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On the basis of theoretical considerations of convectiveradiative heat transfer, a relationship was developed enabling the total convective and radiative heat flux Q_{C+R} emitted from any object at t_{w} and its surroundings at t_{∞} to be calculated from known values of the surface temperature of such an object, i.e., the known temperature difference Δt=t_{w}  t_{∞} and average air temperature T_{av}. This relationship is applied to thermal imaging cameras with the aim of developing appropriate software to enhance their measurement capabilities. They can then be used not only for monitoring and measuring temperature, local overheating, heat losses through insulation materials, thermal bridges, constructional defects, moisture, etc., but also for measuring the heat losses from any object, such walls and buildings. This empirical relationship includes constants relating to the object itself, such as its characteristic dimension l, surface area A, emissivity ε and temperature parameters, which depend on t_{w}, t_{∞}, Δt and T_{av }and on the physical properties of air. Experimental validation of the proposed relationship, performed for two values of the surface emissivity ε, showing the discrepancies ΔQ_{C+R}=1.75% (for ε=0.884) and 4.85% (for ε=0.932), has confirmed its correctness and its practicability.
convectiveradiative heat transfer, infrared camera, experiments, empirical equation, vertical plate
Heat transfer in air is a complex mechanism involving both convection and radiation. Despite this mutual coexistence, these two means of heat transfer are usually considered separately, and the results obtained for either of them, though most often for convection, described by Newton's equation, are corrected by subtracting the radiant heat flux calculated with the StefanBoltzmann equation. On the other hand, when determining heat losses, the calculated radiative heat loss flux is added to the measured convective heat flux. To determine the convective heat flux, it is necessary to know the heat transfer coefficient α, the dimensions l and b of the object under consideration and its surface temperature t_{w}, as well as the ambient air temperature in the undisturbed area t_{∞}. In addition, for the radiant flux the temperature of the surrounding walls t_{ot} ≈ t_{∞} and the surface emissivity ε are also required: this latter quantity should be measured, assumed, or be taken from a set of tables that assign their values to typical surfaces (polished copper, brick, plaster, wood, etc.) [1, 2].
At lower temperatures, convection becomes more important and is of great interest to researchers, even though it involves a much higher level of complexity. This emerges from its physical description, which in the case of free convection requires three conjugate partial differential equations (continuity, NavierStokes and FourierKirchhoff), the mathematical methods of solving them, and carrying out the relevant experimentation. These difficulties, far from discouraging researchers, have inspired them to study convection even more intensively. At the same time, the only reward they could count on was scientific satisfaction, because industry was not interested in this lowintensity mode of heat transfer. Times and attitudes have changed, however: the primary aim nowadays is to conserve energy, for example, by limiting heat losses, mainly caused by free convection, especially in the construction, energy and metallurgy industries.
At higher temperatures, radiative heat transfer is more important, for the description of which the Planck, Wien, StefanBoltzmann and other equations are necessary [3, 4]. In external heat transfer and for simple spatial configurations, especially at low temperatures, radiation can be described only by the StefanBoltzmann equation, but its practical use is troublesome. To determine the radiative heat flux from this equation, it is not enough to know the temperatures of the heating surface t_{w} and the surroundings t_{∞}, and the emissivity ε. One also needs to know the temperature of surrounding objects, as well as the humidity and pressure of the air. In the case of internal heat transfer, however, e.g. in heat exchangers, engine cylinder channels or combustion chambers, when the areas of a heated surface A_{w} and its surroundings A_{∞ }optically interact with each other or take place in optically active media, any considerations of radiation must take into account not only the StefanBoltzmann equation, but Kirchhoff’s and Lambert’s laws [5] as well, with which the equivalent emissivity factor ε_{w∞} and the configuration factor φ=A_{w }/ A_{∞ }can be determined. If A_{w} and A_{∞} are parallel, then ε_{w∞}=1 / (1 /ε_{w }+ 1/ε_{∞}  1), and φ=1 [6]; but when A_{w} is inside A_{∞ }(A_{w }<< A_{∞}), then ε_{w∞} ≈ ε_{w}=ε [7]. Many studies have addressed various configurations of radiative heat transfer surfaces, for example [79].
In order to eliminate the effects of radiation in experimental convection studies, some researchers took various approaches, performing experiments in water [1013], glycerine [1416] or other liquids like ethylene glycol [17], in which radiation does occur but is much smaller than convection in water (assumption Q_{R}=0). Others used polished copper or aluminium plates [1821] assuming that when ε=0, Q_{R} is also equal to 0. The remainder calculated the radiative flux from the StefanBoltzmann equation for given values of ε and subtracted it from the heating power of the tested surfaces.
The following convectiveradiation studies of heat transfer in a confined space are of interest: the numerical examination of the impact of solar radiation on the conversion of heat transfer from conduction to convection in a threedimensional (3D) shallow wedge [22], and an investigation, likewise numerical, of the influence of the absorbingemittingscattering effect in a twodimensional (2D) square cavity on convectiveradiative heat transfer [23]. Similar problems were studied in a partitioned rectangular enclosure with semitransparent walls [24], in a cavity with a porous horizontal layer [25], in a side open cavity [26], and also with regard to heat transfer by conduction in the study [27]. An interesting numerical study examining coupled natural convection and surface radiation through an open fracture in a solid wall facing a reservoir containing isothermal quiescent air was reported in the study [28]. A similar study of coupled free convection and radiative heat transfer in a cavity containing an isothermal vertical plate and filled with carbon dioxide or nitrogen was carried out using holographic interferometry [29]. Another numerical study using a participating medium (like carbon dioxide and water vapour) in the channel between two vertical parallel plates was reported in the study [30]. An experimental study of free convection and radiative heat transfer in air inside rectangular closed cavities of different aspect ratio with two vertical active walls (hot and cold) was described in the study [6]. The results of experimental studies of heat exchange in the channel between two symmetrically heated isothermal vertical walls using a thermal imaging camera, in which the contribution of radiation was limited by the polished surfaces of aluminium plates, were given in the studies [31, 32]. In paper [33] the relationship between radiation and convection from a vertical flat plate was interferometrically and numerically (with the FLUENT programme) investigated. The influence of its emissivity and internal heat conduction were also taken into consideration in it.
Numerical and theoretical considerations of convectiveradiative heat exchange inside square and rectangular enclosures using the commercial software FLUENT 6.3 and Dimensional Analysis were given in the study [34]. The influence of radiation on laminar convective heat transfer determined in optically active media was numerically tested using the Rosseland approximation [35], and with reference to an isothermal vertical plate in the studies [3642].
Determination of the specific heat of a vertical plate material and the emissivity of its surface with the use of transient cooling of the solid system consisting in convectiveradiative cooling in the air, in the temperature range t=42 – 142℃ and Ra=2^{.}10^{6}  2^{.}10^{7}, was described in papers [43, 44]. Experimental studies of convectiveradiative heat transfer from a horizontal cylinder were described in the study [45] and from an isothermal vertical slender cylinder in [46], and different cases that can be found also in the studies [4648]. The same problem, but theoretically using a similarity solution in relation to the vertical plate, was investigated and described in the studies [4952]. The interaction of convection with radiation for a plate inclined at a small angle to the horizontal was analysed using the appropriate transformation and then solving the resulting local nonsimilarity equations numerically [53]. The solution of dimensionless boundarylayer equations of secondorder interactions between radiation and free laminar convection from a vertical, black and isothermal plate to the surrounding grey gas was given in the studies [54, 55]. The effect of radiation on free convection was investigated experimentally in the corner formed by a horizontal plate and a vertical fin of height 30, 50 and 70 mm coated with paints of different colours to change the emissivity from 0.05 to 0.85 [56].
Clearly, there is a growing demand for research results relating, among other things, to combined convectiveradiative heat exchange in construction, industry and in numerous technological problems, including internal combustion engines, furnace design, nuclear reactor safety, fluidized bed pyrolysis, heat exchangers, solar collectors and photo and biochemical reactors. These entities are more interested in the magnitude of the heat flux supplied or emitted from a real heat exchange surface than in the heat transfer mechanisms, the participation of convection and radiation in them, the simplifying assumptions made, the experimental procedures etc. The response to this heightened interest in applications of overall convective and radiative heat transfer can be found in papers dealing with heat losses in construction [5762] and from the housings of industrial devices (motors, pumps, exchangers) [63], and the heat fluxes transferred within specific devices, such as baking ovens [64], internal combustion engine cylinders [65] or glass furnaces [66, 67].
As mentioned above, convection and radiation have usually been considerated separately. But in the case of air, they always coexist and can interact to varying extents, depending on the temperature. The proportion of heat transfer by radiation is higher than by convection and increases as the temperature of the heating surface rises. The study of the interactions of these two heat transfer mechanisms and the search for the possibility of their joint mathematical description in the form of a single empirical equation is the subject of this paper.
2.1 Heat transfer from heated surfaces in air
The density of heat loss from a building, which is the sum of radiation and convection losses q_{c+R}, depends mainly on the temperature difference between the heat transfer surface t_{w} and the ambient temperature t_{∞} (1). These are not only direct losses through the walls of the building, but also indirect losses through nonresidential spaces (staircases, attic and cellars) as shown in the diagram in Figure 1. The mechanism of heat losses from rooms, through walls and ceiling, is connected with convective and radiative heat transfer. From the basement, on the other hand, these losses occur by conduction to the foundations and then to the ground.
Since the temperature difference (t_{w,in}  t_{ground}) as well as the thermal conductivity of the foundations have small values, according to Fourier's law this heat loss flux by conduction was neglected q_{loss},_{floor }≈ 0, and further consideration was focused on the relevant convective q_{c} and radiative q_{r} partial heat loss fluxes and the total one q_{c+R} in generalized terms: for t_{w}, t_{∞} and the vertical surface. The obtained relations for other configurations of planar (horizontal, oblique), cylindrical or conical surfaces can be corrected by substituting the relevant C_{C} and n constants for these configurations in the NusseltRayleigh criterion relationships (1  5).
Figure 1. Types of heat loss fluxes in a model of a building with an indication of how convective heat fluxes can be directly measured, using thermal imaging camera with a grid [19, 68]
During heating or cooling in air, the temperature difference between a heated surface t_{w} and the surroundings t_{∞} causes heat to be exchanged by convection (according to Newton's law, this is proportional to the heat transfer coefficient α_{C}), and by radiation (according to the StefanBoltzmann theory, this is proportional to the coefficient α_{R}). The total heat flux q_{c+R} can be written as:
$\begin{aligned} q_{C+R}=q_C+q_R= & \alpha_C \cdot \Delta t+\sigma \cdot \varepsilon \cdot\left(T_w^4T_{\infty}^4\right)= \alpha_{C+R} \cdot \Delta t\end{aligned}$ (1)
where:
$\alpha_C=\frac{\lambda}{l} \cdot N u_C=\frac{\lambda}{l} \cdot C_C \cdot R a^n$, (2)
And similarly
$\alpha_{C+R}=\frac{1}{l} \cdot N u_{C+R}=\frac{1}{l} \cdot C_{C+R} \cdot R a^n$, (3)
Substituting (2) and (3) in (1) gives:
$\frac{\lambda}{l} \cdot C_C \cdot R a^n+\frac{\sigma \cdot \varepsilon}{\Delta t} \cdot\left(T_w^4T_{\infty}^4\right)=\frac{\lambda}{l} \cdot C_{C+R} \cdot R a^n$ , (4)
And then
$C_{C+R}=C_C+\frac{\sigma \cdot \varepsilon \cdot l}{\lambda \cdot \Delta t \cdot R a^n} \cdot\left(T_w^4T_{\infty}^4\right)$. (5)
Analysis of Eq. (5) suggests the following cases:
C_{C} values are known from the literature, where they are given in the form of the dependence of Nusselt and Rayleigh numbers obtained from experimental, theoretical and numerical research:
$N u_{\mathrm{c}}=C_{\mathrm{C}} \cdot R a^{\mathrm{n}}$. (6)
The value of C_{C+R} can be determined from the measured heating power q and the heat loss fluxes q_{str}, as:
$\alpha_{C+R} \cdot \Delta t=q_{C+R}=qq_{s t r}$, (7)
$C_{C+R}=\frac{QQ_{\text {loss }}}{A \cdot \Delta t}=\frac{U \cdot I \cdot l}{A \cdot \lambda \cdot \Delta t \cdot R a^n}$ (8)
For electric heating with a current power Q=U ∙ I, heat transfer area A, characteristic linear dimension l, and when the heat flux is transferred in its entirety to the heated medium (q_{str=}0).
The emissivity ε of a heated surface is known or can be determined, e.g., with a thermal imaging camera; nonetheless, its value is encumbered with a certain error [69, 70].
Any combination of the three above cases.
2.2 Vertical heated plate
Further considerations of the coexisting mechanisms of heat exchange in the air, i.e., natural convection and radiation, were carried out on the longest tested and best known configuration of the heated surface, i.e. The vertical heated plate. Until recently, the results of theoretical, experimental and numerical research obtained for this were the most frequently published in the scientific literature on free convection, so this configuration of the heated surface is the most representative for validating the correctness of the considerations discussed in this paper.
Figure 2. Draft scheme of a thin, doublesided, heated isothermal vertical plate
In the case of electric heating, the heating power can be written as Q=U ^{. }I, and in the case of a doublesided vertical plate of width b, height h and area A=2^{.}b^{.}h, the heat energy flux q_{c+R }transferred to the environment consisting, according to Figure 2 and Eq. (1), of two convective q_{c}/2 and two radiative q_{r}/2 fluxes, is proportional to the equivalent (convectiveradiative) heat transfer coefficient α_{C+R}, according to the relationship:
$q_{C+R}=\alpha_{C+R} \cdot \Delta t=\frac{QQ_{\text {loss }}}{A}=\frac{U \cdot IQ_{\text {loss }}}{A}$. (9)
Assuming for a thin, doublesided plate that the heat loss Q_{loss}=0 [71], the total heat transfer coefficient α_{C+R }and the Nusselt number Nu_{c+R} can be written as:
$\alpha_{C+R}=\frac{U \cdot I}{A \cdot \Delta t}=\frac{U \cdot I}{2 \cdot b \cdot h \cdot \Delta t}$, (10)
$N u_{C+R}=\frac{\alpha_{C+R} \quad \cdot h}{\lambda}=\frac{U \cdot I}{2 b \cdot \lambda \cdot \Lambda t}$ (11)
From the NusseltRayleigh criterion dependence, analogous to (6) and (2), but for a vertical plate where n=1/4,
$N u_{\mathrm{C}+\mathrm{R}}=C \mathrm{C}+\mathrm{R} \cdot R a^{1 / 4}$, (12)
The constant C_{C+R}, described in detail by (5), can be calculated for specific temperature conditions, determined by the Rayleigh number (16):
$C_{C+R}=\frac{N u_{C+R}}{R a^{1 / 4}}$. (13)
With the results of experimental tests, in the form of known values of t_{w}, t_{∞}, U, I, b, h for temperature conditions (Δt, t_{av}=(t_{w} + t_{∞})/2), the physical properties of air (β, λ, ν and a), and then Nu_{c+R}, Ra and C_{C+R} can be calculated:
$\frac{N u_{C+R}}{R a^{1 / 4}}=\frac{U \cdot I}{2 b \cdot \lambda \cdot \Delta t \cdot R a^{1 / 4}}=C_{C+R} \quad$. (14)
2.2.1 The case where the heat flux is known
In industry, construction and agriculture (breeding, greenhouse crops), where the lowest possible costs of production or exploitation have to be balanced against high energy costs, technical and economic information on flux values and the amount of heat transferred in technological or operational processes is the most important. Without this information, it is hard to take responsible decisions, manage energy optimally and economically, and reduce costs.
When electrical energy is converted into thermal energy, information on the power of heating or cooling devices and the amount of heat energy they transfer can be obtained by direct measurement of the electrical heating power Q=U^{.}I, taking into account the efficiency of conversion, and their operation time $\tau$. In the case of other energy carriers, e.g., heating or cooling media, to determine the amount of transferred thermal energy, the temperature drop t_{in}  t_{out} of the medium and its mass flow rate M have to be known. In the case of steam heating, the measure is the amount of condensate or the decrease in the enthalpy of the heating steam, etc.
In this situation, the value of C_{C+R} and the heat exchange mechanisms are of secondary importance to the user. Nevertheless, a knowledge of C_{C+R}, calculated from a known value of q_{c+R}, may be useful for analysing other cases of convective heat transfer in air. For this purpose, after rearranging Eq. (8), one obtains:
$\begin{aligned} Q_{C+R}=\frac{U \cdot I}{A}=\alpha_{C+R} \cdot \Delta t & =\frac{\lambda}{h} \cdot N u_{C+R} \cdot \Delta t=\frac{\lambda}{h} \cdot C_{C+R} \cdot R a^{\frac{1}{4}} \cdot \Delta t\end{aligned}$, (15)
$C_{C+R}=\frac{U \cdot I \cdot h}{A \cdot \lambda \cdot \Delta t \cdot R a^{1 / 4}}=\frac{U \cdot I}{2 \cdot b \cdot \lambda \cdot\left(T_WT_{\infty}\right)} \cdot R a^{1 / 4}$. (16)
The values of C_{C+R} obtained for specific cases and the heating powers, temperatures, physical properties of air and Ra measured for them, can be used to determine q_{c+R} and Q_{C+R} in other heating or cooling systems, where their direct measurement is impossible. These cases are discussed in the next section.
2.2.2 The case where the heat flux and power are not known
It is not always possible to directly measure the heat flux q_{c+R} transferred by the heating medium. This applies, for example, to the heating of spaces with heat pumps, solar collectors, air heaters, as well as heat losses from buildings, cooling of electronic systems, etc. In these cases, to determine the power and amount of transferred heat, it is necessary to know the values of the coefficients C_{C+R} or C_{C} and C_{R} according to the NusseltRayleigh criteria, which can be used to determine the power Q and the heat flux q=Q/A transferred from the heating device to the environment.
With a known (literature) or experimentally determined value of C_{C+R}, the heat transfer flux from surface A can be calculated from (15) as follows:
$q_{C+R}=\frac{\lambda}{h} \cdot C_{C+R} \cdot \Delta t \cdot R a^{1 / 4}$, (17)
$Q_{C+R}=q_{C+R} \cdot A=2 \cdot b \cdot \lambda \cdot C_{C+R} \cdot \Delta t \cdot R a^{1 / 4}$. (18)
If the heated or cooled object consists not only of a single doublesided heated vertical plate, but also of i such plates or vertical pipes with a diameter d, then the area should be appropriately corrected in formula (18). A=2^{.}b^{.}h^{.}i or A=2^{.}π^{.}d^{.}h^{.}i should be substituted for A =2^{.}b^{.}h, and if such an object contains horizontal components, e.g., a horizontal plate, cuboid or pipe, then the appropriate characteristic linear dimension l and exponent n for Ra should be used in addition.
The value of C_{C+R} can also be determined from Eq. (5), if one knows the literature values of the convective coefficient C_{C} and the calculated radiative heat flux.
The coefficient C_{C} for an isothermal vertical plate has historically taken the following example values: C_{C}=Nu/Ra^{1/4}=0.571 [72],=0.560 [73],=0.540 [74]. In the paper [75] a total of 25 plates of this type were analysed, giving an average value of C_{C,av}=Nu/Ra^{0.252}=0.550. This value, however, is based on the unusual exponent n for a vertical plate that is inconvenient to use, especially when comparing the results. In [71], the coefficients C_{C} and n in the NusseltRayleigh relationship were converted for the assumed values of Ra=10^{6, 7, 8, 9} from n=0.252 to n=0.250, and the following new relationships were obtained:
$\begin{gathered}N u_{\mathrm{c}, \mathrm{av}}=0.550 \cdot R a^{0.252},=0.565 \cdot R a^{0.25}\left(R a=10^6\right), =0.573 \cdot R a^{0.25}\left(R a=10^9\right)\end{gathered}$ (19)
$N u_c=0.569 \cdot R a^{0.25}\left(R a=10^610^9\right)[71]$ (20)
This dependence differs from that given in the study [52] (C_{C=}0.555) by only 2.5% and is therefore used later in this paper.
Knowledge of the second part of Eq. (5), i.e., the radiative heat flux, requires the emissivity of the radiator surface ε to be known or measured in addition to the temperatures T_{w} and T_{∞}. Taking into account both components of the heat flux (convection and radiation), the following relationship can be obtained:
$C_{C+R}=0.569+\frac{\sigma \cdot \varepsilon \cdot l}{\lambda \cdot \Delta t \cdot R a^n} \cdot\left(T_w^4T_{\infty}^4\right)$ (21)
In technical issues, the most important thing is to know the values of α_{C+R} and the heat transfer flux q_{c+R}, which can be calculated from Eq. (21) based on Eqns. (2), (3) and (4):
$\alpha_{C+R}=\frac{\lambda}{l} \cdot 0.569 \cdot R a^{\frac{1}{4}}+\frac{\sigma \cdot \varepsilon}{\Delta t} \cdot\left(T_w^4T_{\infty}^4\right)$ (22)
$\begin{array}{r}q_{C+R}=\alpha_{C+R} \cdot \Delta t=\frac{\lambda}{l} \cdot 0.569 \cdot \Delta t \cdot R a^{\frac{1}{4}}+ \sigma \cdot \varepsilon \cdot\left(T_w^4T_{\infty}^4\right) .\end{array}$. (23)
3.1 Simulation calculations C_{C+R=}f(t_{w}, Δt, l, ε)
Simulation calculations, as well as theoretical, numerical and experimental considerations, require the knowledge or assumption of the temperatures of the heated surface (t_{w}, T_{w}=t_{w}+273.15), the undisturbed area (t_{∞}, T_{∞}=t_{∞}+273.15), the difference between them Δt=ΔT=t_{w }–t_{∞} and the average air temperature T_{av}=(T_{w}+T_{∞})/2, for which its physical properties a, c_{p}, β, λ, μ, ρ, ν should be determined. These properties are now available online; in the past, they had to be read from tables, such as [1, 76].
3.1.1 Calculation of the physical properties of air
For the purposes of this work, based on the study [77], the following formulas were derived for calculating the physical properties of air for a given temperature T_{av} in the range 120 ≤ T_{av} ≤ 480 K [77]:
β =1/T_{av} [1/K], or β=7.17643∙10^{13 }^{.} T_{av}^{4 }– + 2.76969∙10^{10 }^{.} T_{av}^{3 }+ 5.36690∙10^{8 }^{.} T_{av}^{2 } – + 1.29663∙10^{5 }^{.} T_{av} + 3.65078∙10^{3} [1/K], R^{2}=0.9998, (24)
a=7.76593∙10^{14 }^{.} T_{av}^{3 }+ 1.10718∙10^{10 }^{.} T_{av}^{2} + 8.70331∙10^{8 }^{.} T_{av} + 1.3323∙10^{5} [m^{2}/s], R^{2}=0.9999, (25)
ν=1.60765∙10^{13 }^{.} T_{av}^{3 }+ 1.77128∙10^{10}^{.} T_{av}^{2} + 1.25673∙10^{7 }^{.} T_{av} + 1.85135∙10^{5} [m^{2}/s], R^{2}=0.9999, (26)
λ=3.13755∙10^{11 }^{.} T_{av}^{3 }– 4.27648∙10^{8 }^{.} T_{av}^{2} + 7.70091∙10^{5 }^{.} T_{av }+ 2.4048∙10^{2} [W/(m∙K)], R^{2}=0.9999. (27)
In order to perform simulation calculations of heat transfer in air, the following values were assumed: heated surface temperatures t_{w}=90, 80, 70, 60, 50, 40, 30 and 20℃ and the temperature difference Δt=t_{w} – t_{∞}=5, 10, 15, 20, 40 and 60 K, the combination of which determined the temperature of the undisturbed area t_{∞} and the average air temperature t_{av}=(t_{w} + t_{∞})/2.
3.1.2 Introduction of thermodynamic and thermoemission functions
By introducing the function B1, which determines the thermodynamic properties of air, and B2, related to its thermoemission, Eq. (21) can be converted into the form:
$\begin{array}{r}C_{C+R}=C_C+C_R=C_C+\frac{\sigma \cdot \varepsilon \cdot l \cdot\left(T_w^4T_{\infty}^4\right)}{\lambda \cdot\left(T_wT_{\infty}\right) \cdot R a^{1 / 4}}=C_C+B 1 \cdot B 2 \cdot l^{1 / 4} \cdot \varepsilon\end{array}$, (28)
where:
$B 1=\frac{1}{\lambda \cdot\left(\frac{g \cdot \beta \cdot \Delta t}{\gamma \cdot a}\right)^{\frac{1}{4}}}, \mathrm{~m}^{7 / 4 \cdot \mathrm{K} / \mathrm{W}}$, (29)
$B 2=\frac{\sigma \cdot\left(T_w^4T_{\infty}^4\right)}{\Delta t}, \mathrm{~W} /\left(\mathrm{m}^2 \mathrm{~K}\right)$, (30)
$R a=\left(\frac{l^{3 / 4}}{\lambda \cdot R 1}\right)^4=\frac{g \cdot \beta \cdot \Delta t \cdot l^3}{v \cdot a}$. (31)
Convective heat transfer, especially in fluids not accompanied by radiation, is more difficult to study experimentally, but it is easier to understand and obtain reliable and reproducible results in the form of C_{C}=Nu/Ra^{n}. It is simpler to conduct research in air, especially when a surface is heated electrically, but the inconclusive influence of radiation complicates the interpretation of the results. Perhaps by determining the effect of temperature, separately on the functions B1 and B2 and on their product B1·B2, it will be easier to describe convectiveradiative heat transfer and obtain more reliable results.
The measurement uncertainties of the values taken from table data [1, 71, 7678], was assumed at the level of the last significant digit in the data source. For a temperature measurement the accuracy of ±0.1℃ was assumed, but for the temperature differences, according with the study [71], the calculations permit an accuracy of ±0.05 K to be specified. The maximum relative uncertainties of Ra and C_{C}_{+R}, derived from Eqns. (31) and (21) are: δRa_{max=}± 4.7% and δC_{C}_{+R max=}± 8.5%. In order to preserve the clarity of the presentation the uncertainty of measurement in relation to the maximum relative uncertainty δC_{max }are only be given in the following investigation. A detailed analysis of the measurement uncertainties related to this consideration can be found in ref. [71].
Table 1. Calculated values of B1 and B2 for given values of t_{w}, t_{∞} and l
Temperatures 
Physical properties of air 
B1 (Figure 3.a) 
B2 (Figure 3.b) 
B1^{.}B2 (Figure 4) 

t_{w} 
t_{∞} 
t_{,av } 
∆t 
λ ^{. }10^{2} 
b ^{. }10^{3 } 
a^{.}10^{5 } 
n^{.}10^{5} 
(29) 
(35) 
(30) 
(33) 
(34) 
(35) 
(36) 
℃ 
℃ 
℃ 
K 
W/m^{.}K 
1/K 
m^{2}/s 
m^{2}/s 
m^{7/4}K/W 
W/(m^{2}K) 
1/m^{1/4} 

90 
85 
87.5 
5 
3.0439 
2.7843 
3.0758 
2.1734 
0.2748 
0.2768 
10.640 
10.7426 
2.9239 
2.9735 
2.9779 
80 
75 
77.5 
5 
2.9746 
2.8656 
2.9242 
2.0697 
0.2723 
0.2744 
9.7797 
9.7945 
2.6632 
2.6875 
2.6923 
70 
65 
67.5 
5 
2.9042 
2.9501 
2.7754 
1.9678 
0.2699 
0.2720 
8.9667 
8.9302 
2.4199 
2.4287 
2.4341 
60 
55 
57.5 
5 
2.8330 
3.0380 
2.6295 
1.8679 
0.2675 
0.2696 
8.2000 
8.1421 
2.1931 
2.1948 
2.2007 
50 
45 
47.5 
5 
2.7607 
3.1300 
2.4865 
1.7699 
0.2650 
0.2671 
7.4783 
7.4236 
1.9820 
1.9832 
1.9896 
40 
35 
37.5 
5 
2.6875 
3.2269 
2.3467 
1.6738 
0.2626 
0.2647 
6.8002 
6.7684 
1.7859 
1.7918 
1.7988 
30 
25 
27.5 
5 
2.6134 
3.3295 
2.2100 
1.5799 
0.2602 
0.2623 
6.1645 
6.1711 
1.6039 
1.6188 
1.6263 
20 
15 
17.5 
5 
2.5383 
3.4389 
2.0766 
1.4880 
0.2577 
0.2599 
5.5696 
5.6265 
1.4355 
1.4623 
1.4703 
90 
80 
85.0 
10 
3.0266 
2.8043 
3.0377 
2.1473 
0.2306 
0.2306 
10.422 
10.5274 
2.4028 
2.4277 
2.3834 
80 
70 
75.0 
10 
2.9571 
2.8864 
2.8867 
2.0441 
0.2285 
0.2286 
9.5735 
9.5915 
2.1873 
2.1922 
2.1533 
70 
60 
65.0 
10 
2.8865 
2.9717 
2.7386 
1.9427 
0.2264 
0.2265 
8.7721 
8.7389 
1.9863 
1.9793 
1.9454 
60 
50 
55.0 
10 
2.8150 
3.0606 
2.5935 
1.8432 
0.2244 
0.2244 
8.0168 
7.9620 
1.7989 
1.7870 
1.7576 
50 
40 
45.0 
10 
2.7425 
3.1538 
2.4513 
1.7457 
0.2224 
0.2224 
7.3061 
7.2542 
1.6246 
1.6132 
1.5879 
40 
30 
35.0 
10 
2.6690 
3.2519 
2.3122 
1.6501 
0.2203 
0.2203 
6.6387 
6.6094 
1.4627 
1.4563 
1.4346 
30 
20 
25.0 
10 
2.5947 
3.3561 
2.1764 
1.5567 
0.2183 
0.2183 
6.0132 
6.0218 
1.3126 
1.3144 
1.2960 
20 
10 
15.0 
10 
2.5194 
3.4675 
2.0438 
1.4653 
0.2162 
0.2162 
5.4284 
5.4865 
1.1736 
1.1863 
1.1709 
90 
75 
82.5 
15 
3.0093 
2.8246 
2.9997 
2.1213 
0.2079 
0.2073 
10.208 
10.3090 
2.1218 
2.1366 
2.0981 
80 
65 
72.5 
15 
2.9395 
2.9074 
2.8495 
2.0185 
0.2060 
0.2054 
9.3712 
9.3859 
1.9304 
1.9277 
1.8941 
70 
55 
62.5 
15 
2.8687 
2.9936 
2.7021 
1.9176 
0.2041 
0.2035 
8.5814 
8.5455 
1.7518 
1.7391 
1.7100 
60 
45 
52.5 
15 
2.7969 
3.0835 
2.5576 
1.8186 
0.2023 
0.2016 
7.8373 
7.7803 
1.5855 
1.5688 
1.5438 
50 
35 
42.5 
15 
2.7242 
3.1778 
2.4162 
1.7216 
0.2005 
0.1998 
7.1375 
7.0836 
1.4308 
1.4151 
1.3937 
40 
25 
32.5 
15 
2.6505 
3.2774 
2.2779 
1.6266 
0.1986 
0.1979 
6.4806 
6.4493 
1.2872 
1.2763 
1.2582 
30 
15 
22.5 
15 
2.5760 
3.3832 
2.1429 
1.5336 
0.1968 
0.1960 
5.8654 
5.8719 
1.1541 
1.1510 
1.1359 
20 
5 
12.5 
15 
2.5005 
3.4966 
2.0112 
1.4428 
0.1949 
0.1942 
5.2903 
5.3461 
1.0310 
1.0380 
1.0255 
The average discrepancy over the entire tested range of Δt (5–30 K) in relation to (34) 
100% 
99.63% 
99.62% 
Figure 3. Values of B1 calculated from Eqns. (29) and (32) for Δt=5, 10 and 15 K as a function of the heating surface temperature varying in the range t_{w}=20 – 90℃ (a) compared with values of B2 calculated from Eqns. (30) and (33) for t_{∞}= 5, 10 and 15 K (Δt=t_{w} – t_{∞}) as a function of the heating surface temperature varying in the range t_{w}=20 – 90℃ (b)
Figure 4. Comparison of values of B1·B2 calculated from Eqns. (34) (blue circles) and (35) (red squares). Eq. (36), shown in the box, is the result of the approximation of the curves obtained using Eq. (34)
3.1.3 Investigation of the influence of t_{w} and Δt on the values of B1 and B2
Having determined t_{∞} and t_{av} and the thermodynamic properties of air for given values of t_{w}=90, 80, 70, 60, 50, 40, 30 and 20℃, together with Δt=5, 10, 15, 20, 40 and 60 K, the values of B1, B2 and their product were calculated. Some of the results of these simulation calculations, for Δt=5, 10 and 15 K, are summarized in Table 1 and Figure 3.
From the variability of B1 (29) and B2 (30) in the temperature range t_{w}=0 – 90℃ for the set Δt=5, 10, 15, 20, 25 and 30 K, and the resulting values of t_{∞}, t_{av}, the following approximation relationships were determined:
B1=3.503^{.}10^{4 . }Δt ^{0.2315}^{.} t_{w }+ 0.3914 ^{.} Δt ^{0.2661}, (32)
$\begin{aligned} B 2= & \left(2.457 \cdot 10^{2} \Delta t+4.800\right) \cdot \\ & e^{\left(1.418 \cdot 10^{5} \cdot \Delta t+9.168 \cdot 10^{3}\right) \cdot t_w}\end{aligned}$, (33)
The calculated values of B1 from (29) and (32), and B2 from (30) and B2 (33) are compared graphically in Figure 3, in which, as in Table 1, the presentation is limited only to example values obtained for the range t_{w}=20 – 90℃, and Δt=5, 10 and 15 K.
Apart from the values of B1 and B2, on the basis of which relationships (32) and (33) were derived, the values of their product B1·B2 were also calculated (see Table 1). To calculate them, the formulas obtained from the conversions of (29) and (30) as well as (32) and (33) to (34) and (35) were used:
$B 1 \cdot B 2=\frac{1}{\lambda \cdot\left(\frac{g \cdot \beta \cdot \Delta t}{v \cdot a}\right)^{\frac{1}{4}}} \cdot \frac{\sigma \cdot\left(T_w^4T_{\infty}^4\right)}{\Delta t}$, (34)
$\begin{gathered}B 1 \cdot B 2=\left(3.503 \cdot 10^{4} \cdot \Delta t^{0.2315} \cdot t_w+\right. \left.+0.3914 \cdot \Delta t^{0.2661}\right) \cdot(2.457 \cdot \left.10^{2} \Delta t+4.800\right)\cdot e^{\left(1.418 \cdot 10^{5} \cdot \Delta t+9.168 \cdot 10^{3}\right) \cdot t_w} \end{gathered}$ (35)
The values of B1·B2 calculated from these formulas are given in the relevant columns of Table 1 and in Figure 4. Note that Table 1 lists only results obtained for Δt=5, 10 and 15 K, whereas Figure 4 contains all the curves for B1·B2 (t_{w}, Δt). The curves marked with blue lines and circles are a graphic representation of Eq. (34), those with red lines and squares represent Eq. (35).
As a result of the approximations of the B1·B2 (t_{w}) curves shown in Figure 4, obtained on the basis of equation (34) (blue circles and lines) for Δt=5, 10, 15, 20, 25 and 30 K, one universal relationship was obtained:
$\begin{array}{r}B 1 \cdot B 2=\left(2.0461 \cdot \Delta t^{0.3306}\right) \cdot e^{\left(1.008 \cdot 10^{2} \cdot e^{1.426 \cdot 10^{3} \,\, \cdot \Delta t}\right) \cdot t_w},\end{array}$ (36)
The results of the calculations for given t_{w} and Δt obtained with the aid of this universal relationship are listed in the last column of Table 1. Like those resulting from the previous approximation (35), they exhibit a slight (only 0.37 and 0.38%) average discrepancy in relation to the results obtained using the original Eq. (34). This discrepancy concerns the entire scope of the calculations, only partially visualized in Table 1.
3.1.4 Analysis of the influence of t_{w} and Δt on C_{R} values
The dependence on the radiative constant in the NusseltRayleigh criterion relationship C_{R }can be obtained by substituting into (28) one of three equivalent formulas for B1·B2: (34), (35) or (36). In the case of substitution (36) one obtains:
$\begin{aligned} C_R= & B 1 \cdot B 2 \cdot l^{\frac{1}{4}} \cdot \varepsilon=\left(2.0461 \cdot \Delta t^{0.3306}\right) . e^{\left(1.008 \cdot 10^{2} \cdot e^{1.426 \cdot 10^{3} \,\, \cdot \Delta t}\right) \cdot t_w} \cdot l^{1 / 4} \cdot \varepsilon\end{aligned}$, (37)
The linear influence of the surface emissivity factor ε on C_{R}, being obvious and predictable (C_{R}=0 for ε=0 and C_{R}=C_{R,max} for ε=1); hence, by assuming ε=1.0, it was omitted at this stage of the calculations. In contrast, analysis of the influence of the characteristic linear dimension l on C_{R} cannot be omitted, because through B1 (31) it influences the value of the Rayleigh number, by means of which the intensity of convective heat exchange is described. While it is customary in convection that C_{C} ≠ f(Ra)=const, it does not also have to apply to radiative heat transfer, for which it has been neither tested nor proven.
Table 2 shows the values of C_{R} and Ra obtained from relationship (37) for given values of the surface temperature t_{w}=100, 90, 80, 70, 60, 50, 40, 30, 25, 20 and 15℃, three values of the undisturbed area temperature t_{∞}=10, 5 and 0℃, the constant value of the emissivity coefficient ε=1.0, and the following values of the characteristic linear dimension l=0.5, 0.25, 0.15, 0.10, 0.05 and 0.01 m. The values of Ra given in the Table 2 were calculated, as in Table 1, using relationships (14  27) as a function of t_{av}, but in order to keep Table 2 readable, it does not include the calculated values of a, β and ν.
Table 2. Results of C_{R} and Ra calculations for given values of t_{w}, t_{∞} and l and the assumed constant value of ε=1
Temp. 
B1^{.}B2 
l=0.50 m 
l=0.25 m 
l=0.15 m 
l=0.10 m 
l=0.05 m 
l=0.01 m 

t_{w} 
t_{∞} 
(36) 
C_{R}(37) 
Ra^{.}10^{10} 
C_{R}(37) 
Ra^{.}10^{9} 
C_{R}(37) 
Ra^{.}10^{8} 
C_{R}(37) 
Ra^{.}10^{8} 
C_{R}(37) 
Ra^{.}10^{7} 
C_{R}(37) 
Ra^{.}10^{5} 
℃ 
℃ 
1/m^{1/4} 
– 
– 
– 
– 
– 
– 
– 
– 
– 
– 
– 
– 
100 
0 
1.4161 
1.1908 
2.7625 
1.0013 
3.4531 
0.8813 
7.4588 
0.7963 
2.2100 
0.6696 
2.7625 
0.4478 
2.2100 
90 
0 
1.2873 
1.0825 
2.6291 
0.9103 
3.2863 
0.8011 
7.0984 
0.7239 
2.1032 
0.6087 
2.6291 
0.4071 
2.1032 
80 
0 
1.1792 
0.9916 
2.4736 
0.8338 
3.0920 
0.7338 
6.6787 
0.6631 
1.9789 
0.5576 
2.4736 
0.3729 
1.9789 
70 
0 
1.0894 
0.9161 
2.2933 
0.7703 
2.8666 
0.6780 
6.1919 
0.6126 
1.8346 
0.5152 
2.2933 
0.3445 
1.8346 
60 
0 
1.0167 
0.8550 
2.0849 
0.7189 
2.6061 
0.6327 
5.6293 
0.5718 
1.6679 
0.4808 
2.0849 
0.3215 
1.6679 
50 
0 
0.9609 
0.8080 
1.8448 
0.6795 
2.3060 
0.5980 
4.9810 
0.5404 
1.4758 
0.4544 
1.8448 
0.3039 
1.4758 
40 
0 
0.9234 
0.7765 
1.5688 
0.6529 
1.9610 
0.5747 
4.2358 
0.5193 
1.2551 
0.4366 
1.5688 
0.2920 
1.2551 
30 
0 
0.9093 
0.7646 
1.2522 
0.6430 
1.5652 
0.5659 
3.3809 
0.5113 
1.0017 
0.4300 
1.2522 
0.2875 
1.0017 
25 
0 
0.9149 
0.7694 
1.0769 
0.6470 
1.3462 
0.5694 
2.9077 
0.5145 
0.8616 
0.4327 
1.0769 
0.2893 
0.8616 
20 
0 
0.9338 
0.7853 
0.8895 
0.6603 
1.1118 
0.5812 
2.4015 
0.5251 
0.7116 
0.4416 
0.8895 
0.2953 
0.7116 
15 
0 
0.9744 
0.8194 
0.6889 
0.6890 
0.8611 
0.6064 
1.8600 
0.5479 
0.5511 
0.4608 
0.6889 
0.3081 
0.5511 
100 
5 
1.4285 
1.2012 
2.5530 
1.0101 
3.1913 
0.8890 
6.8932 
0.8033 
2.0424 
0.6755 
2.5530 
0.4517 
2.0424 
90 
5 
1.3024 
1.0952 
2.4143 
0.9209 
3.0179 
0.8105 
6.5187 
0.7324 
1.9315 
0.6159 
2.4143 
0.4118 
1.9315 
80 
5 
1.1969 
1.0065 
2.2537 
0.8464 
2.8172 
0.7449 
6.0851 
0.6731 
1.8030 
0.5660 
2.2537 
0.3785 
1.8030 
70 
5 
1.1103 
0.9337 
2.0685 
0.7851 
2.5856 
0.6910 
5.5850 
0.6244 
1.6548 
0.5250 
2.0685 
0.3511 
1.6548 
60 
5 
1.0416 
0.8758 
1.8555 
0.7365 
2.3193 
0.6482 
5.0098 
0.5857 
1.4844 
0.4925 
1.8555 
0.3294 
1.4844 
50 
5 
0.9912 
0.8335 
1.6111 
0.7009 
2.0138 
0.6168 
4.3499 
0.5574 
1.2888 
0.4687 
1.6111 
0.3134 
1.2888 
40 
5 
0.9622 
0.8091 
1.3312 
0.6804 
1.6640 
0.5988 
3.5943 
0.5411 
1.0650 
0.4550 
1.3312 
0.3043 
1.0650 
30 
5 
0.9637 
0.8103 
1.0114 
0.6814 
1.2642 
0.5997 
2.7307 
0.5419 
0.8091 
0.4557 
1.0114 
0.3047 
0.8091 
25 
5 
0.9832 
0.8268 
0.8348 
0.6952 
1.0435 
0.6119 
2.2539 
0.5529 
0.6678 
0.4649 
0.8348 
0.3109 
0.6678 
20 
5 
1.0255 
0.8623 
0.6462 
0.7251 
0.8077 
0.6382 
1.7446 
0.5767 
0.5169 
0.4849 
0.6462 
0.3243 
0.5169 
15 
5 
1.1129 
0.9359 
0.4447 
0.7870 
0.5559 
0.6926 
1.2008 
0.6259 
0.3558 
0.5263 
0.4447 
0.3519 
0.3558 
100 
10 
1.4425 
1.2130 
2.3535 
1.0200 
2.9418 
0.8977 
6.3543 
0.8112 
1.8828 
0.6821 
2.3535 
0.4562 
1.8828 
90 
10 
1.3192 
1.1093 
2.2100 
0.9328 
2.7625 
0.8210 
5.9670 
0.7418 
1.7680 
0.6238 
2.2100 
0.4172 
1.7680 
80 
10 
1.2168 
1.0232 
2.0448 
0.8604 
2.5560 
0.7573 
5.5210 
0.6843 
1.6359 
0.5754 
2.0448 
0.3848 
1.6359 
70 
10 
1.1339 
0.9535 
1.8552 
0.8018 
2.3190 
0.7056 
5.0090 
0.6376 
1.4842 
0.5362 
1.8552 
0.3586 
1.4842 
60 
10 
1.0699 
0.8997 
1.6381 
0.7566 
2.0476 
0.6659 
4.4228 
0.6017 
1.3104 
0.5059 
1.6381 
0.3383 
1.3104 
50 
10 
1.0266 
0.8633 
1.3899 
0.7259 
1.7374 
0.6389 
3.7528 
0.5773 
1.1120 
0.4855 
1.3899 
0.3246 
1.1120 
40 
10 
1.0094 
0.8488 
1.1069 
0.7138 
1.3836 
0.6282 
2.9886 
0.5677 
0.8855 
0.4773 
1.1069 
0.3192 
0.8855 
30 
10 
1.0351 
0.8704 
0.7844 
0.7320 
0.9805 
0.6442 
2.1179 
0.5821 
0.6275 
0.4895 
0.7844 
0.3273 
0.6275 
25 
10 
1.0793 
0.9076 
0.6068 
0.7632 
0.7585 
0.6717 
1.6384 
0.6069 
0.4855 
0.5104 
0.6068 
0.3413 
0.4855 
20 
10 
1.1709 
0.9846 
0.4174 
0.8280 
0.5217 
0.7287 
1.1270 
0.6584 
0.3339 
0.5537 
0.4174 
0.3703 
0.3339 
15 
10 
1.3981 
1.1756 
0.2154 
0.9886 
0.2692 
0.8701 
0.5815 
0.7862 
0.1723 
0.6611 
0.2154 
0.4421 
0.1723 
Figure 5. Graphical interpretation of the influence of t_{w}, t_{∞}, and l on C_{R} expressed by the function C_{R}=f(Δt) assuming ε=1 in relationship (37)
Figure 6. The influence of t_{w}, t_{∞}, and l on C_{R}, calculated from the dependence (37) assuming ε =1, expressed in the form of the function C_{R}=f(Ra)
Figure 7. Diagram of the experimental apparatus for examining: a) free convective heat transfer in water, using the balance method and an isothermal vertical plate, b) total convective and radiative heat transfer in air, obtained by the use of balance method, and the convective only, obtained by the gradient method, with the additional use of IR camera with detection mesh [71]
From among the various possible graphical presentations of the results in Table 2, it was decided to show the relationships C_{R}=f(Δt) in Figure 5 and C_{R}=f(Ra) in Figure 6. In both figures the results for the same temperature of the undisturbed area are indicated for t_{∞}=10℃ by a blue line, for t_{∞}=5℃ by a red one, and for t_{∞}=0℃ by a black one. In Figure 5, the beginnings of the curves related to the heated surface temperature t_{w}=15℃ are on the left side of the graph, those for t_{w}=100℃ on the right. Because these positions are correlated with the values of Δt, they are easier to analyse. In the case of Figure 6, where there is no such correlation, the two outermost dashed blue lines connecting the points with the highest and lowest surface temperatures t_{w}=15 and 100℃ and a third line of minimum C_{R} values for t_{w} ≈ 40℃ have been added, but only for t_{∞}=10℃ these lines were created on the basis of the data shown in bold in Table 2.
Figures 5 and 6 show that the function C_{R} in radiative heat exchange is not constant, as was the case with C_{C}, which was correct for convective heat exchange. The function C_{R }defined in this way depends on the coefficient C_{R }of the emissivity ε, the temperature conditions t_{w} and t_{∞}, the area of the heated surface l, which determine the Rayleigh number, but also on the thermodynamic properties of air a, β and ν, which are included in Ra.
3.1.5 Final solution in the form of C_{C+}_{R} and its application
Knowing the convective constant C_{C} and the radiative function C_{R}, described by Eq. (37), one can determine the value of C_{C+R}=C_{C}+C_{R} from Eq. (28) depending on the criterion dependence (12), which in turn opens up the possibility of determining the total (convectiveradiative) heat flux Q_{C+R} from any vertical surface:
$C_{C+R}=C_C+B 1 \cdot B 2 \cdot l^{\frac{1}{4}} \cdot \varepsilon=\frac{N u_{C+R}}{R a^{\frac{1}{4}}}=\frac{\alpha_{C+R} \quad \cdot l}{\lambda \cdot R a^{1 / 4}} \quad$, (38)
$Q_{C+R}=\frac{\lambda \cdot A}{l} \cdot \Delta t \cdot R a^{1 / 4} \cdot\left(C_C+B 1 \cdot B 2 \cdot l^{\frac{1}{4}} \cdot \varepsilon\right)$. (39)
At first glance, these equations seem complicated and require more time to obtain the results than existing solutions. However, their advantage is that, after measuring t_{w}, t_{∞ }and ε, one can use a thermal imaging camera with appropriate software to directly determine the heat flux from any heated surface of known dimensions l, e.g., a building surface. For this purpose, Eqns. (20), (24)(27), (31), (37), (39) should only include Δt, t_{av}, ε and l, along with the literature value of the constant of free convection from a vertical plate, e.g., C_{C}=0.569 [71]. In the case of horizontal surfaces, this constant and the exponent n will have different values depending on criterion (2) and hence, the entire solution.
In this solution, the heat exchange is assumed by default to be in the laminar range, i.e. from Ra_{cr,I} ≈ 10^{3} to Ra_{cr,II} ≈ 10^{9} (Ra_{cr,air}=2.0^{.}10^{8} and Ra_{cr,water}=3.4^{.}10^{9} [79] for uniform flux) because above and below this range, heat transfer takes place by conduction or transition convection; for the latter, there are other values of the constants in NusseltRayleigh criterion dependence (6) and (20), e.g. C_{C}=0.135 and n=1/3.
4.1 Experimental validation of the solution
At the very beginning of this section, it should be noted that the titular validation concerns only the correctness of the solution for the constant C_{C+R} depending on Nu=C_{C+R}^{.}Ra^{1/4}, describing convectiveradiative heat transfer. The tests of convective heat transfer in water obtained by the use of the balance method (a) and convective heat transfer in air obtained by the gradient method with the use of thermal imaging camera with a detection mesh, as well as combined convective and radiative heat transfer in air obtained by the use of balance method without a camera and mesh (b) were carried out on the test stand shown diagrammatically in Figure 7. Since the details of these studies have already been described in the study [71], they have been omitted here, except for the results. Based on an analysis of the relationships between the measured values of C_{C}, B_{1}^{.}B_{2 }and C_{C+}, these results were used to develop an empirical relationship (39) and to verify it experimentally. On the test stand shown in Figure 7a, the balance method was used to determine the value of C_{C,water}=0.536 for an isothermal vertical plate and free convection in water, while on the test stand illustrated in Figure 6b, the new gradient method of detecting convection with a thermal imaging camera equipped with a detecting mesh was used to determine the value of C_{C,air}=0.579 for air [71]. On the same test stand, but using the balance method, the constant value of C_{C+R}=1.220 was obtained for air, which covers both convection and radiation heat transfer [71].
4.1.1 Research test stand
The main element of the above test stands was a thin doublesided and symmetrically heated vertical plate of height h=0.15 m, width b=0.075 m and thickness s=0.003 m. The plate was made from three layers of copperlaminated glassepoxy composite, each 0.6 mm thick. The outer layers (LAM100X160E0.6), laminated on one side with a copper layer of thickness g=35 mm, consisted of two surface resistance thermometers with resistances 235.1 W and 240.7 W, located on both sides of the plate. The middle layer (LAM100X160ED0.6), laminated on both sides with 18 mm thick copper layers, was a twosided heating resistance heater with 44.8 W and 45.2 W resistances, which was powered by two power supplies with power N=30 W, voltage U=0  10 V and current I=0  3 A.
Both thermometers and heaters were made by etching the resistance paths in copper by photolithography. More information, including dimensions, drawings and photos, on the production and calibration of the heater used in these tests can be found in the study [71]. A similar concept of a doublesided isothermal plate set obliquely was used in the study [80].
4.1.2 The study of radiativeconvective heat transfer in air
The convectiveradiative heat transfer study was carried out in the traditional way, by gradually increasing the heating power of the heaters. Since the heat exchange was symmetrical, as evidenced by practically the same surface temperatures on both sides of the plate, the heater was connected in series. After establishing the thermodynamic equilibrium t_{w}, t_{av} and Δt, the result was recorded, the power increased and another measurement started.
The results of the voltage and direct current variation of the heater I_{i}, A, U_{i}, V, obtained in tabular form, enabled the heat flux Q_{i} transferred to the air from the N_{i} power to be calculated. It was decided not to take into account the heat loss flux at the edges owing to the small surface area (thickness s=3 mm).
Then, based on the values of λ_{i}=f(t_{av}) obtained from (27) and the dimension b=0.075 m of the plate, Nu_{i} was calculated from Eq. (11).
In turn, knowing the temperatures of both sides of the plate surface t_{wI,i }and t_{wII,i} and the air in the undisturbed area t_{∞,i}, the average temperatures of the plate surface and air t_{w,i}, t_{av,i} and Δt_{i} could be calculated, from which the physical properties of the air could be determined from (24), (25) and (26) and consequently, the Rayleigh numbers for h=0.15 m and Δt_{i }from Eq. (31).
The last procedure for processing the experimental data was to calculate the individual values of C_{C+R}=Nu_{C+R}/Ra^{1/4}, some of which (diamonds in Figure 6) along with the mean are given in the experimental part of Table 3. The last row of Table 3 shows the averages of the results of all the research, both experimental and theoretical.
Paper [71] gives identical results, obtained on the same stand for air, but the content of Table 3 has changed. This shows the results of convective transfer, not tabulated in the study [71], and thus the differences in averaging the tabular data. On the other hand, the difference in averaging all the results C_{C+R},_{av}=1.197 (maximum relative uncertainty δC_{exp}.= 8.5%) differs slightly from that given in the study [71]. C_{C+R},_{av}=1.220 results from the use of other formulas to determine the physical properties of air in both papers. The present Eqns. (24)  (27) are correlated with t_{av} and the previous ones with other constants from T_{av}.
In the study [71], apart from the convectiveradiative heat transfer in air, tests of free convection in water, in which radiative heat transfer can be omitted, were carried out on a different stand (see Figure 7a), but with the same heating plate. The criterion dependence obtained in those studies for water takes the following form:
Nu_{C}=(0.536 ± 0.073) ^{.} Ra ^{0.25 }(Ra =2^{.}10^{6} – 8^{.}10^{8}), (40)
This dependence is 5.8% less than (20), but because it was obtained as a result of experimental studies carried out in a similar way and on the same heating plate, we decided to use it to validate the correctness of the solution. Detailed calculations of this validation can be found in the study [78]. The maximum relative uncertainty δC_{exp.=}±13.7% (for the experiment conducted in water) has been included in Eq. (40).
Table 3 also lists the results obtained from processing the experimental data using equations derived from theoretical considerations. The values of function B1·B2, given in the theoretical part of Table 3, were based on Eq. (36), in which the experimental values of t_{w} and Δt were substituted.
Table 3. Results of overall (convective and radiative) heat transfer tests from a vertical plate in air
Heating power N 
Temperature 
Experimental results 
Comparison with theory 

Nu_{C+R} 
Ra ^{.}10^{6} 
C_{C+R} 
B1^{.}B2 
C_{R} 
C_{C+}_{R} 
C_{R} 
C_{C+}_{R} 

ε=0.884 
ε=0.932 

t_{w} 
t_{∞} 
t_{av} 
∆t 
(11) 
(31) 
(13) 
(36) 
(37) 
(41) 
(37) 
(41) 

W 
℃ 
℃ 
℃ 
K 
 
 
 
m^{1/4} 
 
 
 
 
Series I 

1.0 
28.5 
23.9 
26.2 
4.6 
53.207 
1.496 
1.521 
1.642 
0.903 
1.439 
0.952 
1.488 
1.8 
32.8 
24.1 
28.4 
8.7 
53.372 
2.700 
1.317 
1.396 
0.768 
1.304 
0.810 
1.346 
2.9 
37.4 
24.4 
30.9 
13.0 
55.759 
3.900 
1.255 
1.283 
0.706 
1.242 
0.744 
1.280 
4.0 
42.3 
24.9 
33.6 
17.4 
57.362 
5.009 
1.213 
1.228 
0.676 
1.212 
0.713 
1.249 
5.1 
46.7 
24.9 
35.8 
21.8 
57.946 
6.071 
1.167 
1.197 
0.658 
1.194 
0.694 
1.230 
5.9 
50.7 
25.1 
37.9 
25.6 
56.940 
6.920 
1.110 
1.186 
0.652 
1.188 
0.688 
1.224 
6.9 
54.4 
25.2 
39.8 
29.2 
58.694 
7.672 
1.115 
1.183 
0.651 
1.187 
0.686 
1.222 
7.8 
57.7 
25.3 
41.5 
32.4 
59.313 
8.321 
1.104 
1.186 
0.653 
1.189 
0.688 
1.224 
9.1 
61.8 
25.3 
43.6 
36.5 
60.959 
9.095 
1.110 
1.195 
0.658 
1.194 
0.693 
1.229 
Series II 

5.0 
48.5 
25.9 
37.2 
22.6 
54.304 
6.181 
1.089 
1.205 
0.663 
1.199 
0.699 
1.235 
6.1 
52.2 
25.8 
39.0 
26.4 
57.191 
7.015 
1.111 
1.193 
0.656 
1.192 
0.692 
1.228 
6.9 
55.3 
25.6 
40.5 
29.7 
57.480 
7.743 
1.090 
1.188 
0.654 
1.190 
0.689 
1.225 
8.4 
59.2 
25.7 
42.5 
33.5 
61.357 
8.488 
1.137 
1.193 
0.656 
1.192 
0.692 
1.228 
9.2 
63.1 
25.4 
44.3 
37.7 
59.276 
9.303 
1.073 
1.200 
0.660 
1.196 
0.696 
1.232 
10.1 
66.1 
25.2 
45.7 
40.9 
60.120 
9.895 
1.072 
1.210 
0.666 
1.202 
0.702 
1.238 
Average for the 15 results listed in this table 
1.166 
1.246 
0.685 
1.221 
0.723 
1.259 

Average value for all 27 results shown in Figure 6 
1.197 
1.239 
0.682 
1.218 
0.719 
1.255 
Figure 8. The results of the experimental tests, some of which  from Series I (diamonds) and Series II (squares)  are included in Table 3, and the remaining (Series III) experimental points (shown by dots) are not shown in Table 3. The black line shows the approximation of all the experimental results (41). The theoretical solution obtained for ε=0.884 is shown by the blue line, and the one for ε=0.932 by the red line
Further, from Eq. (37), the constant C_{R} was calculated for the two given for experimental plate emissivities ε=0.884 (estimated experimentally) and ε=0.932 (calculated from experimental data) [71] of a plate with a characteristic linear dimension of l=0.15 m. Without enquiring which of these emissivities is closer to reality, both values C_{R} and C_{C+R} were calculated.
The constant C_{C+R} is the sum of the convection constant C_{C} (40) and the radiation constant C_{R}:
$C_{C+R}=C_C+C_R$. (41)
The results of these calculations were also subjected to double averaging with reference to the sample data listed in Table 3 and to all the results shown in Figure 8. The maximum relative uncertainty for the experiment in air (δC_{exp.=}±8.5%) has been adopted here.
At the end of the experimental research and theoretical considerations, it had to be checked whether the results fell within the laminar for air range Ra < Ra_{cr,sir}=2^{.}10^{8} [79]. First, the experimental relationship (40) was checked: since this was obtained for water with the maximum number of Ra_{max} ≈ 8^{.}10^{8} < Ra_{cr,water}=3.4^{.}10^{9 }[79], it also raised no objections.
The results of the experimental studies published in the study [71] were used to validate the solution. Therefore, they cannot be suspected of being biased, even to a minimal degree. It is even more difficult to imagine the possibility of matching the results of theoretical considerations with experimental data. In this situation, the discrepancy between the experimental and theoretical values of C_{C+R},_{av }– 1.75% (for ε=0.884) and 4.85% (for ε=0.932) – offers incontrovertible evidence confirming both the reliability of the experiment and the correctness of the solution of the empirical Eq. (39) with coefficient (36).
Having ensured that the solutions are correct, one can begin to outline its possible practical applications, which may be:
 a simple way of determining the heat loss from any external wall, building envelope, façade of a building, etc., or the heat flux transferred from internal walls inside a building, based on knowledge of: the value of its area A, characteristic linear dimension l (height), temperature t_{w}, the surrounding temperature t_{∞} and the approximate (e.g., tabular) value the surface emissivity ε, which has little effect on the result (ΔQ ≈ 3% for Δε=0.05%).
 if the building surfaces under consideration are not vertical, isothermal or if Ra > Ra_{cr} (C_{C} (20) or (40)), then it suffices to substitute the relevant values of C_{C} in Eq. (39).
 the development of dedicated software for a thermal imaging camera, which is based on the temperature of the heated surface t_{w} and the air in surrounding t_{∞}, which are in thermodynamic equilibrium with the surroundings, and the emissivities of the surface and the surroundings. With such a programme, a properly modified camera, in addition to its current applications in energy audits of buildings [62, 8185], or recently for measuring air velocity [21], net heat flux [86] or heat flux and hot spot temperature in machining process [87], with the use of infrared image sequences, could also measure the total (convectionradiative) energy flux emitted from walls.
This work was supported by the Icelandic Research Fund (Grant No. 184949); the Swedish Research Council (Grant No. 202005110).
a 
coefficient of thermal diffusivity, m^{2}/s 
A 
surface area, m^{2} 
b 
width of the plate, m 
B1 
function, K^{.}m^{7/4}/W (29), (32) 
B2 
function, W/(m^{2.}K) (30), (33) 
c_{p} 
specific heat, J/(kg·K) 
C 
coefficient in the Rayleigh–Nusselt equation 
g 
acceleration due to gravity, m/s^{2} 
h 
height of the plate, m 
I 
current, A 
l 
characteristic length, m 
n 
exponent 
N 
heater power, W 
Nu 
Nusselt number, dimensionless 
Ra 
Rayleigh number, dimensionless 
t,T 
temperature, ℃, K 
q 
flux density, W/m^{2} 
Q 
heat flux, W 
U 
voltage, V 
Greek symbols 

α 
heattransfer coefficient, W/(m^{2}·K) 
β 
coefficient of thermal expansion, 1/K 
$\delta$ 
uncertainty 
ε 
surface emissivity 
σ 
StefanBoltzmann constant, W/(m^{2}·K^{4}) 
Δ 
difference 
λ 
thermal conductivity, W/(m·K) 
$\mu$ 
dynamic viscosity, kg/(m·s) 
ρ 
density, kg/m^{3} 
ν 
kinematic viscosity, m^{2}/s 
Subscripts 

av 
average 
cr 
critical 
C 
convective 
in 
inlet 
max 
maximum 
loss 
losses 
out 
outlet 
R 
radiative 
w 
wall 
∞ 
in surroundings 
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