INFLUENCE OF DIFFERENT ARRANGEMENTS OF THERMAL MASS AND INSULATION ON DYNAMIC THERMAL CHARACTERISTICS OF PLANE WALLS

 

Structure factors for multilayer walls

Relationships Between Structure Factors and Response Factors

Effects of Thermal Mass and Insulation Distribution on Frequency Response of a Wall

 

 

Structure factors for multilayer walls

 

            The thermal structure of a wall is understood as the distribution of thermal resistance and capacity in its volume. Formal relationships, which describe in a quantitative way the effect of structure on the dynamic thermal behavior of walls, follow from the integral formulae for the heat flow across the surfaces of a wall in a finite time interval [6]. They include quantities called thermal structure factors. Relationships between the structure factors, response factors, and z-transfer function coefficients have been derived and analyzed by Kossecka [7, 8] and Kossecka and Kosny [9].

            Thermal structure factors appear in expressions for the asymptotic heat flow across the surfaces of a wall, for boundary conditions independent of time. Consider heat transfer in an exterior building wall of thickness L, separating the room, at temperature T_i, from the external environment, at temperature T_e. Thermophysical properties of the wall—thermal conductivity k, specific heat c_p, and density r— are constant in time, as are surface film resistances R_i and R_e.

            Let q be a dimensionless temperature for the steady-state heat transfer through a wall, with boundary conditions T_i = 0 and T_e = 1. For a plane wall, for which one-dimensional heat conduction conditions are satisfied, the function q(x) is given by

 

  ,                                             (1)

 

where R_i-x and  R_x-e denote the resistance to heat transmission from point x in a wall to the internal and external environment, respectively, and R_T is the total resistance for heat transmission through a wall. R_i-x and R_x-e  can be expressed by the following integrals:

 

  .                                        (2)

 

            Consider now the transient heat transfer process with ambient temperatures held constant for t > 0 at T_i2 and T_e2 and initial temperature in a wall representing the steady state of heat flow for ambient temperatures T_i1 and T_e1. For sufficiently large t, the asymptotic expressions for the total heat flow in the time interval (0, t) across the internal and external surfaces of a wall in the direction from the room to the environment, Q_i(t) and Q_e(t), have the following simple form [6, 7, 8, 10]:

 

  .                                          (3)

  .                                           (4)

   .                                                (5)

 

C is the total thermal capacity of a wall element of the unit’s cross-sectional area:

 

  ,                                                                  (6)

 

whereas  j_ii, j_ie, and j_ee are given by

 

  .                                    (7)

 .                           (8)

  .                                          (9)

           

Dimensionless quantities j_ii, j_ie, and j_ee are called the thermal structure factors. For plane walls, they are determined directly by the thermal capacity and resistance distribution along thickness. In transitions between two different states of steady heat flow, they represent fractions of the total variation of heat stored in the wall volume that are transferred across each of its surfaces as a result of ambient temperature variations. Together with the total thermal resistance R_T and total heat capacity C, they constitute basic thermal characteristics of walls; this is also true in the case of three-dimensional heat transfer [9, 11].

The following identity is a consequence of Eqs. (7) through (9):

 

  .                                                    (10)

 

            Structure factors for a wall composed of n plane homogeneous layers, numbered from 1 to n with layer 1 at the interior surface, are given as follows:

 

   ,                                 (11)

   ,                         (12)

  ,                                (13)

 

where R_m and C_m denote the thermal resistance and capacity, respectively, of the m-th layer; whereas R_i-m and R_m-e denote the resistance to heat transfer from surfaces of the m-th layer to inner and outer surroundings, respectively.

            Structure factor j_ii is comparatively large when most of the total thermal capacity is located near the interior surface x = 0 and most of the resistance is located in the outer part of a wall, near the surface x = L. The opposite holds for j_ee. The following relations are straightforward:  0 <j_ii <1,  0 <j_ee <1.

Structure factor j_ie is comparatively large if most of the thermal capacity is located in the center of a wall and resistance is symmetrically distributed on both sides of it. The following limitations on j_ie result from Eq. (12): for a two-layer wall, 0 < j_ie < 3/16; for an n-layer wall, with n ³ 3, 0 < j_ie < 1/4. For a homogeneous wall with negligibly small film resistances R_i and R_e, j_ii = j_ee = 1/3 , j_ie = 1/6.

The products Cj_ii, Cj_ee, and Cj_ie for a multilayer wall are identified as the thermal mass factors, introduced by the International Organization for Standardization standard [12].

            Structure factors for multilayer walls depend on the arrangement of subsequent layers. To demonstrate this effect, six examples of walls of the same resistance and capacity but of different material configurations were examined. Walls 1 through 6 are depicted in Figure 1. The main part of each wall is a composition of heavyweight concrete layers of total thickness 0.152 m (6 in.) and insulation layers of total thickness 0.102 m (4 in.). The interior layer is 0.013-m (0.5-in.) -thick gypsum plaster; the exterior layer is 0.019-m (0.75-in.) -thick stucco. The total wall thickness is 0.286 m (11.25 in.). Results for the wall with a homogeneous core of the same total thermal resistance and capacity are added for comparison.

            Thermophysical properties of the wall materials are as follows:

 

·      Heavyweight concrete:  k = 1.44 W/mK,  r = 2240 kg/m³,  c_p = 0.838 kJ/kg K;

·      Insulation:  k = 0.036 W/m K,  r = 16 kg/m³,  c_p = 1.215 kJ/kg K;

·      Gypsum board:  k = 0.16 W/m K,  r = 800 kg/m³,  c_p = 1.089 kJ/kg K;

·      Stucco:  k = 0.72 W/m K,  r = 1856 kg/m³,  c_p = 0.838 kJ/kg K.

           

            With surface film resistances of R_i = 0.12 m² K/W and R_e = 0.05 m² K/W, the total thermal resistance for each wall R_T = 3.21 m² K/W, the overall heat transfer coefficient U = 0.312 W/m² K, and the wall thermal capacity C = 329.93 kJ/m²K.

Structure factors j_ii, j_ie, and j_ee for walls 1 through 6 are collected in Table 1. Factor j_ii attains its maximum in wall 3 (all concrete inside) and its minimum in wall 4 (all insulation inside); j_ie atta800080ins its maximum in wall 6 (symmetric insulation) and its minimum in wall 4.

Notice that from the asymptotic formula in Eq. (3) for the heat flow Q_i(t) across the interior surface of a wall, it follows that at a constant exterior temperature (DT_e = 0), storage effects are proportional to Cj_ii. A low value of j_ii for exterior walls thus reduces heating or cooling loads, necessary for a separate (individual) change of the indoor temperature [this phrase is unclear]. This means that for some intermittently used buildings, contrary to the case for continuously used buildings, a configuration with “insulation inside” may be advisable.

 

 

Table 1. Structure factors for walls with cores composed of heavyweight concrete and insulation (shown in Figure 1).

^a 1 in. = 25.4 mm

 

 

 

Relationships Between  Structure Factors and Response Factors

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            The quantities Cj_ii, Cj_ie, and Cj_ee, which in Eqs. (3) and (4) determine the role of storage effects in transitions between different states of steady heat flow, affect particular modes of the dynamic heat flux response of a wall. They appear in the constraint conditions on dynamic thermal characteristics of walls such as the response factors, z-transfer function coefficients, and residues and poles of the Laplace transfer functions [7–9].

            Let X_n, Y_n, and Z_n be the response factors for a wall, corresponding to different heat flux response modes. A response factor with index n represents the heat flux due to the unit, triangular temperature pulse of base width 2d, at the time instant nd [13, 14]. Relationships between response factors X_n, Y_n, Z_n and structure factors j_ii, j_ie, j_ee have the following form:

  .                                                      (14)

   .                                                        (15)

  .                                                      (16)

 

            Analogous conditions are to be satisfied by the response factors for wall elements of complex structure in which three-dimensional heat flow occurs [9]. Equations (14) through (16) refer to the response factors with number n ³ 1, which represent the storage effects in the form of surface heat fluxes after the duration of the temperature pulse. They indicate that, for a given total thermal capacity C, the sum of the products of the response factors multiplied by their indices increases with the appropriate structure factor. This means that the role of response factors with large indices increases with the value of the appropriate structure factor.

            Dimensionless response factors, products of Y_n by R_T, for walls 1 through 6 are represented in Figure 2. The influence of the structure factor j_ie on the character of the variability of Y_n with n is clearly visible.

 

 

Effects of Thermal Mass and Insulation Distribution on Frequency Response of a Wall

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            Like the response factors, responses of walls to periodic temperature excitations are affected by their structural characteristics. This dependence, however, is not represented by constraint equations but appears in the form of significant correlations between the frequency-dependent characteristics of walls and structure factors.

            The solution of a one-dimensional heat transfer problem in a multilayer slab at periodic temperature conditions is presented in several textbooks [14, 15]. The periodic heat flux q_i across the inside surface of a wall is given by

 

  ,                                              (17)

 

where D(iw) and B(iw) are elements of the transmission matrix.

 

Complex numbers 1/B and D/B are called the transmittance and internal admittance of a wall, for angular frequency w. Each of them is represented by its amplitude and phase angle—or time shift.

Dimensionless amplitude is the product of actual amplitude and total resistance R_T. It represents the relationship of the surface (generated by the unit temperature excitation) to the steady state heat flux value (due to the unit temperature difference), proportional to 1/R_T. For the transmittance, it is the decrement factor, DF.

            The effect of structure factors, for given total resistance R_T and capacity C, on a wall’s frequency responses can easily be demonstrated on the simple examples of walls 1 through 6 (Figure 1). Dimensionless amplitude and time shift, for the period of 24 hours, are summarized in Table 2. Results for a wall with a homogeneous core, of the same total thermal resistance and capacity, are added for comparison.

 

 

Table 2. Dimensionless amplitude and time shift of the transmittance and internal admittance for walls with cores

composed of heavyweight concrete and insulation (shown in Figure 1).