2.4 ¿Qué es el control interno y quién lo ejerce?
Art 16 Comité técnico de contrataciones
The building model for the work in this chapter calculates the final energy input, 𝑄𝑓𝑖𝑛𝑎𝑙, required to achieve and maintain a set point temperature for a simple one or two zone house for a single heating event or 24 hour period. The model is based on well-established steady state heat transfer, thermodynamic and geometric equations and uses Microsoft Excel to run. The model is quasi-steady state, as the house is modelled both as it heats up from cold and as it maintains a steady temperature. For most of the work in this chapter, a
single zone house is modelled, and this will be explained first. For modelling spatial variation temperature control, a two zone model is used and will be explained subsequently.
4.4.1
Single zone modelling
Single zone modelling is used to investigate technologies (and key parameters) of conversion device (𝜂𝑏𝑜𝑖𝑙𝑒𝑟), passive system (𝑈𝑒𝑥𝑡 [W/(m2K)]), service level (𝑇
𝑠𝑒𝑡 [°C]) and timer control (𝜏ℎ𝑒𝑎𝑡 [hours]).
The model is designed such that the effect of the values of all parameters can be investigated and therefore the building is parametrised in as flexible a way as possible. The building geometry is determined in the model by assigning values for the floor area (𝐴𝑓 [m3]), number of floors (𝑛
𝑓), height of floors (ℎ𝑠 [m]), length to width ratio of the house (𝜑𝑙𝑤) and number of exposed walls (𝜀) (4 for a detached house, 3 for semi-detached or end terrace, 2 for mid-terrace). The designation of these values allows the external surface area (𝐴𝑒𝑥𝑡 [m2]) and internal volume (𝑉
ℎ [m3]) of the house to be calculated, as in equation (4-9) and (4-10) respectively which are required for the subsequent calculations.
𝐴𝑒𝑥𝑡 =𝐴𝑓 𝑛𝑓+ ℎ𝑓𝑛𝑓𝜀√ 𝐴𝑓 𝜑𝑙𝑤+ 2ℎ𝑓𝑛𝑓√𝐴𝑓𝜑𝑙𝑤 (4-9) 𝑉ℎ= 𝐴𝑓∙ ℎ𝑓 (4-10)
Heat demand is calculated in two parts related to the period in which the house is heating up to set-point temperature, and the period at which this temperature is being maintained. In the first stage, as internal temperature is increasing, heat loss and internal temperature are calculated over every time step using equation (4-11), where values are designated for external thermal transmittance (𝑈𝑒𝑥𝑡 [W/(m2K)]), infiltration rate (𝐼
𝑎𝑐ℎ [air change per hour (ach)], external temperature (𝑇𝑒𝑥𝑡 [°C]) and humidity of air (𝐻 [kgwater/kgdry air]). Equation (4-12) converts a value of infiltration from air change per hour to cubic metres per second and equation (4-13) calculates the heat capacity of air (γair [J/(m3K)]) from value of humidity using values of specific heat capacity (Cp [𝑘𝐽 (𝑘𝑔𝐾⁄ ]) and density (𝜌 [kg/m3]) of dry air and water (values for 20 °C are used as the specific heat capacity and density are not a strong function of temperature at this temperature).
𝑄 𝑙𝑜𝑠𝑠(𝑡) = 𝑈𝑒𝑥𝑡𝐴𝑒𝑥𝑡(𝑇𝑖𝑛𝑡(𝑡) − 𝑇𝑒𝑥𝑡) + 𝐼𝑎𝑖𝑟𝛾𝑎𝑖𝑟(𝑇𝑖𝑛𝑡(𝑡) − 𝑇𝑒𝑥𝑡) (4-11) Where 𝐼 𝑎𝑖𝑟[m3⁄ ] =s 𝑉ℎ[m3] ∙ 𝐼 𝑎𝑐ℎ [h-1] 3600 (4-12) and γair= Cp,air 𝜌𝑎𝑖𝑟∙ 1000 Where 𝑐𝑝,𝑎𝑖𝑟 = (1 − 𝐻) ∙ 𝑐𝑝,𝑑𝑟𝑦 𝑎𝑖𝑟(20℃) + 𝐻 ∙ 𝑐𝑝,𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑢𝑟(20℃) 𝜌𝑎𝑖𝑟= (1 − 𝐻) ∙ 𝜌𝑎𝑖𝑟+ 𝐻 ∙ 𝜌𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑢𝑟 (4-13)
The internal temperature is calculated iteratively via the change in temperature (Δ𝑇 [K]) which is calculated for every time step using equations (4-14) and (4-15). A time step for iteration (𝛿𝑡 [𝑠]) must be chosen, and a value for the maximum output of the heating system, (𝑄 𝑚𝑎𝑥 [W]) must be designated.
Δ𝑇(𝑡) =(𝑄 𝑚𝑎𝑥− 𝑄 𝑙𝑜𝑠𝑠(𝑡)) ∙ 𝛿𝑡 𝑉ℎ∙ 𝛾𝑎𝑖𝑟
(4-14)
𝑇(𝑡 + 𝛿𝑡) = 𝑇(𝑡) + Δ𝑇(𝑡) (4-15)
This iterative process continues from an initial starting temperature (𝑇𝑐𝑜𝑙𝑑 [°C]) until 𝑇(𝑡) is equal to 𝑇𝑠𝑒𝑡. The time at which this point is reached (𝜏𝑟𝑖𝑠𝑒 [hours]) is found using the INDEX – MATCH function of Excel. The heating demand for this time period within which the house is heating up (𝑄𝑟𝑖𝑠𝑒 [kWh]) is calculated by equation (4-16).
𝑄𝑟𝑖𝑠𝑒 =𝑄 𝑚𝑎𝑥× 𝜏𝑟𝑖𝑠𝑒 1000
(4-16)
In the second stage, for the remainder of the heating period, the house is treated as steady state and 𝑄 𝑠𝑢𝑠𝑡 [W] is calculated by the steady state equation (4-17).
𝑄 𝑠𝑢𝑠𝑡 = 𝑈𝑒𝑥𝑡𝐴𝑒𝑥𝑡(𝑇𝑠𝑒𝑡− 𝑇𝑒𝑥𝑡) + 𝐼𝑎𝑖𝑟𝛾𝑎𝑖𝑟(𝑇𝑠𝑒𝑡− 𝑇𝑒𝑥𝑡) (4-17) The heating demand for this period (𝑄𝑠𝑢𝑠𝑡 [kWh]) is calculated as in equation (4-18), where 𝜏ℎ𝑒𝑎𝑡 [hours] is the length of the whole heating period.
𝑄𝑠𝑢𝑠𝑡 =
𝑄 𝑠𝑢𝑠𝑡× (𝜏ℎ𝑒𝑎𝑡− 𝜏𝑟𝑖𝑠𝑒) 1000
(4-18)
Total heat demand for the entire heating period (𝑄ℎ𝑒𝑎𝑡 [kWh]) is the sum of the above two calculated values for heating demand (equation (4-19)), and the demand for final energy is determined via the efficiency of the heating system conversion device, 𝜂𝐶𝐷, as per equation (4-20).
𝑄ℎ𝑒𝑎𝑡= 𝑄𝑟𝑖𝑠𝑒+ 𝑄𝑠𝑢𝑠𝑡 (4-19)
𝑄𝑓𝑖𝑛𝑎𝑙 =𝑄ℎ𝑒𝑎𝑡 𝜂𝐶𝐷
(4-20)
4.4.2
Two zone modelling
For the inclusion of spatial heating control, the building model is developed to include two internal zones at different set-point temperatures. Further parameters which are required to be specified are the proportion of the house which is under-heated (𝜃𝑢ℎ), the number of under-heated zones ( 𝑛𝑢ℎ), and position of under-heated areas within the house (number of internal and external walls), the U-value of internal walls (𝑈𝑖𝑛𝑡 [W/(m2K)]) and air leakage rate from heated zone to under-heated zone (𝐼 𝑎𝑐ℎ,𝑖𝑛𝑡 [ach]). The geometry of the two zone house is represented in Figure 4-2. Heat flows, (𝑄 [W]) and surface area (A [m2]) are calculated for the interfaces of the heated zones with the outside (h-uh), heated zones with the outside (h-o), and under-heated zones with the outside (uh-o).
Figure 4-2 Schematic illustrating the heat transfer (𝑄 [W]) and surface area (A [m2]) of the interfaces between
heated and heated areas (h-uh), heated areas and outside (h-o) and under-heated areas and outside (uh-o). Temperatures (T [°C ]) for the heated area (h), under-heated area (uh) and outside (o) are also shown.
As in the above approach, heat demand is calculated for both the heating up of the house from cold (𝑄𝑟𝑖𝑠𝑒), and for the replacement of heat loss to maintain the internal temperature (𝑄𝑠𝑢𝑠𝑡).
For the calculation of the heating energy required to maintain a steady state within the house, three heat transfers are calculated; from the heated area to the outside (𝑄 ℎ−𝑜 [W]) in equation (4-21), from the under-heated area to the outside (𝑄 𝑢ℎ−𝑜 [W]) in equation (4-22) and from the heated area to the under-heated area (𝑄 ℎ−𝑢ℎ [W]) in equation (4-23).
𝑄 ℎ−𝑜= 𝑈𝑒𝑥𝑡𝐴ℎ−𝑜(𝑇𝑠𝑒𝑡,ℎ− 𝑇𝑒𝑥𝑡) +𝑉ℎ(1 − 𝜃𝑢ℎ)𝐼 𝑎𝑐ℎ,𝑒𝑥𝑡 3600 𝛾𝑎𝑖𝑟(𝑇𝑠𝑒𝑡,ℎ− 𝑇𝑒𝑥𝑡) (4-21) 𝑄 𝑢ℎ−𝑜= 𝑈𝑒𝑥𝑡𝐴𝑢ℎ−𝑜(𝑇𝑠𝑒𝑡,𝑢ℎ− 𝑇𝑒𝑥𝑡) + 𝑉ℎ𝜃𝑢ℎ𝐼 𝑎𝑐ℎ,𝑒𝑥𝑡 3600 𝛾𝑎𝑖𝑟(𝑇𝑠𝑒𝑡,𝑢ℎ− 𝑇𝑒𝑥𝑡) (4-22) 𝑄 ℎ−𝑢ℎ= 𝑈𝑖𝑛𝑡𝐴ℎ−𝑢ℎ(𝑇𝑠𝑒𝑡,ℎ− 𝑇𝑠𝑒𝑡,𝑢ℎ) +𝑉ℎ(1 − 𝜃𝑢ℎ)𝐼 𝑎𝑐ℎ,𝑖𝑛𝑡 3600 𝛾𝑎𝑖𝑟(𝑇𝑠𝑒𝑡,ℎ− 𝑇𝑠𝑒𝑡,𝑢ℎ) (4-23) Although a set-point temperature is specified for the under-heated zone, once the house reaches a steady-state the room may be at an equilibrium temperature higher than the set- point. This higher internal temperature for the under-heated area is experienced if 𝑄 ℎ−𝑢ℎ is greater than 𝑄 𝑢ℎ−𝑜 at the under-heated area designated set-point temperature.
If 𝑸 𝒉−𝒖𝒉 is less than 𝑸 𝒖𝒉−𝒐:
The heating energy required to maintain a steady state within the house (𝑄𝑠𝑢𝑠𝑡 [kWh]) is calculated by equation (4-24).
𝑄𝑠𝑢𝑠𝑡 = (𝑄 ℎ−𝑜+ 𝑄 𝑢ℎ−𝑜) × 𝜏ℎ𝑒𝑎𝑡 (4-24)
If 𝑸 𝒉−𝒖𝒉 is greater than 𝑸𝒖𝒉−𝒐:
An equilibrium temperature (𝑇𝑢ℎ,𝑒𝑞), which is higher than the initial set-point, is maintained instead of the under-heated set-point. The equilibrium exists at the temperature for which heat flow into the under-heated zone from the heated zone (𝑄 ℎ−𝑢ℎ) is equal to the heat loss from the under-heated zone to the outside (𝑄𝑢ℎ−𝑜). This equilibrium temperature is a function of the wall area, U-value and air leakage of the internal and external walls, and is shown in equation (4-25). Equilibrium temperature is independent of the under-heated zone set-point and therefore represents a minimum under-heated temperature for a house.
𝑇𝑢ℎ,𝑒𝑞=𝛽ℎ−𝑢ℎ𝑇ℎ+ 𝛽𝑢ℎ−𝑒𝑥𝑡𝑇𝑒𝑥𝑡 𝛽𝑢ℎ−𝑒𝑥𝑡+ 𝛽ℎ−𝑢ℎ Where 𝛽ℎ−𝑢ℎ= 𝑈𝑖𝑛𝑡𝐴ℎ−𝑢ℎ+ 𝑉ℎ(1 − 𝜃𝑢ℎ)𝐼𝑎𝑖𝑟,𝑖𝑛𝑡 𝛽ℎ−𝑢ℎ= 𝑈𝑒𝑥𝑡𝐴𝑢ℎ−𝑜+ 𝑉ℎ𝜃𝑢ℎ𝐼𝑎𝑖𝑟,𝑒𝑥𝑡 (4-25)
To calculate the heating energy required to maintain a steady state within the house, equations (4-22) and (4-23) are recalculated using values of 𝑇𝑢ℎ,𝑒𝑞 instead of 𝑇𝑠𝑒𝑡,𝑢ℎ. The value of 𝑄𝑠𝑢𝑠𝑡 [kWh] is then determined as in equation (4-26).
𝑄𝑠𝑢𝑠𝑡 = (𝑄 ℎ−𝑜+ 𝑄 ℎ−𝑢ℎ) × 𝜏ℎ𝑒𝑎𝑡 (4-26)
As a simplification, the heating up period, 𝜏𝑟𝑖𝑠𝑒, is assumed to be short enough to not require that the dynamic variation in temperature be included within calculation of 𝑄𝑠𝑢𝑠𝑡 (internal temperature is assumed to be the set-point throughout). For the present model, the heat required to raise the temperature of the house, 𝑄𝑟𝑖𝑠𝑒, is calculated in steady state for the heated and under-heated areas separately as in equation (4-27) and (4-28) respectively, with the total calculated in equation (4-29). The value of 𝑇𝑢ℎ∗ in equation (4-28) is dependent on the equilibrium state of the house, and values 𝑇𝑠𝑒𝑡,𝑢ℎ or 𝑇𝑢ℎ,𝑒𝑞 are used as appropriate based on the above cases.
𝑄𝑟𝑖𝑠𝑒,ℎ = 𝑉ℎ∙ (1 − 𝜃𝑢ℎ) ∙ 𝛾𝑎𝑖𝑟∙ (𝑇𝑠𝑒𝑡,ℎ− 𝑇𝑐𝑜𝑜𝑙) (4-27) 𝑄𝑟𝑖𝑠𝑒,𝑢ℎ= 𝑉ℎ∙ 𝜃𝑢ℎ∙ 𝛾𝑎𝑖𝑟∙ (𝑇𝑢ℎ∗ − 𝑇
𝑐𝑜𝑜𝑙) (4-28)
𝑄𝑟𝑖𝑠𝑒 = 𝑄𝑟𝑖𝑠𝑒,ℎ+ 𝑄𝑟𝑖𝑠𝑒,𝑢ℎ (4-29)
As for the one zone model, the total heating energy and total final energy demand are calculated as in equations (4-19) and (4-20).
4.4.3
Parameters for building modelling
The default values for parameters used in modelling are given in Table 4-3; these values are used unless otherwise stated.
Table 4-3 Default values for model parameters
Symbol Unit Default value Description
𝐴𝑓 𝑚2 98 Floor area of house
ℎ𝑓 𝑚 2.5 Height of one floor of house
𝑛𝑓 − 2 Number of floors of house
𝜑𝑙𝑠 − 1 Ratio of length to width of house (shape descriptor)
𝜀 4 Number of exposed walls
𝛿𝑡 𝑠 10 Time step for temperature iterations
𝑄 𝑚𝑎𝑥 𝑘𝑊 8 Maximum heating power of heating technology
𝑈𝑒𝑥𝑡 𝑊/(𝑚2𝐾) 1.5 Thermal transmittance of external walls 𝑈𝑖𝑛𝑡 𝑊/(𝑚2𝐾) 1 Thermal transmittance of internal walls 𝐼 𝑎𝑐ℎ 𝑎𝑐ℎ 0.75 Infiltration rate of air from outside (one zone) 𝐼 𝑎𝑐ℎ,𝑒𝑥𝑡 𝑎𝑐ℎ 0.75 Infiltration rate of air from outside (two zone)
𝐼 𝑎𝑐ℎ,𝑖𝑛𝑡 𝑎𝑐ℎ 0.2 Internal air exchange rate between rooms (two zone)
𝜏ℎ𝑒𝑎𝑡 ℎ𝑜𝑢𝑟 8 Length of heating period
𝑇𝑒𝑥𝑡 °𝐶 5 External temperature
𝑇𝑠𝑒𝑡 °𝐶 21 Set-point temperature of heated space (one zone)
𝑇𝑠𝑒𝑡,ℎ °𝐶 21 Set-point temperature of heated space (two zone)
𝑇𝑠𝑒𝑡,𝑢ℎ °𝐶 14 Set-point temperature of under-heated space (two zone)
𝑇𝑐𝑜𝑙𝑑 °𝐶 14 Temperature of space before heating
𝐻 𝑘𝑔/𝑘𝑔 0.4 Humidity of air in house
𝑐𝑝,𝑑𝑟𝑦 𝑎𝑖𝑟 𝑘𝐽/(𝑘𝑔𝐾) 1.00 Heat capacity of dry air 𝑐𝑝,𝑤𝑎𝑡𝑒𝑟 𝑘𝐽/(𝑘𝑔𝐾) 1.86 Heat capacity of water vapour
𝜌𝑎𝑖𝑟 𝑘𝑔/𝑚3 1.28 Density of air
𝜌𝑤𝑎𝑡𝑒𝑟 𝑣𝑎𝑝𝑜𝑢𝑟 𝑘𝑔/𝑚3 0.80 Density of water vapour
𝜃𝑢ℎ − 0.6 Proportion of house under-heated
𝑛𝑢ℎ − 1 Number of under-heated spaces