1.2 Modeling Approaches in HVAC Field
1.2.1 Physics-Based Modeling Approach
Physics-based approach is mostly applied to the modeling of HVAC components.
HVAC components include primary and secondary. The primary system mainly includes chillers, boiler, cooling towers, and liquid distribution system; and the secondary system includes air-handling equipment, air-distribution system, and liquid distribution system between the primary system and the building interior. The distribution components are pumps, fans, dampers, valves, ducts, and pipes. They affect the energy flow in buildings by consuming electrical energy which drives pumps and fans and transferring thermal energy to (or from) the workingfluid in all distribution components [7].
1.2.1.1 Chiller Model
A chiller unit basically consists of four individual components, i.e., evaporator, condenser, compressor, and expansion valve, which can be modeled, separately.
The chiller works on the basis of vapor compression cycle in which the phase change of liquid refrigerant in evaporator takes away heat from the air-conditioned space; then, compressor increases the pressure of gas refrigerant making it super- heated and releases it into the condenser where the gas refrigerant is condensed and the condensation heat is rejected to the water or air; afterward, the expansion valve reduces the pressure by releasing the refrigerant in the evaporator in a cool state.
Sami et al. [8] used a lumped-parameter approach to build a set of component models including shell-and-tube condenser and evaporator, air-cooled condenser, direct expansion (DX) evaporator, capillary tube, and thermostatic expansion valve.
The heat exchangers were modeled using a drift-flux model that consists of sepa- rating the vapor and liquid phases and coupling the mass and energy balances of the
individual phases through the evaporation or condensation. The model was then expanded by Sami and his cooperators [9, 10] and used to predict system’s performance.
Lei et al. [11] developed a lumped-parameter dynamic model of a water-cooling refrigeration system based on the component models by using mass and energy balance principles. With the dynamic model, the effects of control inputs such as compressor operational frequency and TEV opening fraction on the output per- formance of the system were investigated.
To simulate the quick start-ups of centrifugal chillers, Schalbart et al. [12]
established a vapor compression cycle model in which the two-phaseflow models were used for modeling the condenser and evaporator of chiller. Meanwhile, the mechanical inertia of thefluid was considered, and all properties were derived from the internal specific energy and density and determined locally by the integration of mass and energy equations applied tofinite volumes of the components: evaporator, condenser, and pipes.
The moving-boundary (MB) formulation is characterized by phase boundaries that move with time within the heat exchanger. Subject to the same assumptions as with thefinite-volume (FV) approach, the MB formulation begins with dividing the heat exchanger volume into variable control volumes that encompass each phase region existing in the heat exchanger [13,14]. Nyers and Stoyan [15] built a model of evaporator with the MB formulation usingfinite differencing within each phase.
This model was used to predict the evaporator’s behavior under step jump, expo- nential saturation, and periodic oscillation of the temperature andflow rate of the secondary fluid, compressor speed, condenser pressure, and throttle coefficient.
The MB formulation was adopted as well by He et al. [16] in the development of a system model of a basic vapor compression refrigeration system for the purpose of studying the effect of multivariable feedback control and by other researchers for the other purposes [17–19]. Bendapudi et al. [20] discussed the FV and MB approaches applied to shell-and-tube heat exchangers. Detailed model formulations of both the FV and the MB approach for shell-and-tube heat exchanger modeling were provided, and stability was demonstrated as components and within a com- plete centrifugal chiller system model. They concluded that the FV formulation would be more robust through start-up and all load-change transients, but execute slower, while the MB method could handle all load-change transients, but start-up stability would be more sensitive to compressor and expansion valve formulations.
1.2.1.2 Cooling Tower Model
Cooling towers are widely used to remove heat from industrial processes and from HVAC systems. Heat rejection in cooling towers is accomplished by heat and mass transfer between hot water droplets and ambient air. Physical models of cooling tower are usually developed using Merkel’s method and Effectiveness-NTU method [21]. In the Merkel’s method, the water loss of evaporation was neglected and the Lewis number was assumed to be one in order to simplify the analysis. Although
the Merkel’s model has been the basis for most modern cooling tower analysis, it does not accurately represent the physics of heat and mass transfer process in the cooling towerfill [22]. The physical model based on the effectiveness-NTU method is relatively accurate compared to that based on the Merkel’s method, but its reliability still depends on the accuracy of geometric information of cooling towers.
The physical models of cooling tower are also developed with the numerical method. Tan and Deng [23] derived a numerical model for the reversibly used water-cooling tower. The model could be used to make a detailed numerical analysis on the air and water states at any horizontal plane along the tower height.
Fisenko et al. [24] presented a mathematical model for the performance of a cooling tower. The model consisted of two interdependent boundary-value problems. The first boundary-value problem described evaporative cooling of water drops in the spray zone of a cooling tower; the second boundary-value problem describedfilm cooling in the pack. In the following years, Fisenko et al. [25] developed a new mathematical model of a mechanical draft cooling tower. The model represented a boundary-value problem for a system of ordinary differential equations, describing a change in the droplets velocity, its radii, and temperature, and also a change in the temperature and density of the water vapor in a mist air in a cooling tower. In addition, the model allowed one to calculate contributions of various physical parameters to the processes of heat and mass transfer between water droplets and damp air and the influence of atmospheric conditions on the thermal efficiency of the tower.
1.2.1.3 Air-Cooling/Heating Coil Model
In a HVAC system, the cooling/heating coil handles the supply air to anticipated conditions. The heat exchanger model can be obtained by the energy and mass balance on the water and air side of the coil. The physics-based approach for heat exchanger modeling results in a large number of governing equations. Solutions to the equations can be grouped into three categories: ① numerical solutions,
②lumped-parameter solutions, and ③analytical solutions.
For the numerical solution, air-cooling coil is physically divided into numerous segments, and the inlet and outlet variables of each segment are calculated in turn [26,27]. With this approach, the distributions of air temperature and humidity in the coil can be obtained, and the coil’s thermal performances can be accurately eval- uated, but it requires large computational cost.
The use of a lumped-parameter model can be, however, more preferred than the use of a numerical solution due to its lower computational costs. In a lumped-parameter model, the enthalpy difference between air and cooling medium is treated as the driving force of simultaneous heat and mass transfer. Such a treatment wasfirstly proposed by Threlkeld [28], based on the assumption of unit Lewis Factor, and has been popularly adopted by many other researchers [29–31].
Analytical solutions may be able to evaluate the thermal performance of dif- ferent types of heat exchangers in an accurate manner but less computational cost.
Bielski and Malinowski [32] used the analytical method to solve a set of partial differential governing equations describing the transient temperature field in a parallelflow three-fluid heat exchanger by using Laplace transform. They compared the results obtained from their analytical solution with that from the numerical and semi-analytical solutions developed in other studies to validate their analytical solution. Yin and Jensen [33] developed a dynamic response model of a heat exchanger by using the integral method. In their model, the temperature distribution in the single-phase fluid was expressed by a combination of the initial and final temperature distributions as well as a determined time function. The model was used to investigate transient temperature responses of the single-phasefluid and the heat exchanger’s shell wall when subjected to a step change in temperature or mass flow rate of inletfluid. Ren and Yang [34] derived an analytical solution model for the coupled heat and mass transfer process in parallel/counter indirect evaporative coolers under real operating conditions. The analytical solution model was vali- dated by comparing its results with those from numerical integrations. Xia et al.
[35] obtained analytical solutions for evaluating the thermal performances of both chilled water wet cooling coils and DX wet cooling coils, respectively, under both unit and non-unit Lewis Factors. With the analytical solutions, the distributions of air temperature and humidity ratio along air flow direction in a wet cooling coil could be predicted, and the differences in the thermal performances of the cooling coils under both unit and non-unit Lewis Factors could be identified. To solve the nonlinear problem existing in the dynamic model of heat exchanger, Yao et.al. [36]
developed a linearized model for air-cooling coil with small disturbances that was solved by means of Laplace transform. The model could be used to study the dynamic influences of inlet variables on the cooling capacities of cooling coil under different initial conditions.
In recent years, the state-space model for air-cooling coil was put forward by Yao et al. [37]. The state-space model is featured by its convenience in describing the dynamic characteristics of a multiple-input-multiple-output (MIMO) system.
The detailed information about the state-space modeling will be introduced in the following chapter (Chap.2).
1.2.1.4 Fan and Pump Model
Fans/pumps are the main power equipment in HVAC system, which consume over 30 % of total energy consumption of a central air-conditioning system [38]. The power of a fan or pump depends on theflow rate, pressure difference between inlet and outlet and efficiency of the fan or pump. Carrado et al. [39] established a dimensionless model for centrifugal ventilator and studied the influence of blade angle and rotate speed on the dimensionless parameters includingfluid massflow rate, pressure head, and input power. The model could assist us to design a con- troller for the control of air flow rate. Wang [40] developed a model for variable frequency pump in which the influence of electric frequency on the pump’s per- formance was considered.
1.2.1.5 Duct/Pipe Model
As transmission channels offluid in HVAC system, the drag loss and energy loss of duct and pipe are the two important factors concerned in the physics-based model development. Shao et al. [41] established a CFD (computationalfluid dynamics) model for duct and analyzed the resistance loss of duct system in HVAC system.
Fisk et al. [42] established a duct system model in large commercial buildings. The duct system model included physical characterization, air leakage, and heat con- duction gains. Rokni and Gatski [43] built a mathematical model of turbulent con- vective heat transfer in duct and investigated the turbulent convective heat transfer in fully developed ductflows. Sugiyama et al. [44] adopted algebraic heatflux models to study turbulent heatflux distributions in a square duct with one roughened wall.
1.2.1.6 Room Model
The air-conditioning system is usually designed with respect to the thermal char- acteristic of the air-conditioned room, and the system control scheme is developed based on the requirement of indoor thermal environment. The physical model for air-conditioned room can be obtained according to energy and mass balance of air within a zone. Heat is transferred to the zone through the supply air, conduction through walls and windows, infiltration and supply air, and internal and external gains [45]. Heat transfer to a zone is commonly modeled by using heat balance method and thermal network zone method [46].
In heat balance method, the energyflow is modeled using thefirst law of energy conservation. For a zone, generally, each heat transfer element (wall, window, ceiling,floor, etc.) and zone air will result in a heat balance equation. These equa- tions are simultaneously solved tofind the unknown temperature of zone and surface of each heat transfer element using the matrix algebra techniques. The heat balance method has been adopted for studying dynamic responses of air temperature indoors.
As early as the 1980s, Metha et al. [47] established a dynamic response model for the indoor air temperatures with the assumption that the air in the room was well mixed and the heat capacity of furniture was included in the air. Borresen et al. [48]
discussed several simplified dynamic room models in which the thermal interaction between room air and surrounding walls were taken into account in different ways, e.g., the influence of the walls on the indoor air temperatures was neglected; the convective heat transfer between the air and the walls was considered, and the walls’ thermal response time was used for the dynamic simulation. The study concluded that the choice of the simplified level employed depended on how closely the long-term responses and steady-state values fit the actual room response. The dynamic room temperature model developed by Tashtoush et al. [49] was obtained by energy balance on the room air, two walls, and the ceiling. The dynamic room model was employed for the control analysis on a HVAC system.
In thermal network zone model, the building is divided into a network of nodes with interconnecting paths through which the energyflows. The implementation of
this method varies based on the selection of nodes on which energy balance is applied. Zone models exist with different levels of complexity: from simple
‘well-mixed’ models with one air node representing the whole air volume in the room to complex CFD models solving the equations of conservation of mass, momentum, and energy [50]. Chen and Peng [51]firstly studied the indoor dynamic temperature distributions by using afixed-flow-field CFD model in which the room thermal response and indoor air distribution were computed considering the outdoor air temperature, solar radiation, indoor heat sources, and other thermal boundary conditions. Although the CFD results have been shown to be accurate, the calcu- lation is too time-consuming. A multi-zone model simplifies the CFD model by dividing a room, which is otherwise modeled in CFD by thousands offinite volumes, into several air zones each of which is assumed to be well-mixed as one air node. Wu et al. [52] developed a nodal room model with four temperature nodes at different heights above thefloor. The model could be used to predict vertical temperature distribution in a typical office room withfloor heating and displacement ventilation.
Yao et al. [53] developed a state-space room model in which the room was divided into three zones in terms of the air momentum and the airflow rate based on the degradation offluid mechanics equations, i.e., supply air zone, working zone, and return air zone. Compared to the single-zone model, the simplified multi-zone room model is much closer to real situations while saving much computational time compared to the CFD model. So, it may be the best choice for studying the room temperature distributions and dynamic thermal characteristics of indoor air.