Empirical Relationship Describing Total Convective and Radiative Heat Loss in Buildings

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 Stefan-Boltzmann 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, Navier-Stokes and Fourier-Kirchhoff), 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 low-intensity 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, Stefan-Boltzmann 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 Stefan-Boltzmann 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 Stefan-Boltzmann 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 [7-9].

In order to eliminate the effects of radiation in experimental convection studies, some researchers took various approaches, performing experiments in water [10-13], glycerine [14-16] 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 [18-21] assuming that when ε=0, Q_R is also equal to 0. The remainder calculated the radiative flux from the Stefan-Boltzmann equation for given values of ε and subtracted it from the heating power of the tested surfaces.

The following convective-radiation 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 three-dimensional (3D) shallow wedge [22], and an investigation, likewise numerical, of the influence of the absorbing-emitting-scattering effect in a two-dimensional (2D) square cavity on convective-radiative heat transfer [23]. Similar problems were studied in a partitioned rectangular enclosure with semi-transparent 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 convective-radiative 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 [36-42].

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 convective-radiative cooling in the air, in the temperature range t=42 – 142℃ and Ra=2^.10⁶ - 2^.10⁷, was described in papers [43, 44]. Experimental studies of convective-radiative 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 [46-48]. The same problem, but theoretically using a similarity solution in relation to the vertical plate, was investigated and described in the studies [49-52]. 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 non-similarity equations numerically [53]. The solution of dimensionless boundary-layer equations of second-order 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 convective-radiative 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 [57-62] 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.

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_av⁴ –  + 2.76969∙10^{-10 .} T_av³ + 5.36690∙10^{-8 .} T_av²  –  + 1.29663∙10^{-5 .} T_av + 3.65078∙10^-3 [1/K], R²=0.9998,               (24)

a=-7.76593∙10^{-14 .} T_av³ + 1.10718∙10^{-10 .} T_av² + 8.70331∙10^{-8 .} T_av + 1.3323∙10^-5 [m²/s], R²=0.9999,       (25)

ν=-1.60765∙10^{-13 .} T_av³ + 1.77128∙10^-10. T_av² + 1.25673∙10^{-7 .} T_av + 1.85135∙10^-5 [m²/s], R²=0.9999,            (26)

λ=3.13755∙10^{-11 .} T_av³ – 4.27648∙10^{-8 .} T_av² + 7.70091∙10^{-5 .} T_av + 2.4048∙10^-2 [W/(m∙K)], R²=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 thermo-emission functions

By introducing the function B1, which determines the thermodynamic properties of air, and B2, related to its thermo-emission, Eq. (21) can be converted into the form:

$\begin{array}{r}C_{C+R}=C_C+C_R=C_C+\frac{\sigma \cdot \varepsilon \cdot l \cdot\left(T_w^4-T_{\infty}^4\right)}{\lambda \cdot\left(T_w-T_{\infty}\right) \cdot R a^{1 / 4}}=C_C+B 1 \cdot B 2 \cdot l^{1 / 4} \cdot \varepsilon\end{array}$,        (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)

$B 2=\frac{\sigma \cdot\left(T_w^4-T_{\infty}^4\right)}{\Delta t}, \mathrm{~W} /\left(\mathrm{m}^2 \mathrm{~K}\right)$,        (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/Raⁿ. 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 convective-radiative heat transfer and obtain more reliable results.

The measurement uncertainties of the values taken from table data [1, 71, 76-78], 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

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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)

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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:

$B 1 \cdot B 2=\frac{1}{\lambda \cdot\left(\frac{g \cdot \beta \cdot \Delta t}{v \cdot a}\right)^{\frac{1}{4}}} \cdot \frac{\sigma \cdot\left(T_w^4-T_{\infty}^4\right)}{\Delta t}$,       (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 Nusselt-Rayleigh 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

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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)

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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)

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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 (convective-radiative) 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 ≈ 10³ to Ra_cr,II ≈ 10⁹ (Ra_cr,air=2.0^.10⁸ and Ra_cr,water=3.4^.10⁹ [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 Nusselt-Rayleigh criterion dependence (6) and (20), e.g. C_C=0.135 and n=1/3.

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, 81-85], 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 (convection-radiative) energy flux emitted from walls.