*Article* **Thermal Scaling of Transient Heat Transfer in a Round Cladded Rod with Modern Dimensional Analysis**

**Botond-Pál Gálfi 1, Ioan Száva 2, Daniela S, ova 2,\* and Sorin Vlase 2,3,\***


**Abstract:** Heat transfer analysis can be studied efficiently with the help of so-called modern dimensional analysis (MDA), which offers a uniform and easy approach, without requiring in-depth knowledge of the phenomenon by only taking into account variables that may have some influence. After a brief presentation of the advantages of this method (MDA), the authors applied it to the study of heat transfer in straight bars of solid circular section, protected but not thermally protected with layers of intumescent paints. Two cases (two sets of independent variables) were considered, which could be easily tracked by experimental measurements. The main advantages of the model law obtained are presented, being characterized by flexibility, accuracy, and simplicity. Additionally, this law and the MDA approach allow us to obtain much more advantageous models from an experimental point of view, with the geometric analogy of the model with the prototype not being a necessary condition. To the best knowledge of the present authors there are no studies reporting the application of the MDA method as it was used in this paper to heat transfer.

**Keywords:** geometric analogy; similarity theory; dimensional analysis; model law; heat transfer; straight bar

#### **1. Introduction**

*1.1. General Considerations*

The idea of dimensional analysis and its practical application dates from the end of the 18th century. The introduction of fundamental units allowed for the creation of some theoretical bases for the application of dimensional analysis in the verification of the correctness of some obtained formulas.

The method of dimensional analysis was conceived and developed in the last century by mathematicians and engineers in order to facilitate experimental investigations of complex structures, as well as difficult to reproduce phenomena, through the easier study of their small-scale models.

This method involves attaching a model (usually scaled down) to the actual structure, called a prototype. The experimental and theoretical study will be carried out/performed on the model, and the results obtained will be transferred to the prototype based on the rigorous application of the model law, specific to dimensional analysis.

The law of the model consists of a finite and well-determined number of dimensionless variables, established by Buckingham's theorem, which have as a starting point precisely the set of variables that intervene in the description of the respective physical phenomenon.

In the classical version (classical dimensional analysis—CDA), obtaining the model law, involves following one of the following paths:

• by the direct application of Buckingham's theorem, presented in detail in the papers mentioned in the paper;

**Citation:** Gálfi, B.-P.; Száva, I.; S, ova, D.; Vlase, S. Thermal Scaling of Transient Heat Transfer in a Round Cladded Rod with Modern Dimensional Analysis. *Mathematics* **2021**, *9*, 1875. https://doi.org/ 10.3390/math9161875

Academic Editor: Sergey Dashkovskiy

Received: 18 June 2021 Accepted: 2 August 2021 Published: 6 August 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).


These ways of obtaining the desired dimensionless groups, which in fact constitute the law of the model, represent quite a difficult and at the same time arbitrary method, which also presuppose the thorough knowledge of the pursued phenomenon.

Compared to these, the method called modern dimensional analysis (MDA) offers a unique and simple way to obtain the model law, requiring only the consideration of all variables that could have an influence on the phenomenon, which is a clear advantage to the MDA. In this case, the complete set of dimensionless groups is obtained, and thus the complete version of the model law.

From this complete variant, based on the exclusion of some physical or dimensional variables irrelevant to the studied phenomenon, will result the model law, which most accurately describes the model–prototype correlation. Thus, based on a unique and simple approach, those correlations will be established, i.e., the model law, which ensures the transfer of the information obtained on the model to the prototype.

In this paper, the authors established that only the law of the model, as shown in paragraph 3.2 (of the variant I studied), can be applied to a concrete case.

A series of papers present the advantages of dimensional analysis [1,2] and the limitations of using this method [3,4]. The basic results in the application of this method have been obtained in recent decades [5–8]. The fundamentals of the method are consistently developed and used in applications [9–13].

From all the fields in which the method of dimensional analysis has been applied, we referred only to its application to heat transfer, which will be the subject of this article.

Some particular cases of heat transfer have been used in the literature. The complexity of a heat transfer problem is significantly reduced using the dimensional analysis method and transforming the problem in a scale-free form. For example, this method is used to study the dimensionless groups in irradiated particle-laden turbulence [14]. For such systems it is concluded that two dimensionless groups are important in the system's thermal response.

An experimental study on the convection heat transfer coefficient and pressure drop values of CO2 led to the use of the dimensional analysis technique to develop correlations between Nusselt numbers and pressure drops [15]. Other example of the dimensional analysis in the case of heat transfer are presented in the literature [16–20].

The complexity and nonlinearity of mechanical or thermal phenomena require a new approach regarding the correlation of experimental results with theoretical data, which requires the development of pertinent mathematical models [21]. The conventional analysis usually involves many trials and diagrams with measurement results.

#### *1.2. Dimensional Modelling, a Design Tool for Heat Transfer Analysis*

Starting from the geometric analogy, a first more efficient approach is given by the similarity theory [22,23], where alongside the prototype, the model—usually a small-scaled model—is defined. The governing equations applied to the prototype are obtained by means of the model's behavior [24,25]. The model must accurately reflect the behavior of the prototype. The similarity between prototype and model is structural or functional. The structural similarity highlights mainly the geometric similarity between prototype and model, while the functional similarity aims to find corresponding equations that describe both prototype and model. Additionally, geometric similarity supposes proportionality between length and angle equality for the prototype and model. Thus, homologous points, lines, surfaces, and volumes of the prototype and model can be defined. Functional

similarity involves similar processes in both systems, prototype and model, that take place at similar times, i.e., the accomplishment of the similarity of all physical properties that govern the analyzed process. This kind of similarity can be kinematic or dynamic, and the phenomena occur so that, in homologous points, at homologous times, each dimension *η* is characterized by a constant ratio between the values corresponding to the model and prototype, *Sη*. These dimensionless ratios, which are constant in time and space, are scale factors of the dimensions involved or similarity ratios. The scale factor *S<sup>η</sup>* is defined as the ratio between the value of the dimension corresponding to the model (*η*2) and the prototype, respectively (*η*1):

$$S\_{\eta} = \frac{\eta\_2}{\eta\_1} \begin{bmatrix} - \end{bmatrix} \tag{1}$$

The reverse of *S<sup>η</sup>* represents the coefficient of transition from the original to the model [21]. There are as many scale factors as dimensions describing the phenomenon. Practically, the mathematical solution of the complex equations that theoretically describe the actual phenomenon is replaced by correlations between dimensionless parameters, which are obtained from the fundamental relations of the phenomenon by a suitable grouping of dimensions, called similarity parameters, such as *Nu, Re, St, Pr*, etc. Therefore, the dimensions are replaced by the corresponding scale factors, multiplied by constants, and by an appropriate grouping, the similarity parameters are obtained, and correlations among them, such as *Nu* = *f*(*Re*, *Pr*, *Gr*, ...), are also obtained. By means of experimental measurements, these correlations simplify the analysis performed and allow a reduction in the number of measurements in order to obtain important parameters of the phenomenon.

Among the basic theorems of similarity, two of them can be highlighted:

	- to be of the same nature;
	- to have the same determinant parameters of similarity;
	- to have the same initial and boundary conditions.

In the case of complex phenomena, the number of dimensionless parameter scales of involved variables and correlations increases very much and therefore the similarity theory must be replaced by a more efficient method that is the dimensional analysis [26]. The main aspects concerning the similarity theory and dimensional analysis are indicated in [27–30].

#### *1.3. Classical Dimensional Analysis (CDA)*

There is in this case a model that will be analyzed instead of the prototype, and as a result of the experiments carried out on the model, by means of dimensionless relations (dimensionless groups *πj*), the behavior of the prototype can be predicted, obviously in conditions of similarity.

By using the *π<sup>j</sup>* groups, *CDA* simplifies very much the experimental investigations and the graphical representations, and the results have a high degree of abstraction and generality. The works [26,29] present in detail the main *π<sup>j</sup>* groups that describe thermal energy processes.

*CDA* is not a substitute for experimental measurements and does not have the purpose of explaining physical phenomena; it aims to simplify and optimize the design of experiments by grouping measurable parameters of a phenomenon in dimensionless groups, defined by Buckingham's *π* theorem. Both model and prototype obey in their behavior the conditions set out in the *π<sup>j</sup>* group.

By using *CDA*, the *π<sup>j</sup>* groups can be set in one of the following ways:


dimensionless quantities and then, by their suitably grouping, the *π<sup>j</sup>* groups are obtained;

• by identifying the full form, but also the simplest equation(s) that describe the phenomenon, which will be transformed into dimensionless forms from which the desired *π<sup>j</sup>* groups will be selected.

According to [24,29] the Buckingham's *π* theorem has the following statement: the required number of independent dimensionless groups formed by combining the variables of a phenomenon is equal to the total number of these quantities minus the number of primary units of measurement that is necessary to express the dimensional relations of the physical quantities.

Consider a process that can be described by a set of independent parameters *yi* , *i* = 1, 2, . . . , *n* by means of the general relation:

$$f(y\_1, y\_2, y\_3, \dots, y\_n) = 0,\tag{2}$$

For describing the *n* quantities, *m* primary units of measurement are required and thus, from Buckingham's theorem, (*n* − *m*) independent *π<sup>j</sup>* dimensionless groups can be formed that are able to describe the considered process. They are in a similar relation:

$$F(\pi\_{1\prime}, \pi\_{2\prime}, \dots, \pi\_{n-m}) = 0,\tag{3}$$

The set of relations is given by:

$$\pi\_{\vec{l}} = \mathbb{F}\_{\vec{l}}(\pi\_1, \pi\_2, \dots, \pi\_{n-m}), \ j = 1, 2, \dots, \ (n-m) \; , \tag{4}$$

The functional relationship among the *π<sup>j</sup>* groups is obtained from trials. As mentioned in [21], *CDA* involves three steps, namely:


Thus, the *π<sup>j</sup>* groups are defined as products of the representative quantities that are involved in describing the phenomenon having unknown exponents (*a*, *b*, *c*,...). From the condition that all the *π<sup>j</sup>* groups are dimensionless (the sum of the exponents of each primary dimension must be zero), a system of equations will be obtained where the unknowns are the exponents. It is a multiple indeterminate system, where convenient values are given from the beginning to the exponents of the primary units, while the rest of the unknown exponents are determined from the solution of the system. Finally, the total number of *π<sup>j</sup>* groups will be obtained.

Unfortunately, all approaches of the *CDA* show several shortcomings. That is why the original method described in [31,32], called modern dimensional analysis (*MDA*), is according to the authors, the most efficient and easy way to approach dimensional analysis.

#### *1.4. Objectives and Purpose of the Paper*

This paper represents a theoretical and experimental study on the implementation of modern dimensional analysis (MDA) in solving the problem of heat transfer, especially to the metal structures used in civil and industrial constructions, protected or unprotected with layers of intumescent paints. A fire protection, in addition to maintaining the flexibility of the original structure, leads both to maintaining the initial load-bearing capacity of the resistance structure for a longer time in case of fire and to increase the guaranteed time for evacuation of persons and property subjected to fire. Other recent studies concerning dimensional analysis are presented in [33–41].

In this article, the authors set out to achieve the following major objectives:


The aim of the manuscript is to apply modern dimensional analysis to the heat transfer in a circular bar. The heat transfer in the bar is transitory. The bar is placed in air; therefore, the boundary condition is convection. The heat transfer coefficients were considered among the other variables in applying *MDA*. As indicated in the manuscript, when using *MDA*, the relations of the model law are correlations among variables that are involved in the phenomenon, and they must not be compared with the physical relations that describe the phenomenon. In contrast with the classical dimensional analysis, MDA considers the variables that might influence the phenomenon, without requiring a thorough knowledge of the phenomenon and the governing relations. The relations of the model law can be extended to bars with tubular section and structures of bars with annular cross-section. This is also an advantage in using *MDA*. To the best knowledge of the authors, the heat transfer in a circular bars described by *MDA* has not been reported before in the literature.

#### **2. Method of Analysis in Modern Dimensional Analysis (***MDA***)**

In a physical relation there is a single dependent variable and a finite number of independent variables. The variables are denoted by (*H*<sup>1</sup> , *H*<sup>2</sup> , *H*<sup>3</sup> , ...), while their dimensions are denoted by (*h*<sup>1</sup> , *h*<sup>2</sup> , *h*<sup>3</sup> , ...). The derived dimensions are obtained from the combination of previously selected primary dimensions, such as *hr*<sup>1</sup> <sup>1</sup> · *<sup>h</sup>r*<sup>2</sup> <sup>2</sup> · *<sup>h</sup>r*<sup>3</sup> 3 · ... · *hrn <sup>n</sup>* (where, *r*<sup>1</sup> , *r*<sup>2</sup> ,*r*<sup>3</sup> ... are the exponents of the primary dimensions, while *n* is the number of the involved primary dimensions). A variable *Hj* has the dimension [*Hj*] = *ϕ<sup>j</sup>* · *h r*<sup>1</sup> *<sup>j</sup>* <sup>1</sup> · *<sup>h</sup> r*2*j* <sup>2</sup> · *<sup>h</sup> r*3*j* <sup>3</sup> · ··, where *<sup>ϕ</sup><sup>j</sup>* is a coefficient.

The author of works [31,32] indicates the following steps for analysis, which were presented in [33]:

• the dimensional matrix (*DM*) is defined; it consists of the exponents of all involved dimensions *hi* that describe all independent variables *Hk* and the dependent one. In the case of four variables, among one is dependent (for instance *H*1), the dimensional relations are:

$$\mathcal{H}\_1 = h\_1^{a\_1} \cdot h\_2^{\theta\_1} \cdot h\_3^{\gamma\_1} \cdot h\_4^{\delta\_1};\\\mathcal{H}\_2 = h\_1^{a\_2} \cdot h\_2^{\theta\_2} \cdot h\_3^{\gamma\_2} \cdot h\_4^{\delta\_2};\\\mathcal{H}\_3 = h\_1^{a\_3} \cdot h\_2^{\theta\_3} \cdot h\_3^{\gamma\_3} \cdot h\_4^{\delta\_3};\\\mathcal{H}\_4 = h\_1^{a\_4} \cdot h\_2^{\theta\_4} \cdot h\_3^{\gamma\_4} \cdot h\_4^{\delta\_4}.\tag{5}$$

The dimensional matrix contains the exponents of these dimensions and is indicated in rel. (6):


Matrix *M*, associated with the dimensional matrix, is:

$$M = \begin{bmatrix} \alpha\_1 & \alpha\_2 & \alpha\_3 & \alpha\_4 \\ \beta\_1 & \beta\_2 & \beta\_3 & \beta\_4 \\ \gamma\_1 & \gamma\_2 & \gamma\_3 & \gamma\_4 \\ \delta\_1 & \delta\_2 & \delta\_3 & \delta\_4 \end{bmatrix} \tag{7}$$

In the general case, there are *N*<sup>V</sup> total variables and *N*<sup>d</sup> primary dimensions that define both the dimensional matrix and the associated one, as a matrix consisting of *N*<sup>d</sup> lines and *N*<sup>V</sup> columns.



$$D \equiv I\_{n \ge n \prime} \tag{8}$$

It should be mentioned that matrix *C* is obtained from the relation:

$$\mathbf{C} = -\left(A^{-1} \cdot \mathbf{B}\right)^{T},\tag{9}$$

Relation (9) is valid if the set of new variables contains only *π<sup>j</sup>* dimensionless quantitates and matrix *D* is a unit matrix.

• the rows *j* = 1, 2, ... , *n* of matrixes *D* and *C* define all *π<sup>j</sup>* dimensionless quantitates. Thus, row *j* of the common matrix (*D* and *C*) contains the exponents that are involved in defining *πj*, which is the product between a dependent variable (from matrix *B*, having the exponent 1) and all involved independent variables (from matrix *A*, having the exponents from the row *j* of matrix *C*). In order to find the model law, the expressions of all *π<sup>j</sup>* dimensionless variables are equal to one. In all products of matrix *D* there is only one dependent variable with exponent 1, while in those of matrix *C* there are all independent variables with the exponents obtained from relation (9).

As mentioned before, in the matrices A, B and C the exponents (*h*1, *h*2, ... , *hm*) of the basic dimensions involved intervene, which helps us to describe the set of variables involved (*H*1, *H*2, *H*3, ..., *Hn*), and in matrix D (which is a unit matrix) these unit values will also represent exponents of dependent variables.

The illustration of how to obtain the elements of the model law is given in Figure 1:

**Figure 1.** The illustration of how to obtain the elements of the model law.

If considering, for example, the dimensionless variable *π*5, on its line there are the exponents of all involved independent variables (*H*9, ... , *H*14), the exponents of the independent variables (*a*<sup>5</sup> , ... , *f*5), as well as the exponent of the dependent variable (*H*5), which is 1, being positioned on the main diagonal of matrix *D*. Consequently, *π*<sup>5</sup> can be written as:

$$
\pi\_5 = \left(H\_5\right)^1 \cdot \left(H\_9\right)^{a\_5} \cdot \left(H\_{10}\right)^{b\_5} \cdot \left(H\_{11}\right)^{c\_5} \cdot \left(H\_{12}\right)^{d\_5} \cdot \left(H\_{13}\right)^{e\_5} \cdot \left(H\_{14}\right)^{f\_5},\tag{10}
$$

As shown before, relation (10) is equal to the unit, and from this equality the dependent variable is expressed (here being *H*5), i.e.,

$$\begin{array}{rcl} \pi\mathfrak{s} = \left(H\mathfrak{s}\right)^{1} \cdot \left(H\mathfrak{s}\right)^{\mathfrak{s}\_{\mathfrak{5}}} \cdot \left(H\_{10}\right)^{\mathfrak{h}\_{\mathfrak{5}}} \cdot \left(H\_{11}\right)^{\mathfrak{c}\_{\mathfrak{5}}} \cdot \left(H\_{12}\right)^{\mathfrak{d}\_{\mathfrak{5}}} \cdot \left(H\_{13}\right)^{\mathfrak{d}\_{\mathfrak{5}}} \cdot \left(H\_{14}\right)^{\mathfrak{c}\_{\mathfrak{5}}} \cdot \left(H\_{14}\right)^{\mathfrak{f}\_{\mathfrak{5}}} = 1 \Rightarrow\\ \Rightarrow \quad H\mathfrak{s} = \frac{1}{\left(H\_{\mathfrak{h}}\right)^{\mathfrak{s}\_{\mathfrak{5}}} \cdot \left(H\_{10}\right)^{\mathfrak{h}\_{\mathfrak{5}}} \cdot \left(H\_{12}\right)^{\mathfrak{d}\_{\mathfrak{5}}} \cdot \left(H\_{13}\right)^{\mathfrak{d}\_{\mathfrak{5}}} \cdot \left(H\_{14}\right)^{\mathfrak{d}\_{\mathfrak{5}}}}{} \;. \end{array} , \end{array} \tag{11}$$

Then, the involved variables (*H*5, *H*9, ... , *H*14) are replaced by the corresponding scale factors (*SHn* ), and finally, the desired expression of the fifth element of the model law is obtained.

Obviously, some of the exponents involved being negative, the relationship obtained will be in the form of an ordinary fraction, where both the numerator and the denominator will have expressions of scale factors at certain powers.

Some observations can be formulated as:


$$\mathbf{C} = -D \cdot \left( A^{-1} \cdot \mathbf{B} \right)^{T},\tag{12}$$

the final expressions of the *π<sup>j</sup>* variables do not change;

	- all parameters that might have an influence upon the phenomenon are considered (total variables of the dimensional set). More information in defining the relevant variables increases the degree of freedom in selecting the properties of the model, and thus a more reliable description of the prototype is possible. Later, based on a careful analysis, the variables that have an insignificant influence can be excluded.
	- the *π<sup>j</sup>* variables can be easily and unitarily determined, which is impossible if *CDA* or the theory of similarity are used. It means that the dimensional set defined by Equation (8) represents the complete set of *π<sup>j</sup>* dimensionless products of variables *Hm*, *m* = *NV*:
	- the calculations required for the arbitrary grouping and analysis used by the two previously mentioned methods, in order to obtain the *π<sup>j</sup>* groups, are

eliminated. They require a thorough knowledge of the phenomenon, thus making *CDA* difficult and inaccessible to many researchers;

