_{1}

^{*}

The removal building heat load and electrical power consumption by air conditioning system are proportional to the outside conditions and solar radiation intensity. Building construction materials has substantial effects on the transmission heat through outer walls, ceiling and glazing windows. Good thermal isolation for buildings is important to reduce the transmitted heat and consumed power. The buildings models are constructed from common materials with 0 - 16 cm of thermal insulation thickness in the outer walls and ceilings, and double-layers glazing windows. The building heat loads were calculated for two types of walls and ceiling with and without thermal insulation. The cooling load temperature difference method,
*CLTD*, was used to estimate the building heat load during a 24-hour each day throughout spring, summer, autumn and winter seasons. The annual cooling degree-day,
*CDD* was used to estimate the optimal thermal insulation thickness and payback period with including the solar radiation effect on the outer walls surfaces. The average saved energy percentage in summer, spring, autumn and winter are 35.5%, 32.8%, 33.2% and 30.7% respectively, and average yearly saved energy is about of 33.5%. The optimal thermal insulation thickness was obtained between 7 - 12 cm and payback period of 20 - 30 month for some Egyptian Cities according to the Latitude and annual degree-days.

The buildings heat load is proportional to the transmission heat through outside walls, ceiling, and glazing windows. The transmission heat load is added to the internal building heat load in summer season and subtracted from it in winter season. The thermal insulation is an effective tool to achieve the minimum heat transmission through walls. The transmission heat is a function of the thermal resistance of composite wall materials which actually are related to the thermal insulation thickness. Obviously, the increasing of thermal insulation thickness will increase the investment cost, but reduce the electrical power consumption. The problem is how we can compromise between the investment of insulation cost and the running cost of electrical power consumption.

Recently, many authors have studied the effect of thermal insulation thickness on the building transmission heat load and power consumption. The degree-day method is one of the well-known and the simplest methods used in the Heating, Ventilating and Air-Conditioning industry to estimate heating and cooling energy requirements. The yearly heating and cooling degree-hours are given both in tabular form and as counter maps [

The effect of external roof color surface with periodic heat flow through a homogeneous flat solid concrete slab by solving the heat conduction equation was studied analytically [

A coupled heat and moisture transfer model is used to estimate the insulation thickness, lifecycle saving, and payback period [

The objective of the present study is to investigate the thermal insulation thickness and energy saving in air-conditioned buildings. The analysis was conducted using expanded polystyrene thermal insulation in two types of outer wall surfaces with and without subjected to solar radiation. The hourly building heat load and power consumption with Cooling Load Temperature Deference, CLTD, method are estimated at various thermal insulation thickness and outside atmospheric conditions. The annual cooling degree-day, CDD, with including the solar radiation on the outer walls is presented to obtain the optimal insulation thickness and payback period.

The transmission heat load is calculated according to the building construction materials. The building model we used was residential buildings or hotels with size of 18 × 18 × 3 m as illustrated in [

The building heat load is the amount of heat removed or added to the internal heat load to maintain the thermal comfort inside the building. The building heat load is two main parts. The first part is the sensible heat load which is a function of dry bulb temperature difference between outside and inside air condition, and radiation heat gain through glazing windows. The second part is the latent heat load which is a function of moisture transfer and vapor content inside the conditioned space. The building heat load in summer is the summation of internal and external heat loads. But, in winter is the difference between the internal and external heat loads. The building heat load was calculated using the Cooling Load Temperature Difference, CLTD method as illustrated in [

The outer walls surfaces are subjected to instantaneous temperature T o ( t ) and solar radiation I ( t ) . The inner faces of walls come in contact with the indoor air which maintained at a fixed temperature of T i , 24˚C to have a better thermal comfort. The one dimensional transient heat conduction model with constant thermal conductivity, density and heat capacity is used for this problem as follows:

∂ T ( x , t ) ∂ t = k ρ C p ∂ 2 T ( x , t ) ∂ x 2 (1)

where k is the thermal conductivity, ρ is the density and C p is the specific heat of the wall material. To solve this problem, two boundary conditions and one initial condition are required. On both sides of the wall, convection boundary conditions are present. On the inner surface, at constant inside convection heat transfer coefficient is:

− k ∂ T ( x , t ) ∂ x | x = L = h i ( T x = L ( t ) − T i ) (2)

Whereas on the outdoor surface of the wall, the boundary condition at constant outside convection heat transfer coefficient can be written as:

− k ∂ T ( x , t ) ∂ x | x = 0 = h o ( T o ( t ) − T x = 0 ( t ) ) (3)

where, h o , h i are the film convection heat transfer coefficient of outside and inside walls of building, and their values are h o = 22 W / m 2 ⋅ K and h i = 9 W / m 2 ⋅ K [

The steady-state solution of the problem at constant inside room temperature, constant inside and outside film convection heat transfer coefficient is taken. The function of outdoor temperature which is assumed to show sinusoidal variations during a 24-hour period, the instantaneous transmission heat transfer rate through walls from outside to inside is defined as,

Q t r a n s ( t ) = A U ( T o ( t ) − T i ) (4)

where, T o ( t ) is the instantaneous outside temperature, and T i is the indoor temperature. In steady state condition, the temperature difference is replaced by Δ T as mention in [

Δ T = ( C L T D + L M ) ⋅ K + ( 25.5 − T i ) + ( T o ( h ) − 29.4 ) (5)

where, CLTD is the correction temperature difference according to latitude and surface orientation, LM is the correction factor of latitude, K is the color factor, T o ( h ) is hourly outside temperature, A is the outer walls area, and U is the overall heat transfer coefficient which is defined as,

1 U = 1 h i + ∑ Δ x k | materials + Δ x k | insulation + 1 h o (6)

To investigate the effect of thermal insulation thickness on the building heat load, we assumed all parameters are constant except thermal insulation thickness. The thermal resistance of composite wall can be written in a linear form as,

R = 1 U = C 1 + C 2 Δ x | insulation (7)

where C 1 and C 2 are constants, and the hourly heat transfer through the walls can be defined as,

Q t r a n s = U A Δ T (8)

In the numerical calculation, some assumptions were made. The thickness of the composite wall is small compared to other dimensions. So one-dimensional temperature variation is assumed. The layers are in good contact; hence the interfacial resistance is negligible. There is no heat generation. The variation of the wall materials thermal properties is negligible. The convection heat transfer coefficient and room temperature is constant.

The hourly outside air conditions of dry bulb temperature and relative humidity are used in this analysis as illustrated in [

The total building heat loads in four season, summer, spring, autumn, and winter with constant internal heat loads are calculated at various thermal insulation thickness, Δ x of 0 - 12 cm. The results of total building heat loads in four season are shown in Figures 4-7. We can see a clear and systematic effect of thermal

insulation thickness, Δ x , on the building heat load. Obviously, the building heat loads decrease with increasing thermal insulation thickness until 6 - 8 cm. Also, the trends of hourly building heat loads in four season are similar to the trends of outside air temperature as shown in

A compression refrigeration machine with air cooled condenser is considered. A Simple refrigeration cycle with DX evaporator of R134a was assumed at constant evaporating temperature of 8˚C and condensing temperature of 10 degrees above outside air temperature. The assumed isentropic compression efficiency is 70% and pressure drop of 0.5 bar in suction and delivery lines. The un-useful superheating is 1˚C in suction line and sub-cooled is 5˚C in liquid line. Cool Packprogram [

It is clear that from

Figures 9-12. We can see a clear effect of thermal insulation thickness, Δ x , on the compressor consumed power, and it decreased with increasing the thermal insulation thickness until 6 - 8 cm. Also, with increasing thermal insulation thickness up to 8 cm, the decrease in compressor power consumption is very small.

The saving percentage in building heat load or compressor consumed power are calculated as the difference between the value without insulation and with insulation to the value without insulation as,

EnergySaving = Valuewithoutinsulation − Valuewithinsulation Valuewithoutinsulation (9)

Because the ratio between building heat load and the compressor consumed power is the coefficient of performance, the saving percentage of building heat load or compressor consumed power is the same value and trend. So, Figures 13-16 show the hourly saving percentage of compressor consumed powers in four seasons.

It is clear that from Figures 13-16, the hourly saving percentage increases from 9:00 AM to 8:00 PM of maximum value, and it decreases again. Also, the maximum saving percentage of consumed power is about 45% in summer, spring and autumn, but it attained to maximum value of 50% in winter.

It is important to determine the insulation thickness that minimizes the total cost, C t , which is the cost of energy consumed plus the insulation cost [

C t = C e n r P W F + C i n s Δ x i n s (10)

where, C e n r , $ / ( m 2 ⋅ year ) is the yearly cost of the electric energy consumed relative to the thermal gains through one square meter of wall. PWF, is the present worth factor. C e n r , $ / ( m 2 ⋅ year ) is the cost of one cubic meter of insulation, and Δ x i n s , ( m ) is the thermal insulation thickness. Energy cost, C e n r , is depending on the yearly thermal gains through the outer surface walls, q T , ( W / m 2 ) . Electrical cost, C e l , ( $ / kWh ) is the price of consumed energy and COP is the coefficient of performance of the refrigeration cycle for air-conditioning system as,

C e n r = q T C e l C O P (11)

The yearly transmission load per unit of wall area is estimated in (J/m^{2}) by the following equation,

q T = 24 / 1000 U C D D = 0.024 U C D D (12)

where, U as Equation (6) and CDD is the annual cooling degree-days which is defined as,

C D D = ∑ 1 365 ( T o − T b ) (13)

where, T o is the outside temperature and T b is the base temperature which is equal to the inside conditioned building temperature which is 24˚C. The data of the annual cooling degree-days, CDD, are picked up from the National Climatic Data Center, meteorological stations of weather data of Egypt [

C D D m o d = ∑ 1 365 ( ( C L T D + L M ) K − 3.9 + ( T o − T b ) ) (14)

We used the absolute difference of ( T o − T b ) to treat the cooling case in summer and heating case in winter. The annual energy cost for unit wall surface area can be rewritten as,

C e n r = 0.024 C D D m o d C e l ( R w t + ( Δ x / k ) i n s ) C O P (15)

where, R w t is the total composite wall thermal resistance except insulation thermal resistance. The lifecycle total cost, C t is the energy cost and insulation cost can be rewrite as,

C t = 0.024 C D D m o d C O P 1 R w t + ( Δ x / k ) i n s C e l P W F + C i n s Δ x i n s (16)

The optimal insulation thickness, Δ x o p t , is the thickness of the insulation layer that corresponds to that minimizing the total cost as,

Δ x o p t = ( 0.024 C D D m o d k i n s C e l P W F C i n s C O P ) 0.5 − k i n s R w t (17)

The lifecycle saving, LCS, is the difference between the saved energy cost over the lifetime and the insulation cost [

L C S = P W F Δ E C − C i n s Δ x i n s (18)

where, Δ E C the saved energy which is the difference between the annual energy consumption per unit area of the wall without and with insulation under cooling condition as,

Δ E C = 0.024 C D D m o d C e l C O P ( 1 R w t − 1 R w t + ( Δ x / k ) i n s ) (19)

And the present worth factor PWF is defined as,

P W F = { 1 + i d − i [ 1 − ( 1 + i 1 + d ) n ] at i ≠ d n 1 + i at i = d (20)

The payback period N P (years) can be obtained by setting LCS to be zero as,

N P = { ln [ 1 − C i n s Δ x i n s Δ E C d − i 1 + i ] ln ( 1 + i 1 + d ) at i ≠ d C i n s Δ x i n c ( 1 + i ) Δ E C at i = d (21)

where, n is the yearly lifecycle, i is the currency inflation rate and d is the interest rate. The parameters used in the calculation of the total cost, the optimum insulating thickness and payback period are given in

Two main parameters are affecting the optimal thermal insulation thickness, the running cost of electrical energy and the investment cost of the thermal insulation volume. The tariff of electrical energy of 0.087 $/kWh and expanded polystyrene thermal insulation cost of 53 $/m^{3} are used in this analysis. The optimal thickness of thermal insulation and payback period is calculated in two cases. The first case is neglecting the effect of solar radiation on the outer walls surfaces and ceiling, and he cooling degree-days CDD as Equation (13). The second case is taking into account the effect of solar radiation. The modified cooling degree-days, CDD_{mod}, as Equation (14) is used only through the day time from sunrise to sunset. But from sunset to sunrise, the CDD is calculated by Equation (13) as the absolute value between outside and inside temperature of ( T o − T b ) to treat the cooling and heating cases.

Without exposed the outer walls surfaces to direct solar radiation,

Parameters | Present data | Reference [ |
---|---|---|

C_{el} ($/kWh) | 0.087 | 0.1102 |

C_{ins} ($/m^{3}) | 53 | 53 |

COP | 3.0 | - |

R_{wt} (m^{2}∙K/W) | 0.5139 | 0.774 |

Inflation rate, i | 1% | 8.53% |

Interest rate, d | 5% | 9% |

Cycle life time, n | 10 years | 15 |

PWF | 8.127 | 8.58 |

the energy cost, thermal insulation cost and total cost as a function of insulation thickness. The running cost of electrical energy is decreased with increasing the thermal insulation thickness, but the cost of thermal insulation increases with increasing the insulation thickness. The thermal insulation thickness is increased gradually from 0 to 16 cm. The total cost is gradually decreasing to a minimum value and increased again. At this point the optimum value of insulation thickness is appointed and it is obtained of 9.12 cm and payback period is 19 months. According to the previous studies [

With including the effect of solar radiation on the outer walls surfaces, the modified annual cooling degree-days, CDD_{mod}, was calculated from Equation (14). The values of correction factor, LM, and the Cooling Load Temperature Difference, CLTD, are illustrated in

Latitude | Months | North | East/West | South | Horizontal |
---|---|---|---|---|---|

December | −2.2 | −2.2 | 7.2 | −5.0 | |

January/November | −2.2 | −2.2 | 6.6 | −3.8 | |

February/October | −1.6 | −1.1 | 3.8 | −2.2 | |

16˚ | March/September | −1.6 | −0.5 | 0.0 | −0.5 |

April/August | −0.5 | −0.5 | −3.2 | 0.0 | |

May/July | 2.2 | −0.5 | −3.8 | 0.0 | |

June | 3.3 | −0.5 | −3.8 | 0.0 | |

December | −2.7 | −3.8 | 7.2 | −7.2 | |

January/November | −2.2 | −3.3 | 7.2 | −6.1 | |

February/October | −2.2 | −1.6 | 5.5 | −3.8 | |

24˚ | March/September | −1.6 | −0.5 | 2.2 | −1.6 |

April/August | −1.1 | −0.5 | −1.6 | 0.0 | |

May/July | 0.5 | 0.0 | −3.3 | 0.5 | |

June | 1.6 | 0.0 | −3.2 | 0.5 | |

December | −2.7 | −4.4 | 6.6 | −9.4 | |

January/November | −2.7 | −4.4 | 6.6 | −8.3 | |

February/October | −2.2 | −2.2 | 6.1 | −5.5 | |

32˚ | March/September | −1.6 | −1.1 | 3.8 | −2.7 |

April/August | −1.1 | 0.0 | 0.5 | −0.5 | |

May/July | 0.5 | 0.0 | −1.6 | 5.0 | |

June | 0.5 | 0.0 | −2.6 | 1.1 |

The average value of CLTD for ceiling 16.38˚C, for walls, North 6.54˚C, East/West 12˚C, South 9.33˚C. The middle color factor, k = 0.65.

wall orientation are calculated by interpolation at a given latitude.

_{mod} for walls and ceiling, the optimal insulation thickness is obtained of 10.89 cm and payback periods of 19 months. We observed that the walls optimum thermal insulation thickness with the effect of solar radiation is increased by 10.31% than walls without solar radiation effect.

It is important to calculate the optimal thermal insulation thickness and payback periods with the effect of solar radiation for some cities in Egypt. So, we used the same procedures for some important cities in Egypt from latitude of 24˚N to 32˚N and the results are illustrated in

12.05 cm with increasing the annual degree-days. Because of the roof is treated as a horizontal slab without suspended ceiling and the CLTD larger than walls. So, the optimum insulation thickness for ceiling is larger than walls by about 20.19%. Also, the payback period is decreased with increasing the optimum insulation thickness as shown in

The optimum insulation thickness of data which sited in reference [

To investigate the effect of insulation materials on the optimum thickness periods, five types of thermal insulation materials are used and its properties are illustrated in

City | Location Latit./Longit. | Object | CDD_{mod}_{ } Degree-days | Dx_{opt}_{ } (cm) | Payback years |
---|---|---|---|---|---|

Alexandria | 31^{o}12'20"N 29^{o}55'28"E | Walls | 2000 | 7.12 | 1.888 |

Ceiling | 3378 | 9.85 | 1.44 | ||

Cairo | 30^{o}01'59"N 31^{o}14'00"E | Walls | 2035 | 7.2 | 1.871 |

Ceiling | 3400 | 9.89 | 1.435 | ||

Assiut | 27^{o}11'00"N 31^{o}10'00"E | Walls | 3805 | 10.58 | 1.355 |

Ceiling | 4743 | 12.05 | 1.210 | ||

Aswan | 24^{o}05'20"N 32^{o}53'59"E | Walls | 3606 | 10.25 | 1.393 |

Ceiling | 4622 | 11.87 | 1.226 |

Insulation materials | k (W/m∙K) | Cost ($/m^{3}) |
---|---|---|

Extruded polystyrene (XPS) | 0.031 | 79 |

Expanded polystyrene (EPS) | 0.039 | 53 |

Glass wool (GW) | 0.04 | 27 |

Polyurethane (PUR) | 0.024 | 118 |

Polyisocyanurate (PIR) | 0.023 | 105 |

insulation material, and it decreases as the price of the insulation materials increased at the same thermal conductivity, Polyurethane (PUR) and Polyisocyanurate (PIR).

The building heat load and energy consumption by air conditioning system are proportional to the environmental conditions and solar radiation intensity. The building heat load was estimated for two models of walls. The walls models are constructed from common materials with 0 - 16 cm of thermal insulation thickness in the outer walls and ceilings, and double-layers glazing windows. The optimal thermal insulation thickness and payback period are estimated for residential building from latitude of 16˚N to 32˚N. The average saving percentages in energy consumption in summer, spring, autumn and winter are estimated as, 35.5%, 32.8%, 33.2% and 30.7% respectively. But the yearly average saving percentage is about of 33.5%. Economic analysis is conducted with the annual cooling degree-days and life cycle total cost to estimate the optimum thickness and payback periods. The optimal thermal insulation thickness for some Egyptian cities is found between 7 - 12 cm for various annual cooling degree-days of 2000 to 5000 and five types of insulation materials. Without including the effect of solar radiation, the optimum insulation thickness of ceiling is larger than walls by about 20%. But with including the effect of solar radiation, the optimal thermal insulation thickness for walls is about 10 cm and ceiling of 11.68 cm, and payback period is found between 15 - 24 months.

The author declares no conflicts of interest regarding the publication of this paper.

Mohamed, M.M. (2020) Optimal Thermal Insulation Thickness in Isolated Air-Conditioned Buildings and Economic Analysis. Journal of Electronics Cooling and Thermal Control, 9, 23-45. https://doi.org/10.4236/jectc.2020.92002

A: wall surface area (m^{2})

C_{el}: energy cost ($/kWh)

C_{ins}: insulation cost ($/m^{3})

CDD: cooling degree-day (˚C/day)

CLTD: cooling load temperature difference (K)

COP: coefficient of performance (−)

h: heat transfer coefficient (W∙m^{−2}∙K^{−1})

K: color factor (−)

k: thermal conductivity (W∙m^{−1}∙K^{−1})

LM: latitude correction factor (−)

Q: heat transfer rate (W)

R: wall thermal resistance (m^{2}∙K∙W^{−1})

T: temperature (K)

t: time (s)

x: thickness (m)

U: overall heat transfer coefficient (W∙m^{−2}∙K^{−1})

enr: energy

el: electrical

o: outside

opt: optimum

i: inside

ins: insulation

trans: transmission

wt: wall materials