on the entire dataset, but on a randomly selected subset of the data, called a minipartition. By using only a subset of the data, SGD significantly reduces the computational cost per iteration, which makes it particularly suitable for large-scale problems [13] .

To solve the problem of calculating the thermal and hydraulic modes of operation of a tubular heater with heat carrier recirculation, the evolutionary search algorithm for the most attractive solutions was used. The basis of the algorithmic approach is the construction of algorithms for evolutionary search of solutions by binary choice relations. This approach is most fully described in [14] .

It is consider a set Ω of elements (decisions) $x=\left\{x^1,x^2,\cdots ,x^n\right\}$, where $x^i$—a scalar parameter (continuous or discrete). It is determined the binary choice relation $R_S$ for elements of the set Ω.

It is meant that there is a rule (algorithm) according to one decision is “better” than another. It is determined that choice relation $R_S$ is a no strictly order relation.

For subset $X$, $X\subset \Omega$ we denote the function of choice in the form

$S\left(X\right)=\left\{x\in X|\forall y\in \left[X\backslash S\left(X\right)\right],x{R}_{S}y\right\}$ (12)

We shall assume that set $S\left(X\right)$ contains the concrete number of elements—N_op.

We shall that for the set Ω it was determined relation $R_G$ with attachment function ${\mu }_{R_G}\left(x,y\right)$: $\Omega \times \Omega \to \left[0,1\right]$. Relation $R_G$ will be termed generation relation.

For subset $X$, $X\subset \Omega$ we denote the function of generation in the form

$G\left(X\right)=X\cup G_H\left(X\right)$,

$G_H\left(X\right)=\left\{y\in \Omega |\exists x\in X,y{R}_{G}x,{\mu }_{R_G}\left(x,y\right)>0\right\}$ (13)

We shall assume that set G(X) contains the concrete number of elements—N_E.

The algorithm to search R_S—optimal solution can be represented as

$X_k=S\left(G\left(X_{k-1}\right)\right)$, $k=1,2,\cdots$ (14)

where X_k—the set of preferred solutions according to the binary choice relation R_S at the iterate step k, X_{k – 1}—this is the same at the iterate step k − 1. G(X)—the function of generation with relation of generation R_G. S(X)—the function of choice with the binary choice relation R_S.

The iterate algorithm (14)—is the general form of evolutionary search.

We will consider the decomposition

$X_k={\displaystyle \underset{j=1}{\overset{N_b}{\cup }}{X}_{jk}}$, $X_{ik}\cap X_{jk}=\varnothing$, $i\ne j$ (15)

where X_jk—the set of preferred solutions according to the binary choice relation R_S at the iterate step k for the branch j of evolutionary search, N_b—the number of branches.

The algorithm (14) takes the form

$X_{jk}=S\left(G\left(X_{jk-1}\right)\right)$, $j=\overline{1,N_G}$, $k=1,2,\cdots$ (16)

These iterate algorithms (14), (16)—are the general form of evolutionary search.

For the task of finding the most attractive solution for a tubular gas heater with recirculation, the binary selection relation can be represented as follows.

The dimensionless efficiency function in the form

$F_1\left(x\right)=\eta$ (17)

The general penalty function in the form of

$F_2\left(x\right)=F_{21}\left(x\right)+F_{22}\left(x\right)$ (18)

where

$F_{21}\left(x\right)$—dimensionless pressure loss throughout the closed heater circuit

$F_{21}\left(x\right)=abs\left[{\displaystyle {\oint }_{heater}\text{d}{p}_i\left(x\right)}/abs\left(p_{\max }-p_{inlet}\right)\right]$ (19)

$F_{22}\left(x\right)$—dimensionless exceeding the maximum permissible heat flux through the outer surface of the heater.

$F_{22}\left(x\right)=abs\left(\Delta Q_{3con}+\Delta Q_{3rad}-\Delta Q_{lim}\right)/\Delta Q_{lim}$ (20)

$\Delta Q_{lim}$—maximum permissible heat flux through the outer surface of the heater.

$\text{d}{Q}_{3con}=\frac{D}{2}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\alpha }_2}\left[T_{wo}\left(l,\varphi \right)-T_o\right]\text{d}\varphi \text{d}l$ (21)

$\Delta Q_{3rad}=\frac{D}{2}{с}_0{\displaystyle \underset{0}{\overset{2\pi }{\int }}{\epsilon }_2}\left[T_{wo}^4\left(l,\varphi \right)-T_o^4\right]\text{d}\varphi \Delta l{10}^{-8}$ (22)

The general relation $R_{SS}$ for choosing between two possible solutions x and y is as follows:

$\begin{array}{c}x{R}_{SS}y=\left[F_2\left(x\right)\le 0\wedge F_2\left(y\right)>0\right]\\ \vee \left[F_2\left(x\right)>0\wedge F_2\left(y\right)>0\wedge F_2\left(x\right)\le F_2\left(y\right)\right]\\ \vee \left[F_2\left(x\right)\le 0\wedge F_2\left(y\right)\le 0\wedge F_1\left(x\right)\ge F_1\left(y\right)\right]\end{array}$ (23)