Optimum of external wall thermal insulation thickness using total cost method *
Jozsef Nyers*, Peter Komuves**
Obuda University Budapest, Becsi ut 96, 1034 Budapest, Hungary ** V3ME, Subotica-Szabadka, M.Corvin 6 E-mail:
[email protected];
[email protected];
[email protected]; EXPRES 2015 March 19-21. 2015 Abstract - This article investigates the energy-economic optimum of thermal insulation thickness for external wall. The investigation was performed by applying the method 'total cost '. An appropriate steady-state mathematical model was developed. The mathematical model consists of energy and economic part. The economic part of the model contains algebraic equations for investment and exploitation. The considered wall was made of brick and covered with polystyrene as thermal insulation material. The energy-economic optimum was obtained by applying an analytical-numerical and graph-numerical method. The optimization criterion was the minimum of total cost. The numerical results obtained by the simulation are presented graphically. The optimum thickness of the thermal insulation layer is shown in the diagrams. In addition, by applying the developed mathematical model and mathematical methods the optimum thickness of thermal insulation layer was obtained for energyeconomic conditions in Serbia in 2015. Keywords- total cost method, energy-economic optimum, mathematical model, thermal insulation, objective function
Nomenclature Heat flux per unit area / Heat per unit area per year / / Convective heat transfer coefficient / / Conductive heat transfer coefficient / / Overall heat transfer coefficient / / Temperature , ∆ Temperature difference , T Time period per year [h/year] Time period [h] Thickness of thermal insulation layer [m] ∆ Difference [-] C Price [€] f Function n Number Subscripts and superscripts i input, investment s savings o output m middle/mean w wall is isolation sc anchor screw
ne fiberglass network gl cement-based adhesive. Glue I.
INTRODUCTION
From the economic-energy efficiency point of view of the buildings, it is essential to use a thermal insulation layer to cover all external surfaces. The thermal insulation layer significantly reduces building heat loss. The reduction of losses depends mainly on the thickness of the thermal insulation layer. Increasing the thermal insulation layer increases investment costs, but reduces the costs of exploitation. The costs for investment and exploitation have opposite tendencies. Thus, there is a technical-economic optimum of thermal insulation thickness. The optimum can be found by applying appropriate mathematical model and an efficient mathematical method of optimization. From the stand point of analysis, the important issue is to apply the appropriate method for displaying the results. The graphic display one of the convenient forms of representation is. The advantage of graphics is that it visually shows the solutions and the trends of solutions. This paper analyses the energy-economic optimum thickness of thermal insulation layer for external brick wall. An appropriate mathematical model is required for the analysis. The model is composed of correlations to describe the cost of investment and exploitation. In the functions, the values are expressed in Euros. The independent variables are the thickness of thermal insulation layers and the payback period of the investment (T). In order to solve the model's equations system a graph-numerical and an analytical-numerical method were applied. The pure analytical method is much more complex than numerical. The numerical solutions are presented in the graphs of the total cost as a function of the thickness. The graphs show the optimum thickness of the thermal insulation layer. In addition by applying the total cost optimization principle and the developed mathematical model, determined the optimum thickness of the thermal insulation layer and adequate payback period. For technical-economic conditions in Serbia, 2015 the obtained results are:
1. If payback period 1year the thickness is 5.25 cm 2. If payback period 2year the thickness is 9.02 cm 3. If payback period 2.2year the thickness is 9.7 cm The obtained results valid if the price of electric energy is 0.08 [€/kWh]. II.
THE PHYSICAL MODEL
The considered physical system for techno-economic optimization consisted of an external wall with thermal insulation. The static part of the wall was made of 25 cm brick while thermal insulation was of polystyrene. From the aspect of thermal calculation of building, only transmission heat losses were taken into account. Both layers generated heat resistance the brick and the thermal insulation layer, too. The ventilation heat losses were not taken into account since they do not affect the energy-economic optimum of the thermal insulation layer thickness.
αi
λw
∑
3
mean temperature difference between the temperature of the internal ti and the mean outside air temperature tmo for the heating season is. 4 ∆ b. Heat demand of the heating season Heat demand per unit area of the heating season per year equal to product of the mean heat flux and the time period per year. ∙ ∙ ∆ 5 ∑
∙
∙
6
IV. ECONOMIC PART OF THE MATHEMATICAL MODEL
λ αo q=const
q=const
Where: overall heat transfer coefficient is.
In economic part of the mathematical model the total cost was defined. Total cost comprises the investment and exploitation cost. A. Investment function
ti
Investment is the sum of wage-pay and material cost. The investment function is: 7
tmo
25 cm
The price of thermal insulation material and the screw anchors depend on of the thermal insulating layer thickness, while other terms can be considered as a constant.
δ
a. Constant part of the investment function
Figure 1. The cross section of the external walls, made of 25 cm brick and thermal insulation layer of polystiren
III. ENERGY PART OF THE MATHEMATICAL MODEL In energy part of the mathematical model the energy losses through external wall was defined. A. Transmission heat losses through the external wall Only transmission thermal losses depend on of the thermal insulation layer thickness, ventilation losses do not. a. Mean heat flux through the external wall The mean heat flux through the external wall per unit area is defined by the mean temperature difference of the heating season and overall heat transfer coefficient. ∙∆ (1) ∑
∙
(2)
Constant part of the investment function makes fibber glass net price, price of glue for polystyrene and labour price, per m2. 8 B. Final form of the investment function The investment function is the sum of independent and dependent terms of thickness. 9 C. Exploitation cost function Exploitation cost is product of the heat losses throu the thermal isulation external wall per unit erea and price of the energy source. ∙ ∙e 10 ∙ ∙e 11 ∙∆
V.
SIMULATION
VI. TOTAL COST OPTIMIZATION METHOD
A. About the simulation A simulation to determine the optimum thickness of thermal insulating layer was done based on the technical-economic situation in Serbia in 2015. B. Data of the simulation Brick wall thickness δ = 0.25 m. Coefficient of heat conduction for the brick wall λ =
0.5 W/m/K. Coefficient of heat conduction of polystyrene λw =
0.05 [W/m/K].
Convection heat transfer coefficient of the indoor air
αi = 8 [W/m2/K]. Convection heat transfer coefficient of the outside air αo = 20 [W/m2/K]. Internal mean temperature ti = 20 [oC]. Mean outdoor temperature of the heating season tmo =4 [oC]. Polystyrene price Cin,m= 40 [€/m2m]. Glue price Cgl=3€/25 kg·7kg/m2 =0.84 [€/kg]. Consumption of glue 7 [kg/m2], Price of the screw anchors for 5cm thickness 0.03 €/piece . C Price of the screw anchors for 10cm thickness 0.043 €/piece . C Number of the screw anchors per m2 n 6. Labour price for the installation of thermal insulating layer 6 [€/m2]. Mean unit price of electric energy in Serbia 2015 e 0.08 €/kWh 0.08 ∙ 10 €/Wh .
The goal of optimizations is to find the optimum thickness of the thermal insulation layer according to minimum cost. A. Objectiv function of total cost method
The implicit objectiv function of total cost method is equal to sum of the investment and exploitation cost. 15 The investment and exploitation cost functions. 16 ∙∆ ∙ ∙ 17 The implicit objectiv function. ∙ ∆ ∙ ∙ 18 Equation (36) contains four independent variables: the insulating layer thickness, the payback period, the mean temperature difference and energy unit price. The equation can be solved by each of the four independent variables. B. Analytical optimization method The mathematical optimum condition is the first derivative of objective function with respect to thickness equal zero.
0
. .
∙ e
12
∙ 16 ∙ T ∙ 0.08 ∙ 10
13
Where: The overall coefficient difference of the heat transfer through the wall. 1 0.05 / . 0.03375 δ .
.
c.
0 2 1.4 9.4 €/
14
6 €/ 29
.
∙ 16 ∙
∙ 0.08 ∙ 10
0
0
∙
,
,
(23) The first derivative of the screw price with respect to thickness is. (24)
∙
∙
25
(39) The first derivative of the overall coefficient with respect to thickness is.
9.4
19
(21) The first derivative of the thermal insulation price with respect to thickness (22) , ∙
28
40 ∙
∙ ∙
∙
Objectiv function with data .
∙ ∙
After derivation
b. Investment function with data 40 ∙ 40 ∙
∙∆
∙∆
a. Exploatation cost function with data ∙
(20)
C. Functions of simulation
∙∆
26
27
28
B. Numerical results The numerical results obtained by simulation are presented graphically in Figures 2.
Objectiv function for determining the optimum thickness of the thermal insulation layer
∙
,
The optimum thermal insulation thickness is 0.09018 m for 2 years i.e. 2 ∙ 4800 h payback period. The total cost 18 [€/m2].
∙∆
0
∙ ∙
29
Objectiv function in ather form ∙
,
1
∙ ∙∆
1
1
0
∙ ∙
30
The obtained optimum equation with respect to thickness can be solved analytically as second-degree algebraic equation. For example 1 0.25 1 8 0.5 0.05 20 1 ∙ 16 ∙ 2 ∙ 4800 ∙ 0.08 ∙ 10 0 0.05 31 The optimum thermal insulation thickness is 0.09018 m for 2 years i.e. 2 ∙ 4800 h payback period. 40
0 ∙
C. The numerical - graphical optimization method The numerical - graphical optimization method is easier performable than the analytical method but not so accurate. The solutions obtained solving the optimum equation are numerically and results are displayed graphically in the coordinate system. The analysis is visual, the graph shows the tendency of total cost variation depend on the thermal insulating layer's thickness. In addition the total cost chart shows the optimum thickness of the thermal insulating layer and the payback period of the investment. The simulation algorithm of optimum function (30) was implemented in the Matlab mathematical package. VII. THE RESULTS A. Analytical resultats Analytical resultats obtained by solving analytically the algebraic objective function. For example:
Figure 2. The total cost functions with optimum thermal insulation thickness for different payback periods in Serbia 2015.
VIII. CONCLUSIONS The aim of this study was to find the optimum thickness of thermal insulation layer for external wall by using the total cost method. The total cost method is based on the condition, minimum total cost. The mathematical model consists of energy and economic part. The energy part includes the equation of heat demand with thermal insulation. The economic part comprises the equation of investment and exploitation. The optimization procedure were analytical-numerical and graphic-numerical. Comparison results obtained applying the analyticalnumerical or graphical-numerical methods, are exactly the same. The graph shows, the optimum polystyrene thickness is: a. 5.25 [cm] for 1 years, total cost 15.1 [€/m2] b. 9.018 [cm] for 2 years, total cost 18 [€/m2] c. 9.8 [cm] for 2.2 year, total cost 18.5[€/m2] Figure 2. All the above data are valid for Subotica-Serbia, for the year 2015.
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