2. Thermal properties in building

Within a building space, heat is distributed or transferred by three fundamental methods which names are:

#### 2.1 Conduction

This method of heat distribution or transfer in building spaces is a resultant effect of kinetic energy transfers at molecular level in any of the three states of matter (solids, liquids and gases). Conduction method of heat distribution or transfer in buildings is naturally validated to flow in the direction of tapering temperature. Such conductive behavior of heat loss in buildings is attractively noticeable in opaque walls during the winter [1]. There are experimental agreements between thermal and electrical conduction pattern in solids. The earliest formal knowledge of heat conduction law through a medium of either solid, liquid or gas was idealized by Joseph Fourier who postulated the law of heat conduction transfer method in the early part of the nineteenth century [2]. We take a congruency of how heat is conveyed through building elements (materials) or its space to be analogous to heat conduction in the building. Firstly, Fourier stated based on experimental verification for a steady conduction that the rate of heat transfer in any medium (inclusive of building elements) by conduction Q is proportional to the temperature difference and the heat flow area impacted by the heat in such a way that the heat conduction rate Q is inversely proportional to the distance through which the conduction traveled [1, 2]. Clearly in mathematical expression Eq. (3), Fourier meant that

$$Q = -KA. \frac{dT}{dx} \tag{3}$$

with K = thermal conductivity of the materials (W/(m.k)); A = area through which heat flow occurred; and dT dx= temperature gradient at any point in x been the space through which the heat flow.

The minus sing indicating a flow from higher point to a lower point. For a recourse to building walls made of brick, block, fiber, paneled steel with known wall thickness, conductivity of heat from outer skin to inner skin can be estimated by:

$$Q = \text{KA.} \frac{T\_1 - T\_2}{\Delta x} \tag{4}$$

with k = thermal conductivity (W/(m.k)); T1 = outer/higher temperature; T2 = inner/lower temperature; A = area through which conduction flowed; and Δx = thickness of materials in which the conduction occurred.

The property of heat causing the differential between outer and inner temperature value is occasioned by the material's resistance to heat which is obtained when the above expression is having the conductivity (k) related to the area (A) as Eq. (5):

$$Q = \frac{T\_1 - T\_2}{\Delta \mathbf{x} / \mathbf{K} \mathbf{A}}, R = \frac{\Delta \mathbf{x}}{\mathbf{K} \mathbf{A}} \text{ as unit thermal resistance} \tag{5}$$

certain rate with a typical building space. The rate of such heat loss brings about the cooling of the environment or space which is proportional to the excess temperature of the building space over the external temperature of its surroundings based on

Within a building space, heat is distributed or transferred by three fundamental

This method of heat distribution or transfer in building spaces is a resultant effect of kinetic energy transfers at molecular level in any of the three states of matter (solids, liquids and gases). Conduction method of heat distribution or transfer in buildings is naturally validated to flow in the direction of tapering temperature. Such conductive behavior of heat loss in buildings is attractively noticeable in opaque walls during the winter [1]. There are experimental agreements between thermal and electrical conduction pattern in solids. The earliest formal knowledge of heat conduction law through a medium of either solid, liquid or gas was idealized by Joseph Fourier who postulated the law of heat conduction transfer method in the early part of the nineteenth century [2]. We take a congruency of how heat is conveyed through building elements (materials) or its space to be analogous to heat conduction in the building. Firstly, Fourier stated based on experimental verification for a steady conduction that the rate of heat transfer in any medium (inclusive of building elements) by conduction Q is proportional to the temperature difference and the heat flow area impacted by the heat in such a way that the heat conduction rate Q is inversely proportional to the distance through which the conduction traveled [1, 2]. Clearly in mathematical expression Eq. (3), Fourier meant that

Q ¼ �KA:

with K = thermal conductivity of the materials (W/(m.k)); A = area through

The minus sing indicating a flow from higher point to a lower point. For a

recourse to building walls made of brick, block, fiber, paneled steel with known wall thickness, conductivity of heat from outer skin to inner skin can be

Q ¼ KA:

Δx = thickness of materials in which the conduction occurred.

<sup>Δ</sup>x=KA , R <sup>¼</sup> <sup>Δ</sup><sup>x</sup>

<sup>Q</sup> <sup>¼</sup> <sup>T</sup><sup>1</sup> � <sup>T</sup><sup>2</sup>

with k = thermal conductivity (W/(m.k)); T1 = outer/higher temperature; T2 = inner/lower temperature; A = area through which conduction flowed; and

The property of heat causing the differential between outer and inner temperature value is occasioned by the material's resistance to heat which is obtained when the above expression is having the conductivity (k) related to the area (A) as Eq. (5):

dT

T<sup>1</sup> � T<sup>2</sup> Δx

dx (3)

(4)

dx= temperature gradient at any point in x been the

KA as unit thermal resistance (5)

forced convection when the excess temperature is small.

2. Thermal properties in building

methods which names are:

Zero and Net Zero Energy

which heat flow occurred; and dT

estimated by:

120

space through which the heat flow.

2.1 Conduction

In practice, the term is commonly referred to as R-value such that Eq. (6)

$$R\_{th} = \frac{\Delta \mathbf{x}}{k} = AR \tag{6}$$

Another pseudo form of measuring thermal conductance in materials is the Uvalue which is expressed as the reciprocal of the R value:

$$U = \frac{1}{R\_{th}}\tag{7}$$

The possibility that a building wall is layered with different materials and different geometries suggests that Fourier laws cannot be restricted to single layer with uniform thermal resistance [1]. HVAC systems carries different insulating and piping materials which calls for Eq. (8) cylindrical examination of Fourier law of steady heat conduction in cylindrical coordinates [3]. Under such consideration,

$$Q = \frac{T\_1 - T\_2}{\ln(r\_o/r\_i)/2\pi kl} \tag{8}$$

with ri = outer radius; ro = inner radius; L = pipe length; and K = thermal conductivity.

As a general rule, to the effect of other geometries, a shape factor (S) is introduced Eq. (9) to accommodate any derived shape for the measurement of heat loss of pipes in buried walls conveying hot fluid as:

$$Q = K \text{S} \Delta T = K \text{S} \left( T\_1 - T\_n \right) \tag{9}$$

Usually shape factors are derivatives of components meant in design to restrain losses which are either isothermal cylinder Eq. (10) with S-value of

$$\mathcal{S} = \frac{2\pi L}{Cosh^{-1}(D/r)}\tag{10}$$

where L ≫ r; and D = buried depth in semi-infinite medium Eqs. (11) and (12).

$$S = \frac{2\pi L}{\text{Im}\,(2D/r)}\frac{L \gg r}{D \le r} \tag{11}$$

$$S = \frac{2\pi L}{\ln\left[\frac{L}{r}\left(1 - \frac{\ln\left(\frac{L}{2d}\right)}{\ln\left(\frac{L}{r}\right)}\right)\right]}\stackrel{D \gg r}{L \gg r} \tag{12}$$

There can also be conduction between two isothermal cylinder Eq. (13) buried in infinite medium with S-value of

$$S = \frac{2\pi L}{Cosh^{-1}\frac{D^2 - r\_1^2 - r\_2^2}{2r\_1r\_2}}\stackrel{L\gg r\_1r\_2}{L\gg D} \tag{13}$$

It can also take the form of conduction through two composite rectangular plane sections with edge section of two adjoining walls having combined k-value and inner/outer surface uniform temperatures S-value of Eq. (14)

$$s = \frac{al}{\Delta x} + \frac{bl}{\Delta x} + 0.54\tag{14}$$

### 2.2 Convention

Unlike heat conduction where the transference of heat is through a body (solid) without visible motion of any part of the body to the naked eyes, convention is a method of heat transfer in fluid by the movement of the fluid itself [4]. Heat convention primarily takes two forms:


Free convention particularly results in density differences in the fluid, occasioned by contact with the surface originating the heat transfer [5]. Free conventions are evident in gentle air circulation in rooms due to solar-warmed windows or walls. On the other hand, forced convention occurs from the effect of an external force. Beside gravity to the problem, fluid moves past a warmer or cooler surface in obedience to Newton's laws of cooling. Under the two considerations, fluid velocities in free convention are considerably lower than fluid velocities in forced convention. Efficiency of heat transferred is a direct consequence of greater mechanical energy consumed in forced flow situation [1, 2]. Forced convention is seen applicable in the heat transfer process from heating and cooling coils. Convention is majorly responsible for cooling of buildings making it a common mode of heat transfers in buildings (Tables 1–3).

As a fall out of Newton's law of cooling which simply states that the rate at which heat is transferred by convention is proportional to the temperature difference and the heat transfer area. Mathematically, Newton by the law Eq. (15) is expressed as:

$$\mathcal{Q} = h\_{con} \mathcal{A} \left( T\_s - T\_f \right) = h\_{con} \mathcal{A} \left( \Delta T \right) \tag{15}$$

The best theoretical approach for analyzing heat convention is attained by parameters of dimensional analysis using mass, length, time and temperature as focal dimensions Eq. (16). For dynamically similar bodies, natural convention is

> l 3 gαP<sup>3</sup> T η !

This expression contains three dimensionless groups which include the Nusselt

bodies involved which serves as equivalents of shape factors (s) in conduction mode. When the convention is not free, i.e., forced, the analyzing equation Eq. (16) takes the

> lvp η � �:F<sup>2</sup>

¼ F<sup>1</sup>

Since it is a forced convention, this expression omits the free component

numbers having reference tables, it makes it easy for HVAC designers to measure.

� � where f1 and f2 is assumed to be dependent on the shapes of the dynamic

: f <sup>2</sup>

Cη λp

> 3 gαP<sup>3</sup>

Cη λp

� � (16)

T=η � � and the Prandtl

� � (17)

η

� �. With all the

hcon:l λT � �

¼ f <sup>1</sup>

� �, the Grashof or free convention number l

hcon:l λT � �

(Grashof) and introduces the Reynolds number to the expression lvp

measured by

Table 1c.

Table 1b.

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

number hcon:<sup>l</sup>

number <sup>C</sup><sup>η</sup>

123

form of Eq. (17)

λT

λp

with hcon = convention coefficient (W/m<sup>2</sup> .k)); A = surface area through which convention occurs; Ts = surface temperature; and Tf = fluid temperature, away from wall.


Table 1a.

Thermal conductivity of common building materials (a, b, c).

#### Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490


#### Table 1b.

2.2 Convention

Zero and Net Zero Energy

convention primarily takes two forms:

restriction of gravity.

gravity current negligibly.

heat transfers in buildings (Tables 1–3).

with hcon = convention coefficient (W/m<sup>2</sup>

Thermal conductivity of common building materials (a, b, c).

from wall.

Table 1a.

122

Unlike heat conduction where the transference of heat is through a body (solid) without visible motion of any part of the body to the naked eyes, convention is a method of heat transfer in fluid by the movement of the fluid itself [4]. Heat

a. Natural (or free) heat convention is when the motion of the fluid is due solely to the presence of the hot body in it giving rise to temperature with a resultant mediums' density gradient causing the fluid to move under the control or

b.Forced heat convention is a process where heat is transferred with relative motion between the hot body and the fluid maintained by some external agency such as draught, making the relative velocity to contribute to the

As a fall out of Newton's law of cooling which simply states that the rate at which heat is transferred by convention is proportional to the temperature difference and the heat transfer area. Mathematically, Newton by the law Eq. (15) is expressed as:

<sup>¼</sup> hcon <sup>A</sup> ð Þ <sup>Δ</sup><sup>T</sup> (15)

.k)); A = surface area through which

Q ¼ hcon A Ts � Tf

convention occurs; Ts = surface temperature; and Tf = fluid temperature, away

Free convention particularly results in density differences in the fluid, occasioned by contact with the surface originating the heat transfer [5]. Free conventions are evident in gentle air circulation in rooms due to solar-warmed windows or walls. On the other hand, forced convention occurs from the effect of an external force. Beside gravity to the problem, fluid moves past a warmer or cooler surface in obedience to Newton's laws of cooling. Under the two considerations, fluid velocities in free convention are considerably lower than fluid velocities in forced convention. Efficiency of heat transferred is a direct consequence of greater mechanical energy consumed in forced flow situation [1, 2]. Forced convention is seen applicable in the heat transfer process from heating and cooling coils. Convention is majorly responsible for cooling of buildings making it a common mode of


#### Table 1c.

The best theoretical approach for analyzing heat convention is attained by parameters of dimensional analysis using mass, length, time and temperature as focal dimensions Eq. (16). For dynamically similar bodies, natural convention is measured by

$$f\left(\frac{h\_{\rm con} \cdot l}{\lambda T}\right) = f\_1\left(\frac{l^3 g a P^3 T}{\eta}\right) \cdot f\_2\left(\frac{C\eta}{\lambda p}\right) \tag{16}$$

This expression contains three dimensionless groups which include the Nusselt number hcon:<sup>l</sup> λT � �, the Grashof or free convention number l 3 gαP<sup>3</sup> T=η � � and the Prandtl number <sup>C</sup><sup>η</sup> λp � � where f1 and f2 is assumed to be dependent on the shapes of the dynamic bodies involved which serves as equivalents of shape factors (s) in conduction mode. When the convention is not free, i.e., forced, the analyzing equation Eq. (16) takes the form of Eq. (17)

$$\left(\frac{h\_{con.l}}{\lambda T}\right) = F\_1\left(\frac{lvp}{\eta}\right).F\_2\left(\frac{C\eta}{\lambda p}\right) \tag{17}$$

Since it is a forced convention, this expression omits the free component (Grashof) and introduces the Reynolds number to the expression lvp η � �. With all the numbers having reference tables, it makes it easy for HVAC designers to measure.


#### Table 2.

Diffusivity of common building materials (a, b).


Table 2b.

A corresponding derivation for R value and thermal resistance exists for convention methods of transfer that serves for both forced and free conventions Eq. (18) as

$$R = \frac{1}{h\_{con}A} \tag{18}$$

Rth <sup>¼</sup> <sup>1</sup> hcon

Heat flows outside of buildings have been a source of heating in the inside of buildings. It is well a good preemptive move to determine heat flows outside of buildings which naturally contributes to the heating in the entire building envelope [1, 3, 5]. Most external heat flows in buildings with forced convention flows are usually regarded as turbulent, but usually again take the form of laminar or

<sup>U</sup> � <sup>1</sup> Rth

Table 3a.

Table 3b.

125

Radiation on surfaces (a, b).

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

(20)

¼ hcon (21)

With <sup>a</sup> heat transfer function of <sup>Q</sup> <sup>¼</sup> <sup>Δ</sup><sup>T</sup> <sup>R</sup> (19)

Resistance to thermal effusion under convention with Rth value and the associated U-value are given by Eqs. (20) and (21)

Table 3a. Radiation on surfaces (a, b).

Table 3b.

A corresponding derivation for R value and thermal resistance exists for convention methods of transfer that serves for both forced and free conventions Eq. (18) as

hconA (18)

<sup>R</sup> (19)

<sup>R</sup> <sup>¼</sup> <sup>1</sup>

With <sup>a</sup> heat transfer function of <sup>Q</sup> <sup>¼</sup> <sup>Δ</sup><sup>T</sup>

Resistance to thermal effusion under convention with Rth value and the associ-

ated U-value are given by Eqs. (20) and (21)

Table 2.

Zero and Net Zero Energy

Table 2b.

124

Diffusivity of common building materials (a, b).

$$R\_{th} = \frac{1}{h\_{\rm con}}\tag{20}$$

$$U \equiv \frac{1}{R\_{th}} = h\_{\text{con}} \tag{21}$$

Heat flows outside of buildings have been a source of heating in the inside of buildings. It is well a good preemptive move to determine heat flows outside of buildings which naturally contributes to the heating in the entire building envelope [1, 3, 5]. Most external heat flows in buildings with forced convention flows are usually regarded as turbulent, but usually again take the form of laminar or

turbulent flow when the convention currents are free [1–3]. Keeping units of measurements in SI units, there are experimental proofs that with air temperature between 19 and 21°C for interior walls and window surfaces having confluence, with exterior surfaces, laminar free convention of air from internal surfaces is given as Eq. (22)

$$h\_{con} = 1.4232 \left(\frac{\Delta T \text{Sin} \beta}{L}\right)^{\frac{1}{4}} \tag{22}$$

[1, 5]. Heat radiations been electromagnetic in nature are originated by thermal movement of particles in matter. At temperatures higher than absolute zero, all matter sends out thermal radiation. The dynamical behavior or movements of particles results in charge acceleration that produces electromagnetic radiation. Thermal emitting bodies at any temperature consist of a wide range of frequencies [3]. Most radiating bodies have dominant frequencies which shift to higher frequencies as the temperature of the source increases. In most thermal radiation situations, the total amount of radiation for all frequencies increases sharply as the temperature rises at a

, with 'T' as the absolute temperature of the body [6, 7]. Consequently,

Q<sup>1</sup>�<sup>2</sup> ¼ A1Eb1F<sup>1</sup>!<sup>2</sup> � A2Eb2F<sup>2</sup>!<sup>1</sup> (28)

� � where 'σ' is the Stefan-Boltzmann

α þ τ þ ρ ¼ 1: (30)

ðτ°

0 n2 (29)

Eb <sup>¼</sup> actual emissive power emissive power of black body

ð Þ τ° E1ð Þ j j τ � τ° θ τð Þ° dτ°

3 5

(31)

the rate of the electromagnetic radiation emitted at a certain frequency is proportional to the amount of absorption that it would experience by the source [3, 6]. Estimation of heat transferred by radiation is called net radiative heat transfer, which is the heat transferred from one surface to another, been the heat leaving the first surface for the other and subtracting the heat arriving from the second surface. For radiating black bodies, the radiation rate from Surface A to Surface B is

where A is surface area, Eb<sup>1</sup> is energy flux, F<sup>1</sup>!<sup>2</sup> is the view factor originating

Since the two surfaces exchange their heat loses, the reciprocity rule holds for the view factors as A1F<sup>1</sup>!<sup>2</sup> ¼ A2F<sup>2</sup>!<sup>1</sup> with heat flux emissive power of black body

constant and T is absolute temperature. Should the value of Q give negative value; it

As a departure for black bodies, two gray surfaces retaining an enclosure have

<sup>Q</sup> <sup>¼</sup> <sup>σ</sup> <sup>T</sup><sup>4</sup>

1�ϵ<sup>1</sup> <sup>A</sup>1ϵ<sup>1</sup> <sup>þ</sup> <sup>1</sup>

where ϵ<sup>1</sup> and ϵ<sup>2</sup> are the emissivity of the surfaces and any <sup>ϵ</sup> <sup>¼</sup> <sup>E</sup>

whose range are over the three properties are calculated as same.

2 4

ð Þ<sup>τ</sup> <sup>θ</sup>4ð Þ� <sup>τ</sup> <sup>1</sup>=<sup>2</sup> β τð ÞE2ð Þþ <sup>τ</sup> β τð Þ° <sup>E</sup>2ð Þþ <sup>τ</sup>° � <sup>τ</sup>

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 � �

Theoretically, the Stefan-Boltzmann law as stated in Eq. 28 above governs radiation emission of a blackbody (ideal radiator). Besides the emissivity of materials, other indices used for computing the rate of radiation heat transfer from surfaces includes absorptivity (α), transmissivity (τ) and reflectivity (ρ) [3, 5, 8]. All radiating surfaces have these three properties Eq. (30) related by the law of conservation of energy as:

However, this relation depends on the nature of the wave-length been radiated which is verifiably true for single wavelengths and gray surfaces. For wavelengths

Temperature absorption questions through building elements with dark boundaries have been given extensive analysis in the works of [7] by integro-differential

<sup>A</sup>1F1!<sup>2</sup> <sup>þ</sup> <sup>1</sup>�ϵ<sup>2</sup> A2ϵ<sup>2</sup>

rate of T4

given in Eq. (28):

means in Eq. (31)

Nr d2 θ τð Þ <sup>d</sup>τ<sup>2</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>

127

from surface 1 to surface 2.

given as Eb <sup>¼</sup> <sup>σ</sup>T<sup>4</sup> then <sup>Q</sup><sup>1</sup>�<sup>2</sup> <sup>¼</sup> <sup>A</sup>1F<sup>1</sup>!<sup>2</sup> <sup>T</sup><sup>4</sup>

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

their heat transfer rate given in Eq. (29):

suggests that net heat transfer is from surface 2 to surface 1.

wherein ΔT equates the temperature difference as Ts � Tf , with L as the length of the horizontal framing member on a vertical stud and titling surface in the direction of the buoyed-driven flow caused by the convention. β is the surface tilt angle with acute properties (30–90°) and the flow condition that L<sup>3</sup> ΔT<1.0.

If L3 ΔT < 1, a turbulent flow occurs for which turbulent free convention from a tilted surface in air gives Eq. (23)

$$h\_{\rm con} = \mathbf{1.3131} \left(\Delta T \text{Sim} \beta\right)^{\frac{1}{5}} \tag{23}$$

For horizontal members' particularly horizontal pipes and cylindrical members in air, laminar free flow convention is estimated from Eq. (24)

$$h\_{con} = \mathbf{1.3131} \left(\Delta T/D\right)^{\frac{1}{4}} \tag{24}$$

with D as the cylinder's outer diameter. Both turbulent and laminar flows have the same standards for test and measurement with those of tilted members and an adjustment for L with D. Notwithstanding, building elements with cylindrical components in air have their turbulent free flow convention computed from Eq. (25)

$$h\_{\rm con} = \mathbf{1.2401 } (\Delta T)^{\frac{1}{3}} \tag{25}$$

However, structural members or surfaces say flat roots having complete 100% exposure horizontally to solar warming without recourse to solar angle have their laminar free flow convention coefficient estimated from Eq. (26)

$$h\_{\rm con} = \mathbf{1.3203 } (\Delta T/L)^{\frac{1}{4}} \tag{26}$$

with L as the average uniform length of the horizontal surface. The flow condition for the above expression is also true for humid or cold surfaces in reversed contact with the sun as obtainable in the surface of a plane skylight in roof tops. Warm surfaces in direct exposure with solar light have their turbulent free convection coefficient computed from Eq. (27) turbulent flow as

$$h\_{con} = \mathbf{1.5214 \ (\Delta T)^{\frac{1}{3}}} \tag{27}$$

warmed surfaces not having direct surface exposures have their laminar convection coefficient reduced owing to stable stratification condition.

#### 2.3 Radiation

This is the process whereby radiant heat energy is transferred from one point to another. It belongs to the class of electromagnetic spectrum between higher and radio waves with a range of wavelength between 740 and 0.3 mm approximately

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

turbulent flow when the convention currents are free [1–3]. Keeping units of measurements in SI units, there are experimental proofs that with air temperature between 19 and 21°C for interior walls and window surfaces having confluence, with exterior surfaces, laminar free convention of air from internal surfaces is

> ΔTSinβ L <sup>1</sup>

4

, with L as the

<sup>3</sup> (23)

<sup>4</sup> (24)

<sup>3</sup> (25)

<sup>4</sup> (26)

<sup>3</sup> (27)

(22)

hcon ¼ 1:4232

length of the horizontal framing member on a vertical stud and titling surface in the direction of the buoyed-driven flow caused by the convention. β is the surface tilt angle with acute properties (30–90°) and the flow condition that L<sup>3</sup> ΔT<1.0.

hcon <sup>¼</sup> <sup>1</sup>:<sup>3131</sup> ð Þ <sup>Δ</sup>TSin<sup>β</sup> <sup>1</sup>

hcon <sup>¼</sup> <sup>1</sup>:<sup>3131</sup> ð Þ <sup>Δ</sup>T=<sup>D</sup> <sup>1</sup>

with D as the cylinder's outer diameter. Both turbulent and laminar flows have the same standards for test and measurement with those of tilted members and an adjustment for L with D. Notwithstanding, building elements with cylindrical components in air have their turbulent free flow convention computed from Eq. (25)

hcon <sup>¼</sup> <sup>1</sup>:<sup>2401</sup> ð Þ <sup>Δ</sup><sup>T</sup> <sup>1</sup>

However, structural members or surfaces say flat roots having complete 100% exposure horizontally to solar warming without recourse to solar angle have their

hcon <sup>¼</sup> <sup>1</sup>:<sup>3203</sup> ð Þ <sup>Δ</sup>T=<sup>L</sup> <sup>1</sup>

reversed contact with the sun as obtainable in the surface of a plane skylight in roof tops. Warm surfaces in direct exposure with solar light have their turbulent free

hcon <sup>¼</sup> <sup>1</sup>:<sup>5214</sup> ð Þ <sup>Δ</sup><sup>T</sup> <sup>1</sup>

warmed surfaces not having direct surface exposures have their laminar con-

This is the process whereby radiant heat energy is transferred from one point to another. It belongs to the class of electromagnetic spectrum between higher and radio waves with a range of wavelength between 740 and 0.3 mm approximately

with L as the average uniform length of the horizontal surface. The flow condition for the above expression is also true for humid or cold surfaces in

For horizontal members' particularly horizontal pipes and cylindrical members

If L3 ΔT < 1, a turbulent flow occurs for which turbulent free convention from a

wherein ΔT equates the temperature difference as Ts � Tf

in air, laminar free flow convention is estimated from Eq. (24)

laminar free flow convention coefficient estimated from Eq. (26)

convection coefficient computed from Eq. (27) turbulent flow as

vection coefficient reduced owing to stable stratification condition.

given as Eq. (22)

Zero and Net Zero Energy

2.3 Radiation

126

tilted surface in air gives Eq. (23)

[1, 5]. Heat radiations been electromagnetic in nature are originated by thermal movement of particles in matter. At temperatures higher than absolute zero, all matter sends out thermal radiation. The dynamical behavior or movements of particles results in charge acceleration that produces electromagnetic radiation. Thermal emitting bodies at any temperature consist of a wide range of frequencies [3]. Most radiating bodies have dominant frequencies which shift to higher frequencies as the temperature of the source increases. In most thermal radiation situations, the total amount of radiation for all frequencies increases sharply as the temperature rises at a rate of T4 , with 'T' as the absolute temperature of the body [6, 7]. Consequently, the rate of the electromagnetic radiation emitted at a certain frequency is proportional to the amount of absorption that it would experience by the source [3, 6]. Estimation of heat transferred by radiation is called net radiative heat transfer, which is the heat transferred from one surface to another, been the heat leaving the first surface for the other and subtracting the heat arriving from the second surface.

For radiating black bodies, the radiation rate from Surface A to Surface B is given in Eq. (28):

$$Q\_{1-2} = A\_1 E\_{b1} F\_{1 \to 2} - A\_2 E\_{b2} F\_{2 \to 1} \tag{28}$$

where A is surface area, Eb<sup>1</sup> is energy flux, F<sup>1</sup>!<sup>2</sup> is the view factor originating from surface 1 to surface 2.

Since the two surfaces exchange their heat loses, the reciprocity rule holds for the view factors as A1F<sup>1</sup>!<sup>2</sup> ¼ A2F<sup>2</sup>!<sup>1</sup> with heat flux emissive power of black body given as Eb <sup>¼</sup> <sup>σ</sup>T<sup>4</sup> then <sup>Q</sup><sup>1</sup>�<sup>2</sup> <sup>¼</sup> <sup>A</sup>1F<sup>1</sup>!<sup>2</sup> <sup>T</sup><sup>4</sup> <sup>1</sup> � <sup>T</sup><sup>4</sup> 2 � � where 'σ' is the Stefan-Boltzmann constant and T is absolute temperature. Should the value of Q give negative value; it suggests that net heat transfer is from surface 2 to surface 1.

As a departure for black bodies, two gray surfaces retaining an enclosure have their heat transfer rate given in Eq. (29):

$$Q = \frac{\sigma \left(T\_1^4 - T\_2^4\right)}{\frac{1 - c\_1}{A\_{1\ell 1}} + \frac{1}{A\_{1\ell 1 \to 2}} + \frac{1 - c\_2}{A\_{2\ell 2}}} \tag{29}$$

where ϵ<sup>1</sup> and ϵ<sup>2</sup> are the emissivity of the surfaces and any <sup>ϵ</sup> <sup>¼</sup> <sup>E</sup> Eb <sup>¼</sup> actual emissive power emissive power of black body Theoretically, the Stefan-Boltzmann law as stated in Eq. 28 above governs radiation emission of a blackbody (ideal radiator). Besides the emissivity of materials, other indices used for computing the rate of radiation heat transfer from surfaces includes absorptivity (α), transmissivity (τ) and reflectivity (ρ) [3, 5, 8]. All radiating surfaces have these three properties Eq. (30) related by the law of conservation of energy as:

$$a + \mathfrak{r} + \rho = \mathbf{1}.\tag{30}$$

However, this relation depends on the nature of the wave-length been radiated which is verifiably true for single wavelengths and gray surfaces. For wavelengths whose range are over the three properties are calculated as same.

Temperature absorption questions through building elements with dark boundaries have been given extensive analysis in the works of [7] by integro-differential means in Eq. (31)

$$N\_r \frac{d^2 \theta(\tau)}{d\tau^2} = n^2(\tau) \theta^4(\tau) - \mathbf{1}/2 \left[ \beta(\tau) E\_2(\tau) + \beta(\tau^o) E\_2(\tau^o - \tau) + \int\_0^\tau n^2(\tau^o) E\_1(|\tau - \tau^o|) \theta(\tau^o) d\tau^o \right] \tag{31}$$

Noting the boundary condition to be

$$
\theta(\mathbf{0}) = \theta\_{2\star} \theta \mathbf{r}^{\bullet} = \mathbf{1}.\mathbf{0}
$$

dθ

in nature.

Fourier Law.

129

<sup>d</sup><sup>τ</sup> <sup>¼</sup> ð Þ <sup>1</sup>=2Nr <sup>ð</sup>βð Þ <sup>0</sup> ½ � <sup>E</sup>3ð Þþ <sup>τ</sup> <sup>E</sup>4ð Þ <sup>τ</sup>° <sup>=</sup>τ° � <sup>1</sup>=3τ°

<sup>τ</sup> ° � <sup>E</sup>3ð Þþ <sup>τ</sup>° � <sup>τ</sup> <sup>1</sup>=3τ° � �

β ¼

ð Þ <sup>τ</sup>° <sup>E</sup>2ð Þþ j j <sup>τ</sup> � <sup>τ</sup>° ð Þ <sup>1</sup>=τ° <sup>½</sup>E3ð Þ� <sup>τ</sup>° � <sup>τ</sup> <sup>E</sup>3ð Þ� <sup>τ</sup>° <sup>g</sup>θ4ð Þ<sup>τ</sup> <sup>d</sup>τ°<sup>Þ</sup> �

j≥1

By securitizing Eq. 36, the steep behavior or temperature gradient of the absorption can be inferred from the dimensionless gradient (β) which satisfies the

> d <sup>θ</sup>T1�T<sup>2</sup> T1�T<sup>2</sup> � �

> > d <sup>τ</sup> τ° � �

In recent times, many simulation techniques have been developed to determine temperature profiles and heat fluxes in building, but many of which are interactive

This brings us to the absorptivity and emissivity of gray surfaces which under

α ¼ ϵ

<sup>τ</sup>° <sup>¼</sup> <sup>L</sup>

Even for non-gray surfaces at a stipulated wavelength, drawing a clue from [5]

<sup>1</sup>=<sup>E</sup> � EL <sup>≪</sup> <sup>1</sup>

This is premised on the notion that the radiant heat flux is not tempered by the material and provided the conductive radiative mechanisms are not acting with each other, the building element will experience a total heat flux Eq. (38) equal to

where qt = total heat flux; qr = radiative heat flux; qc = conductive heat flux—by

ð Þ T<sup>1</sup> � T<sup>2</sup>

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 � �

And radiant heat flux with two infinite parallel plates with temperatures at T1 ,

with reference to Eq. (38), qt and by substitution, reduces to Eqs. (41) and (42)

Such that planar elements with thickness L, having uniform properties and unidirectional steady state heat flow have their values computed from Eq. (39)

qr ¼ kc

qr <sup>¼</sup> r T<sup>4</sup>

1 <sup>ϵ</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup>

and having T2 and emissivites ϵ<sup>1</sup> and ϵ<sup>2</sup> computed by Eq. (40)

þ ð2Nr=τ°½ � θ τð Þ� ° θð Þ 0

<sup>τ</sup> <sup>¼</sup> <sup>0</sup> (37)

<sup>E</sup>) passing through an object with

qt ¼ qc þ qr (38)

<sup>L</sup> (39)

<sup>ϵ</sup><sup>2</sup> � <sup>1</sup> (40)

(36)

<sup>þ</sup> β τð Þ° �E4ð Þ <sup>τ</sup>°

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

Schwartz inequality in Eq. (37)

the Kirchoff's identify are equal, been

thickness, L is related by the formula

experiment that the mean free path of a photon (<sup>1</sup>

�

þ ð<sup>τ</sup>° 0 n2

whereas, [7] showed estimation of the absorbed heat with dimensionless temperature value which has equally been shown to be of value expressed in Eq. (32)

$$\theta(\mathbf{r}^o) = G(\mathbf{r}) = \frac{1}{2N\_r} \Bigg[ \int\_0^\mathbf{r} n^2(\mathbf{r}^o) \left\{ -E\_3(|\mathbf{r} - \mathbf{r}^o|) + E\_3(\mathbf{r}^o) + \frac{\pi}{\tau^o} [E\_3(\mathbf{r}^o - \mathbf{r}) - E\_3(\mathbf{r}^o)] \right\} \theta^4(\mathbf{r}^o) d\tau^o \tag{32}$$

Annotated by Eq. (33)

$$\begin{split} G(\tau) &= \frac{1}{2N\_r} \left( \beta(\mathbf{0}) \left[ -E\_4(\tau) + \frac{\tau}{\tau^o} E\_4(\tau^o) + \frac{1}{3} \left( 1 - \frac{\tau}{\tau^o} \right) \right] \\ &+ \beta(\tau^o) \left[ \left( 1 - \frac{\tau}{\tau^o} E\_4(\tau^o) - E\_4(\tau^o - \tau) + \frac{1}{3} \frac{\tau}{\tau^o} \right] \right. \\ &\left. \left. + 2N\_r \left\{ \theta(\mathbf{0}) + \frac{\tau}{\tau^o} \left[ \theta(\tau^o) - \theta(\mathbf{0}) \mathbf{1} \right] \right\} \right) \end{split} \tag{33}$$

In near real life situation, determination of temperature profiles in buildings have not been successful with closed-form solutions but with numerical methods Eq. (34) to obtain the total building heat flux through its elements as

$$q\_t = \frac{k\_c}{L}(T\_1 - T\_2) + 2\sigma \Big(T\_2^4 \Big[E\_3\tau^o + \frac{1}{\tau^o}E\_4\tau^o - \frac{1}{3\tau^o}\Big]$$

$$+ T\_1^4 \Big[-\frac{1}{\tau^o}E\_4\tau^o - \frac{1}{2} + \frac{1}{3\tau^o}\Big] + \int\_0^\tau n^2(\tau^o)\left\{-E\_2(\tau^o - \tau)\right\}$$

$$+ \frac{1}{\tau^o} \Big[E\_3(\tau^o - \tau) - E\_3(\tau)\big] \{T^4(\tau)d\tau\} + \sigma T\_1^4 - 2\sigma T\_2^4 E\_3(\tau^o)$$

$$- 2\sigma \int\_0^\tau n^2(\tau^o)E\_2(\tau^o - \tau) \Big(T^4\tau\big)d\tau\tag{34}$$

The radiative and conductive fluxes are closely outlined by the terms of Eq. 34 in such a way that the first two terms of Eq. 34 are suggestive of conductive flux, while the last three terms are radiative [8, 9]. Upon the combination of both integrals, Eq. 34 becomes Eq. (35)

$$\begin{split} q\_t &= \frac{k\_c}{L}(T\_1 - T\_2) + 2\sigma \Big( T\_2^4 \Big[ \frac{1}{\tau^o} E\_4 \tau^o - \frac{1}{3\tau^o} \Big] + T\_1^4 \Big[ -\frac{1}{\tau^o} E\_4(\tau^o) + \frac{1}{3\tau^o} \Big] \\ &+ \int\_0^\tau n^2(\tau^o) \frac{1}{\tau^o} [E\_3(\tau^o - \tau) - E\_3(\tau)] \left( T^4 \tau \right) d\tau \Big) \end{split} \tag{35}$$

with such algebraic treatment, Eq. 33 can as well be treated with integral calculus to give Eq. (36)

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

Noting the boundary condition to be

θ τð Þ¼ ° <sup>G</sup>ð Þ¼ <sup>τ</sup> <sup>1</sup>

Zero and Net Zero Energy

2Nr

Annotated by Eq. (33)

ðτ°

0 n2

<sup>G</sup>ð Þ¼ <sup>τ</sup> <sup>1</sup>

qt <sup>¼</sup> kc

<sup>þ</sup> <sup>T</sup><sup>4</sup> <sup>1</sup> � <sup>1</sup> τ°

þ 1

� 2σ ðτ°

<sup>L</sup> ð Þþ <sup>T</sup><sup>1</sup> � <sup>T</sup><sup>2</sup> <sup>2</sup><sup>σ</sup> <sup>T</sup><sup>4</sup>

Eq. 34 becomes Eq. (35)

qt <sup>¼</sup> kc

þ ðτ°

0 n2 ð Þ <sup>τ</sup>° <sup>1</sup> τ°

calculus to give Eq. (36)

128

0 n2

> 2 1 τ°

2Nr

<sup>þ</sup> β τð Þ° <sup>1</sup> � <sup>τ</sup>

<sup>þ</sup> <sup>2</sup>Nr <sup>θ</sup>ð Þþ <sup>0</sup> <sup>τ</sup>

Eq. (34) to obtain the total building heat flux through its elements as

<sup>L</sup> ð Þþ <sup>T</sup><sup>1</sup> � <sup>T</sup><sup>2</sup> <sup>2</sup><sup>σ</sup> <sup>T</sup><sup>4</sup>

<sup>E</sup>4τ° � <sup>1</sup>

� �

θð Þ¼ 0 θ2, θτ° ¼ 1:0

whereas, [7] showed estimation of the absorbed heat with dimensionless temperature value which has equally been shown to be of value expressed in Eq. (32)

τ°

<sup>E</sup>4ð Þþ <sup>τ</sup>° <sup>1</sup>

<sup>E</sup>4ð Þ� <sup>τ</sup>° <sup>E</sup>4ð Þþ <sup>τ</sup>° � <sup>τ</sup> <sup>1</sup>

3

<sup>1</sup> � <sup>τ</sup> τ°

> 3 τ τ°

n o

½ � E3ð Þ� τ° � τ E3ð Þ τ°

<sup>θ</sup>4ð Þ <sup>τ</sup>° <sup>d</sup>τ°

(32)

(33)

ðÞ� <sup>τ</sup>° <sup>E</sup>3ð Þþ j j <sup>τ</sup> � <sup>τ</sup>° <sup>E</sup>3ð Þþ <sup>τ</sup>° <sup>τ</sup>

<sup>β</sup>ðÞ� <sup>0</sup> <sup>E</sup>4ð Þþ <sup>τ</sup> <sup>τ</sup>

τ°

In near real life situation, determination of temperature profiles in buildings have not been successful with closed-form solutions but with numerical methods

> 2 þ 1 3τ°

<sup>τ</sup>° <sup>E</sup>3ð Þ� <sup>τ</sup>° � <sup>τ</sup> <sup>E</sup>3ð Þ�g <sup>τ</sup> <sup>T</sup><sup>4</sup>ð Þ<sup>τ</sup> <sup>d</sup><sup>τ</sup> � � <sup>þ</sup> <sup>σ</sup>T<sup>4</sup>

The radiative and conductive fluxes are closely outlined by the terms of Eq. 34 in such a way that the first two terms of Eq. 34 are suggestive of conductive flux, while the last three terms are radiative [8, 9]. Upon the combination of both integrals,

> <sup>E</sup>4τ° � <sup>1</sup> 3τ°

with such algebraic treatment, Eq. 33 can as well be treated with integral

� �

½ � <sup>E</sup>3ð Þ� <sup>τ</sup>° � <sup>τ</sup> <sup>E</sup>3ð Þ<sup>τ</sup> <sup>T</sup><sup>4</sup><sup>τ</sup> � �dτ<sup>g</sup>

τ°

<sup>τ</sup>° θ τð Þ� ° <sup>θ</sup>ð Þ <sup>0</sup> <sup>1</sup> � � ��

<sup>2</sup> E3τ° þ

þ ðτ°

<sup>þ</sup> <sup>T</sup><sup>4</sup> <sup>1</sup> � <sup>1</sup> τ°

� � �

0 n2

ð Þ <sup>τ</sup>° <sup>E</sup>2ð Þ <sup>τ</sup>° � <sup>τ</sup> <sup>T</sup><sup>4</sup><sup>τ</sup> � �d<sup>τ</sup> (34)

1 τ°

� � �

<sup>E</sup>4τ° � <sup>1</sup> 3τ°

ðÞ� τ° f E2ð Þ τ° � τ

<sup>E</sup>4ð Þþ <sup>τ</sup>° <sup>1</sup>

3τ°

(35)

<sup>1</sup> � <sup>2</sup>σT<sup>4</sup>

<sup>2</sup>E3ð Þ τ°

� � � � �

�� �

$$\begin{aligned} \frac{d\theta}{d\tau} &= (\mathbf{1}/2N\_r)(\boldsymbol{\theta}(\mathbf{0})[E\_3(\tau) + E\_4(\tau^o)/\tau^o - \mathbf{1}/3\tau^o] \\\\ &+ \left(\boldsymbol{\beta}(\tau^o)\left[\frac{-E\_4(\tau^o)}{\tau}\mathbf{e} - E\_3(\tau^o - \tau) + \mathbf{1}/3\tau^o\right] + (2N\_r/\tau^o[\boldsymbol{\theta}(\tau^o) - \boldsymbol{\theta}(\mathbf{0})] \right. \end{aligned} \tag{36}$$

$$\begin{aligned} &+ \int\_0^{\tau^o} n^2(\tau^o) \left\{ E\_2(|\tau - \tau^o|) + (\mathbf{1}/\tau^o)[E\_3(\tau^o - \tau) - E\_3(\tau^o)] \right\} \theta^4(\tau) d\tau^o \end{aligned}$$

By securitizing Eq. 36, the steep behavior or temperature gradient of the absorption can be inferred from the dimensionless gradient (β) which satisfies the Schwartz inequality in Eq. (37)

$$\beta = \frac{d\left(\frac{\theta T\_1 - T\_2}{T\_1 - T\_2}\right)}{d\left(\frac{\pi}{\pi^2}\right)} \overset{\text{|} \ge \mathbf{1}}{\text{  $\mathbf{1}$ }} \tag{37}$$

In recent times, many simulation techniques have been developed to determine temperature profiles and heat fluxes in building, but many of which are interactive in nature.

This brings us to the absorptivity and emissivity of gray surfaces which under the Kirchoff's identify are equal, been

α ¼ ϵ

Even for non-gray surfaces at a stipulated wavelength, drawing a clue from [5] experiment that the mean free path of a photon (<sup>1</sup> <sup>E</sup>) passing through an object with thickness, L is related by the formula

$$
\tau^o = \frac{L}{1/E} - EL \ll 1
$$

This is premised on the notion that the radiant heat flux is not tempered by the material and provided the conductive radiative mechanisms are not acting with each other, the building element will experience a total heat flux Eq. (38) equal to

$$q\_t = q\_c + q\_r \tag{38}$$

where qt = total heat flux; qr = radiative heat flux; qc = conductive heat flux—by Fourier Law.

Such that planar elements with thickness L, having uniform properties and unidirectional steady state heat flow have their values computed from Eq. (39)

$$q\_r = k\_c \frac{(T\_1 - T\_2)}{L} \tag{39}$$

And radiant heat flux with two infinite parallel plates with temperatures at T1 , and having T2 and emissivites ϵ<sup>1</sup> and ϵ<sup>2</sup> computed by Eq. (40)

$$q\_r = \frac{\overline{r} \left( T\_1^4 - T\_2^4 \right)}{\frac{1}{c\_1} + \frac{1}{c\_2} - 1} \tag{40}$$

with reference to Eq. (38), qt and by substitution, reduces to Eqs. (41) and (42)

$$q\_t = k\_c \frac{(T\_1 - T\_2)}{L} + \frac{\sigma \left(T\_1^4 - T\_2^4\right)}{\frac{1}{c\_1} + \frac{1}{c\_2} - 1} \tag{41}$$

kapp ¼ kc þ

qr <sup>¼</sup> <sup>σ</sup> <sup>T</sup><sup>4</sup>

found in the approach of [9] with the condition that T° ≫ 1; then

<sup>1</sup> <sup>þ</sup> <sup>1</sup> ϵ1 � � <sup>þ</sup> <sup>1</sup> ϵ2 � � � <sup>2</sup> h i<sup>Q</sup>

of thickness on apparent thermal properties of insulation. But exact solution is

<sup>Q</sup> <sup>¼</sup> <sup>4</sup>=<sup>3</sup> τ° þ γ

Appropriate approximation to this problem is found in the exponential-kernel as

While that of [8] is consistent with the exponential-kernel approximation, Rennex only introduced a factor in the approximation value of Eq. (48) by proposing that

<sup>3</sup> ½ � factor (48)

<sup>Q</sup> <sup>¼</sup> <sup>4</sup>=<sup>3</sup> <sup>τ</sup>° <sup>þ</sup> <sup>4</sup> 3

<sup>Q</sup> <sup>¼</sup> <sup>4</sup>=<sup>3</sup> <sup>τ</sup>° <sup>þ</sup> <sup>4</sup>

Kapp ¼ kc þ

Due to Rennex, Eq. (50) we have

Provided <sup>ϵ</sup><sup>1</sup> <sup>¼</sup> <sup>ϵ</sup><sup>2</sup> <sup>¼</sup> <sup>ϵ</sup> and <sup>T</sup><sup>4</sup>

131

2 <sup>ϵ</sup> � <sup>1</sup> <sup>þ</sup> <sup>3</sup>

Kapp ¼ kc þ

Kapp <sup>¼</sup> <sup>4</sup>σT<sup>3</sup>

<sup>1</sup>�T<sup>4</sup> ð Þ<sup>2</sup> T1�T<sup>2</sup> � �

2 <sup>ϵ</sup> � <sup>1</sup> <sup>þ</sup> <sup>3</sup>

Factor = 1 + 0.0657 tanh (27°) while addressing the [9] Q-value estimation. A further theory on the effect of elements thickness on the apparent thermal conductivity with the assumption that radiative and conductive heat fluxes do not interact and premised on the computation that total heat flux is equal to the sum of the individual fluxes, [8] obtained the value of Kapp by substitutions in Eq. (49)

> 4σT<sup>3</sup> m

> 4σT<sup>3</sup> m

> > m

2 <sup>ϵ</sup> � <sup>1</sup> <sup>þ</sup> <sup>3</sup>

ffi <sup>4</sup>T<sup>3</sup> m

<sup>4</sup> τ° þ 0:0657

<sup>4</sup> τ°

L

<sup>4</sup> <sup>τ</sup>° <sup>þ</sup> <sup>0</sup>:0657 tan 2ð Þ <sup>τ</sup>° <sup>L</sup> (50)

L (49)

at temperatures T1 and T2 as given in Eq. (47)

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

where γ = 1.4209.

42 nσQ T<sup>4</sup>

Discussing conductivity and heat radiation of building elements with respect to the element's thickness as it affects the apparent thermal properties of insulation has its credit due to [8]. The basic concept of [8] idea is that by the very nature of insulation, conduction and radiation does not occur and their individual heat fluxes are sums. Going by [8] theorem, at heat radiation equilibrium, radiant heat flux between two infinite parallel plates separated by a di-heat gray material or medium

> <sup>1</sup> � <sup>T</sup><sup>4</sup> 2 � �

As stated in heat transfer literature, several methods exist for treating the effect

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 � �

(46)

(47)

3αð Þ T<sup>1</sup> � T<sup>2</sup>

Since ϵ<sup>1</sup> = ϵ<sup>2</sup> = 1 for black plates, qt becomes

$$q\_t = k\_c \frac{(T\_1 - T\_2)}{L} + \sigma \left(T\_1^4 - T\_2^4\right) \tag{42}$$

Besides [5] investigation for the optically thin limit case, the limiting case for the optical thickness limit was investigated by [10] for elements that are large compared to the mean free path of the photon causing the radiation, giving rise to conductive heat transfer process. By experiment, from [10]

$$
\tau^o = \frac{L}{1/E} = EL \gg 1
$$

So that radiant heat flux arising from radiant energy Eq. (43) becomes

$$q\_t = -k\_r \frac{dT}{d\mathbf{x}}\tag{43}$$

And by combining the conductive and radiative heat transfers of the element, the total heat flux for the building element at a steady state for uni-directional heat flow as Eq. (44) becomes

$$q\_t = k\_c \frac{T\_1 - T\_2}{L} + \frac{4\_n^2 \sigma \left(T\_1^4 - T\_2^4\right)}{3aL} \tag{44}$$

where kr = radiative conductivity of a gray medium ffi <sup>16</sup> 3 n2σT<sup>3</sup> α

$$k\_{app} = k\_c + k\_r$$

$$q\_t = -k\_{\mathcal{G}} \cdot \frac{dT}{dx}$$

With this totality conduction, apparent thermal conductivity is obtained by the relationship

$$k\_{app} = q\_t L / (T\_1 - T\_2)$$

With particular reference to the thickness (L) of the material, the conductivity through the optical element at constant temperatures of the plate as Eq. (45) becomes:

$$k\_{app} = k\_c + \frac{\sigma \left(T\_1^4 - T\_2^4\right)}{T\_1 - T\_2}L \tag{45}$$

Experiments have shown that the upper limit of the apparent thermal conductivity of the material greatly depends on the absorption coefficient of the material's thickness and extreme absorption coefficients. With these two conditions, kapp becomes Eq. (46)

Economic Aspects of Building Energy Audit DOI: http://dx.doi.org/10.5772/intechopen.85490

qt ¼ kc

qt ¼ kc

conductive heat transfer process. By experiment, from [10]

qt ¼ kc

where kr = radiative conductivity of a gray medium ffi <sup>16</sup>

Since ϵ<sup>1</sup> = ϵ<sup>2</sup> = 1 for black plates, qt becomes

flow as Eq. (44) becomes

Zero and Net Zero Energy

relationship

becomes Eq. (46)

130

ð Þ T<sup>1</sup> � T<sup>2</sup>

ð Þ T<sup>1</sup> � T<sup>2</sup>

<sup>τ</sup>° <sup>¼</sup> <sup>L</sup>

So that radiant heat flux arising from radiant energy Eq. (43) becomes

qt ¼ �kr

T<sup>1</sup> � T<sup>2</sup> L þ

And by combining the conductive and radiative heat transfers of the element, the total heat flux for the building element at a steady state for uni-directional heat

kapp ¼ kc þ kr

With this totality conduction, apparent thermal conductivity is obtained by the

With particular reference to the thickness (L) of the material, the conductivity through the optical element at constant temperatures of the plate as Eq. (45) becomes:

Experiments have shown that the upper limit of the apparent thermal conductivity of the material greatly depends on the absorption coefficient of the material's thickness and extreme absorption coefficients. With these two conditions, kapp

dT dx

L=ð Þ T<sup>1</sup> � T<sup>2</sup>

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 T<sup>1</sup> � T<sup>2</sup>

qt ¼ �keff :

kapp ¼ qt

kapp <sup>¼</sup> kc <sup>þ</sup> <sup>σ</sup> <sup>T</sup><sup>4</sup>

42 <sup>n</sup>σ T<sup>4</sup>

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 

> 3 n2σT<sup>3</sup> α

<sup>L</sup> <sup>þ</sup> <sup>σ</sup> <sup>T</sup><sup>4</sup>

<sup>L</sup> <sup>þ</sup> <sup>σ</sup> <sup>T</sup><sup>4</sup>

Besides [5] investigation for the optically thin limit case, the limiting case for the optical thickness limit was investigated by [10] for elements that are large compared to the mean free path of the photon causing the radiation, giving rise to

<sup>1</sup>=<sup>E</sup> <sup>¼</sup> EL <sup>≫</sup> <sup>1</sup>

dT

1 <sup>ϵ</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup>

<sup>1</sup> � <sup>T</sup><sup>4</sup> 2 

> <sup>1</sup> � <sup>T</sup><sup>4</sup> 2

<sup>ϵ</sup><sup>2</sup> � <sup>1</sup> (41)

(42)

dx (43)

<sup>3</sup>α<sup>L</sup> (44)

L (45)

$$k\_{app} = k\_c + \frac{4\frac{2}{n}\sigma Q \left(T\_1^4 - T\_2^4\right)}{3\alpha (T\_1 - T\_2)}\tag{46}$$

Discussing conductivity and heat radiation of building elements with respect to the element's thickness as it affects the apparent thermal properties of insulation has its credit due to [8]. The basic concept of [8] idea is that by the very nature of insulation, conduction and radiation does not occur and their individual heat fluxes are sums. Going by [8] theorem, at heat radiation equilibrium, radiant heat flux between two infinite parallel plates separated by a di-heat gray material or medium at temperatures T1 and T2 as given in Eq. (47)

$$q\_r = \frac{\sigma \left(T\_1^4 - T\_2^4\right)}{1 + \left[\left(\frac{1}{c\_1}\right) + \left(\frac{1}{c\_2}\right) - 2\right]Q} \tag{47}$$

As stated in heat transfer literature, several methods exist for treating the effect of thickness on apparent thermal properties of insulation. But exact solution is found in the approach of [9] with the condition that T° ≫ 1; then

$$Q = \frac{4/3}{\pi^{\circ} + \gamma}$$

where γ = 1.4209.

Appropriate approximation to this problem is found in the exponential-kernel as

$$Q = \frac{4/3}{\pi^{\circ} + \frac{4}{3}}$$

While that of [8] is consistent with the exponential-kernel approximation, Rennex only introduced a factor in the approximation value of Eq. (48) by proposing that

$$Q = \frac{4/3}{\tau^{\sigma} + \frac{4}{3}[factor]} \tag{48}$$

Factor = 1 + 0.0657 tanh (27°) while addressing the [9] Q-value estimation. A further theory on the effect of elements thickness on the apparent thermal conductivity with the assumption that radiative and conductive heat fluxes do not interact and premised on the computation that total heat flux is equal to the sum of the individual fluxes, [8] obtained the value of Kapp by substitutions in Eq. (49)

$$K\_{app} = k\_c + \frac{4\sigma T\_m^3}{\frac{2}{\epsilon} - \mathbf{1} + \frac{3}{4}\tau^\sigma + \mathbf{0}.0657}L$$

$$K\_{app} = k\_c + \frac{4\sigma T\_m^3}{\frac{2}{\epsilon} - \mathbf{1} + \frac{3}{4}\tau^\sigma}L\tag{49}$$

Due to Rennex, Eq. (50) we have

$$K\_{app} = \frac{4\sigma T\_m^3}{\frac{2}{e} - 1 + \frac{3}{4}\tau^\circ + 0.0657\tan\left(2\tau^\circ\right)}L\tag{50}$$

Provided <sup>ϵ</sup><sup>1</sup> <sup>¼</sup> <sup>ϵ</sup><sup>2</sup> <sup>¼</sup> <sup>ϵ</sup> and <sup>T</sup><sup>4</sup> <sup>1</sup>�T<sup>4</sup> ð Þ<sup>2</sup> T1�T<sup>2</sup> � � ffi <sup>4</sup>T<sup>3</sup> m

Thickness has its effects on the conductivity of building elements demands [9, 10]. On the whole, optical thickness of elements increases the materials thermal conductivity by asymptotic expansion which tends to a limiting value in such a way that apparent thermal resistance has a linear dependence on element's thickness as the element's thickness approaches infinity. In the same vein, apparent thermal resistivity of the element is equal to the apparent thermal resistance divided by the elements thickness [6, 9, 10].
