Introduction
At present, mathematical modelling is widely used to assess the effectiveness of various constructive solutions. A peculiarity of mathematical modelling is the large amount of computational work; therefore, recently, in terms of accessibility and improvement of the capabilities of computer technology, numerical experiment has become widespread. Mathematical modelling with the use of adequate mathematical models has much in common with the field experiment. This way of research allows simulating the processes that arise in the actual operation of separate equipment and life support systems in general, as well as the impact of various factors thereon. The basis of ma-thematical modelling is the method of differential balance equations [11].
The mathematical modelling of heat processes in passenger cars with heating systems is usually realized in steady regime, when the heat fluxes and parameters of the thermal circuit are constant, do not depend on time. The steady regime refers to the situation in the car, when there is balance between the thermal energy that comes in and that given by fencing structures into the environment. The energy balance of such a system in a steady regime is studied quite well [4-8, 12]. But, any heat exchange is dynamic, and it is not enough to describe a single steady regime. The worse situation is with the analysis of the thermal condition of heated cars in unsteady conditions, in particular, when the heating of high-voltage heating systems is switched off in motion and in parking places with further heating, resulting in transitional heating regimes.
The mathematical model of the car heat condition, when the car is heated with an air heating system, is considered in the works [14, 16, 17], the model, when a car air conditioner operates in the heat pump mode, is considered in the works [3, 13]. The works [2, 15] present a mathematical modelling of unsteady heat exchange processes of a passenger car with air conditioning systems. These mathematical models are similar to each other, and allow the insertion and extraction of additional elements into the calculation scheme rather easily, but the air flow is used for heating and cooling as a coolant. The presented mathematical models do not fully reflect the thermal processes occurring in the car when using a water heating system, where the intermediate coolant is water. There are no such indicators as heat energy accumulation [1].
The works devoted to passenger car heating systems indicate that a car needs the heating system with capacity of 48 kW in the winter period. These requirements were substantiated in the 70-80s of the last century for cars with an effective thermal conductivity of about 1.7 W/ (m2 ∙ K) with the environment temperature of –40°C in winter, with a number of passengers from 32 to 60 people (considered by Kitayev B. N. [6, 7], Kuzmin L. D. [11], Zharikov V. A. [4, 5], Sidorov Yu. P. [10] and other researchers).
Already at the beginning of the 21st century, the car builders of PJSC «Kryukov Railway Car Building Works» reached an effective conductivity in a passenger car of about (0.8 ÷ 1.0) W / (m2 · K), and the installation of double-glazed windows significantly increased its tightness.
Thus, taking into account the car structural changes and the new trends, a more thorough ana-lysis of the car heat regime is required, taking into account the unsteadiness of the process, when heating the passenger car.
Purpose
The purpose of this study was to create a mathematical model of the unsteady heat regime of a passenger car with a water heating system to evaluate the role of unsteady, transitional temperature states of the passenger car, the selection of optimal technical characteristics of heating devices, and constructing an algorithm for their operation control in accordance with operating conditions, and particularly taking into account the manifestation of the car thermal inertia during the transitional operating modes of the heating system.
Methodology
To achieve the set task the author composed a system of differential equations, describing unsteady heat processes during the heating of a passenger car. For the solution of the composed system of equations, the author used the method of elementary balances.
When studying the transitional regimes in the process of cooling and subsequent heating of a passenger car during the operation, the conditions are taken into account when the heat from the TEHs (tubular electric heaters) is perceived by the intermediate coolant and then transmitted to the car. The same is when the car is cooled from the initial temperature to the critical, at which the next heating process begins. The dynamic equation of the temperature process in this case should be solved in two stages: in relation to the intermediate coolant, and from the coolant to the air in the car and then to the outside air.
During formation of the car heat model, physically grounded and experimentally confirmed features of the car heat condition were taken into account, namely:
– temperature of the interior partitions of the car practically coincides with the temperature of the car air;
– difference between the internal partition wall with the temperature tp and the average air tempe-rature ta in the car does not exceed 3 К, since the temperature difference between the external environment with the temperature te and the car air with the temperature tc is mainly damped on its thermal insulation;
– temperature of the air removed from the car through deflectors is equal to the air temperature in the car ta.
– due to increasing the coefficient of heat transfer by convection on the outer surfaces of the partitions, depending on the car speed from 0 to 80 km/h, the heat transfer of the car body increases by 10%, at speeds from 80 to 160 km/h, the coefficient increases by 1%;
– air infiltration volume, depending on the car speed up to 120 km/h, can reach 325 m3/h;
The physical essence of these equations is reduced to the following.
The heat flow is evolved from TEHs QТЕН(τ) in the time interval τ, is transmitted to the intermediate coolant and the metal structure of the heating system. Since the heating devices can not physically transfer the entire heat flow QТЕН(τ) evolved from TEHs, part of this heat is accumulated in the coolant and the metal structure of the heating system Qhs.
In accordance with the energy conservation law (heat balance), the heat flow QТЕН(τ) is consumed on four main components:
, (1)
where Qhs – heat accumulated by heating system; Qpp – heat consumed by heating pipes; Qcl – heat consumed by the coolant to heat the outdoor air; Qbl – heat consumed for water heating for hot water supply, as the boiler does not affect the microclimate in the car and has a slight consumption of heat, this parameter will not be taken into account further.
The listed components are determined by the relationships:
, (2)
, (4)
where Сhs – total heat capacity of the water and the metal structure of the heating system; tin – coolant temperature at the inlet to the heating pipes and heater; tout – coolant temperature at the outlet from the heating pipes is determined by the formula:
, (5)
where
t0
– coolant
temperature
at
the
inlet
to
the
heating
pipes;
ta,
–
room
air
temperature;
l
–
length
of
the
heating
pipes;
–
coefficient
determined
by
the
expression:
, (6)
where
– coefficient
of
heat
transfer
of
heating
pipes;
–
outside
and
inside
diameter
of
the
heating
pipes;
–
coolant
density;
–
coolant
speed
in
the
heating
pipes.
To analyse the heating regime of the car heating system, the equation (1) ÷ (3) must be supplemented by another equation that is used to calculate the heating and cooling of the heating system coolant from a given initial temperature t(0) toa certain final temperature tb for a short period of time τ, at any stage and has the form:
. (7)
The amount of heat entering the car from Qpp(τ), as can be seen from equation (3), depends on the coolant temperature, the heating pipes area, the heat transfer coefficient kpp, the rate of coolant circulation in the heating pipes. The heat Qa(τ), evolved by the air flow VO at the time τ, is uniquely associated with the change in its enthalpy and is determined by the relation:
, (8)
where
– specific
enthalpy,
temperature
and
relative
humidity
of
air
entering
the
car
after
heating
in
calorifer;
–
specific
enthalpy,
temperature
and
relative
humidity
of
air
in
the
car;
VO
–
volume
of
outside
air
supplied
by
the
ventilation
system;
to
determine
the
specific
enthalpy
of
air
I d
– wet
air
diagram
is
used;
–air
temperature
heated
by
the
calorifer
is
determined
by
the
expression:
. (9)
The heat brought by the heated outside air can be determined by another less precise expression:
, (10)
The heat flows Qpp(τ), Qa(τ), Ql(τ), entering the car at the time τ, are absorbed by three components:
, (11)
where Qlos – heat lost by partitions, as well as windows; Qinf – heat consumed for heating of the cold air, which penetrates through the body imperfections and is characterized by Vinf(S), that is, by the volume of infiltrated air, depending on the speed of movement; Qcar – heat consumed for heating the internal air and equipment of the car.
The listed components are determined by the relationships:
, (12)
, (13)
, (14)
where
ko
– coefficient
of
heat
transfer
through
outer
lining;
Fo
– area
of
outer
lining;
ta
– room
air
temperature;
te
–
environment
temperature;
ca
– heat
capacity
of
air;
Vinf
–
volume
of
air
entered
into
the
car
as
a
result
of
infiltration;
Сcar
–
total
heat
capacity
of
all
internal
partitions,
wooden
lining
of
external
car
frame
and
half
heat
capacity
of
the
heat-shielding
layer.
For an integrated analysis of the car heat regime we need one more equation to calculate the heating and cooling of the car air temperature from a given initial temperature t(0) to a certain final temperature ta for a short period of time τ, has the form:
. (15)
It is advisable to highlight several of the most characteristic stages of the car heat regime, within each of which almost constant values of the output parameters are kept:
(16)
With restrictions (16), equation (7) (15) has at its separate stage its own, individual analytical solution of the form:
The equation describing the change in air temperature (15) takes the form of:
, (17)
where the step part depends on both ta, and tin = tb.
So, this is the equation of two variables:
and
The equation describing the temperature of the coolant in the boiler (7) takes the form:
, (18)
where
the
right
side
also
depends
on
.
Consequently, we have a system of two differential equations with two variables:
(19)
The grouping of the right-hand sides of the equations with respect to the variables ta and tb and after transformations has the form:
where
–
indicators
of
car
thermal
inertia
at
the
considered
stage;
–
index
of
car
heat
entropy
at
the
considered
stage:
, (21)
, (22)
, (23)
, (24)
where
–
indicators
of
thermal
inertia
of
the
heating
system
at
the
considered
stage;
–indicator
of
heat
entropy
of
the
heating
system
at
the
considered
stage.
, (25)
, (26)
. (27)
Thus, the initial system of equations has the form:
. (28)
That is, linear equations with constant coefficients.
The linear non-homogeneous second-order equation with third-order coefficients has the form:
, (29)
where
, (30)
, (31)
. (32)
Discriminator of the characteristic equation:
. (33)
The solution of homogeneous equations for the boiler and the car room temperature has the form:
, (34)
. (35)
where R1, R2 are the roots of the characteristic equation:
, (36)
, (37)
, (38)
. (39)
The expressions (34), (35) allow us to estimate not only the temperature of the coolant in the combined electric-coal boiler and the air inside the car, but also to carry out a comprehensive analysis of the heat processes while heating the passenger car, taking into account the structural changes and the unsteadiness of the processes, and evaluate the efficiency of the system «heating system – passenger car». To do this, the initial temperatures tb(0), t(0) of the boiler and inside the car at this stage and the value of the output parameters should be known.
Approbation of the mathematical model when compared with experimental data. The mathematical model described above allowed constructing a calculated model of the car temperature condition, using a water heating system with natural circulation and a discrete two stage high-power heat supply (2 groups of 24 kW).
For simplicity, the infiltration volume was taken from the average speed of movement. The heat capacity of the internal equipment and the heating system is taken in the water equivalent. Heat-efficiency of the combined water-heating boiler is 24+24 kW. The work of the ventilation system was not taken into account; it was not switched on during the experiment.
The dimensions and physical parameters of the elements used to construct the calculation model are given.
The model approbation used the experimental data obtained by the author. The experiment was conducted during train movement, the car number 26487, manufactured at «KVZ» in 1985, overhaul reconditioning on 10.12.2014. Measurement of temperatures was carried out by stationary means, the temperature of the car air was measured by two thermometers located on the boiler and no-boiler side of the car, the temperature of the coolant in the boiler was measured by a regular remote thermometer with a remote sensor.
As can be seen from the data given in Fig. 1, the modelling results quite well coincide with the results of the experiment, that is, the constructed model can be considered rather accurate and used for theoretical studies.