**3. Application of** *MDA* **to the Heat Transfer in a Circular Bar. Case Study**

### *3.1. General Approach*

A metallic (steel) bar with a circular section is considered, being related to the reference system *xGrt* (Figure 2).

**Figure 2.** Bar with circular section.

Generally, the set of variables that govern the transient heat transfer in a bar with circular section that can be further analyzed in terms of dimensions are indicated in Table 1:

**Table 1.** The set of variables that govern the heat transient transfer in a beam with circular section.



#### **Table 1.** *Cont.*


**Table 1.** *Cont.*

\* Heat is numerically equal to the dimension of work; the work is conventionally considered a product between a force having the direction along the bar, *Fx* (*Nx* = *kg*.*mx <sup>s</sup>*<sup>2</sup> ) and the displacement along the same direction *<sup>x</sup>* (*mx*). \*\* where the shear stress *<sup>τ</sup>*<sup>0</sup> has one of the directions, *<sup>x</sup>* or *r*, of the system *xGrt*, the applied force is *F*0, while the surface *A* where it occurs is in a plane that contains the direction of the shear stress; the velocity *w*<sup>0</sup> is normal to the plane where the shear stress is developed; ∇*w*<sup>0</sup> represents its gradient. \*\*\* *this is not suitable for dimensional analysis* (Therefore, it cannot be used in the dimensional analysis).

> Having the dimensions of the variables involved in the transient heat transfer, the MDA was applied as described by Szirtes in [31,32]. Additionally, for acquiring the simplest relations of the model law, according to [31,32], the dimensions were duplicated (in this case, the lengths were duplicated). This will contribute to the reduction in the number of *πj*, *j* = 1, ... , *n* dimensionless variables, once the dimensions of the variables involved increase. Thus, the reduced number of expressions of the Model Law will be obtained.

> According to the principles mentioned in [31,32], the following two sets of independent variables were selected:


which are directly connected with the measurements that were performed and whose magnitude can be controlled during experiments carried out on the model.

These sets are included in matrix *A*; the other quantities, representing dependent variables, form matrix *B*.

It should be noted that the variables contained in matrix A are freely chosen, both for the prototype and for the model. The advantage of choosing these two sets of independent variables lies, inter alia, in the following:

	- accepting convenient and well-determined values for the amount of heat introduced into the system (*<sup>Q</sup>* or . *Q*);
	- setting final temperatures compared to initial ones ( Δ*t*),
	- defining/accepting individual heating times (*τ*) of the prototype and the model;

In the following, the obtained results for these two variants are analyzed.

#### *3.2. First Case Study*

Version I is based on the above-described protocol of the *MDA* and the following quantities were successively obtained:


*cp air*, *Cair*, *Csteel*, *ax air*, *ar air*, *ρair*, *ρsteel*, *λr steel* , *νx air*, *vr air* , *αnx steel* , *αnr steel* , *ηx air*, *ηr air*, *βair*/*steel*

• the dependent variables that are useful for setting convection heat transfer correlations between dimensionless numbers (similarity criteria) *Crit*01, *Crit*02, *Crit*03, Pr*<sup>x</sup>* , *Grx air*, *Fox air*, *For air*, Re*r air*, *Str air* where the mentioned dimensionless numbers are:

$$\begin{aligned} Crit \ 01 = \text{Re}\_{\overline{r}} = Pe\_{\overline{r}} & \stackrel{\overline{m}\_{\overline{x}}}{=} \text{Crit } 02 = Nu\_{\overline{x}} = Pe\_{\overline{x}} = Bi\_{\overline{x}} = \frac{m\_{\overline{x}}^2}{m\_{\overline{r}}^2} \\ Crit \ 03 = Nu\_{\overline{r}} = Bi\_{\overline{r}} & \stackrel{\overline{m}\_{\overline{x}}^2}{=} \end{aligned}$$

• the properties of the paint layer: *ρpaint*, *λx paint*, *λr paint*, *αnr paint*, *δr paint*

The components of the reduced dimensional matrix *(B + A)* are indicated in Tables 2–6, where, as mentioned before, these elements represent exactly the exponents of the dimensions involved in defining those variables.



**Table 3.** The quantities required by experiments (part of matrix B).



**Table 4.** The quantities required by the theoretic analysis (part of matrix B).

**Table 5.** The quantities required by the heat transfer correlations between dimensionless numbers (part of matrix B).


**Table 6.** The properties of the intumescent paint (part of matrix B).


By performing the above-mentioned calculations, the elements of the Dimensional Set were finally obtained, from where all dimensionless *π<sup>j</sup>* expressions were extracted as corresponding lines of the Dimensional Set. In the following, this step-by-step procedure is presented just for the first expression of the model law (related to the dimensionless variable) and for the rest, only the final expressions of the model law are indicated. Thus, the following were obtained:

(a) From experiments on uncoated structures (prototype and model) the following expressions of the Model Law were obtained (that is, the final expressions in which the corresponding scale factors *Sη* of the dependent variables were defined in function of the scale factors of the independent variables):

$$\pi\_1 = \dot{Q} \cdot Q^{-1} \cdot L\_t^0 \cdot \Delta t^0 \cdot \tau^1 \cdot \lambda\_{\text{x+steel}}^0 \cdot \xi^0 = \frac{\dot{Q} \cdot \pi}{Q} = 1 \implies \frac{S\_{\dot{Q}} \cdot S\_{\text{7}}}{S\_Q} = 1 \Rightarrow S\_{\dot{Q}} = \frac{S\_Q}{S\_{\text{7}}} \tag{13}$$

$$
\pi\_2 \colon S\_{A\_{tr}} = \frac{S\_{L\_4}}{S\_\xi} \tag{14}
$$

$$
\pi\_3 \colon S\_{A\_{\rm lat}} = \frac{S\_Q \cdot S\_{L\_t}}{S\_{\rm At} \cdot S\_{\rm \tau} \cdot S\_{\lambda\_{x\,\rm sterl}}} \tag{15}
$$

$$
\pi\_4: \mathcal{S}\_{r\_{cyl}} = \sqrt{\frac{\mathcal{S}\_{L\_t}}{\mathcal{S}\_{\xi}}} \tag{16}
$$

$$\pi\_{\mathfrak{F}} \colon S\_{L\_{\mathfrak{x}}} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\mathfrak{x}} \cdot S\_{\lambda\_{\mathfrak{x}\text{ stel}}}} \tag{17}$$

$$
\pi\_6 \colon S\_{L\_r} = \sqrt{\frac{S\_{L\_t}}{S\_\xi}} \tag{18}
$$

(b) From experiments on coated structures (prototype and model) the set of previous expressions is completed with expressions specific to the coating paint, which are (*π*<sup>31</sup> ... *π*35) . The following set of expressions of the Model Law is obtained (*π*<sup>1</sup> ... *π*6) and (*π*<sup>31</sup> ... *π*35).

$$
\pi\_1 \, S\_{\dot{Q}} = \frac{S\_Q}{S\_\pi} \, \tag{19}
$$

$$
\pi\_2 \colon S\_{A\_{tr}} = \frac{S\_{L\_t}}{S\_{\xi}} \, ^\prime \tag{20}
$$

$$\pi\_3 \colon S\_{A\_{lat}} = \frac{S\_Q \cdot S\_{L\_t}}{S\_{\Lambda t} \cdot S\_{\mathbb{\tau}} \cdot S\_{\lambda\_{x\text{ total}}}},\tag{21}$$

$$
\pi\_4 \colon S\_{r\_{\rm cyl}} = \sqrt{\frac{S\_{L\_t}}{S\_\varsigma}},
\tag{22}
$$

$$\pi\mathfrak{r}\mathfrak{s}:\ S\_{L\_x} = \frac{S\_Q}{S\_{\Lambda t}\cdot S\_{\mathfrak{r}}\cdot S\_{\lambda\_{x\text{ total}}}},\tag{23}$$

$$
\pi\_{\mathfrak{K}} \colon S\_{L\_r} = \sqrt{\frac{S\_{L\_t}}{S\_{\mathfrak{S}\_{\mathfrak{G}}}}} \, \tag{24}
$$

$$\pi\_{\rm 31} : S\_{\rho\_{\rm point}} = \frac{(S\_{\Delta t})^3 \cdot (S\_{\tau})^5 \cdot \left(S\_{\lambda\_{\rm x\, steel}}\right)^3 \cdot S\_{\xi}}{\left(S\_Q\right)^2 \cdot S\_{L\_t}},\tag{25}$$

$$
\pi\_{\text{32}} : \mathcal{S}\_{\lambda\_{\text{x\\_point}}} = \mathcal{S}\_{\lambda\_{\text{x\\_step}}} \tag{26}
$$

$$
\pi\_{\text{33}} \colon S\_{\lambda\_{\text{r\_{\text{point}}}}} = \frac{S\_Q}{S\_{\text{At}} \cdot S\_{\tau}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_{\text{t}}}}} \tag{27}
$$

$$
\pi\_{34} \text{ : } S\_{a\_{nr\ point}} = \frac{S\_{\lambda\_{x\ total}}}{S\_{L\_t}} , \tag{28}
$$

$$
\pi\_{35} \, : \, S\_{\delta\_{r\,\,point}} = \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}}.\tag{29}
$$

(c) For theoretical investigations of parameters dependence (*cp air*, *Cair*, *Csteel*, *ax air*, *ar air*, *ρair*, *ρsteel*, *λr steel* , *νx air*, *vr air* , *αnx steel* , *αnr steel* , *ηx air*, *ηr air*, *βair*/*steel*) on the set of independent variables (of prototype and model), the following set of expressions will be used (*π*<sup>7</sup> ... *π*21) :

$$\pi\_{\mathcal{T}} \colon \mathbb{S}\_{\mathbb{C}\_p \times i\prime} = \frac{\left(\mathbb{S}\_Q\right)^2}{\left(\mathbb{S}\_{\Lambda t}\right)^3 \cdot \left(\mathbb{S}\_{\tau}\right)^4 \cdot \left(\mathbb{S}\_{\Lambda\_{x\,\,\,\,\rm tcl}}\right)^2} \;/\tag{30}$$

$$
\pi\_{\\$} \colon S\_{C\_{\text{air}}} = \frac{S\_{Q}}{S\_{\Delta t}} \,\tag{31}
$$

$$
\pi\_{\mathcal{B}} : S\_{\mathbb{C}\_{\text{total}}} = \frac{S\_{\mathcal{Q}}}{S\_{\Delta t}} \, \tag{32}
$$

$$\pi\_{10}: S\_{a\_{x\
air}} = \frac{S\_{L\_t}}{S\_{\Upsilon} \cdot S\_{\xi}},\tag{33}$$

$$\pi\_{11} \colon S\_{\mathfrak{a}\_{\,\,r\,air}} = \frac{S\_Q}{S\_{\Delta t} \cdot \left(\mathcal{S}\_{\mathbf{r}}\right)^2 \cdot S\_{\lambda\_{\,\,x\,\,slot}}} \,\,\tag{34}$$

$$\pi\_{12} \colon S\_{\rho\_{air}} = \frac{\left(\mathcal{S}\_{\Delta t}\right)^3 \cdot \left(\mathcal{S}\_{\tau}\right)^5 \cdot \left(\mathcal{S}\_{\lambda\_x \text{ total}}\right)^3 \cdot \mathcal{S}\_{\xi}}{\left(\mathcal{S}\_Q\right)^2 \cdot \mathcal{S}\_{L\_t}},\tag{35}$$

$$\pi\_{13} \colon S\_{\rho\_{stol}} = \frac{(S\_{\Lambda t})^3 \cdot (S\_{\tau})^5 \cdot \left(S\_{\lambda\_{x\text{ total}}}\right)^3 \cdot S\_{\xi}}{\left(S\_Q\right)^2 \cdot S\_{L\_t}} \tag{36}$$

$$
\pi\_{14} \colon S\_{\lambda\_{r\text{ stel}}} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\tau}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_{\xi}}}} \tag{37}
$$

$$\pi\_{15} \colon S\_{\nu\_x \text{ } air} \frac{S\_Q}{S\_{\Lambda t} \cdot \left(S\_{\Gamma}\right)^2 \cdot S\_{\lambda\_{x \text{ } stellar}} \cdot S\_{\xi}} \, ^{\prime} \tag{38}$$

$$\pi\_{16} \colon S\_{\nu\_{r\ air}} = \frac{S\_Q}{S\_{\Delta t} \cdot \left(S\_{\tau}\right)^2 \cdot S\_{\lambda\_{x\\_stol}}} \, \tag{39}$$

$$\pi\_{17}:\ S\_{\aleph\_{mx\ total}} = \frac{S\_{\underline{Q}} \cdot S\_{\underline{\xi}}}{S\_{L\_t} \cdot S\_{\Lambda t} \cdot S\_{\underline{\tau}}},\tag{40}$$

$$
\pi\_{18} \, : \, S\_{a\_{nr\ stel}} = \frac{S\_{\lambda\_{x\ stel}}}{S\_{L\_4}},
\tag{41}
$$

$$\pi\_{19} \colon \mathbb{S}\_{\eta\_{x\\_air}} = \frac{\left(\mathbb{S}\_{\Delta t}\right)^2 \cdot \left(\mathbb{S}\_{\tau}\right)^3 \cdot \left(\mathbb{S}\_{\lambda\_{x\\_sat}}\right)^2}{\mathbb{S}\_Q \cdot \mathbb{S}\_{L\_t}},\tag{42}$$

$$\pi\pi\_{20}: S\_{\eta\_{r\ air}} = \frac{(S\_{\Lambda t})^2 \cdot (S\_{\tau})^3 \cdot \left(S\_{\lambda\_{x\\_stel}}\right)^2}{S\_Q} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}}\tag{43}$$

$$\pi\_{21} : S\_{\mathbb{\hat{\beta}\_{vir/stel}}} = \frac{1}{S\_{\Delta t}}. \tag{44}$$

(d) For investigations of the dependence of the parameters on the set of independent variables and for setting of heat transfer correlations between dimensionless numbers based on the expressions of the model law (by combining them favorably), the next set of expressions (*π*<sup>22</sup> ... *π*30) will be used:

$$\pi r 22 : S\_{Crit\ 01} = \frac{S\_{\Delta t} \cdot S\_{\pi} \cdot S\_{\lambda\_{x\ \text{ total}}}}{S\_Q} \cdot \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \tag{45}$$

$$\pi\_{23} \colon S\_{\text{Crit 02}} = \frac{\left(\mathcal{S}\_{\text{Q}}\right)^{2} \cdot \mathcal{S}\_{\text{\textdegree S}}}{S\_{L\_{\text{t}}} \left(\mathcal{S}\_{\text{At}}\right)^{2} \cdot \left(\mathcal{S}\_{\text{\textdegree T}}\right)^{2} \cdot \left(\mathcal{S}\_{\text{\textdegree A}\_{x \text{ stellar}}}\right)^{2}} \tag{46}$$

$$
\pi\_{24} \colon S\_{Crit\ 03} = \frac{S\_{\Lambda t} \cdot S\_{\tau} \cdot S\_{\lambda\_{x\ \text{ star}}}}{S\_Q \cdot S\_{\xi}},
\tag{47}
$$

$$\pi\_{25} \colon S\_{\text{Pr}\_{x\text{ }air}} = \frac{S\_Q}{S\_{L\_t} \cdot S\_{\Lambda t} \cdot S\_{\text{\textpi}} \cdot S\_{\lambda\_{x\text{ }steel}}} \, \tag{48}$$

$$\pi\_{26} \colon S\_{Gr\_{x\\_irr}} = \frac{\left(S\_Q\right)^2 \cdot \left(S\_\xi\right)^2}{\left(s\_{\Lambda t}\right)^2 \cdot \left(S\_\tau\right)^2 \cdot \left(s\_{\lambda\_{x\\_stel}}\right)^2} \tag{49}$$

$$\pi\_{27}: S\_{Fo\_{x\_{\
air}}} = \frac{S\_{L\_t} \cdot \left(S\_{\Lambda t}\right)^2 \cdot \left(S\_{\pi}\right)^2 \cdot \left(S\_{\lambda\_{x\_{\
air}} \text{total}}\right)^2}{\left(S\_Q\right)^2 \cdot S\_{\xi}},\tag{50}$$

$$\pi\_{28} \colon S\_{F0\_{r\
air}} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\Upsilon} \cdot S\_{\lambda\_{x\ \text{ total}}}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}} \tag{51}$$

$$\pi\_{29} \colon S\_{\text{Re}\_{x\text{ }air}} = \frac{S\_Q \cdot S\_\xi}{S\_{\Delta t} \cdot S\_\tau \cdot S\_{\lambda\_{x\text{ }start}}} \, \tag{52}$$

$$\pi\_{\mathfrak{N}0} \colon S\_{St\_{air}} = \frac{1}{\sqrt{\mathcal{S}\_{L\_t} \cdot \mathcal{S}\_{\xi}}}. \tag{53}$$

In order to show how the elements of the model law can be applied for correlating the prototype with the model, the following variables were selected:


These variables are governed by relations (1), (17), and (35) of the model law.

As can be observed, . *Q*<sup>1</sup> is a quantity that refers to the prototype and cannot be measured, since experiments were carried out only on the model, while *Lx* <sup>2</sup> and *δ<sup>r</sup>* <sup>2</sup> *pa*int are corresponding to the model and they can be determined only for the prototype; for the model they are obtained strictly from the elements of the model law.

Considering the set of independent variables, having the dimensions determined for both prototype and model, the scale factors (*SQ* , *SLt* , *S*Δ*<sup>t</sup>* , *S<sup>τ</sup>* , *Sλsteel* , *Sς*) are considered to be known, as well.

In order to obtain . *Q*1, relation (1) is used, where the scale factor *S* . *<sup>Q</sup>* is the ratio between . *<sup>Q</sup>*<sup>2</sup> and . *Q*1. Thus, the following is obtained:

$$
\pi\_1 \, S\_{\dot{Q}} = \frac{S\_Q}{S\_\tau} \Leftrightarrow \frac{\dot{Q}\_2}{\dot{Q}\_1} = \frac{S\_Q}{S\_\tau} \Rightarrow \dot{Q}\_1 = \frac{S\_\tau}{S\_Q} \dot{Q}\_2 \tag{54}
$$

The model length *Lx* <sup>2</sup> is obtained from relation (17), as:

$$\text{Tr}\_{\mathfrak{F}} \colon S\_{L\_x} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\mathfrak{T}} \cdot S\_{\lambda\_{\text{x\\_stul}}}} \Leftrightarrow \frac{L\_{\mathfrak{X}2}}{L\_{\mathfrak{X}1}} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\mathfrak{T}} \cdot S\_{\lambda\_{\text{x\\_stul}}}} \Rightarrow L\_{\mathfrak{X}2} = \frac{S\_Q}{S\_{\Lambda t} \cdot S\_{\mathfrak{T}} \cdot S\_{\lambda\_{\text{x\\_stul}}}} L\_{\mathfrak{X}1} \tag{55}$$

The thickness of the paint layer that covers the model *δ<sup>r</sup>* <sup>2</sup> *pa*int is acquired from relation (29):

$$\text{Tr35}: S\_{\delta\_{r\text{ point}}} = \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \iff \frac{\delta\_{r\text{ 2 point}}}{\delta\_{r\text{ 1 point}}} = \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \implies \delta\_{r\text{ 2 point}} = \delta\_{r\text{ 1 point}} \cdot \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}}.\tag{56}$$

Considering the previous relations, some observations can be made:

	- if the scale factor is the same for all lengths, then *SLt* = *SLx* , and consequently the relation of the fifth element of the model law, *π*<sup>5</sup> can be neglected.
	- if the thickness of the paint is the same for the prototype and model, then the relation of *π*<sup>35</sup> to the model law can be omitted.
	- if it is aimed to conceive a more flexible model, then the model law allows us to consider different scales of the lengths along directions (*x*, *r*, *t*) or different thicknesses of the paint layer, but strictly considering the elements of the model law.

As can be noticed, this is another major advantage of *MDA*, which cannot be obtained if the aforementioned methods are used.

#### *3.3. Second Case Study*

For the second significant version, II, where *<sup>Q</sup>* was substituted by . *Q*, the following significant elements of the dimensional set were obtained, according to Tables 7–11:


**Table 7.** Matrix A, comprising independent variables.

**Table 8.** The quantities required by experiments (part of matrix B).


**Table 9.** The quantities required by the theoretical analysis (part of matrix B).



**Table 10.** The quantities required by the heat transfer correlations between dimensionless numbers (part of matrix B).

**Table 11.** The properties of the intumescent paint (part of matrix B).


The corresponding elements of the model law are:

$$
\pi\_1 \colon \mathcal{S}\_Q = \mathcal{S}\_{\dot{Q}} \cdot \mathcal{S}\_{\tau\_\prime} \tag{57}
$$

$$
\pi\_2 \colon S\_{A\_{tr}} = \frac{S\_{L\_t}}{S\_\xi},
\tag{58}
$$

$$\pi\_3 \colon S\_{A\_{\text{lat}}} = \frac{S\_{\dot{Q}} \cdot S\_{L\_t}}{S\_{\Lambda t} \cdot S\_{\lambda\_{x\_{\text{stcell}}}}},\tag{59}$$

$$
\pi\_{\mathfrak{k}} \colon \mathcal{S}\_{r\_{\text{cyl}}} = \sqrt{\frac{S\_{L\_{\mathfrak{k}}}}{S\_{\mathfrak{k}}}} \, \, \tag{60}
$$

$$\pi\_5: S\_{L\_x} = \frac{S\_{\dot{Q}}}{S\_{\Lambda t} \cdot S\_{\lambda\_{x \text{ total}}}},\tag{61}$$

$$
\pi\_6: S\_{L\_r} = \sqrt{\frac{S\_{L\_t}}{S\_\varsigma}},\tag{62}
$$

$$\tau\tau \colon \mathbb{S}\_{\mathfrak{C}\_{p\\_air}} = \frac{\left(\mathbb{S}\_{\dot{Q}}\right)^2}{\left(\mathbb{S}\_{\Delta t}\right)^3 \cdot \left(\mathbb{S}\_{\tau}\right)^2 \cdot \left(\mathbb{S}\_{\lambda\_{x\\_shol}}\right)^2} \tag{63}$$

$$\pi\_{\mathsf{R}} \colon \mathcal{S}\_{\mathsf{C}air} = \frac{\mathcal{S}\_{\vec{Q}} \cdot \mathcal{S}\_{\mathsf{T}}}{\mathcal{S}\_{\mathsf{A}t}},\tag{64}$$

$$
\pi\mathfrak{g}: S\_{\mathbb{C}\_{\rm steel}} = \frac{S\_{\dot{Q}} \cdot S\_{\mathbb{T}}}{S\_{\Delta t}},
\tag{65}
$$

$$\pi\_{10}: S\_{a\_{\pi\_{\text{sur}}}} = \frac{S\_{L\_t}}{S\_{\pi} \cdot S\_{\mathfrak{g}}},\tag{66}$$

$$
\pi\_{11} \colon S\_{a\_{r\ air}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t} \cdot S\_{\tau} \cdot S\_{\lambda\_{x-stol}}} \cdot \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \,\tag{67}
$$

$$\pi\_{12} \colon S\_{\rho\_{air}} = \frac{(S\_{\Delta t})^3 \cdot (S\_{\pi})^3 \cdot \left(S\_{\lambda\_{\pi\_{sat}}}\right)^3 \cdot S\_{\xi}}{\left(S\_{\dot{Q}}\right)^2 \cdot S\_{L\_{\ell}}} \,\tag{68}$$

$$
\pi\_{13} \colon S\_{\rho\_{stol}} = \frac{\left(\mathcal{S}\_{\Lambda t}\right)^3 \cdot \left(\mathcal{S}\_{\tau}\right)^3 \cdot \left(\mathcal{S}\_{\lambda\_x \\_stol}\right)^3 \cdot \mathcal{S}\_{\mathcal{S}}}{\left(\mathcal{S}\_{\dot{Q}}\right)^2 \cdot \mathcal{S}\_{L\_l}},
\tag{69}
$$

$$
\pi\_{14} \colon S\_{\lambda\_{r\\_stul}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}} \tag{70}
$$

$$\pi\_{15} \colon S\_{\nu\_{x\_{\
uir}}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t} \cdot S\_{\tau} \cdot S\_{\lambda\_{x\_{\
uir}}} \cdot S\_{\xi}} \tag{71}$$

$$
\pi\_{16} \colon S\_{\nu\_{r\\_air}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t} \cdot S\_{\Gamma} \cdot S\_{\lambda\_{x\\_water}}} \cdot \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \tag{72}
$$

$$\begin{array}{ll} \text{ $\pi$ }\_{17} \text{ :  $S\_{\mathfrak{A}\_{\text{max}}}$  } \begin{array}{l} \text{ $\bar{S}\_{\dot{Q}} \cdot S\_{\tilde{\mathfrak{G}}}$ }\\ \text{ $\bar{S}\_{L\_{\text{f}}} \cdot S\_{\Delta t}$ } \end{array} \end{array} \tag{73}$$

$$
\pi\_{18} \colon S\_{a\_{nr\ strel}} = \frac{S\_{\lambda\_x \text{ } \text{ } \text{ } \text{ }}}{S\_{L\_t}} \prime \tag{74}
$$

$$\pi\_{19} \colon \mathbb{S}\_{\eta\_{x\_{\
u}}} = \frac{(\mathbb{S}\_{\Lambda t})^2 \cdot (\mathbb{S}\_{\tau})^2 \cdot (\mathbb{S}\_{\lambda\_{x\_{\
u}} \times t\omega})^2}{\mathbb{S}\_{\dot{Q}} \cdot \mathbb{S}\_{L\_t}},\tag{75}$$

$$\pi\_{20} \colon S\_{\eta\_{r\ air}} = \frac{(S\_{\Delta t})^2 \cdot (S\_{\tau})^2 \cdot \left(S\_{\lambda\_{x\\_stand}}\right)^2}{S\_{\dot{Q}}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}}\tag{76}$$

$$
\pi\_{21} \colon S\_{\mathfrak{f}\_{\text{air}/\text{stol}}} = \frac{1}{S\_{\Delta t}}.\tag{77}
$$

The mentioned dimensionless numbers have the same expressions:

$$\text{Crit } 01 = \text{Re}\_r = \text{Pr}\_r = \frac{m\_X}{m\_r} \text{ } \tag{78}$$

$$\text{Crit } 02 = \text{Nu}\_{\text{X}} = \text{Pe}\_{\text{X}} = Bi\_{\text{X}} = \frac{m\_{\text{X}}^2}{m\_r^{\text{N}}} \tag{79}$$

$$\text{Crit } 03 = \text{Nu}\_r = \text{Bi}\_r = \frac{m\_r^2}{m\_x \cdot m\_t} \text{ } \tag{80}$$

The elements of the model law are:

$$\pi\_{22} \colon S\_{Crit\ 01} = \frac{S\_{\Lambda t} \cdot S\_{\lambda\_x \ \text{stel}}}{S\_{\dot{Q}}} \cdot \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}} \tag{81}$$

$$\pi\_{23} \colon \mathbb{S}\_{Crit\ 02} = \frac{\left(\mathbb{S}\_{\dot{Q}}\right)^2 \cdot \mathbb{S}\_{\mathbb{S}}}{S\_{L\_t} \cdot \left(\mathbb{S}\_{\Delta t}\right)^2 \cdot \left(\mathbb{S}\_{\lambda\_{x\_{\text{strel}}}}\right)^2} \tag{82}$$

$$
\pi\_{24} \colon S\_{Crit\ 03} = \frac{S\_{\Lambda t} \cdot S\_{\Lambda\_x \text{ } stcl}}{S\_{\dot{Q}} \cdot S\_{\xi}},
\tag{83}
$$

$$\pi\_{25} \colon S\_{\text{Pr}\_{x\\_air}} = \frac{S\_{\dot{Q}}}{S\_{L\_t} \cdot S\_{\Delta t} \cdot S\_{\lambda\_{x\\_steel}}},\tag{84}$$

$$\pi\_{2\text{6}} \colon \mathbb{S}\_{Gr\_{x\text{ }air}} = \frac{\left(\mathbb{S}\_{\dot{Q}}\right)^2 \cdot \left(\mathbb{S}\_{\dot{\xi}}\right)^2}{\left(\mathbb{S}\_{\Lambda t}\right)^2 \cdot \left(\mathbb{S}\_{\lambda\_{x\text{ }strel}}\right)^2} \tag{85}$$

$$\pi\pi\text{27}:\ S\_{F\otimes\_{x\text{ }air}} = \frac{S\_{L\_t}\cdot\left(S\_{\Lambda t}\right)^2\cdot\left(S\_{\lambda\_{x\text{ }start}}\right)^2}{\left(S\_{\dot{Q}}\right)^2\cdot S\_{\xi}}\tag{86}$$

$$\pi\_{28} \colon S\_{Fo\_{r\\_air}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t} \cdot S\_{\lambda\_{x\\_stel}}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}} \tag{87}$$

$$\pi\_{29} \colon \mathbb{S}\_{\text{Re}\_{x\\_air}} = \frac{\mathbb{S}\_{\dot{Q}} \cdot \mathbb{S}\_{\text{\\_}}}{\mathbb{S}\_{\Delta t} \cdot \mathbb{S}\_{\lambda\_{x\\_start}}},\tag{88}$$

$$
\pi\_{\text{30}} \colon S\_{St\_{air}} = \frac{1}{\sqrt{S\_{L\_t} \cdot S\_\xi}} \,\,\,\tag{89}
$$

The elements of the model law are:

$$\pi\_{31} \colon S\_{\rho\_{\text{point}}} = \frac{(\mathcal{S}\_{\text{At}})^3 \cdot (\mathcal{S}\_{\text{\tau}})^3 \cdot \left(\mathcal{S}\_{\lambda\_{\text{x\\_starol}}}\right)^3 \cdot \mathcal{S}\_{\text{\xi}}}{\left(\mathcal{S}\_{\dot{Q}}\right)^2 \cdot \mathcal{S}\_{L\_l}},\tag{90}$$

$$
\pi\_{\text{32}} : \ S\_{\lambda\_{\text{x\\_point}}} = S\_{\lambda\_{\text{x\\_stcal'}}} \tag{91}
$$

$$
\pi\_{33} \colon S\_{\lambda\_{r\, p\,int}} = \frac{S\_{\dot{Q}}}{S\_{\Delta t}} \cdot \sqrt{\frac{S\_{\xi}}{S\_{L\_t}}} \tag{92}
$$

$$
\pi\_{\text{34}} : S\_{\text{sl}\_{nr\\_point}} = \frac{S\_{\text{λ}\_{x\\_stcl}}}{S\_{L\_l}},
\tag{93}
$$

$$
\pi\_{\text{35}} \colon S\_{\delta\_{r\text{ point}}} = \sqrt{\frac{S\_{L\_t}}{S\_{\xi}}}.\tag{94}
$$

#### **4. Discussion and Conclusions**

The relations deduced in the paper for the case of the straight bar of the full circular section can be applied without problems to the tubular (ring) bars, both to the resistance structures formed/constituted by them, as well as the reticular structures used in the roofs of industrial halls, gyms, etc.

In these cases, of the structures made of straight bar elements, on the prototype and on the model, the homologous points (and sections) will be identified, with the help of which the thermal stresses on the model will be transferred to the prototype using of the model law.

It is clear that the internationally recognized work and achievements of Sedov [23], as well as other notable scientists [1–5,8,13,22,25–28,30], are not disputed in any way by the authors of this paper. However, a number of difficulties need to be highlighted in addressing the issue of dimensional analysis by them and other illustrious authors compared to the methodology developed by Szirtes, the author of the works [31,32] namely:


the phenomenon of the main measure (dimensions), which takes place in each author according to his own logic, so it is a non-unitary approach to the phenomenon. Thus, based on these approaches, different sets of dimensionless variables may result, which may even represent combinations of those deduced by other authors [36,38–47].


On the contrary, the methodology, called MDA, developed by Szirtes [31,32], represents a unified approach, easy and particularly accessible to any engineer, without requiring deep/grounded knowledge of the phenomenon, but only reviewing all parameters/variables that could have any influence on it.

Here, they are defined, in a unitary and unambiguous way, on the basis of a clear and particularly accessible protocol/procedure:


In the works [36–40,42,46,47] the classical approach is applied to determining the exponents, which will define the dimensionless groups. Thus, they are used either for the normalization of the known differential relations or the evaluation of the main dimensions and later the establishment of some combinations of the variables in order to obtain dimensionless groups.

In the paper [41], the dimensionless groups are arbitrarily defined, based on a combination, according to their own logic.

The only paper in which approaches closer to MDA were found is paper [35], where the determination of exponents was based on the methodology presented in [43], but does not specify how to choose independent or dependent variables, which is a deficiency of the methodology presented in [43] by Langhaar. In contrast, in Szirtes's work, i.e., in [31,32], each time, these independent variables are rigorously chosen, taking into account how an experiment of the model can be conducted more easily, allowing the model to be designed as favorably as possible for the experiments.

The author of the paper [44] uses the choice of independent and dependent variables but applies the standard methodology for determining exponents by solving the system of linear equations, which describes the phenomenon.

The main advantage of MDA in setting the content of these groups of variables is that the elimination of some variables from this whole set does not influence the ones that remain. In other words, the expressions of a certain set will not be influenced if some of the dependent variables are considered or not.

Accordingly, if the whole set of the variables specific to the beam coated with intumescent paint was conceived, representing 35 expressions that define the model law, a certain number of dependent variables can be neglected without affecting the rest of the expressions.

In the above-described protocols, the general cases are indicated, from which several particular cases can be obtained.

Moreover, if for the prototype and model, a certain variable has identical values, then they can be ignored due to the fact that their scale factor became *S<sup>η</sup>* = 1 and consequently one will resolve useful particular cases similarly with the following:


It is also important to mention that, using the MDA, the model can be differently conceived from the prototype (another material, another coat of paint, etc.), which reveals once again the incontestable advantages of the method proposed in [30,31] as compared to the classical dimensional analysis;

Another conclusion is that for tubular sections, where the thickness of the tube is *δr*, the expression of the model law corresponding to length *Lr*, which is identical to *rcyl*, can be applied to the thickness of the tube too. Therefore, the model law is valid also for tubular sections if the same scale is adopted as for *Lr* and *rcyl*.

To the best knowledge of the present authors there are no studies reporting the application of the *MDA* method to the heat transfer in circular bars.

**Author Contributions:** Conceptualization, B.-P.G., I.S. and D.S, .; methodology, B.-P.G., I.S. and D.S, .; software, B.-P.G.; validation, B.-P.G., I.S., D.S, . and S.V.; formal analysis, B.-P.G., I.S. and D.S, .; investigation, B.-P.G., I.S. and D.S, .; resources, B.-P.G., I.S., D.S, . and S.V.; data curation, B.-P.G., I.S. and D.S, .; writing—original draft preparation, I.S.; writing—review and editing, B.-P.G., I.S., D.S, . and S.V.; visualization, B.-P.G., I.S., D.S, . and S.V.; supervision, B.-P.G., I.S., D.S, . and S.V.; project administration, I.S.; funding acquisition, B.-P.G., I.S., D.S, . and S.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding. The APC was funded by the Transilvania University of Brasov.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
