Utilization of Multiple Short Heat Consumption Measurement Method for Diagnostics of Heat Source of Existing Building: Part 1: Theoretical Basis of the Method

FOIT, Henryk; LUBINA, Piotr; TĄTA, Dawid

INTRODUCTION

The thermal diagnostics of heating systems and heat source is connected with the verification of compatibility of the required and actual power of the heat source and heating systems [1, 4]. Determining the required power is one of the tasks of thermal heat source diagnostics. The need to perform such diagnostics periodically is due to national regulations and standards and the directives of the European Parliament and the Council of Europe. It is not always easy to determine the required heat power for existing residential building. The scale of difficulty increases with the increase in the cubic capacity and the number of building users. The lack of construction documentation for the building, lack of knowledge about the composition of the layered construction partitions, forming the outer coating of the heated space, aging of materials, moisture prevents the appropriate calculations. The complete or partial replacement of windows by building users and the introduction of heterogeneous windows of varying degrees of thermal resistance and air tightness make it difficult to correctly assess the heat demand by calculation methods. In residential buildings, especially built more than 15 years ago, natural ventilation is usually used.

In [4] there is included a procedure for determining the required heat source power based on a long-term measurement that includes determining seasonal heat consumption and determining the heat gains and loss of the temperature balance of the building. The problem of determining the heat demand of buildings and its accuracy, also determining the demand (consumption) or the required power of the heat source taking into account the measurements are, among others, reported in works [6, 7, 15, 18-28]. This paper presents the calculation of the heat demand for the central heating installation based on short in-situ measurements in the considered building. The works [8-14, 22-27] present the idea of a proposed method based on defining the loss coefficient by the transmission H_T of the building external envelope elements and the heat loss coefficient for ventilation H_V together with the average ventilation change coefficient of heat loss H_b and the average daily heat gain flux from people and appliances in the building (ϕ_Z,oc+ ϕ_Z,eq). This method uses, among others, multiple short in- situ measurement of daily heat consumption in a building. Determining of H_T and H_V allows to estimate the designed heat loss of a building and further, taking into account the heat demand for hot water preparation, the required heat power of the source and the oversizing of the heating source (heating system).

ASSUMPTIONS CONCERNING HEAT BALANCE EQUATION

The basic assumptions for determining the in-situ short-time measurements of the required heat source power for an existing building are as follows:

the temperatures in the heated rooms are similar to the design values of indoor air temperature,
the settlement of the building is similar to the design, and the individual internal heat gains from people and equipment correspond to the typical values of a residential building,
the current air tightness of the building determines the mass of air flowing into the building under average heating conditions close to the minimum flow resulting from hygienic needs,
measurements are made during the heating season in conditions close to average.

Determining the heat loss transmission and ventilation coefficients, and the number of ventilation air changes n_V and the flow of internal heat gains requires knowledge of the group of values, mainly determined by measurement.

To determine value of measurements’ group, it is necessary to:

make several short (daily) measurements of the heat consumption in the building for the purpose of central heating (using a heat meter, a heat flux meter or gas meter, fuel volume oil meter, mass or volume of solid fuel) and the temperature of the heating factor and, if it is possible, daily course of external air temperature and, if there is a need, internal air temperature, and also one-time measurement of the air temperature in the basement and if necessary in the selected area in the ceiling (eg. on the border of the plaster and the construction layer of the ceiling) or control temperature measurements of the inner surface of the outer wall (walls), as well as the ceiling above top floor.

Measurements should be made for the following values:

–
daily heat consumption by central heating installation ϕ_u, kWh,
–
average daily outdoor air temperature ${\bar{t}}_{e}$ , if necessary, indoor air temperature,
–
average air temperature in the building’s heated space on measured days,
–
average daily temperature in the basement.

These measurements should be supplemented with the results obtained from the nearest meteorological station, measured routinely:

–
hourly daily course of outdoor air temperature t_e ,
–
hourly course of daily wind speed,
–
hourly daily passes of average direct and diffuse solar radiation streams,

or use the results of measurements of hourly outdoor air temperature, wind speed, and solar radiation intensity, determined by the local meteorological station, or use measurements of these values by an individual meteorological station installed at the time of measurements near the considered building.

The basis for the implementation of the method is also to determine the geometrical and structural features of the building and its functional characteristics, such as the number of inhabitants, the type and number of heated rooms, and also the type and parameters of central heating and possibly heat sources. These values are derived from the operational documentation of the building or are obtained by means of rapid thermal diagnostics of the building.

The days chosen for measurements should be these without rain and snow and with moderate wind. As measurement days also very cloudy days should be avoided as well as days with moderate outside temperatures (not above 5 (7)°C, as a midday temperature) and at the same time very sunny days, which would be accompanied by significant overheating of the rooms and / or opening of the windows by inhabitants.

On the basis of the determined measurement data, the daily thermal balances for the building under consideration can be calculated as follows: (1) $\begin{array}{l} ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) = H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + \\ + \frac{n_{V} ({\bar{t}}_{e 1})}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) + \\ + (ϕ_{Z, o c} + ϕ_{Z, e q} + ϕ_{Z, t r} ({\bar{t}}_{e 1})) \cdot η_{e h g} ({\bar{t}}_{e 1}) \end{array}$ where:

t_i – indoor temperature, °C

${\bar{t}}_{e}$ – daily average outdoor temperature, °C,

${\bar{t}}_{e}^{* *}$ – modified external temperature taking into account the temperature in unheated basement of the building, °C,

$H_{T}^{rz}$ – heat loss transmission coefficient, W/K,

ϕ_u – measured, average utility heat flux for considered day, W,

η_trt – efficiency of transmission, regulation and transfer of heat for central heating,

ϕ_sw – average heat flux for heating the ventilation air for considered day, W,

ϕ_z – average internal heat gains flux for considered day, W,

η_ehg – efficiency of use of the internal heat gains,

ϕ_Z,oc – average heat gains flux from people for considered day, W,

ϕ_Z,eq – average heat gains flux from appliances for considered day, W,

ϕ_Z,tr – average for considered day, differential heat flux from solar radiation through transparent components,

n_V – number of air ventilation changes for conditions related to ${\bar{t}}_{e}$ , 1/h,

V_B – volume of the heated area of the building, m³,

c_p – specific heat capacity of air, J/(kg·K),

ρ_p – density of air, kg/m³,

1....j......k – numbers of measurement days.

PRESENTATION OF THE PROPOSED METHOD

Several times (k-fold) daily measurement of heat consumption for purposes of central heating of a building ϕ_u ( ${\bar{t}}_{ej}$ ) gives the possibility to write a system of balance equations for days by increasing temperature ${\bar{t}}_{el}$ : (2) ${\begin{cases} ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) = H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + \frac{n_{V} ({\bar{t}}_{e 1})}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p} . ({\bar{t}}_{i} - {\bar{t}}_{e 1}) \\ + (ϕ_{Z, o c} + ϕ_{Z, e q} + ϕ_{Z, t r} ({\bar{t}}_{e 1})) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) = H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + \frac{n_{V} ({\bar{t}}_{e j})}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) \\ + (ϕ_{Z, o c} + ϕ_{Z, e q} + ϕ_{Z, t r} ({\bar{t}}_{e j})) . η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) = H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + \frac{n_{V} ({\bar{t}}_{e k})}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k} \\ + (ϕ_{Z, o c} + ϕ_{Z, e q} + ϕ_{Z, t r} ({\bar{t}}_{e j})) . η_{e h g} ({\bar{t}}_{e k}) \end{cases}$

In the system of equations (2) the heat loss transmission coefficient through the partitions $H_{T}^{rz}$ is the value as (without reduction coefficients due to the temperature of the unheated space deviating from the actual outdoor temperature): $H_{T}^{r z} = \sum A_{p} \cdot U_{p} + \sum Ψ_{M} \cdot l_{M,}$ where:

A_p – surface of the envelope elements of heated building (external walls, windows and external doors, ceiling above the top floor, ceiling over the unheated basement), m²,

U_p – U coefficients of the envelope elements of the heated building space, W/(m²K),

l_M – length of the linear bridges of the elements of envelope limiting the volume of the building heated, m,

Ψ_M – coefficients of heat transmission through the linear bridges of the partitions of the building envelope limiting the heated volume of building, W/(mK).

Accepting the record $H_{V j}^{r z} = \frac{n_{v} ({\bar{t}}_{e j})}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p}$ system (2) can be presented as follows: (3) ${\begin{cases} H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{V 1}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + \\ + (ϕ_{Z, o c} + ϕ_{Z, e q}) \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{V 1}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e j}) + \\ + (ϕ_{Z, o c} + ϕ_{Z, e q}) \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{V 1}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + \\ + (ϕ_{Z, o c} + ϕ_{Z, e q}) \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$ Assuming that (4) $H_{V j}^{r z} = H_{V}^{r z} + Δ H = H_{V}^{r z} + H_{b} \cdot {\bar{t}}_{e j}$ and marking ϕ_z = (ϕ_Z,oc + ϕ_Z,eq ) it can be written: (5) ${\begin{cases} H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = \\ = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = \\ = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = \\ = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$ where H_b – average ventilation change coefficient of heat loss due to change of ventilation intensity as a function of daily outside temperature, W/K².

Numerous attempts to solve the system for the relevant measurement data (affected by some errors) by method of determinants or linear regression have led to results that are not physically correct.

Therefore, in relation to the system (2), another procedure was used to determine $H_{V}^{rz}$ (and so n_v ) and H_b .

Another form of the system (2) can be obtained if in the system equations $\frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j^{* *}})} \approx c o n s t .$ : (6) ${\begin{cases} [H_{T}^{r z} + H_{V}^{r z} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e 1})}{({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *})}] \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = \\ = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ [H_{T}^{r z} + H_{V}^{r z} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *})}] \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = \\ = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ [H_{T}^{r z} + H_{V}^{r z} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e k})}{({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *})}] \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = \\ = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$ and, after accepting: (6a) $H_{T}^{r z} + H_{V}^{r z} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *})} = H_{T V}^{r z},$ and noticing on the basis of the assumptions made: (6b) $H_{T V}^{r z} \approx c o n s t$ present the system (5) as: (6c) ${\begin{cases} H_{TV}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = \\ = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{TV}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = \\ = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{TV}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = \\ = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + ϕ_{z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$ The condition $\frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j^{* *}})} \approx c o n s t$ is better fulfilled, the difference $({\bar{t}}_{i} - {\bar{t}}_{e j})$ is higher (so the days should be selected with a sufficiently large differences $({\bar{t}}_{i} - {\bar{t}}_{e j})$ ). The lower the outdoor temperature, the temperature ${\bar{t}}_{e j}^{**}$ is usually closer to the temperature ${\bar{t}}_{e j}$ .

Further part presents procedures of designation $H_{TV}^{rz}$ and $H_{T}^{rz}$ , $H_{V}^{rz}$ , H_b and also ϕ_z .

First method

In order to implement it, the terms $H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j})$ should be omitted in the systems (3), (5) or (6c). The omission would be fully justified for ${\bar{t}}_{e j} = 0$ , then the term $H_{b} \cdot {\bar{t}}_{e j} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j^{* *}})} = 0$ . For $| {\bar{t}}_{e j} | ≫ 0$ there appears a certain inaccuracy of the method. As a result of the omission of the terms $H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j})$ in incompitable systems of balance equations (3), (6c), it is possible in further part to determine firstly the auxiliary values $H_{TV}^{rz}$ and further the principal searched values in the form ϕ_Z , $H_{T}^{rz}$ and $H_{V}^{rz}$ . After omitting the terms $H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j})$ and assuming $η_{e h g} \approx c o n s t \approx 1, 0$ , the chosen equations (r..s) of the system (6c) should be subtracted from successive equations [(1 … k) − (r..s)], what results in: (7) ${\begin{cases} {(H_{T V}^{r z})}_{a} = {ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) - ϕ_{u} ({\bar{t}}_{e r}) \cdot η_{t r t} ({\bar{t}}_{e r}) \\ + [ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) - ϕ_{Z, t r} ({\bar{t}}_{e r}) \cdot η_{e h g} ({\bar{t}}_{e r})]} / ({\bar{t}}_{e r}^{* *} - {\bar{t}}_{e 1}^{* *}) \\ d l a {\bar{t}}_{e a} = \frac{| {\bar{t}}_{e 1} | + | {\bar{t}}_{e r} |}{2} \\ . \\ . \\ {(H_{T V}^{r z})}_{f} = {ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) - ϕ_{u} ({\bar{t}}_{e p}) \cdot η_{t r t} ({\bar{t}}_{e p}) + \\ + [ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) - ϕ_{Z, t r} ({\bar{t}}_{e p}) \cdot η_{e h g} ({\bar{t}}_{e p})]} / ({\bar{t}}_{e p}^{* *} - {\bar{t}}_{e j}^{* *}) \\ d l a {\bar{t}}_{e f} = \frac{| {\bar{t}}_{e j} | + | {\bar{t}}_{e p} |}{2} \\ . \\ . \\ {(H_{T V}^{r z})}_{h} = {ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t p} ({\bar{t}}_{e k}) - ϕ_{u} ({\bar{t}}_{e s}) \cdot η_{t r t} ({\bar{t}}_{e s}) + \\ + [ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) - ϕ_{Z, t r} ({\bar{t}}_{e s}) \cdot η_{e h g} ({\bar{t}}_{e s})]} / ({\bar{t}}_{e s}^{* *} - {\bar{t}}_{e k}^{* *}) \\ d l a {\bar{t}}_{e h} = \frac{| {\bar{t}}_{e k} | + | {\bar{t}}_{e s} |}{2} \end{cases}$ Equations occuring in the systems (7), resulting from the subtraction of equations of the output system, will be further referred to as the differential equations.

Subtracting the equations from one another should be done in such a way, that representing of all equations of the system (7) was equal and the differences in temperature ${\bar{t}}_{e}^{**}$ were bigger than 3 K and the equation used to form the differential equation (minuend, subtrahend) always corresponded to the positive value of the temperature difference ${\bar{t}}_{e}$ , e.g for equations j and p: ${\bar{t}}_{e p} - {\bar{t}}_{e j} > 0$ . At the same time, the equations of each pair of equations making up the differential equations should have similar approximate values $H_{TV}^{rz}$ .

The condition of equal representation of equations (6c) in the differential equations system (7) is related to the assumption that individual equations of the system (6c) derive from their “perfect” state. The state of “perfect” equations is the form of thermal balance equation, which corresponds to the fulfillment of the required measurement conditions (fulfillment of the assumed operation of the building on the day of measurement) and the quality of measurements taken into account in the balance of values. Usually, the balance equations are deriving from the “perfect” state. If it is correct to assume that in the considered equation system there is a balance of these deviations, then in the differential equations system (7) representation of individual equations of system (6c) should be equal. If, however, in equation (6c), with the following equations ordered in decreasing temperature difference $({\bar{t}}_{i} - {\bar{t}}_{e}^{**})$ , the last equation would be similar to “perfect”, then the differential equation system (6) can be constructed by subtracting the last equation from succeeding equations. Such a procedure is accompanied by obtaining of the system (7) well replacing system (6c), with differential equations sorted in decreasing difference $({\bar{t}}_{ep}^{**} - {\bar{t}}_{ej}^{**})$ . The equations of each pair of equations creating differential equations should have similar approximate values $H_{TV}^{rz}$ . Values $H_{TV}^{rz}$ should be estimated for successive equations of systems (5) or (6c), after omitting the terms containing H _b.

The linear approximation $[{(H_{T V}^{r z})}_{a}, \dots {(H_{T V}^{r z})}_{f}, \dots {(H_{T V}^{r z})}_{h}]$ in decreasing temperature function ${\bar{t}}_{e a}, \dots {\bar{t}}_{e f}, \dots {\bar{t}}_{e h}$ leads to equation as: ${(H_{T V}^{r z})}_{o} = a_{a p} \cdot {\bar{t}}_{e o} + b_{o}$ , what results in: (7a) $H_{T V}^{r z} = {(H_{T V}^{r z})}_{{\bar{t}}_{e} = 0} = a_{a p} \cdot 0 + b_{o} = b_{o}$

Second method

The condition for applying this procedure is as follows: in equation (6c) there are balance equations with days jj, for which ${\bar{t}}_{jj} \approx 0$ .

From the equations system (5c) are chosen equations with days (m … ..n), for which ${\bar{t}}_{ej} \approx 0$ , what also means $H_{b} \cdot {\bar{t}}_{e j} \cdot \frac{({\bar{t}}_{i} - {\bar{t}}_{e j})}{({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *})} \approx 0$ : (8) ${\begin{cases} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e m}^{* *}) = ϕ_{u} ({\bar{t}}_{e m}) \cdot η_{t r t} ({\bar{t}}_{e m}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e m}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e m}) . η_{e h g} ({\bar{t}}_{e m}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e j}) . η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e n}^{* *}) = ϕ_{u} ({\bar{t}}_{e n}) \cdot η_{t r t} ({\bar{t}}_{e n}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e n}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e n}) . η_{e h g} ({\bar{t}}_{n k}) \end{cases}$

The efficiency η_ehg of using internal gains ϕ_Z takes values close to 1, so it can be assumed that ϕ_Z ⋅ η_ehg = const.

After subtracting from the successive equations [(m … n) − (r..s)] the selected equations (r..s) of the system (8) the result is: (8a) ${\begin{cases} H_{T V}^{r z} \cdot ({\bar{t}}_{e r}^{* *} - {\bar{t}}_{e m}^{* *}) = \\ = ϕ_{u} ({\bar{t}}_{e m}) \cdot η_{p r p} ({\bar{t}}_{e m}) + ϕ_{u} ({\bar{t}}_{e r}) \cdot η_{p r p} ({\bar{t}}_{e r}) + \\ + [ϕ_{Z s p r z e z r} ({\bar{t}}_{e m}) \cdot η_{w y k} ({\bar{t}}_{e m}) - ϕ_{Z s p r z e z r} ({\bar{t}}_{e r}) \cdot η_{w y k} ({\bar{t}}_{e r})] \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{e p}^{* *} - {\bar{t}}_{e j}^{* *}) = \\ = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{p r p} ({\bar{t}}_{e j}) + ϕ_{u} ({\bar{t}}_{e p}) \cdot η_{p r p} ({\bar{t}}_{e p}) + \\ + [ϕ_{Z s p r z e z r} ({\bar{t}}_{e j}) \cdot η_{w y k} ({\bar{t}}_{e j}) - ϕ_{Z s p r z e z r} ({\bar{t}}_{e p}) \cdot η_{w y k} ({\bar{t}}_{e p})] \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{e n}^{* *} - {\bar{t}}_{e s}^{* *}) = \\ = ϕ_{u} ({\bar{t}}_{e s}) \cdot η_{p r p} ({\bar{t}}_{e s}) - ϕ_{u} ({\bar{t}}_{e n}) \cdot η_{p r p} ({\bar{t}}_{e n}) + \\ + [ϕ_{Z s p r z e z r} ({\bar{t}}_{e s}) \cdot η_{w y k} ({\bar{t}}_{e s}) - ϕ_{Z s p r z e z r} ({\bar{t}}_{e n}) \cdot η_{w y k} ({\bar{t}}_{e n})] \end{cases}$

Subtracting the equations from one another should be done in such a way that representing all of the equations of the system (7) in the system (t) was equal, and the temperature differences ${\bar{t}}_{e}^{**}$ big as possible. The equations of each pair of equations that create differential equations should have approximate estimated values $H_{TV}^{rz}$ .

The system (7a) allows to determine the value $H_{TV}^{rz}$ by the matrix account: (9) $\begin{matrix} M = [\begin{array}{l} {\bar{t}}_{e m}^{* *} - {\bar{t}}_{e r}^{* *} \\ . \\ . \\ {\bar{t}}_{e p}^{* *} - {\bar{t}}_{e j}^{* *} \\ . \\ . \\ {\bar{t}}_{e n}^{* *} - {\bar{t}}_{e s}^{* *} \end{array}], \\ W = [\begin{array}{l} ϕ_{u} ({\bar{t}}_{e m}) \cdot η_{t r t} ({\bar{t}}_{e m}) - ϕ_{u} ({\bar{t}}_{e p}) \cdot η_{t r t} ({\bar{t}}_{e p}) + \\ + [ϕ_{Z, t r} ({\bar{t}}_{e m}) \cdot η_{e h g} ({\bar{t}}_{e m}) - ϕ_{Z, t r} ({\bar{t}}_{e p}) . η_{e h g} ({\bar{t}}_{e p})] \\ . \\ . \\ ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) - ϕ_{u} ({\bar{t}}_{e p}) \cdot η_{t r t} ({\bar{t}}_{e p}) + \\ + [ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) - ϕ_{Z, t r} ({\bar{t}}_{e p}) . η_{e h g} ({\bar{t}}_{e p})] \\ . \\ . \\ ϕ_{u} ({\bar{t}}_{e n}) \cdot η_{t r t} ({\bar{t}}_{e n}) - ϕ_{u} ({\bar{t}}_{e s}) \cdot η_{t r t} ({\bar{t}}_{e s}) + \\ + [ϕ_{Z, t r} ({\bar{t}}_{e n}) \cdot η_{e h g} ({\bar{t}}_{e n}) - ϕ_{Z, t r} ({\bar{t}}_{e s}) . η_{e h g} ({\bar{t}}_{e s})] \end{array}], \\ H_{T V}^{r z} = {[M^{T} \cdot M]}^{- 1} \cdot M^{T} \cdot W \end{matrix}$ where:

M^T – transpose of matrix M, [M^T ⋅ M]⁻¹ - inverse matrix of the matrix product M^T i M.

If omitted in the system equations (8a) of the component from H_b causes occurring of an error - its value for the differential equations j and p is δϕ_jp : (10) ${δ ϕ}_{j p} = H_{b} \cdot [t_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) - t_{e p} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e p})]$ and the average value for all equations is δϕ_mm .

If the mean temperature difference is δt_mm : (11) ${δ t}_{m m} = {\begin{cases} {\bar{t}}_{e m}^{* *} - {\bar{t}}_{e r}^{* *} \\ . \\ . \\ {\bar{t}}_{e p}^{* *} - {\bar{t}}_{e j}^{* *} \\ . \\ . \\ {\bar{t}}_{e n}^{* *} - {\bar{t}}_{e s}^{* *} \end{cases}}$ Than the error of designation $H_{TV}^{rz}$ caused by the deviation δϕ_mm takes the form of: $δ H_{T V}^{r z} = | \frac{δ ϕ_{m m}}{δ t_{m m}} |$ .

3.1.

Determination of the coefficient H_b

After returning to the system (6c) and substituting to the system a designated value $H_{TV}^{rz}$ , the following are obtained: (12) ${\begin{cases} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$

The system can be presented as: (13) ${\begin{cases} H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) - ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e 1}) = - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + \\ + ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) - ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) = - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + \\ + ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) - ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e k}) = - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + \\ + ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$

The system gives the solution: (14) $[\begin{array}{l} ϕ_{Z} \\ H_{b} \end{array}] = {[M^{T} \cdot M]}^{- 1} \cdot M^{T} \cdot W,$ for: (15) $\begin{array}{l} M = [\begin{array}{l} {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) & η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) & η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) & η_{e h g} ({\bar{t}}_{e k}) \end{array}], \\ W = [\begin{array}{l} - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) . \\ . \\ . \\ - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) . \\ . \\ . \\ - H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) . \end{array}] \end{array}$

3.2.

Determining of internal heat gains flux ϕ_Z

After substituting to the system (6c) designated values $H_{TV}^{rz}$ and H_b the following values are obtained: (16) ${\begin{cases} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$ and than: (17) ${\begin{cases} ϕ_{Z} = \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) - ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \end{array}}{η_{e h g} ({\bar{t}}_{e 1})} \\ . \\ . \\ ϕ_{Z} = \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) - ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \end{array}}{η_{e h g} ({\bar{t}}_{e j})} \\ . \\ . \\ ϕ_{Z} = \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) - ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{array}}{η_{e h g} ({\bar{t}}_{e k})} \end{cases}$

The system gives the solution: (18) $[ϕ_{Z}] = {[M^{T} \cdot M]}^{- 1} \cdot M^{T} \cdot W,$ for: (19) $\begin{array}{l} M = [\begin{array}{l} 1 \\ . \\ . \\ 1 \\ . \\ . \\ 1 \end{array}], \\ W = [\begin{array}{l} \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) - ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \end{array}}{η_{e h g} ({\bar{t}}_{e 1})} \\ . \\ . \\ . \\ \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) - ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \end{array}}{η_{e h g} ({\bar{t}}_{e j})} \\ . \\ . \\ . \\ \frac{\begin{array}{l} H_{T V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) - ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) - \\ - ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{array}}{η_{e h g} ({\bar{t}}_{e k})} \end{array}] . \end{array}$

3.3.

Determination of heat loss coefficients

H_{T V}^{r z}

H_{V}^{r z}

Condition of use: system (3b, 6c) must contain a suitable number of mutually compatible equations. Designation of $H_{T}^{r z}$ and $H_{V}^{r z}$ should follow from the system of equations, which is not identical with the system from which the simultaneous determination of H_b and ϕ_Z was made.

Designated values $H_{T}^{r z}$ and $H_{V}^{r z}$ will be associated with the values ϕ_Z and H_b accepted to define them.

Using a transformed system (3b, 6c), after substituting H_b and ϕ_Z determined earlier, the result is: (20) ${\begin{cases} H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) = - H_{b} \cdot {\bar{t}}_{e 1} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e 1}) + \\ + ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e 1}) + ϕ_{u} \cdot η_{e h g} ({\bar{t}}_{e 1}) + ϕ_{Z, t r} ({\bar{t}}_{e 1}) \cdot η_{e h g} ({\bar{t}}_{e 1}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = - H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) + \\ + ϕ_{u} ({\bar{t}}_{e 1}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \\ . \\ . \\ H_{T}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}^{* *}) + H_{V}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) = - H_{b} \cdot {\bar{t}}_{e k} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e k}) + \\ + ϕ_{u} ({\bar{t}}_{e k}) \cdot η_{t r t} ({\bar{t}}_{e k}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e k}) + ϕ_{Z, t r} ({\bar{t}}_{e k}) \cdot η_{e h g} ({\bar{t}}_{e k}) \end{cases}$

Determination of $H_{T}^{r z}$ and $H_{V}^{r z}$ follows by solution of the overdetermined system of equations according to: (21) $[\begin{array}{l} H_{T}^{r z} \\ H_{V}^{r z} \end{array}] = {[M^{T} \cdot M]}^{- 1} \cdot M^{T} \cdot W$ where:

M – two-column matrix of coefficients at $H_{T}^{r z}$ and $H_{V}^{r z}$ in system equations,

W – column matrix of free terms in the system equations.

On this basis it is possible to estimate: (21a) $n_{V} = \frac{H_{V}^{r z}}{3600} \cdot V_{B} \cdot c_{p} \cdot ρ_{p}$

Specifying $H_{T V}^{r z}$ and $H_{T}^{r z}$ , $H_{V}^{r z}$ , H_b and z requires the knowledge of the meaningful temperature for measurement conditions ${\bar{t}}_{e}^{* *}$ .

The temperatures ${\bar{t}}_{e}^{* *}$ as averages for the range Δτ{1, 2, …, Δτ} can be derived from the dependence: (22) $\begin{array}{l} {\bar{t}}_{e}^{* *} = t_{i} - t_{i} \cdot (a_{3} + a_{4} + a_{5}) + \\ + \frac{1}{Δ τ} \cdot {\sum_{τ = 1}^{τ = Δ τ} [a_{3} \cdot t_{e} (τ) + a_{4} \cdot ϑ_{i} (τ) + a_{5} \cdot {\bar{t}}_{n} (τ)]} \\ , ° C, \end{array}$ where:

t_i – indoor air temperature, °C,

t_e (τ) – outdoor temperature measured in the hour τ, °C,

${\bar{t}}_{n} (τ)$ – temperature in unheated basement (generally, in an unheated room adjoining heated space), estimated for t_e (τ), °C,

ϑ_i (τ) – average temperature of the inner surface of the partitions of building envelope surrounding heated space of the building, °C,

Coefficients are equal to: (23a) $a_{1} = \frac{A_{w} \cdot U_{w}}{A_{s h a} \cdot U_{s h a}},$ (23b) $a_{2} = \frac{(A_{s h a} - A_{w})}{R_{i} \cdot A_{s h a} \cdot U_{w}},$ (23c) $a_{3} = \frac{A_{w} \cdot U_{w}}{A_{s h a} \cdot U_{s h a} + A_{n} \cdot U_{n}},$ (23c) $a_{4} = \frac{(A_{s h a} - A_{w})}{R_{i} \cdot (A_{s h a} \cdot U_{s h a} + A_{n} \cdot U_{n})},$ (23d) $a_{5} = \frac{A_{n} \cdot U_{n}}{A_{s h a} \cdot U_{s h a} + A_{n} \cdot U_{n}},$ where:

A_sha – the surface of the envelope surrounding the heated space of the building with bypassing the ceiling over the unheated basement,m²,

U_sha – estimated average coefficient U for the envelope surrounding the heated space of the building (including thermal bridges) with bypassing the ceiling over the unheated basement W/(m²K),

A_w – surface of the building windows, m²,

R_i – average thermal resistance by the inner side of the envelope partitions of the heated building space, m²K/W,

A_n – partition area between the heated and unheated space of the building (ceiling above the unheated basement), m²,

U_n – predicted heat transfer coefficient (including thermal bridges) for partitions between heated and unheated building space (ceiling above unheated basement), W/(m²K).

Determination of coefficients, a ₁, a ₂, a ₃, a ₄, a ₅ requires the estimation of the heat loss coefficient for ventilation H_V , the coefficient U_w for the windows in the building and the replacement heat transfer coefficient for the whole envelope of the heated building space and the temperature in the unheated space adjacent to the heated one ${\bar{t}}_{n}$ .

It is recommended to take the time interval Δτ as = 24 h ⋅ 5 = 120 h.

Determining $H_{T V}^{r z}$ , $H_{T}^{r z}$ , $H_{V}^{r z}$ , H_b , ϕ_Z by subtracting two different daily heat requirements is connected with determination of $H_{T V}^{r z}$ , $H_{T}^{r z}$ , $H_{V}^{r z}$ , H_b , ϕ_Z for the averaged daily internal heat gains and ventilation intensity.

Determination of coefficients a₃ , a₄ , a₅ requires, as already mentioned, the estimation of the coefficient U_w for windows in a building in an unheated basement and the replacement heat transfer coefficient for the entire space envelope of the heated building space and the temperature in the unheated basement ${\bar{t}}_{n}$ .

The determined heat loss transmission coefficient was defined in the presented analysis as: (24) $H_{T}^{r z} = A_{s h a} . U_{s h a} + A_{n} \cdot U_{n}, W / K,$ what means that its value in the given relations is constant. In [4] the heat loss transmission coefficient is defined as: (25) $\begin{matrix} H_{T}^{r z} = A_{s h a} . U_{s h a} + b_{t} \cdot U_{n} \cdot A_{n} = A_{s h a} \cdot U_{s h a} + \\ + \frac{t_{i} - t_{n}}{t_{i} - t_{e}} \cdot U_{n} \cdot A_{n} \\ , W / K, \end{matrix}$ where:

A_sha – surface of the external envelope of the building without ceiling surface above the basement, m²,

U_sha – coefficient U of the outer envelope of the building (including thermal bridges) without ceiling area above the basement, W/(m²K),

b_t – coefficient of temperature reduction,

b_{t} = \frac{t_{i} - t_{n}}{t_{i} - t_{e}},

t_i – indoor air temperature, °C,

t_n – air temperature in basement, °C,

t_e – outdoor air temperature, °C,

U_n – coefficient U of the ceiling above the basement, W/(m²K),

A_n – surface of the ceiling above the basement, m².

The average temperature of the inner surface of the envelope partitions surrounding the heated space of the building ϑ_i results from the dependence: (26) $\begin{array}{l} ϑ_{i} (τ) = \\ \sum_{n = 1}^{n = n_{k}} {[h_{e} (n)] \cdot [t_{e} (τ - n) + ({\dot{q}}_{s . z .}^{c . r a d .} (τ - n)) \cdot a_{a b s}^{s . z .} \cdot R_{e}^{'}]]} + h_{i} \cdot t_{i}, \\ ° C, \end{array}$ where:

n_k – number of hours of the time horizon in the past preceding the considered hour τ on the day of measurement,

n – succeeding hour, among hours preceding hour τ,

${\dot{q}}_{s . z .}^{c . r a d .}$ – average hourly flux of total solar radiation on the surface of the building’s outer partitions, W/m²,

$a_{a b s}^{s . z .}$ – coefficient of absorption of total solar radiation on the surface of the building’s outer partitions,

$R_{e}^{'}$ – average heat transmission resistance on the outer envelope surface of the heated space of the building, m²K/W,

h_i i h_e – impulse response functions determined by the EXODUS method [8-13, 22-26].

The assumption of the predicted coefficient U_sha (average for the coating surrounding the heated space of the building excluding the ceiling over the unheated basement) is to be understood as taking the initial value of this coefficient and further correction of it on the basis of the determined value $H_{T}^{r z}$ until the required conformance of these values is achieved.

The next steps in assumption U_sha are:

–
determining the windows area A_w and the characteristics of the windows used (window type (single leaf, double leaf) number of glass, type of glass and type of frame), which allows to determine the coefficient U_w ,
–
determining the surface of the envelope surrounding the heated space of the building omitting the ceiling over the unheated basement A_sha ,
–
determining of the ceiling surface over the unheated basement A_n ,
–
estimation of the U_n coefficient for the ceiling over an unheated basement on the basis of arrays for known conventional ceiling constructions, such as a ceiling containing a reinforced concrete slab, Ackerman ceiling or DZ-3 or, in the case of unknown structures, on the basis of a temperature measurement ${\bar{ϑ}}_{n}$ in the ceiling at a specific location, e.g the borderline of the plaster (on the side of the basement), the elementary ceiling layer and the temperature in the basement ${\bar{t}}_{n}$ and the room above the ceiling of the basement ${\bar{t}}_{i}$ . The temperatures ${\bar{t}}_{n}$ , ${\bar{t}}_{i}$ and ${\bar{ϑ}}_{n}$ , can be treated as constant, so that the heat flow through the ceiling - as determined in time. In the case of the temperature measurements on the borderline of the internal plaster on the basement side and the homogeneous construction layer above, the value U_n is derived from the dependence: (27) $U_{n} = \frac{1}{R_{n}} = \frac{\bar{ϑ_{n}} - \bar{t}}{(R_{i p} + \frac{d_{t w}}{λ_{t w}}) \cdot ({\bar{t}}_{i} - {\bar{t}}_{n})}, W / (m^{2} K),$ where:

d_tw – thickness of the inner plaster layer covering the ceiling above the basement, m,

λ_tw – heat conduction coefficient for the interior plaster covering the ceiling above the basement, W/(mK),

R_ip – heat transmission resistance (on the side of the internal plaster) on the surface of the internal plaster covering the ceiling above the basement, m2K/W,
–
assumption based on the thermal protection requirements in force during the construction of the building or thermal modernization of the external partition coefficient U_sp , except for windows and ceilings over an unheated basement,
–
determining on the basis of assumed assumptions and findings (resulting from e.g the technical documentation of the building or the requirements of thermal protection in force during the building or thermal modernization) of the projected coefficient U_sha as: (28) $\begin{array}{c} U_{p z b s p} = \frac{U_{s p} \cdot (A_{s h a} - A_{w}) + U_{w} \cdot A_{w}}{A_{s h a}} \\ W / (m^{2} K) . \end{array}$

The value U_sha be taken in the following steps is equal to: (29) $\begin{array}{c} U_{s h a} = \frac{a_{M T} \cdot {\bar{H}}_{T}^{r z} - U_{n} \cdot A_{n} - U_{w k} . A_{w}}{A_{s h a}} \\ W / (m^{2} K) \end{array}$ where:

a_MT – participation of heat loss through thermal bridges in heat transmission loss, a_MT ≈ 0,75,

${\bar{H}}_{T}^{r z} - H_{T}^{r z}$ W/K value determined in previous step.

When adopting a new value U_sha for the next calculation step, it is also necessary, in justified cases (large difference U_sha for subsequent steps) to do the correction of coefficients h_i i h_e on the basis of correction of assumed state of the outer partition layer. Correction may consist in determining for a layer with the expected maximum uncertainty of its state (layers of thickness d_i and current coefficient $λ_{i}^{p}$ ) a new value λ_i . The new value of the coefficient U_sp , which replaces in the next step the value of the previous one $U_{sp}^{p}$ , in the case of significant similarity of the layers of the external vertical walls and the ceiling above the highest floor (walls in the form of reinforced concrete slabs and reinforced concrete floors) can be determined from dependency: (30) $U_{s p} = \frac{a_{M T} \cdot {\bar{H}}_{T}^{r z} - U_{n} \cdot A_{n} - U_{w} . A_{w}}{A_{s h a} - A_{n} - A_{w}},W / (m^{2} K) .$

The new value λ_i of layer i is: (31) $λ_{i} = \frac{d_{i}}{{(U_{s p})}^{- 1} - {(U_{s p}^{p})}^{- 1} + \frac{d_{i}}{λ_{i}^{p}}}, W / (m^{2} K) .$

A similar correction can be made for layer d_i thickness. In general, the correction should be based on the coefficient λ_i or thickness d_i , as the most uncertain values of the partition, the state of which in the most significant way influences the meaningful temperature ϑ_i . In the case of significant differences in the structure and U coefficients of the vertical walls and the ceiling above the highest floor, in the presented procedure for determination of $U_{sp}^{p}$ , the U of the vertical walls has to be determined on the basis of the known or well-estimated ceiling coefficient U_sd above the highest floor and ceiling area above the highest floor A_sd . The comments given when determining U_n can be used for determining U_sd coefficient. However, the estimation of U_sd is more difficult and is associated with greater uncertainty as compared with U_sp because of the lack of fully established state on the upper surface of the ceiling under the unheated attic and not fully established state on the surface of the flat roof. The presented method of expressing temperature $t_{e}^{**}$ allows to skip in the dependencies (4)–(20) the average, for the considered measurement days, heat gains flux from solar radiation through the opaque partitions for the measurement cycles associated with ${\bar{t}}_{e j} (ϕ_{Z, t r} ({\bar{t}}_{e j}))$ and the average daily heat accumulation flux in the outer partitions of the building (Δϕ_a ), because ϑ_i (τ) includes the variability of the outside temperature and the intensity of the radiation affecting the outer walls. [8, 9] presents the coefficients h_i and h_e and the values a₃ , a₄ , a₅ for the exemplary partitions.

Designated values ϕ_zj (e.g from (17), (18)) may be the basis for selecting the equations included in the system. The equations (days of measurement), for which ϕ_zj deviate the most from the mean value ϕ_zj , should be omitted in the next approximation step. At least 8- day measurement should be done. After 7 day measurements, the obtained results should be analyzed. For this purpose, the value of the loss coefficient $H_{T V j}^{r z}$ has to be determined for individual measurement days j and the expected average daily gains ϕ_z as: (32) $\begin{array}{c} H_{T V j}^{r z} = \frac{\begin{array}{l} ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) - V_{B} \cdot n_{V} \cdot c_{p} \cdot ρ_{p} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) + \\ + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \end{array}}{({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *})} \\ W / K, \end{array}$

${\bar{t}}_{e j}^{* *}$ – equivalent temperature on the measurement day j, °C,

n_v – number of ventilation air changes estimated for the given day j, 1/h,

V_B – volume of the heated area of the building, m³,

c_p – specific heat capacity of air, J/(kg⋅K),

ρ_p – density of air, kg/m³.

On the basis of the values obtained $H_{T V j}^{r z}$ (j is the number of the succeeding measurement day in question), the arithmetic mean $H_{T V J}^{r z}$ should be determined, and the days with the significant deviation from $H_{T V J}^{r z}$ should be rejected from the days measurements for subsequent days so that the number of days after the elimination of the days with significant deviation from elimination of the days with significant deviation from $H_{T V J}^{r z}$ bigger than 10% was at least 8.

The final verification of the days taken for the final analysis should also be made on the basis of the estimated pre-determined average gains ϕ_Zj , based on the measured data for the considered days and compared to the expected average daily gains.

As the basis for expected returns the indicators given in [16] can be used, and:

in the Regulation of the Minister of Infrastructure of 6 November 2008 on the methodology for calculating the energy characteristic of a building constituting an independent technical and utility unit as well as the way of drawing up and specifying the certificates of their energy characteristic (Journal of Laws 2008 No. 201 item 1240) as 3.2 – 6.0 W/m² for multi-family buildings and 2.5 – 3.5 W/m² for single-family houses
in the Regulation of the Minister of Infrastructure of June 3, 2014 on the methodology for calculating the energy characteristic of a building and a dwelling or a part of a building constituting an independent technical and utility unit and the manner of drawing up and specifying energy characteristic certificates (Journal of Laws of 2014, item 888) as 7.1 W/m² for residential and 1.0 W/m² for staircases in multi-family buildings and 6.8 W/m² for single-family houses.

After omitting in the generalized balance equation of the system (12) (33) $\begin{array}{l} H_{T V j}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}^{* *}) + H_{b} \cdot {\bar{t}}_{e j} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e j}) = \\ ϕ_{u} ({\bar{t}}_{e j}) \cdot η_{t r t} ({\bar{t}}_{e j}) + ϕ_{Z} \cdot η_{e h g} ({\bar{t}}_{e j}) + ϕ_{Z, t r} ({\bar{t}}_{e j}) \cdot η_{e h g} ({\bar{t}}_{e j}) \end{array}$ the component containing H_b (H_b =0) and accepting and proceeding according to (8), (8a) and (9), it is possible to determine the coefficient $H_{T V}^{r z}$ .

By using the set value $H_{T V}^{r z}$ and keeping the assumption H_b = 0, the approximate gains ϕ_Zj for each measurement day can be determined: (34) $ϕ_{Z j} = \frac{H_{T Z}^{r z} \cdot ({\bar{t}}_{i} - {\bar{t}}_{e}^{* *}) - [\begin{array}{l} ϕ_{u} ({\bar{t}}_{e}) \cdot η_{t r t} ({\bar{t}}_{e}) + \\ + ϕ_{Z, t r} ({\bar{t}}_{e}) \cdot η_{e h g} ({\bar{t}}_{e}) \end{array}]}{η_{e h g} ({\bar{t}}_{e})}, kW$

The efficiency of using internal gains can be determined on the basis of the total heat gains of the building: ϕ_u + ϕ_Z + ϕ_Z.tr (from people, appliances and solar radiation) and heat demand for compensation of the heat loss of the building (transmission through the partitions and ventilation) ϕ_den , using the dependence: (35) $η_{u} = 1 - e^{- \frac{ϕ_{d e m}}{ϕ_{u} + ϕ_{Z} + ϕ_{Z, t r}}} 1.$

CONCLUSIONS

The basic stages (steps) of the method of multiple daily measurements in order to determine the values allowing to determine the thermal characteristics of a building are as follows:

–
on the basis of analysis of $H_{T V}^{r z}$ , balance equations should be selected for the system of characteristic equations for the multiple- measurement method,
–
the system of equations characteristic of the multiple-measurement method should be created by at least 8-day balance equations. If during the days of measurements a significant constance of operating conditions of the building was maintained (constant number of people staying in the building, moderate weathering, constant room air temperature) and reliable information has been obtained on the partitions surrounding the heated space of the building, then the number of measurement days can be reduced to the specified lower limit. In case of significant incompatibility of balance equations, steps should be taken to obtain system with more equations greater than 8,
–
it is profitable for a proper determination of the coefficient H_b to cover the daily temperature range $- 10 ° C \leq {\bar{t}}_{e} \leq 5 (7) ° C$ ,
–
among the days of measurement should also be a few days with $- 3 ° C \leq {\bar{t}}_{e} \leq 3 (7) ° C$ ,
–
in systems of differential equations, equations with significant temperature ${\bar{t}}_{e}^{**}$ differences should be created,
–
equations of each pair of equations making up the differential equations should have approximate estimated values $H_{T V}^{r z}$ ,
–
for the determination of the sought values, it is advantageous to take into account the daily heat balances of heat accumulation from solar radiation through the windows in the form of the adoption of the average daily heat gains flux from solar radiation containing ¾ of gains resulting from the radiation transmitted by windows on the considered day and ¼ of the gains from the radiation of the previous day,
–
in the created differential equation system there should be a equilibrium representation of balance equations in the form of the minuends and subtrahends,
–
determined values ϕ_Zj can be the basis for selecting equations considered in the system. In the next approximation step equations (days of measurement), for which ϕ_Zj deviate the most from the average value ϕ_Zj should be omitted,
–
on the basis of the determined values ϕ_Zj , the value $H_{T V}^{r z}$ can be evaluated: if the values ϕ_Zj are negative or very small, this may indicate to low value of the determined $H_{T V}^{r z}$ – than $H_{T V}^{r z}$ should be re-determined for the corrected balance equations system. Correction should consist in the elimination from the system of equations corresponding to the “wrong” values ϕ_Zj , and possibly replacing them with the new balance equations,
–
in the case of significantly different values ϕ_Zj corresponding to individual balance equations (with a maximum value ϕ_Zj : ϕ_{Z max} ), but at the same time values ϕ_Zj close to are expected, it’s been suggested to accept as the value of daily gains ϕ_Z = (0,7 − 0,9) ⋅ ϕ_max . Then $H_{T V}^{r z}$ should be re-determined for the assumed value of gains ϕ_Z = (0.7 − 0.9) ⋅ ϕ_max .
–
if the determined distribution of internal gains covers ϕ_Zj included in the range (0.7 − 1.0). ϕ_{Z max} , the arithmetic mean of the determined values ϕ_Zj can be accepted as value ϕ_Z ,
–
the results obtained with the solution of the overdetermined system of balance equations by means of the matrix method and the determinants, with the course of proceedings presented in the monograph, do not deviate significantly from the results obtained by the multiple regression method. Such a state of results is inter alia the result of the accepted measured uncertainties. The results obtained by the regression method, however, are also accompanied by the determination of their uncertainty,
–
using the meaningful temperature in place of the outside air temperature in a multiple-daily measuring method to determine the values allowing to determine the building’s thermal characteristics leads to a significant change in the results (improvement of their accuracy).

The determined values $H_{T V}^{r z}$ and η_V refer to the average conditions associated with ${\bar{t}}_{e} \approx 0 ° C$ .

Utilization of Multiple Short Heat Consumption Measurement Method for Diagnostics of Heat Source of Existing Building: Part 1: Theoretical Basis of the Method

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