CC BY
4
UDC 620.179.13 DOI: 10.22227/1997-0935.2021.6.720-729
A theoretical and experimental study of transient characteristics of the heat exchange in a thermal control system
Rustam Sh. Mansurov, Yuri E. Voskoboinikov, Vasilisa A. Boeva
Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NSUACE (Sibstrin)); Novosibirsk, Russian Federation
ABSTRACT
Introduction. The "Heater-Blower-Room" thermal control system represents three different interconnected subsystems. It is necessary to study the transient characteristics of the heat exchange process, that is underway in the subsystems, including informative impulse responses, to stabilize the system operation. It is a non-parametric problem, and its solution requires identification algorithms.
Materials and methods. Mathematical models of the subsystems represent the Volterra integral equation of the first kind with an undetermined difference kernel, or an impulse response. An impulse response evaluation is a solution to this equation in respect of registered noisy input and output values. The problem is to evaluate unknown impulse responses for the subsystems where the output of one subsystem is the input of another one. This problem is ill-posed, and features of identification-focused experiments do not allow to apply computational methods of classical regularization algorithms. The co-authors propose two specific non-parametric identification algorithms where impulse responses are evaluated using stable first derivatives by means of smoothing cubic splines through the optimal smoothing parameter selection on the basis of the statistical optimality criterion.
Results. The co-authors solve inverse problems of impulse response identification and direct problems of heat flux reaction prediction. The research results demonstrate a high level of convergence between the evaluated data and observation findings. Both experimental and theoretical results represent the findings of the research performed by the co-authors.
¡y ¡y Conclusions. The results have proven the efficiency of the algorithms proposed for the identification of solutions to the
problems of complex technical systems.
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KEYWORDS: thermal control system, black box model, non-parametric problem of identification, ill-posed problem,
Volterra integral equation of the first kind, stable identification algorithm, transient process, relative excess heat
Acknowledgements. The reported study was funded by RFBR, project No 20-38-90041.
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<0 ф FOR CITATION: Mansurov R.Sh., Voskoboinikov Yu.E., Boeva V.A. A theoretical and experimental study of transient characteristics of the heat exchange in a thermal control system. Vestnik MGSU [Monthly Journal on Construction and
| 3 Architecture]. 2021; 16(6):720-729. DOI: 10.22227/1997-0935.2021.6.720-729 (rus.).
i ?
£ 2 Теоретическое и экспериментальное исследование переходных
о J характеристик теплообмена в системе терморегулирования
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J9 13 Р.Ш. Мансуров, Ю.Е. Воскобойников, В.А. Боева
g <о Новосибирский государственный архитектурно-строительный университет z ■ i
(Сибстрин) (НГАСУ (Сибстрин)); г. Новосибирск, Россия
^ Ю АННОТАЦИЯ
с о
^ о Введение. Система терморегулирования «Обогреватель-вентилятор-помещение» представляет собой три разные
£= взаимосвязанные подсистемы. Для стабилизации работы системы необходимо изучить переходные характеристики
со процесса теплообмена в подсистемах, в том числе информативные импульсные характеристики. Данная задача
о Е является непараметрической, а для ее решения требуется произвести поиск алгоритмов.
^ о
g Материалы и методы. Математические модели подсистем представляют собой интегральное уравнение Вольтер-ры первого рода с неопределенным разностным ядром или импульсным откликом. Оценка импульсного отклика
^ является решением указанного уравнения для установленных входных и выходных шумовых характеристик. Задача
ОТ заключается в оценке неизвестных импульсных откликов в подсистемах, в которых выходная характеристика одной
• . подсистемы служит входной характеристикой другой подсистемы. Задача — некорректна, а эксперименты направ-
у Э ленные на поиск решения не позволяют применить вычислительные методы классических алгоритмов регуляриза-
I- щ ции. Авторы предлагают два особых непараметрических метода идентификации, при которых импульсные отклики
® g оцениваются на основании первых производных с помощью кубических сглаживающих сплайнов на основе выбора
5 X оптимального параметра сглаживания, установленного по критерию статистической оптимальности.
¡Е £ Результаты. Авторы решают обратные задачи идентификации импульсного отклика и прямые задачи прогнози-
¡^ jj рования реакции теплового потока. Исследование показало высокую сходимость расчетных данных с данными на-
U > блюдения. Как теоретические результаты, так и результаты эксперимента представляют собой итог исследований, проведенных авторами настоящей работы.
720 © Rustam Sh. Mansurov, Yuri E. Voskoboinikov, Vasilisa A. Boeva
Распространяется на основании Creative Commons Attribution Non-Commercial (CC BY-NC)
Вывод. Результаты доказали эффективность алгоритмов решения задач сложных технических систем.
КЛЮЧЕВЫЕ СЛОВА: система терморегулирования, модель черного ящика, непараметрическая проблема идентификации, некорректная задача, интегральное уравнение Вольтерра первого рода, стабильный механизм идентификации, переходный процесс, относительное избыточное тепло
Благодарность. Исследование выполнено при финансовой поддержке РФФИ в рамках научного проекта № 20-3890041.
ДЛЯ ЦИТИРОВАНИЯ: Мансуров Р.Ш., Воскобойников Ю.Е., Боева В.А. A Theoretical and experimental study of transient characteristics of the heat exchange in a thermal control system // Вестник МГСУ. 2021. Т. 16. Вып. 6. С. 720-729. DOI: 10.22227/1997-0935.2021.6.720-729
INTRODUCTION
In practical terms, there are thermal control systems that can be considered as a combination of several different simpler subsystems characterized by various physical processes that are underway inside them. The major challenge is to stabilize the climate parameters of such systems [1, 2]. Fluctuating thermal agitations in a system may cause unstable system operation [3]. The system's response to agitations is a transient process that determines the climate parameters and the system stability [4].
In this context, there is a need for an experimental research on the transient characteristics of heat exchange in climate systems [5, 6]. An impulse response is considered to be the most informative characteristic in the automation control theory [3]. The information about an impulse response allows to simulate different operation modes, including the hazardously critical operation of systems, and formulate the optimal system control regularities [4, 5].
This paper addresses the study of the "Heater-Blower-Room" thermal control system that has several interconnected subsystems [6]. The main objective is the non-parametric identification of impulse responses in the thermal control system and its subsystems. The supplementary objectives include:
• the formulation of the mathematical model of the thermal control system;
• the research on inputs and outputs of the subsystems;
• stable non-parametric identification of an impulse response when an input is a step signal;
• stable non-parametric identification of an impulse response when an input is an arbitrary signal.
To attain these objectives, the authors use the methods and algorithms described in [6-9]. This paper addresses the application of these methods and algorithms in the course of the experiments reported in [2, 10].
MATERIALS AND METHODS
Let us explain some fundamentals of the physical processes [11] that are underway in the system and classify the features of inputs and outputs of the subsystems for the successful structural identification of a thermal control system and its subsystems.
Fig. 1 demonstrates the circulation of heat fluxes generated/absorbed by a moist air flux passing though
the elements of the thermal control system [11]. The air mass flow is defined as G = p(T3 )uF, where F, in m2, is the sectional area of an air duct at the points of u velocity sensors and temperature T3 sensors; p(T3), in kg/m3, is the density of dry air at temperature T3. The heat absorbed by the heat flux from the "Heater" element is defined as Qh = caGATh, the heat absorbed from the "Blower" element is defined as Qb = caGATb, the heat absorbed from the "Unit" subsystem is defined as Qu = caGATu, where ca is the thermal capacity of the air; ATh = T2 - T1, ATb = T3 -T2, ATu = T3 - T1. The heat accumulated by the heat flux in the "Room" element is defined as Qr = caGATr, where ATr = T4 - T3. The heat outflow into the environment from the "Room" element is defined as Qe = caGATe, where ATe = T4 - Tx.
In paper [2], relative excess heat Qi is formulated as a similarity parameter that describes transition processes. It allows to identify the trends of transition processes in the moist air flux passing though the elements of the thermal control system. Relative excess heat is defined as:
Q6 =
Qi (т) - 6(0) - Qi (0)|
i = 1... 4,
(1)
where Qi, if i = 1, relative excess heat Qh is generated/absorbed by the moist air flux passing though the "Heater" element; if i = 2, relative excess heat Qrb is generated/absorbed by the moist air flux passing though the "Blower" element; if i = 3, relative excess heat Qrr is assimilated by the heat flux passing though the "Room" element; if i = 4, relative excess heat Q'u is generated/absorbed by the moist air flux passing though
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the "Unit" subsystem; Q.(t) is the heat generated during time t; Q/0) is the heat generated during initial time t = 0; Q.(t.) is the heat generated during final time X, = 500 sec.
As a result of experimental data processing, the evolution of relative excess heat values Qh, Qb, Qrr, Qru in the heat flux, passing through the "Heater-Blower-Room" system, is demonstrated in Fig. 2. The "Blower" operation mode is "0.4", the "Heater" operation mode is "0.01^0.1".
Fig. 3 shows the block diagram of the "Heater-Blower-Room" thermophysical system. The thermal control subsystem "Heater-Blower", previously called the "Unit", is a stationary linear system that can be represented as a combination of two simpler stationary linear subsystems, the "Heater" and the "Blower". Stationary linear properties of these subsystems were demonstrated in the course of the experiments. Physical processes in the "Room" subsystem are characterized as unpredictable and stochastic, so this subsystem is not included in the general identification process. At the same time, it is experimentally proved that the evolution of relative excess heat Qrr is compensated by the evolution of relative excess heat Qru •
Physical processes underway in the thermal control system and its subsystems are too complicated to be adequately described in the language of ma-
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thematics. Therefore, let us assume that the model of this system is a "black box" that has an input and an output. Hence, the model can be formulated in the "input-output" terms in the form of the Volterra integral equation of the first kind to solve the identification problem [12-16]
} 9„ (t-x )kn (x )dx = f (t), t e[0,r], (2)
0
where n is the subsystem indication; kn(z) is the impulse response of a subsystem, or the kernel of the integral equation (2); 9n(T) is the input of the subsystem; fn(t) is the output of the subsystem. In Fig. 3 t ) = 91( x), and it demonstrates the productivity shock at 0.1 as the input of the "Unit" and "Heater" subsystems; f1(t) = Qrh is the output of the "Heater" subsystem, and the input 92(t) of the "Blower" subsystem in operation mode "0.4"; f2 (t) = Qrb is the output of the "Blower" subsystem; f (t) = Q'u is the output of the "Unit" subsystem and the input of the "Room" subsystem. Signals f1(t), f2(t), f(t) are a priori inexact noisy observations contaminated by measurement errors [17] made by the laboratory facility described in [10]. All the system signals are equally spaced, broken down into N = 500 samples with a time step At = 1 sec. As a result of the experimental study, it is discovered that the acceleration of the system elements exceeded 20 sec. Such input-response delays are typical for dy-
0.50
0.00
- 0.50
Time t, sec
Fig. 2. Experimental evolutions of relative excess heat values of the elements of the "Heater-Blower-Room" thermal control system in the flux heating mode
Control subsystem (Unit)
Fig. 3. A block diagram of the "Heater-Blower-Room" thermal control system
namic lags in responding to inputs after some time [18]. Hence, input delays were detected for/1 (t) in 29 sec, for f2(t) — in 33 sec, for ft) — in 55 sec. For the purpose of simulation, all delays were rejected in the course of evaluation of linear impulse responses, and the delay effect was compensated by the time lag element [19].
For the purposes of simulation and operation of the system, described by the mathematical model in form (2), a non-parametric identification problem must be solved. It consists of evaluations of the impulse response obtained from "Unit" subsystem &(t), "Heater" subsystem kj(x), and "Blower" subsystem &2(t), when the productivity shock in the "Heater" element is registered as 0.1 in the process of measuring noisy input and output parameters of the moist air flux. This problem is ill-posed [20, 21] due to the probable solution instability [22] caused by initial data measurement noises [17, 23-25]. In terms of causal relationships, the problem is inverse [21].
A stable solution to the problem can be generated using the authors' non-parametric identification algorithm designated for technical systems reported in [6-8]. In this algorithm, an impulse response is evaluated by smoothing cubic splines with optimal smoothing parameter selection on the basis of the statistical optimality criterion [9].
RESULTS OF THE RESEARCH
Non-parametric identification of the subsystems of a thermal control system can be divided into the following stages.
Stage 1. Impulse response evaluation of the
"Heater" subsystem
Since the constant-amplitude step signal ^(t) is applied to the input of the "Heater" subsystem at time t = 0, impulse response &j(t) can be evaluated by output f(t) differentiation when input ^(t) is represented by the Heaviside function [26] with regard for the amplitude and K = 0.1 coefficient:
K ( t )=K • df ( t t e[°'T ]'
(3)
Smoothing parameter a adjusts the smoothness of smoothing cubic splines, and the measurement noise filtration error as a consequence. The statistical optimality criterion [8, 36] allows to calculate value aW of optimal smoothing parameter a that minimizes the mean-root square value [25, 26, 37]. Let us further assume that smoothing cubic splines are calculated at a = aW. The calculation of smoothing cubic splines coefficients a, b., c, d, i = 1 ... N within set aW is addressed in [27, 29, 38].
Fig. 4 shows impulse response evaluations kk1(t) performed by interpolating splines (the black line), and smoothing cubic splines (the blue line). Smoothing cubic splines with the appropriate smoothing parameter selection procedure, that ensures stable differentiation of noisy data, have proven their efficiency. Differentiation by means of interpolating splines generated undesired results.
vf 0.02
0.01
- 0.01
0.02
100
200 300
Time t, sec
400
where n is the subsystem indication, and for the "Heater" subsystem n = 1.
Despite the established numerical differentiation algorithms, there arises a problem of an unstable solution related to different problem statements [27, 28]. Even low measurement noise in the output signal can cause significant impulse response evaluation errors. To ensure a stable impulse response evaluation, it is necessary to smooth, or filtrate, and differentiate the noisy output /1(t) [9]. Cubic splines [29-33], having natural boundary conditions, are used for this purpose due to their relative implementation simplicity and mathematical software availability [33, 34].
Contrary to interpolating splines [35], smoothing cubic splines are not constructed through the pre-set points of a signal, so they are widely used for noisy data filtration.
Fig. 4. Impulse response evaluations in the "Heater" subsystem
The reason for the initial nonzero values of impulse responses (Fig. 4-6) is the testing environment of the field observations performed in a series of experiments. The pre-set productivity of the "Heater" element was ensured by power controller TTS-25 that generated a sinewave with a pre-set period of 60 sec. In total, eighteen consecutive series of experiments, N = 500 sec each, were conducted. Experiments, that had even numbers, entailed a sequential productivity increase of the "Heater" element, while experiments that had odd numbers, entailed its sequential productivity reduction. After that, the observational data were averaged over nine even heating experiments and nine odd cooling experiments.
The problem of the "Heater" subsystem is the heat flux f(t) reaction prediction by means of agitation ^(t) and the ^evaluated impulse response k1( t). Predicted values f1(tt), i = 1 ... N of the "Heater" subsystem's output f(t) are calculated as follows [12-16, 39]:
/,(0 = -T)<p,(T)A,
(4)
where n is the subsystem indication.
The difference between predicted output values and observational output values is identified as follows:
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The value of 5f calculated by (5) is 1.9 % which evidences a highly qualitative convergence between the calculations and the experiments.
Stage 2. Impulse response evaluation of the
"Blower" subsystem
When evaluating impulse response k2(T), input 92(t) is considered as the arbitrary signal f1(t). By means of differentiating equation (2) with respect to t, and simply transforming the new one, the Volterra integral equation of the second kind [39] can be obtained: 1 r
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0.2
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- 0.6
where n is the subsystem indication, and for the 1 wer" subsystem n = 2.
The solution to integral equation (6) is a well-posed problem [28], but it also encompasses two challenges. Firstly, it is necessary to construct stable first derivatives of noisy inputs and outputs, secondly, the convolution integral must be calculated with minimum integration errors [26] to reduce the systematic error of the identification algorithm. These problems can be solved by smoothing the aforementioned cubic splines.
To evaluate the impulse response, we should calculate smoothing cubic [33, 36, 40] splines of inputs and outputs and construct corresponding first derivatives [7, 8]. Then (N - 1) ' (N - 1) matrix elements $ are formed according to the rules specified in [7]:
0 for i < j
k (t) scs
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20
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tem
40 60 80 100
Time t, sec
5. Impulse response evaluations in the "Blower" subsys-
where i = 1...N, j = 1...N - 1; S^ ,a (x) is the first derivative of the smoothing cubic spline of an input for the pre-set smoothing parameter aW. The exact efficient calculation of the convolution integral is performed using a quadrature formula that has smoothing cubic spline coefficients [29] used to minimize the error of the algorithm. Matrix O allows to approximate the Volterra integral equation of the second kind (6) by a system of linear equations [4]:
Stage 3. Impulse response evaluation of the
"Unit" subsystem
The constant-amplitude 0.1 step signal ^(t), same as 91(t) in the "Heater" subsystem, is applied to the input of the "Unit" subsystem at time t = 0. The impulse response k(t) evaluation algorithm is the same as at Stage 1.
Fig. 6 shows impulse response evaluations k( t) obtained by interpolating splines (the black line), and by smoothing cubic splines (the blue line). The problems of the "Unit" subsystem are the heat flux ft) reaction prediction using agitation ^(t) and the evaluated impulse response k (t). The difference between predicted output values and observational output values 5f calculated using (5) is 1.6 %.
0.04
0.02
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where I is an identity matrix; /„' is a vector of output first derivatives atmesh points calculated by smoothing cubic splines; k„ is the solution vector, and its components are the evaluations for the identified impulse response values k (t); n is the subsystem indication.
0.02
0.04
100
200 300
Time t, sec
400
Fig. 6. Impulse response evaluations in the "Unit" subsystem
0
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Qh
q:
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100
200
300
400
500
Time t, sec
Fig. 7. Theoretical transient evolutions of relative excess heat values of the elements of the "Heater-Blower-Room" thermal control system in a flux heating mode
Following the processing of these problems with the help of evaluated impulse responses, transient evolutions of relative excess heat values Qrh, Qrb, Qrr, Qru in a heat flux, passing through the "Heater-Blower-Room" system, are demonstrated in Fig. 7. The "Blower" operation mode is "0.4", the "Heater" operation mode is "0.01^- 0.1".
The co-authors have identified a high level of convergence between the experimental relative excess heat values, identified in the process of observation (Fig. 2), and calculated relative excess heat values, identified by means of evaluations (Fig. 7). The averaged difference between the observational and evaluated data is 3-4 %, and the maximum difference is 7.8 %, which evidences high qualitative convergence between the calculations and the experiments.
CONCLUSION AND DISCUSSION
A non-parametric identification of complex dynamic systems is a rather complicated and time-consuming problem. On the one hand, this complexity is explained by the ill-posed nature of the solution to the Volterra integral equation of the first kind, and on the other hand, it is caused by different types of inputs and outputs within the identified systems.
The algorithms proposed in the paper allow to perform stable non-parametric identification both for the whole system and its elements with regard for the specific features of the problem in question.
Smoothing cubic splines allow to construct an efficient algorithm for the filtration of various statistical measurement noises interfering with the inputs and outputs of the system. Due to the application of smoothing cubic splines, the value of the identification error depends on differentiation errors of inputs involving matrix ® elements, and differentiation errors of outputs involving vector f components. An optimal smoothing parameter selection, made on the basis of the statistical optimality criterion, means minimum differentiation errors for noisy inputs and outputs, and minimum identification errors as a consequence. Note that in case of anomalous measurements of inputs and outputs, the robust algorithm reported in [8] can be used for a stable non-parametric identification.
This successful solution to the engineering non-parametric identification problem is applicable to the complex thermophysical system "Heater-Blower-Room" if observational data are noise-contaminated. It has proven the efficiency of non-parametric identification algorithms used to solve applied problems. The research results allow to use these algorithms for the identification of impulse responses in other complex dynamic systems.
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36. Lee T.C.M. Smoothing parameter selection for smoothing splines: a simulation study. Computational
Received April 9, 2021.
Adopted in revised form on June 10, 2021.
Approved for publication on June 16, 2021.
Statistics & Data Analysis. 2006; 42(1-2):139-148. DOI: 10.1016/S0167-9473(02)00159-7
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39. Voytishek A.V Classification and applications of randomized functional numerical algorithms for the solution of second-kind fredholm integral equations. Journal of Mathematical Sciences. 2021; 254(5):589-605. DOI: 10.1007/s10958-021-05328-z
40. Crambes C., Kneip A., Sarda P. Smoothing splines estimators for functional linear regression. The Annals of Statistics. 2009; 37(1):35-72. DOI : 10.1214/07-A0S563
Bionotes: Rustam Sh. Mansurov — Candidate of Technical Sciences, Associate Professor, Head of the Department of Heat and Gas Supply and Ventilation; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NSUACE (Sibstrin)); 113 Leningradskaya st., Novosibirsk, 630008, Russian Federation; ID RISC: 377487; rmansurov@inbox.ru;
Yuri E. Voskoboinikov — Doctor of Physical and Mathematical Sciences, Professor, Head of the Department ofApplied Mathematics; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NSUACE (Sibstrin)); 113 Leningradskaya st., Novosibirsk, 630008, Russian Federation; ID RISC: 13547; voscob@mail.ru;
Vasilisa A. Boeva — Graduate Student, Assistant at the Department of Applied Mathematics; Novosibirsk State University of Architecture and Civil Engineering (Sibstrin) (NSUACE (Sibstrin)); 113 Leningradskaya st., Novosibirsk, 630008, Russian Federation; ID RISC: 1027172; v.boyeva@sibstrin.ru.
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ные технологии. 2020. Т. 25. № 3. С. 46-53. DOI: 10.25743/ICT.2020.25.3.006
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Поступила в редакцию 9 апреля 2021 г.
Принята в доработанном виде 10 июня 2021 г.
Одобрена для публикации 16 июня 2021 г.
Об авторах: Рустам Шамильевич Мансуров — кандидат технических наук, доцент, заведующий кафедрой теплогазоснабжения и вентиляции; Новосибирский государственный архитектурно-строительный университет (Сибстрин) (НГАСУ (Сибстрин)); 630008, г Новосибирск, ул. Ленинградская, д. 113; РИНЦ ГО: 377487; rmansurov@inbox.ru;
Юрий Евгеньевич Воскобойников — доктор физико-математических наук, профессор, заведующий кафедрой прикладной математики; Новосибирский государственный архитектурно-строительный университет (Сибстрин) (НГАСУ (Сибстрин)); 630008, г Новосибирск, ул. Ленинградская, д. 113; РИНЦ ГО: 13547; voscob@mail.ru;
Василиса Андреевна Боева — аспирант, ассистент кафедры прикладной математики; Новосибирский государственный архитектурно-строительный университет (Сибстрин) (НГАСУ (Сибстрин)); 630008, г. Новосибирск, ул. Ленинградская, д. 113; РИНЦ ГО: 1027172; v.boyeva@sibstrin.ru.
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