Mathematical Model for Optimum Depth of Thermal Insulating Layer

Received: March 10, 2011 Accepted: March 31, 2011 doi:10.5539/jmr.v3n3p103 Abstract In order to ensure the comfort and health of buildings and resolve contradictory between improvement of thermal insulating function of buildings and reduce of thermosteresis and amount of basic building materials (namely save resourses), heat conduction equation was calculated. This mathematical physics equation which was calculated via finite difference method was attributed to inverse problem of gray distributed parameter systems. The economical model of optimum depth of thermal insulating layer was established. Finally, the optimum materials and depth was determined according to the two models.


Background and analysis of the model
Most of settlement housings in the city of China were flattop buildings, the highest surface temperature of which could reach 75˚C in summer and the lowest surface temperature could reach -40˚C in winter.Many measures were carried out to keep suitable temperature inside buildings.On one hand, heating equipment and conditioner were setting up in buildings; on another hand, the thermal insulation ability of buildings was improving, so that the energy consumption would be decreased.The use of appropriate thermal insulation materials with appropriate depth was very important to ensure the comfort and health of buildings guarantee normal production and living, decrease energy consumption, reduce the weight of the roof system, reduce amount of basic building materials, which would save resources and reduce building costs.

Hypothesis of the model
1) A given thermal insulation material that transfer heat steadily.
2) The rooftop surfaced was considered flat, no significant changes of gradient.
3) The temperature distribution of rooftop flat surface was considered uniform, namely there was no temperature change within any flat surface.Only heat conduction at vertical direction was considered to affect temperature distribution.
4) The optimum depth of thermal insulation layer only related to type of thermal insulation materials, present value factor of life cycle, heating degree days, thermal resistance of non-thermal insulation layer and type of fuel for heating.

Model 1: inversion model of thermal insulation layer depth of flattop buildings
The structure of flat rooftop was showed in Figure 1.

1) Flattop buildings heat conduction equation
From model hypothesis, the heat conduction equation was: λ: Coefficient of heat conductivity of materials; ρ: Dry density of material; c: Specific heat of material; u: Temperature of any site from flattop structure; z: Spatial coordinate which is perpendicular with top flat; l: Depth of flat top; t: Time coordinate.

2) Initial conditions
The initial condition, which was the temperature distribution regularity of various layer of flat top at beginning of calculation, was given for No. 1 equation firstly.The time is 5:00 am.It is supposed that temperature distribution within flat top was uniform, so: φ(z) : Function of temperature change of various layers within flat top at beginning of calculation; the meanings of other symbols were same with above.

3) Boundary conditions
The outside surface of rooftop was calculated according to mixed boundary conditions.Namely, heat discharge of outside surface of rooftop is proportional with rooftop surface temperature and atmospheric temperature, so: λ 0 : Coefficient of heat conductivity of materials; h 0 : Coefficient of heat exchange between rooftop and medium around; α: Thermal radiation absorption coefficient of rooftop outside surface; θ α : Atmospheric temperature of outside rooftop; the meanings of other symbols were same with above.
Further consideration of sun radiation effect: I(t): Sun radiation intensity on surface of rooftop ; the meanings of other symbols were same with above.
The second boundary condition of inside surface of rooftop, namely

4) Additional conditions
As it is about inverse problem of geometrical boundary of heat conduction equation, there were additional conditions in process of inversion calculation of thermal insulation layer depth.
G(t): Function of temperature changes in various levels of inside rooftop at the moment of t.

5) Numerical methods
Finally, simulation of inside surface temperature of flat rooftop was made by finite difference method.Specific difference discrete format deduced from equation (1) ∼ (6) was as following: u i j : Temperature of the i layer of flat rooftop structure at the moment of j; λ i : Coefficient of heat conductivity of the i layer of materials; h i : Coefficient of heat exchange between the i layer of materials and around medium; ρ i : Dry density of the i layer of materials; τ: Time coordinate; the meanings of other symbols were same with above.

Economical model of thermal insulation layer of flat rooftop
Economical model of optimum depth of thermal insulation layer was established on the base of considering of thermal insulation materials type, present value factor of life cycle (PWF), heating degree days (HDD), thermal resistance of nonthermal insulation layer (R wt ) and type of fuel for heating.

1) Calculation of costs within life cycle of thermal insulation layer
The costs within life cycle of thermal insulation layer were composed of investment costs and fuel costs.The cost of thermal insulation materials per unit area was: C in : Price of thermal insulation materials per unit area (Yuan/m 2 ); C i : Price of thermal insulation materials per unit volume (Yuan/m 3 ); d: depth of thermal insulation materials.
Heating fuel costs of outside wall per unit area every year were: C n : Costs of heating fuel per unit area of rooftop (Yuan/m 2 •a).; C f : Price of heating fuel (Yuan/kg); HDD: Heating degree days ( • •d); λ m : Coefficient of heat conductivity of thermal insulation materials; R wt : Sum of thermal resistance of area inside and outside room and thermal resistance of other materials except thermal insulation layer; H: calorific value of fuel (kJ/kg); η: Efficiency of heating system.
The total costs of fuel used for heating in one unit of wall within the lifespan of thermal insulation layer were calculated by cycle value methods.As following: C t : The total costs of fuel used for heating in one unit of wall (Yuan/m 2 ); PWF: Factor of life cycle.
The way of PWF calculation was as following: When g > I: When g < I: So: The annual heating costs of building were composed of investment costs of thermal insulation layer and fuel costs.So the total costs for every one unit of thermal insulation layer within life cycle were: Here, C: Total costs for every one unit of thermal insulation layer within life cycle (Yuan/m 2 ). the meanings of other symbols were same with above.
2) The equation for optimum depth of thermal insulation layer So the optimum depth of thermal insulation layer was descided as following: So the calculation equation of optimum depth of thermal insulation layer was: 4. Test and application of the model

Test of inversion model of thermal insulation layer depth of flattop buildings
u 8. j was calculated by inversion model of thermal insulation layer depth.The equation set ( 7) was programmed by Matlab software.The detailed procedures were as following: (1) The thermal insulation layer depth was estimated as [a, b] according to previous building experience.
(2) The temperature of inside surface was calculated by entering the two endpoint a and b.Then two curves of room temperature versus time were drawn up.
( Figure 2 showed the follow chart. According to previous construct experience, the thermal insulation layer depth was estimated to be about 150 mm∼400 mm in this model, namely, a = 150, b = 400.By inversion model, forward modeling and check of additional conditions, curves produced by program matched with actual curves when the thermal insulation layer depth was in the range of 180 mm∼300 mm (Figure 3).
From Figure 3, we can see that the depth of perlite thermal insulation layer was 180 mm∼300 mm in summer.Similarly, when the depth of perlite thermal insulation layer was 200 mm∼300 mm in winter, curves produced by program matched with actual curves (Figure 4).
From Figure 4, we can see that the depth of perlite thermal insulation layer was 200 mm∼300 mm in winter.From above two cases, we can conclude that the optimum depth of thermal insulation layer was 200 mm∼300 mm.

Test of economical model of thermal insulation layer of flat rooftop
The latest four materials (Polyethylene foam, Styrofoam, Rigid Polyurethane foam and PVC rigid foam) were selected for determination of optimum materials as thermal insulation layer.The parameters of the four materials were listed in Table1.
Figure 5 showed relationship between depths of various thermal insulation layer materials and total costs of per unit area of thermal insulation layer within life cycle.
The depths of various materials with lowest total costs were showed in Figure 6.
The detailed data were listed in Table 2.
Analysis of PWF, HDD and R wt were demonstrated in Figure 7∼9.
From the figures, we can see that the depth of thermal insulation layer of Rigid Polyurethane foam and PVC rigid foam changed too much when Rwt, PWF and HDD changed.So they are not applicable.The suitable depth sof Polyethylene foam and Styrofoam acquired by inversion model were (54, 65) and (43, 55), respectively.The optimum depths of Polyethylene foam and Styrofoam acquired by economical model were 69.3649 mm, 52.8727 mm, respectively.However, 69.3649 mm was not in the range (54, 65) in compare with 52.8727 mm was in the range (43,55).Therefore, the optimum material was Styrofoam and its optimum depth was 52.8727 mm.

Advantages of the model
1) The models established in present study are easy to solve.
2) The knowledge used in this study is primary and the methods to solve problems are easy to understand.
3) Data processing and collecting correspond with national standard.
4) It was very scientific and applicable to consider both heat flow and economic in determination of the optimum thermal insulation layer materials.

Disadvantages of the model
1) Heat Conduction during various lays of rooftop was regarded as linear and unchanged in inversion model.However, this linear relationship did not exist in objective word, because the non-linear relationship such as, mutual restrict, saturation and limitation, could not be evitable.

Figure 4 .
Figure 4. Comparison of temperatures calculated by the model with annual temperatures in winter

Table 1 .
The parameters of four latest materials Material of thermal insulation layer Coefficient of heat conductivity (W/(m • K))

Table 2 .
The corresponding depth of thermal insulation layer with the lowest costs

Table 3 .
Intensity of sun radiation at different moments