A modified zone model for estimating equivalent room thermal capacity

Abstract

The zone model has been widely applied in control analysis of heating, ventilation and air conditioning (HVAC) systems to achieve a high building efficiency. This paper proposed a modified zone model which is much simpler in the HVAC system simulation and has the similar accuracy to the complicated simulation model. The proposed model took into consideration the effect of envelop heat reservoir on the room indoor temperature by introducing the thermal admittance of the inner surfaces of the building enclosure. The thermal admittance for the building enclosure was developed based on the building thermal network analytical theory and transfer function method. The efficacy of the proposed model was demonstrated by comparing it with the complicated model — heat balance method (HTB2 program). The predicted results from the proposed model well agreed with those from the complicated simulation. The proposed model can then make the HVAC system dynamic simulation much faster and more acceptable for control design due to its simplicity and efficiency.

Keywords

room model / thermal network analysis / transfer function / heating / ventilation and air conditioning (HVAC) system simulation

Introduction

An increasing amount of attention has been paid to building energy efficiency on account of energy crisis. Approximately 30% of the total energy is consumed in buildings, with heating, ventilation and air conditioning (HVAC) system accounting for most of the energy consumption. To achieve high building energy efficiency, the designing of an efficient HVAC system becomes very important in building operation. This requires an accurate model which can simulate the HVAC systems for the optimal control in buildings. This model should be sufficiently complex to consider the interaction of system components in real-time-control loops and the influence of room temperature under continuously changing operation conditions.

The zone model has been commonly applied in the simulation and control analysis of HVAC systems. Extensive research was conducted in the last three decades. Boresen [1] discussed four simplified dynamic room models according to the ways of treating the interaction between room air and surrounding walls. In Model I, the influence of the walls was neglected so that the model was pretty simple and could only be used for short-time analysis. In Model II, the overall U-values of the walls were incorporated into the model to account for the correction of the steady-state zone response. Model II was not suitable for dynamics control analysis incorporation the short and long response time elements. Model III considered the effect of temperature variation on room condition on the basis of model II. Model IV further described the thermal dynamics of wall temperature and incorporated wall thermal capacity in the model, making the model most complicated and accurate for control analysis. Roberts et al. [2] used model IV to simulate a single zone environmental chamber for better understanding of the system behavior. The thermal capacity of the wall was considered in the model and the values of the thermal capacity were obtained from the experiments. Glicksman [3] further improved the zone model by using heat balance equations to calculate the temperature of supply cooling air in the zone in order to study the control behavior of the occupants for occupant-controlled heating ventilating and air-conditioning systems. The improved model considered the natural convection of air from the ceiling zone to the ceiling surface together with blackbody radiation from the ceiling surface to the cells. Other inside surfaces were not included in the model of the zone enclosure.

Hung et al. [4] presented a dynamics model which studied the stability of the zone air temperature affected by the thermal storage of the zone interior surface area. However, the calculation of the area increment and the accuracy of the simulation were not provided. Singh et al. [5] proposed a nonlinear model in which the thermal capacity of the air in each zone was considered. The model was incorporated into the adaptive control for a fan-coil heating system (FCH) in two zones. Kasahara et al. [6] described a procedure to derive a dynamic model for an air-conditioned room by using physical laws. A total thermal capacitance zone was introduced and the overall heat capacitance was obtained from the experiments [7]. James et al. [8] applied the heat capacitance of the air and furniture in a zone model and hence developed a methodology to develop and evaluate building thermal control strategies with limited field measurements. However, this zone model only considered the capacitance of the air and furnishings, while the heat capacitance of inside surfaces of zone enclosure was not included. Riederer et al. [9] further extended this model by considering the heat capacitance of air and neglecting the heat capacitance of the zone enclosure and furniture. This new model was called “well-mixed” zone model which was much simpler and could accurately describe the internal zone conditions as claimed by the authors. Recently, Dehra [10] applied the zone model to a section of the photovoltaic solar wall and developed a steady state thermal network nodal equation for the conjugate heat exchanger and heat transport in the solar wall. Goyal [11] proposed a method to reduce the order of dynamic models in multi-zone buildings. The simulation results showed that the prediction of the zone temperatures and humidity ratios by using this reduced model was very close to that obtained from the full-scale model; however, the computation time is reduced by a factor of six percentage points or more. Ruivo et al. [12] proposed the simplified cooling load temperature difference (CLTD) method to cooling and heating loads estimation of rooms with daily and weekend setback and setup thermostats, and developed a transient heat transfer model. This model could predict the thermal behavior of multi layered walls and horizontal roofs. But the internal thermal capacity of the room was assumed negligible in the research. The authors, therefore, suggested further research should be conducted towards the development of a reliable method, taking into consideration the effect of room thermal capacity.

The key issue for the development of a zone model in the control analysis of HVAC systems is that the model should actually reflect the practical operation parameters, the characteristics of the controllers, and the zone temperature. Most of the above-mentioned models ignored the zone air thermal capacity, which reduced the accuracy of the HVAC simulation and hence the controllers. In actual situations, the zone air exchanges the heat with the inner fabric surfaces of the zone enclosure. The amount of thermal energy transferred into the zone air substantially affects the zone air temperature and hence the operation of HVAC controllers. In this paper, the heat capacitance of zone enclosure air is considered in the HVAC simulation so that the model is much closer to the actual zone situation. An analytical method is developed to estimate the zone thermal capacity in the modified zone model which can be implemented in the dynamic model of the HVAC system for better designing and testing of the HVAC controllers.

Thermal network and solution

Zone model

Figure 1 shows a schematic of a zone thermal model. The thermal network, the key in the zone model, contains more information than that extracted from detailed numerical simulation alone [2]. The external walls are assumed to be made of the materials with homogeneous thermal properties. The thermal resistance of the window glazing is assumed to be negligible. Based on this thermal network, the heat transfer model can be simulated as an electrical network. The thermal resistance is treated as electric resistance, and heat flux as electric current.

According to the thermal network theory and transfer function, the modified zone model in the frequency domain can be provided as

(1)\left({\displaystyle \sum _{i=\mathrm{1}}^M{Y}_i}(s)+{\rho }_{a}VC{p}_{a}s\right)T(s)=Q_c(s)+V_s{\rho }_{a}C{p}_a(T_s(s)-T_R(s)),

where Y_i(s) is the heat capacitance of wall i (J/K), which can be applied to multi-walls as detailed in the sections below; ρ_a is the density of air.

Heat capacity of one-layer wall

As illustrated in Fig. 2, the one-layer wall is considered as a simple two-node thermal network which can be extended to the multi-layer wall by analogy. Node 1 corresponds to the outdoor air. T₁ and Q₁ represent the outdoor air temperature and the heat flowing into the room through the wall respectively. Node 2 corresponds to the room air. T₂ and Q₂ represent the room air temperature and the heat flowing out of the room through the wall.

The cascade form of the terminal equations can be expressed in Laplace domains form as

(2)\left[\begin{array}{l}{Q}_o\left(s\right)\hfill \\ Q_i\left(s\right)\hfill \end{array}\right]=\left[\begin{array}{ll}{Y}_1\left(s\right)+Y_2\left(s\right)\hfill & -Y_2\left(s\right)\hfill \\ Y_2\left(s\right)\hfill & -\left(Y_2\left(s\right)+Y_3\left(s\right)\right)\hfill \end{array}\right]\left[\begin{array}{c}{T}_o\left(s\right)\\ T_i\left(s\right)\end{array}\right],

where

\left[\begin{array}{ll}{Y}_1\left(s\right)+Y_2\left(s\right)\hfill & -Y_2\left(s\right)\hfill \\ Y_2\left(s\right)\hfill & -\left(Y_2\left(s\right)+Y_3\left(s\right)\right)\hfill \end{array}\right]

is the admittance matrix. The detailed derivation of the equations can be found in the appendix.

By using the conduction transfer function method [13], the heat flow related to the temperature at both sides of the wall can be expressed as

(3)\left[\begin{array}{c}{Q}_o\left(s\right)\\ Q_i\left(s\right)\end{array}\right]=\left[\begin{array}{ll}{Y}_{11}\left(s\right)\hfill & -Y_{12}\left(s\right)\hfill \\ Y_{21}\left(s\right)\hfill & -Y_{22}\left(s\right)\hfill \end{array}\right]\left[\begin{array}{c}{T}_o\left(s\right)\\ T_i\left(s\right)\end{array}\right]

where

\left[\begin{array}{ll}{Y}_{11}\left(s\right)\hfill & -Y_{12}\left(s\right)\hfill \\ Y_{21}\left(s\right)\hfill & -Y_{22}\left(s\right)\hfill \end{array}\right]

is the transfer matrix.

The admittance matrix and transfer matrix are equal and hence the following equations can be obtained by comparing the two matrixes

(4)\{\begin{array}{l}{Y}_{\mathrm{1}}=Y_{\mathrm{11}}-Y_{\mathrm{12}},\hfill \\ Y_{\mathrm{2}}=Y_{\mathrm{12}}-Y_{\mathrm{21}},\hfill \\ Y_{\mathrm{3}}=Y_{\mathrm{1}},\hfill \end{array}

where, Y₁, Y₂, and Y₃ refer to the outside wall self-admittance, wall heat conductivity, and interior wall self-admittance, respectively. Therefore, the admittance of one-layer wall is obtained.

Heat capacity of multi-layer wall

The heat capacity of the two-layer wall can be obtained from the one-layer wall and hence the method can be extended to obtain the heat capacity of the multi-layer wall. The two-layer wall thermal network is demonstrated in Fig. 3. Node 3 corresponds to the middle part of the wall between two layers. Similarly, the nodal admittance equations for the two-layer wall thermal network can be expressed in matrix form as

(5)\left[\begin{array}{c}{Q}_1\left(s\right)\\ Q_2\left(s\right)\end{array}\right]=\left[\begin{array}{ll}{Y}_1+Y_2-{\displaystyle \frac{Y_2{Y}_2}{Y_2+Y_3+Y_1^{\prime }+Y_2^{\prime }}}\hfill & -{\displaystyle \frac{Y_2{Y}_2^{\prime }}{Y_2+Y_3+Y_1^{\prime }+Y_2^{\prime }}}\hfill \\ {\displaystyle \frac{Y_2{Y}_2^{\prime }}{Y_2+Y_3+Y_1^{\prime }+Y_2^{\prime }}}\hfill & -\left(Y_2+Y_3+{\displaystyle \frac{Y_2^{\prime }{Y}_2^{\prime }}{Y_2+Y_3+Y_1^{\prime }+Y_2^{\prime }}}\right)\hfill \end{array}\right]\left[\begin{array}{c}{T}_1\left(s\right)\\ T_2\left(s\right)\end{array}\right]

In order to obtain the total heat capacitance, the multi-layer wall is assumed as an equivalent one-layer wall thermal network, as depicted in Fig 4. Then the admittance matrix for the multi-layer wall is expressed as

(6)\left[\begin{array}{c}{Q}_1\left(s\right)\\ Q_2\left(s\right)\end{array}\right]=\left[\begin{array}{ll}{Y}_1^{*}\left(s\right)+Y_2^{*}\left(s\right)\hfill & -Y_2^{*}\left(s\right)\hfill \\ Y_2^{*}\left(s\right)\hfill & -\left(Y_2^{*}\left(s\right)+Y_3^{*}\left(s\right)\right)\hfill \end{array}\right]\left[\begin{array}{c}{T}_1\left(s\right)\\ T_2\left(s\right)\end{array}\right].

Comparing Eq. (5) with Eq. (6), Eq. (7) can be obtained.

(7)$$\begin{gathered} Y_{\mathrm{2}}^{*}(s)={\displaystyle \frac{Y_{\mathrm{2}}{Y}_{\mathrm{2}}^{\prime }}{Y_{\mathrm{2}}+Y_{\mathrm{3}}+Y_{\mathrm{1}}^{\prime }+Y_{\mathrm{2}}^{\prime }}}, \\ {Y}_{\mathrm{1}}^{*}(s)=Y_{\mathrm{1}}+{\displaystyle \frac{Y_{\mathrm{2}}(Y_{\mathrm{3}}+Y_{\mathrm{1}}^{\prime })}{Y_{\mathrm{2}}+Y_{\mathrm{3}}+Y_{\mathrm{1}}^{\prime }+Y_{\mathrm{2}}^{\prime }}}, \\ {Y}_{\mathrm{3}}^{*}(s)=Y_{\mathrm{3}}^{\prime }+{\displaystyle \frac{Y_{\mathrm{2}}^{\prime }(Y_{\mathrm{3}}+Y_{\mathrm{1}}^{\prime })}{Y_{\mathrm{2}}+Y_{\mathrm{3}}+Y_{\mathrm{1}}^{\prime }+Y_{\mathrm{2}}^{\prime }}}, \end{gathered}$$

where Y_{\mathrm{3}}^{*}(s)is the self-admittance of the two-layer wall.

For the multi-layer wall, recurrence formula will be obtained by repeating the above calculation. The self-admittance formula is a complicated transcendental hyperbolic function, which can be expanded by power series for simplicity. Therefore, the thermal admittance for external walls, ceiling and floor may be expressed as the ratio of two polynomials in the frequency domain S as

(8)Y(s)={\displaystyle \frac{N(s)}{D(s)}}={\displaystyle \frac{a_{\mathrm{0}}+a_{\mathrm{1}}s+a_{\mathrm{2}}{s}^{\mathrm{2}}+a_{\mathrm{3}}{s}^{\mathrm{3}}+\cdots }{b_{\mathrm{0}}+b_{\mathrm{1}}s+b_{\mathrm{2}}{s}^{\mathrm{2}}+b_{\mathrm{3}}{s}^{\mathrm{3}}+\cdots }}.

Thermal capacitance of inner surface of enclosure

In order to simplify the calculation of the thermal capacitance, only the first order value in the frequency domain is considered. It is assumed that

(9)Y(s)={\displaystyle \frac{a_{\mathrm{0}}+a_{\mathrm{1}}s+a_{\mathrm{2}}{s}^{\mathrm{2}}+a_{\mathrm{3}}{s}^{\mathrm{3}}+\cdots }{b_{\mathrm{0}}+b_{\mathrm{1}}s+b_{\mathrm{2}}{s}^{\mathrm{2}}+b_{\mathrm{3}}{s}^{\mathrm{3}}+\cdots }}=C_{\mathrm{0}}+C_{\mathrm{1}}S.

Expand Eq. (9), then

(10)a_{\mathrm{0}}+a_{\mathrm{1}}s+a_{\mathrm{2}}{s}^{\mathrm{2}}+a_{\mathrm{3}}{s}^{\mathrm{3}}+\cdots =C_{\mathrm{0}}{b}_{\mathrm{0}}+(C_{\mathrm{0}}{b}_{\mathrm{1}}+C_{\mathrm{1}}{b}_{\mathrm{0}})s+(C_{\mathrm{0}}{b}_{\mathrm{2}}+C_{\mathrm{1}}{b}_{\mathrm{1}})s^{\mathrm{2}}+\cdots .

To equalize Eq. (10) equal, it should be assumed that the coefficients of the same order of the polynomial on either side of the equations are equal and the least squares errors of all the functions close to zero.

(11){\displaystyle \frac{d{\displaystyle \sum _{i=\mathrm{1}}^N{[a_i-(C_{\mathrm{0}}{b}_i+C_{\mathrm{1}}{b}_{i-\mathrm{1}})]}^{\mathrm{2}}}}{d{C}_{\mathrm{1}}}}={\displaystyle \sum _{i=\mathrm{1}}^N-\mathrm{2}[a_i-(C_{\mathrm{0}}{b}_i+C_{\mathrm{1}}{b}_{i-\mathrm{1}})]b_{i-\mathrm{1}}}=\mathrm{0.}

Then the heat capacitance of the inner surface can be calculated as

(12)C_{\mathrm{1}}={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^N(a_i{b}_{i-\mathrm{1}}-C_{\mathrm{0}}{b}_i{b}_{i-\mathrm{1}})}}{{\displaystyle \sum _{i=\mathrm{1}}^N{b}_{i-\mathrm{1}}^{\mathrm{2}}}}}.

If only the first order term was considered, i.e. assuming that the first order term of the equations is equal, another heat capacitance formula, Eq. (13) can be obtained:

(13)C_{\mathrm{1}}={\displaystyle \frac{a_{\mathrm{1}}-c_{\mathrm{0}}{b}_{\mathrm{1}}}{b_{\mathrm{0}}}}.

Validation and comparison

The detailed simulation method — heat balance method used by the HTB2 program [14] has been used to validate the modified zone model which considers the heat capacitance of the inner surfaces of the building enclosure in the zone differential equations. The numerical example was an unfurnished room with a dimension of 4 m (width) × 4 m (depth) × 3 m (high), enclosed by concrete walls and slabs at the six sides. The construction materials and properties of the exterior walls and slabs were the same as those given in Table 1. Each wall and slab had three layers. The assumption was made that the room model was adjoined to an identical room in six sides so that no heat was transferred to the outside of the room. The temperature of each wall and slab, as well as the room temperature was assumed to be 0°C initially. Furthermore, the room was assumed to be steadily supplied with a heat input of 640 W and a ventilation air with the temperature of 0°C at a flow rate of 0.052 m³/s. Three models including the heat transfer simulation program HTB2, the simply zone model only considering air heat capacity and the modified zone model were used to predict the room temperature rise in response to the heat input. The simulation results were sketched in Figs. 5 and 6.

The changes in room air temperature with the time obtained from the three different models are shown in Fig. 5. It can be observed that there is a marked difference between the results predicted by using the simplified zone model and those predicted by using the HTB2. When considering the heat losses caused by convection from the room air to the cold fabric surfaces of walls and slabs, the room air temperature predicted by the HTB2 increases much slower than that calculated by using the simplified zone model. The difference is as large as 328.3% by neglecting the heat capacity of the inner surfaces of building enclosure in the simplified zone model. However, the results predicted by using the modified zone model proposed in this paper conform to those predicted by the HTB2.

The cooling load simulated by the three different models was presented in Fig. 6. Once again, a large difference of simulated room cooling load can be observed between the simplified zone model and the HTB2. The cooling load calculated by using the simplified zone model was approximately 0.64 kW with the indoor heat input and the cooling load kept room air temperature at a much higher temperature. The reason for this is that the simplified zone model ignores the wall heat capacity. The cooling load obtained by using the modified zone model is in agreement with that simulated by the HTB2 (complicated model) model with an acceptable error. The results also indicate that the effect of the heat capacity of the building enclosure on the HVAC systems is significant and must be taken into consideration in the development of simulation models.

Conclusions

A modified air zone model that considers the thermal capacity of the walls was proposed. The analytical method was used in the model to derive the thermal capacity while the thermal network theory and heat transfer function analysis was employed in the frequency domain. The modified air zone model was much simpler than the complicated model such as heat balance method used by the HTB2 program. However, it was demonstrated from the numerical example that the modified model has the similar efficacy to that of the complicated HTB2 model. The modified air zone model requires much less computational time which can substantially increase the real-time control accuracy. Further, the simulation from the three models also indicated that the effect of the heat capacity of inner surfaces of the building enclosure on the room air temperature and cooling load was significant. It must be considered in the HVAC system simulation in order to achieve an accurate control.

Notations

References

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